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- Permanent Link:
- https://ufdc.ufl.edu/UFE0021314/00001
## Material Information- Title:
- Developments in the Perturbation Theory of Algebraically Special Spacetimes
- Creator:
- Price, Lawrence Ray, Jr
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2007
- Language:
- english
- Physical Description:
- 1 online resource (180 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Physics
- Committee Chair:
- Whiting, Bernard F.
- Committee Members:
- Klauder, John R.
Muller, Guido Detweiler, Steven L. Groisser, David J. - Graduation Date:
- 8/11/2007
## Subjects- Subjects / Keywords:
- Angular momentum ( jstor )
Distance functions ( jstor ) Einstein equations ( jstor ) Killing ( jstor ) Mass ( jstor ) Mathematical vectors ( jstor ) Scalars ( jstor ) Sine function ( jstor ) Spacetime ( jstor ) Tensors ( jstor ) Physics -- Dissertations, Academic -- UF black, newman, spin - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Physics thesis, Ph.D.
## Notes- Abstract:
- The detection of gravitational waves is the most exciting prospect for experimental relativity today. With ground based interferometers such as LIGO, VIRGO and GEO online and the space based LISA project in preparation, the experimental apparatus necessary for such work is steadily taking shape. Yet, however capable these experiments are of taking data, the actual detection of gravitational waves relies in a significant way on making sense of the collected signals. Some of the data analysis techniques already in place use knowledge of expected waveforms to aid the search. This is manifested in template based data analysis techniques. For these techniques to be successful, potential sources of gravitational radiation must be identified and the corresponding waveforms for those sources must be computed. It is in this context that black hole perturbation theory has its most immediate consequences. This dissertation presents a new framework for black hole perturbation theory based on the spin coefficient formalism of Geroch, Held and Penrose. The two main components of this framework are a new form for the perturbed Einstein equations and a Maple package, GHPtools, for performing the necessary symbolic computation. This framework provides a powerful tool for performing analyses generally applicable to the entire class of Petrov type D solutions, which include the Kerr and Schwarzschild spacetimes. Several examples of the power and flexibility of the framework are explored. They include a proof of the existence of the radiation gauges of Chrzanowski in Petrov type II spaces as well as a derivation of the Teukolsky-Starobinsky relations that makes no reference to separation of variables. Furthermore, a method of determining the non-radiated multipoles in type D spaces is detailed. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2007.
- Local:
- Adviser: Whiting, Bernard F.
- Statement of Responsibility:
- by Lawrence Ray Price.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Price, Lawrence Ray, Jr. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Classification:
- LD1780 2007 ( lcc )
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conjugate(mup) = vector([-1/2*Itexp(-I*S(t, r, theta, phi))*atsin(theta)*2^(1/2)/(r-Ita*cos(thea) 0, 1/2*exp(-I*S(t, r, theta, phi))*2^(1/2)/(r-Itatcos(theta)), -1/2*Itexp(-I*S(t, r, theta, phi))*2^(1/2)/(sin(theta)*(r-Ita*cos(thet)]) mup= vector([1/2*Itexp(I*S(t, r, theta, phi))*atsin(theta)*2^(1/2)/(r+Ita*cos(thea) 0, 1/2*exp(I*S(t, r, theta, phi))*2^(1/2)/(r+I*atcos(theta)), 1/2*Itexp(I*S(t, r, theta, phi))*2^(1/2)/(sin(theta)*(r+Ita*cos(thet)]\} tetdnSB := \{conjugate(mdn) = vector([-1/2*Itexp(-I*S(t, r, theta, phi))*atsin(theta)*2^(1/2)/(r-Ita*cos(thea) 0, -1/2*exp(-I*S(t, r, theta, phi))*(r+I*atcos(theta))*2^(1/2), 1/2*Itexp(-I*S(t, r, theta, phi))*(r^2+a^2)*sin(theta)*2^(1/2)/(r-I* acstea)) mdn= vector([1/2*Itexp(I*S(t, r, theta, phi))*atsin(theta)*2^(1/2)/(r+Ita*cos(thea) 0, -1/2*exp(I*S(t, r, theta, phi))*(r-Itatcos(theta))*2^(1/2), -1/2*Itexp(I*S(t, r, theta, phi))*(r^2+a^2)*sin(theta)*2^(1/2)/(r+I*acste)]) ndn = vector([1/2*2^(1/2)*((r^2-2*M*r+a^2)/((r+acotha)(rIaosh eta))))^(1/2)/B(t, r, theta, phi), 1/2*2^ (1/2)* ((r+I*atcos(theta))*(r-Itatcotha)/r22Mr^)^( /2)/B(t, r, theta, phi), 0, -1/2*atsin(theta)^2*2^(1/2)*((r^2-2*M*r+a2/(+acotha)(rI a*cos(theta))))^(1/2)/B(t, r, theta, phi)]), Idn = vector([1/2*B(t, r, theta, phi)*2^(1/2)*((r ^2-2*M*r+a^2)/((r+I*a tcstea)*rIacsoht)) ^(1/2), -1/2*B(t, r, theta, phi)*2^(1/2)*((r+I*atcos(theta))*(r-Ita*cstea)(^-*~~^)^ CHAPTER 6 THE NON-R ADIATED 1\ULTIPOLES In this chapter we will address the issue of the non-radiated multipoles alluded to in C'!s Ilter 1. The issue is that the metric constructed from a Hertz potential is incomplete in the sense that its multiple decomposition necessarily begins at -e = 2 because the angular dependence of the potential is that of a spin-weight +2 angular function. To see this explicitly, we focus our attention on the IR G metric perturbation (Equation 3-35) in the Schwarzschild spacetime, where the potential, 9, can he decomposed into some radial function, R(r), with exponential time dependence, e-i", and a spin-weight 2 spherical harmonic, -2 Loz(0, 4) (see Appendix D, for details about the spin-weighted spherical harmonics). Ignoring the radial and time dependence, we see that the components of the metric perturbation have angular dependence given by hit ~ 82-2 at = [(e 1) ( + 1)(e + 2)]1/20Ym ine) him ~ -2 Bat = [(-e 1)(e + 2)]1/2-1 z,, (6-2) and similarly for him and hmm. Because the spin-weighted spherical harmonics are undefined for |8| > -e, the above expressions make it clear that the metric perturbation in this gauge has no -e = 0, 1 pieces and therefore provides an incomplete description of the physical spacetime. By continuity, the situation persists in the K~err spacetime. How incomplete is this description? For the n, I iR~~ly of this work, we have focused our attention on gravitational radiation in type D spacetimes. This information is contained in the perturbation of either I',, or ('4, a Tesult established by Wald [16]. In particular, Wald was able to show that well-behaved perturbations of I',. and #'4 determine each other and furthermore that either one characterizes the entire perturbation of the spacetime up to It l i .! perturbations in mass and angular momentum. With I',, and #'4 determined hv the Hertz potential simple expression: {>'9 a'S p'D pb' +t 1 } This is the R< -~---Wheeler equation. We can clean it up a bit by recognizing the object being acted on as 2 j"Ll = P17l; p'8'hln,, the odd-parity piece of the perturbation of I< Furthermore the operator in Equation :321 is the wave operator, 0, in the Schwarzschild background up to a factor of 1/2. Making these identifications, we now have for the R< -~---Wheeler equation: (O + 8t/'2 '--2/3 11L = 0. (:322) A similar equation for ,: = Im( _') was previously derived by Price [51] (whose only relation to the present author is Equation :322), who showed that (modulo angular dependence), Im(t/') is the time derivative of the R< -~---Wheeler variable. Moreover, without reference to Im _', Jezierski [52] arrived at an equation for odd-parity perturbations that is essentially identical to Equation :322, though phrased in more standard language. Additionally, an analysis by Nolan [5:3] who looked at the perturbed Weyl scalars in terms of gauge invariants of the metric perturbation showed explicitly the relation between Imt/' and the gauge invariant quantity associated with the RW variable. Furthermore, Nolan points out that because I', is real in the background, the perturbation of its imaginary part is, when we restrict our attention to odd-parity, gauge invariant in the sense discussed in ChI Ilpter 2. Perhaps more surprisingly, Nolan further asserts that this is true of the perturbations of all the Wevl scalars, which emphasizes the fact that odd-parity perturbations of spherically symmetric spacetimes are obtainable by virtually any means. One thing that sets our treatment of RW apart from others is our sparing use of spherical symmetry. The only place we make explicit use of it is in Equations :315, which defines the background GHP quantities. This certainly simplifies the subsequent calculations considerably, but fails to fully exploit the background symmetry. In [26] S. C'I .1..1 .s-ekhar, Proc. Roy. Soc. Lond. A392, 1 (1984). [27] M. Campanelli, W. K~rivan, and C. O. Lousto, Phys. Rev. D58, 024016 (1998). [28] C. O. Lousto, Class.Quant.Gray. 22, S569 (2005). [29] S. C'I .1..11~ I-ekhar, The M~athematical Theory of Black Holes, Oxford University Press, New York, 1983. [30] A. I. Janis and E. T. N. x.--n! ll. J. Math. Phys. 6, 902 (1965). [31] A. Petrov, Sci. Nat. State University of K~azan 114, 55 (1954). [32] J. Goldberg and R. Sachs, Acta Phys. Polonica, Supp. 13 22, 13 (1962). [33] W. K~innersley, J. Math. Phys. 10, 1195 (1969). [34] M. Walker and R. Penrose, Commun. Math. Phys. 18, 265 (1970). [35] L. P. Hughston, R. Penrose, P. Sommers, and M. Walker, Commun. Math. Phys. 27, 303 (1972). [36] P. Sommers, Killing tensors and type {2, } spacetimes, Ph.D. dissertation, University of Texas at Austin, 1973. [37] M. Demaidiski and M. Francaviglia, J. Phys. A 14, 173 (1981). [38] R. Floyd, The LI;,n of Kerr fields, Ph.D. dissertation, University of London, 1973. [39] R. Penrose and W. Rindler, Sp~inors and Sp~acetime. Volume 2, Cambridge University Press, New York, 1986. [40] B. Carter, Commun. Math. Phys. 10, 280 (1968). [41] B. Carter, Phys. Rev. 174, 1559 (1968). [42] C. D. Collinosn and P. N. Smith, Comm. Math. Phys. 56, 277 (1977). [43] B. P. Jeffryes, Class. Quant. Gray. 4, L17 (1987). [44] H. R. Beyer and I. Craciun, gr-qc/0607070 (2006). [45] A. Held, Comm. Math. Phys. 37, 311 (1974). [46] J. M. Stewart and M. Walker, Proc. Roy. Soc. Lond. A341, 49 (1974). [47] R. Sachs, in R.~ Il.:i. .I;, Group~s and T 'r.~ J.~ -i;, 1964. required NatUrally, One WaS developed. It has been dubbed GHPtools and the Maple code for it is the content of Appendix C. The remainder of this chapter is devoted to explaining its basic use and functionality through a simple Maple worksheet. Every session begins by invoking GHPtools: > restart; > with(GHPtoolsvi); [BD1, BIlc, BIlp, Bllpc, BIS, BI2c, BI21p, BI-~ ., BIS, BI3c, BI31p, BL,l~ BIg BI~c, BI~p, BI~pc, COM1l COM~lc, COM~lp, COM~lpc, COME2, COM~2c, COM~2p, COM )..~ COMS3, COM~3c, COM~3p, COM,;).. DGHP, GHP1 GHPlc, GHP1p?, GHPlpc, GHPB, GHPENP, GHP~c, GHP21p, GHP ).. GHPS, GHP~c, GHP31p, GHP,li.. GHP4 GHP~c, GHP~1p, GHP~pc, GHP5, GHP~c, GHPS1p, GHP~lpc, GHP6, GHP~c, GHPG1p, GHP~lpc, GHPconj, GHPmult, GHPp~rime, NPconj, NPexp~and, NPp~rime, comm, ezcomm, fblw.i;:, getpq, schw, idsimp?, idspec, letcon, tetdnK, tetdnS, tetdnSB, tetupK, tetupS, tetupSB, typed] To begin with, each variable is directly specified by its usual name. For example p would be entered in Maple as conjugate(rho). The primed variables have a '1' appended to the end, so that p' would be entered as conjugate(rhol). The Weyl scalars are recognized as capital W's with the appropriate number, eg. Psi2. The derivatives P, 8, P' and 8' are recognized in Maple as th(), eth(), thp() and ethp(), respectively. GHPtools recognizes the tetrad vectors as labels indicating the position of the index with the actual index in parentheses. For example la and me would be input as lup(a) and conjugate(mdn) (c). Finally, GHPtools contains an arbitrary function, 95 (in Maple: phi), that is quite useful for general calculations. Amongst 4 There is however a series of papers describing rather sophisticated Maple packages that perform some of the manipulations that we want [48, 49], called GHP and GHPII. We stress that GHPtools is no way intended to compete with these or any other Maple packages . elif evalb(expr=Psii) then p:=2; q:=0 elif evalb(expr=Psi2) then p:=0; q:=0 elif evalb(expr=Psi3) then p:=-2; q:=0 elif evalb(expr=Psi4) then p:=-4; q:=0 elif evalb(expr=phi) then p:=pp; q:=pq elif evalb(expr=conjugate(phi)) then p:=pq; q:=pp elif evalb(expr=phil) then p:=-pp; q:=-pq elif evalb(expr=conjugate(phil)) then p:=-pq; q:=-pp else p:=UNKNOWN; q:=UNKNOWN end if; return(p,q); end proc; GHPprime := proc(expr) return(subs({1dn=ndn, lup=nup, ndn=1dn, nup=1up, mdn=conjugate(mdn), mup=conjugate(mup), conjugate(mup)=mup, conjugate(mdn)=mdn, hll=hnn, hnn=hll, hlm=hnmb, hnmb=hlm, hlmb=hnm, hnm=hlmb, hmm=hmbmb, hmbmb=hmm, th=thp, thp=th, eth=ethp, ethp=eth, rho=rhol, conjugate(rho)=conjugate(rhol), rhol=rho, conjugate(rhol)=conjugate(rho), kappa=kappal, conjugate(kappa)=conjugate(kappal), kappai=kappa, conjugate(kappal)=conjugate(kappa), tau=taul, conjugate(tau)=conjugate(taul), taul=tau, conjugate(taul)=conjugate(tau), sigma=sigmal, conjugate(sigma)=conjugate(sigmal), sigmai=sigma, conjugate(sigmal)=conjugate(sigma), epsilon=epsiloni, q*conjugate(beta)),op(1,0p(1,w)))); elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='ethp' and op(0,0p(1,0p(nops(w),w))) ='th') then (p,q):=getpq(op(1,0p(1,0p(nops(w),w))) ); result:=result w + w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w) (op(1,0p(1,0p(nops(w),w)))))*(times((conjuaedla +(p+1)*betal (q+1)*conjugate(beta)),times((DD - p*epsilon q*conjugate(epsilon)), op(1,0p(1,0p(nops(w),w))) ))); elif (op(0,w)='ethp' and op(0,0p(1,w))='th') then (p,q):=getpq(op(1,0p(1,w))); result:=result w + times((conjugate(delta) + (p+1)*betal -(q+1)*conjugate(beta)),times((DD - p*epsilon -q*conjugate(epsilon)),0p(1,0p(1,w)))); elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='ethp' and op(0,0p(1,0p(nops(w),w))) ='thp') then (p,q):=getpq(op(1,0p(1,0p(nops(w),w))) ); result:=result w + w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w) (op(1,0p(1,0p(nops(w),w)))))*(times((conjuaedla +(p-1)*betal (q-1)*conjugate(beta)),times((Delta + p*epsiloni + q*conjugate(epsiloni)), op(1,0p(1,0p(nops(w),w))) ))); elif (op(0,w)='ethp' and op(0,0p(1,w))='thp') then (p,q):=getpq(op(1,0p(1,w))); result:=result w + times((conjugate(delta) + end if; end proc; ETHP := proc(f) local i, rest, temp; if type(f, 'symbol') then return map( ethp, f) elif type(f, 'constant') then 0 elif type( f, list ) then map( ETHP, f) elif type( f, set ) then map( ETHP, f) elif type( f, '=' ) then map( ETHP, f) elif type( f, '+' ) then map( ETHP, f) elif type( f, '*' ) then rest := mul(op(i,f), i=2..nops(f)); ETHP(op(1,f))*rest + op(1,f)*ETHP(rest); elif type( f, '^' ) then op(2,f)*0p(1,f)^(op(2,f)-1)*EETH~P(op(,f) elif type( f, function ) then if op(0,f) = 'th' then return apply(ethp, f); elif op(0,f) = 'thp' then return apply(ethp, f); elif op(0,f) = 'eth' then return apply(ethp, f); elif op(0,f) = 'ethp' then temp:=ETHP(op(f)); return map(ETHP, temp); elif op(0,f) = 'conjugate' then the perturbation from contributing to gravitational radiation. More importantly, this is a sign that our intuition needs adjustment for working in the K~err spacetime. In further contrast to our prior results, the complicated 8 dependence in the tetrad components of the metric perturbation leads to mixing of the (spin-weighted) spherical harmonic e-modes, a complication not previously encountered. Another surprising feature is the fact that the perturbation is complex and thus exhibits both types of 1. .. .ty". Although the static nature of the perturbation guarantees spin-weighted spherical harmonic angular dependence, we must be careful not to speak of parity in the Schwarzschild sense, but rather the real and imaginary parts of the perturbation. In any case the implications of this fact are presently unclear and remain to be determined in future work. 6.2.3 Discussion In this section we discuss in more detail the possible problems with our matching by looking more closely at the assumptions that we made. This will lead naturally to ideas about future work that is beyond our present scope. First off, one may speculate that our requirement of form invariance is perhaps too strict to allow for a proper matching. This does not appear to be the case. A result of Carter [76] implies that, due to stationarity and axial symmetry, the K~err metric (in Bci-; r-Lindquist coordinates) has precisely the minimum number of nonzero components. Having established independently that the mass and angular momentum perturbations preserve theses properties of the background, Carter's result -II_ _- -; that the problem lies elsewhere . This leads us to consider whether the introduction of an infinitesimally thin shell of matter (which is effectively what E, is), necessarily introduces non-K~err perturbations. A shell (of some currently unspecified shape) would presumably be a differentially rotating object. It is unclear whether this disrupts the stationarity or axial symmetry of the exterior spacetime by the introduction of perturbations that we have neglected results of the previous section are general enough to encompass the special case of type D backgrounds, the tetrad choice we made (with -r = 0) is incompatible with the complete integration of the background field equations which is possible in type D spacetimes [45]. The complete integration requires that each of 1" and n" be aligned with one of the two PNDs. In that case we can exploit the full power of the Goldbergf-Sachs theorem and its corollaries to set a = K' = a = a' = ~o = ~1 = ~3 = 4 = 0, while maintaining -r / 0 and -r' / 0. In this section we repeat the previous calculation with this different choice of tetrad. The result of integrating SiI = 0 is the same as in the case of a type II background, given in Equation 4-16. The residual gauge vector, however, now has the following, more complex, form (details of the integration are given in Appendix C): Cl = Clo 1 1 1 P1 1\ 2 2 2 p2 2 [xo go1 1 1 ~ p p 2 p p -[7Op(8 UO + so .48' + o (o em p reo (4-25) o 1 1 o 1 1 p2 P2 1l1 1m=~ ;~ 1 c" o m mo_ o l o P1 p p n~ i~~ I7:0 il(y& th(th(th(th( )))) 4 p th(th(th( ))) Perhaps even more useful is the comm() procedure which commutes derivatives on an expression. It takes two arguments: the first is the term whose first two derivatives will be commuted and the second is the expression into which the result will be substituted. Consider the following examples: > commute_mel := eth(th(ethp(ethp(phi)))) - > th(eth(ethp(ethp(phi)))); commute _mel := eth(th(ethp (ethp () )))) th(eth(ethp (ethp () )))) > DGHP(comm(eth(th(ethp(ethp(phi)))),commuteml) -p eth(ethp (ethp( ))) + r1 th(ethp(ethp( ))) 2 p -1 ethp(ethp( )) -p rl1ethp(ethp( )) pq > commute_me2 := th(th(thp(phi)))-th(thp(th(phi))); conr;;;n;i. _I,,. :' := th(th(thp( ))) th(thp(th( ))) > tdsimp(comm(thp(th(phi)),commute_me2)); th(eth( )) 7 + eth( ) p- eth( ) p-1 th(eth( )) 71l 2 eth( ) p -1 + th(ethp( )) -r +ethp() )p r- ethp() p 1-rl-th(ethp( )) 71l- 2ethp( )p -1+ pp th() )7 71 +: pp p r r1- pp p r11-rlpp th() )W2 3pp p W2 +pqth() )771r +3 pq ~p7 r1 pq p -r171r pq th( ) W2 :3 pq p W2 Computing the perturbed Einstein equations and Weyl scalars necessarily requires the ability to contract various combinations of the tetrad vectors. This functionality is provided by the tetcon() procedure, which also takes two arguments. The first is the expression that contains the uncontracted vectors and the second is a list of the indices to be contracted over. Take the example of computing the trace of the metric: making further use of Equation 2-32, we establish that g~l, = 7 ,,~ = 4~m, = 4~m, = 0. Recall that for any two vectors, A and B, their commutator is given by [A, B] = tAB, which establishes that the K~illingf vectors of the spacetime commute with all of the tetrad vectors. In this light, it is reasonable to expect that we can construct an operator, V, related to the K~illing vector that commutes with all four of the GHP derivatives. Because of the fact that spin- and boost-weights enter explicitly into the commutators (Equations A-1A-3), we would also expect that any such operator would carry spin- and boost-weight dependence. In fact, such an operator can be constructed. By taking as our ansatz: v = ("Be + pA + qB, and computing all of the commutators, we can find explicit expressions for A and B. However, this also requires that Equations 2-36 are satisfied, which implies a K~illing tensor exists. For non-accelerating spacetimes we then have ii = 2-/(r'd T' 'D pP' +~~1 -2 P +,(-5 2 2p' where p and q refer to the GHP type of the object being acted on. This result has been noted by Jeffryes [43], who arrived at it from spinor considerations. If we specialize to the K~err spacetime and the K~innersley tetrad, it is easy to see that it takes the value M~-1/3 e + bM~2/3 ~2 + 2 COS2 H-1, Where b is the boost-weight of the quantity being acted on. Despite this difference between the vector (" and the operator V, we will refer to them interchangeably as a K~illing vector. Similarly, we can follow the same procedure that led where (s,,, and (3,,, are real spin-weight 0 scalars (type {b, b}; b indicating the boost-weight). Thus, given a quantity with boost-weight b and spin-weight 1, the even parity piece is simply ifev,, and the odd parity piece is iifor,,/ Similarly the complex conjugate of such a quantity (same boost weight, but spin-weight -1) has even parity piece a'(ee,, and odd parity piece -id'(oric. The relative minus sign between an odd-parity object and its complex conjugate is a possible source of confusion, so we must he careful when performing parity decompositions. Symmetric, trace-free two-indexed tensors on S also have a simple parity decomposition. It is easy to recognize the (two) components of such tensors as spin-weight +2 scalars. That is, the components are of type {b + 2, b + 2}. We consider the parity decomposition on S by creating the tensor from a vector on S ta, with boost-weight b and spin-weight 0: Xab = nc~ckb + bc~cks Jub~cd~ckd, (3-5) which can in turn he further decomposed into its even and odd parity pieces by applying Equation 3-4 to yield Xub = L( (2(cb) 8c~cd Jub~cd~c(~ d)xever, + 2(cib d8c~dhele which provides us with a means of identifying the even and odd hits of symmetric trace-free tensors on S. This result generalizes quite easily to n-indexed symmetric trace-free tensors (with components of spin-weight in and boost-weight b) on S: 1 This agrees with the correspondence between the even and odd parity vector and tensor spherical harmonics and the spin-weighted spherical harmonics (see Thorne's review [8] for details) (i.e., the "i" comes along for the ride). the description of the exterior spacetime, g b, We choose P+(t) = cft, &'(>=P ro 2M~(68 where, in anticipation of the nr -, W11111 we've chosen the same dimensional constant, P, that we used in the description of the interior spacetime and j > 2. With both metrics specified we now turn our attention to matching the spacetimes. Because both background metrics are the same, it will suffice to match the perturbations only. By imposing [hab] = 0, we arrive at three unique conditions: S+ fo[ca]Yoo + 2 00Yo = 0, (6-39) ro To rifo~ Yoo M~[Q]Yoo ro61M = 0, (6-40) [Q] = 0, (6-41) where we used fo = f(ro). Our choices for Q+ and Q- (6-38,6-37) ensure that the third condition is satisfied. We can solve Equations 6-39 and 6-40 to get equations for [c0] and dQP(i + j) [0][] (6-42) 6M~ = (ro 2M Yoo = -P(i + j)Yoo, (6-43) where we've made use of Equations 6-38 and 6-37. Next we will use the jump conditions to solve for p. Application of the jump conditions (Equations 6-14-618) is simplified by the fact that our metric perturbation is pure spin-0. Thus we only need consider the jump conditions for the spin-0 components of the metric perturbation (hiz, hi,, he, and hmm). For simplicity we will work with Equation 6-15, though it can be directly verified that the We may express this more concisely by introducing ~D = {D, P', a, a'}, so that Suppose now that we have a solution to the Teukolsky equation for Ie',, so that 0 is given by the left side of Equation 3-25 and S is given by the right side of Equation 3-23 (with Tab replaced with Sab) Wald's method then tells us that if Ot9 = 0, then hub = ISt is solution to the perturbed Einstein equations. Using Equations 3-33 we can compute StM: +mamb(P p)(P + 3p)}W + c.c., (3-35) where we've added the complex conjugate (c.c.) to make the metric perturbation real and W remains to be specified. Using Equations 3-33, it is clear that the adjoint of Equation 3-25 is [(p' p')(P + 3p) (8' r)(B + 37r) 31r'_]W = 0, (3-36) which is precisely the equation satisfied by ~!4/3 4' (c.f. Equation 3-29), previously obtained through separability considerations in the K~err spacetime. However, obtaining Equation 3-36 required no reference to separation of variables in a particular spacetime and thus applies to all type D spacetimes. It is important to note that although 9 satisfies the same equation as 4~i/3 4,g 11 iS not1 the perturbation of~ 4 or Ithe metlric~ it generates (Equation 3-35). In Chapter 5 we will explore W's connection to ~4 more Carefully. Though the derivation of Equation 3-35 was quite simple, it fails to yield any information about the gauge in which the metric perturbation exists. In this particular instance, it is fairly straightforward to verify that the metric perturbation we've been led vector([-1/2*Itatsin(theta)*2^(1/2)/(r-Itacstea) 0, -1/2*(r+I*atcos(theta))*2^(1/2), 1/2*I*(r^2+a^2)*sin(theta)*2^(1/2)/(r-I* acstea)) ndn= vector([1/2*2^(1/2)*((r^2-2*M*r+a^2)/((r+acotha)(rIaosh 1/2*2^ (1/2)* ((r+I*atcos(theta))*(r-Itatcotha)/r22Mr^)^( /2), 0, -1/2*atsin(theta)^2*2^(1/2)*((r^2-2*M*r+a2/(+acotha)(rI a*cos(theta))))^(1/2)]), Idn= vector([1/2*2^(1/2)*((r^2-2*M*r+a^2)/((r+acotha)(rIaosh -1/2*2^(1/2)*((r+I*atcos(theta))*(r-Ita*cstea)(^-*~~^)^ 1/2), 0, -1/2*atsin(theta)^2*2^(1/2)*((r^2-2*M*r+a2/(+acotha)(rI a*cos(theta))))^(1/2>)])\; tetupSB := \{1up = vector([1/2*B(t, r, theta, phi)*(r^2+a^2)*2^(1/2)/((r^2-2*M*r+a^2)* (+acotha)(rIaos theta)))^(1/2), 1/2*B(t, r, theta, phi)*2^(1/2)*((r ^2-2*M*r+a^2)/((r+I*a tcstea)*rIacsoht)) ^(1/2), 0, 1/2*B(t, r, theta, phi)*a*2^(1/2)/((r^2-2*M*r+a^2)*(r+I*atcotha)(rIacshe)) ^(1/2)]), nup = vector([1/2*(r^2+a^2)*2^(1/2)/(B(t, r, theta, phi)*((r^2-2*M*r+a^2)*(r+I*atcstea)*rIacos(theta)))^12) -1/2*2^(1/2)*((r ^2-2*M*r+a^2)/((r+I*a tcstea)*rIacsoht)) ^(1/2)/B(t, r, theta, phi), 0, 1/2*a*2^(1/2)/(B(t, r, theta, phi)*((r^2-2*M*r+a^2)*(r+I*atcstea)*rIacos(theta)))^12)) decomposition that takes into account both spin- and boost- weight. In the next chapter we will make some more precise statements in this direction. Recall our expression for the perturbed Einstein equations: 1 1 1 Sab c ~Cc ab ~a b cc ~c(a b)c gab ~c c dd ~c d cd) 2 2 2 By making the replacement V, i 0 and understanding hab aS referring to the tetrad components of the metric perturbation given in Equation 2-63, we arrive at the perturbed Einstein equations in GHP form: 1 1 1 Sab c~c ab -OaOb" cc c8O(a b)c gab c"Oc dd Oc d cd), (2-65) 2 2 2 which (right now, at least) don't look all that different! The tetrad components of Equation 2-65 for an arbitrary algebraically special background spacetime are given in Appendix B. Aside from the obvious cosmetic differences, there are several key distinctions between Equation 2-65 and the standard form of metric perturbation theory worth pointing out. First of all, our form lacks the background Einstein equations present in the standard treatment. Taking their place are the background GHP equations and Bianchi identities. Perhaps more importantly is the inherent coordinate independence. Coupled with the concepts of spin- and boost-weight, this allows for a certain structural intuition not present in coordinate based techniques. This point of view will be stressed throughout. Writing Equation 2-65 is one thing, but actually computing it is another question entirely, which we now turn our attention to. 2.6.2 GHPtools The Details To perform such a computation for an arbitrary background spacetime is no small task, even (or rather especially) in the standard tensor language. For this the aid of Maple was enlisted. Unfortunately, at the time the computation was performed, there were no Maple packages available for performing all such computations at the level of generality arising from a wave equation for the perturbed Riemann tensor, using standard methods [13]. In either case, the result, written here in Bci-;r-Lindquist coordinates, is Teukolsky's master equation (written here in accord with [14]) d 8 1 8 8 2 d (T2+ 2) +ta_ s(r-M)) -4s(r+iacos0) 8 8 1 8 8 +n" sin2 8 (Sin2 H iS COS 8 8 cos 8 8 cos 8 si2 x As/2 i' __ s~a/2C, s17 where s: = +2 corresponld to the W~eyl scalars ,',, anld I#2-4/3 4, TSpectively. Th'le Weyl scalars are perturbations of the extremal spin components of the curvature tensor. The significance of the Weyl scalar ~4 is that far away from the source of gravitational radiation ~4 N h+ ix, 18 where h+ and hx are the two polarizations of outgoing gravitational radiation in the transverse traceless gauge. Similar results hold for I',, and incoming radiation. For other values of s, solutions correspond to fields of other spin: s = 0 is the massless scalar wave equation, a = +1/2 the Weyl neutrino, a = +1 the Maxwell field, a = +3/2 the Rarita-Schwinger field, and so on. Note that angular separation necessarily involves time separation for a / 0. Separated solutions to Equation 1-17 are of the form ~, = e-iwqeim*,R(r),S(aw,1 8) (omitting the e, m and w subscripts). The angular functions, sS(aw, 8), are generally referred to as "spin weighted spheroidal harmonics". In the limit that aw = 0,, s,,em( reduce to the standard spin weighted spherical harmonics (cf. Appendix D), which are interrelated by the spin raising and lowering operators, a and 8' [15], developed in the following chapter. For aw / 0, solutions correspond to functions of different spin weight, but the ,S(aw, 8) no longer share common eigenvalues. Thus a metric reconstruction based on spin weight +2 functions would be incompatible with one based on spin weight 0 BI3c := proc() return(GHPconj(BI3())); end proc; BI3pc:= proc() return(GHPconj (GHPprime (BI3()))); end proc; BI4 := proc() return(DGHP(th(Psi4)-ethp(Psi3)-ethp(Phi2)tpPi0=sgmls2- *taul*Psi3+rho*Psi4-2*kappal*Phil0+2*sigma*hl +conjugate(rhol)*Phi20-2*conjugate(tau)*Ph2+ojgt~im)Pi2) end proc; BI4p := proc () return (GHPprime (BI4 ())) ; end proc; BI4c := proc () return(GHPconj (BI4())); end proc; BI4pc:= proc() return (GHPconj (GHPprime (BI4 ()))); end proc; The first step is to introduce new derivative operators P/ and 8I = 8 such that they commute with P when acting on quantities that P annihilates,3 [9, 1 P]xo = [P, 8]xo 0, [P, 8 ]xo 0, (4-11) where [a, b] denotes the commutator between a and b. The explicit form of the operators is given in Appendix C. The next step, the heart of Held's method, is to exploit the GHP equation Pp = p2, and its complex conjugate, Pp = p2, to express everything as a polynomial in terms of p and p, with coefficients that are annihilated by P. Held's method is then brought to completion by choosing four independent quantities to use as coordinates [56, 62]. In this work, we will not take this extra step. For type II spacetimes (and the accelerating C-metrics), this step has not been carried out, while for all remaining type D spacetimes, it has been carried through to completion [45, 46]. In a spacetime more general than type II, there is no possibility of having a repeated PND. When a repeated PND exists, we can appeal to the Goldberg-Sachs theorem [32] and set is = o- = Wo = ~1 = 0 in Equations B-1-B-7. Following Held's partial integration of Petrov type II backgrounds [56], we also perform a null rotation (keeping la fixed, but changing n") to set -r = 0. As a consequence, it follows from the GHP equations that -r' = 0. Now we are in a position to address the question of when the full IRG can be imposed. First we apply the 1- & gauge conditions in Equations 4-5 to Equations B-1-B-7. While most of the perturbed Einstein equations depend on several components of the metric perturbation, after imposing Equations 4-5, the expression for SiI depends only on hmm and the ll-component of the perturbed Einstein tensor simply becomes {(pD p p) + 2pp~hmm {((9 2p)(P + p p)}hmm = 8xri, (4-12) 3 Such quantities are denoted with the degree mark, o, as in Pxo = g P:=P-4; elif has (lo, conjugate (eta2)) then Q:=Q-4; elif (has (lo, xil) and not (has (lo, conjugate (xii)))) then P:=P+4; elif has (lo, conjugate (xii)) then Q:=Q+4; elif (has (lo, xi2) and not (has (lo, conjugate (xi2)))) then P:=P-4; elif has (lo, conjugate (xi2)) then Q:=Q-4; elif has(lo,h) then P:=P+0; Q:=Q+0; elif has(lo,hl) then P:=P+1; Q:=Q+1; elif has(lo,hn) then P:=P-1; Q:=Q-1; end if ; return(DGHP(subs(subs({p=P,q=Q,z=10},commsep); end proc; GHPcomm := proc (whichcom, solvef or, whichvar) local a,b; (a,b) :=getpq(whichvar); CHAPTER 1 INTRODUCTION Einstein's theory of general relativity, introduced in 1915, to this one of the final frontiers of fundamental physics. Since its inception progress in the field has been largely theoretical because of the tremendous difficulty inherent in making gravitational measurements. In particular, one of the most exciting and fundamental predictions of general relativity-the existence of gravitational waves-has remained elusive. Not for long. With ground based interferonieters such as LIGO, VIRGO and GEO online and the space based LISA mission in preparation, the detection of gravitational waves is all but inininent. These experiments bring with them the task of analyzing the data they collect. For some of the promising sources of gravitational waves, the collision of two black holes, the method of choice for data analysis, known as matched filtering, requires knowledge of the expected waveforms. In the past two years the field of numerical relativity has undergone a revolution and promises to provide the most accurate waveforms for situations involving the collision of two black holes of comparable nmasses-situations that require the use of full nonlinear general relativity. There is however, one promising source of gravitational waves that is currently out of reach for numerical relativity-the situation where the larger black hole is roughly a million times more massive than the smaller one, known as an extreme mass ratio inspiral, or EAIRI. This problem lies squarely in the realm of perturbation theory, the subject of the present work. In particular, the "solution" of the EAIRI problem requires moving beyond the test mass approximation of general relativity to describe the motion of the small black hole (treated as a particle in the spacetinle of the larger black hole because of the huge mass difference)-one must account for the first order corrections to the motion of the small black hole, due to self-force. The appropriate equations of motion have been determined in general by Mino, Sasaki and Tanaka [1] and Quinn and Wald [2] and are referred to as the has a solution given by 6M =[] (6-81) [0]~ =9 (6-82) [P] [Ca], (6-83) dS a2a Sin2 8 -[0], (6-84) d# (T2 82 a2 which is again easily seen to reduce to the Schwarzschild result in the appropriate limit. From these equations we can see clearly the issues involved in choosing a matching surface. First, because the left sides of Equations 6-81-684 are all constant, this must be reflected in the right sides as well, which currently exhibit dependence on both r and 0. Presumably, some choice of r = r (0) will enforce this, though it is currently unclear what that choice might be. Note that because of this, r = constant surfaces do not appear to be good for matching. What we have encountered appears to be an instance of a longstanding problem with matching the K~err solution to a source [74, 75]. Namely, there is no known matter solution that correctly reproduces the multiple structure of the full K~err geometry. In our problem, we're trying to force the issue by specifying both the metric and the source. On the other hand, because we're not matching the entire source, which includes quadrupole and higher moments, but only the non-radiated multipoles that merely take us from one K~err solution to the next, it is not clear that the matching (in this instance) should fail. Though we are unable to perform the matching here, we maintain that nothing forbids it. Most authors faced with this issue turn to the -lei-- rotation" approximation and keep only terms linear in a. In this approximation the K~err metric can be viewed as the first order perturbation of the Schwarzschild solution to the K~err solution. That is, the background is given by Schwarzschild plus a term identical to that in Equation 6-59. It end proc; tetupK := {1up = vector([(r^2+a^2)/(r^2-2*M*r+a^2), 1, 0, a/(r^2-2*M*r+a^2)]), mup = vector([1/2*Itatsin(theta)*2^(1/2)/(r+I*acstea) 0, 1/2*2^ (1/2)/(r+I*atcos(theta)), 1/2*I*2^(1/2)/(sin(theta)*(r+Ita*cos(thet)]) nup= vector([1/2*(r^2+a^2)/((r+I*atcstea) rIacos(theta))) -1/2*(r^2-2*M*r+a^2)/((r+I*atcstea)*rIacos(theta))) 0, 1/2*a/((r+I*atcos(theta))*(r-Ita*cos(thet)]) conjugate(mup) = vector([-1/2*Itatsin(theta)*2^(1/2)/(r-Itacstea) 0, 1/2*2^ (1/2)/(r-Itatcos(theta)), -1/2*I*2^(1/2)/(sin(theta)*(r-Ita*cos(thet)))} tetdnK := \{mdn = vector([1/2*Itatsin(theta)*2^(1/2)/(r+I*acstea) 0, -1/2*(r-Itatcos(theta))*2^(1/2), -1/2*I*(r^2+a^2)*sin(theta)*2^(1/2)/(r+I~acstea)) ndn= vector([1/2*(r^2-2*M*r+a^2)/((r+I*atcos(tea)(-acotha)) 1/2, 0, -1/2*a*(r^2-2*M*r+a^2)*sin(theta)^2/((r+Iacste))(-acoth