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## Material Information- Title:
- The spacetime manifold of a rotating star
- Creator:
- Ahmad, Ejaz, 1963-
- Publication Date:
- 1996
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- English
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- viii, 124 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
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Critical points ( jstor ) Distance functions ( jstor ) Harmonic functions ( jstor ) Horizon ( jstor ) Killing ( jstor ) Laplace equation ( jstor ) Multipoles ( jstor ) Polynomials ( jstor ) Schwarzschild metric ( jstor ) Astrophysics ( lcsh ) Dissertations, Academic -- Physics -- UF Physics thesis, Ph. D - Genre:
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## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1996.
- Bibliography:
- Includes bibliographical references (leaves 118-122).
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- Typescript.
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- Vita.
- Statement of Responsibility:
- by Ejaz Ahmad.
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THE SPACETIME MANIFOLD OF A ROTATING STAR By EJAZ AHMAD A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 UNIVERSITY OF FLORIDA LIBRARIES Copyright 1996 by Ejaz Ahmad ACKNOWLEDGEMENTS The author expresses his sincere appreciation to all members of his committee: to Professor Ipser for teaching him how to simulate neutron stars numerically; to Professor Detweiler for many discussions on rotating black holes; and to Professor Whiting for his generous assistance in all mathematical matters. The author would like to express his thanks to the Relativity Group at the University of Chicago for a weeklong visit in the Summer of 1993. The author benefited greatly from the encouragement and insight of Professor Subrahmanyan Chandrasekhar. It is much to the author's regret that Professor Chandrasekhar passed away in August 1995 before this work could be completed. The author would like to thank Dr. Bob Coldwell for his assistance on numerous numerical matters. The author would also like to thank Mr. Chandra Chegireddy, the Departmental System Manager, for his assistance on all computer related matters both big and small. Finally the author would like to thank his parents, and his extended family, and numerous friends for their continued support. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS .............. ABSTRACT . . . . . . . . . . . . . . . . . . . . . CHAPTERS . . . . . . . . . . . . . . . . . . . . . 1 INTRODUCTION ................ 2 THE KILLING VECTORS ............ 2.1 The Field Equations in Covariant Form . . . . 2.2 The Killing Bivector . . . . . . . . . . . 2.3 The Electrostatics Analogy . . . . . . . . . 3 WEYL'S HARMONIC FUNCTION . . . . . . . . 3.1 The Schwarzschild Solution . . . . . . . . . 3.2 The Kerr Solution ............. 3.3 Weyl's Harmonic Function for Rational Solutions ...... III ..... vii . . . . . vi1 ......1 ......8 . . . . . . 11 ~8 ~8 ~9 . . . . . 24 . . . . . . 26 . . . . . . 29 . . . . . . 37 3.3.1 Determination of Critical Points in the Harmonic Chart 3.3.2 Determination of Critical Points in the Isotropic Chart 3.4 The Zipoy-Vorhees and the Tomimatsu-Sato Solutions . .. 3.5 Weyl's Harmonic Function and Global Structure ..... 3.5.1 The Kerr Solution . ..... ................ 3.5.2 The Double Kerr Solution . . . . . . ...... 3.5.3 The Manko et al., 1994 Solution ......... 4 STATIONARY AXISYMMETRIC FIELDS IN VACUUM . . 4.1 The Conformal Invariance of the Field Equations in Vacuum 4.2 The Existence of Global Charts . . . . . . . . . . . ... 4.3 Practical Aspects of Coordinate Charts . . . . . . . . . . 76 4.4 Asymptotic Expansions for the Metric Functions . . . . . . 77 4.4.1 The Complete Nonlinear Vacuum Field Equations . . . 78 4.4.2 The Linearized Vacuum Field Equations . . . . . . . 78 4.4.3 Expansions for the Metric Functions . . . . . . . . . 80 4.5 Equivalence to Thorne Multipole Moments . . . . . . . . . 87 4.5.1 Thorne's Definition of Vector Spherical Harmonics . 87 4.5.2 Thorne's Expansion for the Metric Functions ...... 88 4.5.3 The Multipole Moments of the Kerr Solution . . . . . 89 5 CONCLUSION . . .. .. .. .. . . . .. . . . . . . . 94 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . 98 A GAUGES AND COORDINATES FOR STATIONARY AXISYMMETRIC SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . 98 A.1 The Radial Gauge . . . . . . . . . . . . . . . . . . . 99 A.2 The Isothermal Gauge . . . . . . . . . . . . . . . . 100 A.3 Two Global Coordinate Charts . . . . . . . . . . . . . 102 A.4 The Three Types of Coordinates . . . . . . . . . . . . 104 B VARIOUS FORMS OF THE SCHWARZSCHILD METRIC . . 107 C VARIOUS FORMS OF THE KERR METRIC . . . . . . . . 112 D THE MANKO et al., 1994 SOLUTION . . . . . . . . . . . 116 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . 118 BIOGRAPHICAL SKETCH ...................122 LIST OF FIGURES Figure Page 2.1 Two-dimensional Cross-section of the Schwarzschild Solution . . 17 2.2 Two-dimensional Cross-section of the Double Kerr Solution . . 18 2.3 Two-dimensional Cross-section of a Solution with a Toroidal Killing H orizon . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 The isometric embedding of the two-surface S orthogonal to the group orbits in R3. This is an inverted Flamm's paraboloid in isotropic coordinates . . . . . . . . . . . . . . . . . . . ... . . 21 3.1 D2(0, z) and B(0, z) for the Kerr Solution . . . . . . . . . . 54 3.2 D2(0, z) and B(0, z) for the Double Kerr Solution . . . . . . 58 3.3 D2(0, z) and B(0, z) for the Manko et al. 1994 Solution . . .. 63 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE SPACETIME MANIFOLD OF A ROTATING STAR By EJAZ AHMAD August 1996 Chair: James R. Ipser Major Department: Physics Stationary Axisymmetric Systems in General Relativity are characterized by two Killing vector fields. The norm of the Killing bivector satisfies a two-dimensional Poission equation on the two-surfaces orthogonal to the group orbits. For perfect fluids, the source term in the Poission equation is the fluid pressure. In the vacuum region the Poission equation reduces to the Laplace equation, and the norm of the Killing bivector is a harmonic function. The harmonic property of the norm of the Killing bivector was noted by Herman Weyl in 1917 for static axisymmetric systems. Although subsequent studies of stationary axisymmetric systems have utilized the harmonic properties of the function in question, surprisingly there is no discussion of the boundary data for the Laplace equation in vacuum, or the connection between sources and the Killing bivector via the Poission equation. The contemporary literature makes references to Weyl's harmonic function; however, there is apparent confusion regarding the number and location of the critical points of this harmonic function. In this dissertation the boundary data for Laplace's equation for various sources of astrophysical interest is discussed for the first time. In particular, vii the critical points of Weyl's harmonic function are determined for the Kerr solution by generalizing Weyl's results for the Schwarzschild solution. Next, an algorithm for the determination of the critical points for arbitrary rational stationary axisymmetric solutions is given. In the study of stationary axisymmetric systems two global coordinate charts have been used in the isothermal gauge: the harmonic chart and the isotropic chart, both having been introduced by Weyl in 1917. These two coordinate systems are related to each other by complex conformal transformations. In this study it is shown that the isotropic chart is asymptotically Cartesian mass centered(ACMC) as defined by Thorne (1980). Therefore it is possible to simply 1 read off the mass the mass-i moment from the part of gtt, and the angular 1 momentum I-pole moment from the r7 part of gto once a solution is expressed in isotropic coordinates. This prescription does not work in Weyl's harmonic coordinates which are not ACMC. Further, the coordinate vector fields associated with harmonic coordinates are not smooth at the critical points of Weyl's harmonic function. CHAPTER 1 INTRODUCTION In the last two decades there has been considerable progress in the theory of nonlinear partial differential equations particularly in two dimensions. In two dimensions many important equations have been shown to be exactly soluble by the Riemann-Hilbert method of classical complex analysis. These mathematical techniques have been applied successfully to the theory of General Relativity in spacetimes with high symmetry. In particular the technique of inverse scattering transform may be applied whenever a spacetime possesses two Killing vector fields. Using these techniques it has been possible to generate exact solutions to the Einstein field equations for stationary axisymmetric systems, inhomogeneous cosmological models, and colliding plane waves. Rapidly rotating neutron stars, which is the subject of this treatise, fall under the category of stationary axisymmetric systems. Generalizing the work of Buchdahl(1954) and Ehlers(1957), Geroch(1972) constructed the symmetry group of the stationary axisymmetric vacuum field equations; Kinnersley and Chitre(1978) determined the subgroup of the Geroch group which preserves asymptotic flatness. Using the method of the Riemann-Hilbert problem, Hauser and Ernst(1979) proved the conjecture of Geroch that any asymptotically flat stationary axisymmetric solution may be generated from Minkowski space using group theoretical methods. At the present time various algorithms exist for generating exact solutions for 2 stationary axisymmetric systems. A plethora of exact solutions has been generated purporting to represent the exterior of a rapidly rotating neutron star. Notwithstanding the abovementioned developments, very little is known about the spacetime of a rotating star. Theoretical studies of the global properties of the spacetime of a star and it causal structure have not been undertaken. It is known from the numerical simulations of Wilson(1972), and Butterworth and Ipser(1975) that some rapidly rotating stars possess ergotoroids. However, it is not known whether or not the vacuum solution of the star possesses an event horizon or a Killing horizon. Therefore, whether or not the exterior star solution possess a Killing horizon is a key question in the study of stationary axisymmetric spacetimes in General relativity. Apart from its theoretical significance in terms of understanding the vacuum solutions to Einstein's equations, this question has practical consequences. For example, it is known that some of the simpler exact solutions do possess Killing horizons, some do not, while the question is unanswered for some of the more complicated solutions. More precisely, two separate questions arise: First, does the real star, which in this case could be numerically generated, possess a Killing horizon ? Second, given an exact solution, what is the procedure for determining whether or not it possesses a Killing horizon. This question is being asked here for the first time. Some tools for analysing this issue will be developed in this study. Once these two questions are answered the task of comparing exact solutions with numerically generated exterior vacuum solutions may be undertaken. In a stationary axisymmetric system the Killing horizon is determined by the determinant of the (t, q) part of the metric. In the vacuum region the determinant satisfies the Laplace equation, and is therefore a harmonic function. In order to understand the global properties of the star, it is important to understand the behavior of the determinant of the (t, q) part of the metric. More precisely it is important to determine the boundary data for the harmonic function under consideration in the vacuum region. In a stationary axisymmetric system, all quantities are functions of two variables, and the Laplace equation in question is two-dimensional. In the interior of the star the determinant is no longer a harmonic function and it satisfies the Poission equation with pressure as the source. The answer to the key question, whether or not the exterior vacuum solution possesses a Killing horizon, will be determined by the boundary conditions of the Laplace equation in the exterior and the Poission equation in the interior. Hence the global properties of the star solution-i.e., the presence or absence of a Killing horizon-is determined directly by the pressure distribution, and only indirectly by the energy density. This important point has not been noted in the literature. It has been noted in the literature that a harmonic function exists in stationary axisymmetric systems. However, the boundary data for this harmonic function has not been determined for the Kerr solution or the rotating star. The Laplace equation is conformally invariant in two dimensions, and the critical points are conformal invariants (Walsh, 1950). In the present study the boundary data for Laplace's equation will be determined in terms of the critical points of the harmonic function in question for the Kerr solution and for the rotating star. For the star solution, the critical points of the harmonic function will be determined by the pressure distribution. The clarification of boundary data for the two-dimensional Laplace equation is one of the goals of this study. 4 As is often the case in General Relativity, the important issues outlined above are obscured by issues of gauge and coordinate choices. One of the results of the present investigations is a unified treatment of gauges, charts and coordinates used in the study of stationary axisymmetric systems presented in Appendices A, B, and C. Theoretical work is conducted in the harmonic chart with the metric in the Papapetrou form in prolate spheroidal coordinates, while numerical studies are conducted in the isotropic chart with the metric functions in the Bardeen form in cylindrical polar coordinates (Appendix A). In the harmonic chart the determinant of the (t, q) part of the metric and its conjugate harmonic function are used as coordinates. Once the harmonic function D is used as a coordinate, its properties are obscured. So much so that several authors have commented that once a transformation is made to the harmonic coordinates, "all information about the harmonic function is lost" (Vorhees, 1969, p. 2120). Others have commented that there is an arbitrariness in terms of the boundary data inherent in stationary axisymmetric vacuum solutions (Quevado, 1990). These statements are incorrect. First, the determinant of the (t, ï¿½) part of the metric is an invariant quantity: it is the magnitude of the Killing bivector constructed from the two Killing vectors that define the symmetry of the spacetime. Second, even if a coordinate transformation is made to harmonic coordinates, information about the harmonic function is contained in the conformal factor of the metric on the (p, z) space part of the metric. The critical points of a harmonic function define global topological properties of the solutions to Laplace's equation: two stationary axisymmetric solutions with different number of critical points are topologically distinct. It will be shown in Section 3.5 that solutions with different number of critical points have diffeent global properties in the full 5 four-dimensional spacetime. One of the important technical issues resolved in the present study is the procedure for determining the critical points of the harmonic function D in harmonic coordinates, as well as in isotropic coordinates. Boundary data for the Laplace equation for D can be extracted from the metric regardless of gauge choice or coordinate chart. The quantity D is an invariant quantity (Carter, 1969); it satisfies the field equations. Therefore every stationary axisymmetric spacetime has unique boundary data for D. Once the appropriate boundary data are determined for a solution, coordinate transformations can be made from the harmonic chart to the isotropic chart, and a comparison can be made between the exact solution in the harmonic chart and the numerical solutions in the isotropic chart. Once the boundary data for D is determined, information about the degrees of freedom of the exterior vacuum solution becomes transparent. There are four coordinate invariant quantities that can be constructed from the Killing vectors: V = gtt, X = goo, W = gto, and D2 = VX + W2. As these quantities are algebraically related, at each point of the spacetime three algebraically independent quantities are present. In the numerical computations in the isotropic chart three functions of the Bardeen form of the metric are computed: e2v = V + W2/X = D2/X, WB = -W/X, and D = piB. The invariant definition of D is not present in the pioneering work of Bardeen (Bardeen, 1970, 1973). The invariant definition of D as the magnitude of the Killing bivector, D2 = VX + W2, is given in the works of Carter(1969,1973). In the numerical studies of rotating neutron stars, three metric functions are computed in the exterior vacuum region. Asymptotic expansions for 6 the metric functions were determined by Butterworth and Ipser(1976) to establish boundary conditions for their numerical algorithm for constructing rotating stars. There are three sets of parameters in these expansions. The invariant definitions of these parameters have not been given. It is shown in the present study that two sets of parameters corresponds to the multipole moments defined by Geroch for stationary systems in terms of the norm and twist of the timelike Killing vector-i.e., V and W. The third set of parameters correspond to the function D. These parameters arise due to the presence of the additional symmetry-the axisymmetry-of the spacetime. The invariant definition of D in terms of the magnitude of the of the Killing bivector involves two Killing vector fields. The third set of parameters is a not part of the invariant multipole moments defined by Geroch and Hansen(1974) as they assume only stationarity and not axisymmetry. Therefore they do not contain any information about the magnitude of the spacelike Killing vector field that defines axisymmetry X or the magnitude of the Killing bivector D. In addition to the invariant quantities defined above in term of the Killing vector fields, there is another invariant quantity which arises due to the two-dimensional nature of the problem. This quantity is the Gaussian curvature of the (p, z) two surface. Its value at each point is determined by the field equations in terms of the gradients of V and W. In terms of coordinates it is determined by the conformal factor, eC, of the two surface S. Due to the conformal invariance of the field equations in vacuum, the conformal factor is not present in the field equations in the vacuum region. Therefore it is possible to solve for the three functions V, W, and D first. As the field equations are not all independent, the solution for the conformal factor may be obtained afterwards. 7 In the present study the asymptotic expansion for e is derived. The conformal factor e is expressed in terms of the three sets of parameters mentioned above. Therefore it is shown that the three sets of parameters defined by Butterworth and Ipser(1976) determine all four metric functions in the vacuum region. CHAPTER 2 THE KILLING VECTORS 2.1 The Field Equations in Covariant Form A stationary axisymmetric spacetime has two Killing vector fields: a timelike Killing vector field ( and a spacelike Killing vector field 71. The trajectories of the spacelike Killing vector 7 are closed, and 77 vanishes on the rotation axis. In the neighborhood of the rotation axis the normalization condition for this Killing vector is X,aX ,a 01, 4X where X = 7a7a. At large distances X -+ r2 sin2 0. The normalization condition for the timelike Killing vector field ( is -(a = V -+ 1 at large distances from the source. If an ergosurface exists in the spacetime, then ( is null on the ergosurface, and spacelike in the ergoregion. In an asymptotically flat spacetime these conditions are sufficient to determine the Killing vectors uniquely (Exact Solutions, Section 17.1). If coordinates are chosen such that = , and 7 = , then the metric can be written as ds2 = e2U(dt - wpdO)2 + e-2U (YMNdxM dxN + D2d 2). In a stationary axisymmetric spacetime all quantities are independent of the coordinates t and q, and are functions of the coordinates (x1, x2) which parametrize the two surface S orthogonal to the group orbits. The functions W, U, and wp 8 transform as scalars under coordinate transformations on the two surface S. Their invariant definitions are as follows: VG= -V = -e2U a 2Ur2 - 2UW2 77 7a = X = -e-2UD2 2UP (ava = W = e-2U)p -2[ab] ab = D2 = VX + W2. The scalar D2 is the norm of the Killing bivector. Bivectors are antisymmetric tensors of second order, or two forms (Exact Solutions, Section 3.4). In terms of the metric components, the scalar D2 is equal to minus the determinant of the two-dimensional metric on the group orbits. In vacuum the covariant form of the field equations for the three invariant scalars are (Kramer and Neugebauer, 1968; Exact Solutions, Section 17.4): (DUM);M =- De-4Uw,MWM (De-4Uw,M);M = 0 (D,M);M = 0. In the above equations w is the twist scalar of the Ernst potential, $ = e2U + iw, defined as follows: wM = w,M. It is related to the Papapetrou metric function u)p as follows: wp,M = De-4U EMNW N; EMN is the Levi-Civita tensor on the two surface S. These equations are covariant with respect to transformations of the two-dimensional metric YMN on the two surface S. The last equation shows that the magnitude of the Killing bivector satisfies the two-dimensional Laplace equation on the two surface S; the scalar D is therefore a harmonic function on the two surface S. The remaining field equations determine the Gaussian curvature of the two surface S: e2U ,2U D,M;N _ e'(M N) + W,(MWO,N) KG'YMN D 2e2U 10 2.2 The Killing Bivector As discussed in the previous section, in a stationary axisymmetric spacetime there are three independent coordinate invariant scalars that can be constructed from the Killing vectors: V, X, and W. A fourth coordinate invariant quantity which is a combination of the previous three is the norm of the Killing bivector: D2 = VX + W2. The scalar D2 is positive outside the Killing horizon, and negative inside the Killing horizon (Hawking and Ellis, p. 167). D2 vanishes trivially on the axis as 7 vanishes on the axis. The region off the axis where the D2 vanishes is the Killing horizon where $[ab] 0 0, but its norm vanishes (Carter 1969). Therefore the quantity D contains important global information which is not carried by either of the three other coordinate invariant quantities, V, X, or W, individually. The field equation for D is remarkably simple: D satisfies the Laplace equation on the two surface S orthogonal to the group orbits. Therefore D is a harmonic function with well defined complex analytic properties. For example, it cannot have any maxima or minima in the interior, and its critical points are isolated. Although it is mentioned in the literature that D is a harmonic function, the boundary data for the two-dimensional Laplace equation have not been given in the literature. It is mentioned that the critical points of D are isolated; however, the number and location of these critical points are not specified for any stationary axisymmetric solution (e.g. Geroch and Hartle, 1982, Appendix A; Wald, 1984, Section 7.1). Let the harmonic function D represent the determinant of the two metric on the group orbits and E be its conjugate harmonic function. Then D and E can be represented as the real and imaginary parts of a complex analytic function W - D + iE. By the properties of complex analytic functions, D and 11 E will satisfy the Laplace equation in two dimensions. The function W will henceforth be referred to as Weyl's harmonic function after Hermann Weyl who defined it for the first time in 1917 for the Schwarzschild solution (Weyl, 1917, p. 140). For the Schwarzschild solution m2 W= Z--, 4Z' where W = D + iE and Z = Pi + izi represents isotropic coordinates. The use of the harmonic function W itself to parametrize the two surface S leads to Weyl's harmonic coordinates. (Further details of coordinates and gauges may be found in Appendix A.) For stationary axisymmetric solutions consisting of rational functions, Weyl's harmonic function for the Schwarzschild solution can be generalized to bl b3 bn S= Z - + 3 + Zn The generalization given above satisfies the boundary condition that W -+ Z at large distances. The function W is defined only on the half space since the cylindrical coordinate pi is defined for pi > 0. Complex functions arising in cylindrical coordinates must satisfy the relation: [W(Z)]* = -W(-Z*), where the * symbol denotes complex conjugation. This condition is satisfied if the coefficients bn are real numbers. For rotating stars the coefficients bi are 2D pressure multipole moments arising from the 2D Poission equation. 2.3 The Electrostatics Analogy The goal of this section is to build intuition about Weyl's harmonic function by developing an analogy with two-dimensional electrostatics. In this analogy the lines of constant D represent equipotential lines while the level curves of the conjugate harmonic function E represent the lines of flux of the electric 12 field. If VE = -VD : 0, then the two sets of curves are orthogonal. The electrical field E - VD is tangent to the flux lines of constant E. For simplicity this analogy is developed first assuming that the two surface S is flat. At the end of the section, the curvature of S will be taken into account. The most general form of Weyl's harmonic function given above corresponds to a non-truncating Laurent series expansion about infinity. It is convergent outside a region containing the origin, and a small circle excising the pole at infinity. (This expansion will be valid if there are no other charges present. If there are additional charges, then the radius of convergence of the Laurent series will be different.) For example, for a stationary axisymmetric spacetime of a rotating star, the excluded region surrounding the origin could represent the two-dimensional cross section of the interior of the star. The Laurent series expansion for a complex analytic function has a simple interpretation in terms of two-dimensional electrostatics. Morse and Feshbach write: "That part of the Laurent series with negative powers of z, which is convergent outside the inner circle of convergence, therefore corresponds to the statement that the field from the charge distribution within the inner circle of convergence may be expressed as a linear superposition of fields of a sequence of multipoles at the point the origin." (Morse and Feshbach, 1953, vol. 1, p. 379) The general term z-n represents a two-dimensional multipole of order 2n-1. From the above discussion it is clear that determinant of the (t, 0) part of the metric D can be regarded as a two-dimensional potential defined on the two surface S and the coefficients bn are two-dimensional multipole moments. It can be regarded as that potential which generates a field consisting of pure multipole fields located at the origin superimposed upon a uniform field of unit magnitude 13 pointing along the p axis of cylindrical coordinates, i.e., along the equator. In terms of complex analysis, Weyl's harmonic function and its derivative have the following Laurent series expansions: OO W = Z + -(-1)nIn n=1 00 dW _b dZ= 1 + (-1)n+l nbn dZ Zn+1l n=1 where W = D + iE and Z = pi + izi represents isotropic coordinates. Equatorial symmetry has been assumed in the above expressions. The above rational expansion will be valid when the two-dimensional surface S is simply connected. In terms of the 2D electrostatic analogy, the above expressions correspond to the multipole expansions for the potential and its gradient-the electrical field E. In terms of stationary axisymmetric solutions, the above expansions represent the determinant of the 2D metric of the group orbits and its gradient on the two surface S orthogonal to the group orbits-i.e., the determinant of the (t, 0) part of the metric and it gradient in terms of (pi, zi) which parametrizes the two surface S. The critical points of the complex analytic function W defined above can be obtained by determining the points at which = 0. In terms of electrostatics, the electrical field vanishes at the points of the equilibrium of the electrical field. The physical significance of the critical points of Weyl's harmonic function W will be discussed in Section 3.5. Next the boundary conditions of the problem are considered. The Killing bivector is zero on the axis since the rotational Killing vector is zero on the axis. Therefore the norm of the Killing bivector D2 is zero on the axis and the axis is always at zero potential. The Killing horizon is also determined by the norm of the Killing bivector vanishing. Therefore, in addition to the axis 14 the Killing horizons will also be at zero potential. The boundary conditions on D are as follows: (1) D approaches the flatspace cylindrical coordinate p at large distances; (2) D = 0 on the rotation axis since the axisymmetric Killing vector q vanishes on the rotation axis; and (3) D = 0 on the Killing horizon since the Killing horizon is defined by the conditions (i) ([ajb] 0, and (ii) -26[ab]ab = D2 -= VX + W2 =- 0. It is assumed that D is regular in the interior. Next the boundary data for Laplace's equation in two dimensions will be discussed. For a rational solution, the Laurent series expansion for Weyl's harmonic function will truncate. In this case, the field in question will consist of a superposition of a uniform field and a finite number of multipoles. The expression for the field will be a polynomial in Z. The roots of this polynomial determine the critical points of Weyl's harmonic function W. If the degree of the polynomial is n, there will be n roots. In terms of the electrostatics analogy, the critical points of Weyl's harmonic function represent equilibrium points of the two-dimensional field, i.e., points where E = 0. Therefore, if it is known that the polynomial in question has n roots, the degree of the polynomial will also be n. Similarly, if it is known that the field in question has n critical points, the number of multipoles present will become known as well. The precise details of the location of the equilibrium points will be determined by the precise values of the multipole moments bn. As the coefficients bn in the polynomial expression for the field are real, the roots of the polynomial will either be real, or occur in complex conjugate pairs. Under the assumption of equatorial symmetry, there will be an even number of roots which will correspond physically to an even 15 number of critical points in Weyl's harmonic function, and an even number of equilibrium points in the field. It is clear that fields with different number of critical points are physically distinct. For example, a dipole field has two lobes, while an octupole field has six lobes. The number of critical points determine topological properties of the solution in two dimensions. From the complex analysis point of view, they correspond to polynomials of different degree, and are therefore distinct. In Chapter Three stationary axisymmetric solutions corresponding to different number of critical points will be discussed. Solutions with different number of critical points are physically distinct in four dimensions as well. For example a solution with two critical points on the axis has one Killing horizon, while a solution with four critical points on the axis will have two Killing horizons. Mathematically multipole moments bn are simply the eigenfunction expansion coefficients of the associated Strum-Louville eigenvalue problem. Therefore knowledge of the precise values of these coefficients constitute boundary data for the Laplace Equation. Once the precise values of the two-dimensional multipole moments bn are given, the boundary data for the Laplace equation in two dimensions is uniquely determined. The field in question is a superposition of point multipoles at the origin and a uniform field. Once the multipole moments are determined, the values of D and its derivative are also determined over suitable curves on the plane. For example, the value of D on a semicircle of radius a is D(a, ) = acos - cos 0 + - cos 30 - ... + b- cos nO. a a an 16 In two dimensions the potential is a complex analytic function. For the class of potentials under consideration, the field is a polynomial. Once the coefficients of the polynomial are known, it is uniquely defined on the entire complex plane. Next, some examples of stationary axisymmetric fields are considered. These are simple solutions with Killing horizons where D2 = VX + W2 = 0. Figures 2.1-3 shows the two-dimensional cross-section of several stationary axisymmetric fields. Rotation Axis Killing Horizon Figure 2.1 Two-dimensional Cross-section of the Schwarzschild Solution The first example is the Schwarzschild solution(Fig. 2.1). The determinant of the (t, q) part of the metric is zero on the axis and on the horizon represented by the semicircle. There are two critical points of this field located at the points where the axis intersects the horizon. Rotation Axis ?) Killing Horizon 1 I I Killing Horizon 2 I I I Figure 2.2 Two-dimensional Cross-section of the Double Kerr Solution The second example consists of two Kerr Black holes revolving about the same axis (Kramer and Neugebauer, 1980). This configuration has two horizons each surrounding a single Kerr black hole. There are four critical points where the two horizons intersect the axis(Fig 2.2). Rotation Axis Cross Section of Toroidal Killing Horizon Figure 2.3 Two-dimensional Cross-section of a Solution with a Toroidal Killing Horizon The third example represents a stationary axisymmetric solution with a toroidal Killing Horizon(Carter, 1973). In terms of the two-dimensional boundary value problem, the axis is at zero potential. The circle represents the cross section of the toroidal Killing Horizon which is again at zero potential(Fig 2.3). It should be noted that the Killing horizon may not necessarily be an event horizon. Hence the precise interpretation of the third solution is not clear. According to Carter, this configuration will necessarily have a critical point between the horizon and the axis (Carter, p. 188, and Fig. 10.2). As harmonic functions do not admit maxima or minima in the interior, this critical point must be a saddle point. 