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RADIATION REACTION ON MOVING PARTICLES IN GENERAL RELATIVITY By EIRINI MESSARITAKI 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 2003 ACKNOWLEDGMENTS First and foremost I would like to thank Dr. Steve Detweiler, Professor of Physics at the University of Florida, who was my advisor during my graduate studies. He has always been very supportive of my work, very willing to help and extremely patient with me. I feel very lucky to have worked with him during the past 5 years. I would also like to thank Dr. Bernard Whiting for his help on writing this dissertation and for his contributions to the research. Many thanks go to the other members of my committee: Dr. Jim Fry, Dr. An drew Rinzler and Dr. Vicki Sarajedini, as well as to Dr. Richard Woodard and Dr. Luz Diaz, for their useful comments on the dissertation. I would also like to thank my brothers, my family and my close friends. Their help and support has always been very important and I could not have come this far without them. My research was partly supported by the Institute of Fundamental Theory at the University of Florida and I am grateful for that. TABLE OF CONTENTS ACKNOWLEDGMENTS . ... ABSTRACT . . . . . ii . . v CHAPTER 1 INTRODUCTION . ........... 2 RADIATION REACTION IN FLAT SPACETIME . 2.1 Definitions of the Fields ............. 2.2 Derivation of the Equations of Motion ...... 2.3 Radiation Reaction ................. . . . 3 RADIATION REACTION IN CURVED SPACETIME . . 14 3.1 Nonlocal Quantities ................... ....... 14 3.2 Green's Functions ................... ....... .. 18 3.2.1 Green's Functions for a Scalar Field ....... ....... 18 3.2.2 Green's Functions for a Vector Field ...... ........ 23 3.2.3 Green's Functions for a Gravitational Field . . .... 27 3.3 Motion of a Charged Particle in Curved Spacetime . .... 32 3.3.1 Equations of Motion .................. .. .. 32 3.3.2 Calculation of the Fields .................. . 34 3.3.3 Conservation of Energy and Momentum . . 37 3.3.4 Radiation Reaction .................. ..... 40 4 SELFFORCE .. ......... 4.1 Scalar SelfForce .. .......... 4.1.1 Direct and Tail Fields ...... 4.1.2 The SField and the RField . 4.2 Electromagnetic SelfForce ........ 4.3 Gravitational SelfForce .. ....... 5 SINGULAR FIELD FOR SCHWARZSCHILD GEODESICS 5.1 ThorneHartleZhang Coordinates .. .......... 5.2 Scalar Field of a Charged Particle .. .......... 5.3 Scalar Field of a Dipole .. ............... 5.4 Electromagnetic Potential of a Charged Particle . 5.5 Electromagnetic Potential of an Electric Dipole . 5.6 Electromagnetic Potential of a Magnetic Dipole . 5.7 Gravitational Field of a Spinning Particle ........ 5.8 GRTENSOR Code ............... . . ... 77 6 REGULARIZATION PARAMETERS FOR THE SCALAR FIELD 80 6.1 Regularization Procedure ................... ...... 81 6.2 Order Calculation of the Scalar Field ....... .......... 84 6.3 Scalar Monopole Field ................... ...... 88 7 CALCULATION OF THE RETARDED FIELD . . 102 7.1 Analytical Work .......... ................... .. 102 7.2 Numerical Code .......... ................... .. 110 8 APPLICATIONS AND CONCLUSIONS . . . 111 8.1 Equations of Motion ................... ....... 111 8.2 Effects of the Scalar SelfForce ....................... 112 8.2.1 Calculation of the SelfForce ....... ............ 115 8.2.2 Change in Orbital Frequency ... . . 116 8.3 Effects of the Gravitational SelfForce .... . . 117 A GRTENSOR CODE FOR THE SINGULAR FIELDS . . 119 B DECOMPOSITION OF fXQ .. ....... . . 134 C NUMERICAL CODE FOR THE RETARDED FIELD . . 143 REFERENCES . . . . . . . 163 BIOGRAPHICAL SKETCH . . . . . 166 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 RADIATION REACTION ON MOVING PARTICLES IN GENERAL RELATIVITY By Eirini Messaritaki August 2003 Chairman: Steven L. Detweiler M.,. iP r Department: Physics A particle in the vicinity of a Schwarzschild black hole is known to trace a geodesic of the Schwarzschild background, to a first approximation. If the interaction of the particle with its own field (scalar, electromagnetic or gravitational) is taken into account, the path is no longer a background geodesic and the selfforce that the particle experiences needs to be taken into account. In this dissertation, a recently proposed method for the calculation of the self force is implemented. According to this method the selfforce comes from the inter action of the particle with the field 9R .= t ~s for a scalar particle; with the electromagnetic potential AR = A' As for a particle creating an electromagnetic field; or with the metric perturbation hb = hrb hb for a particle creating a gravita tional field. First, the singular fields ys, As and hSb are calculated for different sources moving in a Schwarzschild background. For that, the ThorneHartleZhang coordi nates in the vicinity of the moving source are used. Then a modesum regularization method initially proposed for the direct scalar field is followed, and the regularization parameters for the singular part of the scalar field and for the first radial derivative of the singular part of the selfforce are calculated. Also, the numerical calculation of the retarded scalar field for a particle moving on a circular geodesic in a Schwarzschild spacetime is presented. Finally, the selfforce for a scalar particle moving on a circular Schwarzschild orbit is calculated and some results about the effects of the selfforce on the orbital frequency of the circular orbit are presented. CHAPTER 1 INTRODUCTION The General Theory of Relativity, developed by Albert Einstein at the beginning of the twentieth century, has been successfully used by scientists for many years to explain 1.1,.i .,1 phenomena and to make predictions about ].li ,i, .1l systems. One of the most exciting predictions of General Relativity is the existence of gravitational ra diation, which can be thought of as a wavelike distortion of spacetime. It is expected that gravitational radiation from remote .Ir I1. .li,i .i1 systems will yield very impor tant information about those systems, thus answering many questions that scientists currently have. Even though the detection of gravitational waves has proved to be a challenging task, recent developments in technology have made scientists confident that gravitational waves will indeed be detected in the near future. Both earthbased detectors (such as LIGO and VIRGO) and spacebased detectors (such as LISA) are expected to start operating in the next decade or two. For the data collected by these detectors to be useful, accurate information about the sources of gravitational radiation is required. Only if scientists know what the gravitational waves emitted by specific systems should look like, can they compare them with the patterns observed and draw conclusions about the pi, i. .,1 characteristics of those systems. That is why there has been a lot of interest lately in predicting the gravitational radiation emitted by different 1.i 1 1% i. ., systems. One .iI1 ,,,li,i1 .I1 system for which gravitational waves are expected to be de tected, specifically by spacebased detectors, is the binary system of a small neutron star or black hole (of mass equal to a few times the mass of the Sun) and a super massive black hole (of mass equal to a few million times the mass of the Sun) [1]. Supermassive black holes are believed to exist at the centers of many galaxies, in 2 eluding our own. It is also believed that their strong gravitational field can capture smaller stars and black holes, which then move toward the supermassive black hole, until they are absorbed by it. The exact evolution of the system with time can give the pattern of the gravitational radiation that the system is expected to emit. Know ing that pattern is crucial in distinguishing whether the gravitational waves measured by the detector come from such a system. It can also help extract information about many characteristics of the system, such as the masses and the angular moment of the two components of the binary system. To determine how the system evolves, the path of the smaller star or black hole needs to be calculated. That path is largely determined by the influence of the gravitational field of the supermassive black hole on the small star or black hole; it is also affected by the interaction of the small star or black hole with its own gravitational field. This latter interaction, commonly referred to as the selfforce, is responsible for the radiation reaction effects, which cause the decay of the small star's or black hole's orbit toward the supermassive black hole. Even though my motivation for studying the selfforce effects stems from the need to know the evolution of this system, radiation reaction effects are present in other systems that are easier to deal with mathematically. One can think, for example, of a particle of a certain scalar charge, which moves in a Schwarzschild background spacetime and creates its own scalar field. There is also the case of a particle that carries an electric charge and creates an electromagnetic field as it moves in spacetime. The scalar field of the particle in the first case and the electromagnetic field of the electric charge in the second case will affect the motion of each particle, causing its worldline to differ from what it would be if radiation reaction effects were not present. It is useful to predict the evolution of such systems, mainly because they can give an idea of how the more difficult system can be handled successfully and not because they are realistic systems, since they are not expected to be observed in nature. In the past, various efforts have been made to develop a concrete scheme for including the radiation reaction effects for scalar, electromagnetic and gravitational fields in the equations of motion. One approach that has been used extensively in volves calculating the flux, both at infinity and at the event horizon of the central supermassive black hole, of quantities (such as the energy and momentum of the par ticle) that are constants of the motion when radiation reaction is not present [29]. That flux is then associated with the rate of change of those quantities at the location of the particle and the evolution of the particle's motion can be predicted. However, this approach is generally not applicable in the case in which the central black hole is a rotating black hole, except under some special circumstances [10]. One reason is that the evolution of the Carter constant [11], which is a constant of the motion that is not derived by a Killing vector, cannot be calculated using the flux at infinity and at the event horizon. In addition, this approach does not take into account the very significant nondissipative effects of the selfforce that the particle's field exerts on the particle [12]. That is why it is important for the selfforce, and not just its radiation reaction effects, to be calculated. The calculation of the selfforce is the subject of this dissertation. In Chapters 2 and 3 of this dissertation, I present the innovative work of Dirac [13] for the radiation reaction effects on an electron moving in flat spacetime and the theoretical generalization of it to the case of curved spacetime by DeWitt and Brehme [14]. In Chapter 4, I present different methods for practical calculations of the selfforce that have been proposed in the past, with emphasis on one in particular. In Chapters 58 of this dissertation, I present an implementation of that particular method for calculating the radiation reaction effects for a scalar, an electromagnetic and a gravitational field. CHAPTER 2 RADIATION REACTION IN FLAT SPACETIME The first successful attempt to provide an expression for the radiation reaction effects on a particle in special relativity was made by Dirac in 1938 [13]. In his famous paper, he studied the motion of an electron moving in flat spacetime by using the concept of conservation of energy and momentum along the electron's worldline. In this chapter I present the decomposition of the electromagnetic field and the derivation of the equations of motion, which include the radiation damping effects on the electron. The scheme for calculating the radiation reaction effects in curved spacetime presented in this dissertation extends Dirac's work. For the equations to look as simple as possible, the speed of light is set equal to 1 in this chapter. It is noted that the metric signature used by Dirac is (+ ) and that is what is used in this chapter, in order to keep the results identical to those derived by Dirac. In all subsequent chapters, however, the signature is changed to ( + ++), in order to adhere to the currently widely used convention. Thus, some of the equations given in this chapter change in subsequent chapters. Since none of the results described in this chapter are used in the chapters that follow, that should not cause any confusion at all. 2.1 Definitions of the Fields Dirac's analysis starts by assuming that an electron of electric charge q is moving in flat spacetime, so that the background metric is the Minkowski metric r',. The worldline of the electron is denoted by 2P, = z,, () (2.1) where s is the proper time along the worldline. The electromagnetic 4vector potential that the electron creates is assumed to satisfy the Lorentz gauge condition V'A = 0 (2.2) and obeys Maxwell's equations V2A 4'jr. (2.3) It is wellknown that the Lorentz gauge condition leaves some arbitrariness for the electromagnetic potential. In Equation (2.3), j, is the chargecurrent density vector of the electron, namely Sqj (xo zo) 6(x Z) 6(X2 J 2) 6(x3 Z) ds. (2.4) The electromagnetic field associated with a 4vector potential A" is, in general, given by the equation F^ = V"A" V^A. (2.5) It is clear that, unless the boundary conditions of the problem are specified, Equations (2.2) and (2.3) do not have a unique solution. In fact, adding any solution of the homogeneous equation V2A, 0 (2.6) (which represents a sourcefree radiation field and obeys the Lorentz gauge condition) to a solution of Equation (2.3), gives a different solution of Equation (2.3) that also obeys Equation (2.2). One solution of interest for the problem at hand is the retarded electromagnetic field F'"" created by the electron, which is derived by the wellknown LienardWiechert potentials Aet that obey Equations (2.2) and (2.3). Assuming that there is also a radiation field Fi incident on the electron, the actual field in the neighborhood of the electron is the sum of those two fields FCt = Fret F (2.7) Another interesting solution is the one given by the advanced potentials Adv. It is rea sonable to expect that the advanced potentials play a role symmetrical to the retarded potentials in this problem. If the outgoing radiation field leaving the neighborhood of the electron is Fo(, then FV = Fv + Fo.V (2.8) The radiation emitted by the electron is given by the difference of the outgoing and the incoming radiation F r wd  Fut F. (2.9) Using Equations (2.7) and (2.8) for the actual field, an alternative expression for the radiation field can be obtained F = F F (2.10) The important implication of Equation (2.10) is that, because the retarded and the advanced fields are uniquely determined by the worldline of the electron, so is the radiation emitted by the electron. It becomes clear from the analysis of Section (2.2) that the radiation field is responsible for the deviation of the electron's worldline from a background geodesic, because that field shows up in the equations of motion, in the radiation damping term. It is significant to realize that Equation (2.10) for the radiation field is consistent with what one would expect for the radiation of an accelerating electron. To under stand that, one should recall that the radiation produced by an accelerating electron can be calculated using the retarded field F/, that the electron creates, at very large distances away from the electron and at much later times than the time when the acceleration takes place. At those distances and times the advanced field vanishes and Equation (2.10) gives the correct result. The advantage of Equation (2.10) is that it gives the radiation produced by the electron at any point in spacetime, so it can be used for the radiation in the neighborhood of the electron. A field that is needed for the analysis of Section (2.2) is the difference of the average of the retarded and advanced fields from the actual field, denoted by f"" f "" Fjj (fe~ + F ). (2.11) 2 ac r ret advi) Using Equations (2.7) and (2.8) with Equation (2.11) f" = (F~ l + F."). (2.12) 2 Because the incoming and outgoing radiation fields are derived from potentials that satisfy the homogeneous Equation (2.6), the field fl" is sourceless. It is also free from singularities at the worldline of the electron. 