ta)))]), Idn = vector([1, -(r+I*atcos(theta))*(r-Itatcos(theta))/(r^-*~~^) 0, -atsin(theta)^2]), conjugate(mdn) = vector([-1/2*Itatsin(theta)*2^(1/2)/(r-Itacstea) 0, CHAPTER 5 THE TEUK(OLSK(Y-STAROBINSK(Y IDENTITIES Having established the conditions for the existence of the radiation gauges, we will use the corresponding metric perturbations to establish some useful relationships between the perturbed Weyl scalars known generally (and quite loosely) as the Teukolsky-St arohinsky identities. Because Hertz potentials are solutions to the Teukolsky equation, these identities have immediate relevance for metric reconstruction in the IRG, both in the time-domain approach of Lousto and Whiting [25] and the frequency domain approach of Ori [2:3]. The original analysis of Teukolsky [11, 12] was based on the .I- i-i np u'tic form of the solutions of the separated angular and radial functions in the K~err spacetime as well as a theorem due to Starohinsky and Churilov [64]. Only later did C'I .!1.4 I-ekhar provide a full analysis, which is nicely summarized in his book [29]. Our analysis, however, will be entirely symbolic, involving only GHP quantities. This approach has the advantage not only of applying to a larger class of spacetimes, but displaying the structure inherent in the identities in a much more obvious way. A similar analysis of some of the identities we will discuss was previously undertaken in the NP formalism by Torres del Castillo [65] and later translated into GHP hy Ortigoza [66]. These prior analyses made use of the most general type D spacetime and translated back and forth between coordinate-based and coordinate-free expressions. In contrast, our approach will not make any reference to the choice of coordinates or a tetrad (other than requiring it to be aligned with the principal null directions). Because of this, our approach will showcase one of GHPtools' greatest strengths-the ability to commute several derivatives with relative ease. Our starting point is the (source-free) IR G metric perturbation given by P:=P+1; Q:=Q+1; elif has(10,conjugate(rho)) then P:=P+1; Q:=Q+1; elif (has(lo,rhol) and not(has(10,conjugate(rhol)))) then P:=P-1; Q:=Q-1; elif has(10,conjugate(rhol)) then P:=P-1; Q:=Q-1; elif (has(lo,kappa) and not(has(10,conjugate(kappa)))) then P:=P+3; Q:=Q+1; elif has(10,conjugate(kappa)) then P:=P+1; Q:=Q+3; elif (has(lo,kappal) and not(has(10,conjugate(kappal)))) then P:=P-3; Q:=Q-1; elif has(10,conjugate(kappal)) then P:=P-1; Q:=Q-3; elif (has(lo,tau) and not(has(10,conjugate(tau)))) then P:=P+1; Q:=Q-1; elif has(lo,conjugate(tau)) then the help of GHPtools) to the Teukolsky equations. They are [( -4p )(' ')- 8 4 -f'(' 7) S'_,', = 4xrTo, (3-25) [(D'~~~~~~~~ p '( ) 8 7 )( )-3']4 T4, (3-26) where, in terms of the components of the metric perturbation ,,- (8 -T')8 -f')zz (P p)(P p)hmm [(P P)(B 2r') +(di 7')(> 2p)]him (3-27) (D' p'( 27)h + (p' 7)(p' 2p')]h~), (-8 and where the parentheses, (), around the tetrad indices denote symmetrization. It is both interesting and important to note that, in the K~err spacetime, the coordinate description of Equation 3-26 does not lead to the separable equation discussed in (I Ilpter 1 (Equation 1-17). To obtain a separable equation, an extra factor of --4/3" muSt be brought in, resulting in the following expression: [(p' p')(P + 3p) (a' -r)(a + 37r) 3/' _]1' _4/3 4 --4/;:3:34. (3-29) Below we will see the same expression arising from very different considerations. 3.4 Metric Reconstruction from Weyl Scalars The solutions of the Teukolsky equation lead quite naturally to a metric perturbation in several different v- .--s. The original result, due to Cohen and K~egeles [20] used spinor methods. Shortly after that, C'!,l~!!. i .---1:! [54] obtained essentially the same result using factorized Green's functions. Some time later, Stewart [21] entered the game and provided a new derivation rooted in spinor methods. Eventually, Wald [55] introduced a Let us begin by considering a small perturbation, hab, of the Schwarzschild geometry. Thus our spacetime metric is 9ab = gb a Lb, (11) where gfbd ads b I _1)t- 2Ii _'d ~ 28 in2 tii' 2 _2) is the Schwarzschild metric in Schwarzschild coordinates. Putting Equation 1-1 into the Einstein equations and keeping only terms linear in hub leads us to the perturbed Einstein equations : 1 1 1 ~ab ~c c b ~a b cc c ~(a b)c + ab ~c c dd ~c d cd) = 0, 2 2 2 (1-3) where V, is the derivative operator compatible with the background geometry 1-2 and the indices are raised and lowered with the background metric. Henceforth we will refer to Sab aS the Einstein tensor, and the expression to the right of it as the Einstein equations (dropping the qualifier "perturbed" for brevity). Essentially every perturbative analysis of the Schwarzschild spacetime makes extensive use of its spherical symmetry. The first step in this direction is to decompose the components of the metric perturbation into scalar, vector and tensor harmonics. Heuristically, we write 'U2 respectively and the subscripts 81 S2 'Ul hab 2 3 'U2 vl v2 t+4 vl v2 t where s, v and t stand for scalar, vector and tensor, distinguish between the various scalars and vectors. Consider the metric of the two-sphere: _15) yABd A XB d2 Sin2 d2 return(DGHP(GE2)); end proc; GHP2p := proc () return(DGHP(GHPprime(GE2))); end proc; GHP2c := proc () return(DGHP (GHPconj (GE2))); end proc; GHP2pc:= proc() return(DGHP (GHPconj (GHPprime (GE2)))); end proc; GHP3 := proc () return(DGHP(GE3)); end proc; GHP3p := proc () return(DGHP(GHPprime(GE3))); end proc; GHP3c := proc() return(DGHP (GHPconj (GE3))); end proc; BIOGRAPHICAL SKETCH Larry was born in 1978, in El Paso, Texas. He is the eldest child of (the elder) Larry Price and Pamela Villa. At last count, he has approximately 6 siblings. Fr-om the ages of about five to twelve, he attended a funny sort of school where the students were all forced to dress the same and gather on Fil 1 .- to listen to a man in a dress read from a big book. He was treated well there, but his entry into the Texas public school system in the fifth grade proved to be a good move. In middle school, Larry realized he understood algebra much better than his teacher (who happened to also be the school's basketball coach), a point that he made clear in class at every opportunity. It goes without ?iing that his initial desire to publicly humiliate jocks subsequently grew into a much deeper interest in mathematics and physics. These interests were furthered in high school, where Larry explored other areas as well. Among these is the theater. Few people are aware that Larry has performed in leading roles in several musicals, as well as an operetta. Upon graduating high school in 1997, Larry decided that it would be best to get as far .li.-- ., from El Paso as he could. To this end, he attended a small liberal arts school named Reed College in Portland, Oregon, where he spent some of the best years of his life. Reed provided a valuable opportunity for Larry to further pursue the sciences and read some really great books at the same time. It also gave him the opportunity to interact with many interesting people from widely different backgrounds. It was there that Larry came in contact with Nick Wheeler, a truly unique individual who remains a trusted mentor. Alas, all good things must come to an end, and so Larry graduated from Reed with a B.A. in physics in 2001. With his path uncertain at the time, Larry decided to stay in Portland for the following year. There Larry tried his hand as a computational chemist for Schroidinger, Inc. The people there were fantastic and the p l.lllhacks weren't bad, but he need more which also defines Ro, a quantity annihilated by P. Then we can rewrite Equation 4-16 as hm =1ao+aoP P a + bo] (p + p). (4-21) 2 pp 2 In a similar fashion, we rewrite Equation 4-18 as (4-22) (8~'d + aa' "b" p"o pro a in which each coefficient in big square brackets is purely real. Now, suppose we have a particular solution for SiI = 0 (i.e., ao, ao and be are fixed) and our task is to solve for the components of the gauge vector which removes this solution. By comparing Equations 4-21 and 4-22 we see that, for any given (mo and (mo, we can fix (to (up to a solution of D (to = 0) via p1 0 o (a + ao -I(mo + igo~), (4-23) 2 2 and we can fix (no by setting 1 1, 1~~" -a~0 1 6 o= (o a.)n + be g' g o plol" -t /m m o (4-24) 2 2 2 to completely eliminate the nonzero hmm, thus imposing the full IRG while still leaving two completely unconstrained degrees of gauge freedom, (mo and (mo. Once in the IRG, Equations 4-23 and 4-24, with ao, ao and be set to zero and (mo and (mo arbitrary, give the remaining components of a gauge vector preserving the IRG. It is currently unclear how to take advantage of this remaining gauge freedom to simplify the analysis of perturbations in the full IRG. 4.4 Imposing the IRG in type D Type D background metrics are of considerable theoretical and observational interest since they include both the Schwarzschild and K~err black hole spacetimes. K~innersley first obtained all type D metrics by integrating the N. ein-! lIs-Penrose equations [33]. While the Rewriting the P piece and using Equation C-11 with p = 1 leads to 19(pim) + -r (1 + poft Cl = 0, (C-17) p p which, after substituting Equation C-3, the complex conjugate of Equation C-4 and Equation C-15 along with some re I1 llpil_ yields Integration then gives us m mol~ _I 0 lo a lo _~ ) lo _C19) p p and the solution for (m then follows from complex conjugation m7 = (mo ~o a_ o.(-0 p p Finally, we are in a position to deal with (n, by writing P'e + Des + (-r + -r')(m + (-r + -r')(m = 0, (C-21) in terms of Held's operators (Equations C-1, C-3 and C-4) as en>(I t~rt-I f- -6 P~+ ~-rllp p1)E (C-22) +4P ), ~+ (7 + v'r)(m + (7 + -y')(m = 0. Substituting Equations C-3, C-4, C-5, C-15, C-19 and C-20, rearranging terms and letting the dust settle leads to ~;=-p~ 1 1 -a o 1 1\ 2 2 pp ,I 1 1, (C- 23) [xro 8 + cto a or( a +l~ 2xo~mo + 2 o mo p p 2 2 > gdn := Idn(a)*ndn(b) + Idn(b)*ndn(a) mdn(a)*conjugate(mdn) (b) - > mdn(b)*conjugate(mdn) (a); > gup := subs({1dn=1up, ndn=nup, mdn=mup},gdn); gdn := Idn(a) ndn(b) + Idn(b) ndn(a) mdn(a) mdn(b) mdn(b) mdn(a) gup := lup (a) nup (b) + lup (b) nup (a) mup (a) m up (b) mup (b) m up (a) > tetcon(gdn*gup, {a,b}); Finally, GHPtools provides some functionality for translating expressions into NP expressions that can subsequently be converted to ordinary coordinate expressions. This functionality is provide by the aptly named procedure GHP2NP (), which takes as its input a GHP expression. The functionality provided by the procedure is limited to expressions involving at most two derivatives. Furthermore, the derivatives must appear in a specified order according to the following rules: (1) a and 8' must ahr-7i- appear to the left of P and P', (2) B must appear to the left of a' and (3) P must appear to the left of P'. Take the following example: > GHP2NP(th(thp(hln))+eth(ethp(hln))); DD(a(hln)) + E A(hln) + E A(hln) + 6(6(hln)) + /36(hln) + p316(hln) In order to aid in the conversion of such quantities into coordinate expressions, GHPtools contains, as lists of arrays, some commonly used tetrads in the K~err spacetime. They are: the K~innersley tetrad with indices up tetupK and down tetdnK, the symmetric tetrad (tetupS, tetdnS) and the symmetric tetrad boosted by a function B(t, r, 8, 4) and spun by a function S(t, r, 8, 4) (tetupSB, tetdnSB). These are called simply by invoking their names: *(times((delta p*beta + q*conjugate(betal)),op(1,0p(nops(w),w)))) elif(op(0,w)='eth') then (p,q):=getpq(op(1,w)); result:=result w + times((delta p*beta + q*conjugate(betal)),op(1,w)); elif(op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='ethp') then (p,q):=getpq(op(1,0p(nops(w),w)) ); result:=result w + w/op(0,0p(nops(w),w))(op(1,0p(nops(w),w)) *(times((conjugate(delta) +p*betal - q*conjugate(beta)),op(1,0p(nops(w),w)))); elif(op(0,w)='ethp') then (p,q):=getpq(op(1,w)); result:=result w + times((conjugate(delta) + p*betal q*conjugate(beta)),op(1,w)); else result :=result; end if; end do; return(NPexpand(result)); end proc; GE1:=th(rho)-ethp(kappa)=rho^2+sigmatconugesim)cngaekp) *tau-taul*kappa+Phi00; GE2:=th(sigma)-eth(kappa)=sigma*(rho+conugerh)-aptaucng (Equations 5-1 and 5-2) this begs the question of why we should concern ourselves with such trivialities. The answer is, in part, that these trivial perturbations represent the largest contribution to the self-force, as shown by Detweiler and Poisson [17]. Although it is unclear if such contributions persist in all gauge invariant quantities of interest, such as certain characterizations of the orbital motion of the particle [68], there is in fact a more compelling reason to be concerned with the non-radiated multipoles. In recent work, K~eidl, Fr-iedman and Wiseman [69] have looked at the problem of computing the self-force in a radiation gauge in the context of a static particle in the Schwarzschild spacetime. In their calculation, they found the perturbations of mass and angular momentum arising in the construction of a Hertz potential. Thus, although the Hertz potential cannot he used to determine these perturbations, it must still 1:0.0~.--" about them and they must he determined by some other means. In this chapter we will present a general prescription for computing the non-radiated multipoles. 1\ore specifically, we will consider the problem of computing the shifts in mass and angular momentum due to a point source in a circular (geodesic), equatorial orbit around a black hole. Specifically we are after expressions for 61M and 6a, the shifts in mass and angular momentum, in terms of the orbiting particle's mass, p, and orbital parameters. The idea is quite simple: match an interior spacetime, (g,, Af-), to an exterior spacetime, (g91, Af*), differing only in mass and angular momentum, on a hypersurface (of codimension 1), E,,, containing the perturbation. The basic conditions for a good matching are (1) that the metric is continuous across 27, and (2) the first derivatives of the metric are continuous except where the source is infinite. These conditions are compatible with Israel's quite general junction conditions [70]. Before we can do any ]rce Ib t.11r We must first determine the geometry of 27,. In spherically symmetric spacetimes, the obvious choice is the simplest-the (round) 2-sphere, as we'll see below in our calculation in Schwarzschild. For the K~err spacetime, be written her pE 2r, sin 8 sin(4 Rt)G(r ro), her = 2 pE r sin 8 cos(4 Rt)G(r ro), p-E bro = ~r, cos v cos(4/ -- Ot6(rV ro0), Note that the singular nature of this metric perturbation inherently excludes it from our analysis, as it destroys the continuity of the metric perturbation across E,. It is well known [7] that the gauge transformation leading to this description can be interpreted as a transformation from a non-inertial frame tethered to the central black hole to the center of mass reference frame. 6.2 Kerr In contrast to the situation in the Schwarzschild background, mass and angular momentum perturbations in the K~err background are much more complicated. There is, however, one simplifying feature of the mass and angular momentum perturbations. Namely, the fact that both perturbations are stationary. Therefore the angular dependence is not given by the spin-weighted spheroidal harmonics, sS(aw, 8, 4), but rather their aw = 0 limit-the spin-weighted spherical harmonics. The primary issue with treating the non-radiated multipoles in the context of matched spacetimes is the choice of the matching surface, E,. Most of our discussion will be focused on this issue. 6.2.1 Mass Perturbations In place of Birkhoff's theorem there is Wald's theorem [16], described earlier, assuring us that infinitesimal mass perturbations of the K~err solution lead to other K~err solutions (with infinitesimally different masses, of course) because such perturbations do not contribute the perturbations of I',, or tb4 (Which we will verify shortly). Thus we have the defines a null tetrad. It is important to note that there is some ambiguity implicit in the above assignment, e.g. we can swap the roles of z' and x" (or y") in the above definitions without changing the character (real or complex) of the null vectors or modifying their inner products. We will return to this issue later in this section. For simplicity, we introduce the following notation for our tetrad (borrowed from C'I 1.in b I-iekhar [29]): e*, = (1", na, m", ma), where the tetrad index (i) = {1, 2, 3, 4} = {1, n, m, m}. In a further attempt to avoid confusion we'll take spacetime indices from the beginning of the alphabet (a, b, c...) and tetrad indices from later in the alphabet (i, j, k...). Just as the vector index can be raised or lowered with the spacetime metric e" gab = 6i~b and e(ij,gab =6t), we may introduce a similar object for raising and lowering tetrad indices For a properly normalized (Equation 2-1) null tetrad 0 1 0 0 0 0 0 -1 0 0 -1 0. It then follows that we can express our spacetime metric as where 1(,nb) a ~l~ b + bna). and op(0,0p(1,0p(nops(w),w))) ='th') then (p,q):=getpq(op(1,0p(1,0p(nops(w),w))) ); result:=result w+ w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w) (op(1,0p(1,0p(nops(w),w)))))*(times((delt- (p+1)*beta +(q+1)*conjugate(betal)),times((DD - p*epsilon-q*conjugate(epsilon)), op(1,0p(1,0p(nops(w),w)))))); elif (op(0,w)='eth' and op(0,0p(1,w))='th') then (p,q):=getpq(op(1,0p(1,w))); result:=result w + times((delta (p+1)*beta + (q+1)*conjugate(betal)),times((DD p*epsilon - q*conjugate(epsilon)),0p(1,0p(1,w)))); elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='eth' and op(0,0p(1,0p(nops(w),w))) ='thp') then (p,q):=getpq(op(1,0p(1,0p(nops(w),w))) ); result:=result w + w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w) (op(1,0p(1,0p(nops(w),w))))) *(times((delta - (p-1)*beta +(q-1)*conjugate(betal)),times((Delta + p*epsiloni+q*conjugate(epsiloni)), op(1,0p(1,0p(nops(w),w)))))); elif (op(0,w)='eth' and op(0,0p(1,w))='thp') then (p,q):=getpq(op(1,0p(1,w))); result:=result w + times((delta (p-1)*beta + (q-1)*conjugate(betal)),times((Delta + p*epsiloni +q*conjugate(epsiloni)),0p(1,0p(1,w)))); comm:=ethp(th(z))=solve(COM2c(),ethp(th(z)) elif (dl='thp') and (d2='ethp') then comm:=thp(ethp(z))=solve(COM2p(),thp(ethp~)) elif (dl='ethp') and (d2='thp') then comm:=ethp(thp(z))=solve(COM2p(),ethp(thp~)) elif (dl='thp') and (d2='eth') then comm:=thp(eth(z))=solve(COM2pc(),thp(ethz); elif (dl='eth') and (d2='thp') then comm:=eth(thp(z))=solve(COM2pc(),eth(thpz); elif (dl='eth') and (d2='ethp') then comm:=eth(ethp(z))=solve(COM3(),eth(ethpz); elif (dl='ethp') and (d2='eth') then