20 From the previous examples it is clear the every stationary axisymmetric solution has an associated 2D electrostatics problem. For pure vacuum solutions the boundary conditions are determined by the determinant of the (t, ï¿½) part of the metric vanishing on the Killing Horizon. For solutions to stationary axisymmetric systems with perfect fluid sources, for example, the spacetime surrounding a rotating star, the determinant of the (t, 0) part of the metric has a two-dimensional multipole expansion since it satisfies Laplace's equation in the vacuum region. Corresponding to the Laplace's Equation in vacuum, there is also a Poission equation for D inside perfect fluid sources: V2 D = 167rp-f-g, where p is the fluid pressure, and g is the volume element of the four dimensional metric. (Trumper 1967, Eqn. 5; Bardeen, 1971, Eqn II.14; Chandrasekhar and Friedman, 1972, Eqn. 74 and 75; Butterworth and Ipser, 1976, Eqn. 4c). For a rotating star the source term for the harmonic function D in the exterior region will be the fluid pressure in the interior of the star. Hence the coefficients bi of the harmonic expansion for D in the exterior region are simply two-dimensional pressure multipole moments. The multipole moments bi can be evaluated as volume integrals of the pressure inside the star. The procedure for determining the coefficients bn for rational solutions given in harmonic coordinates will be given in Chapter Three. One of the main results of the present work is to point out that by comparing the coefficients bn, numerical stars solutions and exact analytical solutions may be compared. Up to this point the two surface S has been regarded as a flat plane. In reality the two surface S is curved. How does the curvature of S affect the electrostatics analogy ? Two-dimensional electrostatic fields on curved surfaces have been discussed by Riemann (Riemann 1892, Klein 1893). Electrostatics fields on 21 Riemann surfaces have qualitatively similar properties in terms of sources, field lines, and equilibrium points. The global structure of a multipole field does not change on a curved surface: for example, an octupole field will have a six-lobed structure regardless of whether or not it is on a curved surface. So the global structure of the norm of the Killing bivector remains the same when the curvature of the two surface S is taken into account. The only complication is the precise definition of coordinates. That is where precisely the critical points are located. In Chapter Four it will be shown that field equations are conformally invariant. The metric functions V, W, D2, and X can be solved first in flat space. The two surface S is built afterwards. The coordinates on this surface can be chosen uniquely. The two surface S is known very well. At every point on S, the Gaussian curvature of the two surface is determined from the field equations: RG1212 = -2e- 2 (-Vv VV + p2 e-4Vw.Vw). (911922 - g122) In the above expression, v and w, are the Bardeen metric functions; their invariant definitions may be found in Appendix A. Therefore at every point on the two surface S three coordinate invariant quantities are known: KG, D, and E. For an arbitrary local coordinate chart these variables are not directly related. However, there is a global boundary condition along the axis. Spacetime is flat near the axis. This fact, along with the requirement that a single global chart cover the entire two surface S, fixes global coordinates on S uniquely. The uniqueness of global coordinates on S is demonstrated in greater detail in Section 4.2. As the curvature of S is known at every point, the two surface S can be embedded in three dimensional Euclidean space(Fig. 2.4). More precisely, if Figure 2.4 The isometric embedding of the two-surface S orthogonal to the group orbits in R3. This is an inverted Flamm's paraboloid in isotropic coordinates. the Gaussian curvature is negative definite, it is possible to make an isometric embedding of a surface in three-dimennsions (Romano and Price, 1995). This can be done by solving the Darboux equations of classical differentical geometry. The Darboux equation belongs to the class of Monge-Ampere equations and may be solved by the method of characteristics (Courant and Hilbert, Vol. II, pg. 495-499). In the case of the Schwarzschild solution, the 3D embedding is simply the Flamm's paraboloid (Flamm, 1916; MTW Section 23.8, Fig. 23.1). Therefore for every stationary axisymmetric solution there will be a unique curved surface in three dimensional space representing the curvature of the two surface S orthogonal to the group orbits. Weyl's harmonic function can be visualized as representing an electrical field on this curved two surface. The critical points of 23 Weyl's harmonic function will correspond to equilibrium points of the electrical field, where E = 0. CHAPTER 3 WEYL'S HARMONIC FUNCTION In the relativity literature there is considerable confusion regarding Weyl's harmonic function. In the textbook by Bergmann(1942) it is stated that on the basis of symmetry alone the static axisymmetric metric can be brought into the Weyl form (p. 206). This error was noted by Synge(1960) who pointed out that the existence of Weyl coordinates in vacuum was a consequence of the field equations (Weyl, 1917, p. 137; Synge, 1960, p. 310, Note 1). As noted in the previous chapter, for stationary axisymmetric systems D, the determinant of the (t, q) part of the metric, satisfies the Laplace equation on the two surface S orthogonal to the group orbits. The important question: What is the boundary data for this field equation? has not been discussed in the literature. In this chapter the boundary data for the Laplace equation will be determined for several stationary axisymmetric solutions based on Weyl's harmonic function 1W. The real and imaginary parts of Weyl's harmonic function satisfy the Laplace equation in two dimensions. Therefore, determining Weyl's harmonic function for a particular stationary axisymmetric solution is equivalent to determining boundary data for the Laplace equation for the stationary axisymmetric solution in question. In this chapter the boundary data for the Laplace equation will be determined first for the Schwarzschild solution and then for the Kerr solution in terms of Weyl's harmonic function. A general prescription will be given for 25 determining the harmonic function W for the recently obtained rational solutions to the stationary axisymmetric field equation generated by the application of soliton theory to General Relativity. Weyl's harmonic function will then be determined for the Tomimatsu-Sato series of solutions. Finally, the relation between Weyl's harmonic function and the structure of the Killing horizon will be discussed for the Double Kerr solution and the Manko et al., 1994 solution. In the previous chapter it was shown that the magnitude of the Killing bivector, -2[aOb]6ab = D2 = VX + W2, satisfies the two-dimensional Laplace equation on the two surface S orthogonal to the group orbits: (D,M);M 0. The magnitude of the Killing bivector D2 is also the determinant of the (t, 4) part of the metric. The boundary conditions on D are as follows: (1) D approaches the flatspace cylindrical coordinate p at large distances; (2) D = 0 on the rotation axis since the axisymmetric Killing vector q vanishes on the rotation axis; and (3) D = 0 on the Killing horizon since the Killing horizon is defined by the conditions (i) ([a7b] 0 0, and (ii) -24[a7b]a b = D2 = VX+W2 = 0. Therefore the question arises: What is the boundary data for D that is different for distinct solutions ? The answer to this question lies in the harmonic properties of D, more specifically, in the critical points of the harmonic function W = D + iE defined on the two surface S. It will be shown in this chapter that the harmonic function W has critical points, where VD = VE = (0, 0), at the intersections of the Killing horizons and the rotation axis. The Kerr solution has two critical points on the axis. The double Kerr solution of Kramer and Neuguebauer has four critical points on the axis. The general N-Kerr solutions of Yamazaki, which are nonlinear superpositions of N separate Kerr-like solutions on the axis, 26 has 2N critical points along the axis where the Killing horizons of each Kerrlike solution intersects the axis. Therefore, the number of critical points of the harmonic function W constitute topological boundary data for the Laplace equation. The knowledge of these critical points permits the differentiation of one stationary axisymmetric solution from another. If the critical point does not lie on the axis, as in the solution with a toroidal Killing horizon, the physical significance of the critical point is not clear, as the complete solution for the field equations for the toroidal Killing horizon has not yet been constructed. In this case, according to Carter(1973), there will be a critical point in the interior region between the toroidal horizon and the rotation axis (Fig. 2.3). Also, a toroidal Killing horizon does not intersect the rotation axis. The critical points of all known exact solutions do lie on the axis. Therefore, the consideration of off-axis critical points of the harmonic function W is an issue of principle relegated to future investigations. In the existing literature on stationary axisymmetric systems in General Relativity although it is mentioned that D is a harmonic function, the boundary data for the two-dimensional Laplace equation that D satisfies has not been given for any of the known solutions. It is mentioned that the critical points of D are isolated; however the number and location of these critical points is not specified for any stationary axisymmetric solution (e.g. Geroch and Hartle, 1982, Appendix A; Wald, 1984, Section 7.1). The discussion of this important issue begins with the critical points of the simplest known solution-the Schwarzschild solution-which is discussed in the next section. 27 3.1 The Schwarzschild Solution The boundary data for the Laplace equation for the Schwarzschild solution may be obtained very easily from Weyl's classic paper of 1917. Weyl introduced the use of the determinant of the (t, ï¿½) part of the metric (D = Ph) and its conjugate function(E = Zh) as coordinates for the first time in 1917. He noted that in such coordinates the spherically symmetric Schwarzschild solution corresponds to a line segment of length 2m, with constant linear mass density, along the axis of symmetry (Weyl, 1917, p. 140). As D vanishes on the horizon as well as on the axis, both curves are represented by D = Ph = 0 in Weyl's harmonic coordinates, and both become part of the axis in Weyl's harmonic coordinates defined by Ph = 0. Weyl also introduced isotropic coordinates for the Schwarzschild solution in the same paper (Weyl, 1917, p. 132, Eqn. 12). The key to unravelling the boundary data for the Schwarzschild solution is the equation used by Weyl to map the Schwarzschild solution from isotropic coordinates to harmonic coordinates. This conformal transformation describes the scalar D in a chart which makes its complex analytic properties transparent. Weyl's harmonic function for the Schwarzschild solution is m2 W= Z 4Z' where W = D + iE = Ph + izh are harmonic coordinates, and Z = pi + izi represents isotropic coordinates (Weyl, 1917, p. 140). This conformal transformation maps the horizon which is a semicircle of radius m/2 in isotropic coordinates to a line segment of length 2m in harmonic coordinates. In this form the boundary data for Laplace equation is quite apparent. In terms of two-dimensional multipole moments discussed in Section 2.3 this 28 map represents the superposition of a uniform field and a dipole field. This map known, as the Joukowski map in fluid dynamics, represents a uniform flow along the z direction past an infinite cylinder. Very far from the cylinder the flow is uniform. The scalar D approaches ordinary cylindrical coordinate p at large distances from the source. The fluid dynamics analogy can be extended further to determine the critical points of Weyl's harmonic function. In fluid dynamics when a flow encounters a barrier, or corner, the fluid velocity must vanish; such points are known as stagnation points. In the case of the Joukowski map, the stagnation points are at the two points where the flow encounters the cylinder head on-i.e., at the two points where the circle representing the cylinder intersects the axis. Therefore, in the case of the Schwarzschild solution the critical point of the harmonic function W are at the North and South poles where the horizon intersects the axis. Next the critical points of the harmonic function W for the Schwarzschild solution is determined by direct computation. At the critical points of a complex harmonic function, its first derivative vanishes; if higher derivatives vanish, the critical points are termed degenerate. For the Joukowski map, the critical points are determined by the roots of the equation dW m2 W=i+ =0 dZ I 4W2 The roots of this equation are Z = 0 + im/2, which correspond to the North and South poles. From the elementary properties of the Joukowski map, it follows that VD = VE = (0, 0) at the poles for the Schwarzschild solution. These are the isolated critical points of the harmonic function W. Since the derivative is a 29 second degree polynomial in , it has only two roots; therefore the Schwarzschild solution has only two critical points. Hence, the two critical points at the poles are the only critical points of Weyl's harmonic function W for the Schwarzschild solution. The boundary data for the Laplace equation for the Schwarzschild solution has now been determined. The determinant of the (t, q) part of the metric, D2, has not been discussed in the literature as an important invariant quantity. Next this important quantity is expressed in several different coordinate charts for the Schwarzschild metric. The scalar D is the real part of Weyl's harmonic function W = D + iE. Therefore, considering the real part of the Joukowski map given above, an expression is obtained for D2: D2 = VX = 2 = ph in harmonic coordinates by definition rsin9, - f? 12 in isotropic coordinates tk 4r?)' = r sin 1 - in Schwarzschild coordinates. S 8 ( r, 3.2 The Kerr Solution The Kerr solution is considered next: in particular, the boundary data for the Laplace equation for the Kerr solution will be determined. The data for the Kerr metric is not quite apparent at first; however, proceeding in analogy with the Schwarzschild solution, the result in the end turns out to be simple and qualitatively similar. The key issue is the number of critical points of the harmonic function W for Kerr solution. In the static limit the number must reduce to the result for the Schwarzschild solution derived in the previous section-two critical points at the poles. Since rotation does not alter the topology of the 30 horizon the result remains the same for the rotating Kerr solution. An example where the critical points change due to a change in the topology of the horizon is given by Carter(1973): in the case of a toroidal Killing horizon, a saddle point exists between the torus and the axis (Fig. 3.1). Next, the critical points of the harmonic function W will be obtained by the methods of potential theory. As discussed in Section 2.3, D is a potential function in two dimensions, and curves of D = constant are equipotential curves. For the Kerr solution, the horizon and the axis intersects at right angles at the two poles. If the intersection were not at right angles, there would be a kink in the horizon which would destroy the smoothness properties of the horizon at the pole. From the point of view of potential theory, the axis and the horizon are two equipotential curves with D = 0 that intersect at right angles. By the elementary properties of solutions to the Laplace equation, it follows that VD -+ (0, 0) smoothly at the poles (Jackson, 1975, Section 2.11). In two dimesions, near a right angle corner, the solution to the Laplace equation for the potential is given by D = DoR2 sin 2E, where R, E are defined in a small chart which is flat in a sufficiently small neighborhood of the North pole (Jackson, 1975, Section 2.11, Fig. 2.13). The components of the field near a right angle corner are E = -VD = ( -2DoRsin 26, -2DoRcos 2E) where Do is a constant. The field varies linearly with distance(R) from the pole, and vanishes at the pole: VD = (0, 0) at the poles. By the Cauchy-Riemann relations VE = (0, 0) as well. The gradient of the determinant of the (t, ï¿½) part 31 of the metric and its conjugate harmonic function both vanish at the poles. The fact that the gradient of the determinant of the (t, ï¿½) part of the metric vanishes at the poles for the Kerr solution has not been noted in the literature. The elementary properties of solutions to Laplace equation in two dimensions lead to nontrivial conclusions in the theory of rotating black holes. It has been shown that the gradient of D must vanish at the poles due to purely geometric boundary conditions at the poles. Next it is shown that VD = (0, 0) at the poles using the constancy of the surface gravity boundary condition on the horizon. In the Bardeen form of the stationary axisymmetric metric, the function e2v = V + W2/X = D2/X. On the horizon of a Kerr black hole e2" = 0. Bardeen has shown that the gradient of ev approaches a limiting value, the surface gravity NH, as the horizon is approached: e-P(ev),r -KH; and e-/(ev)o -+ 0 since the surface gravity is constant on the horizon (Bardeen, 1973, Eqn. 2.18-20). The conformal factor el - 1 near the poles (Appendix C). Therefore, the gradient Vev is bounded and well behaved near the horizon; its components in any coordinate basis are finite; Ve" - (KH, 0) in polar coordinates (ri, Oi) while in cylindrical coordinates (pi, zi), Ve" is bounded by the vector A = (KH, NH) near the horizon. Next the magnitude of the rotational Killing vector is considered. In the neighborhood of the rotation axis the normalization condition for the rotational Killing vector is X aXa -+1, X'" 4X where X = ?aa; X - 0 near the axis. By arguments similar to the above, it follows that the gradient vector r is well behaved. Near the axis it is bounded X1 32 by the vector B = (2, 2) in cylindrical coordinates (ps, zi), since at large distances X - r2 sin2 0, and V -+ (2 sin 0, 2 cos 9) in polar coordinates (ri, 9i). X1 Now that it is has been established that the gradients of X and ev are bounded near the poles, where the horizon intersects the rotation axis, it is straightforward to calculate the gradient of D directly at the poles 1 D = e XI VX 1 VD= -ev + X Vev. 21 Along the axis the second term in the gradient drops out since X = 0 on the axis, and Ve" is bounded by the vector A. Therefore along the axis VX VD = le" VX X'2 As the North pole is approached along the axis, e" -+ 0 smoothly and is X2 bounded by the vector B. Hence as the North pole is approached along the axis VD -+ (0,0). Since e' approaches zero smoothly, VD -+ (0, 0) smoothly as well. Next the North pole is approached along the horizon. Along the horizon the first term in the gradient drops out since e" = 0 on the horizon, and X is X7 bounded by the vector B. Therefore along the horizon VD = X 2Ve". As the North pole is approached along the horizon, X, - 0 smoothly and Vev is bounded by the vector A. Hence as the North pole is approached along the horizon VD -+ (0, 0). 33 Since X1 -+ 0 smoothly, VD -+ (0, 0) smoothly. Hence the limit exists and VD = (0, 0) at the North pole. It is established that VD = (0, 0) at the two points where the horizon intersects the axis. Thus Weyl's harmonic function has two critical points for the Kerr solution: one at each pole. This result has been established by three different methods:(1) Topological method: since the topology of the horizon is not altered by rotation, the number of critical points of the Kerr solution must be the same as those of the Schwarzschild solution. (2) Potential theoretic method: the field goes to zero linearly with distance near a right angle corner. (3) The method employing the horizon boundary conditions: e2v -> 0 smoothly near the horizon; its gradient normal to the horizon is the surface gravity KH; due to the constancy of the surface gravity on the horizon, its gradient has no tangential component. Once the number of critical points of Weyl's harmonic function is determined for the Kerr solution, the exact form of the harmonic function W can be determined uniquely. Weyl's harmonic function for the Kerr solution is therefore given by: m2 - a2 W=Z 4Z This is similar to the Schwarzschild case except the mass has been replaced by the quantity k = -/m2 a2. This value is determined by making a conformal transformation from Weyl's harmonic chart in prolate spheroidal coordinates to Weyl's isotropic chart in spherical polar coordinates. The complete form of the Kerr metric in Weyl's isotropic coordinates is given in Appendix C. The parameter k is related to the surface area of the horizon A and the surface gravity KH as follows: k = - (Carter, 1973, p. 197, Eqn 10.55). If the angular momentum satisfies the condition a < m, a Killing horizon will 34 exist for the Kerr solution. In isotropic coordinates, the horizon is a semicircle with constant ri coordinate, ri = k/2 (Bardeen, 1973, p. 251, Eqn. 2.8). By the Joukowski map given above, it will be mapped to a line segment of length 2k in Weyl's harmonic coordinates. In prolate spheroidal coordinates, this is the line segment between the two foci (Appendix C). It should be noted that even though the horizon is determined by curve with a constant coordinate value r = k/2 in isotropic coordinates, this does not imply that the horizon is geometrically spherical since the metric on the two surface S in the Kerr case has angular dependence. Similarly, even though the conformal transformation is the Joukowski map for both the Schwarzschild and the Kerr solution, the interpretation of the coordinates on the two surface S is different. Finally the norm of the Killing bivector D2 is examined for the Kerr solution in various coordinate charts. D2 = -24[afib] ab = VX + W2 vx2 = P in harmonic coordinates by definition sin 2 m2 _ a_22 = r sin 0 1 - in isotropic coordinates = (rjL - 2mrBL a2) sin2 0 in Boyer - Lindquist coordinates. In Boyer-Lindquist coordinates, 1 (r= L- 2mrBL a2) sin 0 D = (rBL - m) sin0 (r BL - 2mrBL + a2) cos0] VD [(rL - 2mrBL + a2) ' rBL - [0, (rBL - 2mrBL + a2) along the axis -0 along the axis rBL = 0, 0] only at the poles sneA=(221 = +r2 2) i since A = (rL-2mrBLa2) = 0 only at the poles where rBL = m+(m2-a 2. The above calculation is not completely valid since the denominator 1 (r2L - 2mrBL + a2) 2 becomes zero on the horizon. To get around this difficuly, the following technique due to Professor Whiting is employed: Instead of evaluating VD at the poles, its magnitude is evaluated. M - vVD - VD = /grrD,rD,r + gOOD,oD,e (r2L - 2mrBL + a2) (rBL - M)2 sin2 0 (r2L - 2mrBL + a2) cos2 0 - + B r 2 COS2 0 (r 2 L r 2 COS2 0 rBL + a2c (r L - 2mrBL + a2) rB ï¿½ a2 cos2 9 (rBL - m)2 sin 20 (r2L - 2mrBL + a2) cos2 0 r + a2 COS2 rBL 2 cos2 9 The magnitude M does not blow up anywhere: the first term in M vanishes along the axis, while the second term vanishes along the horizon. It is clear that M goes to zero smoothly as the pole is approached. Hence, VD goes to zero smoothly as the pole is approached in Boyer-Lindquist coordinates. As before the pole is shown to be a critical point of the harmonic function D. It has been stated inThe Mathematical Theory of Black Holes(MTB) that the above expression for A for the Kerr solution is "A solution . compatible with the requirements of regularity on the axis and convexity of the horizon . . . " (MTB, p. 279). The function D = A sin 0 satisfies the Laplace equation for D = e = A1/2f(0) which is a part of the Einstein field equations (MTB, Eqn 6.43). [A1/2(1/2),r],r + f- 1f(),r,r = 0. Therefore for a particular stationary axisymmetric solution, such as the Kerr solution, the solution for D, which in this case is A sin 9, must be unique. The question therefore becomes what is the boundary data for the Laplace Equation (MTB, Eqn 6.43) ? It has been shown above that the determinant D for the 36 Kerr solution has two critical points at the poles. As shown above by direct computation VD= V(rBL - 2mrBL + a2) sinO= (0,0) at the poles. Therefore the function A sin 0 satisfies the correct boundary conditions for the problem in question. Hence D = (rL - 2mrBL ï¿½ a2) sin 0 is the unique solution for the Kerr metric. The boundary data for the function D is transparent in the isothermal gauge. In other gauges it is not so clear since the metric on the two surface S and determinant of the (t, q) part of the metric become intertwined. The various gauges that have been used in the study of stationary axisymmetric solutions are discussed in Appendix A. Before proceeding further, it is important to note the following points. The determinant D2 carries global information since D2 = VX + W2 = 0 on the horizon. None of the three coordinate invariant quantities V, X, or W, singly by themselves carry global information about the horizon that is carried by their combination D2 = VX + W2. Therefore, from the physical point of view the determinant D has important global properties. From the mathematical point of view, function D also has global properties since D is a harmonic function, e.g. its critical points are isolated, it has no maxima or minima in the interior, etc. Although the quantities V, X, and W, satisfy a system of coupled quasi-linear second order elliptic partial differential equations, none of them are harmonic functions; only the combination D2 = VX + W2 is harmonic with well defined global properties. This confluence of global physical properties and global mathematical properties is quite remarkable, and at the same time necessary. These global properties also have practical consequences. The fact that Weyls' harmonic function V has critical points at the poles has been derived quite easily 37 using potential theory which embodies the global harmonic properties of W. In contrast, deriving the same result using local properties only of the functions e' and X, was less insightful and tedious. 3.3 Weyl's Harmonic Function for Rational Solutions In this section Weyl's harmonic function will be further investigated for rational soloutions of the stationary axisymmetric solutions. As most of the known solutions fall into this category, this is not too restrictive. The form of the metric in the isothermal gauge is preserved under conformal transformations. Several authors have incorrectly commented that once a conformal transformation has been made to Weyl's harmonic coordinates (Ph, zh), all information about the harmonic function W = D + iE = Ph + izh is lost. Lewis(1932) writes: "the canonical [Weyl's harmonic] coordinates serve to remind one of the degree of arbitrariness involved in our solutions" (Lewis, 1932, p. 177). Similarly Zipoy(1970) writes "Since p [Ph] is initially an arbitrary harmonic function of (7, () [Pi, zi], there will clearly be no intrinsic means of determining what the coordinates represent physically" (Zipoy, 1970, p. 2120). Evidently D, the determinant of the (t, q) part of the metric, is a coordinate invariant quantity. It will have a unique value at every point on the 2-surface S. The confusion stems from treating Ph = D as "just another coordinate", and ignoring its invariant definition. It should be pointed out that the invariant definition of D as the norm of the Killing bivector did not exist prior to Carter's work on Killing horizons (Carter, 1969). In the previous section Weyl's harmonic function was determined for the important solutions of Schwarzschild and Kerr. Additional stationary axisymmetric solutions have been obtained recently. These exact solutions are 38 determined in Weyl's harmonic coordinates where the explicit form of the harmonic function remains obscure. The critical points of Weyl's harmonic function for these solutions have not been determined. Therefore it would be highly desirable to obtain a general procedure for identifying the critical points of a harmonic function for any rational stationary axisymmetric solution. Once this is done the determinant of the (t, q) part of the metric could be determined as a function of Weyl's isotropic coordinates on the two surface S. Once a solution is expressed in Weyl's isotropic coordinates the topology of the horizon, the boundary data for Laplace equation, and the critical points of the harmonic function all become transparent. As pointed out in the introduction, many authors have erroneously claimed that once Weyl's harmonic coordinates are chosen, all the information about the harmonic function 14W is lost. The information in question is not lost: it is hidden in the conformal factor e27. A procedure for extracting Weyl's harmonic function 14W from the conformal factor e2y is described next. This procedure may be applied when the number of critical points is finite, and the functions involved are rational. The prescription begins with a recapitulation. In Weyl's harmonic coordinates the metric on the 3D space is 2-y 2 2 2 2. ds D = e (dp + dz) + pd2. The flatness boundary condition on the conformal factor is e27 -+ 1 on the axis. In Weyl's isotropic coordinates the metric on the 3D space is: ds2D = e2(dp? + dz2) + p2B2dï¿½2. The flatness boundary condition on the conformal factor is e2( - B on the axis. Upon first examination it appears that the isotropic metric has two functions 39 in the 3D space-e24 and B-whereas the harmonic metric has only one function e2". Upon closer examination and by reference to the Kerr metric in isotropic and harmonic coordinates, it is apparent that while e2( and B are polynomials, e27 is a rational function-being the ratio of two polynomials. Under a conformal coordinate transformation from isotropic coordinates Z = pi + izi to harmonic coordinates, W = D + iE = Ph + izh, the conformal factor transforms as A2 A2 dWi 2 dZ In isotropic coordinates the determinant of the (t, 0) part of the metric D = piB, OB OB VD = (B + pi , pi ). Opi ozi On the axis pi = 0, and VD = (B, 0). By the Cauchy-Riemann relations VE = (0, B). By definition, 14W = D + iE; and therefore along the axis A2 dV_2 A --+ B2, dZ and the axis boundary condition is satisfied e2( e2 1 e27 e -- --+ A2 B2 3.3.1 Determination of Critical Points in the Harmonic Chart The critical points of Weyl's harmonic function are defined by d = 0. At such points, VD = VE = (0, 0); therefore, at the critical points along the axis B = 0. Further, by the flatness condition near the axis, eC = B = 0 also. 40 Therefore by the conformal transformation given above, at the critical points the conformal factor in Weyl's harmonic coordinates e2- e2( 0 e2"7 --, A2 -2 0 If the critical points are not degenerate, i.e., if d2W 0, then by L'Hospital's rule Be e- = -+ 1. OB The above considerations are valid regardless of whether of not e7, e, and B are transcendental or rational functions. At the critical points of Weyl's harmonic function the numerator and denominator will both approach zero. If the conformal factor e27 is a transcendental function there could be an infinite number of critical points. In the case of rational solutions to stationary axisymmetric field equations, the conformal factor e2- in Weyl's harmonic coordinates is given by ratio of two polynomials. In this case the number of critical points is finite, being determined by the degree of the polynomials involved. This is not an additional restriction in terms of known solutions, as all known solutions are of this form. For rational solutions e2^y is the ratio of polynomials in two variables (Ph, Zh) e2-y F(ph, Zh) G(ph, zh) Along the axis the numerator and denominator polynomials become identical, and e27 -4-+ 1. Along the axis, let F(0, zh) -+ G(O, zh) ~ P(zh); then 2-y F(0, zh) P(zh) e2 =- -+ + 1. G(0, zh) P(zh) At the critical points along the axis e27 -+ 0. Therefore, 2 P(zh) 0 e P(zh) -+ P(zh) 0' 41 The critical points of VW are determined very simply by the roots of the polynomial P(zh). The critical points of Weyl's harmonic function for a rational solution are determined by the roots of the polynomial P(zh), where P(zh) is defined by F(0, Zh) -+ G(0, Zh) -+ P(zh), and e2 = F(phzh) A polynomial GFph,ZhJ of degree n has n roots. Due to the equatorial symmetry of the problems under consideration, the roots will occur in pairs +a and -a, and n will be an even number. The critical points will occur in pairs one above and one below the equatorial plane. Next we verify the above prescription by applying it to the Kerr solution. The conformal factor e2/ in Weyl's harmonic coordinates consists of quotients of quadratic polynomials for both the Schwarzschild and the Kerr solution. Hence they must have the same number of critical points which is determined by the degree of the polynomials. Since quadratic polynomials have two roots, both the Kerr and Schwarzschild solution have two critical points. The conformal factor for the Schwarzschild solution is (Appendix B): - x2 - 1 x2 - y2 ' while the conformal factor for the Kerr solution is (Appendix C): p22 ~2 2 e27 2 p2(x2 _ y2) where x and y are prolate spheroidal coordinates. In such coordinates, along the axis y = 1, and the desired polynomial P(x) = x2 - 1. The critical points are determined by P(x) = x2-1 = 0-i.e., x = +1. In prolate spheroidal coordinates these are also the points where the horizon intersects the axis-i.e.,the North and South poles. In cylindrical coordinates, zh = kxy = kx along the axis. Therefore in cylindrical harmonic coordinates the two critical points at the poles are located 42 at zh = ï¿½m for the Schwarzschild case, and at Zh = =1/m2 - a2 for the Kerr case. It should be noted that the denominator polynomial is the same for both cases, while the numerator has changed. The conformal factor for the static Schwarzschild and the rotating Kerr solutions are different. However, Weyl's harmonic function has a similar form for both cases. 3.3.2 Determination of Critical Points in the Isotropic Chart The key to unravelling the transformation from harmonic coordinates to isotropic coordinates is the fact that for rational solutions the numerator and denominator are both polynomials. The critical points are determined by the roots of the polynomials. The conformal factor will vanish at the critical points, which are the same points on the two surface S regardless of the which coordinates isotropic or harmonic that is being used. Harmonic and isotropic coordinates are defined in Section A.3. The algorithm for extracting the boundary data of the Laplace equation will be first illustrated for a three term harmonic function W with two free parameters bi and b3 and four critical points W Z _b b3 Z Z3' Physically this represents the superposition of a dipole field(bi) and an octupole field(b3) upon a uniform field of unit magnitude in two dimensions. For the Schwarzschild solution, b, = m2/4, and b3 = 0. The critical points of this transformation are determined by the four roots of the quartic polynomial dW bi 3b3 = 1+ dZ Z2 Z4" 43 In general the roots of this equation will be complex. In all known cases, however, the critical points lie on the axis, and all the roots are pure imaginary quantities. By definition, W = D + iE and D _ piB(pi, zi). Along the axis, VD = (B, 0) and VE = (0, B). By the Cauchy-Riemann relations, D OE dW1 b 3b3 = _ = B(0, zi) = Q(zi) = T(-d) = 1 - + -4 where Q(zi) is a polynomial in 1. For the Schwarzschild solution Q(zi) m2 1- At the critical points, VD = (0, 0) and VE = (0, 0). Hence the critical points along the axis are determined by the roots of the polynomial Q(zi): bl 3ba B(O, zi) = Q(zi) = 1 - - + - = 0. z? zi z I For the Schwarzschild solution the roots are zi = +m/2. Due to the equatorial symmetry of the problem, the roots always occur in pairs + one above and one below the equatorial plane. For the above quartic, the roots will be ï¿½/31 and ï¿½i2 where 01 and /32 are given by Sb, + b - 12b3 1/= 2 Sb - b - 12b3 32 = 2 In Weyl's harmonic coordinates, these points along the axis are mapped according to b1 b3 zh - l(0) = zi + - zi z Let /31 and /32 in isotropic coordinates map to a1 and a2 in harmonic coordinates. Then bl b3 = 8 1 + /3 1 b_ b3 a2 = 32 + - 3' 32 /302 The solution in question has four critical points along the axis. In Weyl's harmonic coordinates these points are represented as: (0, ai), (0, a2), (0, -a2), and (0, -al). In Weyl's isotropic coordinates these same points are represented as (0, 1i), (0,132), (0, -032), and (0, -01). For the Schwarzschild solution 3 = m/2 in isotropic coordinates, transforms to ca = m in harmonic coordinates. At the critical points, e2- P(zh) 0 P(z) O The critical points are determined by the roots of P(zh). In the case under consideration, the polynomial P(zh) will vanish at ï¿½aj and ï¿½a2. Hence the limiting value of the conformal factor along the axis will be determined uniquely by the fourth degree polynomial P(zh) with roots at ï¿½al and ï¿½a2. That is: P(zh) = (zh - a )(z - a2) z4 - 2 Zh a2zh + ao. Substituting a, and a2 in the definitions of ao and a2, and then substituting l1 and 32 for al and a2, the following relations are obtained: 16(b2 + 4b3)2 a0 = 1 27b3 16b1 4b a2 = -+ . 