2.2 Derivation of the Equations of Motion The main concept used by Dirac in deriving the equations of motion for the electron moving in flat spacetime is the conservation of energy and momentum. The first step is to surround the worldline of the electron by a thin cylindrical tube of constant radius. This radius is assumed to be very small, specifically smaller than any length of ] li, i .1l significance in the problem. The objective is to calculate the flow of energy and momentum across the threedimensional surface of the tube, using the stressenergy tensor Tl, calculated from the actual electromagnetic field. The stressenergy tensor is given by 4 7 T9p = Fact ,V Fact + 1 pp Fact ac3 F`a (2.13) P 4 act. The flow of energy and momentum out of the surface of the tube is equal to the difference in the energy and momentum at the two ends of the tube. In the following, dots over quantities denote differentiation with respect to the proper time s. To simplify the notation, (dz,/ds) is set equal to v,,. One then obtains the equations vv = 1, (2.14) v., = 0, (2.15) v, *: + .,,'' = 0. (2.16) To calculate the stressenergy tensor, the electromagnetic 4vector potential and the electromagnetic field need to be calculated first. The retarded potential at a point x, generated by an electron moving on the worldline z,(s) is given by Aretp q (2.17) cl t (xe z v) calculated at the retarded proper time, which is the value of s that solves the equation (x, z,) (x" z") 0. (2.18) An equivalent expression for the retarded potential, obtained from Equation (2.17) by using elementary properties of the 6 function, is Aet. = 2 q f 6[(x, z) (xv z)] ds (2.19) where Tint is a proper time between the retarded and advanced proper times. By differentiating Equation (2.19) and using Equation (2.5), the retarded electromagnetic field is obtained Fret, 2 q q i d ip) 6[(x z) ( )] ds. (2.20) Fret / (y T y[ \ ) By using again some of the elementary properties of the 6 function, an equivalent expression for the retarded field is derived Fret p, d z (xzj z( z )] (2.21) A (x ^ zA ^ ds 1i (x" z) (2.21) Again, all quantities are calculated at the retarded proper time. Since the goal is to calculate the stressenergy tensor at the two ends of the worldtube surrounding the worldline of the electron, it can be assumed that the point x, is very close to the worldline. Specifically x = z/(so) + 7/' (2.22) where the 7t's are very small. Then, the fields can be Taylorexpanded in the 7,'s. In the expansions that follow, all the coefficients are taken at the proper time so. It can be assumed that the retarded proper time is so a, where it is reasonable that a is a small positive (Ii.ml, il' of the same order as 7,. The detailed calculations needed to obtain the Taylor expansions of the electro magnetic fields are tedious but straightforward. They can be found in Dirac's paper; here I present only the results of that calculation. Since 7 is a spacelike vector, it can be assumed that 7y,7 = 2, where E is a positive number. Taylorexpanding the retarded field and keeping only the terms that do not vanish in the limit E 4 0 gives Fretpv q (1 j)V [ v) > 1 (v __ v) (1  7VA) + 1 68 15%v (vs'> Vv7p) 2 6 1/ 2 2 + 2 KQV '4v)]. (2.23) The advanced field can be obtained by changing E to E and changing the sign of the whole expression. By using the retarded and advanced fields in Equation (2.10) the radiation field on the worldline is obtained 4q Frad v 3 ( v V &,vj). (2.24) The actual field then becomes Fact p = fiv + q [1 7Av^] ( E 1 7EU) 1 1 + (1 +2 V)( v, (2.25) 1 ( J_ vv)j/ + E1 To calculate the energy and momentum flow out of the tube, the stressenergy tensor needs to be calculated. In fact, only its component along the direction of y, is necessary. Substituting Equation (2.25) into Equation (2.13): 4 i T ,,p F.atpu F 't vp F. ,3 F13a Sa t p act t q2 (1 7 )1[ 1 4 + = q2 1 (2 E 2 \VA) 'Y I E(1 +t 3 ]Av) t 2 3 + qE lvf4. (2.26) The flow of energy and momentum out of the surface of the tube is given by integrating this component of the stressenergy tensor over the surface of the tube. The result is ( 12 E1 2 q v" f, ) ds where the integration is over the length of the tube. As mentioned earlier, this flow of energymomentum depends only on the con ditions at the two ends of the tube. That means that the integrand must be a perfect differential, namely 1 2 1 c q Vf" f = B, (2.28) for some B,. Equation (2.15) and the fact that f"" is antisymmetric in its indices put a restriction on B., specifically 12 ,PB q2 E1 , v/* / 5Qev q v. f" 0. (2.29) Thus, the simplest acceptable expression for B. is BP, kvg (2.27) (2.30) for some k independent of s. From Equation (2.28) one obtains k = 2 21 (2.31) where m must be a constant independent of E in order for Equations (2.28) and (2.29) to have a welldefined behavior in the limit E > 0. Finally, the equations of motion for the electron are m i,.= q v, (2.32) and m plays the part of the rest mass of the electron. This result is used in Section (2.3) to derive an expression for the radiation damping effects on the moving electron. 2.3 Radiation Reaction The field f" defined in Section (2.1) allows the equations of motion (2.32) to be expressed in a simple form. However, in practical applications one would prefer to have the incident radiation field Fi~" in the equations of motion, as Fi~ is usually given. By substituting Equations (2.7) and (2.10) into Equation (2.11), an expression for f^' is obtained, that involves the incident field and the radiation field: f'"= F' + Ftad" (2.33) Then, the equations of motion become 1 m t,, =q2 vvFad, (2.34) The first term of the righthand side of Equation (2.34) involves the incident radiation field and gives the work done by that field on the electron. The second term of the righthand side involves the radiation field emitted by the electron and is the term of particular interest when one considers radiation reaction. This term gives the effect of the electron radiation on itself and is present even if there is no external radiation field, namely even if Fi = 0. The fact that the radiation reaction effects on the electron can be described by using the electromagnetic field F` is one of the important results of Dirac's work and is the idea used when motion of particles in curved spacetime is considered, to calculate radiation reaction effects on those particles. It is worth writing the equations of motion in terms of the characteristics of the worldline of the electron and the external radiation field, which can be achieved by using Equation (2.24) for the radiation field of the electron m ', = q v, Fi,,;, + 2 q 2 + 2 q2 tV" v. (2.35) It is an important and a very interesting feature of Equation (2.35) that, in addition to the first derivative of vI, its second derivative shows up as well. A discussion of that fact and of some of its implications is presented in [13], but that is beyond the scope of this dissertation. CHAPTER 3 RADIATION REACTION IN CURVED SPACETIME Dirac's study of the radiation reaction effects on an electron gave a relatively simple result, because the electron was assumed to be moving in flat spacetime. The analysis becomes significantly more complicated for particles moving in curved space time. DeWitt and Brehme [14] were the first to study the motion of an electrically charged particle in a curved spacetime using Green's functions. Later, Mino, Sasaki and Tanaka [15] generalized that analysis for the case of a particle creating a grav itational field while moving in curved spacetime. Since Green's functions are used extensively in subsequent chapters to determine the selfforce effects on different par ticles, it is appropriate to present the analyses of DeWitt and Brehme and of Mino, Sasaki and Tanaka before proceeding. In this and all subsequent chapters, geometrized units are used, meaning that Newton's gravitational constant and the speed of light are both set equal to 1. Note also that the signature of the metric is changed to ( + ++) from now on, so minor changes in some of the equations previously mentioned should not be surprising. 3.1 Nonlocal Quantities One of the main characteristics of any Green's function is that it connects two points in spacetime. It does that by propagating the effect of the source, from the point where that source is located (source point) to the point where the field needs to be calculated (field point). Since Green's functions are inherently nonlocal quantities, the discussion about them can be facilitated if some more elementary nonlocal quantities are introduced first. The most general class of nonlocal quantities is the class of bitensors, which are simply tensors whose indices refer to two points in spacetime. In the following, x denotes the field point and z denotes the source point. In order for the indices for each point to be easily identifiable, all unprimed indices refer to the field point x and all primed indices refer to the source point z. A very important (1ll..1l il' for the study of nonlocal properties of spacetime is the biscalar of geodesic interval s(x, z). It is the magnitude of the invariant distance between the points x and z as measured along a geodesic that joins them and is a nonnegative quantity. It is defined by the equations V VVs = V's V,,s 1, (3.1) lim s 0, (3.2) where V" denotes covariant differentiation with respect to the background metric g"". It is obvious that s must be symmetric under interchange of its two arguments, namely s(x, z) = s(z, x). (3.3) With the signature of the metric being ( + ++), the interval s is spacelike when the + sign holds in Equation (3.1) and timelike when the sign holds. The points x for which s = 0 define the null cone of z. It is more convenient to use a different qi(111.m il to measure the invariant distance between the source point and the field point, a (qil.,li i l' that is, however, related to s(x, z). That quantity is Synge's [16] world function and is defined by (x, z) s 2(x,z). (3.4) 2 By using the defining equations for s one can deduce that a has the properties 1 1 V'u VcUa VO'a V/,a 2 2 (3.5) (3.6) limr a 0. XZ Also, a is positive for spacelike intervals and negative for timelike intervals. Another very significant nonlocal q(1.,,l il'v is the bivector of geodesic parallel displacement, denoted by g,0,(x, z). The defining equations for it are V3g",' V3T 0, V3'g0' Va (T 0 lim g9' XZ (3.7) (3.8) S"' Oa Equation (3.7) signifies that the covariant derivatives of gaa' are equal to zero in the directions tangent to the geodesic joining x and z. Equation (3.8) expresses the fact that g,,, is equal to the Kronecker6 when x = z. The bivector of geodesic parallel displacement also has the property that 9aa' (x, z) ga'a(z,x). (3.9) In the following, the determinant of ga', is denoted by g I ga, I, 6 1 9 . (3.10) (3.11) The effect of applying the bivector of geodesic parallel displacement to a local vector A"' at z is a parallel transport of that vector from point z to point z, along the and geodesic that connects the two points. The result is a local vector A" at point x. In general, g,,, can be used to transform any bitensor that has indices referring to the two points x and z to a tensor whose indices refer to one point only, for example 9, 9, T3' 1T36 T T (3.12) How useful the bivector of geodesic parallel displacement is in studying nonlocal pror erties of spacetime can be understood if expansions of a bitensor about one point are considered. In order for a bitensor to be expanded about a certain point in spacetime, all its indices must refer to that specific point. For bitensors for which that is not the case, their indices must first be homogenized by applying g'', and then the expansion about the specific point can be taken. A bivector that is very useful for the Hadamard [17] expansion of the Green's functions is defined by Daa'(x, z) = VaV, T(x, z) (3.13) and a biscalar relating to it is its determinant D= D,  (3.14) DeWitt and Brehme proved that lim D ',, (x, z) = gaa (z) (3.15) which shows that the biscalar D is i'ii, v.iilii. at least when x and z are close to each other. In fact, D is the Jacobian of the transformation from the set of variables {zW', X"}, which specify the geodesic between x and z in terms of its two end points, to the set of variables {z", Vc'a}, which specify the geodesic in terms of one of its end points and the tangent to the geodesic at that end point. DeWitt and Brehme also showed that D obeys the differential equation D1 V0(D Va) =4. (3.16) Instead of the biscalar D, a different biscalar is used in the Hadamard expansions of the Green's functions. That biscalar is denoted by A and is defined as A(x, z) =[g(x, z)]D(x, z). (3.17) DeWitt and Brehme proved that A can be expanded in terms of derivatives of a and that the expansion is A 1+ R Voa Va + 0(s3), for x _ z (3.18) where R" is the background Ricci tensor. 3.2 Green's Functions The goal of the work of DeWitt and Brehme was to study the radiation damping effects on a particle of a given electric charge that moves in curved spacetime. To do that it is necessary to study the vector field that represents the electromagnetic potential created by the particle. However, the equations for the scalar field are less complicated and thus easier to deal with. Also, the lack of indices makes the results more transparent. For those reasons the scalar Green's functions and their corresponding fields are discussed first. 