comm:=ethp(eth(z))=solve(COM3(),ethp(ethz); else error "Can't commute X1 and X2", di, d2; end if; # add up p and q values from the components of the metric perturbation P:=0 + 2*(numboccur(10,hll) + numboccur(lo,hlm) + numboccur(lo,hmm)) 2*(numboccur(10,hnn) + numboccur(lo,hnmb) + numboccur(lo,hmbmb)) + numboccur(lo,th) + numboccur(lo,eth) - numboccur(lo,thp) numboccur(lo,ethp); Q:=0 + 2*(numboccur(10,hll) + numboccur(lo,hlmb) + numboccur(lo,hmbmb)) 2*(numboccur(10,hnn) + numboccur(lo,hnm) + numboccur(lo,hmm)) + numboccur(lo,th) + numboccur(lo,ethp) - numboccur(lo,thp) numboccur(lo,eth); # now add up p and q values from all other objects # this is where we can modify the procedure to recognize new things if (has(lo,rho) and not(has(lo,conjugate(rho)))) then how = 0, (3-11) sin 08oe(sin Ohte + 8444)> = 0, (3-12) sin 08o(sin Ohro + 84&,4) = 0, (3-13) as the mode-independent expression of the RW gauge. Now we can transform this description into GHP language. It is a relatively straightforward process now to write the tetrad components of the metric perturbation (hiz, hi,, etc.) in terms of the coordinate components of the metric perturbation (htt, h,,, etc.) and invert the relations. With this knowledge in hand, it becomes evident that Equations 3-10 and 3-11 are simply combinations of Amm =0 and hm = 0. The effect of these conditions is to remove the spin-weight +2 pieces from the metric perturbation. After a quick look at the coordinate form of the a and 8' operators, we note that Equations 3-12 and 3-13 are combinations of B'hlm + c7l;.. = 0 and allt, + B'hm = 0, which restricts the form of the spin-weight +1 parts of the metric perturbation. Note that the essence of the RW gauge lies in the fact that all of the information about gravitational radiation gets pushed into the spin-1 components of the metric perturbation. In this language, it is natural to generalize these conditions to more general type D spacetimes on the basis of spin-weight considerations. The spirit of the RW gauge -II- -_ -r ;that we keep the requirement that no spin 2 components enter the metric perturbation. The requirement on the spin 1 components is easily generalizable by putting in pieces proportional to -r and -r' which both vanish in the Schwarzschild background. The Ricci tensor is represented by the following ten scalars: 1 1 oo0 = R1, 21 -24 , 2 2 = -(R1 R34), ~02 2 R33, 1 1 oi 2R13, 22 2 -R22, (2-12) 1 1 ~12 2 R23, 20 2 -R44 > to =R14, 1 ' 2 24 The field equations then follow from Equations 2-9 and 2-10. A full set of equations for the NP formalism is composed of the commutators, the equations involving dependence on matter, and the Bianchi identities. This is given in Appendix A. 2.2 GHP In 1973 Geroch, Held and Penrose (GHP) [9] introduced some convenient modifications of the NP formalism. Specifically, they identified the notions of spin and boost weight and make explicit use of an inherent discrete symmetry of the NP equations. In the NP formalism, there is an implicit invariance under a certain interchange of the basis vectors which GHP have built on through the introduction of the prime (') operation, defined by its action on the tetrad vectors: (2-13) A glance at Equations 2-6 and 2-7 -11- -- -; the adoption of a change in notation: and similarly for the directional derivatives of Equation 2-8 D' = a and 6' = 6. (2-15) to obeys P~hab = 0, (3-37) gab ab = 0, (3-38) which is known in the literature as the ingoing radiation gauge (IRG), an unfortunate name because ingoing radiation is carried by 1" and Equation 3-37 tells us that the metric perturbation is completely orthogonal to 16. Thus there is only outgoing radiation in the ingfoingf radiation gauge! Obtaining the gauge conditions in Equations 3-37 and 3-38 is more natural in the approaches of Cohen and K~egeles [20] and Stewart [21]. One startling aspect of the gauge conditions is that there are five of them. This being the case, we must be concerned about the circumstances under which the metric perturbation in the IRG is well-defined. This is the subject of the next chapter. Our derivation began with the Teukolsky equation for Ie',, Had we instead started with the Teukolsky equation for 4,/3 ~4, We WOuld be led to a metric~ perturbation in terms of a Hertz potential, 9', that satisfies the Teukolsky equation for Ie',, The resulting metric perturbation and gauge conditions are then simply the GHP prime of Equations 3-35, 3-37 and 3-38, respectively. In this case, the metric perturbation exists in the so-called outgoing radiation gauge (ORG). For the remainder of this work, we will focus our attention on the IRG metric perturbation, but all the results hold for the ORG perturbation as well. On a final note we remark that the Teukolsky equation for Ie',, (Equation 3-25) actually exists in the more general type II spacetimes, without its companion for ~4- I this case, Wald's method also leads to metric perturbation (in the IRG, no ORG exists here), with a potential, 9, satisfying the adjoint of Equation 3-25, which, in this instance, is not the equation for the perturbation of #4- return apply(ethp, f); elif op(0,f) = 'T' then return apply(ethp,f); elif op(0,f) = 'ln' then return ETHP(op(1,f))/op(1,f); else error "routine not built to handle that function: X1", op(0,f); end if; else error "routine not built to handle that type: %1", whattype(f); end if; end proc; DGHP := proc(expr) local result; result:=subs({th=THORN,thp=THORNP,eth=ETHeh=TPsubexr) return(expand(eval(result))); end proc; D_delta := proc(f) local i, rest, temp; if type(f, 'symbol') then return map( delta, f) elif type(f, 'constant') then 0 elif type( f, list ) then map( D_delta, f) elif type( f, set ) then map( D_delta, f) comparatively simple derivation of the same result. This is the approach we will follow here. Wald's method is centered around the notion of adjoints. Consider some linear differential operator, that takes n-index tensor fields into m-index tensor fields. Its adjoint, Lt, which takes ni-index tensor fields into n-index tensor fields is defined by no ...a, (/3)a,...a,, (gtCa)bl...b, /3by...b,, = aa, (3_30) for some tensor fields c1 "l" and /3bl...b" and some vector field s". If Lt = then L is self-adjoint. An important property of adjoints is that for two linear operators, L and M./1 (MZ/)t = M2/t t. Now let 8 = S(hub) denoted the linear Einstein operator, S the operator that gives either of the Teukolsky equations from 8 (Equation :32:3 or :324), O = O( ~,, or and T = T(hub) the operator that acts on the metric perturbation to give I',, or (Equation :32:3 or :324). Then the Teukolsky equations can he written concisely as SE = OT. (:3-31) It follows by taking the adjoint that 'St atS =a ItOt, (:332) where we have used the fact that the perturbed Einstein equations are self-adjoint. Thus, if W satisfies OtW = 0, then StW is a solution to the perturbed Einstein equations! This remarkably simple and elegant result holds for any system having the form of Equation :3-31, whenever 8 is self-adjoint. In order to apply this result to the Teukolsky equation we note that scalars are all self-adjoint and the adjoints of the GHP derivatives are given by elif (has(lo,phil) and not(has(10,conjugate(phil)))) then P:=P-pp; Q:=Q-pq; elif has(10,conjugate(phil)) then P:=P-pq; Q:=Q-pp; elif (has(lo,chil) and not(has(10,conjugate(chil)))) then P:=P+4; elif has(10,conjugate(chil)) then Q:=Q+4; elif (has(10,chi2) and not(has(10,conjugate(chi2)))) then P:=P-4; elif has(10,conjugate(chi2)) then Q:=Q-4; elif (has(lo,omegal) and not(has(10,conjugate(omegal)))) then P:=P+4; elif has(10,conjugate(omegal)) then Q:=Q+4; elif (has(10,omega2) and not(has(10,conjugate(omega2)))) then P:=P-4; elif has(10,conjugate(omega2)) then Q:=Q-4; elif (has(lo,etal) and not(has(10,conjugate(etal)))) then P:=P+4; elif has(10,conjugate(etal)) then Q:=Q+4; elif (has(10,eta2) and not(has(10,conjugate(eta2)))) then functions. This incompatibility does not arise for Schwarzschild, where reconstruction from solutions of the RW equation can translate into comparable metric reconstruction from the Weyl scalars, since there is a unique way of representing tensors on the sphere. The spin weighted spherical (and spheroidal) harmonics fail to be defined for e < |8| and thus the Teukolsky equation can give us no information about the -E = 0, 1 modes. This is not a surprise since Iel, and #'4 are comporterts of the curvature tensor, which carries information about the quadrupole (and higher multiple) generated gravitational waves. In fact, Wald has shown [16] that for vacuum perturbations each of I',, and ('4 is SUffleient to characterize the perturbation of the spacetime, up to shifts in mass and angular momentum. In Schwarzschild, these lower multiple moments can he expressed appropriately in terms of spherical harmonics using the RW formalism, but any comparable expressions for the K~err case would be incompatible with metric coefficients constructed from spin weight +2 functions (i.e., they would be expressed in different bases). Yet, these low--A multiple moments are urgently sought, since they convey information about the energy and both the axial and non-axial components of the angular momentum of a particle in orbit around the black hole. 1\oreover, in recent calculations demonstrating the precise relation of the -E = 0, 1 multipoles in Schwarzschild to shifts in the mass and angular momentum, Detweiler and Poisson [17] emphatically point out that such shifts are just as important as the radiating multipoles for describing the motion of a small black hole orbiting a supermassive black hole. The non-radiated multiple moments are the subject of C'!s Ilter 6. Solutions of the Teukolsky equation lead quite naturally to metric perturbations through the use of Hertz potentials which solve Equation 1-17. We now turn our attention to this subject. 1.3 Metric Perturbations of Black Hole Spacetimes The first explicit solutions for metric perturbations given in terms of Hertz potentials were written down by C'!,l~!!. 1,.---1:! [18] and Cohen and K~egeles [19]. This work was either alone or in conjunction with various other techniques, will help to clarify the problem enormously. Another avenue worth pursuing is the commuting operator associated with the K~illing tensor due to Beyer [44] (cf. ('! .pter 1 ). Recall that Beyer's operator commutes with the scalar wave equation in K~err. It is very tempting to think that such an operator would exist for the Teukolsky equation as well. The GHP formalism, and GHPtools (of course), provide the ideal environment in which to study such questions. Furthermore, in the context of work performed by Jeffryes [79] concerning the implications of the existence of the K~illing spinor (which includes a discussion of the Teukolsky-Starobinksy identities), it is natural to think that such an operator may in fact shed some new light on the Teukolsky-Starobinsky identities in the form presented in ('! .pter 5. Additionally, the existence of a generalization of Beyer's operator carries with it the possibility of new decomposition of functions in the K~err spacetime--just as the existence of the K~illing vectlors and lead to separation in t and cf according to e-ime and e""m* (respectively),l the eigenfunctions of a generalized Beyer operator may provide a new separation of variables in the K~err spacetime. This is certainly a possibility worth pursuing. Finally, both GHPtools and our form of the perturbed Einstein equations are entirely general and ready for use by researchers interested in more general (or even more specialized) backgrounds than Petrov type D. In particular, the class of type II spacetimes seems a likely candidate for further analysis, especially with the aid of the integration technique of Held. We have only begun to scratch the surface of the wide v-1I r ii of problems these tools can help solve. 1. +-{(( 9 p')(' r'' + F'a) +2pa m 1. +- { (- + p )8 p) ) (8 1 + 2-' -r) + 2f '(D 2p)h, - 8 7 7' p 2p'r knr,B4 1. { (-a(' p' r) s + a'( + R'e' } ha(r'--r) 1. +- { (9 p + ) (8 2) p(P' 27' )p + 2'(P p) }h~a- 2z 2('-3rB+-'21-- +4r)--(1-2r), 1. +-P { (-(' p' p) (a' + ) + (P' r') + (a) p-' + ( 7~ + 2 ')a' (('- 2p'a' +5 ( 9 2p'f }hy a +{'(p'~ 2p' -'r') 2-r)r + a'4 Int)hl +(('- {( ')(> 2p') + -s(B 27 + 27')p + -r('(8 47' + 27) p)~ +-{-r '( 8' 27 + -r ( 2p+ p -2 ('- }hm These results are to be expected because of the axisymmetric nature of both the perturbations and the background spacetime. We now turn our attention to the mass and angular momentum perturbations in the Schwarzschild background. 6.1 Schwarzschild The Schwarzschild spacetime provides the perfect tested for our technique. 1\oreover, because of the spherical symmetry of the background, matching the spacetime is quite straightforward. In this case we can ah-li-s choose the matching hypersurface, Ez>, to be a (round) 2-sphere and exploit the orthogonality and completeness of the spin-weighted spherical harmonics to smear out the delta source on Ez,. The only caveat is that we must choose Ez, outside of the innermost stable circular orbit. If the location of Ez, is ro, then this amounts to requiring ro > 6Af. 6.1.1 Mass perturbations Our first task is to construct a suitable description of source-free mass perturbations of the Schwarzschild spacetime. We will then glue two such spacetimes together, as described above. We will write the Schwarzschild metric as d~s2 = d2 f-1 2 ,2 d2 Sin2 8d 2) (628) where f = 1 2Af/r. According to Birkhoff's theorem, the only static, spherically symmetric solution to the Einstein equations is the Schwarzschild solution. Thus, we are assured from the outset that perturbing the mass will simply lead us to another Schwarzschild spacetime with a mass At + 61f. The nonzero components of the corresponding metric perturbation are given by htt -26M r (6-29y) which is easily obtained hv linearizing a mass perturbation of Equation 6-28. In order to characterize mass perturbations more generally, we will introduce more freedom by Equations 4-5 require l"(hab ~(a;b)) = 0, (4-8) where the semicolon denotes the covariant derivative. In terms of components this reads 2P61 = hul, (4-9) (P + p)(m + (B + ')ll = him, (P + P)(m~ + (8' + 7')(: = him. Similarly, for the trace condition in Equation 4-6 to be satisfied by the gauge transformed metric, we require a'(m + am + (P' + pl)(1 + (P + P)(n = hmm. (4-10) Any extra gauge transformation that satisfies l"~((;b) = 0-solves the homogeneous form of Equation 4-9preserves the four 1- h gauge conditions in Equations 4-5. This is what is meant by residual gauge freedom. We will explicitly use this residual gauge freedom to impose the 1- h and trace conditions simultaneously, thus establishing the IRG. We will find that some gauge freedom still remains, as explained in Section 4.3. Now, we turn our attention to the general case of type II background spacetimes. 4.2 Imposing the IRG in type II In order to show that residual gauge freedom can be used to impose the IRG, we need to solve for the residual gauge freedom as well as examine any perturbed Einstein equation that might impede the imposition of the trace condition of the IRG. For this, we turn to a coordinate-free integration method develop by Held. Rather than give a detailed explanation, we present the basics and refer the interested reader to the literature for an in-depth account [45, 46]. The resulting proposal for a generalized RW gaugfe is hmm = 0, hmm = 0, (3-14) (B + atr + b-r')hlm + (8' + atr + b-r')hlm = 0, (8' + b-r + a-r')hm + (B + b-r + a-r')hm = 0, where a and b are (generally complex) constants that must be determined by some other means. Note that the form of Equations 3-14 is restricted by requiring the gauge restrictions to be invariant under both prime and complex conjugation. The full utility of the generalized RW gauge remains to be explored, but it is clear that any simplification it brings will apply uniformly to all type D spacetimes. 3.2.2 The Regge-Wheeler Equation With the pieces in place, we turn our attention to the odd-parity perturbations of the Schwarzschild spacetime. Starting with the description of the background, we have p = p, p' = p', and '_= ,(3-15) with all other background quantities vanishing, so the situation is immediately simplified. Next we proceed with the parity decomposition by writing the components of the metric perturbation as, for example, him = h +Ib, him h""" ihgg, etc. Note the relative minus signs between the odd-parity bits and their complex conjugates. From here on we will specialize to odd-parity and thus drop the "odd" labels and factors of i since no confusion can arise. With this specialization, our gauge conditions now read: hmm 0 kmm = 0 (3-16) a'hlm c7le.. = 0 B'hm cll, = 0. The field equations are obtained from the splitting of the Riemann tensor into a trace-free part and its traces according to 1 1 Rabcd Cabcd + @ac bd + bd ac gbc ad gad bc) acgbd gbcgad) R. (2-9) 2 2 where Cabcd, abcd, ab and R denote the Weyl tensor, Riemann tensor, Ricci tensor and Ricci scalar, respectively. Since both the Ricci tensor and the Ricci scalar vanish in the absence of sources, the Weyl and Riemann tensors are identical in source-free spacetimes. In that sense the Weyl tensor represents the purely gravitational degrees of freedom. The Riemann tensor is then expressed purely in terms of the spin coefficients and their derivatives by contracting all four vector indices with e )'s and making use of the Ricci identity, (Ve~b Vb a = Rabcd~d = abcd~d, (2-10) where vd is an arbitrary vector. In four dimensions the Riemann tensor has twenty independent components and the Ricci tensor has ten, leaving the Weyl tensor with ten independent components. In the NP formalism, this translates into five complex scalars: n = Cabcd a blc d I = -Cabcd a blc d ',_=-Cabcdla blc d + a b c d), (2-11) = Cabcdanb cnd ~4 -abcd Ra b c d 40 by the addition of the term V" b b and therefore, in the Lorentz gauge, contributes nothing to the fields. In practice, Equation 1-20 is used to reduce the Hertz hivector potential to a single complex (or two real) scalar potential(s). Herein lies the power of the method. However, moving to curved-space naturally complicates things. While the wave equations are modified to include curvature pieces, the transformation in Equation 1-20 is retained (see Cohen and K~egeles [20] and Stewart [21]). As a result, the field equations are still satisfied and the six components of Hab are still reduced to two, but the transformation in Equation 1-20 explicitly breaks the Lorentz gauge because derivatives no longer commute. In this way a new gauge is introduced that brings with it complications for the inclusion of sources. The necessary and sufficient conditions for the existence of this gauge are the subject of C'!s Ilter 4. 