3 27b3 Here, the coefficients a2 and ao are expressed in terms of b, and b3. Therefore a relation has been established between the polynomial P(zh) in harmonic coordinates and the polynomial Q(zi) in isotropic coordinates. Given the polynomial 45 P(zh) in harmonic coordinates, it is possible to construct the polynomial Q(zi) in isotropic coordinates, and vice versa. The transformation for six critical points has been computed explicitly using a Mathematica program. The algebraic procedure given above can be extended to eight critical points located symmetrically with respect to the equatorial plane. The conformal factor in harmonic coordinates will be a polynomial for rational solutions. At the critical points of the harmonic function, the polynomial P(zh) will vanish and the conformal factor e27 -+ 0. The degree of the polynomial P(zh) determines the number of roots of the polynomial P(zh), and therefore the number of critical points located along the axis present in the solution. Once the number of critical points is known, the number of terms in the polynomial Q(zi) which determines the same critical points in isotropic coordinates are also known. For a solution with 2n critical points, the highest power of zh in P(zh) will be 2n and the highest power of zi in Q(zi) will be -2n. The number of free parameters ai in P(zh) will be the same as the number of free parameters bi in Q(zi), both being equal to n. Therefore a general form of Q(zi) can be assumed with the correct number of terms and free parameters bi. The location of the 2n critical points in isotropic coordinates which are the roots ï¿½ 3 of the polynomial Q(zi) can be expressed in terms of the n coefficients bi. Next these roots can also be transformed to harmonic coordinates, since this map is determined also in terms of the coefficients bi. Hence the n roots ï¿½ai of the polynomial P(zh) will be determined in terms of the n coefficients bi. Finally the n coefficients ai of the polynomial P(zh) could be expressed in terms of the n coefficients bi. The transformation is now completely determined, explicit relations between the coefficients of the polynomial P(zh) and Q(zh) having been 46 identified. Therefore, given a rational solution in Weyl's harmonic coordinates, it is possible to extract the harmonic function W in isotropic coordinates. This could be done algebraically up to eight critical points. For rational solutions it will always be possible to identify the finite number of critical points ï¿½ai numerically given the parameters bi. Before proceeding to the next section, the key points are summarized. In a stationary axisymmetric system, the magnitude of the Killing bivector, -24[aTb]a7b = D2 = VX + W2, is a harmonic function on the two surface S orthogonal to the group orbits. In isotropic coordinates, D = piB, and VD = (B, 0) = (Q(zi), 0) along the axis. The critical points of the harmonic function, VD = (0, 0) are determined by the roots of the polynomial Q(zi) in isotropic coordinates. In harmonic coordinates, e2 P( = z- - 1 along the axis. At the critical points e27 -+ 0. The critical points of W = D + iE = Ph + izh are determined by the roots of the polynomial P(zh). If the coefficients ai of P(zi) are known, then the coefficients bi of Q(zi) can be determined and vice versa, since the roots ai of P(zh) are related to the roots 3 of Q(zh) by the conformal transformation bl b3 ai = #i + b/ b3 + ... + (-1)j+l bj Every stationary axisymmetric solution has a unique set of critical points. These critical points provide boundary data for the Laplace equation for D. Contrary to the claim of Zipoy, that boundary data for D cannot be determined in harmonic coordinates, it has been shown here that the data can be determined in any coordinate chart. 47 3.4 The Zipoy-Vorhees and the Tomimatsu-Sato Solutions In previous sections Weyl's harmonic function was discussed for the Schwarzschild and the Kerr solutions. In this section the generalization of the static Schwarzschild to a more general static solution-the Zipoy-Vorhees(ZV) series of metrics-will be discussed. Unlike the Schwarzschild solution, these metrics are not spherically symmetric. They are only axisymmetric. In the rotating case the Kerr metric has been generalized to the Tomimatsu-Sato (TS) series of metrics. These metrics have a ring singularity on the equator, and do not represent black hole solutions. However, they could represent other situations where the singularity is covered by matter. The Tomimatsu-Sato solutions reduce to the Zipoy-Vorhees solutions in the static limit. In the limit a = m, they reduce to the extreme Kerr solution for all values of 6. For odd values of 6 there are no directional singularities at the poles, the metric on the Killing horizon is spacelike, and the Killing horizon is also an event horizon (Sato, 1982). The static axisymmetric metric in the isothermal gauge in Weyl's harmonic coordinates (Ph = D and zh = E) is: ds2 = _e2Udt2 + e-2U [e2 (dp2 + dz2) + p2d ]. The field equations are : v2 1 VDU = UPP + -Up +Uzz = 0, P p = p(U2 - U ), 1z = 2p(UpUz). If U and y are solutions to the field equation, then so are (Zipoy, 1969; Vorhees, 1970; Cuevado, 1990) U' - U 7' * 62 48 New solutions can be generated by the above transformation. This is a highly nonlinear solitonic transformation and it can only be performed in the Weyl's harmonic coordinates (ph,Zh). If this transformation is applied to the Schwarzschild solution in Weyl's harmonic chart in prolate spheroidal coordinates (Appendix B), the Zipoy-Vorhees metric is obtained: ds2 = _(X- )6dt2 + k2 X+)[X2-1)62{(X2 _ Y2 1 + dy2 Xz+1 (X- [x2_- 2) 3(- + 2 1 - y 2} +(X2 - 1)(1 - Y2)d2] where e2U (X ) (X _6 = ( + + 1 x+1 z+1 e 2 _2 2 ' with 6 an integer; 6 = 1 for the Schwarzschild solution. It should be noted that the apparent absence of spherical symmetry for the Schwarzschild solution in Weyl's harmonic coordinates is critical to the nontrivial nature of the Z-V transformation. If attention is paid to the conformal factor e27 it is apparent that the degree of the polynomials has increased by 62. The critical points are determined by P(x) = (x2 - 1)62 = 0-i.e., x = ï¿½1. The roots of the polynomial are all equal. The conformal factor is related to the derivative of Weyl's harmonic function. Therefore, in isotropic coordinates Weyl's harmonic function for the Zipoy-Vorhees metrics may be obtained by taking the derivative of the harmonic function for the Schwarzschild solution and raising it to the power 62: dW + k2 )62 dZ 4Z2 W= ( + 4 2 dZ, 49 where k2 - (m 2 - a2)/62. In the rotating case, degree of the polynomials are the same as in the static case. As in the Kerr case, the numerator polynomial is different for the static and the rotating case. The conformal factor and the curvature of the two surfaces are different for the static ZV(6) and the corresponding rotating case TS(6). However, the corresponding harmonic function has a similar structure: For example for the TS(2) solution: dW k2 4 dZ Z2) k2 3k4 k k8 + 2 +Z4 + 16Z6 +256Z8 k2 3k4 k k W= Z Z 24Z3 80Z5 - 1792Z7' which shows that the series for W is rapidly convergent. Due to the multiple roots the critical points of Weyl's harmonic function for the ZV(6) and TS(6) solutions are degenerate. Since all 62 derivatives vanish at the poles, the degree of the degeneracy is also 62. 3.5 Weyl's Harmonic Function and Global Structure In the previous section a procedure was given for determining boundary data for the Laplace equation for rational solutions. In particular the critical points of the harmonic function W could be determined. In this section the relation between these critical points and global structure of the solution will be explored. First the relations between the critical points, the magnitude of the Killing bivector, and the Killing horizon will be reviewed for the Kerr metric. Next, two complicated exact solutions will be discussed: the double Kerr solution, and the Manko et al., 94 solution. Wheras the Kerr solution has two critical points (Section 3.3), the last two solutions have four critical points. 50 In a stationary axisymmetric spacetime there are three independent coordinate invariant scalars that can be constructed from the Killing vectors: afa = gtt = -V, ga7a = goo = X, and (aa = gt = W. A fourth coordinate invariant quantity which is a combination of the previous three is the magnitude of the Killing bivector: -2[ab] ab = D2 = VX + W2. The scalar D2 is also the negative of the determinant of the (t, 0) part of the metric. The scalar D2 is positive outside the Killing horizon, and negative inside the Killing horizon (Hawking and Ellis, p. 167). D2 vanishes trivially on the axis as 77 vanishes on the axis. The region off the axis where the D2 vanishes is the Killing horizon where [al7b] 0, but its magnitude vanishes (Carter 1969). Therefore D2 = VX + W2 contains important global information about the Killing horizon which is not carried by either of the three other coordinate invariant quantities, V, X, or W, alone. PROPOSITION A proposition concerning Killing horizons and critical points of Weyl's harmonic function will be established next. At large distances from the source the determinant of the (t, 0) part of the metric, D, approaches the flat space cylindrical coordinate p. The coordinate vector fields associated with the flatspace cylindrical coordinates (p, z) do not vanish anywhere-i.e., Vp 5 (0,0) and Vz 5 (0, 0) at every point in the half plane. Isotropic coordinates have this property-i.e., Vpi and Vzi do not vanish anywhere. (However, the coordinate vector fields associated with the harmonic coordinates (Ph, Zh) vanish at the critical points of Weyl's harmonic function.) Therefore isotropic coordinates are chosen for these computations. 51 Let the scalar D be decomposed in isotropic coordinates as follows: D = piB. The isotropic cylindrical coordinate pi vanishes only on the rotation axis where the spacelike Killing vector field vanishes: X = a9 = 0. The isotropic radial coordinate represents the true axis. The metric function B vanishes only on the Killing horizon where the magnitude of the Killing bivector vanishes, -24[ab]6a b = D2 -= VX + W2 = 0, but the Killing bivector itself ([a b] $ 0. The metric function B represents the true horizon. Let the point P(0, zl) represent the intersection of a Killing horizon and the rotation axis. By definition, the function B = 0 on the horizon. Also by definition, D = piB; therefore its gradient is VD = (B, 0) along the axis where Pi = 0. Therefore, VD = (B, 0) = (0, 0) at the point P, since B = 0 at the point P. As the gradient of D vanishes at P, therefore the point P is a critical point of Weyl's harmonic function W. The harmonic function W = D + iE; by the Cauchy-Riemann relations, VD = (0, 0) = VE = (0, 0). Therefore it is proved that the intersections of the Killing Horizons and the axis are critical points of Weyl's harmonic function. Next the converse proposition will be established. Let the point Q(0, z2) be a critical point of Weyl's harmonic function that lies along the axis. By the definition of a critical point VD = (0, 0). If the critical point is degenerate, higher derivative of W could vanish as well. Along the axis, VD = (B, 0). Therefore B = 0 at the point Q. As B = 0 only on the Killing horizon, the point Q is therefore the point of intersection of a Killing horizon and the axis. Hence, it is proved that the critical points of Weyl's harmonic function that lie on the axis represent intersections of Killing horizons and the axis. 52 Therefore it is proved that a point along the axis represents the intersection of a Killing horizon and the axis if and only if it is also a critical point of Weyl's harmonic function. QED. The practice of identifying global information using the gradient of the magnitude of the Killing bivector is a standard procedure for spacetimes with two Killing vector fields (Verdaguer,1993). In the Exact Solutions book it is stated that: "The gradient of W [D] provides an invariant characterization of the corresponding spacetimes and determines essentially the character and global properties of a solution." (Exact Solutions, p. 176). The gradient of the magnitude of the Killing bivector was used by Gowdy to distinguish different regions of an inhomogeneous cosmological solution. (Gowdy, 1971, 1974). The Gowdy solutions possess two spacelike Killing vector fields. They have no cross terms in the metric, and are analogous to static axisymmetric Weyl solutions. Kitchingham has generalized the Gowdy solutions to metrics with cross terms (1984,1986). An analogy between Kithchingham's inhomogeneous cosmological model and some newly generated stationary axisymmetric solutions will be presented in Section 3.5.3. The solutions of Manko et al. 1994, and Kitchingham were generated using the same mathematical technique-the solution to the Riemann-Hilbert problem. 3.5.1 The Kerr Solution The global structure of the Kerr metric is well known (Hawking and Ellis, Section 5.6). The determinant of the (t, 0) part of the metric, D2 = VX + W2, is positive everywhere outside the Killing horizon while on the horizon it is zero; inside the horizon, it is negative. The determinant D2 vanishes on the axis 53 since the rotational Killing vector vanishes on the axis (Fig 3.1). Just slightly off the axis, D2 is positive outside the horizon, while it is negative inside the horizon (Fig 3.1). In terms of the two Killing vectors of the stationary axisymmetric system-the timelike Killing vector field ( and the spacelike Killing vector field 77: The 2-surface of transitivity (group orbit) spanned by the two Killing vector fields ( and q is spacelike or timelike respectively when the magnitude of the Killing bivector [apob] is positive or negative. Horizon D2(0, z) and B(0, z) for the Kerr Solution Figure 3.1 55 The gradient of D along the axis is given by VD = (0, B); VE = (B, 0); where B is a scalar function of the isotropic coordinate zi. Along the axis, the segment BN represents the part of the axis that extends above the North Pole, outside the horizon. Similarly the segment AM represents the part of the axis that is below the South Pole. The segment of the axis AB lies inside the horizon. The gradient function B is positive outside the horizon on the segments BN and AM; it is negative inside the horizon on all of AB (Fig. 3.1). The points A and B where the gradient scalar B vanishes are the critical points of D and E since VD = VE = (0, 0) at these two points. Therefore the behavior of the gradient scalar B along the axis makes it possible to determine where D2 changes sign-i.e., the intersection of a Killing horizon and the axis. It is not possible to determine this information from D2 directly since D2 is always zero on the axis. Close to the axis, D2 is positive outside the horizon, and negative inside (Fig. 3.1). The principle of identifying Killing horizons by the critical points of D and E will be applied to two new solutions with four critical points next. Before proceeding, the following observations are made. While the scalar function B is defined in isotropic coordinates, the critical points along the axis identified by B = 0 do not depend on any choice of coordinates or gauge. The gradient of a scalar is a well defined vector. The points at which both components of the gradient vector vanish are invariantly defined. Furthermore, these points are the critical points of the harmonic function D. From complex variable theory it is known further that these points are isolated and carry global information. Second, at the point S which is the origin of isotropic coordinates the function B diverges. The physical significance of the point S is not clear since it is not identical to the disk identified by the rBL = 0 in Boyer-Lindquist 56 coordinates. For the purposes of this discussion, as long as B does not change sign at the point S, it will not be a critical point, and not affect the global structure. 3.5.2 The Double Kerr Solution Next we consider the double Kerr solution (Kramer and Neugebauer, 1980). This solution consists of the nonlinear superposition of two Kerr black holes along the axis. This solution and its properties are reviewed in great detail by Dietz and Hoenselaers(1985). For the present purposes, it will be sufficient to examine the conformal factor given in Weyl coordinates and extract the critical point information by the procedure given in the previous section. The conformal factor is a rational function e2y F(ph, zh) G(ph, Zh) ' where F(ph, Zh) and G(ph, zh) are quartic functions of (Ph, zh). Along the axis, 2y F(0, Zh) P(zh) G(0, zh) P(zh) The gradient polynomial P(zh) in harmonic coordinates is P(zh) = zh - (m2 + k2- 2)z + [(k2 - m2 + a2)2 - 4a2n2] = 0 for two eqal mass m, equal angular momentum a objects separated by a distance k (Dietz and Hoenselaers, 1985, Eqn. 2.51c, Eqn 2.52.a, and 4.11). The roots of P(zh), which determine the critical points along the axis, are ï¿½al and ï¿½a2, where: al = (K+i + K-), = (K- 57 S= [(m2 + k2 a2 + d)/2]1/2 _ = [(m + k2 _ a2 - d)/2]1/2 d = [(k2 - M )2+ a - 4a2m2]1/2. As this is a quartic polynomial, from the detailed example in Section 3.4, it is clear that Weyl's harmonic function will have three terms and two parameters bl, b3 when expressed in isotropic coordinates. Physically this represents the superposition of a dipole field(bi) and an octupole field(b3) upon a uniform field of unit magnitude in two dimensions. As shown in Section 3.4, it is possible to express bl, and b3 in terms of a2 and ao. The critical points of Weyl's harmonic function are located along the axis as follows: two critical points A and B will determine the intersections of the horizon of one black hole and the axis; the other two critical points C and D will represent the intersection of the other black hole with the axis (Fig. 3.2). As before the magnitude of the Killing bivector D2 will vanish on the axis. Close to the axis D2 is positive above the North pole of the top black hole near the segment DF and below the South pole of the lower black hole near the segment AE. In this case there is also the segment of the axis BC between the two black holes near which D2 is positive (Fig. 3.2). In three dimensions the segments AE, BC, and DF represent the boundary of a single connected region in space which is the region outside the two horizons of the two black holes. The segments AB and CD represent regions interior to the two black holes inside the Killing horizons. The axis of the double Kerr spacetime has three distinct physical regions separated by two Killing horizons. Horizon 1I A B --A B/ Horizon 2 C D C D Figure 3.2 D2(0, z) S1 S2 and B(O, z) for the Double Kerr Solution 59 Based upon the analogy with a single Kerr black hole, the behavior of the gradient of the Killing bivector, VD and VE is shown schematically on Fig. 3.2. The function B will vanish at the four critical points A, B, C and D. Along the segments DF and AE the function B will be positive since B -+ 1 at infinity. The function B will be negative inside the Killing horizon along the axis along the segments AB and CD. As the function B changes sign at the critical points B and C, it must be positive on the part of the axis that lies between the two black holes outside the Killing horizons on the segment BC. Thus the critical points of Weyl's harmonic function W permit the identification of intersections of Killing horizons and the axis. The nonzero component of the gradient of D determines whether or not D2 is positive or negative. The magnitude of the Killing bivector is positive where B is positive at infinity, and alternates in sign with B near the axis. In the literature, several authors have gone to great lengths to identify the various regions of the double Kerr Spacetime. In Weyl coordinates, all three segments of the axis are defined by Ph = D = 0. The authors take great pains to identify the Killing horizons without the use of critical points, or the gradient of the magnitude of the Killing bivector. The horizons were identified numerically by showing that gtt and e2Y are negative along the AB and CD. Further, it was shown numerically that w was constant on AB and CD. It was found necessary to evaluate these quantities numerically for selected choice of parameters as the form of the metric is exceedingly complicated. The global structure of the horizons is easily identified in isotropic coordinates as the coefficients b, and b3 can be expressed in terms of the coefficients of P(Zh). The schematic figure (Fig. 3.2) is in isotropic coordinates. An exact 60 map could be produced for suitable choices of parameters such that the solution possesses two horizons. The behavior of the function B at the origin O, and the two black hole centers S1 and S2 requires further examination. As before as long as B does not change sign, the critical points of the solution will be unaffected, and the global properties of the solution will be unchanged. 3.5.3 The Manko et al., 1994 Solution In the previous example, the information about the horizon structure could be obtained with some difficulty even though there were three separate regions and two Killing horizons. The next example under consideration, the Manko et al., 1994 solution, has not been analyzed for horizon structure. Based on the two previous examples, a possible interpretation for this solution will be suggested. The conformal factor is a rational function e2-y F(ph, zh) G(ph, Zh)' where F(ph, Zh) and G(ph, Zh) are quartic functions of (Ph, zh) (Appendix D). Along the axis, 2y F(0, zh) P(zh) e = -+ - 1. G(0, zh) P(zh) The critical points are determined by the roots of the polynomial P(zh) which in this case is P(zh) = z4 - (mn2 - a2 - q2 - 2b)z2 + (b2 + c2) = 0. The roots of P(zh) are ï¿½al and +a2, where: al = ++ K-, 61 al2 = + --_, = [m2 _ a2 q2 + 2(d - b)]1, _ = [m2 _ a2 _ q2 + 2(-d - b)]1/2 d = (b2 + C2)1/2. As this is a quartic polynomial, it will have four critical points. By the procedure given above, the parameters bj and b3 for the solution in isotropic coordinates can be obtained. Next the four critical points and the two horizon curves can be obtained. The critical points in isotropic coordinates are given by the roots of the equation: bi 3b3 Q(z) = 1 - z? z4 Off the axis the Killing horizon is determined by the metric function B in isotropic coordinates since D = pjB(pi, zi) = 0 on the Killing horizon. As D is a harmonic function, the function B can be expressed in polar coordinates as: B(ri, ) = 1 - b 3b3 sin(30i) r? 4r sin(90) For each value of 0, there are two roots for the above polynomial. Hence D will vanish on two curves for this solution. Based on the location of the critical points along the axis, it is clear that one B = 0 curve will enclose the other B = 0 curve. There will be four real roots if b? - 12b3 > 0. This condition could be translated into a condition on a2 and ao, using the relations between the bi and the ai for solutions with four critical points given in Section 3.3. There are five free parameters determining the ai in this case: m, a, q, d, and b. It is clear that 62 with so many free parameters, keeping the mass fixed, and changing the other four parameters, a2 and ao could be given a wide range of values, so that the above condition on the bi may be satisfied. And the conjectured global structure with four critical points becomes possible. The authors of the solution describe the solution as that belonging to a single source, and the solution being symmetrical about the equator. Assuming that the four roots are real and located along the axis, the outer roots at oal are located at C and D, and the inner roots at ï¿½a2 are located at A and B. Based on this structure, it is conjectured that in the Manko, et al. 1994 solution one horizon lies inside the other (Fig. 3.3). In the double Kerr solution, there are two horizons also; however, one horizon does not lie inside another. In the double Kerr solution there are two regions where the magnitude of the Killing bivector is negative. In this case the magnitude of the Killing bivector is negative inside the outer Killing horizon, however it turns positive after crossing the inner Killing horizon. Therefore it is conjectured that the Manko et al., 1994 solution has two disconnected regions where magnitude of the Killing bivector is positive. Possible justification for this conjecture is provided below. It is clear that the solution has four critical points. At these critical points the gradient of the Killing bivector must vanish. The function B must positive along DF and CE. At the critical points C and D it will change sign (Fig. 3.3). Hence D2 will be negative near the axis along CA and BD. At the critical points A and B, the function B changes sign again. Consequently, D2 will become positive near the axis (Fig. 3.3). B D Outer Horizon B A B / I B(O, z) for the Manko et al. C D D C Figure 3.3 D2(0, z) and 1994 Solution 64 The authors of this solution looked at this solution in Weyl coordinates only. In Weyl Coordinates the nature of this solution is difficult to decipher. Along CA and BD, gtt and e27, will be negative. However, they will be double valued along AB. Furthermore, wp will have the same value along AC and BD, but will be double valued along AB. It might be erroneously believed that the spacetime consists five disconnected regions: one region outside CD; one region outside CA and another along BD; and finally two regions along AB since Wp is double-valued. Fortunately no interpretation of this solution has been made. A problem with Weyl coordinates is therefore identified from this proposed configuration. If one Killing horizon encircles another, the horizon structure will be rather difficult to interpret since several distinct horizon curves will be represented along the axis overlapping one another. The connectivity of the regions in three dimensions is also obscured. The possible interpretation given above of Manko et al., 1994 solution in terms of having several disconnected regions, where the magnitude of the Killing bivector is positive, is motivated by the the work of Kithchingham(1986). Kithchingham generalized the Gowdy Universes to universes with several disconnected components using the inverse scattering transform technique. It should be pointed out that Manko et al., generated the solution discussed above using a variant of the inverse scattering transform. Finally in conclusion it should be pointed out that Yamazaki has given closed form formulas in terms of determinant for the N-Kerr solutions. In this case the interpretation of the solutions is straight forward: each black hole has a horizon inside of which the magnitude of the Killing bivector is negative. Based on the interpretation of the Manko et al., 1994 solution it is conjectured that 65 solutions with N overlapping horizons might also exist where the sign of D2 would change, every time a Killing horizon is crossed. The structure of such solutions will be transparent in isotropic coordinates: there will be 2N critical points, and N Killing horizons curves enclosing each other where the function B goes to zero. CHAPTER 4 STATIONARY AXISYMMETRIC FIELDS IN VACUUM In the previous chapters, certain global invariants of stationary axisymmetric solutions were introduced. In the present chapter the metric functions will be discussed in detail. In particular, expansions will be developed for all four metric functions. The relations between the expansions parameters and the relativistic multipole moments of the solution will be obtained. 4.1 The Conformal Invariance of the Field Equations in Vacuum The stationary axisymmetric spacetime metric contains four functions: V, W, D, and e2p. For the metric in the isothermal gauge in the Bardeen form: ds2 = - e2vdt2 + e2ï¿½(dq - wdt)2 + e2p(dr2 + dz2) The invariant definitions of the various functions are: e2" = V + W2/X e20 = X e20+2v~= VX + W2 = D2 w = W/X The field equations with sources are as follows (Seguin 1975): 2'_+ l+ e20-2vVw Vw = -81re2 [E + P P e- V2ev +Vv"Vï¿½- e2-2vVw" Vwï¿½= 4re2P [ (E +P)lv + 2P] 4 - 1 - V 2 2 e-"V 2 ev + Vv V - le2 -2"Vw ï¿½ Vw = +4e2p (, + P)+2 + 2P 1 - v2 2 e-vV2v + V2 _e 20-2vVw Vw = ï¿½8re2p U( + p) 1 2 + P 1 - Pv2 V[e a-Vw] = -16 e2 +2p 2 v2 DO+v+2p p 1eDeO+v = 167re[+1+2 p All operators used in the above equations are defined with respect to the twodimensional flat space metric ds2 = dr2 + dz2. If sources are present, the conformal factor e2p is present in every single equation. However, in vacuum, if no sources are present, all the terms in the right hand side vanish. All the terms involving the conformal factor drop out of the field equations. The field equations without sources are as follows: e- V2eO + V2p + le2-2vVw ï¿½ Vw = 0 e-vV2 ev + Vv. V - Ie2 -2"Vw. Vw = 0 e-vV2ev + V2 - 2-2"Vw Vw = 0 V[e 3-"Vw] = 0 v De+V = 0 The conformal factor e2P decouples from the other three variables. The three equations for V, W, and D do not involve e2 . Therefore in vacuum, it is possible to solve the three equations for them first. Therefore, if the field equations are satisfied for the three variables, the fourth will be automatically be satisfied. The field equations are conformally invariant in vacuum, in the sense that they do not depend on the conformal factor. Further, the conformal factor is determined by the field equations, once the field equations are satisfied. 68 This property will be exploited in terms of the expansions of the metric functions in section 4.3. The expansions will be derived for the three independent variables-V, W, and D-first; the expansions for e2j" will be determined afterwards. The conformal invariance of the field equations is transparent in the isothermal gauge. It is not so in other gauges. For example, in any other gauge, let the metric of the two surface S be ds2 = e22dx + e2p3dx2, and A = e/3-2 then the equation for D = e takes the form: [ 2 ),2+ -12 ),3,3 = 0. This is the general form of the two-dimensional Laplace equation in a diagonal metric. The metric on the two surface S is present in all the vacuum field equations in the other gauges (MTB, p. 274, Eqn. 5-10). An important practical consequence, is that in other gauges four functions have to solved for in the vacuum. In the isothermal gauge the functions of the (t, q) part of the metric are solved first, and the metric on the two surface S is determined afterwards. In analytical studies in the harmonic coordinates in the isothermal gauge, the Ernst potential is determined first, while the metric on the two surface S is determined afterwards. In numerical studies in isotropic coordinates, the three metric functions of the (t, q) part of the metric-eV, w, and B-are solved first, and then the metric on the two surface S, i.e., the conformal factor e. 4.2 The Existence of Global Charts Stationary axisymmetric systems are two-dimensional problems: all the relevant functions are defined on the two surfaces orthogonal to the group orbits. As discussed in section 2.3 the two surface in question can be visualized 69 as a metal plate with known Gaussian curvature at each point. The metal plate extends from the axis to infinity where it becomes flat. The two surface S is diffeomorphic to the half plane. It is known from the work of Resetnjak, that if the Gaussian curvature of a surface is bounded, and if the region is diffeomorphic to the half plane, then it will be possible to cover it with a single isothermal chart (Morgan and Morgan, 1970). As discussed in section 2.2 the Gaussian curvature of the two surface S is determined by: G R1212 = -2e-2(-VV _ Vp + 3p2e-4Vw Vw). (911922 - 912 where (Xl, x2) are coordinates parametrizing the two surface S. The first term corresponds to the magnitude of the acceleration of a ZAMO while the second term corresponds to the shear of the ZAMO world lines (Bardeen, 1973, p. 247) These are well behaved and bounded functions. They remain finite even on the black hole horizon. The first term reduces to a constant value, the surface gravity, KH (Bardeen, p. 252). Therefore the Gaussian curvature of the two surface is bounded (of order one). Hence by the Theorem of Resetnjak it will be possible to find a single isothermal chart to cover the entire two surface S. The choice of an isothermal form for the metric on the two surface corresponds to choosing the isothermal gauge conditions. As discussed in Appendix A, there is no gauge freedom-in terms of freely specifiable functions-in the isothermal gauge. The various coordinate systems that preserve the isothermal form of the metric are related to one another by well defined complex analytic coordinate transformations. Under such coordinate transformations, the conformal factor of the metric is altered as follows: e2"72 e271 A2 A2 A2 I I dZ2112 dZz 70 where the new coordinates Z2 = x2 + iy2 are complex analytic functions of the old coordinates Z1 = xl + iy. This coordinate transformation will be well defined everywhere except points where - 0. Different choices of coordinate systems on S correspond to different choices of the conformal factor. The conformal factor is related to the Gaussian curvature as follows (Exact Solutions, pg. 197): -( 2 2 G = -2e-2 2 Therefore the Einstein field equation for the conformal factor is: O2l + = -Vv - Vv + D2e-4vVw. Vw Ox Mx 4 This is a second order elliptic partial differential equation. Therefore, the complete specification of the conformal factor requires two boundary conditions. The first boundary condition is that the metric on the two surface S becomes identical to the two-dimensional flat space metric at infinity, i.e., e2 - 1 at infinity. The second boundary condition is that the three dimensional space near the axis be flat: if a small circle is drawn surrounding the axis, its circumference must equal 27r times the radius. Two global charts have been used in the study of stationary axisymmetric systems: Weyl's harmonic chart and Weyl's isotropic chart. In both these charts a single global coordinate system covers the entire portion of the two surface S outside the Killing horizon. In Weyl's harmonic coordinates the metric on the 3D space is: dsD = e2"( dp + dz) + phdo2 The flatness boundary condition on the conformal factor is e2 -4 1 on the axis. In Weyl's isotropic coordinates the metric on the 3D space is: ds=D 2 (dp? + dz?) + B2 2 The flatness boundary condition on the conformal factor is e --+ B. Under a coordinate transformation from isotropic coordinates Z = pi + izi to harmonic coordinates, W = D + iE = Ph + izh, the conformal factor transforms as: e27 2 A2 A2 dZW 2 In isotropic coordinates the determinant of the (t, 0) part of the metric D satisfies D = piB OB OB VD = (B + Pi , Pi). O~ zi On the axis pi = 0, and VD = (B, 0). By the Cauchy-Riemann relations VE = (0, B). By definition, W = D + iE; therefore along the axis 2 dW 2 2 dZ and the axis boundary condition is satisfied e2 e2 1 e2 - - -+ 1. A2 B2 Once the flatness boundary condition along the axis is imposed, a single and conformal factor is defined globally. The conformal factor satisfies a second order PDE: once two boundary conditions are imposed, it becomes unique. Once the field equations are solved, subject to the above conditions, a single, unique conformal factor is determined globally. Different choices of coordinates on S 72 correspond to different conformal factors. Therefore a unique global conformal factor selects a unique, global system of coordinates on the two surface S. In isotropic coordinates the conformal factor e2( is uniquely determined for a particular solution. In harmonic coordinates the conformal factor e27 is uniquely determined for a particular solution; and they are related to each other by the conformal coordinate transformation defined above. At the critical points of Weyl's Harmonic function the gradient of D and E vanish. Along the axis the critical points are determined by B = 0, since along the axis VD = (B, 0), VE = (0, B). However, their gradients do not vanish unless the critical point is degenerate. At the critical points both B and e5 = B vanish. It should be noted that the equation e2( = 0, does not imply that the Gaussian curvature KG = 0. In Weyl's harmonic coordinates, e27 -+ 0/0. However, if the critical point is not degenerate and higher derivatives are not zero as well, e27 -_+ 1 by L'Hospital's Rule. Hence the spacetime is well behaved near the critical points of Weyl's harmonic function if the critical points are not degenerate. The flatness condition near the axis is a highly non-trivial condition: it connects the (p, z) part of the metric to the (t, ï¿½) part of the metric. It connects the conformal factor of (p, z) part of the metric to the determinant of the (t, q) part of the metric. It connects the Gaussian curvature of the two surface S to the magnitude of the Killing bivector. The determinant of the (t, 0) part of the metric sets boundary conditions on the conformal factor on the (p, z) part of the metric as follows: along the axis, VD = (B, 0); VE = (0, B). Since D and E, are harmonic conjugates, their partial derivatives are related by the CauchyRiemann relations. Along the axis one component of the gradient is zero, the 73 only non-vanishing function is B. Therefore, the flatness condition near the axis, which is e'(0, zi) = B(O, zi) = Q(zi) in isotropic coordinates, and ef =P(zh) P(zh) in harmonic coordinates, couples the Gaussian curvature of the two surface S to the non-zero component of the gradient of Weyl's harmonic function along the axis. The uniqueness of the gradient function B is apparent in the case of rational functions. In this case the number of critical points of Weyl's harmonic function is finite, and along the axis the function B is a polynomial in 1/zi: bl 3b3 nbn B(O, zi) Q(zi) = 1 - i+ + ... + n . zzz I t t A rational solution to the Einstein field equations will have a specific number of critical points. The number of critical points will determine the number of terms in the polynomial Q(zi). The precise location of the critical points in isotropic coordinates will be determined by the specific values of the coefficients bi. The critical points will be determined by the roots of the polynomial Q(zi) in isotropic coordinates. As discussed in Section 3.4, the location of the critical points in Weyl's harmonic coordinates are determined by the roots of the polynomial P(zh) with coefficients ai. The coordinate transformation from isotropic to harmonic coordinates is uniquely determined by the coefficients bi. Hence the coefficients ai and the coefficients bi are related algebraically. As harmonic coordinates are invariantly defined, Ph = D and zh = E, the coefficients of the 74 polynomial P(zh), ai, are invariantly defined. Therefore, since bi the coefficients of the polynomial Q(zi) are related algebraically to the coefficients ai, they are also unique. Invariantly, the critical points where VD = (0, 0) is determined by the roots of the polynomial P(zh) in harmonic coordinates and by the roots of the polynomial polynomial Q(zi) in isotropic coordinates. Both these polynomials are unique. Hence they define unique global charts: Weyl's harmonic chart and Weyl's isotropic chart, respectively. In the case of an infinite number of critical points, the above series for B will not truncate, and Q(zi) will not be a polynomial with finite number of roots. In this case the uniqueness of the coefficients bi can be established by considering the complex analytic properties of Weyl's harmonic function. In isotropic coordinates Weyl's harmonic function W = D + iE has a Laurent series expansion about infinity: 00 W = Z + (-1)n b Zn n=1 This series converges everywhere outside a certain region containing the origin and outside a small circle excising the simple pole at infinity. The Laurent series coefficients bi are unique. Further by considering the gradient of Weyl's harmonic function: dW 00b = 1 + (-1)n+l n dZ n=En+1 n=1 it is clear that B(0, zi) = Q(zi) has a Taylor Series expansion about infinity, iT dW) + "0(ln nbn Q(zi) d 1 + z(-1)n 1 n=11 which is well defined and unique. Q(zi) is a real function of the one variable. The derivative < 1 exterior to the unit circle. If there are no critical points in the region under consideration, dW > 0. (As discussed in section dZ 3.5 the critical points of Weyl's harmonic function corresponds to intersections of Killing horizons with the axis.) Therefore B is very smooth and well defined function in the regions where it is defined, e.g., in the exterior of a star. It has been shown that on the axis of symmetry, pi = 0, VD = (B, 0); VE = (0, B). It has been assumed that V is the covariant derivative associated with the metric on the two surface S. However, the gradient function B could also be defined in terms of Lie derivatives which do not involve the metric. Let A and B define two smooth vector fields: A = , and B = , then the Lie derivative of Weyl's harmonic scalars along the axis defined by pi = 0 are: A (D) = (B, 0) B (E) = (0, B). For a specific solution to the stationary axisymmetric field equations, the gradient of Weyl's harmonic function W = D + iE is uniquely determined by its complex analytic properties. This defines the scalar function B uniquely along the axis, without any reference to the metric on the two surface S. The Lie derivative does not involve the metric or assume the existence of an affine connection on the two surface S. The flatness condition near the axis may now be used to set the conformal factor e6 = B along the axis. Finally the Gaussian curvature at each point on S will define e4 uniquely everywhere for that particular solution. As mentioned, different choices for the conformal factor correspond to different sets of coordinates on the two surface S. Therefore, every stationary axisymmetric solution has a unique global isotropic chart in the vacuum region defined by the harmonic properties of D; where D is invariantly defined by -2f[a7b]1a7b = D2 = VX + W2 76 The above prescription does not work for Weyl's harmonic coordinates. In Weyl's harmonic coordinates Ph corresponds to both the axis and the Killing horizon on which Ph = D = 0. In isotropic coordinates the equation pi = 0 corresponds to the the true axis determined by the flat space cylindrical coordinate p =0. Furthermore the vector fields C = , and D = are not smooth aPh andot at the points of intersection of the horizon and the axis. The vector fields A and B are smooth everywhere like the vector fields corresponding to r = and aa z where p and z correspond to ordinary flat space cylindrical coordinates at large distances. The difference between an ordinary partial derivative and the Lie derivative is explained by Hawking and Ellis: "the ordinary partial derivative is a directional derivative depending only on a direction at the point in question . . . the Lie derivative LxTIp depends not only on the direction of the vector field X at the point p, but also on the direction of X at neighboring points." (Hawking and Ellis, p. 30) In this regard the covariant derivative is similar to the ordinary partial derivative. It is important to note that the vector fields associated with Weyl coordinates will not be smooth at the intersection of the Killing horizons and the axis. 4.3 Practical Aspects of Coordinate Charts A second important issue is the physical interpretation of solutions given in Weyl's harmonic coordinates. Solutions to Laplace equation in 3D cylindrical polar coordinates generate the static Weyl solutions. However, the physical interpretation of these solutions remain obscure. Weyl introduced the use of the determinant of the (t, ï¿½) part of the metric (D) and its conjugate function(E) as coordinates for the first time in 1917. He noted that in such coordinates the spherically symmetric Schwarzschild solution corresponds to a line 77 segment of length 2m, with constant linear mass density, along the axis of symmetry. (Weyl, 1917, p. 140). Weyl also introduced isotropic coordinates for the Schwarzschild solution in the same paper. This raises a second important question: Given a solution in Weyl's harmonic coordinates, is it possible to transform it to some generalization of "isotropic" coordinates, where the physical interpretation might be more transparent? For a stationary axisymmetric solution the Killing bivector vanishes trivially on the axis since the rotational Killing vector vanishes on the axis. However, D also vanishes on the horizon, as the horizon is defined by the norm of the Killing bivector being zero (Carter, 1969). In Weyl's harmonic coordinates the horizon which is topologically a sphere is mapped to a line segment along the axis. This difficult does not arise in Weyl's isotropic coordinates where the horizon is represented by a sphere of coordinate radius m/2. In this treatise it has been shown that it is possible to generalize Weyl's isotropic coordinates to solutions which are not spherically symmetric by the procedure given in section 3.3. Given any stationary axisymmetric solution in Weyl's harmonic coordinates, it is possible to uniquely determine the corresponding solution in Weyl's isotropic coordinates. 4.4 Asymptotic Expansions for the Metric Functions In this section asymptotic expansions will be given for all four metric functions of a stationary axisymmetric vacuum solution. For the reasons discussed in the previous section, isotropic coordinates will be employed. The metric functions will be in the Bardeen form. The results presented here are a generalization of the expansions of Butterworth and Ipser(1976) where expansions were given up to the fourth order Legendre Polynomial. Based on Professor 78 Ipser's notes from 1975, a symbolic algebra program was written in Mathematica to carry out the expansions to arbitrary order. All four metric functions were obtained to the fourteenth order Legendre polynomial. The results were validated by checking the Kerr metric to this order. 4.4.1 The Complete Nonlinear Vacuum Field Equations Let N e2v and r sin 0 B = e+v. Then the vacuum field equations given in Section 4.1 may be written as follows: V2N-N-1 VNVN+B-1 VBVN- -1 r2sin2 0B2Vw - Vw = 0 V(r2 sin2 0 Vw) + r2 sin2 0 Vw- 3 VB -2 ] =0 B N v2 D -2 3-2 2 Si2 p2 VW . VW = 0 V2D+ N VN -VN -N r sin2O2wV= 4 4 VD r sin 0 B = 0. The gradient operator used above is the flat space cylindrical coordinate operator defined in three dimensions.. There is considerable confusion in the literature regarding the above equations. Some authors have failed to note that the Laplacian operator in the equations for ( and B are two-dimensional and not three-dimensional. 4.4.2 The Linearized Vacuum Field Equations The equation for B is linear. In order to derive expansions for the four metric functions linearized versions of the above field equations will be generated by dropping the nonlinear terms: V2N = 0 V(r2 sin2 0 Vw) = 0 VD =0 V2D rsin 0 B = 0. 79 The above equations will be used next to generate linearized "zeroth" order approximations to the metric functions. The eigenfunction expansions for the functions N, (, and B can be written down by inspection since they are solutions to Laplace's equation in three and two dimensions: N(r, 0) = 1 + E ' P(cos0) r1+1 =) Z, cos(10) r B sin(10) r sin 0 B(r, 0) = B sin() r Bo B sin(10) S B(r, ) = 1 - r2 1+1 sin ' where P(cos 0) is the Legendre polynomial of order 1. There is no logarithmic term in the solutions to the two dimensional Laplace equation. At large r, S--+ 0, and B --+ 1. The equation for w may be rewritten in the following form: 02(rsin Ow) 2 8(r sin Ow) 1 02(rsin Ow) 1 Or2 +- +-- [cot 0(r sin Ow)] = 0. rr Or r2 092 r2 jo The above equation is in the standard form for the equation of the magnetic vector potential for a current loop in the xy plane: 82AO, 2 A4 1 02AO 1 a -VxVx A =- - + r + (cot 0Aï¿½) =0. Or' r ar r (902p The solution for the above equation is: A(r, 0) z At P1 (cos 0) rl+l where P1 (cos 9) is the associate Legendred polynomial with m= 1. Hence, the solution for the metric function w is: W P11 (cos O) w(r,O) = Zr+2 sin 9 W dP(p) r1+2 d1i since d P ( p ) P1(cos0) = sin 0 dP () dp where p = cos0. The boundary condition on w has been given by Papapetrou(1948): 2J ,r3 where J is the angular momentum. For the associated Legendre polynomial P(-p) = (-1)'*+P (pj). The coefficitents W, must vanish for even values of I to preserve symmetry about the equatorial plane. 4.4.3 Expansions for the Metric Functions When the angular eigenfunctions are substituted into the three nonlinear field equations, the following equations are obtained: d2N 2 dN 1(1 + 1) NB1 d2N +dN N -N-1VN VN + BVB VN dr2 dr r2 -N-1 r2 sin2 0B2 Vw Vw = 0 d2w 4 dw (1 - 1)(l + 2) w + 3B1Vw VB 2N-1V VN dr2 r dr r d2( ld( 12 1-2 _ -2 r2 2 2 VW d2+rdr r2(+ 4N2VN.VN- 4 2 N sin20B2Vw-Vw=0 dr rdr r2 4 4 Due to the conformal invariance of the field equations, as discussed in Section 4.1, e2( decouples from the equations for e2' and w. Therefore it is possible to derive expansions for e2v and w first in terms of the three sets of multipole moments (BI, N, Wt). Afterwards the expansion for e2( can be derived in terms of the same three sets of parameters. There are four coordinate invariant quantities in a stationary axisymmetric system, V, X, W, and D2 = VX + W2, out of which three are algebraically independent. Therefore it is not surprising that three sets of parameters are sufficient to describe the exterior vacuum region of a star. N_ NJ N2 N N N6 N6 NoN No' + 03 + 0+ N0 N=1+-+-~ï¿½+-++++rï¿½ N ~ 1T +" 2 03 r4 r5 r6 77 N2 1 N22 N 3 4 r r r5 6 7 N[ N4 N1 2 +P4 + + 6 + N6 +PAs + 7 W, Wl W2 W3 W4 5 W, W1 T3 ï¿½4 ï¿½5 6r77ï¿½78 9 +4 + + + + r9 2~1W ] +p + + W51 + 95 I r zo z4 z4 4 z04 z (= + + + + + Z0 [0 r 2 3 4 5 6 r' r r3 r r r Z2osZ21 Z2 Z3 Z4 + cos 20 + Z2 + + 6 + + + cos 40 + Z4 + 1 + r4 r5 r6 + cos 60 + ] First the above expansions for N and w are substituted in the field equations for N and w. All the angular functions are expressed in terms of the Legendre polynomials. Next all the terms in the field equations are expanded out, and a two-dimensional eigenfunction expansion is obtained in terms of the Legendre polynomials and powers of . In order to expand the above equations it is necessary to expand all the functions in terms of the basis polynomials, which in this case are the Legendre polynomials. It is also necessary to expand the derivative of a Legendre polynomial as a sum of Legendre polynomials. Gradshteyn and Ryzhik(1980) give the following identity for expanding the derivative as the sum ( p. 1026, Eqn. 8.915.2 ): Pn = E (2n - 4k - 1) Pn-2k-1k=0 The summation is cutoff at the first term with a negative subscript. To expand the product of two Legendre polynomials as the sum of Legendre polynomials, the following identity is used ( p. 1026, Eqn. 8.915.5 ): m 2m+(2 2n -4k + I am-kakan-k) Pm+n2k, Pm Pn = : E m+2n-2k+l) am+n-k / k=O where (2k - 1)!! ak -- 1_1 83 and n > m. Next the terms in the field equations are grouped together and set equal to zero: Nm (N + terms = 0 rm 4 Wm + terms =0 The two field equations reduce to a number of new non-linear algebraic equations. However, each term is linear in Nl and W1 and these can be solved for in terms of the leading terms Nm and Win. Once these algebraic relations are substituted back into the expansions for N and w, the metric functions e2v and w can be expressed entirely in terms of (BI, N, W). Next the same procedure is applied to the field equation for e2( , and it too can be expressed in terms of three sets of parameters. The axis flatness condition e( = B along the axis is imposed. The final form of the metric functions are as follows: The metric function v has the following form: M + Bo J2 -3B2 +B2 - 36J2 4j2 (-7 Bo + 12M2) L= - i-1+-+ + + r 3r2 +M r3 + 15r4 15 Mr5 15 Bj - 10 Bo B2 + 3 B4 + 580 Bo 2 - 320 J2 M2 - 12 B2 N2 - 12 J2 N2 17 + 0 0( 105 r6 r M P2(p) N2 2J2 32B2 + 144J2-9Bo0N2 J2(56Bo - 84 M2 + 45W3) r M r + 42r2 + 21Mr3 1 + 1264 (-120 Bo B2 + 24 B4 - 848 Bo j2 +448 j2 M2 + 18 B2 N2 - 27 B2 N2 +168 J2N2 - 540 J2W3) + O(i )) M P4(p) N4 36 J2 W3 768 B4 - 192 B2 N2 - 1512 J2 N2 - 175 Bo N4 + 5400 J2 W3+13 + + +o( )6+ 1 + rs 2 7Mr 770r2 +M Po (p) (No ) + --+ o() 85 The metric function w has the following form: 2J 3M_ 3(3Bo-8M2) M (100 Bo - 80 M2 - 3 N2) r3 r 5r2 15r3 90 B2 + 9 B2 + 84 j2 - 440 Bo M2 + 160 M4 + 24 M2 N2 + 35 r4 M + M (-1512 B2 - 144 B2 - 1824 J2 + 2240 Bo M2 -448 M4 + 72 Bo N2 140 r5 - 168 M2 N2 + 45 N2 W3) + 1 (-3150 B - 756Bo B2 + 27B4 - 8316Bo J2+21448 B0 M2 + 1944B2M2 +34704 J2 M2 - 14560 Bo M + 1792 M6 - 1896 Bo M2 N2 + 1344 M N2 + 252 M2 N2 - 1080 B2 W3 - 1620 j2W3 -810 M2N2 W3) + O(-)) J W dP M (9 N2 - 25 W3) -32 B2 - 32 J2 -42 M2 N2 - 25 Bo W3 + 50 M2 W3 + 3d3(z 1 + + + r5 10 W3r 15 W3 r2 M + (2432 B2 + 3104 j2 - 621 Bo N2 + 1428 M2 N2 - 70 N4 + 1645 Bo W3 315 W3 r -980 M2 W3 + 105 N2 W3) + 1 (20664 Bo B2 + 1512 B4 + 22176 Bo J2 - 49408 B2 M2 - 84016 j2 M2 + 3465 W3 r4 +25110 Bo M2 N2 -17472 M4 N2 + 567 M2 N22 + 2660 M2 N4 + 7875 B2 W3 - 5355 B2 W3 +13860 j2W3 - 28910 Bo M2 W3 +7840 M4 W3 -3045 M2 N2 W3) + O(1)) J W dP5 () 1+M (35 N4 + 33 N2 W3 - 147 W) + W a(,)1 + r' 63 We r 11 1 (-6912 B4 + 1134M2 N22 - 7000 M2 N4 + 4095 Ws r2 - 5760 B2 W3 - 18720 J2 W3 - 5790 M2 N2 W3 - 6615 Bo W5 + 11760 M2 W) + 0(1 3) J W7 dP1) 1 + r9 The metric function C has the following form: -M2 54J2 + 8 Bo M2 +3M2 N2 2 2J2 M 4Tr2 + 64r4 r5 1 -3Bo J2 B2 M2 + B2M2 18J2M2 BoM2N2 15M2N2 + 5M2N4 r- 2 12 14 + 7 28 512 256 15 J2 W3)+ () + 64 Cos(29) ( M2 3 (-6 j2 + M2 N2) 2 J2 M + r2 4 16r2 r 1 j2 j2M: 189 2 2+15M2 N4 + 1 (756 Bo j2 + 80 B2 M2 - 1152 j2 M2 - 54Bo M2 N2 189 M2 N 175M 945 J2 W3 15) Cos(49) -Bo 9 J2 Bo M2 15 M2 N2 -9 M2 N2 + 70 M2 N4 -1080 J W3 + 8 + 3 B2 +2 T4 2 32 8 64 512 r2 + ( )) Cos(60) 3 Bo J2 B2 M2 B2 M2 5 Bo M2N2 75M2 2 0 - 3 BoB 2 + 5B + 2 75 2 r6 3 16 12 4 32 1024 105 M2 N4 105 J2 W3 1 ) 512 + 128 r 87 All four metric functions have been expanded in terms of three sets of parameters (BI, N, WI). Only terms of the order P6 have been given above. The Mathematica program for expanding the metric functions may be obtained from the author. The expansions of fourteenth order generate over 10 megabytes of output, therefore they have not been printed out. All metric functions have been verified to the fourteenth order for the Kerr metric. 4.5 Equivalence to Thorne Multipole Moments In the previous section, it was shown that all four metric functions can be expanded in terms of the three sets of parameters (BI, N, WI). The question arises what are the precise physical meanings of these parameters. Considering the pure vacuum region first, it will be shown that the parameters are multipole moments: The parameters N are mass multipole moments in the Newtonian limit; the parameters W are current (angular momentum) multipole moments in the Newtonian limit (Thorne, 1980, Section V.C). The parameters BI are exact two dimensional pressure multipole moments since the equation for B is linear (Butterworth and Ipser, 1976, Eqn. 15). Corresponding to the Laplace's Equation in vacuum, there is also a Poission Equation for D inside sources: V2DD = 16rpviZg, where p is the fluid pressure, and g is the volume element of the four dimensional metric (Trumper 1967, Eqn. 5; Bardeen, 1971, Eqn II.14; Chandrasekhar and Friedman, 1972, Eqn. 74 and 75; Butterworth and Ipser, 1976, Eqn. 4c). Hence the BI are termed 2D pressure multipole moments. It will be shown the parameters N and the parameters W correspond to the multipole moments defined by Thorne (1980) for stationary systems. The parameters B arise due to the presence of the additional symmetry, namely axisymmetry, that is present. 88 4.5.1 Thorne's Definition of Vector Spherical Harmonics In his 1980 review paper "Multipole Expansions of Gravitational Radiation", Professor Thorne defines the following pure spin vector spherical harmonics (Thorne(1980), Eqn 2.18a-c): yE,lm = [1(1 + 1)]-1/2 r VYlm = -n x yB,lm yB,lm = i[l(1 + 1)]-1/2 L ylm = n X yE,m yR,lm - n ylm The vector harmonics yE,lm and yB,lm are purely transverse; yR,lm is purely radial. The yE,lm and yR,lm have "electric-type parity" r = (-1)1; yB,lm has "magnetic-type parity" r = (-1)1+1. For the stationary axisymmetric case, m = 0. The electric vector spherical harmonic reduces to: yE,O = [1(1 + 1)]-1/2 r VY10 = [1(1 + 1)]-1/2 r VP(cos 8) dP(cos9) = [1(1+ 1)]-1/2 0, dPdO ,s) 0 The magnetic vector speherical harmonic is: yB,/0 = n x yE,lm = [1(1 + 1)]l/2dPl(co - [1(1 + 1)]-1/2 sin 0 dPI(p dO d 4.5.2 Thorne's Expansion for the Metric Functions Professor Thorne has given expansions for stationary sources from which the multipole moments may easily be read off (Thorne, 1980, Section XI). His expansions for the metric functions are given in an "Asymptotically Cartesian and Mass Centered" (ACMC) coordinate system. In an ACMC coordinate system the metric coefficients gtt and gto have a simple structure in terms 1 of - and the spherical harmonics (Thorne, Eqn 11.1). For the axisymmetric case r the expansions in terms of the multipole moments reduce to the following: 2M 2M2 I10pl(pu) + (1 - 1)pole + ... + (Opole) 9tt = -1 + - r2 rl+1 2Jsin2 () slo dPj I) sin2 (9) + (1 - 1)pole + ... + (Opole) gto = -d 2J si (0)dp r The above expressions are obtained by substituting axisymmetric angular eigenfunction P(p) for the spherical harmonic Y10O, and the angular eiqenfunction dPp) for the ï¿½ component of the vector spherical harmonic yB,0 in Thorne's Equation 11.4. (Note that there is a factor of rsin(9) difference with Thorne due to the normalized basis. Thorne, Eqn 11.22) The normalization constants have been omitted in the above equations. They will be dealt with in the next section. Professor Thorne's procedure for extracting the multipole moments is 1 to read off the mass-i moment from the - part of gt, and the current -pole 1 moment from the -7 part of gto. By comparing the above expansions for gtt and gto with the expansions for v and w, it is clear that the expansions given by Butterworth and Ipser (1976) for the metric functions are in the same form as given by Thorne (1980). Hence the isotropic coordinates introduced by Bardeen (1970) are ACMC. Further, the coefficients N and W, introduced by Butterworth and Ipser (1976) are equivalent to the Thorne multipole moments. (The normalization will be worked out exactly by comparing the multipole moments for the Kerr metric.) Gursel(1983) has shown that the Thorne multipole moments are equivalent to the invariant Geroch-Hansen multipole moments. Hence, the NJ and the W1 are also equivalent to the invariant multipole moments. 90 4.5.3 The Multipole Moments of the Kerr Solution Various authors have calculated the multipole moments of the Kerr metric: Newman and Janis (1965), Hernandez(1967), Hansen(1974), and Thorne(1980). They have all obtained the same results up to normalization. The even mass multipole moments are mal, while the odd ones vanish. The odd current multipole moments are mat while the even ones vanish. However, these authors have not specified the normalization. Thorne has given the exact normalization for the first three multipole moments in terms of the trace free symmetric multipole moments (Pirani, 1964, Section 2.3). The "z" components of the trace free symmetric part of the multipole moments are: 22 1zz = --2ma2 3 Szz = -4ma2 15 (Thorne, Eqn 11.28). It has been shown by Hansen(1974), that in the axisymmetric case, the multipole moments are completely determined by the "z" components of the trace free symmetric part. ( See also Landau and Lifshitz, 1975, Eqn. 41.7, and 41.8) During the course of the present study the following general forms for the "z" components of symmetric trace free multipole moments were obtained. In particular the general forms for the normalization constants for the Kerr metric multipole moments were obtained: M[k] = (2k)!! (2k- 1)!!2k J[k] = (2(k - 1))!! (2k- 1)!!2k-2" 91 Hence the first four nonvanishing mass multipole moments of the Kerr metric are: 221 Izz = ma2 = N2 3 2 8 4 1 ,zzzz = +-ma4 _ 1N4 35 2 16 6 1 zzzzzz - ma 6= -N6 231 2 128 1 Izzzzzzz = +2 ma = -N8. 6435 2 The factor of arises in the above definitions since N = e2v, and the mass multipole moments are defined with respect to v which corresponds to the gravitational potential in the Newtonian limit. The first four nonvanishing current(angular momentum) multipole moments of the Kerr metric are: Szzz = -4ma3 = W3 15 16 Szzzzz = +- mas = W5 315 32 Szzzzzzz - 32ma = W7 - 3003ma7 = 256 Szzzzzzzzz = + ma9 = W9. 109395 92 The above relations were verified by examining the metric functions for the Kerr metric in isotropic coordinates: 2V _ _ )2 +k 2+ 2 +a2 Co 2o e 2v =D 2/X 4P 2 P2-M S2 2os 012 { ( k )2 + a2}2 a2sin20(1 k22 2am + k2 + 2 W = -W/X = r-- -7T4r 1 k2 + 2 a2 2 2 sin2(l k2 2 112JI1+2~ J acos2 0 1- k 2+2 2 2 2 2 C 2 2 S k2 + ) 2 212 a2sin20(1- k22 D k2 r sin 0 4r2 where k2 = 2 - a2. The above expressions for the metric functions in isotropic coordinates are generalizations of the expressions given by Bardeen and Wagoner(1971) for the extreme Kerr solution (VIII.2-5). The above metric functions were expanded at the poles and at the equator for the specific value of angular momentum parameter a = M/2, and the following expressions were obtained: e2v (r,) 2M 2M2 9M3 M4 211M5 325M6 5187M7 110 e(r,3 0) =6 + +7 Sr 2 8r3 4r4 + 128r5 128r6 2048r r 2v 7r 2M 2M2 13M3 9M4 633M5 1243M6 32325M7 10 e2 (r, -) = 1 - - + . . .+ + +O 2 r r2 -8r3 + 4r4 - 128r5 + 128r6 - 2048r7 O(r) M2 3M3 79M4 21M5 351M6 329M7 18571M8 112 4 r,0)-+ + +__ ___ + O ( )_r r4 16r5 4r6 + 128r + 128r8 2048r9 +O(r) 7r M2 3M3 83M4 7M5 1169M6 1739M7 46419M8 1 12 W(r, -) - + -+ + O( 2 r r 16r5 r6 128r 128r8 2048r9 +O(r 3M2 9M4 1 12 e2(r, 0) = 1 - r2+ 256r4 +O(r ) 1 5M2 265M4 2M5 19M6 19M7 795M8 369M9 112 2(r ) 8r2 + 256r4 - r5 + 8r6 - 8r7 + 256r8 64r9 + O(r) The tracefree multipole moments given above were substituted in the expansions for the metric functions given in Section 4.4.3. The results were |

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PAGE 1 THE SPACETIME MANIFOLD OF A ROTATING STAR By EJAZ AHMAD A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 UNIVERSITY OF FLORIDA LIBRARIES PAGE 2 Copyright 1996 by Ejaz Ahmad PAGE 3 ACKNOWLEDGEMENTS The author expresses his sincere appreciation to all members of his committee: to Professor Ipser for teaching him how to simulate neutron stars numerically; to Professor Detweiler for many discussions on rotating black holes; and to Professor Whiting for his generous assistance in all mathematical matters. The author would like to express his thanks to the Relativity Group at the University of Chicago for a weeklong visit in the Summer of 1993. The author benefited greatly from the encouragement and insight of Professor Subrahmanyan Chandrasekhar. It is much to the author's regret that Professor Chandrasekhar passed away in August 1995 before this work could be completed. The author would like to thank Dr. Bob Coldwell for his assistance on numerous numerical matters. The author would also like to thank Mr. Chandra Chegireddy, the Departmental System Manager, for his assistance on all computer related matters both big and small. Finally the author would like to thank his parents, and his extended family, and numerous friends for their continued support. iii PAGE 4 TABLE OF CONTENTS Page ACKNOWLEDGEMENTS iii ABSTRACT vii CHAPTERS 1 1 INTRODUCTION 1 2 THE KILLING VECTORS 8 2.1 The Field Equations in Covariant Form 8 2.2 The Killing Bivector 9 2.3 The Electrostatics Analogy 11 3 WEYL'S HARMONIC FUNCTION 24 3.1 The Schwarzschild Solution 26 3.2 The Kerr Solution 29 3.3 Weyl's Harmonic Function for Rational Solutions 37 3.3.1 Determination of Critical Points in the Harmonic Chart 39 3.3.2 Determination of Critical Points in the Isotropic Chart . 42 3.4 The Zipoy-Vorhees and the Tomimatsu-Sato Solutions .... 46 3.5 Weyl's Harmonic Function and Global Structure 49 3.5.1 The Kerr Solution 52 3.5.2 The Double Kerr Solution 56 3.5.3 The Manko et ai, 1994 Solution 60 4 STATIONARY AXISYMMETRIC FIELDS IN VACUUM ... 66 4.1 The Conformal Invariance of the Field Equations in Vacuum . 66 4.2 The Existence of Global Charts 68 iv PAGE 5 4.3 Practical Aspects of Coordinate Charts 76 4.4 Asymptotic Expansions for the Metric Functions 77 4.4.1 The Complete Nonlinear Vacuum Field Equations ... 78 4.4.2 The Linearized Vacuum Field Equations 78 4.4.3 Expansions for the Metric Functions 80 4.5 Equivalence to Thome Multipole Moments 87 4.5.1 Thome's Definition of Vector Spherical Harmonics ... 87 4.5.2 Thome's Expansion for the Metric Functions 88 4.5.3 The Multipole Moments of the Kerr Solution 89 5 CONCLUSION 94 APPENDICES 98 A GAUGES AND COORDINATES FOR STATIONARY AXISYMMETRIC SYSTEMS 98 A.l The Radial Gauge 99 A.2 The Isothermal Gauge 100 A.3 Two Global Coordinate Charts 102 A.4 The Three Types of Coordinates 104 B VARIOUS FORMS OF THE SCHWARZSCHILD METRIC . . 107 C VARIOUS FORMS OF THE KERR METRIC 112 D THE MANKO et ai, 1994 SOLUTION 116 REFERENCES 118 BIOGRAPHICAL SKETCH 122 V PAGE 6 LIST OF FIGURES Figure Page 2.1 Two-dimensional Cross-section of the Schwarzschild Solution . .17 2.2 Two-dimensional Cross-section of the Double Kerr Solution . . 18 2.3 Two-dimensional Cross-section of a Solution with a Toroidal Killing Horizon 19 2.4 The isometric embedding of the two-surface S orthogonal to the group orbits in R^. This is an inverted Flamm's paraboloid in isotropic coordinates 21 3.1 D2(0,z) and 5(0,2) for the Kerr Solution 54 3.2 D2(0, z) and B{0, z) for the Double Kerr Solution 58 3.3 i:)2(0,z) and 5(0, 2) for the Manko a/. 1994 Solution .... 63 vi PAGE 7 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE SPACETIME MANIFOLD OF A ROTATING STAR By EJAZ AHMAD August 1996 Chair: James R. Ipser Major Department: Physics Stationary Axisymmetric Systems in General Relativity are characterized by two Killing vector fields. The norm of the Killing bivector satisfies a two-dimensional Poission equation on the two-surfaces orthogonal to the group orbits. For perfect fluids, the source term in the Poission equation is the fluid pressure. In the vacuum region the Poission equation reduces to the Laplace equation, and the norm of the Killing bivector is a harmonic function. The harmonic property of the norm of the Killing bivector was noted by Herman Weyl in 1917 for static axisymmetric systems. Although subsequent studies of stationary axisymmetric systems have utilized the harmonic properties of the function in question, surprisingly there is no discussion of the boundary data for the Laplace equation in vacuum, or the connection between sources and the Killing bivector via the Poission equation. The contemporary literature makes references to Weyl's harmonic function; however, there is apparent confusion regarding the number and location of the critical points of this harmonic function. In this dissertation the boundary data for Laplace's equation for various sources of astrophysical interest is discussed for the first time. In particular, vii PAGE 8 the critical points of Weyl's harmonic function are determined for the Kerr solution by generalizing Weyl's results for the Schwarzschild solution. Next, an algorithm for the determination of the critical points for arbitrary rational stationary axisymmetric solutions is given. In the study of stationary axisymmetric systems two global coordinate charts have been used in the isothermal gauge: the harmonic chart and the isotropic chart, both having been introduced by Weyl in 1917. These two coordinate systems are related to each other by complex conformal transformations. In this study it is shown that the isotropic chart is asymptotically Cartesian mass centered(ACMC) as defined by Thorne (1980). Therefore it is possible to simply read off the mass the mass-/ moment from the -^p^part of gu, and the angular momentum /-pole moment from the -K part of gu once a solution is expressed r in isotropic coordinates. This prescription does not work in Weyl's harmonic coordinates which are not ACMC. Further, the coordinate vector fields associated with harmonic coordinates are not smooth at the critical points of Weyl's harmonic function. viii PAGE 9 CHAPTER 1 INTRODUCTION In the last two decades there has been considerable progress in the theory of nonlinear partial differential equations particularly in two dimensions. In two dimensions many important equations have been shown to be exactly soluble by the Riemann-Hilbert method of classical complex analysis. These mathematical techniques have been applied successfully to the theory of General Relativity in spacetimes with high symmetry. In particular the technique of inverse scattering transform may be applied whenever a spacetime possesses two Killing vector fields. Using these techniques it has been possible to generate exact solutions to the Einstein field equations for stationary axisymmetric systems, inhomogeneous cosmological models, and colliding plane waves. Rapidly rotating neutron stars, which is the subject of this treatise, fall under the category of stationary axisymmetric systems. Generalizing the work of Buchdahl(1954) and Ehlers(1957), Geroch(1972) constructed the symmetry group of the stationary axisymmetric vacuum field equations; Kinnersley and Chitre(1978) determined the subgroup of the Geroch group which preserves asymptotic flatness. Using the method of the Riemann-Hilbert problem, Hauser and Ernst(1979) proved the conjecture of Geroch that any asymptotically flat stationary axisymmetric solution may be generated from Minkowski space using group theoretical methods. At the present time various algorithms exist for generating exact solutions for 1 PAGE 10 2 stationary axisymmetric systems. A plethora of exact solutions has been generated purporting to represent the exterior of a rapidly rotating neutron star. Notwithstanding the abovementioned developments, very little is known about the spacetime of a rotating star. Theoretical studies of the global properties of the spacetime of a star and it causal structure have not been undertaken. It is known from the numerical simulations of Wilson(1972), and Butterworth and Ipser(1975) that some rapidly rotating stars possess ergotoroids. However, it is not known whether or not the vacuum solution of the star possesses an event horizon or a Killing horizon. Therefore, whether or not the exterior star solution possess a Killing horizon is a key question in the study of stationary axisymmetric spacetimes in General relativity. Apart from its theoretical significance in terms of understanding the vacuum solutions to Einstein's equations, this question has practical consequences. For example, it is known that some of the simpler exact solutions do possess Killing horizons, some do not, while the question is unanswered for some of the more complicated solutions. More precisely, two separate questions arise: First, does the real star, which in this case could be numerically generated, possess a Killing horizon ? Second, given an exact solution, what is the procedure for determining whether or not it possesses a Killing horizon. This question is being asked here for the first time. Some tools for analysing this issue will be developed in this study. Once these two questions are answered the task of comparing exact solutions with numerically generated exterior vacuum solutions may be undertaken. In a stationary axisymmetric system the Killing horizon is determined by the determinant of the {t, 4>) part of the metric. In the vacuum region the determinant satisfies the Laplace equation, and is therefore a harmonic function. PAGE 11 3 In order to understand the global properties of the star, it is important to understand the behavior of the determinant of the {t, (f>) part of the metric. More precisely it is important to determine the boundary data for the harmonic function under consideration in the vacuum region. In a stationary axisymmetric system, all quantities are functions of two variables, and the Laplace equation in question is two-dimensional. In the interior of the star the determinant is no longer a harmonic function and it satisfies the Poission equation with pressure as the source. The answer to the key question, whether or not the exterior vacuum solution possesses a Killing horizon, will be determined by the boundary conditions of the Laplace equation in the exterior and the Poission equation in the interior. Hence the global properties of the star solution-?, e., the presence or absence of a Killing horizon-is determined directly by the pressure distribution, and only indirectly by the energy density. This important point has not been noted in the literature. It has been noted in the literature that a harmonic function exists in stationary axisymmetric systems. However, the boundary data for this harmonic function has not been determined for the Kerr solution or the rotating star. The Laplace equation is conformally invariant in two dimensions, and the critical points are conformal invariants (Walsh, 1950). In the present study the boundary data for Laplace's equation will be determined in terms of the critical points of the harmonic function in question for the Kerr solution and for the rotating star. For the star solution, the critical points of the harmonic function will be determined by the pressure distribution. The