3.2.1 Green's Functions for a Scalar Field A point particle of scalar charge q which is moving on a worldline F : z"'(r), where T is the proper time along the geodesic, creates a scalar field y which obeys Poisson's equation: V2 = 4 7 Q. (3.19) In Equation (3.19), Q is the source function for the point particle. Specifically L(y) =q (g) 64(y z(r)) dr (3.20) where y is some point in spacetime. It is desired to express the different solutions of Equation (3.19) as integrals containing Green's functions, namely (3.21) so the properties of the various Green's functions are emphasized in this section. One function of importance is the symmetric Green's function, Gsym(x, z), which satisfies the inhomogeneous differential equation (3.22) V2Gsym(x, z) The Hadamard form [17] of this function is 1 Gsym(x, z) [U(x, z) 6(a) V(x, z) ((a)] 87 (3.23) where 6 is the step function that equals 1 if the argument is greater than zero and equals 0 otherwise. U(x, z) and V(x, z) are biscalars that are free of singularities and symmetric under the interchange of x and z, namely U(x, z) = U(z, x) (x) 4 I (g) G(x, y) Q(y) dy4 J spacetime 4 q/ G[x, z(T)] dr, (3.24) (g) 64(X z). and V(x, z) V(z, x). (3.25) They can be determined by expanding the solution of Equation (3.22) in powers of a, in the vicinity of the geodesic F. The biscalar U(x, z) satisfies the differential equation 1 U1(x, z) VoU(x, z) = A1(x, z) VoA(x, z) (3.26) 2 and the boundary condition lim U(x,z) 1 (3.27) XZ and is given by U(x,z) =[A(x,z)) 1 (3.28) =1 + R, Var Va + O(S3), x  F 12 where s is the proper distance from the point x to F measured along the spatial geodesic which is orthogonal to F. The biscalar V(x, z) satisfies the homogeneous differential equation V2V(x, z) 0 (3.29) and is given by 12 V(x, z) Rz) + 0(s), x > P. (3.30) It is noteworthy that the symmetric Green's function vanishes for a > 0, that is for spacelike separation of the points x and z. Also, because both U(x, z) and V(x, z) are symmetric under the interchange of x and z, so is Gsym: Gsym(x, z) =Gsym(z, x). (3.31) If & = d((x, z(T))/dT, the scalar field ,''r(x) associated with the symmetric Green's function is W x) = [U(x, z) 6(a) V(x, z) G(1)] dr 2 j(3.32) [qU(x, z) [qU(x,z) q( et +j ( 3. L 2 Jret+ 2 ] ,v 2 _v and consists of two parts. The first part is the one that contains the biscalar U(x, z) and the 6function 6(a) and is referred to as the direct part. This is the term that corresponds to a = 0, which is the part of the field that comes from the retarded and advanced proper times (Tret and Tadv respectively), namely the proper times that correspond to the intersection of the geodesic F with the past and future null cone of the point x. In other words, the direct part has the same singularity on the null cone that the symmetric scalar field has in flat spacetime. The second part is the one that contains the biscalar V(x, z) and the stepfunction O(a) and is referred to as the tail part. This term gives the part of the field that corresponds to a < 0, which is the part that comes from the interior of the past and future null cone of x. It is the part of the field that is due to the curvature of spacetime and vanishes in flat spacetime. The symmetric Green's function can be separated into the retarded and ad vanced parts, which constitute two very important Green's functions themselves. Specifically it can be written as 1 Gsym(x, z) = [Gret(, z) + Gadv(, z) (3.33) 2 where the retarded and advanced Green's functions are given by Gret(x, z) = 2 [f(x), z] Gsym(x, z), (3.34) Gadv(x, z) = 2 [, Z(x)] Gsym(x, z). (3.35) In Equations (3.34) and (3.35), E(x) is any spacelike hypersurface that contains x. The stepfunction O[Z(x), z] = 1 O[z, Z(x)] is equal to 1 when z lies to the past of E(x) and vanishes when z lies to the future of E(x). Both the retarded and the advanced Green's functions satisfy the inhomogeneous differential equation V2Gret(x, z) V2Gadv(,) _(g) 64(x, z). (3.36) They also have the property that Gret(x, z) Gadv(z, x). (3.37) The retarded and advanced scalar fields, both solutions of the inhomogeneous differential Equation (3.19), can also be expressed as integrals of the respective Green's functions '(x) = 4 q Gret [x, z(r)] d = 2 z) q V(, z) dr, (3.38) 1 2 J(r 1} r+00 adv (x) = 4 r q Gadv[x, Zr) dr 2 q z) q V(x, z) d. (3.39) 1 2( I dv Jdv The retarded field '/" (x) is the actual scalar field that results from the scalar particle and is singular at the location of the particle. The last scalar Green's function that is significant for the analysis that follows is the radiative Green's function Grad, which is defined in terms of the retarded and advanced Green's functions in a manner analogous to the radiation field defined by Dirac in flat spacetime, namely Grad(x, z) = Get(x, z) Gadv(x, z). (3.40) It is obvious by this definition that Grad(x, z) satisfies the homogeneous differential equation V2Grad(x, z) =0. (3.41) By Equation (3.37) one can infer that Grad(x, z) is antisymmetric under the inter change of its arguments Grad(x, z)= CGad(z, x). (3.42) The corresponding radiation field equals the difference between the retarded and advanced fields ."() = ,.r (x) dv(x) 4 i q Grad[x, z(r)] dr (3.43) and is a sourceless field because it satisfies the homogeneous differential equation V2 1,'r (x) 0. (3.44) Consequently, ,' is smooth and differentiable everywhere in space. The Green's functions and the corresponding scalar fields that were mentioned in this section are used extensively in Chapter 4, where the selfforce on scalar particles is discussed. 3.2.2 Green's Functions for a Vector Field Now that the scalar Green's functions and their properties have been discussed, it is straightforward to define analogous Green's functions for the electromagnetic vector potential A" generated by an electrically charged particle moving on a geodesic F: z"'(r). Assuming that A" satisfies the Lorentz gauge condition V7A0 = 0, (3.45) Maxwell's equations for it become V2Ac R`A3 4 Jo, (3.46) and, again, the goal is to express the vector potential as an integral that contains a Green's function and the source J", specifically AX) s4 peacetime (g)i GC,(x, y) J(y) d4y. (3.47) The symmetric Green's function for a vector field obeys the equation V2Gasym (, z) R 3 Gsym' (x, ;) 9 2 g ,,a 64(, z) and the Hadamard form of it is 1 [uI ,(x, z) 6(a) 87 V1, (x, z) G( The biscalar U,,, is given by the differential equation (2 V3U,, + 00, A1 V3A) V,3a with the boundary condition lim U", (x, z) = g0, (z). xz As shown in [14], the solution is unique Ua,(x, Z) A /( Z) g00'(x, Z) S[1+ R3'7' V7a V7y + O(s3)] g,, for x z 12 (3.48) Gsym (x, z) aa' ^ (3.49) (3.50) (3.51) (3.52) where Equation (3.18) was used to derive the final expression. For the bivector V,,, DeWitt and Brehme prove that 1 1 lim V0 '(x, z) = g (I R gg R) + 0(s), for x > F. (3.53) xz 2 6 The symmetric electromagnetic vector potential is calculated by A (x) 47 (g) Gs(x, y) J(y) d4y i() g [A(x, y) ga(x, y) 6(a) V0(x,y) ()} J((y) d4y (3.54) Just as in the case of the symmetric scalar field, the symmetric electromagnetic poten tial also consists of the direct part, that contains the 6function and comes from the retarded and advanced proper times, and the tail part, that contains the Ofunction and is the contribution from within the past and future null cone. An interesting fea ture is the appearance of the bivector of geodesic parallel displacement in the direct part. It signifies that the electromagnetic radiation is parallel propagated along the null geodesic that connects the points x and z. The symmetric Green's function can again be separated into the retarded and advanced parts Gsym(x, ) I [G ret(, (advx, z) (3.55) aa' \*r z! 2 9 [^aa'^i z) aa' \ 0.00 which are given by the equations ret (x, z) 2 6[E(x), z] Gsy(x, z) (3.56) ~7LLYI~L ~1 IV ii~~l /i U~~y \LU ,i (3.57) Gadv, z) = 2 8[z, (x)] Gsym(, (z). They are solutions of the inhomogeneous differential equation V72G e(x, zI ret,(, z) V2C /redv/ (x R adv/, z) G Ilaail y z) l\a L v Gaa' z) a Ra3Ga' {ZL  g 2 ga(, z)J (x, z) (3.58) and have the property that (3.59) The retarded and advanced electromagnetic potentials are then given by A t (x)= 4 p J spacetime (gGre (x, y) J (y) d4 y (3.60) Av (x) 4 (g GC (x,y) J3(y) dy (3.61) / spacetime and Art(x) is the actual potential generated by the moving charged particle. The radiative Green's function is defined in terms of the retarded and advanced Green's functions by the equation rtad/ )\ _tret /(X, Z)\ adv \ aa' X, (xz) a' )z) Gaa (, z) and satisfies the homogeneous differential equation (3.62) V2Gad (, z) G,,, (X, Z) 3 rad (x, z) 0. a 03a'. Using the symmetry expressed in Equation (3.59) for the retarded and advanced Green's functions it can be inferred that Girad (X, Z) = rad ) aLrI\X, Z ,,,Z, X. (3.64) (3.63) G ret (X, ) Gadv( x) The corresponding radiation vector potential is equal to the difference between the retarded and advanced vector potentials Ad(x) A t(x) A"ad() (3.65) Sg)l x, y) j(y) d4 (3.65) Sspacetirme and is a sourcefree field, because it satisfies the homogeneous differential equation V2A d ) R aA;d(x) = 0. (3.66) The important quantities for the motion of a charged particle are not the vec tor potentials but rather the electromagnetic fields. The electromagnetic field that corresponds to a vector potential A0 is calculated by: F,3(x) = VA,(x) V7AP(x). (3.67) For completeness, the definitions of the symmetric and radiation fields are given, in terms of the retarded and advanced fields: sym /(Frt +F adv (3.68) Frad Fret Fadv (3.69) 3.2.3 Green's Functions for a Gravitational Field Even though the work of DeWitt and Brehme did not cover the motion of a massive particle creating its own gravitational field, the Green's functions for a gravitational field are presented here, for completeness. A more detailed description is given by Mino, Sasaki and Tanaka in [15] and it is that analysis that is followed here. Assuming that a massive pointlike particle is moving on a geodesic F : z"'(r) and is inducing a perturbation h,,(x) on the background, the tracereversed metric perturbation is (3.70) where h(x) and h(x) are the traces of hP"(x) and of hP"(x) respectively. It is assumed that the tracereversed metric perturbation obeys the harmonic gauge condition Vh"(x) = 0. (3.71) In this gauge, the linearized Einstein equations [18] become: V2h + 2Rs3 h76 167rTa (3.72) to first order in the metric perturbation. The Green's functions of interest in this case are bitensors and are used to express the tracereversed metric perturbation in integral form h3(x) = 167 time / spacetime (g) G':S(x, y) T, (y) d4y. The symmetric Green's function G03'y' (x, z) satisfies the differential equation V2GSM (x, z) +2 Rs 0 (x) Gy 6 (x, z) Psym / V sym Y 2 g'((x, z) g3'(x,z) 64 Z) (3.74) (g)2 and the Hadamard form for it is yG'('x, ) [Uay's'(x, z)6(a~) Vapy's'(x, z)()].8 sym 8x (3.73) (3.75) hy,(x) = hy,(x) g[v,,(x)h(x) The bitensor U8Y'3'' (x, z) is the solution of the homogeneous differential equation [2 V7 U`y3' (x, z) + py' (, z)] VAo A (3.76) with the boundary condition (3.77) Mino, Sasaki and Tanaka [15] prove that the solution is 2 g7'((x, z) g)' (x, z) A( z). (3.78) The bitensor V3P'6' (x, z) is divergencefree V3Vo'~''(x, z) 0 and satisfies the homogeneous differential equation V2Vy'5' (, z) + 2 c Vvy7''(x, z) (3.79) (3.80) with the boundary condition (3.81) It is proven in [15] that the solution is 9g' R,1 ~ ,'a'(z) + 0(r), for x + F. limn U,37'5' (x, z) lim 2g"(' (x, z) g9)"'(x, z). Xzz Ucpy'Gs'(x, z) lim VP,7's' (, z) o(>0 Voap,3y' (x, z) (3.82) The symmetric gravitational field is obtained by hpm(x) = 167 g (g G's' (x, y) T6 ^(y) d4y Sspacetime = 2 (9) [Uo^(x, y)6(a) VoyS'(x, y) (T)] T7s(y) d4y J spacetime (3.83) Its direct part is the part that contains the 6function and gives the contribution from the null cone of point x. Its tail part is the part that contains the Ofunction and gives the contribution that comes from within the null cone of point x. The symmetric Green's function can be separated to the retarded and advanced Green's functions a 'S'(,, z) = [G%,t'(x, z) + G '6 (x, z)1 sym G' t *v ) which are given by the equations (3.84) G '(x, z) = 2 e[Z(x), z] G`s'y(x, z) Gad"''(x, z) = 2 e[z, Z(x)] G'7'(x, z). (3.85) (3.86) They are solutions of the inhomogeneous differential equation V2G'/y' s(x, z) +2 RC3 (x) G"' '(x, z) V2G Cd' (x, z) + 2 R ,(x) G"d'v (x, z) 2 g((x, Z) g)1'(X, Z) 64  (9g) (3.87) and have the property that (3.88) G'ret (x, z) G" 6(z) The retarded and advanced gravitational fields are given by hret = 6 (g G (x, y) Ts(y) y (3.89) spacetime 16r ( g) C v(x,T y) T7 (y) d4y (3.90) spacetime and hr' is the actual tracereversed gravitational perturbation induced by the moving particle. The radiative Green's function is defined in terms of the retarded and advanced Green's functions by the equation Ga'(x, z) G'y''(x, z) G`p' (x, z) (3.91) "rad ' ret  adv r , and satisfies the homogeneous differential equation V2G'5'(x, z) + 2 R (x) G^""'(x, z) 0. (3.92) Using the symmetry expressed in Equation (3.88) for the retarded and advanced Green's functions, a symmetry property for the radiative Green's function can be derived, namely Goy6'(, z) G Y' (z, x). (3.93) Finally, the radiative gravitational perturbation is given by the difference be tween the retarded and advanced gravitational perturbations ha13 (X h a,(X ha,3 (X'i ra".d) ret) adv(W 2 ((3.94) =167 (g) 'G ,)T(t)d4 Jspacetime and is a solution of the homogeneous differential equation V2r7/ ) 2 R d (x) = 0. (3.95) 3.3 Motion of a Charged Particle in Curved Spacetime Following DeWitt and Brehme's analysis, I present the equations of motion and the radiation reaction effects for a particle carrying an electric charge e. Studying the vector case is useful in comparing the result derived in curved spacetime with Dirac's result for flat spacetime. Similar analyses can be done for the scalar case and the gravitational case. Those analyses can be found in a recent review of the subject by Poisson [19]. 