1.3.2 The Inversion Problem for Gravity The formulation of the gravitational Hertz potential proceeds analogously to that of its (flat space) electromagnetic counterpart, with a few differences. For one, the result is a metric perturbation in one of two complimentary gauges. Additionally, the potential itself is a solution to the Teukolsky equation for s = +2 (or s = -2; the choice of the sign of s determines which gauge the metric perturbation is in), though it is not the curvature perturbation of the metric perturbation it generates. In analogy to the electromagnetic example above, the components of the metric perturbation are given by two derivatives of the potential. The natural language in which to express the metric perturbation arising from the Hertz potential is again the spin coefficient formalism of Newman and Penrose [22], or its modification due to Geroch, Held and Penrose [9]. Thus we postpone the formal development of the subject until ('I Ilpter 3, when the necessary formalism is in place, and instead offer an overview of the general process and documented research on the topic of reconstructing the metric perturbation from solutions to the Teukolsky equation (assuming the form of metric perturbation is prescribed), which we will refer to as the inversion problem. P:=P-1; Q:=Q+1; elif (has(lo,taul) and not(has(10,conjugate(taul)))) then P:=P-1; Q:=Q+1; elif has(10,conjugate(taul)) then P:=P+1; Q:=Q-1; elif (has(lo,sigma) and not(has(10,conjugate(sigma)))) then P:=P+3; Q:=Q-1; elif has(10,conjugate(sigma)) then P:=P-1; Q:=Q+3; elif (has(lo,sigmal) and not(has(10,conjugate(sigmal)))) then P:=P-3; Q:=Q+1; elif has(10,conjugate(sigmal)) then P:=P+1; Q:=Q-3; elif (has(10,Psi0) and not(has(10,conjugate(Psi0)))) then P:=P+4; elif has(10,conjugate(Psi0)) then Q:=Q+4; elif (has(lo,Psil) and not(has(10,conjugate(Psii)))) then P:=P+2; elif has(lo,conjugate(Psii)) then APPENDIX C: INTEGRATION A LA HELD We provide details of the integration that lead to Equation 4-17 and 4-25. As it turns out, the type II calculation is actually much simpler than the the type D calculation because it uses a tetrad in which -r = -r' = 0. Therefore we will work out the type D calculation in detail and the type II result mostly follows by setting certain quantities to zero, as indicated below. We will need some results (and their complex conjugates) from the integration of the type D background: 8 p -o_ o 2 C1 i~p 1 1 o 2 -0 2_ 2 2 1 1 1 11 +Tto "~2 2iT 0T~ 0t 2 22p +ToO, (C-2) -r = -Wo" cto o rpp, (C-3) 92 0 3. (C-5) As noted in the text, xo" / 0 leads to the accelerating C-metrics, which we include for full generality. Henceforth the corresponding quantities in type II spacetimes can be obtained by setting -ro a r = cto 4 0 and Wo 420~1 in the type D result. Thus, in type II 1 This arises from the fact that in type D spacetimes there is only one non-vanishing Weyl scalar, 92. Ill type II spacetimes, however, both 93 and 94 arT in genOTra alSO nonzero. Though we do not refer to any of the other Weyl scalars in this work, we would like maintain agreement with the standard conventions. elif type( f, '=' ) then map( THORN, f) elif type( f, '+' ) then map( THORN, f) elif type( f, '*' ) then rest := mul(op(i,f), i=2..nops(f)); THORN(op(1,f))*rest + op(1,f)*THORN(rest); elif type( f, '^' ) then op(2,f)*0p(1,f)^(op(2f)-1)*THRNORN(op(,f) elif type( f, function ) then if op(0,f) = 'th' then temp:=THORN(op(f)); return map(THORN, temp); elif op(0,f) = 'thp' then return apply(th, f); elif op(0,f) = 'eth' then return apply(th, f); elif op(0,f) = 'ethp' then return apply(th, f); elif op(0,f) = 'conjugate' then return apply(th, f); elif op(0,f) = 'T' then return apply(th,f); elif op(0,f) = 'ln' then return THORN(op(1,f))/op(1,f); else error "routine not built to handle that function: %1", op(0,f); end if; this subject in OsI Ilpter 3. Continuing in our cartoon language (Equation 1-4), we now consider the two sectors of the metric perturbation independently, writing 00v dd yodd d 0 0 vodd Uodd ( 0 odd ?Ll todd todd odd ?Ll todd todd and 81 82 even Deven even 82 'U3 U] even even even 84 even even .. even even S4- The final step before appealing to the Einstein equations consists of choosing a gauge. Equation 1-3 is invariant under the transformation hub i ab ( @)~ab = ab a~b + b a, (1-12) where (a is an arbitrary vector and 4~ is the Lie derivative. Taking the odd-parity sector as an example, the R;- ear--Wheeler gauge vector takes the form (" = (0, 0, AeAB BY~) Im)3 where A is a function chosen so that the odd parity part of the metric perturbation 1-10 takes the form 0 0 0 v dd 0 0 O r 0 00 0 vodd ?L" 0 0. Similar simplifications arise in the even-parity sector. and op(0,0p(1,0p(nops(w),w))) ='th') then (p,q):=getpq(op(1,0p(1,0p(nops(w),w))) ); result:=result w + w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w) (op(1,0p(1,0p(nops(w),w)))))*(times((DD -(p+1)*epsilon -(q+1)*conjugate(epsilon)),times((DD p*epsilon - q*conjugate(epsilon)),0p(1,0p(1,0p(nops(w)w))) elif (op(0,w)='th' and op(0,0p(1,w))='th') then (p,q):=getpq(op(1,0p(1,w))); result:=result w + times((DD (p+1)*epsilon (q+1)*conjugate(epsilon)),times((DD - p*epsilon q*conjugate(epsilon)),0p(1,0p(1,w)))); elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='thp' and op(0,0p(1,0p(nops(w),w))) ='thp') then (p,q):=getpq(op(1,0p(1,0p(nops(w),w))) ); result:=result w + w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w) (op(1,0p(1,0p(nops(w),w)))))*(times((Delt + (p-1)*epsiloni+(q-1)*conjugate(epsiloni))imsDet +ptepsiloni+q*conjugate(epsiloni)), op(1,0p(1,0p(nops(w),w)))))); elif (op(0,w)='thp' and op(0,0p(1,w))='thp') then (p,q):=getpq(op(1,0p(1,w))); result:=result w + times((Delta + (p-1)*epsiloni+(q-1)*conjugate(epsiloni))imsDet +ptepsiloni + q*conjugate(epsiloni)),0p(1,0p(1,w)))); elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='th' REFERENCES [1] Y. Minor, M. Sasaki, and T. Tanaka, Phys. Rev. D55, 3457 (1997). [2] T. C. Quinn and R. M. Wald, Phys. Rev. D56, 3381 (1997). [3] S. Detweiler and B. F. Whiting, Phys. Rev. D67, 024025 (2003). [4] B. F. Whiting and L. R. Price, Class. Quant. Gray. 22, S589 (2005). [5] T. R;--- and J. A. Wheeler, Phys. Rev. D108, 1063 (1957). [6] F. Zerilli, Phys. Rev. Lett. 24, 737 (1970). [7] F. J. Zerilli, Phys. Rev. D2, 2141 (1970). [8] K(. Thorne, Rev. Mod. Phys. 52, 299 (1980). [9] R. Geroch, A. Held, and R. Penrose, J. Math. Phys. 14, 874 (1973). [10] S. A. Teukolsky, Astrophys. J. 185, 635 (1973). [11] W. Press and S. Teukolsky, A-r ~1inph--s. J. 185, 649 (1973). [12] S. Teukolsky and W. Press, Astrophys. J. 193, 443 (1974). [13] M. P. Ryan, Phys. Rev. D10, 1736 (1974). [14] B. F. Whiting, J. Math. Phys. 30, 1301 (1989). [15] J. N. Goldberg, A. J. MacFarlane, E. T. N. i.--us! lIs, F. Rohrlich, and E. C. G. Sudarshan, J. Math. Phys. 8, 2155 (1967). [16] R. M. Wald, J. Math. Phys. 14, 1453 (1973). [17] S. Detweiler and E. Poisson, Phys. Rev. D69, 084019 (2004). [18] P. L. Cl!!. .1,....---1:!, Phys. Rev. D11, 2042 (1975). [19] J. M. Cohen and L. S. K~egeles, Phys. Rev. D10, 1070 (1974). [20] L. S. K~egeles and J. M. Cohen, Phys. Rev. D19, 1641 (1979). [21] J. M. Stewart, Proc. Roy. Soc. 367, 527 (1979). [22] E. Newman and R. Penrose, J. Math. Phys. 3, 566 (1962). [23] A. Ori, Phys. Rev. D67, 124010 (2003). [24] L. Barack and A. Ori, Phys. Rev. D64, 124003 (2001). [25] C. O. Lousto and B. F. Whiting, Phys. Rev. D66, 024026 (2002). interior perturbation is then characterized by ~1Y10 = r sin 0. (6-57) Finally, it is easy to set kg ~8,S 2r-S =0, by imposing S(t, r) = T2S(t). Note that because of the quadratic dependence on r, we cannot perform this gauge transformation in the exterior spacetime if we wish to preserve .I-i-m!hlli' c flatness. This is not a problem because the angular momentum perturbation provides the necessary freedom for matching. Finally, the piece in h4 is proportional to the time derivative of S(t, r), which -II__- -is we choose S(t) = yt, to keep the perturbation static. In summary, we have for the interior and exterior metric perturbations h.,o =72 Sin 8 11, (6- 58) 26aM~ sin2 8 "t4 (6-59) with all other components vanishing. Continuity of the metric perturbation ([hab] = 0) requires yra =Yl (6-60) 2M~ sin O' where we've used the equality of azY~o to expand Y,$. As before, the radial jump conditions will determine y. In this case we'll use the odd-parity (imaginary) part of Equation 6-17. The relevant tetrad components are given by: iyr2 1 10 him- him = hm him- 12(-1 &t h = ht h' ir YIO II (6-62) Im Im am am p~2 1/2 ' 752 (Yi"ilG~pL 16xr Tim Ti =r0/ O)s COS 8)6(4 Ot). (6-63) one should in fact expect a relative sign because of the occurrence of 1',, 4 and its complex conjugate in the same expression. 1\oreover, this must occur even in the K~err spacetinle, where we have the real-intaginary separation instead of the parity separation. Such a consideration makes clear the obvious advantage of treating the Teukolsky-Starohinsky identities in terms of the fundamental GHP quantities. Beginning at this level and then performing the separation of variables allows for no ambiguity in the resulting expressions. Appendix A. Generally speaking, (" is complex, and its real and imaginary parts satisfy Equation 2-28 independently [36], so all type D spacetimes possess two independent Killing vectors. These two K~illing vectors each give rise to a constant of motion along a geodesic. In other words, if u" is tangent to a geodesic (Ub bUa = 0), then (su" is conserved along u": =0, (2-29) where the first term vanishes as a consequence of (K~illing's) Equation 2-28 and the second because u" is tangent to a geodesic. In addition to the existence of two K~illingf vectors, the K~illingf spinor also gives rise to the conformal K~illing tensor [35, 37]: Pub XABXA'B' -" 2 T -1/3 (lanb) M mbOm)), (2-30) which also exists in every type D background. The conformal K~illing tensor is alternatively defined as a solution to V(cPub)= 09(ab d c)d. (2-31) Conformal K~illing tensors are useful because they give rise to conserved quantities along null geodesics. If k" is tangent to a null geodesic (kb bk" = 0 and k'k, = 0) then the quantity Pubk'kb is COnSerVed along k": keVe(Pubk'kb) = k'kbkeVePub + 2Pubkek("Vekb) =k'kbkeV~cPub) (k, k")kcV bPbc of the perturbation (6Q) and the background (4~Q). Quantities that satisfy 4~Q = 0 for every (0 are therefore called gauge invariant. It is straightforward to see that the perturbation of Q is gauge invariant if and only if: (1) Q vanishes in the background, (2) Q is a constant scalar in the background or (3) Q is a constant linear combination of K~roenecker deltas. This is a result originally due to Sachs [47]. A direct consequence of this fact is that the metric perturbation, arguably the most fundamental quantity we deal with, fails to be gauge invariant. Fortunately, type D spacetimes come equipped with two gauge invariants, I,, and tb4, Which have simple expressions in terms of the components of the metric perturbation. As we will see, appropriate use of gauge freedom simplifies our computations tremendously. 2.6 GHPtools A New Framework for Perturbation Theory With the basic formalism in place, we are ready to present the tools that form the basis of the subsequent chapters. The motivation for our framework comes from two places: (1) the desire to take advantage of gauge freedom in standard metric perturbation theory and (2) the success of the GHP formalism in perturbation theory. As mentioned in the previous chapter, gauge freedom proved absolutely crucial for the RW analysis and that of Cohen & K~egeles [20], C!!. I.1, i.---1:! [18], and Stewart [21], and it will certainly pll li- a central role in any future description of metric perturbations. The second ingredient, the GHP formalism comes with several advantages. First of all, the inherent coordinate independence and notational economy makes calculations in general spacetimes tractable. Furthermore, by virtue of the Goldberg-Sachs theorem, we can deal with the entire class of type D spacetimes at once. Additionally, spin- and boost- weights provide useful bookkeeping and, as we'll see, a useful context for understanding the roles that various quantities pll li-. Last but not least, the use of a spin coefficient formalism has proved absolutely crucial for studying perturbations of anything other than spherically symmetric spacetimes. We will put these ideas together to compute the perturbed Einstein equations in a mixed tetrad-tensor form. This is the heart of our work. type D spaces in C'!s Ilter 1. Perhaps the best example of this is our proof of the existence of radiation gauges in sourcefree regions of spacetime. Our form of the Einstein equations and Held's integration technique is a powerful combination that allowed us to prove the result in arbitrary type II backgrounds, where the background integration isn't even complete. Finally, our treatment of the non-radiated multipoles demonstrates the power of our framework when combined with existing techniques. Our results in the K~err spacetime represent the first attempt at treating this part of the perturbation. Though we were unable to obtain the description in terms of a matched spacetime, we nevertheless provided a perturbation suitable for use in metric reconstruction. 7.2 Future Work For all the generality inherent in the framework we developed, the applications we presented were narrowly focused around the problem of metric reconstruction in the K~err spacetime. This leaves many problems to be explored, both within the realm of metric perturbations of K~err and otherwise. We detail some of these below. Perhaps most pressing is the generalization of our result for the non-radiated multipoles in the K~err spacetime to encompass more general orbits. In particular, orbits not lying in the equatorial plane are of particular interest. Such orbits necessarily contain off-axis angular momentum, which in turn are widely thought to be related to Carter's constant (associated with the K~illing tensor). For such orbits the K~omar formulae fail to completely characterize these off-axis angular momentum components, so it is clear that we must look elsewhere for a solution. One potential avenue for progress is the Einstein equations themselves. As we noted in the previous chapter, mass and angular momentum perturbations are both stationary perturbations with angular dependence characterized by the spin-weighted spherical harmonics. The simplifications this brings for working with the Einstein equations is immense and may prove to make the problem tractable, without recourse to purely numerical methods. In any case, it seems clear that our framework, which make it straightforward to see that for spacetimes without acceleration Equation 2-27 is real up to a complex phase (e2ic). NOte als0 that ( ,= -(e. What happened to the other (linearly independent) K~illing vector? It is given by Ob a ab 1 / ( 7 -13 ic -1/3 2 b ;*r [e-'ic -1/3: eCic -:1/32 /'mb- Tmb) (2 -43) Proving that this expression satisfies K~illingf's equation in general is a bit involved, and since we'll have no direct use for Equation 2-43 in subsequent chapters, we refer the interested reader elsewhere [36] for details. Once again, using Equations 2-42, it is straightforward to see that Equation 2-43 is real up to a phase. Using the K~innersley tetrad in the K~err spacetime, Equation 2-43 becomes rib -b ~ b (2-44) where t" is the timelike K~illing vector and *" is the axial K~illing vector. Because rib is proportional to a, it clearly vanishes in the Schwarzschild spacetime. This can also been seen by noting that, in the Schwarzschild spacetime, -r = -r' = 0 and thus comparisons of Equations 2-27 and 2-43 reveal that the two K~illing vectors are not linearly independent [42]. In [36] it is shown how one can infer spherical symmetry from this fact. 2.3.3 Commuting Operators An important property of K~illing vectors is the fact that they commute with all of the tetrad vectors: 4)gab = 2V(aib) = 0 = 24~(l~anb) m(amb)) where the first line follows from the definition of the K~illing vector and the second and third from Equation 2-3. By contracting the last line with each of the tetrad vectors and Integration then results in 1 1= 1 $ P1 1\ 2 2 2 pp2/ [xo go ,1 1 1 ~; -(8+0) -(8 +o) ro+- -+- DIr p p 2 p p , (C 24) P P 1 1 1l .1 and our task is complete. 1/2), 0, -1/2*B(t, r, theta, phi)*atsin(theta) ^2*2^ (1/2)*((r^2-2*M*r+a2/(rIacstha) r- a*cos (theta))))^ (1/2>)])\; end module: (p-1)*betal -(q-1)*conjugate(beta)),times((Delta + p*epsiloni + q*conjugate(epsiloni)),0p(1,0p(1,w)))); elif(op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='th') then (p,q):=getpq(op(1,0p(nops(w),w)) ); result:=result w + w/op(0,0p(nops(w),w))(op(1,0p(nops(w),w)) *(times((DD p*epsilon -q*conjugate(epsilon)), op(1,0p(nops(w),w)) )); elif(op(0,w)='th') then (p,q):=getpq(op(1,w)); result:=result w + times((DD p*epsilon - q*conjugate(epsilon)),0p(1,w)); elif(op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='thp') then (p,q):=getpq(op(1,0p(nops(w),w)) ); result:=result w + w/op(0,0p(nops(w),w))(op(1,0p(nops(w),w)) *(times((Delta + p*epsiloni+ q*conjugate(epsiloni)),0p(1,0p(nops(w),w))) elif(op(0,w)='thp') then (p,q):=getpq(op(1,w)); result:=result w + times((Delta + p*epsiloni + q*conjugate(epsiloni)),0p(1,w)); elif(op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='eth') then (p,q):=getpq(op(1,0p(nops(w),w)) ); result:=result w + w/op(0,0p(nops(w),w))(op(1,0p(nops(w),w)) is no surprise, then, that the resulting background geometry possesses enough spherical symmetry to allow for a straightforward treatment of the problem. It can he directly verified that such a procedure would remove the 8 dependence in Equations 6-81-684 and allow for a matching on r = constant surfaces (which are round 2-spheres in this case). Because this approach fails to shed new light on the situation in the full K~err spacetime, we will not follow it here. Instead, we will focus on Equations 6-66-6-69, which we know to be correct. Let's review the situation. We have established that the metric perturbation in Equations 6-66-669 is a perturbation towards another K~err solution with differing mass. Furthermore, we previously established that 6M~ = pE (Equation 6-26). The problem is that we are currently unable to perform the matching. In practice, the relevant portion of the spacetime is the exterior where gravitational radiation and the non-radiated multipoles are observed far away from the source. Because of this, we contend that considerations from the K~omar formula and Wald's theorem together provide the correct perturbation in the exterior spacetime, independently of any matching considerations. Thus our result is likely useful in the EAIRI problem even though we lack the metric perturbation everywhere in the spacetime. Moreover, the perturbation is still simple to interpret and .I-i-.npind'' ;cally flat, so it is amenable to some analysis. This being the case, we remark that mass perturbations of the K~err background remain confined to the s = 0 sector of the perturbation. It is likely that this is true in general (at least in type D), but a general proof of this remains elusive. Furthermore, contrary to what one might expect in the K~err spacetime, the mass perturbation does not mix spherical harmonic -modes, but is purely -e = 0. We now turn our attention to angular momentum perturbations. 