clarification of boundary data for the two-dimensional Laplace equation is one of the goals of this study. PAGE 12 4 As is often the case in General Relativity, the important issues outlined above are obscured by issues of gauge and coordinate choices. One of the results of the present investigations is a unified treatment of gauges, charts and coordinates used in the study of stationary axisymmetric systems presented in Appendices A, B, and C. Theoretical work is conducted in the harmonic chart with the metric in the Papapetrou form in prolate spheroidal coordinates, while numerical studies are conducted in the isotropic chart with the metric functions in the Bardeen form in cylindrical polar coordinates (Appendix A). In the harmonic chart the determinant of the {t, PAGE 13 5 four-dimensional spacetime. One of the important technical issues resolved in the present study is the procedure for determining the critical points of the harmonic function D in harmonic coordinates, as well as in isotropic coordinates. Boundary data for the Laplace equation for D can be extracted from the metric regardless of gauge choice or coordinate chart. The quantity D is an invariant quantity (Carter, 1969); it satisfies the field equations. Therefore every stationary axisymmetric spacetime has unique boundary data for D. Once the appropriate boundary data are determined for a solution, coordinate transformations can be made from the harmonic chart to the isotropic chart, and a comparison can be made between the exact solution in the harmonic chart and the numerical solutions in the isotropic chart. Once the boundary data for D is determined, information about the degrees of freedom of the exterior vacuum solution becomes transparent. There are four coordinate invariant quantities that can be constructed from the Killing vectors: V = gu, X = g^^, W = gt^, and = VX + W^. As these quantities are algebraically related, at each point of the spacetime three algebraically independent quantities are present. In the numerical computations in the isotropic chart three functions of the Bardeen form of the metric are computed: e^"" = V + W^/X = D^/X, UB = -W/X, and D = piB. The invariant definition of D is not present in the pioneering work of Bardeen (Bardeen, 1970, 1973). The invariant definition of D as the magnitude of the Killing bivector, = VX + W^, is given in the works of Carter(1969,1973). In the numerical studies of rotating neutron stars, three metric functions are computed in the exterior vacuum region. Asymptotic expansions for PAGE 14 6 the metric functions were determined by Butterworth and Ipser(1976) to establish boundary conditions for their numerical algorithm for constructing rotating stars. There are three sets of parameters in these expansions. The invariant definitions of these parameters have not been given. It is shown in the present study that two sets of parameters corresponds to the multipole moments defined by Geroch for stationary systems in terms of the norm and twist of the timelike Killing vector-ie., V and W. The third set of parameters correspond to the function D. These parameters arise due to the presence of the additional symmetry-the axisymmetry-of the spacetime. The invariant definition of D in terms of the magnitude of the of the Killing bivector involves two Killing vector fields. The third set of parameters is a not part of the invariant multipole moments defined by Geroch and Hansen(1974) as they assume only stationarity and not axisymmetry. Therefore they do not contain any information about the magnitude of the spacelike Killing vector field that defines axisymmetry X or the magnitude of the Killing bivector D. In addition to the invariant quantities defined above in term of the Killing vector fields, there is another invariant quantity which arises due to the two-dimensional nature of the problem. This quantity is the Gaussian curvature of the (p, z) two surface. Its value at each point is determined by the field equations in terms of the gradients of V and W. In terms of coordinates it is determined by the conformal factor, e^, of the two surface S. Due to the conformal invariance of the field equations in vacuum, the conformal factor is not present in the field equations in the vacuum region. Therefore it is possible to solve for the three functions V, W, and D first. As the field equations are not all independent, the solution for the conformal factor may be obtained afterwards. PAGE 15 7 In the present study the asymptotic expansion for is derived. The conformal factor is expressed in terms of the three sets of parameters mentioned above. Therefore it is shown that the three sets of parameters defined by Butterworth and Ipser(1976) determine all four metric functions in the vacuum region. PAGE 16 CHAPTER 2 THE KILLING VECTORS 2.1 The Field Equations in Covariant Form A stationary axisymmetric spacetime has two Killing vector fields: a timelike Killing vector field ^ and a spacelike Killing vector field r]. The trajectories of the spacelike Killing vector 77 are closed, and t] vanishes on the rotation axis. In the neighborhood of the rotation axis the normalization condition for this Killing vector is 4X ' where X = rjÂ°"qaAt large distances X -> sin^ 9. The normalization condition for the timelike Killing vector field ^ is -^"^a = F ^ 1 at large distances from the source. If an ergosurface exists in the spacetime, then ^ is null on the ergosurface, and spacelike in the ergoregion. In an asymptotically flat spacetime these conditions are sufficient to determine the Killing vectors uniquely {Exact Solutions, Section 17.1). If coordinates are chosen such that ^ = and 77 = then the metric can be written as In a stationary axisymmetric spacetime all quantities are independent of the coordinates t and (f), and are functions of the coordinates {xi,X2) which parametrize the two surface S orthogonal to the group orbits. The functions W, U, and up 8 PAGE 17 transform as scalars under coordinate transformations on the two surface S. Their invariant definitions are as follows: n^r^a X = -e-2^D2 e'^^ul ^Â«r7a = W = e-^^cop -2^[aVb]ev' = D' = VX + W'. The scalar is the norm of the Killing bivector. Bivectors are antisymmetric tensors of second order, or two forms {Exact Solutions, Section 3.4). In terms of the metric components, the scalar is equal to minus the determinant of the two-dimensional metric on the group orbits. In vacuum the covariant form of the field equations for the three invariant scalars are (Kramer and Neugebauer, 1968; Exact Solutions, Section 17.4): (De-^VM)'"" = 0 (Am)'"" = 0. In the above equations u is the twist scalar of the Ernst potential, S Â— e^^ + iu>, defined as follows: um = i^,MIt is related to the Papapetrou metric function ujp as follows: up^M = De~'^^ tMN^^\ ^MN is the Levi-Civita tensor on the two surface S. These equations are covariant with respect to transformations of the two-dimensional metric ^mn on the two surface S. The last equation shows that the magnitude of the Killing bivector satisfies the two-dimensional Laplace equation on the two surface S; the scalar D is therefore a harmonic function on the two surface S. The remaining field equations determine the Gaussian curvature of the two surface S: PAGE 18 10 2.2 The Killing Bivector As discussed in the previous section, in a stationary axisymmetric spacetime there are three independent coordinate invariant scalars that can be constructed from the Killing vectors: V, X, and W. A fourth coordinate invariant quantity which is a combination of the previous three is the norm of the Killing bivector: = VX + W'^. The scalar D'^ is positive outside the Killing horizon, and negative inside the Killing horizon (Hawking and Ellis, p. 167). vanishes trivially on the axis as rj vanishes on the axis. The region off the axis where the D"^ vanishes is the Killing horizon where ^[aVb] 0, but its norm vanishes (Carter 1969). Therefore the quantity D contains important global information which is not carried by either of the three other coordinate invariant quantities, V, X, or W, individually. The field equation for D is remarkably simple: D satisfies the Laplace equation on the two surface S orthogonal to the group orbits. Therefore D is a harmonic function with well defined complex analytic properties. For example, it cannot have any maxima or minima in the interior, and its critical points are isolated. Although it is mentioned in the literature that Z) is a harmonic function, the boundary data for the two-dimensional Laplace equation have not been given in the literature. It is mentioned that the critical points of D are isolated; however, the number and location of these critical points are not specified for any stationary axisymmetric solution (e.g. Geroch and Hartle, 1982, Appendix A; Wald, 1984, Section 7.1). Let the harmonic function D represent the determinant of the two metric on the group orbits and E be its conjugate harmonic function. Then D and E can be represented as the real and imaginary parts of a complex analytic function yV = D + iE. By the properties of complex analytic functions, D and PAGE 19 11 E will satisfy the Laplace equation in two dimensions. The function W will henceforth be referred to as Weyl's harmonic function after Hermann Weyl who defined it for the first time in 1917 for the Schwarzschild solution (Weyl, 1917, p. 140). For the Schwarzschild solution where W = D + iE and Z Â— pi + izi represents isotropic coordinates. The use of the harmonic function W itself to parametrize the two surface S leads to Weyl's harmonic coordinates. (Further details of coordinates and gauges may be found in Appendix A.) For stationary axisymmetric solutions consisting of rational functions, Weyl's harmonic function for the Schwarzschild solution can be generalized to The generalization given above satisfies the boundary condition that W ^ Z at large distances. The function W is defined only on the half space since the cylindrical coordinate pi is defined for Pi > 0. Complex functions arising in cylindrical coordinates must satisfy the relation: [W(Z)]* = Â—W[Â—Z*), where the * symbol denotes complex conjugation. This condition is satisfied if the coeflficients 6Â„ are real numbers. For rotating stars the coefficients hi are 2D pressure multipole moments arising from the 2D Poission equation. 2.3 The Electrostatics Analogy The goal of this section is to build intuition about Weyl's harmonic function by developing an analogy with two-dimensional electrostatics. In this analogy the lines of constant D represent equipotential lines while the level curves of the conjugate harmonic function E represent the lines of fiux of the electric PAGE 20 12 field. If VE = -VD ^ 0, then the two sets of curves are orthogonal. The electrical field E ~ VD is tangent to the flux lines of constant E. For simplicity this analogy is developed first assuming that the two surface S is flat. At the end of the section, the curvature of S will be taken into account. The most general form of Weyl's harmonic function given above corresponds to a non-truncating Laurent series expansion about infinity. It is convergent outside a region containing the origin, and a small circle excising the pole at infinity. (This expansion will be valid if there are no other charges present. If there are additional charges, then the radius of convergence of the Laurent series will be different.) For example, for a stationary axisymmetric spacetime of a rotating star, the excluded region surrounding the origin could represent the two-dimensional cross section of the interior of the star. The Laurent series expansion for a complex analytic function has a simple interpretation in terms of two-dimensional electrostatics. Morse and Feshbach write: "That part of the Laurent series with negative powers of z, which is convergent outside the inner circle of convergence, therefore corresponds to the statement that the field from the charge distribution within the inner circle of convergence may be expressed as a linear superposition of fields of a sequence of multipoles at the point the origin." (Morse and Feshbach, 1953, vol. 1, p. 379) The general term 2"" represents a two-dimensional multipole of order 2"~^ From the above discussion it is clear that determinant of the {t, (f)) part of the metric D can be regarded as a two-dimensional potential defined on the two surface S and the coefficients 6Â„ are two-dimensional multipole moments. It can be regarded as that potential which generates a field consisting of pure multipole fields located at the origin superimposed upon a uniform field of unit magnitude PAGE 21 13 pointing along the p axis of cylindrical coordinates, i.e., along the equator. In terms of complex analysis, Weyl's harmonic function and its derivative have the following Laurent series expansions: where W = D + iE and Z = pi + izi represents isotropic coordinates. Equatorial symmetry has been assumed in the above expressions. The above rational expansion will be valid when the two-dimensional surface S is simply connected. In terms of the 2D electrostatic analogy, the above expressions correspond to the multipole expansions for the potential and its gradient-the electrical field E. In terms of stationary axisymmetric solutions, the above expansions represent the determinant of the 2D metric of the group orbits and its gradient on the two surface S orthogonal to the group orbits-i.e., the determinant of the (t, ) part of the metric and it gradient in terms of (pi, zi) which parametrizes the two surface S. The critical points of the complex analytic function W defined above can be obtained by determining the points at which ^ = 0. In terms of electrostatics, the electrical field vanishes at the points of the equilibrium of the electrical field. The physical significance of the critical points of Weyl's harmonic function W will be discussed in Section 3.5. Next the boundary conditions of the problem are considered. The Killing bivector is zero on the axis since the rotational Killing vector is zero on the axis. Therefore the norm of the Killing bivector D"^ is zero on the axis and the axis is always at zero potential. The Killing horizon is also determined by the norm of the Killing bivector vanishing. Therefore, in addition to the axis >V = 2 + 5](-l) 'n PAGE 22 14 the Killing horizons will also be at zero potential. The boundary conditions on D are as follows: (1) D approaches the fiatspace cylindrical coordinate p at large distances; (2) D = 0 on the rotation axis since the axisymmetric Killing vector 77 vanishes on the rotation axis; and (3) D = 0 on the Killing horizon since the Killing horizon is defined by the conditions (i) ^[aVb] 0, and (ii) -2^[a^6]r^'' = = VX + = 0. It is assumed that D is regular in the interior. Next the boundary data for Laplace's equation in two dimensions will be discussed. For a rational solution, the Laurent series expansion for Weyl's harmonic function will truncate. In this case, the field in question will consist of a superposition of a uniform field and a finite number of multipoles. The expression for the field will be a polynomial in The roots of this polynomial determine the critical points of Weyl's harmonic function W. If the degree of the polynomial is n, there will be n roots. In terms of the electrostatics analogy, the critical points of Weyl's harmonic function represent equilibrium points of the two-dimensional field, i.e., points where E = 0. Therefore, if it is known that the polynomial in question has n roots, the degree of the polynomial will also be n. Similarly, if it is known that the field in question has n critical points, the number of multipoles present will become known as well. The precise details of the location of the equilibrium points will be determined by the precise values of the multipole moments 6Â„. As the coefficients 6Â„ in the polynomial expression for the field are real, the roots of the polynomial will either be real, or occur in complex conjugate pairs. Under the assumption of equatorial symmetry, there will be an even number of roots which will correspond physically to an even PAGE 23 15 number of critical points in Weyl's harmonic function, and an even number of equilibrium points in the field. It is clear that fields with different number of critical points are physically distinct. For example, a dipole field has two lobes, while an octupole field has six lobes. The number of critical points determine topological properties of the solution in two dimensions. From the complex analysis point of view, they correspond to polynomials of different degree, and are therefore distinct. In Chapter Three stationary axisymmetric solutions corresponding to different number of critical points will be discussed. Solutions with different number of critical points are physically distinct in four dimensions as well. For example a solution with two critical points on the axis has one Killing horizon, while a solution with four critical points on the axis will have two Killing horizons. Mathematically multipole moments 6Â„ are simply the eigenfunction expansion coefficients of the associated Strum-Louville eigenvalue problem. Therefore knowledge of the precise values of these coefficients constitute boundary data for the Laplace Equation. Once the precise values of the two-dimensional multipole moments 6Â„ are given, the boundary data for the Laplace equation in two dimensions is uniquely determined. The field in question is a superposition of point multipoles at the origin and a uniform field. Once the multipole moments are determined, the values of D and its derivative are also determined over suitable curves on the plane. For example, the value of Â£> on a semicircle of radius a is Â£>(a,^) = acos^Â—cos ^ + -^ cos 3^ -... + Â— cos n^. PAGE 24 16 In two dimensions the potential is a complex analytic function. For the class of potentials under consideration, the field is a polynomial. Once the coefficients of the polynomial are known, it is uniquely defined on the entire complex plane. Next, some examples of stationary axisymmetric fields are considered. These are simple solutions with Killing horizons where = VX + = 0. Figures 2.1-3 shows the two-dimensional cross-section of several stationary axisymmetric fields. PAGE 25 17 Rotation Axis Killing Horizon Figure 2.1 Two-dimensional Cross-section of the Schwarzschild Solution The first example is the Schwarzschild solution (Fig. 2.1). The determinant of the {t, (j)) part of the metric is zero on the axis and on the horizon represented by the semicircle. There are two critical points of this field located at the points where the axis intersects the horizon. PAGE 26 18 Rotation Axis Killing Horizon 1 I I I Killing Horizon 2 I I I I Figure 2.2 Two-dimensional Cross-section of the Double Kerr Solution The second example consists of two Kerr Black holes revolving about the same axis (Kramer and Neugebauer, 1980). This configuration has two horizons each surrounding a single Kerr black hole. There are four critical points where the two horizons intersect the axis(Fig 2.2). PAGE 27 19 Rotation Axis Cross Section of Toroidal Killing Horizon Figure 2.3 Two-dimensional Cross-section of a Solution with a Toroidal Killing Horizon The third example represents a stationary axisymmetric solution with a toroidal Killing Horizon (Carter, 1973). In terms of the two-dimensional boundary value problem, the axis is at zero potential. The circle represents the cross section of the toroidal Killing Horizon which is again at zero potential(Fig 2.3). It should be noted that the Killing horizon may not necessarily be an event horizon. Hence the precise interpretation of the third solution is not clear. According to Carter, this configuration will necessarily have a critical point between the horizon and the axis (Carter, p. 188, and Fig. 10.2). As harmonic functions do not admit maxima or minima in the interior, this critical point must be a saddle point. PAGE 28 20 Prom the previous examples it is clear the every stationary axisymmetric solution has an associated 2D electrostatics problem. For pure vacuum solutions the boundary conditions are determined by the determinant of the {t, 4>) part of the metric vanishing on the Killing Horizon. For solutions to stationary axisymmetric systems with perfect fluid sources, for example, the spacetime surrounding a rotating star, the determinant of the {t, (p) part of the metric has a two-dimensional multipole expansion since it satisfies Laplace's equation in the vacuum region. Corresponding to the Laplace's Equation in vacuum, there is also a Poission equation for D inside perfect fluid sources: ^Id^ ~ IStt^^/^, where p is the fluid pressure, and g is the volume element of the four dimensional metric. (Trumper 1967, Eqn. 5; Bardeen, 1971, Eqn IL14; Chandrasekhar and Friedman, 1972, Eqn. 74 and 75; Butterworth and Ipser, 1976, Eqn. 4c). For a rotating star the source term for the harmonic function D in the exterior region will be the fluid pressure in the interior of the star. Hence the coefficients bi of the harmonic expansion for D in the exterior region are simply two-dimensional pressure multipole moments. The multipole moments hi can be evaluated as volume integrals of the pressure inside the star. The procedure for determining the coefficients 6Â„ for rational solutions given in harmonic coordinates will be given in Chapter Three. One of the main results of the present work is to point out that by comparing the coefficients hn, numerical stars solutions and exact analytical solutions may be compared. Up to this point the two surface S has been regarded as a flat plane. In reality the two surface S is curved. How does the curvature of S affect the electrostatics analogy ? Two-dimensional electrostatic fields on curved surfaces have been discussed by Riemann (Riemann 1892, Klein 1893). Electrostatics fields on PAGE 29 21 Riemann surfaces have qualitatively similar properties in terms of sources, field lines, and equilibrium points. The global structure of a multipole field does not change on a curved surface: for example, an octupole field will have a six-lobed structure regardless of whether or not it is on a curved surface. So the global structure of the norm of the Killing bivector remains the same when the curvature of the two surface S is taken into account. The only complication is the precise definition of coordinates. That is where precisely the critical points are located. In Chapter Four it will be shown that field equations are conformally invariant. The metric functions V, W, and X can be solved first in flat space. The two surface S is built afterwards. The coordinates on this surface can be chosen uniquely. The two surface S is known very well. At every point on S, the Gaussian curvature of the two surface is determined from the field equations: Â«G = 7 ^^^^^ 2 N = -2e-2^(-Vi/ Â• Vu + |p2e-4^va; Â• Va;). (^11^22 512) ^ In the above expression, u and uj, are the Bardeen metric functions; their invariant definitions may be found in Appendix A. Therefore at every point on the two surface S three coordinate invariant quantities are known: kq, D, and E. For an arbitrary local coordinate chart these variables are not directly related. However, there is a global boundary condition along the axis. Spacetime is flat near the axis. This fact, along with the requirement that a single global chart cover the entire two surface S, fixes global coordinates on S uniquely. The uniqueness of global coordinates on S is demonstrated in greater detail in Section 4.2. As the curvature of S is known at every point, the two surface S can be embedded in three dimensional Euclidean space(Fig. 2.4). More precisely, if PAGE 30 22 Figure 2.4 The isometric embedding of the two-surface S orthogonal to the group orbits in R^. This is an inverted Flamm's paraboloid in isotropic coordinates. the Gaussian curvature is negative definite, it is possible to make an isometric embedding of a surface in three-dimennsions (Romano and Price, 1995). This can be done by solving the Darboux equations of classical differentical geometry. The Darboux equation belongs to the class of MongeAmpere equations and may be solved by the method of characteristics (Courant and Hilbert, Vol. II, pg. 495-499). In the case of the Schwarzschild solution, the 3D embedding is simply the Flamm's paraboloid (Flamm, 1916; MTW Section 23.8, Fig. 23.1). Therefore for every stationary axisymmetric solution there will be a unique curved surface in three dimensional space representing the curvature of the two surface S orthogonal to the group orbits. Weyl's harmonic function can be visualized as representing an electrical field on this curved two surface. The critical points of PAGE 31 23 Weyl's harmonic function will correspond to equilibrium points of the electrical field, where E = 0. PAGE 32 CHAPTER 3 WEYL'S HARMONIC FUNCTION In the relativity literature there is considerable confusion regarding Weyl's harmonic function. In the textbook by Bergmann(1942) it is stated that on the basis of symmetry alone the static axisymmetric metric can be brought into the Weyl form (p. 206). This error was noted by Synge(1960) who pointed out that the existence of Weyl coordinates in vacuum was a consequence of the field equations (Weyl, 1917, p. 137; Synge, 1960, p. 310, Note 1). As noted in the previous chapter, for stationary axisymmetric systems D, the determinant of the {t, (j)) part of the metric, satisfies the Laplace equation on the two surface S orthogonal to the group orbits. The important question: What is the boundary data for this field equation? has not been discussed in the literature. In this chapter the boundary data for the Laplace equation will be determined for several stationary axisymmetric solutions based on Weyl's harmonic function W. The real and imaginary parts of Weyl's harmonic function satisfy the Laplace equation in two dimensions. Therefore, determining Weyl's harmonic function for a particular stationary axisymmetric solution is equivalent to determining boundary data for the Laplace equation for the stationary axisymmetric solution in question. In this chapter the boundary data for the Laplace equation will be determined first for the Schwarzschild solution and then for the Kerr solution in terms of Weyl's harmonic function. A general prescription will be given for 24 PAGE 33 25 determining the harmonic function W for the recently obtained rational solutions to the stationary axisymmetric field equation generated by the application of soliton theory to General Relativity. Weyl's harmonic function will then be determined for the Tomimatsu-Sato series of solutions. Finally, the relation between Weyl's harmonic function and the structure of the Killing horizon will be discussed for the Double Kerr solution and the Manko et al., 1994 solution. In the previous chapter it was shown that the magnitude of the Killing bivector, -2^[aVb]C^^ = D"^ = VX + W'^, satisfies the two-dimensional Laplace equation on the two surface S orthogonal to the group orbits: (D^m)'^ = 0. The magnitude of the Killing bivector D"^ is also the determinant of the {t, (j)) part of the metric. The boundary conditions on D are as follows: (1) D approaches the fiatspace cylindrical coordinate p at large distances; (2) Z) = 0 on the rotation axis since the axisymmetric Killing vector r] vanishes on the rotation axis; and (3) D = 0 on the Killing horizon since the Killing horizon is defined by the conditions (i) ^[a^] ^ 0, and (ii) -2e[a%]C"^'' = D"^ ^ VX^W"^ = 0. Therefore the question arises: What is the boundary data for D that is different for distinct solutions ? The answer to this question lies in the harmonic properties of D, more specifically, in the critical points of the harmonic function W = D + iE defined on the two surface S. It will be shown in this chapter that the harmonic function W has critical points, where VD = VE = (0,0), at the intersections of the Killing horizons and the rotation axis. The Kerr solution has two critical points on the axis. The double Kerr solution of Kramer and Neuguebauer has four critical points on the axis. The general A^-Kerr solutions of Yamazaki, which are nonlinear superpositions of separate Kerr-like solutions on the axis. PAGE 34 26 has 2N critical points along the axis where the Killing horizons of each Kerrlike solution intersects the axis. Therefore, the number of critical points of the harmonic function W constitute topological boundary data for the Laplace equation. The knowledge of these critical points permits the differentiation of one stationary axisymmetric solution from another. If the critical point does not lie on the axis, as in the solution with a toroidal Killing horizon, the physical significance of the critical point is not clear, as the complete solution for the field equations for the toroidal Killing horizon has not yet been constructed. In this case, according to Carter(1973), there will be a critical point in the interior region between the toroidal horizon and the rotation axis (Fig. 2.3). Also, a toroidal Killing horizon does not intersect the rotation axis. The critical points of all known exact solutions do lie on the axis. Therefore, the consideration of off-axis critical points of the harmonic function W is an issue of principle relegated to future investigations. In the existing literature on stationary axisymmetric systems in General Relativity although it is mentioned that D is a harmonic function, the boundary data for the two-dimensional Laplace equation that D satisfies has not been given for any of the known solutions. It is mentioned that the critical points of D are isolated; however the number and location of these critical points is not specified for any stationary axisymmetric solution (e.g. Geroch and Hartle, 1982, Appendix A; Wald, 1984, Section 7.1). The discussion of this important issue begins with the critical points of the simplest known solution-the Schwarzschild solution-which is discussed in the next section. PAGE 35 27 3.1 The Schwarzschild Solution The boundary data for the Laplace equation for the Schwarzschild solution may be obtained very easily from Weyl's classic paper of 1917. Weyl introduced the use of the determinant of the {t, (j)) part of the metric [D = ph) and its conjugate function(Â£' = Zh) as coordinates for the first time in 1917. He noted that in such coordinates the spherically symmetric Schwarzschild solution corresponds to a line segment of length 2m, with constant linear mass density, along the axis of symmetry (Weyl, 1917, p. 140). As D vanishes on the horizon as well as on the axis, both curves are represented hy D Â— = 0 in Weyl's harmonic coordinates, and both become part of the axis in Weyl's harmonic coordinates defined by ph Â— 0. Weyl also introduced isotropic coordinates for the Schwarzschild solution in the same paper (Weyl, 1917, p. 132, Eqn. 12). The key to unravelling the boundary data for the Schwarzschild solution is the equation used by Weyl to map the Schwarzschild solution from isotropic coordinates to harmonic coordinates. This conformal transformation describes the scalar D in a chart which makes its complex analytic properties transparent. Weyl's harmonic function for the Schwarzschild solution is where W = D + iE = ph + izh are harmonic coordinates, and Z = pi + izi represents isotropic coordinates (Weyl, 1917, p. 140). This conformal transformation maps the horizon which is a semicircle of radius m/2 in isotropic coordinates to a line segment of length 2m in harmonic coordinates. In this form the boundary data for Laplace equation is quite apparent. In terms of two-dimensional multipole moments discussed in Section 2.3 this PAGE 36 28 map represents the superposition of a uniform field and a dipole field. This map known, as the Joukowski map in fluid dynamics, represents a uniform flow along the z direction past an infinite cylinder. Very far from the cylinder the flow is uniform. The scalar D approaches ordinary cylindrical coordinate p at large distances from the source. The fluid dynamics analogy can be extended further to determine the critical points of Weyl's harmonic function. In fluid dynamics when a flow encounters a barrier, or corner, the fluid velocity must vanish; such points are known as stagnation points. In the case of the Joukowski map, the stagnation points are at the two points where the flow encounters the cylinder head on~i.e., at the two points where the circle representing the cylinder intersects the axis. Therefore, in the case of the Schwarzschild solution the critical point of the harmonic function W are at the North and South poles where the horizon intersects the ax;is. Next the critical points of the harmonic function W for the Schwarzschild solution is determined by direct computation. At the critical points of a complex harmonic function, its first derivative vanishes; if higher derivatives vanish, the critical points are termed degenerate. For the Joukowski map, the critical points are determined by the roots of the equation dZ ~ 4VV2 ~ The roots of this equation are Z = 0Â±im/2, which correspond to the North and South poles. From the elementary properties of the Joukowski map, it follows that VÂ£) = VE = (0, 0) at the poles for the Schwarzschild solution. These are the isolated critical points of the harmonic function W. Since the derivative is a PAGE 37 29 second degree polynomial in it has only two roots; therefore the Schwarzschild solution has only two critical points. Hence, the two critical points at the poles are the only critical points of Weyl's harmonic function W for the Schwarzschild solution. The boundary data for the Laplace equation for the Schwarzschild solution has now been determined. The determinant of the {t, (j)) part of the metric, D^, has not been discussed in the literature as an important invariant quantity. Next this important quantity is expressed in several different coordinate charts for the Schwarzschild metric. The scalar D is the real part of Weyl's harmonic function W = D + lE. Therefore, considering the real part of the Joukowski map given above, an expression is obtained for D^: = VX = Ph in harmonic coordinates by definition 2 /i2/^i m'^x^ = r,smUi \1 Â— j m isotropic coordinates i = Tg sin dgll 1 in Schwarzschild coordinates. 