3.3.1 Equations of Motion Let's assume that a particle of bare mass mo and electric charge e is moving in a curved spacetime of metric g,, on a worldline described by F : z(r), where T is the proper time. In the following, the notation u' = (dz"'/dr) = z"' is used. Also, the symbol (D/dr) is used to denote absolute covariant differentiation with respect to the proper time r along the worldline. The particle generates an electromagnetic vector potential A"(x) and an elec tromagnetic field Fo3(x) given by Equation (3.67). The action of the system contains three terms; the first term comes from the particle, the second from the electromag netic field and the third from the interaction between the two: S Sparticle Sfield Sinteraction, (3.96) where Sparticle =Mo j [g0,a' (z) U' U' d (3.97) Sfield Fp Fp3 g d4x (3.98) 167 Sinteraction = e A, (z) ua' dT. (3.99) In determining the equations of motion, the stationary action principle can be applied to S for variations in the potential A" and for variations in the worldline z"' separately. Varying A" gives V3Fc3 4 7J" (3.100) where J" is the current density defined as JP(y) e g,(y, z) u'64(y,z) d. (3.101) The electromagnetic field Fcp is invariant under a gauge transformation of the form An  An + VcA(x), where A(x) is any scalar function of x. For convenience, A(x) can be chosen in such a way that the electromagnetic potential satisfies the Lorentz gauge condition VoA" 0 (3.102) in which case Equation (3.100) for the electromagnetic field becomes V2AS(x) R]A3(x) = 4Jc(x). (3.103) Equation (3.103) allows one to determine the electromagnetic field, once the worldline of the charged particle is known. On the other hand, varying the worldline zc' gives Dua' mo = eF c (z) u '. (3.104) (IT(ZU Equation (3.104) allows one to calculate the worldline of the particle, once the elec tromagnetic field is specified. As is wellknown, the electromagnetic field given by Equation (3.103) is diver gent at the location of the particle and cannot be used in Equation (3.104). An alternative method for studying the motion of the particle is to take advantage of the conservation of energy and momentum for the particle and the field, the same method that was used by Dirac in his analysis of the motion of an electron in flat spacetime. 3.3.2 Calculation of the Fields It becomes obvious in Section (3.3.3) that in order for the conservation of energy and momentum to be implemented along the worldline of the charged particle, it is necessary for the retarded and advanced fields generated by the particle to be calculated. A brief description of the calculation is given here; the details can be found in [14]. In the following, Tret and Tadv are the proper times along the worldline of the particle, where the past and future null cones of x respectively intersect that worldline. They are referred to as retarded and advanced proper times. Also, re is the proper time at the point where the hypersurface E(z) intersects the worldline of the particle. All these points are shown in Figure (3.1). 'ladv x Z(x) Sret null cone Z Figure 3.1: Retarded and advanced proper times Starting with Equation (3.60) for the retarded potential, and using the Hada mard expansion for the Green's function, a simple expression for A"t can be obtained A" (x) 47 d4y (g)i G (xy) J3(y) 47re d4 y( g) 2 e[Y(x), y] [Ua3(x, y) (T) Va (x, y) ()] x J ~8r x dT g ,(y, z) ua' 64(yz) =e JX Uo (x, z) 6(a) u' dT ef V0&(x, z) 0( ) un' dT nil 1' f1 Tret 7 Tr et e Vl,'(x, z) uC dT. LV pOU J ret Joo (3.105) By a similar calculation, starting with Equation (3.61), the advanced potential can be calculated av UC' r+o00 A (dv e, e Va (x, ) u dr. (3.106) LVa'o T v dv Equations (3.105) and (3.106) give the covariant LienardWiechert potentials. The retarded and advanced electromagnetic fields can be calculated by Equation (3.67) if the respective potentials are differentiated. The result is a f Du8' FUVet/adv J ,V u' (Y'u + u7 V Du d ) (V77 uT')3 [V,(U.,Va UpV,a) uCu + (UviV UU U Va) ] dT x (V77,( u')2 + (VUc, VUlc, + V'CV ,a ,Va) n' (V/,a T ')1 STret / dv SeI T (V,,V ,V,, ,) u'/ dT. 7'3et /adv (3.107) In Equation (3.107), the lower of the two signs corresponds to the retarded field and the upper sign corresponds to the advanced field. This is an important expression because it gives the electromagnetic field at point x in terms of the characteristics of the worldline of the charged particle. The significance of this becomes clear when the final equations of motion of the charged particle are written down. As becomes obvious in Section (3.3.3), when calculating the balance of energy and momentum between the particle and the field along the worldline, the values of the fields very close to the worldline need to be known. Expanding the retarded and advanced fields in powers of a and its derivatives is a tedious but straightforward calculation, which yields a complicated expression for the two fields, Equation (5.12) of [14]. Following Dirac's ideas, the total field in the vicinity of the charged particle can be written as the sum of the retarded field created by the particle plus the external incoming field F,= F + F, (3.108) which, by using Equations (3.68) and (3.69), can be written in terms of the radiation and symmetric fields SF + + F sm. (3.109) This breaking up of the total field is very important because, after implementing the conservation of energy and momentum along the worldline of the particle, only the radiation field F rd and the external incoming field Fc, show up in the equations of motion of the charged particle. Since the radiation field is the difference between the retarded and advanced fields, only two terms of Equation (3.107) are of interest here. Specifically, it is sufficient to write Fret/adv C (gpag '31) 2  ( 2 D ' f 1, ( (T), (T')) d (3.110) 2 J, + Ftra + (higher order terms) where, again, the lower sign corresponds to the retarded field and the upper sign corresponds to the advanced field. In Equation (3.110), the notation fa0',,(X, z) = Vlrl(Xz, z) V Val ,(, z) (3.111) is used. The term Fetr is, in fact, a sum of terms and is the same for the retarded and the advanced fields, so it vanishes when the difference of the two is taken. 3.3.3 Conservation of Energy and Momentum The first step, when considering the conservation of energy and momentum, is the construction of a cylindrical surface that surrounds the worldline of the charged particle. That surface is referred to as the worldtube from now on. As in Dirac's analysis, the radius c of the worldtube is very small, because the balance of energy and momentum between the particle and the field needs to be calculated in the vicinity of the particle's trajectory. DeWitt and Brehme [14] explain the procedure for the construction of the worldtube in great detail. For the needs of the present calculation, the worldtube can be thought of as the threedimensional surface that is generated by the particle if the particle is surrounded by a sphere of a very small radius and is then left to move along its worldline, starting at proper time Tr and ending at proper time T2. In the following, the cylindrical surface is denoted by E, its two end caps are denoted by Ei (corresponding to the earliest proper time Ti) and E2 (corresponding to the latest proper time 72). The total volume enclosed by the worldtube is denoted by Vt. The stressenergy tensor T"' of the system consists of one part that comes from the particle and one part that comes from the field To T + T where TP =mof g cg u"'u' 64 dr (3.113) 1 1 T (F4 F, g.'3FF6) (3.114) 47 7 4 where Fp is the total field in the vicinity of the particle. It is the sum of the field generated by the particle plus any external incoming field that is present. The conservation of energy and momentum for the system can be expressed as VTc3 0. (3.115) When integrating Equation (3.115) over the worldtube, special attention should be given to the fact that the integral (f VpTc3d4x) is not a vector in the curved spacetime that is being considered here. That means that Gauss's theorem cannot be used as usual, to convert the integral over the volume Vt to an integral over the surface of the worldtube. The natural way to overcome this difficulty is to consider the integral f g.'VpTP, d4x instead. In this integral, the bivector of geodesic parallel displacement is used, so that any contributions to the integral by the point x are referred back to the fixed point z, which is assumed to correspond to proper time T. That integral is, then, a contravariant vector at point z, so Gauss's theorem can be used. That gives (f 12 Tt d f (v +if) T3 d4x = 0 (3.116) where dYE is the surface element along the surface of the worldtube. The limit of the radius of the worldtube going to zero is taken next. Also the integral over the surface Y of the worldtube is expressed as a double integral, over the proper time and over the solid angle. In addition, the proper times Tr and T2 are let to approach r. If their infinitesimal difference is denoted by dr, Equation (3.116) becomes mo dr + lim go'T"3dE = 0. (3.117) It is in the calculation of the integral of the second term that the symmetric field and the radiation field mentioned in Equations (3.68), (3.69) and (3.109) are very useful. As was mentioned earlier, the result contains only the incoming external field, the radiation field and the tail field and is Duo' 1 1 ' mD = eFin a +eFrad, a' [12 z(T)) '(T')dT' (3.118) dT 2 2o where m is the renormalized observed mass of the charged particle m mo+ lim Ie 1 (3.119) eo 2 An equivalent expression is derived by using Equation (3.110) and the definition of the radiation field given in Equation (3.69), and it is Duo' 2 D2 U' Du Dup, m = eFin ,' + e2 U d1r 3 dr2 dr dr (3.120) +2 fu f ,(Z(T), Z(T')) ')d'. ~I J0o Equation (3.120) contains only the ].li',i .,1 characteristics of the worldline of the charged particle and the incoming external field. 3.3.4 Radiation Reaction A short discussion about Equations (3.118) and (3.120) is in order. First it is significant to notice that, except for the last term that contains the (Iil.,,I il' fa0a', these equations are identical to Equations (2.34) and (2.35) derived by Dirac for flat spacetime. Since that term contains the biscalar V,,,, it comes from the tail part of the retarded electromagnetic field. Also, it is interesting that the integral that shows up in these two equations needs to be calculated from oo to the proper time T, meaning that knowledge of the entire past history of the particle is required. That is something that renders the use of Equation (3.120) impractical and is a point that is also discussed in the following chapters. CHAPTER 4 SELFFORCE In general, the selfforce can be due to a scalar, an electromagnetic or a gravita tional field. Two different methods for the selfforce calculation are presented in this chapter. The analysis is given in detail for the scalar selfforce, due to the simplicity of the notation. The results are also given for the electromagnetic and gravitational selfforce. The first method, in which the direct and tail parts of the retarded fields are used, was described by DeWitt and Brehme in [14] for the scalar and electromagnetic fields and by Mino, Sasaki and Tanaka in [15] for the gravitational field. An axiomatic approach to this method was also presented by Quinn [20] for the scalar field and by Quinn and Wald [21] for the electromagnetic and gravitational fields. The second method, in which the singularsource part and the regular remainder of the retarded fields are used, was proposed by Detweiler and Whiting in [22], where they described it for the scalar, electromagnetic and gravitational fields. As in Chapter 3, z denotes a point along the worldline of the moving particle and x any point in spacetime. The limit of x > F is the coincidence limit. The retarded and advanced proper times, Tret and Tadv, are the proper times at the points where the null cone of x intersects the worldline F (see Figure (3.1)). Primed indices refer to z and unprimed ones refer to x. 4.1 Scalar SelfForce A particle of scalar charge q is assumed to be moving in a background spacetime, described by the background metric gab. For simplicity it can be assumed that there is no external scalar field and consequently, if the scalar charge q is small, the lowest order approximation for the worldline F : z''(T) of the particle is a background geodesic, where T is the proper time. But the scalar field y generated by the particle interacts with the particle, inducing a selfforce on it. The selfforce gives F an acceleration, so, to higher order, the particle does not trace a background geodesic. 4.1.1 Direct and Tail Fields The discussion of the scalar field selfforce in [14] and [20] shows that the self force on the particle can be calculated by I 7 particle (4.1) where .,~" is the scalar field that interacts with the particle and is given by SqU(x z) q V(x (T)) d. (4.2) L 2 \7ret 0_ It can be inferred from the discussion of the scalar Green's functions of Section (3.2.1) that the first term of the righthand side of Equation (4.2) comes from the direct part of the retarded field generated by the moving particle. This first term is finite and differentiable at the location of the particle and its contribution to the selfforce is the curvedspacetime generalization of the AbrahamLorentzDirac force of flat spacetime. This contribution results from the moving particle's acceleration, if the worldline F is not a geodesic, and can be expressed in terms of the acceleration of F and components of the Riemann tensor [14, 15, 20, 21]. The integral term of the righthand side of Equation (4.2) comes from the tail part of the retarded field of the moving particle. Its contribution to the selfforce represents the result of the scattering of the retarded field of the particle, due to the curvature of spacetime. Taking the derivative of the tail part of ", gives one term that comes from the implicit dependence of the retarded proper time on x and one term that comes from the dependence of the integrand on x, as was calculated by Quinn in [20]: STrrtet = q V(x, z(Tret)) [Varret(x)] q (V, V(x, z(r))} dr 7t (4.3) =q V 'rVa o q rV, V(x, z(7)) } dr ( Iret J_ O c q R(x\ ret S(x z) +O(r) q {Va V(x, z(r)) } dr. In Equation (4.3), r is the proper distance from x to F measured along the spatial geodesic that is orthogonal to F. The spatial part of the first term of the derivative calculated in Equation (4.3) is not well defined when x is on F, unless R(x) = 0 there. That makes this approach problematic since, for the selfforce to be calculated, the derivative must be calculated at the location of the particle. To overcome this difficulty, V.' " can be averaged over a small spatial 2sphere surrounding the particle, thus removing the spatial part. Then the limit of the radius of the 2sphere going to zero can be taken and a finite contribution to the selfforce is obtained. Another complication comes from the fact that, in order for the contribution of the tail term to the selfforce to be calculated, knowledge of the entire past history of the moving particle is necessary, as is clear from the integral term in Equation (4.3). It is also important to note that the field just like the field ',' is not a llvi, illyy wellunderstood field. That is because it is not a solution to any particular differential equation. 