6.2.2 Angular Momentum Perturbations Our lack of success in matching mass perturbations extends to angular momentum perturbations in precisely the same way, though the expressions involved are more S2007 Larry R. Price elif evalb(expr=hlmb) then p:=0; q:=2 elif evalb(expr=hlm) then p:=2; q:=0 elif evalb(expr=hmbmb) then p:=-2; q:=2 elif evalb(expr=hmmb) then p:=0; q:=0 elif evalb(expr=hmm) then p:=2; q:=-2 elif evalb(expr=rho) then p:=1; q:=1 elif evalb(expr=conjugate(rho)) then p:=1; q:=1 elif evalb(expr=rhol) then p:=-1; q:=-1 elif evalb(expr=conjugate(rhol)) then p:=-1; q:=-1 elif evalb(expr=kappa) then p:=3; q:=1 elif evalb(expr=conjugate(kappa)) then p:=1; q:=3 elif evalb(expr=kappal) then p:=-3; q:=-1 elif evalb(expr=conjugate(kappal)) then p:=-1; q:=-3 elif evalb(expr=tau) then p:=1; q:=-1 elif evalb(expr=conjugate(tau)) then p:=-1; q:=1 elif evalb(expr=taul) then p:=-1; q:=1 elif evalb(expr=conjugate(taul)) then p:=1; q:=-1 elif evalb(expr=sigma) then p:=3; q:=-1 elif evalb(expr=conjugate(sigma)) then p:=-1; q:=3 elif evalb(expr=sigmal) then p:=-3; q:=1 elif evalb(expr=conjugate(sigmal)) then p:=1; q:=-3 elif evalb(expr=conjugate(Psi0)) then p:=0; q:=4 elif evalb(expr=conjugate(Psii)) then p:=0; q:=2 elif evalb(expr=conjugate(Psi2)) then p:=0; q:=0 elif evalb(expr=conjugate(Psi3)) then p:=0; q:=-2 elif evalb(expr=conjugate(Psi4)) then p:=0; q:=-4 elif evalb(expr=Psi0) then p:=4; q:=0 ACKENOWLED GMENTS The task of writing acknowledgements necessarily comes the task of forgetting to acknowledge everyone who deserves it. My apologies to anyone I've forgotten. First of all, I owe a great deal to my advisor, Bernard Whiting for his patient guidance and all his support. It has been a pleasure to worth with him for the past five years. I would like to thank Steve Detweiler for useful providing useful comments and perspective throughout the years. My friends throughout the years deserve a great deal of thanks for making life in Gainesville bearable: Josh McClellan, Flo Courchay, Wayne Bomstad, Ethan Siegel, Scott Little, Aaron Manl I1 li li-, lan Vega, K~arthik Shankar and anyone I've forgotten. I owe a very special thanks to Lisa Danker both for putting up with and making life easier for me during the creation of this document. All of my parents-Pam Villa and Larry and Audrey Price-deserve more thanks than I can give them for their continued support throughout the years. Finally, thanks go the Alumni fellowship program and Institute for Fundamental Theory at the University of Florida for financial support over the years. 6.1.2 Angular momentum perturbations Treating angular momentum perturbations is a bit more involved. One reason for this is the fact that it inherently changes the form of the metric. From Equation 6-23, it is clear that our metric perturbation will acquire an At4 component. Realizing this as a perturbation towards the K~err spacetime, we will write it as 26aM~ sin2 8 has (6-51) which is just the linearization about a = J/M~ of the corresponding component of the (background) K~err metric. Because of this, there will be nonzero contributions to him, hm and their complex conjugates which means that we must now take parity into consideration. To that end we will introduce a gauge vector with components it = P(t, 7) m(0, ) (6-52) (r= Q(t, r) m,(0, ) (6-53) 1 i le = [R(t, r)--(a +') +S(t, r) (a a')]em(e, ~) 2 2 sm0 =R(t, r) ~t(0, ~) + S(t, r) (6-54) sin 8 i sin 0 (4= [R(t, r) (a a') S(t, r)> (a + a')]Nem(8, 4) 2 2 =R(t, r)Y m(0 ) Slt, r) sin 0@(0,4) (6-55) where we've defined Q' =i (8T + T')L = (1m+-1m nd T- = ~(n 8') m = ~(1Ye -1 m,), where 1%m are the spin-weight +1 spherical harmonics discussed in Appendix D. This form of the gauge vector was obtained by considering (a = (ma + (als (mh (mm, and making use of the parity decomposition discussed in ChI Ilpter 3. This makes it easy to see that P, Q and R represent the even-parity degrees of gauge freedom and S represents the only odd-parity gauge freedom available. A natural question to ask is what parity the perturbation in Equation 6-51 has. For an answer, we look to the source terms. A quick computation reveals that m, = Im = m = -Im, from which it follows our notation) which exists in every type D background. The trace part becomes Va Pub ,bK = 0. (2-34) The existence of a K satisfying this condition is both necessary and sufficient for the existence of the K~illing tensor. By making the appropriate substitution (V, 8 ,), using Equation 2-30 and taking components with respect to the tetrad vectors, we are led to the followingf: PK = (I<'_j )-1/3(p +p), S =-<_ -/ (2-35) P'K = ( 2 2 -1/3(p / pt), S (22-/ By applying all the commutators in Appendix A to K and making use of Equation 2-35, we arrive at a series of relations which we compactly write (following C'I .!1.4 I-ekhar [29]) p p' 7 ' (2-36) These integrability conditions are both necessary and sufficient for the existence of a Kt satisfying Equation 2-34 and thus provide necessary and sufficient conditions for existence of the K~illing tensor in a type D background. They are satisfied for every non-accelerating type D spacetime. These relations are the primary result of this section. It is straightforw~ardl to verify: that K = (e- <;/3 --/3), Whe~re e"ic 1S a phase factor whose origins will be described below in Equation 2-41. It follows that the K~illing tensor may be expressed as Kab~~ ~ 22-/(ab- ic -1/3 F(-ic 13 2ab. (2-37) Historically, the K~illing tensor was discovered by Carter [40, 41] while considering the separation of the Hamilton-Jacobi equation in the K~err background. The constant of motion derived from the K~illing tensor is thus known as the Carter constant. In a non-acceleratingf spacetime, where the full K~illingf tensor is available, the K~illingf vector in Equation 2-27 is real up to a complex phase. If we specialize to the K~err which represents the largest class of gaugfe transformations consistent with form invariance. This requirement also restricts S(t, 4) = pt + S(4), (6-74) while stationarity again necessitates P(t) = a~t. (6-75) Next we turn our attention to the matching problem. In order to clarify the issues involved in the matching problem, we'll take a look at the matching conditions themselves. Suppose we've chosen some E,, but have yet to specify it explicitly. That is, we have not yet written (or imposed) r = something. The full set of matching conditions now take the form (6-76) (6-77) (6-78) (6-79) htt [Ca](p2 + 2rM~) + 2[Plamr sin2 0 2r61M = 0, 44 : 2 [ca]amr [P] (a2a COS2 H 2 2 + 2) + 2amr) +2amr dS- 2ar61M = 0, h,, dQ r6M~ her Q = 0, & : a2T Sin2 ObM~ (a2 COS2 2 2 2) + 2amr) = 0i, (6-80) where a = T2 2M~r + a2 and 752 = 2 + 2 COS2 8 aS before and we have imposed the condition in Equation 6-79 in the others. Note that this reduces to the Schwarzschild result in Equations 6-39-641 by taking a 0 and setting r = ro. This set of equations background. Now suppose we've identified some geometric quantity of interest (could be scalar, vector, tensor, etc., for simplicity we write it with no indices), Q = Q(A), and we are interested in its first order perturbation, 6Q, towards the physical spacetime, evaluated in the background. Before we can compute anything we must confront the issue of how to relate quantities on two different curved manifolds. One can imagine introducing a (suitably well-behaved) vector field, (", that connects points in the physical spacetime to points in the background. Then, to compute 6Q, we evaluate Q at some point p + 6p in the physical spacetime, pull the result back along (" to the background spacetime, subtract from it the value of Q at a point p in the background, divide by 6p and take the limit as 6p 0 The mathematical apparatus for performing this task is the Lie derivative. Thus, the first order perturbation, 6Q, to a quantity, Q, evaluated in the background spacetime is given by 6Q = 4Q(A)~= (o (261) The important point about this prescription is the fact that (" not only fails to be unique, but there is, in general, no preferred choice for it. A choice of (" is more commonly known as a choice of gauge. According to Equation 2-61, the difference between 6Q computed with (" and rl" is given by sQg sQ, = 4_~,Q, and so we define 6Q, the gauge transformation of 6Q by 6Q = 6Q1 4Q3. (2-62) Note that a gauge transformation in this sense represents a change in the way we identify points in the physical spacetime with points in the background. This is to be distinguished from a coordinate transformation, which changes the labeling of coordinates in both the physical and background spacetimes. The significance of Equation 2-62 is that unless 4~Q = 0 for every (", there is some ambiguity in identifying the perturbation-we can't differentiate between the contributions GHP5c := proc () return(DGHP (GHPconj (GE5))); end proc; GHP5pc := proc () return(DGHP (GHPconj (GHPprime (GE5)))); end proc; GHP6 := proc () return(DGHP(GE6)); end proc; GHP~p := proc() return(DGHP(GHPprime(GE6))); end proc; GHP~c := proc() return(DGHP (GHPconj (GE6))); end proc; GHP~pc := proc () return(DGHP (GHPconj (GHPprime (GE6)))); end proc; COM1 := proc () GHP3pc := proc () return(DGHP (GHPconj (GHPprime (GE3)))); end proc; GHP4 := proc () return(DGHP(GE4)); end proc; GHP4p := proc () return(DGHP(GHPprime(GE4))); end proc; GHP4c := proc () return(DGHP (GHPconj (GE4))); end proc; GHP4pc := proc () return(DGHP (GHPconj (GHPprime (GE4)))); end proc; GHP5 := proc () return(DGHP(GE5)); end proc; GHP5p := proc () return(DGHP(GHPprime(GE5))); end proc; to Equation 2-45 to obtain a similar operator associated with rl" (Equation 2-43): P 21/ a -ic -1/3]( l pi -/ 2/ [e-ic -1/3 eic -1/3 2( / /) +2(pv q)pp' 2-1/3e' 71 --/ -,1/3 (2-46) 2(pT +I q/ '2-136 -13 -/ 1pe ,/3i ?-4ic 221/3 1/3 ;2/3 -2/3) / 4ic 2 Y/3 1/ -2Y/32/ which also commutes with all four GHP derivations. On a final note, we remark that in recent work Be o;r [44] obtained an operator related to K~illing tensor that commutes with the scalar wave equation. The operator has the feature that it is first order in time. In this context it is tempting to ask if there exists an operator analogous to those defined for the K~illingf vectors that commutes with each of the GHP derivatives. The answer is currently unclear and so we leave it for future investigation. 2.4 The Simplified GHP Equations for Type D Backgrounds With Equations 2-36 in hand, we are now in a position to completely simplify the GHP equations for the special case of type D backgrounds. Our starting point is the GHP equations and Bianchi identities adapted to a Type D background: Pp = p2 (2-47) Dr =p~r 7')(2-48) Sp = -r(p -p) (2-49) B7 = 72 (2-50) D'p a'r = pp' -rf 2a (2-51) and op(0,0p(1,0p(nops(w),w))) ='thp') then (p,q):=getpq(op(1,0p(1,0p(nops(w),w))) ); result:=result w + w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0p~np~)w) (op(1,0p(1,0p(nops(w),w)))))*(times((DD -(p-1)*epsilon -(q-1)*conjugate(epsilon)),times((Delta + p*epsiloni + q*conjugate(epsiloni)),0p(1,0p(1,0p(nopsw)))); elif (op(0,w)='th' and op(0,0p(1,w))='thp') then (p,q):=getpq(op(1,0p(1,w))); result:=result w + times((DD (p-1)*epsilon (q-1)*conjugate(epsilon)),times((Delta + p*epsiloni + q*conjugate(epsiloni)),0p(1,0p(1,w)))); elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='eth' and op(0,0p(1,0p(nops(w),w))) ='eth') then (p,q):=getpq(op(1,0p(1,0p(nops(w),w))) ); result:=result w +w/op(1,0p(0,0p(nops(w),w)))(op(0,0p(1,0pnpo)w) (op(1,0p(1,0p(nops(w),w))))) *(times((delta (p+1)*beta + (q-1)*conjugate(betal)),times((delta p*beta + q*conjugate(betal)),op(1,0p(1,0p(nops(w),w))) elif (op(0,w)='eth' and op(0,0p(1,w))='eth') then (p,q):=getpq(op(1,0p(1,w))); result:=result w + times((delta (p+1)*beta + (q-1)*conjugate(betal)),times((delta p*beta + q*conjugate(betal)),op(1,0p(1,w)))); elif (op(0,w)='*' and op(1,0p(0,0p(nops(w),w))) ='eth' As a consequence of Equation :3-35, the actual perturbed Weyl scalars follow directly from Equations :327 and :328.1 The expressions are at first sight quite complicated, but by commuting derivatives so that they appear in a standard order and using the fact that the potential satisfies the Teukolsky equation, they become: < 94, (5-1) ('414 r_ 4/ /37'a 87a p'P fp 24'2)]W .) (5 2) The term in square brackets [] in Equation 5-2 is actually just the operator form of the (generally complex) K~illing vector (acting on 9, which has type {-4,0}) discussed in C'!s Ilter 2. We can further combine the relations in Equations 5-1 and 5-2 to eliminate a~ny reference to\ the po~tent~ials. Tphe firt s~tep is to\ act o~n Equaition 5-2 with > #'2-4/3 which gives us p4 -,-4/:% 4 4 -/ 4 _qpV 53 Commuting the eight derivatives on the first term (using GHPtools, of course) yields the useful identity p394 -- 149 __ i14 -- 4!p4W, (5-4) which we will have occasion to exploit again. Commuting the derivatives in the second term of Equation 5-3 poses no problem because V commutes with everything. Now it is a simple matter to identify the resulting expression with the terms in Equations 5-1 and 5-2 to arrive at the following p4 -4: % 4 1 -4/ 7,, (5-5) p/4 -,-4/3 __i ~4 --4/:% 4 + V4, (5-6) i We thanlk Joh~n Friedmanl and Toby Keidl for noting missing factors of in? several earlier papers. Stewart [21] and C'!,l~!!. 1,.---1:! [18] have these factors correct, the latter with different sign conventions. these variables, GHPtools computes the primes and complex conjugates through the procedures GHPprime () and GHPconj (): > GHPconj (GHPprime (rho+conjugate (rho))); pl + pl > GHPconj (Psi2); > GHPprime (conjugate(mdn) (a)); mdn(a) The {p, q} type of any quantity may be obtained by the use of the getpq function, which returns p and q, in that order: > getpq(Psi2); 0, O > getpq(rhol); -1, -1 > getpq(phi); 1pp, pq Note that 4 is given the general type pp, qq. Before any computation begins it is often useful to specify the spacetime in which subsequent computations are to take place by specifying the value of the global variable spacetime: > spacetime := typed; spacetime := {@20 =0, #21 = 0, a = 0, #10 = 0, #22 = 0, II= 0, al = 0, #12 = 0, E = 0, 93 = 0, 94 = 0, 91 = 0, #11 = 0, 90 = 0, N1 = 0, a = 0, 000 = 0, 001 = 0, 002 = 0, O = 0, #12 = 0, #20 = 0, #11 = 0, #21 = 0, #22 = 0, al = 0, a = 0, a = 0, 002 = 0, #10 = 0, N1 = 0, 90 = 0, 91 = 0, 93 = 0, 94 = 0, 000 = 0, 001 = 0} + -((' p') ( pr) + (B -r) r') -r(8' + p) -r( ) + p)2 mm +{((P-2p)a' +(r + ') + (r -')2 non, (B-6) 1. {D'(pD p' p') + 2p'p' 'r-7) 'h-7) ,U}i 1. --{D(B pr + p)r + 2pp')hl,z 1. - { (9+p' 2p') (8 pr + rp) ( + 2p) + 2p ( + p'8 p')a a'2 t(8'- )(P 2p(a f r') + -r' -(P 2p 2)' f (28 + 4-'p)h -27-(8 2p)( + 27r'f + pp'}hi 2p-2p 2- +4r'z 1. 1. - 2f'p' 2r'( p)( r')- 3}him a'h, -{-(2P 2p(P' 27')p + '( 2p~ 2p)P 2pf + 47' p'(h+ pm 2B7 return(DGHP(th(thp(z))-thp(th(z))=(conjugetu)au)thz +(tau-conjugate(taul))*ethp(z)-p*(kappalkpp-utu+PiPhlPIz -q*(conjugate(kappal)*conjugate(kappa)-cougetu)cngaeau+ conjugate(Psi2)+Phill-PI)*z)); end proc; COMlp := proc() return(GHPprime(COM1())); end proc; COMic := proc() return(GHPconj(COMi())); end proc; COMlpc := proc() return(GHPconj(GHPprime(COM1()))); end proc; COM2 := proc() return(DGHP(th(eth(z))-eth(th(z))=conjugaeroetz)sgathz- conjugate(taul)*th(z)-kappatthp(z)-p*(rhokpp-ulig+Pi)z q*(conjugate(sigmal)*conjugate(kappa)-conuaerocnugetu) +Phi01)*z)); end proc; COM2p := proc() CHAPTER 2 NEW TOOLS FOR PERTURBATION THEORY In this chapter we develop the basic formalism we will be working within for the remainder of this work. We begin with a description of the spin coefficient formalism of N. i.--us! Ia and Penrose [22] and introduce the modifications of it due to Geroch, Held and Penrose [9]. Within the latter formalism, we develop the properties of the general class of spacetimes with which we will be working. Included is a discussion of gauge and the general framework of relativistic perturbation theory. The chapter ends with the introduction to the framework we will exploit in subsequent chapters. 2.1 NP The ?-. i.--us! lIs-Penrose (henceforth NP) formalism has its roots in the spinor formulation of General Relativity. Despite the great beauty and generality of the spinor approach, we will approach the subject as a special case of the tetrad formalism. In this view, the NP formalism is developed by (1) introducing a basis of null vectors for the spacetime and (2) contracting everything in sight with unique combinations of the aforementioned basis vectors. We begin by introducing an orthogonal tetrad of null vectors, 16, n", m" and m", with la and n" being real and m" and m" being complex conjugates. We will impose a relative normalization lnan = -mema = 1, (2-1) with all other inner products vanishing. As an example to keep in mind, consider an orthonormal tetrad on Minkowski space, (t", x", y, za), such that t"t, = -x"x, = -y"y, -zaza = 1. Since the vectors are properly normalized, it is easy to verify that 1 1 la (t" + z), na (t" z") (2-2) 1 1 m" = (xa iya), m" (Xa iya), Finally, we remark that scalars naturally arising from contractions of tensors in the spacetime with various combinations of la and if have no components in S and are thus all of even-parity. Note that such objects necessarily have zero spin-weight. This provides enough information to characterize the parity of arbitrary objects. In practice, we are generally given some spin- and boost-weighted scalar, q' (and/or its complex conjugate), and we merely want to identify the even- and odd-parity pieces without explicitly decomposing it according to Equation :37. In this case Equation :37 allows us to do so by simply writing In the context of a spacetime where 1" and if are fixed by considerations other than being orthogonal to of and of (e.g. Petrov type D, where we would like them aligned with the principal null directions), but of and of fail to form a closed 2-surface (the K~err spacetime provides one such example; this can he seen by noting that B and a' don't commute), the question arises of whether or not something like Equation :38 is still useful to consider. It appears so. In such a case the decomposition theorems (the first lines of Equations :35 and :34) fail to be true, but this isn't a serious issue. Because a~b and Feb still allow us to decompose tensors into their "proper" and pI-, ud.I" pieces, in place of Equation :37 we have 70..0, = (-1)nn![n s .. z ,,3 tee, i .,,)+ i ,. .ni,,3 sve + ir ,,) (:39) where i..