3.2 The Kerr Solution The Kerr solution is considered next: in particular, the boundary data for the Laplace equation for the Kerr solution will be determined. The data for the Kerr metric is not quite apparent at first; however, proceeding in analogy with the Schwarzschild solution, the result in the end turns out to be simple and qualitatively similar. The key issue is the number of critical points of the harmonic function W for Kerr solution. In the static limit the number must reduce to the result for the Schwarzschild solution derived in the previous section-two critical points at the poles. Since rotation does not alter the topology of the PAGE 38 30 horizon the result remains the same for the rotating Kerr solution. An example where the critical points change due to a change in the topology of the horizon is given by Carter (1973): in the case of a toroidal Killing horizon, a saddle point exists between the torus and the axis (Fig. 3.1). Next, the critical points of the harmonic function W will be obtained by the methods of potential theory. As discussed in Section 2.3, is a potential function in two dimensions, and curves oi D = constant are equipotential curves. For the Kerr solution, the horizon and the axis intersects at right angles at the two poles. If the intersection were not at right angles, there would be a kink in the horizon which would destroy the smoothness properties of the horizon at the pole. From the point of view of potential theory, the axis and the horizon are two equipotential curves with D Â— 0 that intersect at right angles. By the elementary properties of solutions to the Laplace equation, it follows that -> (0,0) smoothly at the poles (Jackson, 1975, Section 2.11). In two dimesions, near a right angle corner, the solution to the Laplace equation for the potential is given by D = DqE? sin 29, where i?, 9 are defined in a small chart which is flat in a sufficiently small neighborhood of the North pole (Jackson, 1975, Section 2.11, Fig. 2.13). The components of the field near a right angle corner are E = -VD = ( -2Doi?sin29, -2Doi?cos29 ) where Dq is a constant. The field varies linearly with distance (i?) from the pole, and vanishes at the pole: VD = (0, 0) at the poles. By the Cauchy-Riemann relations VE = (0, 0) as well. The gradient of the determinant of the {t, (p) part PAGE 39 31 of the metric and its conjugate harmonic function both vanish at the poles. The fact that the gradient of the determinant of the {t, (j)) part of the metric vanishes at the poles for the Kerr solution has not been noted in the literature. The elementary properties of solutions to Laplace equation in two dimensions lead to nontrivial conclusions in the theory of rotating black holes. It has been shown that the gradient of D must vanish at the poles due to purely geometric boundary conditions at the poles. Next it is shown that VD = (0, 0) at the poles using the constancy of the surface gravity boundary condition on the horizon. In the Bardeen form of the stationary axisymmetric metric, the function e^'' = F + W'^/X = D'^/X. On the horizon of a Kerr black hole e^^ Â— 0. Bardeen has shown that the gradient of e*^ approaches a limiting value, the surface gravity kh-, as the horizon is approached: e~f^{e'^)r KH\ and e~^^{e'')0 -> 0 since the surface gravity is constant on the horizon (Bardeen, 1973, Eqn. 2.18-20). The conformal factor ef" ~ 1 near the poles (Appendix C). Therefore, the gradient Ve" is bounded and well behaved near the horizon; its components in any coordinate basis are finite; Ve*^ Â— >Â• {kjj, 0) in polar coordinates (rj, di) while in cylindrical coordinates {pi, Zi), Ve" is bounded by the vector A = {kh, i^h) near the horizon. Next the magnitude of the rotational Killing vector is considered. In the neighborhood of the rotation axis the normalization condition for the rotational Killing vector is AX ' where X = Tj^rja; X ^0 near the axis. By arguments similar to the above, it follows that the gradient vector ^ is well behaved. Near the axis it is bounded PAGE 40 32 by the vector B = (2, 2) in cylindrical coordinates {pi, Zi), since at large distances X Â— > r sin e, and ^ ^ (2 sin ^, 2 cos ^) in polar coordinates {ri,9i). Now that it is has been established that the gradients of X and are bounded near the poles, where the horizon intersects the rotation axis, it is straightforward to calculate the gradient of D directly at the poles D = e''x'2 Along the axis the second term in the gradient drops out since X = 0 on the axis, and Ve'^ is bounded by the vector A. Therefore along the axis ' xV As the North pole is approached along the axis, -) 0 smoothly and is bounded by the vector B. Hence as the North pole is approached along the axis VD -> (0,0). Since approaches zero smoothly, VD -)> (0, 0) smoothly as well. Next the North pole is approached along the horizon. Along the horizon the first term in the gradient drops out since = 0 on the horizon, and is X2 bounded by the vector B. Therefore along the horizon As the North pole is approached along the horizon, 0 smoothly and Ve'' is bounded by the vector A. Hence as the North pole is approached along the horizon VD^>(0,0). PAGE 41 33 Since X5 0 smoothly, VD Â— )Â• (0,0) smoothly. Hence the limit exists and VD = (0, 0) at the North pole. It is established that VD = (0, 0) at the two points where the horizon intersects the axis. Thus Weyl's harmonic function has two critical points for the Kerr solution: one at each pole. This result has been established by three different methods: (1) Topological method: since the topology of the horizon is not altered by rotation, the number of critical points of the Kerr solution must be the same as those of the Schwarzschild solution. (2) Potential theoretic method: the field goes to zero linearly with distance near a right angle corner. (3) The method employing the horizon boundary conditions: e^"^ 0 smoothly near the horizon; its gradient normal to the horizon is the surface gravity kh; due to the constancy of the surface gravity on the horizon, its gradient has no tangential component. Once the number of critical points of Weyl's harmonic function is determined for the Kerr solution, the exact form of the harmonic function W can be determined uniquely. Weyl's harmonic function for the Kerr solution is therefore given by: This is similar to the Schwarzschild case except the mass has been replaced by the quantity k = Vm^ o?. This value is determined by making a conformal transformation from Weyl's harmonic chart in prolate spheroidal coordinates to Weyl's isotropic chart in spherical polar coordinates. The complete form of the Kerr metric in Weyl's isotropic coordinates is given in Appendix C. The parameter k is related to the surface area of the horizon A and the surface gravity kh as follows: k = (Carter, 1973, p. 197, Eqn 10.55). If the angular momentum satisfies the condition a < m, a Killing horizon will PAGE 42 34 exist for the Kerr solution. In isotropic coordinates, the horizon is a semicircle with constant rj coordinate, rj = k/2 (Bardeen, 1973, p. 251, Eqn. 2.8). By the Joukowski map given above, it will be mapped to a line segment of length 2k in Weyl's harmonic coordinates. In prolate spheroidal coordinates, this is the line segment between the two foci (Appendix C). It should be noted that even though the horizon is determined by curve with a constant coordinate value r = k/2 in isotropic coordinates, this does not imply that the horizon is geometrically spherical since the metric on the two surface S in the Kerr case has angular dependence. Similarly, even though the conformal transformation is the Joukowski map for both the Schwarzschild and the Kerr solution, the interpretation of the coordinates on the two surface S is different. Finally the norm of the Killing bivector is examined for the Kerr solution in various coordinate charts. in harmonic coordinates by definition in isotropic coordinates = i^BL~ ^mrsL + a^) sin In Boyer-Lindquist coordinates. in Boyer Lindquist coordinates. (jBL sin 6 (r|^ 2mrBL + a^) 5 cos ^ .(4^-2mrBi + a2)5 1 ' irlL-2mrBL + a^y^ along the axis 0, 0 only at the poles since A = {r%L-2mrBL-\-a?)^2 = o only at the poles where tbl = m+( The above calculation is not completely valid since the denominator PAGE 43 35 1 ('''bl ~ ^J^^BL + a^^ becomes zero on the horizon. To get around this difficuly, the following technique due to Professor Whiting is employed: Instead of evaluating VD at the poles, its magnitude is evaluated. M = VVD VD ^ {rBL 2mrgL + Q^) {rBL rn)^ sin^ 6 (r|^ 2mrBL + Q^) cos^ 6 V ^BL + ^^^^ ^ ('^BL ~ ^^"^BL + a^) + a2 cos2 e _ /(rfiz, m)2 sin^ ^ (r^^ 2mrBi + a^) cos^ ^ V ^BL + ^^^^ ^ ^BL + ^ The magnitude M does not blow up anywhere: the first term in M vanishes along the axis, while the second term vanishes along the horizon. It is clear that M goes to zero smoothly as the pole is approached. Hence, VD goes to zero smoothly as the pole is approached in Boyer-Lindquist coordinates. As before the pole is shown to be a critical point of the harmonic function D. It has been stated inThe Mathematical Theory of Black Holes{MTB) that the above expression for A for the Kerr solution is "A solution . compatible with the requirements of regularity on the axis and convexity of the horizon ..." (MTB, p. 279). The function D = Asin^ satisfies the Laplace equation for D = = A^/^/C^) which is a part of the Einstein field equations (MTB, Eqn 6.43). [AV2(aV2)_^]_^ + ;-1^(^)_^^^^0 Therefore for a particular stationary axisymmetric solution, such as the Kerr solution, the solution for D, which in this case is A sin 9, must be unique. The question therefore becomes what is the boundary data for the Laplace Equation (MTB, Eqn 6.43) ? It has been shown above that the determinant D for the PAGE 44 36 Kerr solution has two critical points at the poles. As shown above by direct computation VD= V(r|j^ -2mrBi + 0^)5 sin ^= (0,0) at the poles. Therefore the function A sin 9 satisfies the correct boundary conditions for the problem in question. Hence D = (r|^ Â— 2mrBL + o^)^ sin^ is the unique solution for the Kerr metric. The boundary data for the function D is transparent in the isothermal gauge. In other gauges it is not so clear since the metric on the two surface S and determinant of the (t, (j)) part of the metric become intertwined. The various gauges that have been used in the study of stationary axisymmetric solutions are discussed in Appendix A. Before proceeding further, it is important to note the following points. The determinant carries global information since = VX + W"^ = 0 on the horizon. None of the three coordinate invariant quantities V, X, or W, singly by themselves carry global information about the horizon that is carried by their combination D"^ = VX ^W"^. Therefore, from the physical point of view the determinant D has important global properties. From the mathematical point of view, function D also has global properties since D is a harmonic function, e.g. its critical points are isolated, it has no maxima or minima in the interior, etc. Although the quantities V, AT, and VF, satisfy a system of coupled quasi-linear second order elliptic partial differential equations, none of them are harmonic functions; only the combination = VX + W"^ is harmonic with well defined global properties. This confluence of global physical properties and global mathematical properties is quite remarkable, and at the same time necessary. These global properties also have practical consequences. The fact that Weyls' harmonic function W has critical points at the poles has been derived quite easily PAGE 45 37 using potential theory which embodies the global harmonic properties of W. In contrast, deriving the same result using local properties only of the functions and X, was less insightful and tedious. 3.3 Weyl's Harmonic Function for Rational Solutions In this section Weyl's harmonic function will be further investigated for rational soloutions of the stationary axisymmetric solutions. As most of the known solutions fall into this category, this is not too restrictive. The form of the metric in the isothermal gauge is preserved under conformal transformations. Several authors have incorrectly commented that once a conformal transformation has been made to Weyl's harmonic coordinates {ph,Zh), all information about the harmonic function W = D + iE = + izh is lost. Lewis (1932) writes: "the canonical [Weyl's harmonic] coordinates serve to remind one of the degree of arbitrariness involved in our solutions" (Lewis, 1932, p. 177). Similarly Zipoy(1970) writes "Since p [ph] is initially an arbitrary harmonic function of (^)C) [Pi,Zi], there will clearly be no intrinsic means of determining what the coordinates represent physically" (Zipoy, 1970, p. 2120). Evidently D, the determinant of the {t, (j)) part of the metric, is a coordinate invariant quantity. It will have a unique value at every point on the 2-surface S. The confusion stems from treating ph = D as "just another coordinate", and ignoring its invariant definition. It should be pointed out that the invariant definition of D as the norm of the Killing bivector did not exist prior to Carter's work on Killing horizons (Carter, 1969). In the previous section Weyl's harmonic function was determined for the important solutions of Schwarzschild and Kerr. Additional stationary axisymmetric solutions have been obtained recently. These exact solutions are PAGE 46 38 determined in Weyl's harmonic coordinates where the explicit form of the harmonic function remains obscure. The critical points of Weyl's harmonic function for these solutions have not been determined. Therefore it would be highly desirable to obtain a general procedure for identifying the critical points of a harmonic function for any rational stationary axisymmetric solution. Once this is done the determinant of the {t, (p) part of the metric could be determined as a function of Weyl's isotropic coordinates on the two surface S. Once a solution is expressed in Weyl's isotropic coordinates the topology of the horizon, the boundary data for Laplace equation, and the critical points of the harmonic function all become transparent. As pointed out in the introduction, many authors have erroneously claimed that once Weyl's harmonic coordinates are chosen, all the information about the harmonic function W is lost. The information in question is not lost: it is hidden in the conformal factor e^'^. A procedure for extracting Weyl's harmonic function W from the conformal factor e'^'^ is described next. This procedure may be applied when the number of critical points is finite, and the functions involved are rational. The prescription begins with a recapitulation. In Weyl's harmonic coordinates the metric on the 3D space is The flatness boundary condition on the conformal factor is e^'^ Â— > 1 on the axis. In Weyl's isotropic coordinates the metric on the 3D space is: dsljj = e^^idp] + dzf) + p}B''dct>\ The flatness boundary condition on the conformal factor is e^^ B on the axis. Upon first examination it appears that the isotropic metric has two functions PAGE 47 39 in the 3D space-e^^ and 5-whereas the harmonic metric has only one function e^"^ . Upon closer examination and by reference to the Kerr metric in isotropic and harmonic coordinates, it is apparent that while e^^ and B are polynomials, e^'^ is a rational function-heing the ratio of two polynomials. Under a conformal coordinate transformation from isotropic coordinates Z = pi + izi to harmonic coordinates, yV Â— D + lE = ph -\izh, the conformal factor transforms as ~ A2 dZ In isotropic coordinates the determinant of the {t, (j)) part of the metric D = piB , opi azi On the axis pi = 0, and VD = {B,0). By the Cauchy-Riemann relations VE = (0, B). By definition, W = D + iE; and therefore along the axis dW dZ and the axis boundary condition is satisfied e27 = ^ = ^ A2 S2 3.3.1 Determination of Critical Points in the Harmonic Chart The critical points of Weyl's harmonic function are defined by ^ = 0. At such points, VD = VE = (0,0); therefore, at the critical points along the axis B = 0. Further, by the fiatness condition near the axis, = 5 = 0 also. PAGE 48 40 Therefore by the conformal transformation given above, at the critical points the conformal factor in Weyl's harmonic coordinates e2^ = A2 Q'2-1 If the critical points are not degenerate, i.e., if ^ ^ ^ 0, then by L'Hospital's rule dB ^ ^ The above considerations are valid regardless of whether of not , e^, and B are transcendental or rational functions. At the critical points of Weyl's harmonic function the numerator and denominator will both approach zero. If the conformal factor e^'^ is a transcendental function there could be an infinite number of critical points. In the case of rational solutions to stationary axisymmetric field equations, the conformal factor e^"^ in Weyl's harmonic coordinates is given by ratio of two polynomials. In this case the number of critical points is finite, being determined by the degree of the polynomials involved. This is not an additional restriction in terms of known solutions, as all known solutions are of this form. For rational solutions e^T is the ratio of polynomials in two variables {ph,Zh) Along the axis the numerator and denominator polynomials become identical, and ^ 1. Along the axis, let F(0, Zh) ^ G{0, Zh) ^ P{zh); then G{0,Zh) P{zh) At the critical points along the axis e^T Therefore, g27 ^ . 0 PAGE 49 41 The critical points of W are determined very simply by the roots of the polynomial P{zh). The critical points of Weyl's harmonic function for a rational solution are determined by the roots of the polynomial P{zh), where P{zh) is defined by F{0,Zh) G{0,Zh) -> P{zh), and e^T = ^j^^'^^j . A polynomial of degree n has n roots. Due to the equatorial symmetry of the problems under consideration, the roots will occur in pairs +a and Â—a, and n will be an even number. The critical points will occur in pairs one above and one below the equatorial plane. Next we verify the above prescription by applying it to the Kerr solution. The conformal factor e^"^ in Weyl's harmonic coordinates consists of quotients of quadratic polynomials for both the Schwarzschild and the Kerr solution. Hence they must have the same number of critical points which is determined by the degree of the polynomials. Since quadratic polynomials have two roots, both the Kerr and Schwarzschild solution have two critical points. The conformal factor for the Schwarzschild solution is (Appendix B): while the conformal factor for the Kerr solution is (Appendix C): where x and y are prolate spheroidal coordinates. In such coordinates, along the axis y = 1, and the desired polynomial P{x) = x^ 1. The critical points are determined by P{x) =x^-l = 0-i.e., x = Â±1. In prolate spheroidal coordinates these are also the points where the horizon intersects the axis-Â«.e.,the North and South poles. In cylindrical coordinates, Zh = kxy = kx along the axis. Therefore in cylindrical harmonic coordinates the two critical points at the poles are located PAGE 50 42 at Zh = Â±m for the Schwarzschild case, and at Zh = Â±\/m? Â— c? for the Kerr case. It should be noted that the denominator polynomial is the same for both cases, while the numerator has changed. The conformal factor for the static Schwarzschild and the rotating Kerr solutions are different. However, Weyl's harmonic function has a similar form for both cases. 3.3.2 Determination of Critical Points in the Isotropic Chart The key to unravelling the transformation from harmonic coordinates to isotropic coordinates is the fact that for rational solutions the numerator and denominator are both polynomials. The critical points are determined by the roots of the polynomials. The conformal factor will vanish at the critical points, which are the same points on the two surface S regardless of the which coordinates isotropic or harmonic that is being used. Harmonic and isotropic coordinates are defined in Section A. 3. The algorithm for extracting the boundary data of the Laplace equation will be first illustrated for a three term harmonic function W with two free parameters h\ and 63 and four critical points Physically this represents the superposition of a dipole field (61) and an octupole field(63) upon a uniform field of unit magnitude in two dimensions. For the Schwarzschild solution, 61 = m2/4, and 63 = 0. The critical points of this transformation are determined by the four roots of the quartic polynomial _&1__363 dZ Z2 ^4 PAGE 51 43 In general the roots of this equation will be complex. In all known cases, however, the critical points lie on the axis, and all the roots are pure imaginary quantities. By definition, W = D + lE and D = piB{pi, Zi). Along the axis, VD = (5, 0) and VE = {0,B). By the Cauchy-Riemann relations, where Q{zi) is a polynomial in Jr. For the Schwarzschild solution Q{zi) 2 1 At the critical points, VD = (0, 0) and VE = (0, 0). Hence the critical points along the axis are determined by the roots of the polynomial Q{zi): B{0,Zi) = Q{zi)^l-% + ^^0. For the Schwarzschild solution the roots are Zi = Â±m/2. Due to the equatorial symmetry of the problem, the roots always occur in pairs Â±P one above and one below the equatorial plane. For the above quartic, the roots will be Â±/?i and Â±/?2 where /3i and /?2 are given by P2 bl + yjb\ I2h \ bl yjbj 1263 2 In Weyl's harmonic coordinates, these points along the axis are mapped according to Zi zf Let pi and /?2 in isotropic coordinates map to ai and 0:2 in harmonic coordinates. Then 6]_ _ 63 PAGE 52 44 The solution in question has four critical points along the axis. In Weyl's harmonic coordinates these points are represented as: (0, ai), (0, 0:2), (0, Â—0(2), and (0, Â—Oil). In Weyl's isotropic coordinates these same points are represented as (0,A), (0,/32), (0,-^2), and {0,-pi). For the Schwarzschild solution (3 = m/2m isotropic coordinates, transforms to a; = m in harmonic coordinates. At the critical points, P{zh) 0' The critical points are determined by the roots of P{zh)In the case under consideration, the polynomial P{zh) will vanish at Â±ai and Â±012. Hence the limiting value of the conformal factor along the axis will be determined uniquely by the fourth degree polynomial P{zh) with roots at Â±ai and Â±0:2. That is: PizH) = {zl-al){zl-al) = 4(Â«1 + 0^2)4 + "1Â«2 = 4a2zl + ao. Substituting ai and 0:2 in the definitions of ao and 02, and then substituting Pi and P2 for ai and 012, the following relations are obtained: 16(62 + 463)2 "Â° = 2763 I661 46? Â«2 = ^7+ ^ 3 2763 Here, the coefficients 02 and cq are expressed in terms of 61 and 63. Therefore a relation has been established between the polynomial P{zh) in harmonic coordinates and the polynomial Q{zi) in isotropic coordinates. Given the polynomial PAGE 53 45 P(z/i) in harmonic coordinates, it is possible to construct the polynomial Q{zi) in isotropic coordinates, and vice versa. The transformation for six critical points has been computed explicitly using a Mathematica program. The algebraic procedure given above can be extended to eight critical points located symmetrically with respect to the equatorial plane. The conformal factor in harmonic coordinates will be a polynomial for rational solutions. At the critical points of the harmonic function, the polynomial P{zh) will vanish and the conformal factor e^''' ^ ^. The degree of the polynomial P{zh) determines the number of roots of the polynomial P{zh), and therefore the number of critical points located along the axis present in the solution. Once the number of critical points is known, the number of terms in the polynomial Q{zi) which determines the same critical points in isotropic coordinates are also known. For a solution with 2n critical points, the highest power of Zh in P{zh) will be 2n and the highest power of Zi in Q{zi) will be -2n. The number of free parameters in P{zh) will be the same as the number of free parameters bi in Q{zi), both being equal to n. Therefore a general form of Q{zi) can be assumed with the correct number of terms and free parameters bi. The location of the 2n critical points in isotropic coordinates which are the roots Â±pi of the polynomial Q{zi) can be expressed in terms of the n coefficients bi. Next these roots can also be transformed to harmonic coordinates, since this map is determined also in terms of the coefficients bi. Hence the n roots Â±ai of the polynomial P{zh) will be determined in terms of the n coefficients bi. Finally the n coefficients a, of the polynomial P{zh) could be expressed in terms of the n coefficients bi. The transformation is now completely determined, explicit relations between the coefficients of the polynomial P{zh) and Q{zh) having been PAGE 54 46 identified. Therefore, given a rational solution in Weyl's harmonic coordinates, it is possible to extract the harmonic function W in isotropic coordinates. This could be done algebraically up to eight critical points. For rational solutions it will always be possible to identify the finite number of critical points Â±Q!,numerically given the parameters 6j. Before proceeding to the next section, the key points are summarized. In a stationary axisymmetric system, the magnitude of the Killing bivector, -'^^[aVb]^Â°"n'' = D'^ = VX + jg a harmonic function on the two surface S orthogonal to the group orbits. In isotropic coordinates, D = piB, and VD = {B,0) = {Q{zi),0) along the axis. The critical points of the harmonic function, VD = (0, 0) are determined by the roots of the polynomial Q{zi) in isotropic coordinates. In harmonic coordinates, e^^ = 1 along the axis. At the critical points e^T ^. The critical points of W = D + iE = ph + iz^ are determined by the roots of the polynomial P{zh). If the coeflicients aj of P{zi) are known, then the coefficients bi of Q{zi) can be determined and vice versa, since the roots a,of P{zh) are related to the roots pi of Q{zh) by the conformal transformation Every stationary axisymmetric solution has a unique set of critical points. These critical points provide boundary data for the Laplace equation for D. Contrary to the claim of Zipoy, that boundary data for D cannot be determined in harmonic coordinates, it has been shown here that the data can be determined in any coordinate chart. PAGE 55 47 3.4 The ZippyVorhees and the Tomimatsu-Sato Solutions In previous sections Weyl's harmpnic functipn was discussed fpr the Schwarzschild and the Kerr solutions. In this section the generalization of the static Schwarzschild to a more general static solution-the ZippyVprhees(ZV) series of metrics-will be discussed. Unlike the Schwarzschild solution, these metrics are not spherically symmetric. They are only axisymmetric. In the rotating case the Kerr metric has been generalized to the Tomimatsu-Sato (TS) series of metrics. These metrics have a ring singularity on the equator, and do not represent black hole solutions. However, they could represent other situations where the singularity is covered by matter. The Tomimatsu-Sato solutions reduce to the ZipoyVorhees solutions in the static limit. In the limit o = m, they reduce to the extreme Kerr solution for all values of S. For odd values of S there are no directional singularities at the poles, the metric on the Killing horizon is spacelike, and the Killing horizon is also an event horizon (Sato, 1982). The static axisymmetric metric in the isothermal gauge in Weyl's harmonic coordinates {ph Â— D and Zh = E) is: ds' = -e'^dt' + e-'^ [ e'^dpl + dzl) + pld^^ ]. The field equations are : 7z = 2p{UpU,). If U and 7 are solutions to the field equation, then so are (Zipoy, 1969; Vorhees, 1970; Cuevado, 1990) U' ^5U 7' -4(^^7 PAGE 56 48 New solutions can be generated by the above transformation. This is a highly nonlinear solitonic transformation and it can only be performed in the Weyl's harmonic coordinates {ph,Zh)If this transformation is applied to the Schwarzschild solution in Weyl's harmonic chart in prolate spheroidal coordinates (Appendix B), the Zipoy-Vorhees metric is obtained: + (x2-l)(l-y2)rf<^2 where 2U fX-\\ ^X-W \x + \) \x-\-l) \x^ Â— y / \x^ Â— y ' with b an integer; (5 = 1 for the Schwarzschild solution. It should be noted that the apparent absence of spherical symmetry for the Schwarzschild solution in Weyl's harmonic coordinates is critical to the nontrivial nature of the Z-V transformation. If attention is paid to the conformal factor e"^"^ it is apparent that the degree of the polynomials has increased by 5^. The critical points are determined by P{x) = (x2 l)-^^ = 0-i.e., x = Â±1. The roots of the polynomial are all equal. The conformal factor is related to the derivative of Weyl's harmonic function. Therefore, in isotropic coordinates Weyl's harmonic function for the Zipoy-Vorhees metrics may be obtained by taking the derivative of the harmonic function for the Schwarzschild solution and raising it to the power S'^: PAGE 57 49 where = {rn? Â— a^)/5'^. In the rotating case, degree of the polynomials are the same as in the static case. As in the Kerr case, the numerator polynomial is different for the static and the rotating case. The conformal factor and the curvature of the two surfaces are different for the static ZV(5) and the corresponding rotating case TS((5). However, the corresponding harmonic function has a similar structure: For example for the TS(2) solution: _ fc^ 3fc^ k^ ~ Z2 ^ 8^4 ^ 16^ + ~ Z ~ 24^3 ~ 80^5 ~ 1792^7' which shows that the series for W is rapidly convergent. Due to the multiple roots the critical points of Weyl's harmonic function for the ZW{5) and TS((5) solutions are degenerate. Since all 6"^ derivatives vanish at the poles, the degree of the degeneracy is also 5"^. 3.5 Weyl's Harmonic Function and Global Structure In the previous section a procedure was given for determining boundary data for the Laplace equation for rational solutions. In particular the critical points of the harmonic function W could be determined. In this section the relation between these critical points and global structure of the solution will be explored. First the relations between the critical points, the magnitude of the Killing bivector, and the Killing horizon will be reviewed for the Kerr metric. Next, two complicated exact solutions will be discussed: the double Kerr solution, and the Manko et al, 94 solution. Wheras the Kerr solution has two critical points (Section 3.3), the last two solutions have four critical points. PAGE 58 50 In a stationary axisymmetric spacetime there are three independent coordinate invariant scalars that can be constructed from the Killing vectors: ^"'U = 9tt = -V, rjÂ°'r]a = 94,4, = X, and ^"r/a = gt^, = W. A fourth coordinate invariant quantity which is a combination of the previous three is the magnitude of the Killing bivector: -2^[aVb]^''v'' = = VX + W^. The scalar is also the negative of the determinant of the {t, (j)) part of the metric. The scalar is positive outside the Killing horizon, and negative inside the Killing horizon (Hawking and Ellis, p. 167). vanishes trivially on the axis as 77 vanishes on the axis. The region off the axis where the vanishes is the Killing horizon where ^[aVb] ^ 0, but its magnitude vanishes (Carter 1969). Therefore = VX + W^ contains important global information about the Killing horizon which is not carried by either of the three other coordinate invariant quantities, V, X, or W, alone. PROPOSITION A proposition concerning Killing horizons and critical points of Weyl's harmonic function will be established next. At large distances from the source the determinant of the {t,(f)) part of the metric, D, approaches the flat space cylindrical coordinate p. The coordinate vector fields associated with the flatspace cylindrical coordinates {p,z) do not vanish anywhere-z.e., Vp 7^ (0,0) and Vz ^ (0,0) at every point in the half plane. Isotropic coordinates have this property-i.e., Vpi and Vzi do not vanish anywhere. (However, the coordinate vector fields associated with the harmonic coordinates {ph,Zh) vanish at the critical points of Weyl's harmonic function.) Therefore isotropic coordinates are chosen for these computations. PAGE 59 51 Let the scalar D be decomposed in isotropic coordinates as follows: D = piB. The isotropic cylindrical coordinate pi vanishes only on the rotation axis where the spacelike Killing vector field vanishes: X = rfr]a = 0. The isotropic radial coordinate represents the true axis. The metric function B vanishes only on the Killing horizon where the magnitude of the Killing bivector vanishes, -2^[a'nh]i'''rf' ^ D"^ = VX + W'^ = but the Killing bivector itself ^[aVb] 7^ 0The metric function B represents the true horizon. Let the point P(0,zi) represent the intersection of a Killing horizon and the rotation axis. By definition, the function B = 0 on the horizon. Also by definition, D = piB; therefore its gradient is VD = {B,0) along the axis where Pi = 0. Therefore, VD = {B, 0) = (0, 0) at the point P, since B = 0 at the point P. As the gradient of D vanishes at P, therefore the point P is a critical point of Weyl's harmonic function W. The harmonic function W = D + iE; by the Cauchy-Riemann relations, VD = (0,0) VE = (0,0). Therefore it is proved that the intersections of the Killing Horizons and the axis are critical points of Weyl's harmonic function. Next the converse proposition will be established. Let the point (5(0, ^2) be a critical point of Weyl's harmonic function that lies along the axis. By the definition of a critical point VD = (0,0). If the critical point is degenerate, higher derivative of W could vanish as well. Along the axis, VD = (P,0). Therefore P = 0 at the point Q. As P = 0 only on the Killing horizon, the point Q is therefore the point of intersection of a Killing horizon and the axis. Hence, it is proved that the critical points of Weyl's harmonic function that lie on the axis represent intersections of Killing horizons and the axis. PAGE 60 52 Therefore it is proved that a point along the axis represents the intersection of a Killing horizon and the axis if and only if it is also a critical point of Weyl's harmonic function. QED. The practice of identifying global information using the gradient of the magnitude of the Killing bivector is a standard procedure for spacetimes with two Killing vector fields (Verdaguer,1993). In the Exact Solutions book it is stated that: "The gradient of W [D] provides an invariant characterization of the corresponding spacetimes and determines essentially the character and global properties of a solution." {Exact Solutions, p. 176). The gradient of the magnitude of the Killing bivector was used by Gowdy to distinguish different regions of an inhomogeneous cosmological solution. (Gowdy, 1971, 1974). The Gowdy solutions possess two spacelike Killing vector fields. They have no cross terms in the metric, and are analogous to static axisymmetric Weyl solutions. Kitchingham has generalized the Gowdy solutions to metrics with cross terms (1984,1986). An analogy between Kithchingham's inhomogeneous cosmological model and some newly generated stationary axisymmetric solutions will be presented in Section 3.5.3. The solutions of Manko et al. 1994, and Kitchingham were generated using the same mathematical technique-the solution to the Riemann-Hilbert problem. 3.5.1 The Kerr Solution The global structure of the Kerr metric is well known (Hawking and Ellis, Section 5.6). The determinant of the {t,(j)) part of the metric, = VX-\W"^, is positive everywhere outside the Killing horizon while on the horizon it is zero; inside the horizon, it is negative. The determinant vanishes on the axis PAGE 61 53 since the rotational Killing vector vanishes on the axis (Fig 3.1). Just slightly off the axis, is positive outside the horizon, while it is negative inside the horizon (Fig 3.1). In terms of the two Killing vectors of the stationary axisymmetric system-the timelike Killing vector field ^ and the spacelike Killing vector field 77: The 2-surface of transitivity (group orbit) spanned by the two Killing vector fields ^ and 7/ is spacelike or timelike respectively when the magnitude of the Killing bivector i[aVh] positive or negative. PAGE 62 D B PAGE 63 55 The gradient of D along the axis is given by VD = (0, B);VE = {B,0); where 5 is a scalar function of the isotropic coordinate Zi. Along the axis, the segment BN represents the part of the axis that extends above the North Pole, outside the horizon. Similarly the segment AM represents the part of the axis that is below the South Pole. The segment of the axis AB lies inside the horizon. The gradient function B is positive outside the horizon on the segments BN and AM; it is negative inside the horizon on all of AB (Fig. 3.1). The points A and B where the gradient scalar B vanishes are the critical points of D and E since VD Â— VE = (0, 0) at these two points. Therefore the behavior of the gradient scalar B along the axis makes it possible to determine where changes sign-z.e., the intersection of a Killing horizon and the axis. It is not possible to determine this information from directly since is always zero on the axis. Close to the axis, is positive outside the horizon, and negative inside (Fig. 3.1). The principle of identifying Killing horizons by the critical points of D and E will be applied to two new solutions with four critical points next. Before proceeding, the following observations are made. While the scalar function B is defined in isotropic coordinates, the critical points along the axis identified by B = 0 do not depend on any choice of coordinates or gauge. The gradient of a scalar is a well defined vector. The points at which both components of the gradient vector vanish are invariantly defined. Furthermore, these points are the critical points of the harmonic function D. From complex variable theory it is known further that these points are isolated and carry global information. Second, at the point S which is the origin of isotropic coordinates the function B diverges. The physical significance of the point S is not clear since it is not identical to the disk identified by the rsi, = 0 in Boyer-Lindquist PAGE 64 56 coordinates. For the purposes of this discussion, as long as B does not change sign at the point S, it will not be a critical point, and not affect the global structure. 