4.1.2 The SField and the RField In order to provide an alternative scalar field that gives the selfforce on the moving particle, Detweiler and Whiting [22] took advantage of the fact that adding a homogeneous solution of a differential equation to an inhomogeneous solution of the same differential equation, gives a new inhomogeneous solution. Specifically, since the biscalar V(x, z) is a symmetric homogeneous solution of the scalar differential Equation (3.29), it can be added to the symmetric Green's function GsYm(x, z) to give a new symmetric inhomogeneous solution GS(x, z) of Equation (3.22) Gs(x, z) = Gsy(, ) + V(, z) 87 [U(x, z)6(a) V(x, z)O(a) + V(x, z)] (4.4) 87r 1 [U(, z)6(a) + V(x, z)e(a )], 87 which obeys V2GS(x, z)= (g)4(z ). (4.5) The superscript S is used to indicate that this Green's function obeys the differential equation that contains the Source term. This new symmetric Green's function has support on the null cone, coming from the term that contains the 6function, and outside of the null cone, at spacelike separated points, coming from the term that contains O(a). It has no support within the null cone, as shown in Figure (4.1). The local expansion of the biscalar V(x, z) given in Equation (3.30) is sufficient for the purposes of this dissertation, since the singular Green's function is only used for points x close to the worldline of the particle. The RegularRemainder Green's function is defined in terms of the retarded and singular Green's functions as GR(x, z) = Get(x, z) Gs(x, z) = I {2e[E(x), z][U(x, z)6(a) V(x, z)O(a)] [U(x, z)6(a) + V(x, z)0(a)]}, (4.6) Tadv Gail direct  and GS Tret null cone Z Figure 4.1: Support of Gdirect, Gtail and Gs. and it is obvious that, by construction, it obeys the homogeneous differential equation V2GR(x, z) =0. (4.7) Clearly, GR(x, z) has no support inside the future null cone of x. Using the new Green's functions, the fields ys and 9R can be defined. First, the singular field is s(x) = 4q fGS[x, z(T)] dT [qU(x,z) [q U(xz) dv q ad 2(a o ,et r 2o e Tqtdv and obeys Poisson's equation (4.8) V(z, z) dr, V2yS(x) = 4(L. (4.9) It is noteworthy that the singular field does not depend on the entire past history of the moving particle, but only on its motion between the retarded and advanced proper times. The regularremainder is defined as (Wx)=,~X) S(Wx) [qU(x,z) Tdv 2 7etro (4.10) FTret f1 Tadv q + V(x, z) dr, 0 2 2Tret and by definition obeys the homogeneous differential equation V2R(x) = 0. (4.11) Since the field yR is a sourcefree field, it is smooth and differentiable at any point in spacetime and consequently at any point along the worldline of the particle as well. That is the most significant property of it, as far as the calculation of the selfforce is concerned. To compare the scalar fields ,, and R, the difference of the two is calculated from Equations (4.10) and (4.2) R ) `(X) q /JTadv 2 t V(x, z) dr. (4.12) Using Equation (3.30) for the biscalar V(x, z) to expand the above integrand in the coincidence limit, the difference of the two fields becomes R(x) + O(r)]dT 2 1ret 12 12 (4.13) R(.) ,.  (X) where the fact that the difference between the advanced and the retarded proper times is equal to Tadv Tret 2r + 0(r2), for x > F (4.14) was used. Taking the derivative gives VaR V. .' + q Va[rR(x)] + (terms that vanish as x F). (4.15) 12 The second term of the righthand side of Equation (4.15) gives an outwardly pointing spatial unit vector and cancels the first term of the right side of Equation (4.3). Con sequently, the fields 9R and I'" give the same selfforce. In addition, the averaging procedure that was necessary for the calculation of the selfforce using , '" is not required when 9R is used. A selfforce calculation that has been demonstrated in [23] showed an additional benefit of calculating the selfforce using the difference of the retarded and singular fields. The retarded field can be computed numerically and, since the singular field depends only on the motion of the particle between the retarded and advanced proper times, the knowledge of the entire past history of the particle is not required. That is a significant advantage of this method compared to the one that involves the direct and tail fields. Additionally, it is important that all the scalar fields used in this analysis are specific solutions of the homogeneous or inhomogeneous Equation (3.19). That makes them welldefined pir, i .,1 fields, a property that the direct and tail scalar fields do not have. For those reasons, in this dissertation the selfforce is calculated using . = qV i 'p (4.16) 4.2 Electromagnetic SelfForce A detailed analysis for the selfforce in terms of the direct and tail electromag netic potentials was done in [14] and was presented in Chapter 3. In this section, the calculation of the selfforce using the singularsource and the regular remainder fields is presented. It is assumed that a particle of electric charge q is moving on a worldline F : za'(r) in a background described by the metric gab. For the purposes of this section it can be assumed that there is no external electromagnetic field and the lowest order approximation to the particle's motion is a background geodesic. The particle creates an electromagnetic potential AP and an electromagnetic field Fab, which interact with the particle, causing a selfforce to act on it and making its worldline to deviate from a background geodesic to order q2. In the Lorentz gauge VaAa 0, (4.17) the electromagnetic potential generated by the particle can be calculated by using Maxwell's equations V2A R" Ab 4Ja (4.18) and the different solutions of interest were described by the Green's functions of Section (3.2.2). If a potential A"f, analogous to ,", is used to calculate the selfforce in this case, its tail part is Aself (tail) (x) q Vab (X, Z()) Ub' dr. (4.19) OO Taking the derivative of it gives S'Tret (tail) q LVab Ub' VcT ret x q f (VbUb)dT q (b'd' b'd') b q 2 6 r  r o (4.20) 7c[Vab'Ub'] dr, where Equation (3.53) for the bivector Vab' was used to derive the last expression. It is obvious that the same problems that were encountered in the scalar case are encountered here as well. The first term is not welldefined unless (Rab' ga'b'/R)u b is zero at the particle's worldline, the entire past history of the moving particle needs to be known and the potential used in the selfforce calculation is not a solution of Maxwell's equations. To overcome those difficulties, the singular Green's function in the neighborhood of the particle is defined analogously to the scalar case Ga,(x, z): GsY(x, z) + VI (x,z) (aalx, 8) Vaa, (X, Z) 1 [Uaa(X, Z)((T) + Vaa (X, Z)G()]. 87r (4.21) It obeys the inhomogeneous differential equation V2G a,(2, Z) RabGsa, z) (4.22)  gaa' g 29aa164(2, z) and gives the singular electromagnetic potential As 47 (g) GS (,y)b(y)d4y, (4.23) which is a solution of the inhomogeneous Maxwell's Equations (4.18). By definition, both Gsb, and As have no support inside the past and future null cone of x. The V7Aself 7c, Aa regular remainder is defined by A(x) A t(x) Aa(x). (4.24) It has no support within the future null cone of x and it obeys the homogeneous Maxwell's equations. Consequently, it is smooth and differentiable everywhere in space, including every point along the worldline of the particle. The electromagnetic selfforce can be calculated by the equation = qg" (VA VbA) b (4.25) and is welldefined at the particle's location. It is stressed, again, that both As and A4 are welldefined solutions of Maxwell's equations. 4.3 Gravitational SelfForce It is now assumed that a particle of mass m is moving on a worldline F : za (r), in a background described by the metric gab. The particle causes a metric perturbation hab on the background metric. This perturbation obeys the harmonic gauge Vahab 0 (4.26) where hab is the tracereversed version of the metric perturbation 1 hab hab 2gabhc. (4.27) The linearized Einstein equations for it are V2hab 2Rcbdhcd= 16Tab. (4.28) This metric perturbation interacts with the particle, causing a selfforce to act on it and forcing its worldline to deviate from a background geodesic. If hs'f, which is not a solution of Einstein's equations, is used to calculate the selfforce, its tail part contains the bitensor Vabc'd' given in Equation (3.82). The contribution of this tail term to the derivative comes partly from the dependence of the proper time Tret on the point x. Specifically T et V f (tail) LM 'Vabc/d/'d'' Vfr] ie ri Vf [Vabc'd'U'ud] dT Ir f (4.29) M[g7' ag b a (f Z f)j m t Vabre VII/, ]dT.(4 L J Xr f _O Just like before, the first term is not welldefined at the particle's worldline, unless Rc'a'd'b'U'cUd' equals zero along the worldline. Also, knowledge of the entire past history of the particle is required in order to calculate the integral term. The singular Green's function Gabc'd'(x, z) in the neighborhood of the particle can be defined as S bc'd'(X z) = G(sm z) + Vabc'd'(X, Z) 1 8r (4.30) [Uabc'd' (, Z)6(T) Vabc'd' (, Z) (o)1. It gives the singular field h b which is an inhomogeneous solution of Equation (4.28) and has no support inside the past or future null cone of x hSb = 16 l(g)Gbd(x, y)Td(y)d4y. (4.31) The regular remainder is defined as h b(x) h x h (t). ab ab ha'~b Wl (4.32) It obeys the homogeneous differential equation V2 + 2RaCb d 0, (4.33) which means that it is smooth and differentiable everywhere in spacetime. Also, it has no support within the future null cone of x. Then the selfforce can be calculated by S= (gb_ + iab)icd(vR 5Vbhh) (4.34) which is welldefined at the location of the particle. It is important that both hR, and hs, are solutions of specific differential equations. A significant conclusion can be drawn at this point. As befits the problem, the selfforce is causing the particle to move on a geodesic of the metric (gab + hR), to order m2. This metric (gab + hR) is, itself, a homogeneous solution of the Einstein equations. An extended discussion of that point is given in [22] and [24] and is also presented in Chapter 8 of this dissertation. CHAPTER 5 SINGULAR FIELD FOR SCHWARZSCHILD GEODESICS It was seen in Chapter 4 that the singular field is important in calculating the selfforce. It is the part that must be subtracted from the retarded field to give the regular remainder, which is then differentiated to give the selfforce. It is also clear that, since the singular field obeys the inhomogeneous Poisson, Maxwell or Einstein equations, it depends on the worldline of the moving particle and on the background spacetime. The coordinates that are used in this chapter to caclulate the singular field are the ThorneHartleZhang coordinates, abbreviated from this point on as THZ coordinates. Those coordinates were initially introduced by Thorne and Hartle [25] and later extended by Zhang [26]. A short discussion on them is presented in Section (5.1) of this chapter. The detailed calculation of the singular field of a scalar charge q for geodesics in a Schwarzschild background was presented in [23]. A brief discussion about that field is given here for completeness, since the regularization parameters associated with it are calculated in Chapter 6. The singular field is also calculated for geodesics in a Schwarzschild background, for a dipole generating a scalar field, for an electric charge, an electric dipole and a magnetic dipole each generating its own electromagnetic field and for a spinning particle generating a gravitational field. For all these calculations, MAPLE and GRTENSOR were used extensively. The GRTENSOR code is explained in Section (5.8) and presented in Appendix A. In Sections (5.3) through (5.7), only the results that the code gives are presented. 5.1 ThorneHartleZhang Coordinates The Poisson, Maxwell and Einstein equations for the singular field assume a relatively simple form when they are written in a coordinate system in which the background spacetime looks as flat as possible. In the following, it is assumed that the particle is moving on a geodesic F in a vacuum background described by the metric gab. Also, R is a representative length scale of the background geometry, the smallest of the radius of curvature, the scale of inhomogeneities of the background and the time scale of curvature changes along F. A normal coordinate system can always be found so that, on the geodesic F, the metric and its first derivatives coincide with the Minkowski metric [18]. Such a normal coordinate system is not unique. The one used here is the THZ coordinate system and is used only locally, close to the worldline of the particle. Specifically it is assumed that the background metric close to the worldline of the particle can be written as gab = lab + Hab 4 (5.1) r ab + 2Hab + 3Hab + 0( ), where ]ab is the flat Minkowski metric in the THZ coordinates (t, x, y, z) and p2 2 +2 2. (5.2) Also 2HbdXadXb = xijj(dt2 + dxl) + 4kpqBqZPXidtdXk 2Habd a db__i 2(kpq k 20 2 + 5 21 L zxJxk #2ikX k [Xijpq k P pqI (5.3) p3 p4 3HabdXadXb O( )abdX adb + 0(R )idxidx. (5.4) In this and the following, a, b, c and d denote spacetime indices. The indices i, j, k, I, n,p and q are spatial indices and, to the order up to which the calculations are performed, they are raised and lowered by the 3dimensional flat space metric ij. The dot denotes differentiation with respect to the time t along the geodesic. Also, Qijk is the 3dimensional flat space antisymmetric LeviCivita tensor. If Hab consists of the terms given in Equation (5.3), the coordinates are second order THZ coordinates and are well defined up to the addition of arbitrary functions of O(+3). If Hab also includes the terms given in Equation (5.4), the coordinates are thirdorder THZ coordinates and are well defined up to the addition of arbitrary functions of O( ). The tensors S and B are spatial, symmetric and tracefree and their components are related to the Riemann tensor on the geodesic F by E Rtitj (5.5) ,i ~PiRpqqjt. (5.6) 2 They are of O(L) and their time derivatives are of O(). For the calculation of the singular fields that follows, only the first two terms of Equation (5.3) are included in gab. The remaining two terms are of 0() and must be included in a higherorder calculation which must also take into account terms coming from the 3Hab part of the metric. The 'gothic' form of the metric is also defined as gab Jgab g gga(.7 (5.7) and its difference from the Minkowski metric is Hb a g ab ab For the lowest order terms of Hab it can be found that 4 Hab 2Hab 3Hab (+ o( 4 where 2Htt 2i jxi 2 Htk 210 [ X iXk k i 2] 2H5 2 (ij)pq qk k I ep(i 8) X 2 2` Ox )C qkXpX q P. 2Ht o ( P3 0 ( P4 R 3 RO(4) s ^ O( ) . (5.8) (5.9) (5.10) (5.11) (5.12) (5.13) Using the simple symmetry properties of the tensors S and B, the following relationships can be shown for their components and the spatial THZ coordinates (x, y, z): k ki k Cijk l + Ceikl j ijk + ikl (5.14) (5.15) Cjkl i and (ijkSxl i+ ijkknk (eijk~lna + tijkaln k  eilkXj + k Xli) _ikl B Xj + Ckl i)X iklIknX + Cjkl1k in al (5.16) (5.17) These relationships are used in the following sections when calculating the singular fields for different sources to simplify the expressions for those fields. The calculation of the regularization parameters shown in Chapter 6 requires one to know the exact relationship between the THZ coordinates and the background Schwarzschild coordinates. A general procedure for finding that relationship for a given geodesic is to try to satisfy the three basic properties of the second order THZ coordinates: On F: the coordinate t measures the proper time