~ 1 and "odd" are written in quotes to emphasize the fact that they really refer to real and imaginary in this context and the bar over tau indicates the proper spin- and boost-weight. Clearly, Equation :39, lacks the advantage present in Equation :37 of being able to put all of the angular dependence into B and a' and regard the entire tensor as arising from the two real scalars -rev,, and -r,., Nevertheless it provides a useful decomposition of spin- and boost-weighted scalars, without separation of variables, that While the metric is invariant under a Lorentz transformation, the tetrad vectors are not. In the null tetrad formalism, a Lorentz transformation, which in general is described by six parameters, is broken up into three classes of tetrad rotations. We will consider only a tetrad rotation of Type III herel In the language of our Minkowski space example, this amounts to a boost in the z t plane and a rotation in the x y7 plane. Under such a transformation za via (1 v2 1/2 ' to Vza (1 v2 4 x" = cos Oxa sin Of , y"a = sin Oxa + cos Of ~, which translates to (2-16) ~ a iB a where r = J(1 v)/(1 ) The two transformations can be combined into one using (2 = reie. Then Equation 2-16 may be summarized by (2-17) A quantity, X, is then said to be of type {p, q} if, under Equation 2-17, X ("(97. Alternatively [9], we may ;?i that X possesses spin weight s = (p q)/2 and boost weight b = (p + q)/2. The p and q values for the tetrad vectors can be read off from Equation 2-17. They allow one to determine the spin and boost weights of the spin 1 Descriptions of the other types of tetrad rotation can be found in [30] or [29]. where we've omitted the terms that will vanish in the limit as a result of requiring .I-noi-nd ic flc atness. Similarly, we define angular momentum by J = lim (Kb-K b bS,62) where we have introduced the extrinsic curvature, Kab, of E, and the rotational K~illingf vector a". For a generic metric perturbation of the K~err spacetime, we have 1 r2xr rx 6J =lim r 7sin 86,4 r2 Sin 8 r 4 ded#. (6-23) Though these definitions provide the most general prescription for computing the mass and angular momentum, for stationary and axially symmetric spacetimes (those containing both timelike and axial K~illing vectors), the K~omar formulae [73] evaluated at infinity allow us to compute the value of the perturbationS2 of M~ and J. though not the entire perturbation in the interior and exterior spacetime. The formulae are given by 6M = (Lb- 9a)?a b 3x,~ (6-24) 6J Lb 9b)na b 3x (6-25) where E is spacelike hypersurface that extends to infinity, n" is the unit normal to it, to and #" are the timelike and axial K~illing vectors and 2/7d3Z is the volume element on E. Because our stress-energy tensor is confined to a spacelike hypersurface, Ep, at r = ro, to compute the ADM mass we must take the limit as To oo. In this limit, with the source given by Equations 6-7-6-12, the K~omar formulae give (for the K~err spacetime) sM = pE, (6-26) 6J = p-L. (6-27) 2 We thank John Friedman for -II--- _t h-r;! the use of the K~omar formulae. (A-10) APPENDIX A THE GHP RELATIONS In this appendix, we give the GHP commutators, field equations and Bianchi identities, as well as the derivatives of the tetrad vectors. The full set of equations is obtained by applying to those listed prime, complex conjugation or both. When acting on a quantity of type {p, q}, the commutators are: [P, P'] (7 T')8 + (7' f')' p(s'~ Tr'a + 2 11 (A-1) -q(a'R pfr' + @ol), (P' p')P + (p p)P' + p(pp' -q pp'- aa' + ', 1 -I n. r- 8' = (p p)a ( -r p') - p's =p2 _/ - P'a -p'a a'p + 'r + k -- a'-r = pp' + aa' -- -rf -- KK (A-2) [a, a'] - ee' + 011 II) (A-3 The GHP equations are: Bp Pp -r 87 D'p ~ 1 + @ol, CK 02, ;' I' 2II. The Bianchi identities are given by: P 1 a'l',, p@ol + 8 oo -r'l',, + 4pl 1,' l~_ + -r'@oo 2p~ol -2a410 + 2n~11 + s@02, where the second expression follows from taking the prime of the first. We will refer to these relations as the first form of the Teukolsky-Starobinsky identities. Note that the use of V as a commuting operator restricts the validity of these relations to non-accelerating type D metrics. In the an~ ll-k- of Torres del Castillo [65] and Ortigoza [66], where explicit coordinate expressions were used, Equations 5-5 and 5-6 both appear to be true. This fact appears to be coincidental since it is unclear how it follows in general from the fundamental equations of perturbation theory. The remainder of the identities we will present have not appeared in the literature in this form and we can only claim they hold for non-accelerating type D spacetimes. Before we continue, we'll take a look at the content of Equations 5-5 and 5-6 in the context of th~e Ker~r spacetimre. If wve write i',, ~ R+2(r)S+2(0, Q) and 2 4"/3 4 , R-2(r)S-2(0, 4) and understand the time dependence of each to be given by e-ist, then Equation 5-5 tells us: (1) the result of four radial derivatives on R+2 is proportional to R-2 and (2) the result of four angular derivatives on S-2 is proportional to S+2. The same is true of Equation 5-6 with the +'s and -'s swapped. Note that Equations 5-1 and 5-2 (and their primes in the ORG) ;?i essentially the same thing with the subtle difference that the angular and radial functions are not obviously solutions to the same perturbation. No such ambiguity arises in Equations 5-5 and 5-6. Remarkably, we can actually take things a step further and arrive at expressions for I'n, and ~4 independently. We begin by acting /4 7-4/3 o qain55 p14/4-4/34 -434 pl4 -4/3 14 1,-4/3 i 3Pl4 -4/3 ,,; (5-7) By recalling that Ir has. th same~,,,. type,. as 2-4/3 i4 '; Car~rieS HO Weight), weit can simply take the prime and conjugate of Equation 5-4, and use it to commute the derivatives on conjugate(epsiloni)=epsiloni, conjugate(beta)=betal, conjugate(betal)=betal, Psi0=0, conjugate(Psi0)=0, Psii=0, conjugate(Psii)=0, conjugate(Psi2)=Psi2, Psi3=0,conjugate(Psi3)=0, Psi4=0, conjugate(Psi4)=0, Phi00=0, conjugate(Phi00)=0, Phi01=0, conjugate(Phi01)=0, Phi02=0, conjugate(Phi02)=0, Phil0=0, conjugate(Phil0)=0, Phill=0, conjugate(Phill)=0, Phil2=0, conjugate(Phil2)=0, Phi20=0, conjugate(Phi20)=0, Phi21=0, conjugate(Phi21)=0, Phi22=0, conjugate(Phi22)=0, PI=0, conjugate(0)=0}; typed:={Psi0=0, conjugate(Psi0)=0, Psi4=0, conjugate(Psi4)=0, kappai=0, conjugate(kappal)=0, sigmai=0, conjugate(sigmal)=0, kappa=0, conjugate(kappa)=0, sigma=0, conjugate(sigma)=0, epsilon=0, Psii=0, conjugate(Psii)=0, Psi3=0,conjugate(Psi3)=0, Phi00=0, conjugate(Phi00)=0, Phi01=0, conjugate(Phi01)=0, Phi02=0, conjugate(Phi02)=0, Phil0=0, conjugate(Phil0)=0, Phill=0, conjugate(Phill)=0, Phil2=0, conjugate(Phil2)=0, Phi20=0, conjugate(Phi20)=0, Phi21=0, conjugate(Phi21)=0, Phi22=0, conjugate(Phi22)=0, PI=0, conjugate(0)=0}; flatxyz:={kappai=0, conjugate(kappal)=0, sigmai=0, conjugate(sigmal)=0, kappa=0, conjugate(kappa)=0, sigma=0, conjugate(sigma)=0, epsilon=0, conjugate(epsilon)=0, taul=0, conjugate(taul)=0, tau=0, conjugate(tau)=0,rhol=0, conjugate(rhol)=0, rho=0, conjugate(rho)=0, Psii=0, epsiloni=0, conjugate(epsiloni)=0, beta=0, conjugate(beta)=0, betal=0, conjugate(betal)=0, Psi0=0, conjugate(Psi0)=0, Psii=0, conjugate(Psii)=0, Psi2=0, is the so-called K~innersley tetrad [33], which takes the form l' = ,r 1, 0, (2-23) Ra 2r 12, -a, 0, a) (2-24) 2 (r2 + 2 COS2 H meL (ia sin 8, 0, 1, i/ sin 8) (2-25) Z(r +ia cos 8) Clearly, Equations 2-22 help simplify the GHP equations tremendously. However, type D spacetimes are so special that their description in terms of the GHP formalism is even further simplified. Such simplification is due in large part to the existence of various objects satisfying suitable generalizations (and specializations) of K~illing's equation. 2.3.2 The Killing Vectors and Tensor Virtually all of the ... I,!c" that happens when one considers type D spacetimes can be traced back to the existence of a two-index K~illing spinor. Without delving into the world of spinors we remark that a two index K~illing spinor [34-36], XAB = X(AB), iS a solution tO2 VA'(AXBC) = 0, (2-26) where A and A' are spinor indices and the parentheses denote symmetrization. The first consequence of the existence of XAB iS that the quantity (" = VA'Bi~ A __ --/3 qla at /J1 a'I1 am), (2-27) is a K~illing vector--( satisfies V(aib) = 0. (2-28) The proof of this in spinor language can be found in [36], and the GHP expression can be verified directly by making the replacement V, i 0 and utilizing the expressions in 2 Equation 2-26 is also known as the twister equation, which provides a different means of understanding its relevance. CHAPTER 3 REGGE-WHEELER & TEUK(OLSK(Y As a first application of our framework, we will provide a more detailed discussion of the R;- -ar--Wheeler and Teukolsky equations. This leads quite naturally to a discussion of the metric perturbation generated from a Hertz potential, which will phIi-. a ill l.) .r role in subsequent chapters. Our starting point is a general discussion of parity that does not assume either spherical symmetry or angular separation from the outset. 3.1 Parity Decomposition of Spin- and Boost-Weighted Scalars One crucial feature of the R;- ear--Wheeler analysis is the identification of even and odd-parity modes. In the context of spherically symmetric backgrounds, where angular dependence can be separated off using spherical harmonics, it is sufficiently simple to achieve this decomposition by considering the behavior of the spherical harmonics under a parity transformation directly. For (scalar, vector or tensor) functions defined on more general 2-surfaces, this task can be cumbersome, if not outright impossible. Furthermore, narrowing our focus to angular functions obfuscates the fact that there is something more fundamental happening. It is the goal of this section to provide a more general description of the parity decomposition, applicable to more general 2-surfaces without appealing to separation of variables. We will also see that the GHP formalism is uniquely suited to this description. The decomposition theorems we make use of are proven by Detweiler and Whiting [50]. Our first assumption is that our spacetime manifold, M~, admits a spacelike, closed 2-surface, S, topologically a 2-sphere, with positive Gaussian curvature and a positive definite metric given by where m, and m, are two members of a null tetrad. For a spherically symmetric background Fab is proportional to the metric of the (round) 2-sphere and m" and m" can be directly associated with the background metric. More generally, we allow for the equations. Note that under prime, {p, q}' { -p, -q}, and under complex conjugation, {p, q} { q, p}. A basic set of the GHP equations is given in Appendix A. 2.3 Killing Tensors and Commuting Operators 2.3.1 Specialization to Petrov Type D In this section we provide a brief explanation of why the NP and GHP formalisms are so specially equipped to handle problems in black hole space-times. For an arbitrary space-time there are precisely four null vectors, k", that satisfy kbkekleCabcb~dkyl = 0, (2-21) where Cabcd is the Weyl tensor introduced in Equation 2-9 and the square brackets [] denote anti-symmetrization. The vectors k" define the so-called principal null directions of the space-time. For some space-times, one or more of the principal null vectors coincide. The general classification of space-times based on the number of unique principal null directions of the Weyl tensor was given in 1954 by Petrov [31] and bears his name. It turns out that all the black hole solutions of el-r mphli--;cal interest-including Schwarzschild, K~err and K~err-Newman-are of Petrov type D, meaning they possess two principal null vectors, each with degeneracy two. According to the Goldberg-Sachs theorem [32] and its corollaries, for a space-time of type D with 1" and n" aligned along the principal null directions of the Weyl tensor, the following hold (and reciprocally): a = s' = a = a' = I<'n = ~1 = i' = ~4 = 0. (2-22) This is equivalent to the statement that both 1" and n" are both geodesic and shear-free. Thus, in the NP and GHP formalisms, all black hole space-times are on equal footing. In the K~err spacetime, the commonly used tetrad (aligned with the principal null directions) a'l',, 27r' I + 3p 2 2m 3 00p~o -2-r001 27010o + 2p@l 02~o, (A-11) 2a' I 3-r'a 2 2p 3 ~4 2p'910 +27r'911 + -r'@20 2p@21 22~a, (A-12) 3 'l', 4'' .+ 4 2n' 10 + 2a'@ 1 +p'ao2 2-r021 + 22~. (A-13) P9 2 8'1 01 00, +p~, 2PII pi. 2 21~ 20~, a, 28'II P9 4 .l' 21 +120 Finally, the derivatives of the tetrad vectors are given by Oalb BOmb - a irmb T-mb) na Emb mb) +ma(@mb ; "IiI.) + Eii, mb) - a Klb + ~b) na ~b nb) +m,(p'lb ; "';,.) m('b +Ob) (A-14) (A-15) where u" =. ( 0, 0, J) ;is the four-elocitty of the particle parameterizednr by proper time (7), ro0 is the radius of the orbit and 02 = ~. For circular equatorial geodesics r = ro, (6-8) 0 = (6-9) dt (r, + a2) r + a(L aE), (6-10) Odr a ad4 aT r = aE + (6-11) Sd-r a) with T = (r, + a2 E aL, (6-12) where E = E/p and L = L/p are the energy and angular momentum per unit mass, respectively. We can recover the corresponding result for the Schwarzschild spacetime by simply taking a 0 Because the integration in Equation 6-6 is purely radial, it is clear that the only terms that actually participate in the integral on the left side are those involving two radial derivatives. This is where our form of the perturbed Einstein equations comes in. While it is generally quite tedious and impractical to compute the perturbed Einstein tensor for a background more general than Schwarzschild and pick out the terms involving two derivatives, it is a quite trivial task for the Einstein equations in GHP form. All we need to do is pick out the pieces involving two of P and P' (a mindless task with the aid of GHPtools), plug in our favorite tetrad and voila! Note that these conditions on the second derivatives are generally invariant with respect to choice of tetrad. Because of this, we will write the jump conditions out in the symmetric tetrad, which is obtainable from the K~innersley tetrad by a simple spin-hoost (Equation 2-16) 2.6.1 Einstein's New Clothes The main idea behind our framework is to reorganize the tensors of interest into their tetrad components. The metric perturbation, for example, has the decomposition hub = ~'.. "' A b un a~~b + 2hlul(anb) + 2hmmm(,m b) 21hmnja~b) 2htlm76iamb) 2humll~,mb) 2hu-mljamlb) (2-6(3) + mmmemb mmmammb, so that, for example, hit = hub a b. In order for this to be valid within the GHP formalism, each component of Equation 2-63 must have a well-defined spin- and boost- weight. Because the background metric (Equation 2-3) is invariant under a spin-boost (Equation 2-17) it has type {0, 0}, which must also be the type of the metric perturbation, hab. Therefore the type of the individual components of the metric perturbation are determined by their tetrad indices: hit : {2, 2} a {2 2 him : {2, 0} hm : {-2, 0} him : {0, 2} hm : {0, -2} (2-64) kmm : {2, -2} kmm : {-2, 2} hi, : {0,0 kmm,,: {0, 0}. All of the vectors and tensors we will concern ourselves with can be treated in this way. It is worthwhile to stop here and take a look at what Equation 2-63 really means. Comparing with our treatment of Schwarzschild (Equation 1-4), we note that the scalar parts of the metric are "mixed up" in hu, hin and h,,, all of which have spin weight zero but differ in boost weight. Similarly, the vector parts are given by him, hm and their complex conjugates and likewise the tensor pieces are given here by hmm, hmm and hmm. However, these identifications are completely independent of the background spacetime. Thus, in a certain sense, Equation 2-63 provides a generalization of the RW mode else error "routine not built to handle that type: \%1", whattype(f); end if; end proc; THORNP := proc(f) local i, rest, temp; if type(f, 'symbol') then return map( thp, f) elif type(f, 'constant') then 0 elif type( f, list ) then map( THORNP, f) elif type( f, set ) then map( THORNP, f) elif type( f, '=' ) then map( THORNP, f) elif type( f, '+' ) then map( THORNP, f) elif type( f, '*' ) then rest := mul(op(i,f), i=2..nops(f)); THORNP(op(1,f))*rest + op(1,f)*THORNP(rest); elif type( f, '^' ) then op(2,f)*0p(1,f)^(op(2,f)-1)*THRNPORP(op1,) elif type( f, function ) then if op(0,f) = 'th' then return apply(thp, f); elif op(0,f) = 'thp' then temp:=THORNP(op(f)); return map(THORNP, temp); elif op(0,f) = 'eth' then return apply(thp, f); complicated. This being the case, we will focus our attention on the general features of the angular momentum perturbation that can he obtained independently of a good matching. We begin by noting that the nonzero components on the metric perturbation are given by 4M~ar cos2 86a htt= (6-85) (r2 2Afr + a2 2' 2a(r2 Sin2 0 + 2rM~ cos2 8)6 her (6-86) (r2 2Af~r + a2 2 2M~ar sin2 8(2 a2 cos2 8 hte (6-87) (T2 + 2 COS2 H 2 hoo = -2a Cos2 86a (688) 2a sin2 8 [2 p2 2 a"COS2 :)+3(r. + 2M~ sin2 8 16a he (6-89) (r2 + 2 COS2 H 2 The corresponding tetrad components (in the symmetric tetrad) are given by aba[(r2 a2) Sin2 0 2Afr(cos2 8 p hit = h,z (6-90) p"2 aba sin2 8 hi,z (6-91) aba(cos2 8 p h,waz= (6-92) -iba(a2 Iff) Sin2 bi,~ = h,z,>= (6-93) (r + i cos H) ii aba sin2 8 h,,n = (6-94) (r + ia cos 8)2 where we have omitted the complex conjugates. Though it is not immediately obvious, this perturbation makes no contribution to I~,, or 2/4, enSuring that this is a valid angular momentum perturbation. In light of relatively straightforward results for mass perturbations, the nontrivial form of Equations 6-90-694 comes as a surprise. Unlike mass perturbations, angular momentum perturbations are not confined to a single s sector, whereas one might expect them to be exclusively s = 1, as intuition from working in the Schwarzschild background would lead us to believe. Note that although the perturbation appears in the s = +2 sector of the metric, the vanishing of the s = +2 components of the Weyl curvature keep |