3.5.2 The Double Kerr Solution Next we consider the double Kerr solution (Kramer and Neugebauer, 1980). This solution consists of the nonlinear superposition of two Kerr black holes along the axis. This solution and its properties are reviewed in great detail by Dietz and Hoenselaers(1985). For the present purposes, it will be sufficient to examine the conformal factor given in Weyl coordinates and extract the critical point information by the procedure given in the previous section. The conformal factor is a rational function G{ph,Zh)' where F{ph, Zh) and G{ph, Zh) are quartic functions of (p/j, Zh). Along the axis, 2^ ^ F(0,z,) ^ P{zh)_ G(0,zft) P{zh) The gradient polynomial P{zh) in harmonic coordinates is P{zh) = zl{n? + k^a^)zl + [{k^ -m^ + 0^)2 iaV] = 0 for two eqal mass m, equal angular momentum a objects separated by a distance k (Dietz and Hoenselaers, 1985, Eqn. 2.51c, Eqn 2.52.a, and 4.11). The roots of P{zh), which determine the critical points along the axis, are Â±ai and Â±012, where: ai = (/Â«+ + K_), 0:2 = {k+ Â«-), PAGE 65 57 K+ = [{rr? + e-a^ + d)l2]^l'^, K= [{m^ + -a^ d)/2]V2, d = m2 + a2)2-4a2mY/2. As this is a quartic polynomial, from the detailed example in Section 3.4, it is clear that Weyl's harmonic function will have three terms and two parameters 61, 63 when expressed in isotropic coordinates. Physically this represents the superposition of a dipole field(6i) and an octupole field(63) upon a uniform field of unit magnitude in two dimensions. As shown in Section 3.4, it is possible to express bi, and 63 in terms of 02 and oq. The critical points of Weyl's harmonic function are located along the axis as follows: two critical points A and B will determine the intersections of the horizon of one black hole and the axis; the other two critical points C and D will represent the intersection of the other black hole with the axis (Fig. 3.2). As before the magnitude of the Killing bi vector will vanish on the axis. Close to the axis is positive above the North pole of the top black hole near the segment DF and below the South pole of the lower black hole near the segment AE. In this case there is also the segment of the axis BC between the two black holes near which is positive (Fig. 3.2). In three dimensions the segments AE, BC, and DF represent the boundary of a single connected region in space which is the region outside the two horizons of the two black holes. The segments AB and CD represent regions interior to the two black holes inside the Killing horizons. The axis of the double Kerr spacetime has three distinct physical regions separated by two Killing horizons. PAGE 66 Horizon 1 Horizon 2 PAGE 67 59 Based upon the analogy with a single Kerr black hole, the behavior of the gradient of the Killing bivector, VD and V^' is shown schematically on Fig. 3.2. The function B will vanish at the four critical points A, B, C and D. Along the segments DF and AE the function B will be positive since J5 Â— >^ 1 at infinity. The function B will be negative inside the Killing horizon along the axis along the segments AB and CD. As the function B changes sign at the critical points B and C, it must be positive on the part of the axis that lies between the two black holes outside the Killing horizons on the segment BC. Thus the critical points of Weyl's harmonic function W permit the identification of intersections of Killing horizons and the axis. The nonzero component of the gradient of D determines whether or not D"^ is positive or negative. The magnitude of the Killing bivector is positive where B is positive at infinity, and alternates in sign with B near the axis. In the literature, several authors have gone to great lengths to identify the various regions of the double Kerr Spacetime. In Weyl coordinates, all three segments of the axis are defined by p/j = D = 0. The authors take great pains to identify the Killing horizons without the use of critical points, or the gradient of the magnitude of the Killing bivector. The horizons were identified numerically by showing that gu and e^'^ are negative along the AB and CD. Further, it was shown numerically that u was constant on AB and CD. It was found necessary to evaluate these quantities numerically for selected choice of parameters as the form of the metric is exceedingly complicated. The global structure of the horizons is easily identified in isotropic coordinates as the coefficients hi and 63 can be expressed in terms of the coefficients oiP{zh). The schematic figure (Fig. 3.2) is in isotropic coordinates. An exact PAGE 68 60 map could be produced for suitable choices of parameters such that the solution possesses two horizons. The behavior of the function B at the origin 0, and the two black hole centers 5i and S2 requires further examination. As before as long as B does not change sign, the critical points of the solution will be unaffected, and the global properties of the solution will be unchanged. 3.5.3 The Manko et a/., 1994 Solution In the previous example, the information about the horizon structure could be obtained with some difficulty even though there were three separate regions and two Killing horizons. The next example under consideration, the Manko et al, 1994 solution, has not been analyzed for horizon structure. Based on the two previous examples, a possible interpretation for this solution will be suggested. The conformal factor is a rational function ^ F{ph, Zh) where F{ph,Zh) and G{ph,Zh) are quartic functions of {ph,Zh) (Appendix D). Along the axis, 27 ^ F{0,zh) P{zh)_ G{Q,Zh) P{zh) The critical points are determined by the roots of the polynomial P{zh) which in this case is Pizh) = zi(m2 a2 g2 _ 2b)zl + [b" + c^) = 0. The roots oi P{zh) are Â±ai and Â±Q!2, where: Q;i = + PAGE 69 61 n_ = [m'-a'-q' + 2{-d-b)]'/\ As this is a quartic polynomial, it will have four critical points. By the procedure given above, the parameters hi and 63 for the solution in isotropic coordinates can be obtained. Next the four critical points and the two horizon curves can be obtained. The critical points in isotropic coordinates are given by the roots of the equation: Q[zi) = 1 ^ + Â— . zzOff the axis the Killing horizon is determined by the metric function B in isotropic coordinates since D = piB{pi,Zi) = 0 on the Killing horizon. As D is a harmonic function, the function B can be expressed in polar coordinates as: rf rf sm(^i) For each value of 9, there are two roots for the above polynomial. Hence D will vanish on two curves for this solution. Based on the location of the critical points along the axis, it is clear that one 5 = 0 curve will enclose the other B = 0 curve. There will be four real roots if bf I263 > 0. This condition could be translated into a condition on 02 and ao, using the relations between the bi and the Oj for solutions with four critical points given in Section 3.3. There are five free parameters determining the in this case: m, a, q, d, and b. It is clear that PAGE 70 62 with so many free parameters, keeping the mass fixed, and changing the other four parameters, 02 and oq could be given a wide range of values, so that the above condition on the bi may be satisfied. And the conjectured global structure with four critical points becomes possible. The authors of the solution describe the solution as that belonging to a single source, and the solution being symmetrical about the equator. Assuming that the four roots are real and located along the axis, the outer roots at Â±ai are located at C and D, and the inner roots at Â±0:2 are located at A and B. Based on this structure, it is conjectured that in the Manko, et al. 1994 solution one horizon lies inside the other (Fig. 3.3). In the double Kerr solution, there are two horizons also; however, one horizon does not lie inside another. In the double Kerr solution there are two regions where the magnitude of the Killing bivector is negative. In this case the magnitude of the Killing bivector is negative inside the outer Killing horizon, however it turns positive after crossing the inner Killing horizon. Therefore it is conjectured that the Manko et a/., 1994 solution has two disconnected regions where magnitude of the Killing bivector is positive. Possible justification for this conjecture is provided below. It is clear that the solution has four critical points. At these critical points the gradient of the Killing bivector must vanish. The function B must positive along DF and CE. At the critical points C and D it will change sign (Fig. 3.3). Hence D"^ will be negative near the axis along CA and BD. At the critical points A and B, the function B changes sign again. Consequently, will become positive near the axis (Fig. 3.3). PAGE 71 Figure 3.3 D^{0,z) and B{0,z) for the Manko et al. 1994 Solution PAGE 72 64 The authors of this solution looked at this solution in Weyl coordinates only. In Weyl Coordinates the nature of this solution is difficult to decipher. Along CA and BD, gtt and e'^'^, will be negative. However, they will be double valued along AB. Furthermore, up will have the same value along AC and BD, but will be double valued along AB. It might be erroneously believed that the spacetime consists five disconnected regions: one region outside CD; one region outside CA and another along BD; and finally two regions along AB since ujp is doublevalued. Fortunately no interpretation of this solution has been made. A problem with Weyl coordinates is therefore identified from this proposed configuration. If one Killing horizon encircles another, the horizon structure will be rather difficult to interpret since several distinct horizon curves will be represented along the axis overlapping one another. The connectivity of the regions in three dimensions is also obscured. The possible interpretation given above of Manko et ai, 1994 solution in terms of having several disconnected regions, where the magnitude of the Killing bivector is positive, is motivated by the the work of Kithchingham(1986). Kithchingham generalized the Gowdy Universes to universes with several disconnected components using the inverse scattering transform technique. It should be pointed out that Manko et al., generated the solution discussed above using a variant of the inverse scattering transform. Finally in conclusion it should be pointed out that Yamazaki has given closed form formulas in terms of determinant for the N-Kerr solutions. In this case the interpretation of the solutions is straight forward: each black hole has a horizon inside of which the magnitude of the Killing bivector is negative. Based on the interpretation of the Manko et al., 1994 solution it is conjectured that PAGE 73 65 solutions with A'^ overlapping horizons might also exist where the sign of would change, every time a Killing horizon is crossed. The structure of such solutions will be transparent in isotropic coordinates: there will be 2N critical points, and N Killing horizons curves enclosing each other where the function B goes to zero. PAGE 74 CHAPTER 4 STATIONARY AXISYMMETRIC FIELDS IN VACUUM In the previous chapters, certain global invariants of stationary a:xisymmetric solutions were introduced. In the present chapter the metric functions will be discussed in detail. In particular, expansions will be developed for all four metric functions. The relations between the expansions parameters and the relativistic multipole moments of the solution will be obtained. 4.1 The Conformal Invariance of the Field Equations in Vacuum The stationary axisymmetric spacetime metric contains four functions: V, W, D, and e'^^^. For the metric in the isothermal gauge in the Bardeen form: The invariant definitions of the various functions are: e^'' = V + W^/X ^2rP+2u ^VX + W^ = D'' UJ = W/X The field equations with sources are as follows (Seguin 1975): 66 PAGE 75 67 e-V'v2e^ + V2/i + ie2'/'-2''Va; Â• Vu; = -%'ae^^' \ P * L 1 2 r 7)2 e-^ V^e" + V V |e2^-2'' Vw Â• Vw = ^^-ne^^ (e + P) ^ + P f (e + P)t;^ 4' V2^e'^+'^ = 167re'^+''+2'^P All operators used in the above equations are defined with respect to the twodimensional flat space metric ds^ = dr^ + dz^. If sources are present, the conformal factor e2'^ is present in every single equation. However, in vacuum, if no sources are present, all the terms in the right hand side vanish. All the terms involving the conformal factor drop out of the field equations. The field equations without sources are as follows: e-Vv2eV' + + {e^^-'^'^Vu Â• Vo; = 0 e-'^V^e" + VvV^ie2V'-2'^Va; Â• Vo; = 0 e-'^V^e'' + V^/i f e2'/'-2''Va; Vo; = 0 Vfe^^-^Vo;] = 0 Vi^e^+'' = 0 The conformal factor e^^^ decouples from the other three variables. The three equations for V, W, and D do not involve e^f^. Therefore in vacuum, it is possible to solve the three equations for them first. Therefore, if the field equations are satisfied for the three variables, the fourth will be automatically be satisfied. The field equations are conformally invariant in vacuum, in the sense that they do not depend on the conformal factor. Further, the conformal factor is determined by the field equations, once the field equations are satisfied. PAGE 76 68 This property will be exploited in terms of the expansions of the metric functions in section 4.3. The expansions will be derived for the three independent variables-l^, W, and D-first; the expansions for e^^^ will be determined afterwards. The conformal invariance of the field equations is transparent in the isothermal gauge. It is not so in other gauges. For example, in any other gauge, let the metric of the two surface S be ds"^ = e^'^^dxl + e'^t'^dxl, and A = e''3-/^2 then the equation for D = takes the form: [A'f\e%U + [A-'/'{e^U, = 0. This is the general form of the two-dimensional Laplace equation in a diagonal metric. The metric on the two surface S is present in all the vacuum field equations in the other gauges (MTB, p. 274, Eqn. 5-10). An important practical consequence, is that in other gauges four functions have to solved for in the vacuum. In the isothermal gauge the functions of the {t, 0) part of the metric are solved first, and the metric on the two surface S is determined afterwards. In analytical studies in the harmonic coordinates in the isothermal gauge, the Ernst potential is determined first, while the metric on the two surface S is determined afterwards. In numerical studies in isotropic coordinates, the three metric functions of the (t,^) part of the metric-e", u, and B-are solved first, and then the metric on the two surface S, i.e., the conformal factor e^. 4.2 The Existence of Global Charts Stationary axisymmetric systems are two-dimensional problems: all the relevant functions are defined on the two surfaces orthogonal to the group orbits. As discussed in section 2.3 the two surface in question can be visualized PAGE 77 69 as a metal plate with known Gaussian curvature at each point. The metal plate extends from the axis to infinity where it becomes flat. The two surface S is diffeomorphic to the half plane. It is known from the work of Resetnjak, that if the Gaussian curvature of a surface is bounded, and if the region is diffeomorphic to the half plane, then it will be possible to cover it with a single isothermal chart (Morgan and Morgan, 1970). As discussed in section 2.2 the Gaussian curvature of the two surface S is determined by: KG = , ^^^^^ , , = -2e-2^(-Vi/ Â• Vi/ + |p2e-4''Va; Â• Va;). where (xi, X2) are coordinates parametrizing the two surface S. The first term corresponds to the magnitude of the acceleration of a ZAMO while the second term corresponds to the shear of the ZAMO world lines (Bardeen, 1973, p. 247) These are well behaved and bounded functions. They remain finite even on the black hole horizon. The first term reduces to a constant value, the surface gravity, kh (Bardeen, p. 252). Therefore the Gaussian curvature of the two surface is bounded (of order one). Hence by the Theorem of Resetnjak it will be possible to find a single isothermal chart to cover the entire two surface S. The choice of an isothermal form for the metric on the two surface corresponds to choosing the isothermal gauge conditions. As discussed in Appendix A, there is no gauge freedom-in terms of freely specifiable functions-in the isothermal gauge. The various coordinate systems that preserve the isothermal form of the metric are related to one another by well defined complex analytic coordinate transformations. Under such coordinate transformations, the conformal factor of the metric is altered as follows: A2 2 A2 = dZ2 PAGE 78 70 where the new coordinates Z2 Â— X2 + iy2 are complex analytic functions of the old coordinates Zi Â— xi + iyi. This coordinate transformation will be well defined everywhere except points where = 0. Different choices of coordinate systems on S correspond to different choices of the conformal factor. The conformal factor is related to the Gaussian curvature as follows {Exact Solutions, pg. 197): Therefore the Einstein field equation for the conformal factor is: This is a second order elliptic partial differential equation. Therefore, the complete specification of the conformal factor requires two boundary conditions. The first boundary condition is that the metric on the two surface S becomes identical to the two-dimensional flat space metric at infinity, i.e., e^^ = 1 at infinity. The second boundary condition is that the three dimensional space near the axis be flat: if a small circle is drawn surrounding the axis, its circumference must equal 27r times the radius. Two global charts have been used in the study of stationary axisymmetric systems: Weyl's harmonic chart and Weyl's isotropic chart. In both these charts a single global coordinate system covers the entire portion of the two surface S outside the Killing horizon. In Weyl's harmonic coordinates the metric on the 3D space is: PAGE 79 71 The flatness boundary condition on the conformal factor is Â— ) 1 on the axis. In Weyl's isotropic coordinates the metric on the 3D space is: The flatness boundary condition on the conformal factor is ^ Under a coordinate transformation from isotropic coordinates Z Â— pi + izi to harmonic coordinates, W = D + iE = ph + izh, the conformal factor transforms as: ,2e A2 dZ In isotropic coordinates the determinant of the {t, (p) part of the metric D satisfies D^PiB VD = {B + piÂ—, PiÂ—). opi dzi On the axis pi = 0, and VD = (5,0). By the Cauchy-Riemann relations VE = (0, B). By definition, yv = D + iE; therefore along the axis A2 = dW dZ B' and the axis boundary condition is satisfied ''=A^ = B^-''Once the flatness boundary condition along the axis is imposed, a single and conformal factor is defined globally. The conformal factor satisfies a second order PDE: once two boundary conditions are imposed, it becomes unique. Once the field equations are solved, subject to the above conditions, a single, unique conformal factor is determined globally. Different choices of coordinates on S PAGE 80 72 correspond to different conformal factors. Therefore a unique global conformal factor selects a unique, global system of coordinates on the two surface S. In isotropic coordinates the conformal factor e'^^ is uniquely determined for a particular solution. In harmonic coordinates the conformal factor e^'^ is uniquely determined for a particular solution; and they are related to each other by the conformal coordinate transformation defined above. At the critical points of Weyl's Harmonic function the gradient of D and E vanish. Along the axis the critical points are determined by B = 0, since along the axis VD = (B,0), = (0,5). However, their gradients do not vanish unless the critical point is degenerate. At the critical points both B and = B vanish. It should be noted that the equation Â— 0, does not imply that the Gaussian curvature kq = 0. In Weyl's harmonic coordinates, e^'^ 0/0. However, if the critical point is not degenerate and higher derivatives are not zero as well, e^^ ^ 1 by L'Hospital's Rule. Hence the spacetime is well behaved near the critical points of Weyl's harmonic function if the critical points are not degenerate. The flatness condition near the axis is a highly non-trivial condition: it connects the (p, z) part of the metric to the {t, (j)) part of the metric. It connects the conformal factor of (p, z) part of the metric to the determinant of the {t, (j)) part of the metric. It connects the Gaussian curvature of the two surface S to the magnitude of the Killing bivector. The determinant of the (f, (j)) part of the metric sets boundary conditions on the conformal factor on the (p, z) part of the metric as follows: along the axis, VD = (B,0); VE = (0,S). Since D and E, are harmonic conjugates, their partial derivatives are related by the CauchyRiemann relations. Along the axis one component of the gradient is zero, the PAGE 81 73 only non-vanishing function is B. Therefore, the flatness condition near the axis, which is e^{0,Zi)=B{0,Zi) = Q{zi) in isotropic coordinates, and in harmonic coordinates, couples the Gaussian curvature of the two surface S to the non-zero component of the gradient of Weyl's harmonic function along the axis. The uniqueness of the gradient function B is apparent in the case of rational functions. In this case the number of critical points of Weyl's harmonic function is finite, and along the axis the function B is a polynomial in l/zii B{0,zi) = Qizi) = !-+ _ + ... + i i A rational solution to the Einstein field equations will have a specific number of critical points. The number of critical points will determine the number of terms in the polynomial Q{zi). The precise location of the critical points in isotropic coordinates will be determined by the specific values of the coefficients bi. The critical points will be determined by the roots of the polynomial Q{zi) in isotropic coordinates. As discussed in Section 3.4, the location of the critical points in Weyl's harmonic coordinates are determined by the roots of the polynomial P{zh) with coefficients a^. The coordinate transformation from isotropic to harmonic coordinates is uniquely determined by the coefficients 6j. Hence the coeflacients a,and the coefficients bi are related algebraically. As harmonic coordinates are invariantly defined, ph = D and Zh = E, the coefficients of the PAGE 82 74 polynomial P{zh), ai, are invariantly defined. Therefore, since bi the coefficients of the polynomial Q{zi) are related algebraically to the coefficients Oj, they are also unique. Invariantly, the critical points where VD = (0, 0) is determined by the roots of the polynomial P{zh) in harmonic coordinates and by the roots of the polynomial polynomial Q{zi) in isotropic coordinates. Both these polynomials are unique. Hence they define unique global charts: Weyl's harmonic chart and Weyl's isotropic chart, respectively. In the case of an infinite number of critical points, the above series for B will not truncate, and Q{zi) will not be a polynomial with finite number of roots. In this case the uniqueness of the coefficients fej can be established by considering the complex analytic properties of Weyl's harmonic function. In isotropic coordinates Weyl's harmonic function W = D + iE has a Laurent series expansion about infinity: n=l ^ This series converges everywhere outside a certain region containing the origin and outside a small circle excising the simple pole at infinity. The Laurent series coefficients hi are unique . Further by considering the gradient of Weyl's harmonic function: n=l it is clear that B(0, 2,) = Q{zi) has a Taylor Series expansion about infinity. o(.).i(^) = i+D-i)"^ n=l which is well defined and unique. Q{zi) is a real function of the one variable. The derivative < 1 exterior to the unit circle. If there are no critical PAGE 83 points in the region under consideration, 75 > 0. (As discussed in section dZ 3.5 the critical points of Weyl's harmonic function corresponds to intersections of Killing horizons with the axis.) Therefore B is very smooth and well defined function in the regions where it is defined, e.g., in the exterior of a star. It has been shown that on the axis of symmetry, pi = 0, VZ) = {B, 0); VE = (0,5). It has been assumed that V is the covariant derivative associated with the metric on the two surface S. However, the gradient function B could also be defined in terms of Lie derivatives which do not involve the metric. Let A and B define two smooth vector fields: A = -M-, and B = then the Lie opi ' dzi ' derivative of Weyl's harmonic scalars along the axis defined by pi = 0 are: Ca{D) = {B,0) Cb{E) = {0,B). For a specific solution to the stationary axisymmetric field equations, the gradient of Weyl's harmonic function W = D + iE is uniquely determined by its complex analytic properties. This defines the scalar function B uniquely along the axis, without any reference to the metric on the two surface S. The Lie derivative does not involve the metric or assume the existence of an affine connection on the two surface S. The flatness condition near the axis may now be used to set the conformal factor = B along the axis. Finally the Gaussian curvature at each point on S will define uniquely everywhere for that particular solution. As mentioned, diff'erent choices for the conformal factor correspond to different sets of coordinates on the two surface S. Therefore, every stationary axisymmetric solution has a unique global isotropic chart in the vacuum region defined by the harmonic properties of D; where D is invariantly defined by -2([aVb]^''v^ ^D^ = VX + W^ PAGE 84 76 The above prescription does not work for Weyl's harmonic coordinates. In Weyl's harmonic coordinates ph corresponds to both the axis and the Killing horizon on which ph = D = 0. In isotropic coordinates the equation pi Â— 0 corresponds to the the true axis determined by the flat space cylindrical coordinate p Â— 0. Furthermore the vector fields C = ^PÂ—, and D = jPÂ— are not smooth at the points of intersection of the horizon and the axis. The vector fields A and B are smooth everywhere like the vector fields corresponding to r = ^ and z = ^ where p and z correspond to ordinary flat space cylindrical coordinates at large distances. The diff'erence between an ordinary partial derivative and the Lie derivative is explained by Hawking and Ellis: "the ordinary partial derivative is a directional derivative depending only on a direction at the point in question . . . the Lie derivative >CxT|p depends not only on the direction of the vector field X at the point p, but also on the direction of X at neighboring points." (Hawking and Ellis, p. 30) In this regard the covariant derivative is similar to the ordinary partial derivative. It is important to note that the vector fields associated with Weyl coordinates will not be smooth at the intersection of the Killing horizons and the axis. 4.3 Practical Aspects of Coordinate Charts A second important issue is the physical interpretation of solutions given in Weyl's harmonic coordinates. Solutions to Laplace equation in 3D cylindrical polar coordinates generate the static Weyl solutions. However, the physical interpretation of these solutions remain obscure. Weyl introduced the use of the determinant of the (t, (j)) part of the metric {D) and its conjugate function(Â£') as coordinates for the first time in 1917. He noted that in such coordinates the spherically symmetric Schwarzschild solution corresponds to a line PAGE 85 77 segment of length 2m, with constant linear mass density, along the axis of symmetry. (Weyl, 1917, p. 140). Weyl also introduced isotropic coordinates for the Schwarzschild solution in the same paper. This raises a second important question: Given a solution in Weyl's harmonic coordinates, is it possible to transform it to some generalization of "isotropic" coordinates, where the physical interpretation might be more transparent? For a stationary axisymmetric solution the Killing bivector vanishes trivially on the axis since the rotational Killing vector vanishes on the axis. However, D also vanishes on the horizon, as the horizon is defined by the norm of the Killing bivector being zero (Carter, 1969). In Weyl's harmonic coordinates the horizon which is topologically a sphere is mapped to a line segment along the axis. This difficult does not arise in Weyl's isotropic coordinates where the horizon is represented by a sphere of coordinate radius m/2. In this treatise it has been shown that it is possible to generalize Weyl's isotropic coordinates to solutions which are not spherically symmetric by the procedure given in section 3.3. Given any stationary axisymmetric solution in Weyl's harmonic coordinates, it is possible to uniquely determine the corresponding solution in Weyl's isotropic coordinates. 4.4 Asvmptotic Expansions for the Metric Functions In this section asymptotic expansions will be given for all four metric functions of a stationary axisymmetric vacuum solution. For the reasons discussed in the previous section, isotropic coordinates will be employed. The metric functions will be in the Bardeen form. The results presented here are a generalization of the expansions of Butterworth and Ipser(1976) where expansions were given up to the fourth order Legendre Polynomial. Based on Professor PAGE 86 78 Ipser's notes from 1975, a symbolic algebra program was written in Mathematica to carry out the expansions to arbitrary order. All four metric functions were obtained to the fourteenth order Legendre polynomial. The results were validated by checking the Kerr metric to this order. 4.4.1 The Complete Nonlinear Vacuum Field Equations Let N = e^'^ and r sinO B = e^'^^ . Then the vacuum field equations given in Section 4.1 may be written as follows: V'^N N-^ VN-VN + B-^ VB-VN N'^ sin^ 9 B'^Vu-Vu = {) V(r^ sin^ 9 Vu;) + sin^ 9 Va; ' VB ^ VN 3^-2 B N 0 VoD C + VN-VN-N-^ sin^ ^ Va; Â• Va; = 0 ^^ 4 4 V2Â£,rsin^B = 0. The gradient operator used above is the flat space cylindrical coordinate operator defined in three dimensions.. There is considerable confusion in the literature regarding the above equations. Some authors have failed to note that the Laplacian operator in the equations for C and B are two-dimensional and not three-dimensional. 4.4.2 The Linearized Vacuum Field Equations The equation for B is linear. In order to derive expansions for the four metric functions linearized versions of the above field equations will be generated by dropping the nonlinear terms: V^iV = 0 V{r^sm'^9Vuj) = 0 C = 0 V2Â£) rsin^ B = 0. i .1 PAGE 87 79 The above equations will be used next to generate linearized "zeroth" order approximations to the metric functions. The eigenfunction expansions for the functions A'^, C) s-^id B can be written down by inspection since they are solutions to Laplace's equation in three and two dimensions: r Bi sin(/^) rsin^B(r,^) = ^ r ^ ^ ^ r'+^ sm^ where P/(cos^) is the Legendre polynomial of order /. There is no logarithmic term in the solutions to the two dimensional Laplace equation. At large r, C Â— ) 0, and 5 Â— > 1. The equation for uj may be rewritten in the following form: ^^(rsin^a;) 2a(rsin^a;) 1 d'^irsinOu) Id, ^ 2 + Â— + -0 ^.9 + cot^ rsinM = 0. The above equation is in the standard form for the equation of the magnetic vector potential for a current loop in the xy plane: 0. -V X V X (pA^ = -(j) The solution for the above equation is: where P/(cos^) is the associate Legendred polynomial with m = 1. Hence, the solution for the metric function u is: = E .1+2 PAGE 88 80 since dfjL where /i = cos^. The boundary condition on u> has been given by Papapetrou(1948): 2J where J is the angular momentum. For the associated Legendre polynomial P^^(Â— //) = (Â— l)'"'"^P;^(/i). The coefficitents Wi must vanish for even values of / to preserve symmetry about the equatorial plane. 4.4.3 Expansions for the Metric Functions When the angular eigenfunctions are substituted into the three nonlinear field equations, the following equations are obtained: d^N 2dN 1(1 + 1) 1 1 -N-'^ sin^ 6 B^ Vuj -Vlo = 0 dr^ r dr -^ d'^u Aduj (/-!)(/ + 2) T 1 Â— 2 + -Â— '-^ '-UJ + W-^Vuj VP 2iV-^ Va; Â• ViV = 0 dr r dr r d% IdC r 1 , 9 0 0 o dr r dr A A Due to the conformal invariance of the field equations, as discussed in Section 4.1, e^^ decouples from the equations for e^" and uj. Therefore it is possible to derive expansions for e^" and uj first in terms of the three sets of multipole moments (P/, A'^;, Wi). Afterwards the expansion for e^^ can be derived in terms of the same three sets of parameters. There are four coordinate invariant quantities in a stationary axisymmetric system, V, X, and = VX + W"^, out of which three are algebraically independent. Therefore it is not surprising PAGE 89 81 that three sets of parameters are sufficient to describe the exterior vacuum region of a star. /p ep'2, ^3 y4 ^6 rpl ^3 f"^ 7*^ 7**^ Â„ r N4 N} Ni +P2 +^6 + UJ = + Wi wl w? w? wf wP wP Â— H H H H H yÂ»4 ^5 ^6 ^9 W} yÂ»8 y 9 ^7 ^8 ^9 + W7 PAGE 90 82 c = + COS 2d First the above expansions for N and u are substituted in the field equations for A'^ and u). All the angular functions are expressed in terms of the Legendre polynomials. Next all the terms in the field equations are expanded out, and a two-dimensional eigenfunction expansion is obtained in terms of the Legendre polynomials and powers of ^ . In order to expand the above equations it is necessary to expand all the functions in terms of the basis polynomials, which in this case are the Legendre polynomials. It is also necessary to expand the derivative of a Legendre polynomial as a sum of Legendre polynomials. Gradshteyn and Ryzhik(1980) give the following identity for expanding the derivative as the sum ( p. 1026, Eqn. 8.915.2 ): The summation is cutoff at the first term with a negative subscript. To expand the product of two Legendre polynomials as the sum of Legendre polynomials, the following identity is used ( p. 1026, Eqn. 8.915.5 ): P^ = X;(2n-4A;-l)PÂ„_2fc_i. fc=0 P m Pn Â— ^ ^ 2m + 2n 2A; + 1 2m + 2n 4A; + 1 a. "mÂ—k^k(^nÂ—k ^m+nÂ—k Tn+nÂ—2ki where o-k = (2A:1)!! A;! PAGE 91 83 and n>m. Next the terms in the field equations are grouped together and set equal to zero: The two field equations reduce to a number of new non-linear algebraic equations. However, each term is linear in A^^ and W^^ and these can be solved for in terms of the leading terms Nm and WmOnce these algebraic relations are substituted back into the expansions for A'^ and u, the metric functions e^^ and u can be expressed entirely in terms of {Bi^Ni^Wi). Next the same procedure is applied to the field equation for e^^ , and it too can be expressed in terms of three sets of parameters. The axis flatness condition Â— B along the axis is imposed. The final form of the metric functions are as follows: PAGE 92 84 The metric function u has the following form: _M ( Bo -3Bl + B2-36J^ 4J^-7Bo + l2AP) r 15 B^ 10 Bo 52 + 3 54 + 580 Bo 320 12 52 12 P Ar2 ^ ^ 1 ' 105 r6 M P2(/i) /'iV2 2 J2 32 52 + 144J2-9 5oiV2 7^56 50 84 + 45 V^s) ;3 1 2 Mr^ 42r2 21 Mr-3 + _i_ (-120 Bo 52 + 24 54 848 Bo + 448 + 18 5^ iV2 27 B2 N2 126 r" + 168 J2 iV2 540 J2 + O(-) MPi(^ f 1^ _ 36J^_W3_ 768 54 192 52 N2 1512 TV; 175 5o iV4 + 5400 J^Ws 1 7 Mr ^ 770r2 3 PAGE 93 85 The metric function uj has the following form: _2J ( 3M 3(3go-8M^) M (lOOBp 80M^ SiVz) 90^g + 9^2 + 84 J2 -440goM2 + 160 JVf* + 24 ATa 35 + Â— ^ (-1512 144 52 1824 + 2240 Bq 448 M"* + 72 Bq A^q 140 ^ + ;rr^ (-3150 756 Bq B2 + 27 B4 8316 Bq + 21448 B^ + 1944 B2 945 r" 168 M^iVj + 45 A^zW^s) + 34704 7^ 14560 Bq M'* + 1792 1896 Bq N2 + 1344 N2 + 252 iV| 1080 BzW^s1620 J^W^s810 M^ATjW^s) +0(^) ^ JW^3(iP3(M) A M (9iV2 -25W3) -32 B2 32 42 25 Bp W3 + 50 W3 + ^^-^ (2432 B2 + 3104 J2 621 Bq N2 + 1428 A^2 70 + 1645 Bq W3 315Vy3H 980 1^3 + 105iV2M^3) -^ 3465W^(^Â°^^^^Â°^+ 25110 Bo N2 17472 M" N2 + 567 A/j^ + 2660 + 7875 B^ ^^3 5355 B2 1^3 1 ^\ + 13860 J2 W3 28910 Bp + 7840 M'' ^^3 3045 N2 H^s) + O(-) (20664 Bp B2 + 1512 B4 + 22176 Bp 49408 B2 84016 3465 vTs JWsdPsifi) ( M (35 7V4 + 33 N2 W3 147 W5) 5760 B2 1^3 18720 7^ W3 5790 TVj W3 6615 Bp H^s + 11760 W's) + 0( J)^ j + 1 + 0(1) r PAGE 94 86 The metric function ( has the following form: -M2 54j2 + 8BoM2 + 3M2iV2 2J^M 4J.2 64 2 ~ 12 14 7 28 512 256 Cos{26) ( ^ 3 (-6J2 + M2yV2) 2 M H ^ Â— I Bq H Â— : Â— I 77Â—2 1 -5 Â— 4 16r 189M2 7V2 llhW^Ni 1 189 M'' iv'^ + Â—^-(756 Bo + 8052 1152 54Bo iVj Â— ^ + 448 r* 16 Cos(46i) (-El Â„ 9J2 BoM2 15 M^iVz -9 7V| + 70 1080 + "7^ [T-^^Â°^'^^^'~^^^~T2 32 + ^024~ lOSM^AT^ 105J2W^3 r.A^\ H + O(-) 512 128 PAGE 95 87 All four metric functions have been expanded in terms of three sets of parameters {Bi, Ni, Wi). Only terms of the order Pq have been given above. The Mathematica program for expanding the metric functions may be obtained from the author. The expansions of fourteenth order generate over 10 megabytes of output, therefore they have not been printed out. All metric functions have been verified to the fourteenth order for the Kerr metric. 