along the geodesic and the spatial coordinates x, y and z are equal to 0. Also, gab = lab and the first derivatives of the metric vanish there. At linear, stationary order: Hi = 0(x3). The coordinates satisfy the harmonic gauge: ,gab = 0(x2). The details of this procedure are described in Appendix A of [23]. For circular orbits in a Schwarzschild background of mass M, the THZ coor dinates have been calculated in Appendix B of [23] using specific properties of the spherically symmetric background, which is much simpler than following the proce dure just described. Their functional relationships with the Schwarzschild coordinates are given here in order to facilitate the discussion of the calculation of the singular fields and the regularization parameters. The Schwarzschild coordinates are (ts, r, 0, q) and the Schwarzschild metric is ds2 = ( 1 ) dt2 (1 )Mdr12 r2d2 r2 sin2 Od2. (5.18) r F The circular orbit given by = Qts has orbital frequency equal to 2 = (Mro 3), at Schwarzschild radius ro. 58 First, the functions X, y, Z, which are Lie derived by the Killing vector =  + Q2, are chosen X (5.19) (1 2M y r sin(O) sin(O ts) o (5.20) ro 3M Z rocos(0). (5.21) Then the functions x, y, z, also Lie derived by the Killing vector ", and the function t are defined in terms of X, y, Z r sin 0 cos(9 Qts) ro (1 2M) ro X2 Y2o 3M z2 2 + ro 2Mj *2 M2 2 + y2( o 3A/ 2r(5 2M)(r, 3M) M [ (r2 r + 3M2) r5(1 2Mro 3M) 8(ro 2M) S(2 o 114r 8 123M2) 142 *  (28r2 114r . + 123 M2) + (14r2 28 014 0 + _(3r3 7 56(ro 2M )2 M2y2Z2(7ro 18M) 4(ro 2M) y rsin0sin(O 14 L S[ 2X M )(3ro 8M) + 3Z2 (r 2M)2] 48r .1 + 33M2) 4r2.1 + 337r .1 2 430M3) MZ4 (3r0o + 22M), 56 ro 3M) F024 2n(\ (5.22) (5.23) +2 o  3M+1 72 (ro 2M/ y 2 2 0M o 2(14r( 69r), 3M2) L 2Z2(r +y2(14r 69r .fT + 89M2) + 2Z2(ro 15M) 2M)(7ro 24M)], M r2a(1 2M SF V MZ z rcos0+ 2 [M) X2(2r o 3M) + 2(r 3M) + Z2(r 2M)] 2r (r o 3M) MXZ +14rM(1 2M)[MX2(13ro 19M) (5.24) 14r5(1 2M (ro 3M) y2(14r 36r .I + 9M2 )+ Z2(o 2M)(14ro 15M)], 3M i rQy To Qti F 2 ro0 3M MZ2l (ro M) + )/y2 o + M r(1 2M (ro 3M) 2 3(ro 2M) (5.25) QMXy QMX X2(r 2 11r _t + 11M2) 14r (r 2M)(ro 3M) r y2(13r2 45r .1t + 31M2) + Z2(13ro 5M)(ro 2M)]. Finally x x= cos(Qtts) sin(Qtts) (5.26) y = J sin(Qtts) + ycos(Qtts) where Qt Q 3M There are two coordinate systems of interest. The first system is (t, x, y, z) which is a noninertial coordinate system that corotates with the particle, meaning that the x axis always lines up the center of the black hole and the center of the particle. The y axis is always tangent to the spatially circular orbit and the z axis is always orthogonal to the orbital plane. As was already mentioned, for the spatial coordinates of this system it holds that: I x= Iy =~ Iz = 0. The second system is (t, x, y, z) and is a locally inertial and nonrotating system in the vicinity of F. However, when viewed far away from F these coordinates appear to be rotating due to Thomas precession, as is clear from the terms involving the sine and the cosine of (Qtts) in Equations (5.26). It is also noted that: L p2 IJ z L Vt = 0, but Lx, I~y and I~t are not equal to zero. This second system is used to calculate the singular fields. In order to avoid confusion of the field point x and the source point z with the THZ coordinates, the notation needs to be changed. The field point is denoted as p and its coordinates as xa and the source point is denoted as p' and its coordinates as x$,. 5.2 Scalar Field of a Charged Particle The singular field generated by a particle that carries a scalar charge q and is moving on a geodesic F : p'(r) (where, as usual, r is the proper time) of the Schwarzschild background must obey Poisson's equation V2S = 47 (5.27) where the V2 is written in the THZ coordinates and the source term is Q= q ( g) 4 p p'()) (dr. (5.28) That singular field is derived in [23] and is equal to is + O( ). (5.29) Here, instead of following the exact calculation of the singular field, which is performed in detail in [23] following a similar derivation in [27], I prove that it does indeed satisfy the scalar field equation to the order specified in Equation (5.29). The differential operator of the scalar field equation becomes, in THZ coordi nates: gV7aV 'iS a a, ab .'S) aa(Hb .)(5.30) (5.30) = rh.. ,.~j: Huiaa9s 2Hta(i at)ys Haitc.,t.. If the field s = (q/p) is substituted into this equation, the first term gives the expected 6function singularity and the last two terms vanish since p does not depend on the time t. An explicit calculation shows that for the second term, 2H3 x Dc rA; is equal to zero. The remainder 3H" gives a term that scales as O()). So )Vavv() = 4iq6"(x) + O( ), for 0. (5.31) p R4 R Consequently, for the remainder to be removed a term of O(1) must be added to q/p. So s q/p + O() is an inhomogeneous solution of the scalar wave equation and the error in this approximation is C2. 5.3 Scalar Field of a Dipole The calculation of the singular scalar field generated by a dipole moving on a geodesic in a Schwarzschild background is presented in this section. The dipole moment is assumed to have a random orientation and its THZ components are denoted as K = (0, K, Ky, K). In this and the following sections, the subscripts (or superscripts) (0), (1) and (2) are used to indicate the order of significance of each term or component. The subscript (0) refers to the most dominant contribution, the subscript (1) refers to the next most significant correction, which is calculated for the singular fields, and the subscript (2) refers to the next correction, the order of which is predicted for the singular fields. It is important to realize that multiplying two terms of order (1) does not necessarily give a term of order (2), because of the fact that there is no O() correction to the metric (see Equation (5.1)) and the fact that the first correction to the metric is of O() while the second is of O( ). The scalar field of a dipole can be thought of as having the form s ts o0) + 1) 2) (5.32) and it obeys the differential equation V(0+1+2) QS V+1+2) ( ) + 1) ,2)) 47ro (5.33) where the source term is given by K [64v (p p'())] g dr. (5.34) The zerothorder term is the scalar field generated by a dipole that is stationary at the origin of a Cartesian coordinate system and it obeys the lowest order differential equation Vo7o o) T 4 (5.35) It is equal to s Kx' Kx + Ky Kz () 3 (2 +2 + 2)3/2 (536) The firstorder term obeys the differential equation derived from Equation (5.33) (V2 () + 72 lS o V2 )I/1S 2 )TS (5.37) V(O) (1) (1IO) =0 = ) V (o)1) 1) 0)(7) which means that the Vt) of the zerothorder part of the field is the source term in the scalar differential equation for the firstorder part of the field. In general, that source term is expected to contain the S's and the B's and to give the firstorder correction coming from the dipole's motion on the Schwarzschild geodesic. In this case, Equation (5.37) gives that o2 a2 O2 (a2 2 4 2 (1>)= 0 (5.38) so the first order correction to the field can be set equal to zero. The source for the next order correction comes from the part of the V2 that is of O( i) acting on the zerothorder scalar field so) The differential equation is of the form ata J O) o( x aai o) O( 4 ). (5.39) That means that the next order term must be of O( ). Finally, the singular scalar field of a dipole moving on a Schwarzschild geodesic is equal to Ts Kx + Kyy + Kzz 3S + O( ). (5.40) P3 R3 5.4 Electromagnetic Potential of a Charged Particle In this section, the singular electromagnetic potential generated by a charge q moving on a Schwarzchild geodesic is calculated. Its general form is A = A +(o) + A(1)+ AS(2). (5.41) Since in the vacuum background the Ricci tensor is Rab = 0, (5.42) this electromagnetic potential, as well as all those calculated in this chapter, obey the vacuum Maxwell's equations in curved spacetime o+1+2) A (0+1+2) 47a. (5.43) The zerothorder term obeys the differential equation (5.44) V2 a = 47 J '(O)AS(O) 4d the source term being J (q (g [p_ p'()] dr, 0, 0, 0). (5.45) The solution of this differential equation is the wellknown Coulomb electromagnetic potential AS(o) =(, 0, 0, 0). (5.46) The firstorder correction to this electromagnetic potential obeys the differential equation derived from Equation (5.43) V72 o)A V A2 I (5.47) The firstorder part of V2 (which contains the S's and the B's) acting on the zeroth order electromagnetic potential is the source term for the firstorder correction. Sub stituting the THZ components of A'I() A() (A(l), A(1), A(), A()) (5.48) into the differential equation results in four differential equations, one for each one of these components. Each equation relates a specific sum of second derivatives of a component to a sum of terms of the form: qx f or the tcomponent, 5 (5.49) qB xx a for the spatial components, P3 where the dots denote appropriate indices. Solving these four equations is straightforward, once one notices that the solu tion should have the form Stcicj' for the tcomponent, p (5.50) qcp.B' xkxi for the pspatial component, P each term multiplied by an appropriate algebraic factor. Substituting these expres sions into the differential equations gives a set of simple algebraic equations for these factors, which can be easily solved to give the final expression for the firstorder cor rection: A() q Ceijxij, i jiBkXJk, q k yijXLik XJk, zijLikX X (5.51) The next order correction comes from the part of the V2 that is of O() acting on the zerothorder electromagnetic potential. It gives a differential equation for each component of A~(2) of the form a a S,,As(2) =0( ( ) x O .A(o) O( (5.52) which indicates that the next order correction must be of O(). It is noteworthy that the firstorder correction Ad(1) does not appear in the equation for the secondorder correction. That is because it only shows up in terms that involve the O() part of the metric, which are of the form 0( ) x2A O(A)( (5.53) and must be included in a higherorder calculation. Finally, the singular electromagnetic potential of a charge q that is moving on a geodesic in a Schwarzschild background is equal to !([I IijxixJ1, X (5.5x 4) A= 2 [ Jk 2 k 2 + k O( ). (5.54) 5.5 Electromagnetic Potential of an Electric Dipole The calculation of the singular electromagnetic potential of an electric dipole is presented in this section. The dipole moment is assumed to point at some random direction and its THZ components are qa = (0, q", qY, q). The singular electromagnetic potential can be written as A = As(o) + A(1) + AS(2) (5.55) and obeys the vacuum Maxwell's equations V(o+1+2) A(o+1+2) 47 (5.56) where the source is ja ( /qi[64(p p'())] gdr, 0, 0, o). (5.57) The zerothorder term is the electromagnetic potential generated by an electric dipole that is stationary at the origin of the Cartesian coordinate system, so it obeys the differential equation (o)As(o) = 47J" (5.58) The solution to this equation is wellknown and has only a tcomponent A () 0, 0, (5.59) The firstorder correction to the electromagnetic potential obeys the differential equation V )(1) 7(1) A(o) (5.60) If its THZ components are assumed to be A ) (A(), A ), A A )) (5.61) and are substituted into Equation (5.60), the differential equation for A~,) becomes a set of four secondorder differential equations for these components. Each differential equation relates a sum of second derivatives of a component to a sum of terms of the form q xx for the t component, Sp (5.62) q xx x for the spatial components, P7 where, again, the dots denote appropriate indices. These differential equations indicate that the solution should be equal to a sum of the terms qijkX i xjx k qi i XkX k SJk and (5.63) for the tcomponent and a sum of the terms P ei LJ3 k P i k P k Cifq 1xxkx ciqI3klXXX 3 q ikXXlXl S(5.64) eP jqk flk xkxl Eijk qiL3PjXkX1 c ijk qiL3jxp k X 1 P3 P3 3 for the pspatial component, each term multiplied by an appropriate numerical co efficient. In fact, using Equations (5.15) and (5.17), the last two terms that are expected to show up in the solution for the pcomponent can be eliminated in favor of the remaining four. Substituting these expressions into the differential equations gives simple abgebraic equations for the coefficients. The final expressions for the components of A(,) are At 1 qiSJkx i x k (l) 2 p3 2 *(5 .6 5 ) p i 1ik JP k & j A % + + s(1) 2p 2p3 2p3 The order of the next term in the expansion of the electromagnetic potential can be predicted. It is the solution of the differential equation whose source term comes from the 0( ) part of the V2 acting on Apo. That differential equation has the form ap aOA (2) = O( ) A O( ). (5.66) 3 JAOPR Consequently, the secondorder correction must be of O(). The terms that involve the firstorder correction A4(1) do not contribute to the equation for the secondorder correction, because they involve the O(L) part of the metric and that results in terms of0( ). Finally, the singular electromagnetic potential for an electric dipole moving on a Schwarzschild geodesic is equal to At qxi iq xI xxk + 0( ), p L +(5.67) A4 3 e qP x'k +X k kP 1 k kPli xxkl] + 0( ). 5.6 Electromagnetic Potential of a Magnetic Dipole In this section, the singular electromagnetic potential of a magnetic dipole mov ing on a geodesic of the Schwarzschild background is calculated. The magnetiza tion m' is assumed to point at some random direction and its THZ components are ma = (0, m", mZ). The singular electromagnetic potential can be written as AS AS(o) + AS() + A(2) (5.68) and obeys the vacuum Einstein equations (+1+2) A(o+1+2) 47Ja (5.69) where the source term is Jo =0, i m 4 (5.70) J m p p'(T))] g d. The zerothorder term is the electromagnetic potential generated by a magnetic dipole that is stationary at the origin of a Cartesian coordinate system and is the solution of the differential equation VO)As(O) 47J. (5.71) Its THZ components are A(o) (0, 't P3 j )3 (5.72) 70 The firstorder term obeys the differential equation derived from Maxwell's Equations (5.69) 72 o)A V2 A) (5.73) which indicates that the source term for AS() is the VI) of the zerothorder part of the potential. The THZ components of A are A() = (A( A, A( s(1), A()) (5.74) and when substituted into Equation (5.73) the result is a set of four secondorder differential equations, one for each one of those four components. Each equation relates a sum of second derivatives of one component to the source term which consists of terms of the form: mB xxx ..5 for the tcomponent, p (5.75) ...X 7 for the spatial components, p7 where the dots denote the appropriate indices. Solving these differential equations is tedious but not difficult, because each equation involves only one component of AS(1) and only one of the two tensors S and B. A careful look at the equations indicates that the solution should be a sum of the terms miB X k k mijkX i XjX k 3x and 3JX (5.76) p3 p3 for the tcomponent and a sum of the terms eJmi Sxkxklxl Emifklxk1 1 kP ?kik Xi1X P P3 P 3 p3 p(5.77) el m Xk x jk 1 tijkMi Epj k xl1 ijkmi Xk Xlxp p3 p3 p3 p p p3? for the pspatial component, with appropriate numerical factors in front of each term so that the equations are satisfied. Using Equations (5.15) and (5.17), the first and last terms expected to show up in the final expression for the pcomponent can be eliminated, since they can be expressed as linear combinations of the remaining four terms. Substituting these expressions into the differential equations gives a system of four algebraic equations for those factors. Solving these algebraic equations is trivial. The result is that the firstorder components of the electromagnetic potential are: A[ iBjk XiX k BijiXjxkXk] [ 3 (5.78) 1 1 1 . A() l [P kl J kX Pijjk ilJkX ijkMiS x] 8(1 p3 2 22 The order of the next correction to the singular electromagnetic potential can be predicted. It is the solution of the differential equation that has the O(&) part of the V2 acting on the zerothorder electromagnetic potential as the source term. Specifically, it looks like p 1 aOjAS(2) = 0( ) x ijA A(o) 0( (5.79) meaning that the Ag(2) correction is of O(). As in the previously studied cases, the terms that involve the firstorder correction AS(1) do not contribute to this equation, since they involve the O() part of the metric which results in terms of O(h). Finally, the singular electromagnetic potential for a magnetic dipole moving on a Schwarzschild geodesic is equal to A t [mBjkX i XX k 2_ iik k] + O( ) S 3 16 3 R3 A mC x rx P mi kl X3xkx 1 mkil X kl 1 miSpjXkXlXl ( S = +0 + 3. 2 3) (5.80) 5.7 Gravitational Field of a Spinning Particle The calculation of the singular gravitational field of a spinning particle mov ing on a Schwarzschild geodesic is presented in this section. The particle is as sumed to have a small angular momentum pointing at some random direction A (0, A', AY, A) in THZ coordinates. The singular gravitational field can be written as hs ab = h h() h(2) (5.81) h Sab a Sab S ab and obeys the linearized Einstein equations a2 (+1+2) c dh(O+l+2) 16r Tb (5.82) (0+1+2) Sab + 2~(0+1+2) ab 'Scd 16 7ab (582) where hsab =hsab 9Sgabhs (5.83) is the tracereversed version of hsab. The zerothorder part of the singular field is the gravitational field generated by a particle with angular momentum A" that is stationary at the origin of a Cartesian coordinate system. For the angular momentum pointing along the zaxis, that is the wellknown Kerr solution with the mass set equal to zero. Since the angular momentum is assumed to be small and the effects of the mass of the particle are not taken into account, only the terms of the Kerr metric that are linear in the angular momentum need to be considered. In the BoyerLindquist coordinates (tBL, r, 0, q) around the spinning particle, hjab (where the superscript Z denotes that this is the part of the zerothorder gravitational field coming only from the zcomponent of the angular momentum) is equal to 0001 S0 0 00 h4b A sin2 0 0 0 0 (5.84) 0 0 0 0 1 0 0 0) 1000 Using GRTENSOR, this expression can be easily converted into the equivalent expres sion in THZ coordinates 0 y x 0 h ab 2Ay 0 (5.85) P x 0 0 0 0 0 0 0 0000 The relationships between the BoyerLindquist and the THZ coordinates used for this conversion are the usual relationships between the spherical and the Cartesian coordinates, namely t =tBL r = V2 + y2 + z2 r22 (5.86) 0 = arctan (5 z = arctan . This is sufficient because the corrections to these relationships that involve the angular momentum would give terms of higher order in the angular momentum and must be ignored in this analysis, since only firstorder terms in the angular momentum are kept. The components of the angular momentum along the x and y axes must be treated separately, because of the axial symmetry of the Kerr metric. The analyses 74 for the angular momentum being along the x and y axes are very similar to that for the angular momentum being along the z axis and the only change comes from the different form that the relationships (5.86) have. Specifically, when the angular momentum points along the x axis, the axial symmetry is around the x axis and the relationships used are t =tBL r = 2 + y2 + 0 = arctan V2 x (5.87) = arctan . y For the angular momentum pointing along the y axis, the axial symmetry is around the y axis and the relationships are t = tBL r 2 + y2 + z2 a a 2 z 0 arctan  Y (5.88) x = arctan . z Adding all the contributions that result from this analysis, the zerothorder singular gravitational field becomes 0 (Ayz + Azy) (A'z Azz) S(A"y + Ax) (A'z + Azy) (A'z Axz) (A"y + Ayx) 0 0 0 0 0 0 0 0 0 (5.89) h(0) 2 S ab 3 p The firstorder correction to this gravitational field obeys the differential equa tion derived from Equation (5.82): 72 ) h d() c 2 (o) R2 hC d((O) o) hS ab+ (0) a b S cd (1)S ab (1) a b S cd (5.90) so the firstorder V2 and the firstorder Riemann tensor acting on the zerothorder solution give the source for the firstorder correction. The firstorder correction must be a symmetric tensor so it is assumed to be equal to J j(t1) Q1) hil) Q1) (1) 1) h(1)x h(1)xy h(1)xz S ab ) () h)(5.91) "ty xy h2y hj"yz h(l) h,(l) () h(l) Substituting it into the firstorder equation results in 10 differential equations. There is one set of four differential equations for the four diagonal components, each equation containing all four diagonal components. There is also one differential equation for each one of the t i components and one differential equation for each one of the i j components, for i / j. In each equation, a sum of second derivatives of components is related to a sum of terms of the form A.B x.x xxx AB xx for the t t and p q components, (5.92) for the t p components, p7 where the dots denote the appropriate indices for each term. Solving the differential equations in this case is slightly more complicated than in the previous cases, mainly because of the fact that four of them involve all diagonal components rather than only one of them. Still, the process becomes significantly easier if one notices that the t t component must be an appropriate sum of terms of the form AiLjkX i xk p3 A'BijXjXkXk and A 3j P3 each t p component must be a sum of terms of the form EpijAiS xlXkXk EpijA' l xlxk EpijAl~zxZxkxk P3 P3 P Epi^1 x Il fk 1icflA ` Xk Ck i31 A 'S3k p l k p3 P3 P3 c AlgSxixlxk iA lAig~8x'XkXk cijAiS"xPxlxk p3 P 3 P 3 and each p q spatial component must be a sum of terms of the form ApBqkXkXlXl Aq,3pkXkXlXl Ap3klXqXkXl Aq,3klXpXkXl p3 P 3 3 3 Ak3pqX k Xx11 Ak klXpXqXl pgqAiBjkXixJxk 1pqA'BL3ijXXkXk P3 P3 P3 P3 Ak pkXqxlxl AkBqkXpXlXl AkBplXqX kX AkBqlXpXkXl p3 p3 p3 p3 (5.93) (5.94) (5.95) with appropriate numerical coefficients in front of each term. Equations (5.15) and (5.17) can again be used to eliminate the first and last terms in favor of the remaining four, for the expression for the t p components. Substituting these sums into the differential equations gives algebraic equations for the coefficients, which are fairly easy to solve. The result is that the firstorder correction to the gravitational field has components SAjBjk xixjxk 2 j P3 i [pjAjS xxlxk cpjA'Skxlxk+ 3cj,A SpjXlXkXk 3 [2AiX'X + A'BiXpXqXj + 2A'2 ,,,, A p 3 A(pBq)ix'xx A(pXq)Bijx'x + q, (Aijki A'B3ijXjXk (5.96) h(1) S tt h(0) S tp h(S) S pq The order of the next correction can be predicted. That correction is the solution of the differential equation 772 h (2) d (2) Y2 h(0) R C da (0) 0) hab + 2 (0) b S cd (2) S ab 2(2) a b S cd (5.97) where V2) and R(2) b come from the 0( ) part of the metric. So the equations look like a, i bjh O( ) x aijh(O)b O( ) (5.98) and the second correction to the gravitational field is of O(). Again, h)b does not show up in the equations for the second correction, since it relates to the 0( _) part of the metric and results in terms of O(R). Finally, the singular gravitational field due to a spinning particle moving on a Schwarzschild geodesic is equal to hS A Bjk i k 0( p3 Rp hs=~ 2 c (pij A hstp 2 + 1 [pA3 XXlXk EpjA'SXXlXk k +pi cjAjX'1Xk k] +0() R3 hspq [2AiB pqxXj + A'I3ijXpXqXj + 2A'B,,. rj AiB3j(pXq)X 3 A(pB)X'XX 3A(pXq)ijX'Xa + q,, (AiL3jkeik A'3ijXrjkk)] +04(). (5.99) 5.8 Grtensor Code The GRTENSOR code (running under MAPLE) used to derive the differential equations for the first order correction to the singular field is given in Appendix A. Since the case of the gravitational field is the most complicated one, the analysis is presented for the gravitational field generated by a spinning particle with the angular momentum pointing along the THZ zaxis. The analyses for the scalar fields and the electromagnetic potentials are very similar, and can be easily deduced from that for the gravitational field. An effort was made to keep the symbols in the code in accordance with the ones used in this chapter for the various quantities. In the situations where that is not the case, the comments in the code should make the notation clear enough for the reader to follow. The parameter e is used to keep track of the order of each term in the com ponents of the tensors S and B and is set equal to 1 at the end. Throughout the calculation, only first order terms are kept. Specifically, the Christoffel symbols and the components of the Riemann and Ricci tensors are calculated first and all their terms that are of order higher than 1 are set equal to zero. Doing that makes the sub sequent analysis significantly simpler and the running time of the code significantly shorter. After the various quantities associated with the problem are calculated, the test tensor hbartest(a,b), whose exact dependence on the THZ coordinates is not specified, is used as a trial solution in the linearized Einstein equations and the source term coming from the zerothorder solution h0(a,b) is examined. That helps identify the terms that should be expected to show up in each component of the solution hbar(a,b). Specifically, it helps determine which tensor's components, S (denoted as EE in the code) or B (denoted as BB), should show up in each component of hbar(a,b) and gives an idea of how they should be contracted to the spatial THZ coordinates x, y, z. The terms that result from this analysis are multiplied by algebraic factors and the appropriate sum is substituted into the linearized Einstein equations. The 79 result is a simple system of algebraic equations which can be easily solved to give the values of the algebraic factors. That completes the solution. The last component of this analysis is a simple confirmation performed for the gravitational field that was calculated, that the algebraic coefficients obtained do, indeed, give the required solution. The confirmation is simply done by replacing the initially unknown algebraic coefficients with their exact values in the expression for the solution and substituting that expression into the Einstein equations. Despite the fact that it was not explicitly mentioned in Sections (5.3)(5.7), that confirmation was performed for all singular fields and potentials that were calculated. CHAPTER 6 REGULARIZATION PARAMETERS FOR THE SCALAR FIELD A modesum regularizarion procedure for the scalar singular field is presented and implemented in this chapter. This regularization procedure was first proposed by Barack and Ori in [28], where they described the calculation of the regularization pa rameters for the direct part of the selfforce on a particle carrying a scalar charge. The procedure was later implemented by different groups for the calculation of the regular ization parameters for the direct part of the selfforce on a scalar charge on different geodesics [2833] and also for nongeodesic motion [31, 33] around a Schwarzschild black hole. The calculation of the regularization parameters has also been performed for the direct part of the electromagnetic selfforce [34] and for the direct part of the gravitational selfforce [3436], for arbitrary geodesics around a Schwarzschild black hole. Even though the regularization procedure was initially described for the contri bution of the direct part of the scalar field to the selfforce, it can be used equally successfully for the contribution of the singular scalar field to the selfforce, as was demonstrated in [23]. In that paper, the regularization parameters for the selfforce on a scalar charge in circular orbit around a Schwarzschild black hole were calculated and the selfforce results ended up being in excellent agreement with the results that were derived using the direct scalar field [2831]. This chapter begins with an outline of the regularization procedure for the scalar selfforce. The description closely follows that given by Barack and Ori [28] for the direct selfforce but is presented here for the singular selfforce instead. Then, the regularization parameters are calculated for the singular scalar field (rather than the scalar selfforce) of a charged particle that moves on an equatorial circular orbit in a Schwarzschild background and the results of [23] for the scalar selfforce are repro duced. Finally, the regularization parameters for the first derivative of the singular part of the selfforce are also calculated. 