4.5 Equivalence to Thorne Multipole Moments In the previous section, it was shown that all four metric functions can be expanded in terms of the three sets of parameters {Bi, Ni, Wi). The question arises what are the precise physical meanings of these parameters. Considering the pure vacuum region first, it will be shown that the parameters are multipole moments: The parameters Ni are mass multipole moments in the Newtonian limit; the parameters Wi are current (angular momentum) multipole moments in the Newtonian limit (Thorne, 1980, Section V.C). The parameters Bi are exact two dimensional pressure multipole moments since the equation for B is linear (Butterworth and Ipser, 1976, Eqn. 15). Corresponding to the Laplace's Equation in vacuum, there is also a Poission Equation for D inside sources: ^IdD Â— 16Kp^/^, where p is the fluid pressure, and g is the volume element of the four dimensional metric (Trumper 1967, Eqn. 5; Bardeen, 1971, Eqn 11.14; Chandrasekhar and Friedman, 1972, Eqn. 74 and 75; Butterworth and Ipser, 1976, Eqn. 4c). Hence the Bi are termed 2D pressure multipole moments. It will be shown the parameters Ni and the parameters Wi correspond to the multipole moments defined by Thorne (1980) for stationary systems. The parameters Bi arise due to the presence of the additional symmetry, namely axisymmetry, that is present. PAGE 96 88 4.5.1 Thome's Definition of Vector Spherical Harmonics In his 1980 review paper "Multipole Expansions of Gravitational Radiation" , Professor Thorne defines the following pure spin vector spherical harmonics (Thorne(1980), Eqn 2.18a-c): Y^''"* = [/(/ + l)]-i/2 r Vr'"* = -n X Y^-'"* Y^'^"* = + l)]-^/2 L y''" = n X Y^-''" The vector harmonics Y-^'''" and Y^''"* are purely transverse; Y-^-'"* is purely radial. The Y^-'"* and Y^-'"* have "electric-type parity" tt = (-1)'; Y^''"Â» has "magnetic-type parity" tt = (-1)'+^ For the stationary axisymmetric case, m = 0. The electric vector spherical harmonic reduces to: Y^-'O = [/(/ -H 1)]-V2 r VF'Â° = [/(/ + 1)]-V2 r VPi(cos^) = [/(/-hl)]-^/2 ^ dPA cos 9) The magnetic vector speherical harmonic is: 4.5.2 Thome's Expansion for the Metric Functions Professor Thorne has given expansions for stationary sources from which the multipole moments may easily be read off (Thorne, 1980, Section XI). His expansions for the metric functions are given in an "Asymptotically Cartesian and Mass Centered" (ACMC) coordinate system. In an ACMC coordinate system the metric coeflJicients gu and gt^ have a simple structure in terms PAGE 97 89 of and the spherical harmonics (Thome, Eqn 11.1). For the axisymmetric case the expansions in terms of the multipole moments reduce to the following: = _1 + ^ _ ^ + /^Â°PK/^) + 1 We + + j Opole) 2Jsin2(^) , ^5'0^5zn2(^) + (/-l)po/e + ... + (0po/e) The above expressions are obtained by substituting axisymmetric angular eigenfunction P;(//) for the spherical harmonic r'Â°, and the angular eiqenfunction ^^^^ for the 0 component of the vector spherical harmonic in Thome's Equation 11.4. (Note that there is a factor of rsin(6') difference with Thorne due to the normalized basis. Thorne, Eqn 11.22) The normalization constants have been omitted in the above equations. They will be dealt with in the next section. Professor Thome's procedure for extracting the multipole moments is to read off the mass-/ moment from the part of gtu and the current /-pole moment from the \ part of g^^. By comparing the above expansions for ga and gt^ with the expansions for V and w, it is clear that the expansions given by Butterworth and Ipser (1976) for the metric functions are in the same form as given by Thorne (1980). Hence the isotropic coordinates introduced by Bardeen (1970) are ACMC. Further, the coefficients A^, and Wi introduced by Butterworth and Ipser (1976) are equivalent to the Thorne multipole moments. (The normalization will be worked out exactly by comparing the multipole moments for the Kerr metric.) Gursel(1983) has shown that the Thorne multipole moments are equivalent to the invariant Geroch-Hansen multipole moments. Hence, the A/ and the Wi are also equivalent to the invariant multipole moments. PAGE 98 90 4.5.3 The Multipole Moments of the Kerr Solution Various authors have calculated the multipole moments of the Kerr metric: Newman and Janis (1965), Hernandez(1967), Hansen(1974), and Thorne(1980). They have all obtained the same results up to normalization. The even mass multipole moments are ma', while the odd ones vanish. The odd current multipole moments are ma' while the even ones vanish. However, these authors have not specified the normalization. Thorne has given the exact normalization for the first three multipole moments in terms of the trace free symmetric multipole moments (Pirani, 1964, Section 2.3). The "z" components of the trace free symmetric part of the multipole moments are: T 2 2 J^zz = --ma 3 4 15' (Thorne, Eqn 11.28). It has been shown by Hansen(1974), that in the axisymmetric case, the multipole moments are completely determined by the "z" components of the trace free symmetric part. ( See also Landau and Lifshitz, 1975, Eqn. 41.7, and 41.8) During the course of the present study the following general forms for the "z" components of symmetric trace free multipole moments were obtained. In particular the general forms for the normalization constants for the Kerr metric multipole moments were obtained: (2A;)!! M[k] J[k] PAGE 99 91 Hence the first four nonvanishing mass multipole moments of the Kerr metric are: 2 2 1.. '^zzzz = = 00 z 16 1 ^zzzzzz = = 128 o 1 T-zzzzzzzz = +^7^"^a = 7;^sThe factor of ^ arises in the above definitions since A'^ e^", and the mass multipole moments are defined with respect to v which corresponds to the gravitational potential in the Newtonian limit. The first four nonvanishing current (angular momentum) multipole moments of the Kerr metric are: Szzz = -t^ma^ = W3 10 16 . Ozzzzz = = OLD 32 7 ma W7 'zzzzzzzc 256 9 Â•^"^ = +109395"^^ =^^- PAGE 100 92 The above relations were verified by examining the metric functions for the Kerr metric in isotropic coordinates: .2 N 2 r UB = -W/X = ^Â— ^ Jg2^ ^ g2/x+2j/ r sin 6 Ar"^ where ]fp= m'c? . The above expressions for the metric functions in isotropic coordinates are generaUzations of the expressions given by Bardeen and Wagoner(1971) for the extreme Kerr solution (VIII.2-5). The above metric functions were expanded at the poles and at the equator for the specific value of angular momentum parameter a = M/2, and the following expressions were obtained: e2^(^0) = lÂ— + Â— Â— 211M5_ 325M6 5187M^ 1 ^2u(^ ![n 1 _ ^ , ^ _ 9M^ _ 633M5 1243M6 32325M7 1 1Â° 2^ r r2 8r3 +4r4 128r5+ 128r6 ~ 2048r7 ^^^-^ 10 a;(r 0) = Â— Â— + ^^^^ _ 21^^ , 351M6 329M7 _ 18571M8 1 12 ^3 r4 16r5 4r6 ^ 128r7 + 128r8 2048r9 ^^7^ a;fr -) = Â— Ml! . 83M4 _ 7M5 1169M6 _ 1739M7 46419M8 l 12 2i( HA 1 3M2 9M4 ^ 1 12 e2^rr -) = 1 ^ + 265M4 _ ?ill! , IQM^ _ igM^ 795M8 369M9 1 12 ^'2^ 8r2 256r4 + 8r6 "s^T" + + O(-) The tracefree multipole moments given above were substituted in the expansions for the metric functions given in Section 4.4.3. The results were PAGE 101 93 verified to 14th order in using a Mathematica program. The following conclusions were may be drawn from the above checks: (1) The generalizations of the expansions for the stationary axisymmetric metric functions given by Butterworth and Ipser(1976) are checked. (2) The generalizations of the expressions for the Kerr metric functions in isotropic coordinates for arbitrary values of the angular momentum are checked. (3) Generalizations of the trace free symmetric multipole moments of the Kerr metric given by Thome are checked. If there was an error in any one of the three generalizations mentioned above, the expansions obtained by substituting the trace free multipole moments and the expansions obtained by expanding the metric functions directly would not likely be in agreement. Based on the results obtained in this section a new procedure can be given for extracting the multipole moments of stationary axisymmetric systems. Unlike Weyl's harmonic coordinates, Weyl's isotropic coordinates are ACMC as defined by Thome (1980, Section XI). Therefore once a metric is expressed in an isotropic coordinate system, the multipole moments can be read of directly. To transform from Weyl's harmonic coordinates to Weyl's isotropic coordinates, the algorithm given in Section 3.3.2 should be used. PAGE 102 CHAPTER 5 CONCLUSION The problem of the boundary data for Laplace equation is solved. One of the applications of these boundary data is the comparison of exact analytic solutions with numerical simulations. The first step in this process consists of ascertaining the gauge and coordinate conditions that is used in theoretical studies and numerical simulations. This has been presented in Appendix A. The material presented there is original in that all the presentation is coordinate invariant. All the important quantities have been identified in the Bardeen form used by the numerical simulations and the Papapetrou form used by the theorists. Hence the translation from one form to another is simple once the coordinate invariant definitions are used. As a by product of examining things in a coordinate invariant manner, certain algebraic identities were shown to be a consequence of the physics. In particular the identities of Chandrasekhar and Tomimatsu-Sato, are simply algebraic expressions of the invariant relation: = VX + W^. The form of the Kerr metric in Bardeen coordinates given in Appendix C, will be of use in the numerical simulations where Bardeen himself has only given the form for the extreme Kerr solution. The metal plate analogy of Chapter two is mathematically accurate: the Gaussian curvature of the plate is invariant therefore the shape of the plate is uniquely determined by field equations. The temperature of the plate is D, which is also invariantly defined therefore it is also unique. The value of this analogy 94 PAGE 103 95 lies in the fact two metric functions, e'^^ and B obtain physical representations which make their properties clear. For example, the flatness condition on the axis implies that there exists a relation between the temperature of the metal plate along the axis, and the Gaussian curvature along the axis. The next phase in this inquiry will be to to build the metal plate numerically for the Kerr solution. The most interesting result in Chapter three is the critical points of the Kerr metric. It is stated in many places in the literature that stationary axisymmetric systems have a harmonic function associated with them. Where are the critical points of this harmonic function? This question was answered in Section 3.2 by extending Weyl's work from 1917. The next result of Chapter Three is somewhat technical in nature; however, it is very important for practical applications. Obtaining the information about the harmonic function in Weyl's harmonic coordinates is a very important result. It is essential to be able to extract the bi parameters for rational solutions. The 6,parameters are known for the stars. Therefore numerical simulations and theoretical predictions can be compared. The next phase in this inquiry would be to make a list of all known exact solutions that purport to represent the exterior of a rotating star, determine their bi parameters, and compare them to the sets of parameters emanating from actual rotating stars. In this fashion, certain exact solutions may be found to be unsuitable to represent the exterior of a rotating star. As mentioned in the introduction, very little is known about the horizon and singularity structure of the star solutions. It is known that some of them possess ergotoroids. The parameters bi determine the norm of the Killing bivector, which in turn determines the Killing horizon. Therefore knowledge PAGE 104 96 of the pressure multipole moments yields information about the causal structure of the exterior solution. The suggested interpretation of the Manko et al. 1994 solution is new. The link between Kitchingham's work on cosmology and Manko's work on stationary axisymmetric systems in terms of their origin-the Riemann-Hilbert problem-and their interpretation-in terms of several disconnected regions-is identified in Section 3.5. In terms of the results presented in Chapter Four, the significant point is the simplicity of isotropic coordinates. The vector fields associated with isotropic coordinates are smooth. Isotropic coordinates are ACMC ( Asymptotically Cartesian Mass Centered ) as defined by Professor Thorne. Hence, the multipole moments of a solution may be read off directly in isotropic coordinates. In Weyl's harmonic coordinates a long involved procedure given by Hoenselaers is used. The exact coefficients of the Kerr metric multipole moments are given. The customary results are stated as (constant) x ma', without specifying what the constants are. The mathematical properties of definition of the system is a system of nonlinear elliptic partial differential equations. It is highly curious that the polynomials P{z) and Q{z) should arise in this context. What is the explanation for the existence of these polynomials for rational solutions? It is known that certain polynomials arise in the Riemann-Hilbert problem for two KiUing vector fields. If a relation could be established between the Riemann-Hilbert problem and polynomials defined in Chapter 3, a relation would be obtained concerning the full nonlinear structure of the problem under consideration. The pressure multipole moments bi could then be related to the Geroch Group. The Geroch-Hansen invariant multipole moments Ni and Wi embody Newtonian like properties of PAGE 105 97 stationary axisymmetric systems. The bi would be related to the nonlinear symmetries of the problem under consideration. As mentioned in Chapter 2, the Killing horizon is determined by the bi which shows that these parameters contain important relativistic information. Therefore, the most difficult and the most important problem that arises from this study is to understand the relationship between the boundary data of a rotating star and the Riemann-Hilbert problem. PAGE 106 APPENDIX A GAUGES AND COORDINATES FOR STATIONARY AXISYMMETRIC SYSTEMS In a general spacetime four of the ten components of the metric tensor can be given arbitrary values by the four degrees of freedom to make coordinate transformations (Hawking and Ellis, p. 74). In a spacetime with two commuting Killing vector fields, there are only two degrees of freedom left. The metric is then of the form given in Section 2.1. There are no free functions on the group orbits, i.e., on the {t, 4>) part of the metric. There are two degrees of freedom to make coordinate transformations on the two surface S that is orthogonal to the group orbit. One degree of freedom is used to eliminate the cross term (Bardeen, 1971, Eqn. 35). The remaining degree of gauge freedom may be used to choose either the radial gauge or the isothermal gauge. Once this choice is made no free functions remain in either gauge (MTB, p.68, Eqn. 2.11). All remaining metric functions are determined uniquely apart from trivial coordinate transformations. A detailed discussion of gauges and coordinates for axisymmetric situations in the ADM 3 + 1 formalism may be found in Bardeen and Piran (1983). The invariant form of the two-dimensional Laplace Equation for D, the determinant of the (t, (^) part of the metric is 98 PAGE 107 99 If this equation is written in a diagonal metric (MTB, p. 273, Eqn. 1), where D = e^, the equation for D takes the following form There is freedom to impose one gauge condition on //2 and ^3. In the isothermal gauge //2 = A'aj and the Laplace equation takes the form: [ie%h + [(e^),3],3 = 0. The conformal factor drops out of the equation as it should since the Laplace equation is conformally invariant in two dimensions (Wald, 1984, Appendix D, p. 447, Eqn. D.ll). The metric of the two surface S does not enter into the Laplace equation. This amounts to solving Laplace's equation in flat space. If any other gauge condition is imposed, let e^^'^"^ = A, the conformal invariance of the Laplace equation is obscured, [AV2(e/^),2],2 + [A-V2(e^)^3]^3 = 0. The boundary data for the function D is transparent in the isothermal gauge. In other gauges it is not so clear since the metric on the two surface S and determinant of the (t, 0) part of the metric become intertwined. A.l The Radial Gauge The radial gauge is defined by the condition: g^(j,gee = r'^ sin^ 9. In this gauge the area of an r = constant surface element is sin^ 0ddd(t) (Bardeen and PAGE 108 100 Piran, 1983). The radial gauge was used by Hartle and Sharp (1967) to study slowly rotating neutron stars. Recently Bonnazola et ai, have shown that the radial gauge is not global except in the case of spherical symmetry (Bonnazola, 1993, Appendix A). The metric in the radial gauge develops problems at the origin. A. 2 The Isothermal Gauge The isothermal gauge is defined by the condition: Qpp = gzzThe metric has a conformal form in this gauge. The metric preserves its conformal form under complex analytic coordinate transformations, which change the conformal factor, while preserving the form of the metric. The isothermal gauge has been used for all exact relativistic computations of rotating neutron stars. The stationary axisymmetric metric in the isothermal gauge is ds'^ = gttdt^ + 2gt^dtd(j) + g^^d^'^ + e'^^'{dxl + dxl) = -Vdt^ + Wdtd PAGE 109 101 In the Papapetrou form the metric functions are defined as follows ds^ = -Vdt'^ + Wdtd(f) + #2 ^ e2M(^^2 ^ ^^2) = -V(dt yd4>y + ^d(t>'^ + e'^f'idxl + dxj) n2 = -/{dt ujpd(j)f + +y ^^ + e^^'idxl + dxl) = -(^Â—yt^+x(d(l>-Â—dt) +e^t'{dxl + dxl) = -e^^dt^ + ^(d(l>^dtV + e'^^'{dxi + dxj) e ^ X ) = -e^^'dt' + ^ [D\d(l> uJBdtf + e2C(dr2 + dz^) The fact that the magnitude of the spacelike Killing vector, X = r]Â°"na, was being replaced by the magnitude of the Killing bivector, = VX + W'^ -2C[a%]C^^) has not been noted in the literature (Papapetrou, Bardeen, T and S, MTB); it is only noted in the Exact Solutions book. The same symbol u is used in the literature for the metric function associated with gt^p even though it represents two diff'erent functions with difl^erent dimensions and units. The invariant definitions are as follows: Papapetrou's ujp = W/V = gt4,/gtt ; Bardeen's ub = -W/X = -gt4>/ 9 PAGE 110 102 These simple algebraic relations cannot be implemented in practice without the nontrivial identities given by Tomimatsu and Sato (1973, Eqn. 4.9) which generalize the Kerr metric identity given in MTB (p. 289, Eqn. 137) for the TS{6) metrics. A. 3 Two Global Coordinate Charts The isothermal form of the metric is preserved under conformal coordinate transformations. Therefore locally it is possible to choose different coordinate charts and still preserve the conformal form of the metric. However, if a single global coordinate chart is desired, then the choices are severely restricted. Different choices of coordinates correspond to different choices of the conformal factor. The conformal factor satisfies the field equations. Once appropriate boundary conditions are imposed, and the field equations satisfied, the conformal factor becomes unique and defines a unique global coordinate system. As discusses in Section 4.2, the conformal factor satisfies a second order elliptic equation and requires two boundary conditions. The first boundary condition is that the conformal factor go to unity at infinity and the isothermal coordinates match on to ordinary flat space coordinates. The second boundary condition is the flatness of the 3D metric in the neighborhood of the axis. Once these two conditions are imposed the global chart becomes unique. Further details on the uniqueness of global charts may be found in Section 4.2. In the study of stationary axisymmetric systems in the isothermal gauge, two global charts have been used: Weyl's harmonic chart and Weyl's isotropic chart both of which are defined in Weyl's classic 1917 paper. In the Weyl's harmonic chart one of the coordinates is the magnitude of the Killing bivector: -2([^r)b]ev^ = = VX + W^; is also the determinant of the PAGE 111 103 {t, (j)) part of the metric. According to the field equations D satisfies the Laplace equation on the two surface S that is orthogonal to the group orbits. The scalar D is therefore a harmonic function on the two surface S; let E be its conjugate harmonic function. In Weyl's harmonic chart the harmonic conjugates, D and E, are chosen as cylindrical coordinates (p/^, Zh)\ Â— D, and Zh = E. These coordinates are well defined everywhere except at the isolated critical points of the harmonic function where its derivative vanishes and VD = VE = (0,0). It is shown in Chapter 4 that Schwarzschild solution has two critical points at the poles. As these critical point lie on the boundary, they appear not to cause any serious problems. In this context it will be assumed that the two surface S to be parametrized is simply connected. For example in the case of the solution with a toroidal Killing horizon discussed in Section 2.2 (Fig 2.3) the two surface S is not simply connected. According to Carter (1973) there will be a critical point in the interior for this solution. If there are any isolated critical points in the interior, a single global Weyl chart will not be admissible. Weyl's harmonic coordinates are related to Weyl's isotropic coordinates by the conformal transformation , hi 363 nhn = 1 + ^ ^ + dZ 2:2 Z4 where W = ph + izh, and Z = pi + izi. The critical points where harmonic coordinates break down may be identified in isotropic coordinates by setting = 0. For the Schwarzschild solution the transformation from isotropic coordinates to harmonic coordinates was given by Weyl in 1917 PAGE 112 104 The coordinate vector fields associated with the flatspace cylindrical coordinates {p,z) do not vanish anywhere-?, e., Vp ^ (0,0) and Vz ^ (0,0) at every point in the plane. Isotropic coordinates have this property-i.e., Vpi and Vzi do not vanish anywhere. However, the coordinate vector fields associated with the harmonic coordinates {phiZu) vanish at the critical points of Weyl's harmonic function. In some cases, for example, the Zipoy-Vorhees solutions and the Tomimatsu-Sato solutions, the critical points may be degenerate, in which case higher derivatives will also vanish. The Morse indices of the coordinate vector fields at the degenerate critical points have not yet been calculated. In Weyl's isotropic coordinates the Papapetrou form of the metric is: In Weyl's harmonic coordinates the Papapetrou form of the metric is: = -j{dt Upd(t>f + y [e2T(dp2 + dzl) + p^#2' where D"^ = p\ Â— pfB"^. The flatness boundary condition on the conformal factor is e^i' 1 on the axis. The flatness boundary condition on the conformal factor is e^^ ^ B. Under a coordinate transformation from isotropic coordinates to harmonic coordinates the conformal factor e^^ transforms as: e27 = A2 ^1 on the axis dW 2 B on the axis dZ since VD = {B, 0) and VE = (0, B) on the axis; andW = D + iE. PAGE 113 105 A. 4 The Three Types of Coordinates In two dimensions three types of coordinates are used: cartesian, polar, and elliptic. These are simply the two-dimensional cross sections of the three dimensional coordinates: cylindrical polar coordinates {p,z, PAGE 114 106 Unlike Cartesian coordinates or polar coordinates, elliptic coordinate systems have an intrinsic length scale associated with them-A;, the distance from the center to the focus. In the case of the Kerr black hole, the constant k is related to the surface area of the horizon (Carter, 1973, p. 197, Eqn. 10.55). PAGE 115 APPENDIX B VARIOUS FORMS OF THE SCHWARZSCHILD METRIC In this appendix various forms of the Schwarzschild metric will be derived. First the standard form in prolate spheroidal coordinates will be derived. This form is used in the Zipoy-Vorhees transformations to generate static axisymmetric solutions from the Schwarzschild solution (Section 3.5). The transformation from prolate spheroidal coordinates to Schwarzschild coordinates and to isotropic coordinates will also be given. These serve as illustrations of the transformations used in the next appendix, where the various forms of the Kerr metric are derived from the standard form in prolate spheroidal coordinates. The first exact solution to Einstein's field equations was given by Karl Schwarzschild in 1916. This solution is spherically symmetric. The following year Herman Weyl found a series of static axisymmetric solutions. The static metric in the isothermal gauge in Weyl's harmonic chart in cylindrical coordinates {ph = D and Zh Â— E) is The static axisymmetric field equations for the above metric are Vix,C/ = Upp + -Up + C/^, = 0 7p = p{Ul Ul) lz = 2p{UpU,). 107 PAGE 116 108 The static axisymmetric Weyl solutions are given by solutions to the Laplace equation in three dimensions in cylindrical coordinates. For the spherically symmetric Schwarzschild solution the two functions e^^ and e'^'^ are given in cylindrical coordinates as follows (Weyl, 1917, p. 140) Â„2[/ f], +r2 Â— 2m ''I + ^2 + 2m ^27 _ in + r2 2m) {r I + r2 + 2m) 4rir2 where rf = + {zh m)^ and rl = {zh + m)^. The quantities n and r2 are distances from the two points (0,m) and (0, -m). These two points are the critical points of Weyl's harmonic function in the harmonic chart-the North and South poles of the horizon. To transform to prolate spheroidal coordinates {x, y) the North pole (0, +m) and the South pole (0, -m) are taken to be the foci of the ellipses as follows: ri + r2 X = y 2m ri r2 2m ri+r2 2m a; Â— 1 = 2m ri+r2 + 2m a: + 1 2m 3;2 _ 2 ^ nr2 In two dimensions the curve x = constant represents the locus of points such that the sum of their distances from the two foci at (0, Â±m) remain constant, i.e., an ellipse with foci at (0,Â±m). In three dimensions the surface x = constant represents a prolate spheroid aligned along the z axis. Similarly, in two dimensions the curve y = constant represents the locus of points such that the difference of their distances from the two foci at (0, Â±m) remains constant, i.e., PAGE 117 109 an hyperbola with foci at (0, Â±m). In three dimensions the surface y = constant represents a hyperboloid of one sheet surrounding the z axis. Employing the transformations mentioned above, the Schwarzschild metric is obtained in prolate spheroidal coordinates: The mass m is the intrinsic length scale associated with the Schwarzschild solution in prolate spheroidal coordinates. The two static metric functions are g2t/ 1 \x + lJ (a;2 Â— u'^) ^x^ The form given above will be termed the standard form, where the metric functions and the flat space two-dimensional metric (the portion enclosed in curly brackets) are separate. The reduced form given below will be useful only for the purpose of transforming to Schwarzschild coordinates and the isotropic coordinates ds^ + dd^ + sin^ ddcfp_x^ 1 The substitution y = cos9 has been made in the above reduction. To transform the reduced form to Schwarzschild coordinates, the following relation between the dimensionless elliptic coordinate x and the Schwarzschild radial coordinate may be employed: x = 1. =^ x + l = '-Â±; x-l = '-Â±-2and ^2 _ ^ ^ r^jr, -2m) m m ' m2 PAGE 118 110 With the above substitutions, the Schwarzschild metric in Schwarzschild coordinates is obtained: 2 2m\ ,.9 2m\~i ds' To transform the reduced form to isotropic coordinates, the following relation between the dimensionless elliptic coordinate x and the isotropic radial coordinate rj may be employed: x = ^ ^1 + =^ x + l = Â— 1 + Â— ; a; 1 = Â— 1 ; mV 2riJ m\ 2ri) m V 4r/ c/x^ _ drf x"^ Â— 1 rf With the above substitutions, the Schwarzschild metric in isotropic coordinates is obtained A my (^,mY \ ^2^J drf + rfde"^ + r? sin^ Odcj)^ 2 The flatness condition near the axis, = 1 ^ = 5, is satisfied in isotropic Ar coordinates. Unlike the transformation to Schwarzschild coordinates, the transformation to isotropic coordinates is a conformal transformation which preserves the isothermal form of the metric. In terms of complex quantities W = p/^ + izh and Z = pi-\izi, the transformation from isotropic coordinates to harmonic coordinates is given by the Joukowski map: W = Z-^ AZ PAGE 119 Ill The Schwarzschild metric in harmonic coordinates and in isotropic coordinates was given by Herman Weyl in 1917. The complex transformation between the two coordinates is also given in the same paper (Weyl, 1917, p. 132, p. 140). PAGE 120 ds^ = -f{dt ujpd(f))^ + J APPENDIX C VARIOUS FORMS OF THE KERR METRIC In this section the Kerr metric will be transformed from harmonic coordinates to isotropic coordinates. For illustration the metric will be transformed to Boyer-Lindquist coordinates first. The stationary axisymmetric metric in the Papapetrou form in prolate spheroidal coordinates has the following form: In harmonic coordinates the metric functions for the Kerr metric have been given by Ernst(1968). The Tomimatsu-Sato metrics are also given in the above form (Tomimatsu and Sato, 1973). A detailed discussion along with the field equations in prolate spheroidal coordinates may be found in the review paper of Reina and Treves (1976) and in the monograph of de Felice and Clark (1990, Chapter 11). The metric functions for the Kerr metric are: J. _ p^x^ + q^y^ 1 UJp (jpx + 1)2 + q^y"^ -2q{l-y'^){px^\) p^x^ + cp-y"^ Â— 1 p^{x'^ Â— J/2) y _ (1 V'mv^ + 1)^ + q^? pWx^ 1)(1 y2)] {px + 1)2 -I^2^2 D'=pl^k\x^-V){l-y^) 112 PAGE 121 113 where k = \/m^ Â— ; p = k/m and q = a/m, so that + = 1. The constant A; is the intrinsic length factor associated with the elliptic coordinates. The parameter k is related to the surface area of the horizon A and the surface gravity kh as follows: k = (Carter, 1973, p. 197, Eqn 10.55). In harmonic coordinates the conformal factor e'^'*' satisfies the flatness condition near the axis, y = Â±i: _ p'^x^ + q'^y^ 1 _^ p'^x^ + 1 ^ p2(^2 1) ^ ^ p2(2;2_y2) p'^{x^-l) p'^{x^-l) To transform the Kerr metric in the isothermal gauge in prolate spheroidal coordinates to the radial gauge in Boyer-Lindquist coordinates, the following relation between the dimensionless elliptic coordinate x and the Boyer-Lindquist radial coordinate r may be employed: x = ^ , and y = cos 6; r r Â— m px + l = Â—; px = . m m The following expressions occurring in the metric are evaluated first: Â„2 2 , ^2Â„2 T _ {r mf a^cos^e 2mr + cos^ 9 y ^ q y JÂ— 2 ' 9 ^ Â— o ) I , in2 , 2 2 +a^cos^^ [px + \y + q'y' = 2 > m ^ ^ = Â— A = 9 k^ k^ 2f 2 2n ^ I {r k^ cos^ e p (x^ -y^)=^^ ^ > 2mr + sin^ 61-1-0^ cos^ 6 m2 PAGE 122 114 The metric functions of the Papapetrou form in Boyer-Lindquist coordinates take the following form: p^x^ + (^y^ 1 2mr + cos^ Q f = Up {px + 1)2 + g2j^2 ^2 _|_ ^2 pQg2 Q -2q{l y'^){px + 1) -2amrsin^e p2a;2 + q2y2 _i ^2 _ ^ ^2 ^Qg2 ^ ' 27 _ P^^^ + 9^2/^ ~ 1 _ ~ 2mr + cos^ 0 p2(3,2_y2) ~ -2mr + m2sin2^ + a2cos2^' = k^{x^ 1)(1 = (r^ 2mr + a^) sin^ 6. {px + 1)2 + g2y2 _ {(r2 + a2)2 _ Q2(r2 _ 2mr + a2) sin^ 9} sin^ 9 + 0^ cos^ ^ To transform the Kerr metric from harmonic coordinates to isotropic coordinates a conformal transformation is made, which preserves the isothermal form of the metric. The same conformal transformation that was used for Schwarzschild is used for the Kerr metric since the conformal factor e^'^ consists of quadratic polynomials in both cases. After this transformation is made the correct boundary condition along the axis = B is satisfied. The conformal transformation in question is obtained by replacing m by A; in the transformation used for the Schwarzschild metric: x = ^(l + and y = cos 9; x2 Â— I f2 PAGE 123 115 The following expressions occuring in the metric are evaluated first: 2 2, 22 1 m + (px + 1)' + = m 2 r 1 2 a cos ^ 1 + 4r' m + + m a cos Q r / The metric functions of the Papapetrou form in isotropic coordinates take the following form: / p^x^ + q^y"^ 1 (px + 1)2 + ^2^2 2 ^ ^ ^ cos^ ^ _ UJp = 4r 2g(l-y2)(p3; + i) 2am sin^ g f 1 + + r J e27 = P'^x2 + g2y2 _ J j92(2;2 Â— 2 2/1 9 Q COS 6 m (-5) , cos^ ^ ' 1 T fcicos^^ X = (1 y'^MpX + 1)^ + q'^Y p2g2(^2 _ _ ^2)J (j9X + 1)2 + qf2y2 r2 sin^ 9 {(i + ^ + 4r^ mV + g^ cos^ 9 ^V-i)(i-y') 4r^ PAGE 124 APPENDIX D THE MANKO et al., 1994 SOLUTION The solution given by Manko et ai, (1994) was generated using SibgatuUin's Algorithm (Sibgatullin, 1991; Manko and Sibgatullin, 1993) for solving the Riemann-Hilbert problem for stationary axisymmetric systems. The solution has five free parameters: mass m, angular momentum a, charge q, mass quadrupole moment b, and magnetic dipole moment c. In this case the conformal factor, AA* BB* + CC* 16[(m2 a2 ^2)2 _ 4b'^]2R_R_^r-r+ AcÂ± = [m2 a2 g2 + 2{Â±d b)]^/\ The Papapetrou metric function. / = -gtt = AA* BB* + CC* {A + B){A* + B*) 116 PAGE 125 117 The functions A, B, and C are defined as follows in the simplified case when the mass quadrupole moment 6 = 0 (Manko,1993): A = -aqb){R-r+ R+r+) + iK+{a + q){R_rR+r+)] +aq + 6)(i?_r+ + R+r-) + iÂ«;_(a q){R-r+ R+r-)] 4b{aq + b){R-R+ + r-r+) B = mK-K+{{m^ -a^ q^){r+ r+ R^ R+) + K-K+{r+ r+ + /?_ + R+) + iq[{K+ + Â«_)(r_ r+) + {k+ R_)]} C = K-K+ilqirn^ q^) 2ab]{R+ R+ r+) qK-K+{R+ R^ + r-+ r+) + + b){R_ -R+ + r+r_) + K-{q'^-b){R+-R_ + r+-r-)]}. No physical interpretation has been given for this solution by its authors in terms of its horizon structure. The authors do not state whether it possesses a Killing horizon or an event horizon. 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After graduating from St. Gregory's High School in Dhaka, he attended Notre Dame College in eleventh and twelfth grades where he was "Man of the Year" in 1980 for his extracurricular activities. In 1981 Ejaz Ahmad was awarded a full scholarship to attend Bowdoin College in Brunswick, Maine, USA. The author was awarded the Hall Physics Prize and the Smythe Mathematics Prize at Bowdoin College. He was elected to Phi Beta Kappa in 1983. The author graduated summa cum laude from Bowdoin College 1984 with a double major in mathematics and physics. Ejaz Ahmad was a graduate student at the University of Chicago for two years starting in 1984. He was awarded the Gregor Wentzel Prize for his teaching of physics to undergraduates. At Chicago, he interacted with the Relativity Group, and earned his stipend doing computer simulations with the Solar Energy Group. After receiving his master's Degree in physics, he worked as a scientific programmer and system manager in the Department of Geophysics. During this time the author made movies of the Greenhouse Effect from NASA satellite data for Professor V. Ramanathan. Ejaz Ahmad came to the University of Florida in 1989. He began working with Professor Ipser in 1990 in relativistic astrophysics. He also worked under Professor Paul Avery writing computer programs for an object oriented simulation package in high energy physics. He was the computer system manger for the Astrophysics Group from 1991 to 1996. In addition to the above mentioned activities, the author was the President of the Bangladeshi Student Association, the Chair of the International 123 PAGE 132 Student Council, and a member of the Strategic Planning Committee of the Graduate School. Ejaz Ahmad was also a physics tutor for the University of Florida Athletic Association where he tutored many athletes, including members of the Florida Gators football team. The author's interests include global political economy and postmodern philosophy-particularly Michel Foucault and Paul Feyerabend. 124 PAGE 133 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Cjocterof Philosophy. James tR. Ipser, Chair Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Kwan-Yu Chen Professor of Astronomy I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Steven L. Detweiler Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Henry E. Kandrup Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Christopher J. Stanton Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Bernard F. Whiting Associate Professor of Physics PAGE 134 This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1996 Dean, Graduate School |