6.1 Regularization Procedure As was shown in Chapter 4, the selfforce on a particle that carries a scalar charge q can be calculated from the equation S q lim VR q liImV (, ) (6.1) pp' pp' where p' is the point along the worldline of the charged particle on which the selfforce needs to be calculated and p is a point in the vicinity of p'. It is assumed that the charge q is moving in a Schwarzschild background of mass M and the Schwarzschild coordinates are (ts, r, 0, y). For the calculation of the retarded field, the source term in Poisson's Equation (3.19) can be decomposed in terms of spherical harmonics and the retarded field can be written as 00 1 1=0 m=1 Then, the Imcomponents of '"t can be calculated numerically. That calculation is discussed in great detail in Chapter 7. Here it is just noted that the important property of the "';.'s and of their first rderivatives is that they are finite at the location of the particle, even though tt is singular there. If the spherical harmonic decomposition of the singular field is also considered, Equation (6.1) becomes 00 1 R= q lim V, 1 .m q lim a (6.3) 1=0 m=1 l,m It is helpful to define the multiple 1modes of the two contributions to the selfforce, which result after performing the msummation of each term individually in Equation (6.3). Specifically _Flret q77,, 1.n, In (6.4) iFl qVa 7, Y ,, in which gives for the selfforce .T lim, F t ) (6.5) In Equation (6.5), the difference in the multiple 1modes must be taken before the summation over I is performed. From this point on, the discussion of the regularization procedure becomes spe cific to the problem of the scalar field (q/p), since more detailed results are available for this case. However, a similar analysis can be done for any other scalar field. The goal is to find a function hl, such that the series (ZF hl,) (6.6) converges. When such a function is found, the selfforce can be written as = (EFt hl,) E, (6.7) where Ea0 lim (.Fs hi.). (6.8) p___1p , Because of its definition, the function hi, should be calculated by investigating the .1 Inil.litic expansion of 17t for large 1. On the other hand, because the selfforce is known to be welldefined, FJt and Fs,~ are expected to have the same large/ behavior, so ha, can be determined by the ..ii.mptotic behavior of FS, instead. The singular part Fs of the selfforce consists of terms of different order in the limit p  p' and it has been shown [23] that, in principle, only the first three of those terms are expected to give nonzero contributions, for the field (q/p). However, for reasons that will become clear shortly, the next order terms are included and S= S(A) S(B) S(C) (D) + ) (6.9) The superscripts A, B, C, D indicate the different orders, A coming from the most dominant term, B from the next more dominant and so on. The superscript E refers to all terms of order higher than that of F (D). The modesum regularization procedure amounts to performing the spherical harmonic decomposition of each such term, which results in an expression of the form .S \ A n ln + lm i lm lrn\1 __ = (A$ +B~ + nl + D + Em)Y (6.10) I,m with A'" corresponding to F. A), etc. For the simple case of a scalar charge moving in a Schwarzschild background and generating the scalar field (q/p), the parameters A B C~m and DI" have been shown [23] to have a very simple form so that, when the explicit expression for the spherical harmonics Yj, is substituted into Equation (6.10), the summation over m can be performed and the result is an expression of the form 1 1 Da E+4)] ( = [A, ( + + B + Ca E,7,((127 1)+ 2 + (+l (21 1) (21+ 3) where the regularization parameters Aa, Ba, Ca, Da and Ea are 1independent quan tities which do depend on the background geometry and the characteristics of the orbit. Finally, the selfforce can be calculated by 10 .FR [ lim Tl = (6.12) 1 1 D_ Aa(1 + ) B Ca Ea 2 (1 + 1) (21 1)(21 + 3) One important point that should be made is that the infinite sum over 1 must be performed, in order for the selfforce to be calculated. Notice, however, that the contributions for large I get less significant for the terms containing Da and Ea. The reason for including these terms is now clear. Even though the sum over I of each of these two terms is exactly equal to 0, including these terms improves the convergence of the sum. An additional benefit of including these terms is that the approximation to F R becomes more differentiable, as is explained in [23]. 6.2 Order Calculation of the Scalar Field It should be obvious from the analysis of Section (6.1) that, in order to calculate the regularization parameters for the singular field generated by a charge q, it is necessary to have an expression of (1/p) in which the order of each term is known. The derivation of such an expression is presented in this section for an equatorial circular orbit of radius ro in the Schwarzschild background. It is noted that the results were derived using MAPLE extensively. It is also noted that the derivation was presented in [23] where the results of it for the radial derivative ,( () were given. For simplicity, the scalar charge q is set to 1. In order for the calculation of the selfforce regularization parameters to be made easier, the Schwarzschild coordinates can be rotated, as explained in [30]. Specifically, new angles 6 and D can be defined in terms of the usual Schwarzschild angles by the equations sin 0 cos( ts) cosO (6.13) sin 0 sin(q ts) = sin cos D (6.14) cos0 = sin sin D (6.15) so that the coordinate location of the particle is moved from the equatorial plane, where 0 = to a location where sin = 0, for a specific ts. Such a coordinate rotation preserves the index 1 of any spherical harmonic Ym,(O, q). That means that any Yim(0, 9) is mapped into a linear combination of spherical harmonics Ym( (b), where m' = 1,...,1. Consequently, each 1multipole mode of the field or the self force that results after summation over m is the same, regardless of which angles, (0, q) or (6, D), are used for calculating it. The benefit of this coordinate rotation for the calculation of the regularization parameters for the singular part of the selfforce can be understood if one remembers that in the limit p  p' the angle 6 is equal to 0. That means that Yi(0, D) has to be used, for which Ym(0, ) 0, for m / 0 (6.16) m(0,) = ,for m 0. (6.17) So the sum over m can, after this coordinate transformation, be replaced with just the m = 0 term. However, for the regularization parameters of the singular field, the limit p  p' must not be taken, so the Yim's for all m's must be taken into consideration. The m = 0 spherical harmonic must be considered only when the regularization parameters for the selfforce are derived from the ones for the scalar field and when the regularization parameters for the first rderivative of the selfforce are calculated. A comment on the order of each term needs to be made at this point. The parameter is used to indicate a term of order x" in the coincidence limit p p'. In that limit, r or and 6 0. That means that the factor (r ro) is of order e and the factor (1 cos 6) is of order e2. At the end of the calculation, the parameter e can be set equal to 1. The relationships between the Schwarzschild coordinates (ts, r, 0, q) and the THZ coordinates (t, x, y, z) for a circular orbit on the equatorial plane of a Schwarzschild background that were given in Section (5.1) are used for this calculation. As was mentioned earlier, p2 = 2 y2 + z2 in terms of the spatial THZ coordinates x, y and z. However, it is clear from Equation (5.26) that the relationship x2 2 2 + 2 (6.18) holds between {x, y} and {x, y}, so the sum (2 + y2) is used to calculate p2. The Equations (5.22), (5.23) and (5.24) are substituted into the expression for p2, with the Schwarzschild angles 0 and ) replaced by the new angles 6 and D. That substitution gives that the lowest order term for p2, denoted by p2, is 2 roA2 + o 2o 2M o 2 2r2 3 cos (), (6.19) where A r ro (6.20) and M x 1 sin2 I. (6.21) ro 2M Clearly, p2 is of order e2 in the coincidence limit, as should have been expected. Next the variables 0, D and r are eliminated from the order expression of p2 in favor of the variables A, p and X, by using Equations (6.19), (6.20) and (6.21). Finally, the result is inverted and the square root is taken, in order to obtain the order expression of (1/p). The result of this calculation is a very long expression. Here, I only give the terms of this expression that are necessary to calculate the regularization parame ters of the singular field, the singular selfforce along the radial direction and the rderivative of the singular selfforce along the radial direction. Exactly how that is determined will become clear shortly, when the general term A"~'q, where a, p and q stand for integers, will be discussed. The terms of interest are 1 _1 + P P +o\ r0o 3M 1 1 A [ 2r 3M r, 3M 1] A3 L2ro(ro 2M) ro7 + 2r,(r, 2M) 2r,(r, 2M) M if r, 3M 1 ro M 1 r (ro 2M) x ro X2] 2ro 3M 5r 22r +21M2 1 5r 22r .+ 21M2 1 A2 + 2r(ro 2) 4(ro 2M)2 8ro(r 2M)2 X A4 A6 +o( , + [_ M(r 2M) (ro M)(ro 4M) 1 2r4 (ro 3M) 8(r 2M)r4 x (r 3M)(5r 7r .[ 14M2) 1 3(r + M)(r 3M)2 1 16r(ro 2M)2 X 16r2(ro 2M)2 x X3 (A3 A5 A7 A9 } +0( 7, ' + O(3). (6.22) 6.3 Scalar Monopole Field As is clear from the previous calculation of (1/p), the angular dependence of every term shows up in factors of the form p"fq, where p is an odd integer and q = 0,1, 2,.... The spherical harmonic decomposition of that factor, which is necessary when calculating the regularization parameters for the scalar field, is given in detail in Appendix B. The result is 2r 2(r 2MA)p/2 ] q M x'" ro : EP.2 )Y e,) (6.23) l 0 n 1 rm=  where 2 (r 3(6.24) 2r0(r 2M)2 and the coefficients E'72, ,M) are given in Equation (B.41). It is stressed that all the rdependence of pP9 resides in the sum Z:o m2 and the term (72)i+n+l, in Equation (B.41) for E'f, and is always proportional to powers of A (r ro). At this point, a note on the term (72)2 for an odd integer /3 is in order. For that term, the square root of 72 must be considered. One might think that 1 1 r, 3M 2 r 3M 2 (y2)i o 2 o_. (6.25) (2r2) oo 2M)2 2roo 2M)2 (. But that implies that being on the equatorial plane with O 0 and approaching the particle by taking the limit r ro, would give for the leading term p in the expansion of p: 2M 1 172 2ro 3M (6.26) ro 2 A. ro 2M According to this, the sign of the leading order term of p could be either positive or negative, depending on whether r > ro or r < ro. That is clearly not correct, since by definition p = 2+ y2 z2, which is always positive. For that reason, taking the square root of 72 always implies r 3M 2 2 A. (6.27) [2r0(r0 2M)2 j For the selfforce along the radial direction to be calculated, the rderivative of (1/p) and the limit r  ro have to be taken. That means that any term of order (r ro)2 or higher gives, after the limit is taken, no contribution to the selfforce. However, as has already been mentioned, it is desired to calculate the first derivative with respect to r of the selfforce, which is the second derivative with respect to r of the field, in the limit r  ro. Consequently, terms of order (r ro)2 have to be retained, because they do give a contribution at that limit, while any terms of order (r ro) or higher can be disregarded. As is clear from Equation (B.41) of Appendix B, the general term AOPX in Equation (6.22) has two pieces that contain the rdependence. The first piece comes from the sum over s in Ef', and gives terms proportional to a+2s. Such a term should be kept only for a + 2s < or 5 a < (6.28) 4 2 Since s > 0, the a+2s contribution can be immediately disregarded for terms with a > while for terms with a <  a limited number of values of s have to be 2' 2 retained in the sum. The second piece that contains rdependence comes from the term (_2) +'+1 and is proportional to 1p+a+2n+2. Such a term should be kept only forp + a n 2 + < or n< a (6.29) 4 2 Since n > 0, all terms of Equation (6.22) for which p + a > 1 do not contribute through the 7p+a+2n+2 piece. Combinations of p and a for which p + a < should be examined individually and a specific number of n's must be retained. The terms of the expansion (6.22) for (1/p) for which just the order (and not the explicit expression) is given are terms that fall in both categories that according to this analysis can be ignored. I present now the calculation of the regularization parameters coming from the terms of different order, for the singular field (1/p). The abbreviation used for the hypergeometric function is FA 2F1(,A++1; ), (6.30) 2' 2 2M and it is also noted that the hypergeometric function 1 M F,' 2F1 (, 2 1; ) (6.31) 2' r, 2M is denoted by Fa, as is done in [23]. Also, for the three sums over k, v and A the abbreviation ( ) 21+1 F(l ( 1 l m+1) ( 2 [( m+1)F(l +m+ 1)]+ F r(1k + 2 r(I + 1) S( 2 F(k +i)F(2k Iml + ) F(+ 1)F( + 1) Iml I(1 )A2 F(A +1)F( A+1) (6.32) that is used in appendix B is also used here. In addition, the upper limit of the nsummation is denoted by N = I m 2k+ 2v. (6.33) For the lowest order term of Equation (6.22) 11 T(1) C  (6.34) the exponents of A" X q are a 0, p 1, q 0. Equations (6.28) and (6.29) give that only the s = 0 and s = 1 terms of the first piece and only the n 0 term of the second piece of E ,'q need to be kept. Consequently (6.35) where, using Equation (6.23) Am 2\ ro 2 1M) , A2, (r ,m 1 2r0 o 2M) I o 2ro 2M) (6.36) S( 1)1+22n+i F(N ) [ ? n (2n +)!! F(N n + 1/2 n=0 (n FA,m 2] (T )3/2 J or, when the explicit expression for 72 and its square root are substituted r f 4 r 13Ma3r m(ro 2 AM) kvA 2ro( 2M)2 S(1) +n22n+i F(N ) [, O (2n + 1)!! F(N n 1) L ( n=0 IA Tr03o 3M Y 2 3/2 2ro(ro 2 M)2 j (6.37) T(1_) 1 Y AYl (', () 2(q)2 F1A (Y){ kfeA ' As is explained in Appendix B, only the even m's should be included in the sum of Equation (6.35). The regularization parameter A, given in [23] for the selfforce can be easily obtained from this result. First the m = 0 parameter is considered, for reasons that were explained earlier. That makes the sums over A and over v equivalent to the terms with A = 0 and v = 0. For the hypergeometric function F1' it can easily be proven that F '0 F (6.38) using Equations 15.3.3 of [37]. Then the first derivative of Am with respect to r is taken. That makes the term proportional to F1 2 in Equation (6.37) vanish, while the term proportional to F, gives a factor of A which vanishes when the coincidence limit is taken. For the first term the derivative of A gives sgn(A). Finally, the spherical harmonic Yo(0, 1) is substituted with 21. This procedure gives 2 [o( 3d)]d 1 1 2(sgn(A)) T r 2M1) (+ 2) (6.39) 12 2k F( k +) k0 F(k + 1)F( 2k + 1) The sum over k can be easily calculated for any value of 1, using MAPLE. A general proof that it is equal to rT for any 1 cannot be given. However, for every value of 1 that was tried, the sum ended up being /7, which gives S[ro(r 3M)]i 1 T_) pp' 2( sgn(A)) o(o ) (6.40) Sr(ro 2M) ( + 2) which is the result of [23]. Now, the second derivative of the term T(_1) with respect to r is taken, which is equal to the contribution of that term to the first derivative of the selfforce. The m = 0 component is considered since the coincidence limit must be taken, so A = 0 and v 0 as well. Only the term proportional to F, 2 survives the differentiation, since it is proportional to A2. The result is 02T(1) 0r2 p>p' 3 d2Alo 21+ 1 dr2 47 0 0 (ro 3M) 2 1 1 e F3 / 2 >2 ( k F(1 k + 1) 12k 22(n ) 0 F(k+ 1) F(I 2k n 1)(2n +1)! (6.41) The zerothorder term contribution that is considered in Equation (6.22) is SC[ ro 3M 1 T(o) L2ro(ro 2M) X S2rr3M ro 3M ] A3 (6.42) ro p 2ro(ro 2M) 2ro(ro 2M) 3 (.2 For the piece that is proportional to Ap1 the exponents are a =1, p 1, q = 1 for the term that contains X and a =1, p = 1, q = 0 for the term that does not contain X. In both cases, Equations (6.28) and (6.29) indicate that the s = 0 term of the first piece and the n = 0 term of the second piece of ET1,q should be kept. For the piece that is proportional to A3"3 the exponents are a = 3,p = 3, = 0 for the term that does not contain X and a = 3, p = 3, = 1 for the term that does contain X. In this case, Equations (6.28) and (6.29) give that none of the sterms and only the n =0 term of the second piece of E3,q need to be kept. That gives (6.43) T(o) coy BimYm(,, m) imT where, from Equation (6.23) B r, 1F rM 3 , m A 2r2(r, 2M) 2r,(r, 2M) ^i [2r (ro. 2M)] [ 2ro 3,0 2r(ro 2M) EIz SE, 7,0 ~ ro 3M E3,11 2r(r 2M) 'm 1 Substituing the explicit expressions of the coefficients Ef"' into Equation (6.44) for Bim results in R ro 3M 2r(ro k,v,A A = (1)+n22n+ 2 (N +1) roo 3M S0 (2n + 1)!! (N n 1) (r 3/2 vA (ro 3M) ro 3M F,,. r (ro 2M) 2ro(ro 2M) A3 (ro ,3M) r2ro 3M fA IA /2r (ro 2M)L ro 3M Fo 1 (6.45) F0 It is easy to recognize that A3 ^1  (6.46) so the last two terms in the last expression for BRi can be combined to give a signifi cantly simpler expression, namely Sro 3M 2rB (r 2M) ) A : (_1)i+ 22n+ F(N ) 3M Am =0 (2n +l)!! F(N n +1) 2ro(ro2M) 3/2 1 AA (ro 3M) 2r (ro 2M) 2(r 3M)(ro M) (4r (ro 2M) AI rAm+ 1ia F0 (o 1/2 ) o 3M)F1 (6.47) (6.44) 