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PAGE 1 1 PAGE 2 2 PAGE 3 Thetaskofwritingacknowledgementsnecessarilycomesthetaskofforgettingtoacknowledgeeveryonewhodeservesit.MyapologiestoanyoneI'veforgotten.Firstofall,Ioweagreatdealtomyadvisor,BernardWhitingforhispatientguidanceandallhissupport.Ithasbeenapleasuretoworthwithhimforthepastveyears.IwouldliketothankSteveDetweilerforusefulprovidingusefulcommentsandperspectivethroughouttheyears.MyfriendsthroughouttheyearsdeserveagreatdealofthanksformakinglifeinGainesvillebearable:JoshMcClellan,FloCourchay,WayneBomstad,EthanSiegel,ScottLittle,AaronManalaysay,IanVega,KarthikShankarandanyoneI'veforgotten.IoweaveryspecialthankstoLisaDankerbothforputtingupwithandmakinglifeeasierformeduringthecreationofthisdocument.Allofmyparents|PamVillaandLarryandAudreyPrice|deservemorethanksthanIcangivethemfortheircontinuedsupportthroughouttheyears.Finally,thanksgotheAlumnifellowshipprogramandInstituteforFundamentalTheoryattheUniversityofFloridafornancialsupportovertheyears. 3 PAGE 4 page ACKNOWLEDGMENTS ................................. 3 ABSTRACT ........................................ 6 CHAPTER 1INTRODUCTION .................................. 8 1.1PerturbationsofSphericallySymmetricSpacetimes ............. 9 1.2PerturbationsofKerrBlackHoleSpacetimes ................. 13 1.3MetricPerturbationsofBlackHoleSpacetimes ................ 15 1.3.1HertzPotentialsinFlatspace ..................... 16 1.3.2TheInversionProblemforGravity ................... 17 1.3.2.1Ori'sconstructionforKerr .................. 18 1.3.2.2TimedomaintreatmentforSchwarzschild ......... 19 1.3.2.3WorkingintheRegge-Wheelergauge ............ 20 1.4ThisWork .................................... 21 2NEWTOOLSFORPERTURBATIONTHEORY ................. 23 2.1NP ........................................ 23 2.2GHP ....................................... 27 2.3KillingTensorsandCommutingOperators .................. 30 2.3.1SpecializationtoPetrovTypeD .................... 30 2.3.2TheKillingVectorsandTensor .................... 31 2.3.3CommutingOperators ......................... 36 2.4TheSimpliedGHPEquationsforTypeDBackgrounds .......... 38 2.5IssuesofGaugeinPerturbationTheory .................... 40 2.6GHPtools-ANewFrameworkforPerturbationTheory ........... 42 2.6.1Einstein'sNewClothes ......................... 43 2.6.2GHPtools-TheDetails ......................... 44 3REGGE-WHEELER&TEUKOLSKY ....................... 52 3.1ParityDecompositionofSpin-andBoost-WeightedScalars ......... 52 3.2Regge-Wheeler ................................. 56 3.2.1TheRegge-WheelerGauge ....................... 56 3.2.2TheRegge-WheelerEquation ..................... 58 3.3TheTeukolskyEquation ............................ 61 3.4MetricReconstructionfromWeylScalars ................... 62 4THEEXISTENCEOFRADIATIONGAUGES .................. 66 4.1TheRadiationGauges ............................. 66 4.2ImposingtheIRGintypeII .......................... 69 4 PAGE 5 ........................... 72 4.4ImposingtheIRGintypeD .......................... 73 4.5Discussion .................................... 76 5THETEUKOLSKY-STAROBINSKYIDENTITIES ................ 78 6THENON-RADIATEDMULTIPOLES ...................... 84 6.1Schwarzschild .................................. 91 6.1.1Massperturbations ........................... 91 6.1.2Angularmomentumperturbations ................... 96 6.1.2.1Odd-parityangularmomentumperturbations ....... 97 6.1.2.2Even-paritydipoleperturbations .............. 99 6.2Kerr ....................................... 100 6.2.1MassPerturbations ........................... 100 6.2.2AngularMomentumPerturbations ................... 104 6.2.3Discussion ................................ 106 7CONCLUSION .................................... 108 7.1Summary .................................... 108 7.2FutureWork ................................... 109 APPENDIX ATHEGHPRELATIONS .............................. 111 BTHEPERTURBEDEINSTEINEQUATIONSINGHPFORM ......... 113 CINTEGRATIONALAHELD ............................ 117 DSPIN-WEIGHTEDSPHERICALHARMONICS .................. 121 EMAPLECODEFORGHPTOOLS ......................... 123 REFERENCES ....................................... 175 BIOGRAPHICALSKETCH ................................ 179 5 PAGE 6 Thedetectionofgravitationalwavesisthemostexcitingprospectforexperimentalrelativitytoday.WithgroundbasedinterferometerssuchasLIGO,VIRGOandGEOonlineandthespacebasedLISAprojectinpreparation,theexperimentalapparatusnecessaryforsuchworkissteadilytakingshape.Yet,howevercapabletheseexperimentsareoftakingdata,theactualdetectionofgravitationalwavesreliesinasignicantwayonmakingsenseofthecollectedsignals.Someofthedataanalysistechniquesalreadyinplaceuseknowledgeofexpectedwaveformstoaidthesearch.Thisismanifestedintemplatebaseddataanalysistechniques.Forthesetechniquestobesuccessful,potentialsourcesofgravitationalradiationmustbeidentiedandthecorrespondingwaveformsforthosesourcesmustbecomputed.Itisinthiscontextthatblackholeperturbationtheoryhasitsmostimmediateconsequences. ThisdissertationpresentsanewframeworkforblackholeperturbationtheorybasedonthespincoecientformalismofGeroch,HeldandPenrose.ThetwomaincomponentsofthisframeworkareanewformfortheperturbedEinsteinequationsandaMaplepackage,GHPtools,forperformingthenecessarysymboliccomputation.ThisframeworkprovidesapowerfultoolforperforminganalysesgenerallyapplicabletotheentireclassofPetrovtypeDsolutions,whichincludetheKerrandSchwarzschildspacetimes. Severalexamplesofthepowerandexibilityoftheframeworkareexplored.TheyincludeaproofoftheexistenceoftheradiationgaugesofChrzanowskiinPetrovtype 6 PAGE 7 7 PAGE 8 Einstein'stheoryofgeneralrelativity,introducedin1915,tothisdayremainsasoneofthenalfrontiersoffundamentalphysics.Sinceitsinceptionprogressintheeldhasbeenlargelytheoreticalbecauseofthetremendousdicultyinherentinmakinggravitationalmeasurements.Inparticular,oneofthemostexcitingandfundamentalpredictionsofgeneralrelativity|theexistenceofgravitationalwaves|hasremainedelusive.Notforlong.WithgroundbasedinterferometerssuchasLIGO,VIRGOandGEOonlineandthespacebasedLISAmissioninpreparation,thedetectionofgravitationalwavesisallbutimminent.Theseexperimentsbringwiththemthetaskofanalyzingthedatatheycollect.Forsomeofthepromisingsourcesofgravitationalwaves,thecollisionoftwoblackholes,themethodofchoicefordataanalysis,knownasmatchedltering,requiresknowledgeoftheexpectedwaveforms.Inthepasttwoyearstheeldofnumericalrelativityhasundergonearevolutionandpromisestoprovidethemostaccuratewaveformsforsituationsinvolvingthecollisionoftwoblackholesofcomparablemasses|situationsthatrequiretheuseoffullnonlineargeneralrelativity.Thereishowever,onepromisingsourceofgravitationalwavesthatiscurrentlyoutofreachfornumericalrelativity|thesituationwherethelargerblackholeisroughlyamilliontimesmoremassivethanthesmallerone,knownasanextrememassratioinspiral,orEMRI.Thisproblemliessquarelyintherealmofperturbationtheory,thesubjectofthepresentwork. Inparticular,the\solution"oftheEMRIproblemrequiresmovingbeyondthetestmassapproximationofgeneralrelativitytodescribethemotionofthesmallblackhole(treatedasaparticleinthespacetimeofthelargerblackholebecauseofthehugemassdierence)|onemustaccountfortherstordercorrectionstothemotionofthesmallblackhole,duetoself-force.TheappropriateequationsofmotionhavebeendeterminedingeneralbyMino,SasakiandTanaka[ 1 ]andQuinnandWald[ 2 ]andarereferredtoasthe 8 PAGE 9 3 ].Ineithercase,thefundamentalobjectofinterestisthemetricperturbation,hab,introducedbytheparticleonthelargeblackhole'sspacetime.ThereforetheEMRIproblemalsorequiresustocomputethemetricperturbation,beforewecancomputetheself-forceontheparticle.Thisisthepieceoftheproblemtowhichthepresentworkaimstocontribute.Determiningthemetricperturbationisataskthatdependsquitesensitivelyonthespacetimebeingperturbed.Forsphericallysymmetricbackgrounds,thisproblemiswellunderstoodandmostoftheremainingproblemsarecomputationalinnature.However,forthemoreinterestingandastrophysicallyrelevantsituationwherethelargerblackholeisrotating,ourunderstandingisnotquitecomplete.Itisonthismoregeneralsituationthatwefocus.Beforewecontinue,wenotethatalloftheastrophysicallyinterestingspacetimes,includingtheKerrandSchwarzschildmetrics,possesscurvaturetensorswiththesamebasicalgebraicstructure.Wewillelaborateonthismorefullyinthenextchapter,butfornowwemerelypointoutthatthesespacetimesbelongtothelargerclassofalgebraicallyspecialspacetimes. Theremainderofthischapterisdevotedtoprovidingareviewoftheliterature[ 4 ].Everyattempthasbeenmadetophrasethecurrentdiscussioningenerallyaccessiblelanguage.Manyoftheseresultswillbeexploredinfurtherdetailinlaterchapters,aftertheappropriateformalismhasbeendeveloped. 5 ](henceforthRW),whoprovidedananalysisofrstorderperturbationsoftheSchwarzschildsolution(whichwaslatercompletedbyZerilli[ 6 7 ]).Thefactthatthebackgroundissphericallysymmetriciscrucialtotheiranalysis.Thebasicswillbepresentedhere.Amorecompletediscussion,inaverydierentlanguage,isprovidedinChapter3. 9 PAGE 10 where rdt212M r1dr2r2(d2+sin2d2)(1{2) istheSchwarzschildmetricinSchwarzschildcoordinates.PuttingEquation 1{1 intotheEinsteinequationsandkeepingonlytermslinearinhableadsustotheperturbedEinsteinequations: 2rcrchab1 2rarbhcc+rcr(ahb)c+1 2gab(rcrchddrcrdhcd)=0;(1{3) whereraisthederivativeoperatorcompatiblewiththebackgroundgeometry 1{2 andtheindicesareraisedandloweredwiththebackgroundmetric.HenceforthwewillrefertoEabastheEinsteintensor,andtheexpressiontotherightofitastheEinsteinequations(droppingthequalier\perturbed"forbrevity). EssentiallyeveryperturbativeanalysisoftheSchwarzschildspacetimemakesextensiveuseofitssphericalsymmetry.Therststepinthisdirectionistodecomposethecomponentsofthemetricperturbationintoscalar,vectorandtensorharmonics.Heuristically,wewrite wheres;vandtstandforscalar,vectorandtensor,respectivelyandthesubscriptsdistinguishbetweenthevariousscalarsandvectors. Considerthemetricofthetwo-sphere: 10 PAGE 11 whererAisthederivativecompatiblewithAB(Equation 1{5 ).Theothervectoristheodd-parity(pseudo-)vector whereABisjustthestandardLevi-Civitasymbol.Todenetensorharmonics,weessentiallyjusttakeonemorederivativeofEquations 1{6 and 1{7 .Theevenparitytensorsaregivenby andtheodd-parity(pseudo-)tensorby Evenparityobjectspickupminussignsunderaparitytransformation(!;!+)accordingto(1)`,andoddparityobjectspickupminussignsaccordingto(1)`+1.Forthisreasontheevenparitypartsaresometimesreferredtoas\electric"andtheoddparityparts\magnetic"intheolderliterature.Becauseparityisaninherentsymmetryofsphericallysymmetricbackgrounds,itprovidesanaturalwayofdecouplingthetwodegreesoffreedomofthegravitationaleld.Note,however,thatparityisnotagoodsymmetryinevenslightlylesssymmetricspacetimes(e.g.Kerr).Wewillreturnto 8 ]forthestandardtensorharmonicsandtheirrelationtovariousotherrepresentationsofthesphere,orAppendixDforthespin-weightedsphericalharmonicswhichprovideanotheralternativefortheangulardecomposition. 11 PAGE 12 1{4 ),wenowconsiderthetwosectorsofthemetricperturbationindependently,writing and ThenalstepbeforeappealingtotheEinsteinequationsconsistsofchoosingagauge.Equation 1{3 isinvariantunderthetransformation whereaisanarbitraryvectorand$istheLiederivative.Takingtheodd-paritysectorasanexample,theRegge-Wheelergaugevectortakestheform whereisafunctionchosensothattheoddparitypartofthemetricperturbation 1{10 takestheform Similarsimplicationsariseintheeven-paritysector. 12 PAGE 13 wheretheletters'o'and'e'standforoddandeven,respectively,r=r+ln(r where~2=r2+a2cos2,=r22Mr+a2,Misthemassanda=J=Mistheangularmomentumpermassoftheblackhole.ThespincoecientformalismofGeroch,HeldandPenrose[ 9 ]developedinthenextchapterhasprovedtobefundamentalinvirtuallyeveryperturbativedescriptionoftheKerrspacetime. TherstsuccessfulperturbationanalysisoftheKerrgeometrywasperformedbyTeukolskyinaseriesofpapersbeginningin1973[ 10 { 12 ].TeukolskytookashisstartingpointtheperturbedBianchiidentitiesinaspincoecientformalism.Eachquantityisperturbedawayfromitsbackgroundvalueandonlyrstordertermsarekept.Equivalently,thoughwithconsiderablymoreeort,Teukolsky'sresultcanalsobeseenas 13 PAGE 14 13 ].Ineithercase,theresult,writtenhereinBoyer-Lindquistcoordinates,isTeukolsky'smasterequation(writtenhereinaccordwith[ 14 ]) @r@ @r1 n(r2+a2)@ @t+a@ @s(rM)o24s(r+iacos)@ @t+@ @cossin2@ @cos+1 sin2nasin2@ @t+@ @+iscoso2)s=2s=4s=2Ts; wheres=2correspondtotheWeylscalars0and4=324,respectively.TheWeylscalarsareperturbationsoftheextremalspincomponentsofthecurvaturetensor.ThesignicanceoftheWeylscalar4isthatfarawayfromthesourceofgravitationalradiation whereh+andharethetwopolarizationsofoutgoinggravitationalradiationinthetransversetracelessgauge.Similarresultsholdfor0andincomingradiation.Forothervaluesofs,solutionscorrespondtoeldsofotherspin:s=0isthemasslessscalarwaveequation,s=1=2theWeylneutrino,s=1theMaxwelleld,s=3=2theRarita-Schwingereld,andsoon.Notethatangularseparationnecessarilyinvolvestimeseparationfora6=0. SeparatedsolutionstoEquation 1{17 areoftheforms=ei!teimsR(r)sS(a!;)(omittingthe`,mand!subscripts).Theangularfunctions,sS(a!;),aregenerallyreferredtoas\spinweightedspheroidalharmonics".Inthelimitthata!=0,sS`m()reducetothestandardspinweightedsphericalharmonics(cf.AppendixD),whichareinterrelatedbythespinraisingandloweringoperators,and0[ 15 ],developedinthefollowingchapter.Fora!6=0,solutionscorrespondtofunctionsofdierentspinweight,butthesS(a!;)nolongersharecommoneigenvalues.Thusametricreconstructionbasedonspinweight2functionswouldbeincompatiblewithonebasedonspinweight0 14 PAGE 15 Thespinweightedspherical(andspheroidal)harmonicsfailtobedenedfor` PAGE 16 20 ]: Asusualavectorpotential,Aa,isintroducedandtheLorentzgauge,raAa=0,isimposedsothattheMaxwellequationsleaddirectlyto2Aa=0. ThenaHertzpotentialHabisintroducedviaAa=rbHab,whereHab=Hba,sothattheMaxwelleld,Fab,isobtainablebytwoderivativesofHab.However,Habisonlydeneduptoatransformationofthetype whereMcabiscompletelyantisymmetricand2Ca=0.Itiseasytoseethatinatspacetime,wherederivativescommute,thetransformationEquation 1{20 onlychanges 16 PAGE 17 1{20 isusedtoreducetheHertzbivectorpotentialtoasinglecomplex(ortworeal)scalarpotential(s).Hereinliesthepowerofthemethod.However,movingtocurved-spacenaturallycomplicatesthings.Whilethewaveequationsaremodiedtoincludecurvaturepieces,thetransformationinEquation 1{20 isretained(seeCohenandKegeles[ 20 ]andStewart[ 21 ]).Asaresult,theeldequationsarestillsatisedandthesixcomponentsofHabarestillreducedtotwo,butthetransformationinEquation 1{20 explicitlybreakstheLorentzgaugebecausederivativesnolongercommute.Inthiswayanewgaugeisintroducedthatbringswithitcomplicationsfortheinclusionofsources.ThenecessaryandsucientconditionsfortheexistenceofthisgaugearethesubjectofChapter4. 22 ],oritsmodicationduetoGeroch,HeldandPenrose[ 9 ].ThuswepostponetheformaldevelopmentofthesubjectuntilChapter3,whenthenecessaryformalismisinplace,andinsteadoeranoverviewofthegeneralprocessanddocumentedresearchonthetopicofreconstructingthemetricperturbationfromsolutionstotheTeukolskyequation(assumingtheformofmetricperturbationisprescribed),whichwewillrefertoastheinversionproblem. 17 PAGE 18 20=DDDDIRG;and (1{21) 244=1 4LLLLIRG1232@tIRG; whereL=[@+scoticsc@']+iasin@tandD=1[(r2+a2)@t+@r+a@]denederivativesin(orthogonal)nulldirections,=(riacos)1andIRGisthepotential.Whileforapotentialsatisfyingthes=+2Teukolskyequation,wehaveaperturbationintheoutgoingradiationgauge(ORG),where 244=2bbbb2ORG;and (1{23) 20=1 4LLLLORG+1232@tORG; whereb=1 2[(r2+a2)@t@r+a@]andL,thecomplexconjugateoftheoperatordenedabove,arealsoderivativesinnulldirections(mutuallyorthogonaltoeachotherandthosedenedbytheoperatorsintheIRG).ThesearetheequationswewouldliketoinvertforthepotentialsIRGandORG.Oncethisisdone,thepotentialmaythenbeusedtoconstructthemetricperturbation.Wenowlookatseveraldierentapproachestothisproblem. 1{21 or 1{22 (ortheirORGcounterparts).Ori[ 23 ]hasrecentlyperformedthistask|integratingEquation 1{21 inordertondthepotentialIRGintermsof0. 18 PAGE 19 1{17 withs=2,soitdoesprovideacompletesolutioninthefrequencydomain. Forincorporatingsources,OricontinuestotakeEquation 1{21 ascorrect,wherenow0isasource-dependent,non-vacuumsolution.Equation 1{21 allowsthefreedomtoaddtoIRGanyfunctionthatiskilledbythefourderivativesthere.Oriutilizesthisfreedomtochoosefunctionsthatreproducethediscontinuityatthesourceand,byextension,0.However,Equation 1{17 nolongerappliesfor,nordoesEquation 1{22 for4intheformithashere. 24 ]suggeststhatgaugefreedommayplayaroleinresolvingtheseissues. 25 ]havechosentoworkinthetimedomain.Becauseofthischoicetheirresultisonlyvalidinthe 5{2 arosebyrepeateduseoftheTeukolskyequationinquiteacomplicatedexpression,initiallygivencorrectlybyStewart[ 21 ],andalsoobtainablefromtheresultsofChapters2and3here.Thefullformoftheexpressionmaystillapplyhere. 19 PAGE 20 OneunexpectedfeatureofLoustoandWhiting'sworkishowalgebraicallyspecialfrequenciesemergeinafundamentalway.Algebraicallyspecialsolutionsarisewhenoneof0or4iszerowhiletheotherisnot,andthenonlyforspecic(complex)frequencies.Whilethisisinherentlyafrequencydomainphenomena,itplaysacrucialroleinthistimedomainapproach.ThealgebraicallyspecialequationherehasasourcetermdependingontheinitialdatafortheHertzpotential|thistermeectivelycorrespondstothatwhicharisesforaLaplacetransform.FortheSchwarzschildbackground,allthealgebraicallyspecialfrequenciesareknownandthealgebraicallyspecialsolutionshavebeenfoundexplicitly[ 26 ],sotheequationsforthisanalysiscouldbesolvedbyquadrature[ 25 ].AttemptstogeneralizethistechniquetotheKerrbackgroundhavetodateremainedunsuccessful. 27 ].Lousto[ 28 ]hasrecentlychosentoworkwithsuchaformulation,ratherthanwithaHertzpotentialformulation.Thisimmediatelygiveshimfreedomovergaugechoiceanditcircumventstheproblemspreviouslyencounteredwiththeintroductionofsources.Havingcalculatedexplicitlythedependenceonsources,andknowingalsohowtorepresentallrelevant 20 PAGE 21 Loustoactuallyusesboth0and4inhisconstruction.Forconcretenessandforaccesstoavastbodyofpriorexperience,LoustoalsochosetoworkinagaugeknownastheRWgauge.NotethatEquations 1{21 and 1{22 areonlyvalidintheIRG.However,0and4areeasilyexpressibleintermsofanarbitrarymetricperturbation,whichallowsthemtobewrittenintermsoftheRWvariablesforanychoiceofgauge.IntheRWgauge,0and4becomealgebraicintheevenparitysectorandrstorderoperatorsintheoddparitysector.ToprovideenoughconditionstosolveforallthecomponentsofthemetricperturbationintermsoftheWeylscalars,LoustomustturntotheEinsteinequations(withsources),alsointheRWgauge.Itisinthiswaythatreconstructionwithsourcesisaccomplished. Theidenticationofgaugeinvariantquantities,beyond0and4,isvirtuallynonexistentintheKerrspacetimeandaspointedoutseveraltimesbefore,theangulardecompositionthereisnotasrobustasthatavailableinsphericallysymmetricbackgrounds.Inshort,Lousto'sworkisquitenotableforitsinclusionofsources,butitsrelianceonRWtoolsandtechniquesmakeitdiculttoseehowtoextendthemethodtotheKerrbackground. 21 PAGE 22 InChapter2,wewilldeveloptheformalismnecessaryforbuildingourframework.Additionally,theframeworkwillbepresented,whichincludesanewformfortheperturbedEinsteinequationsaswellasaMaplepackagethataidsnotonlyintheirapplication,butanycomputationintheformalismofGeroch,HeldandPenrose.Chapter3thenprovidesafurtherdiscussionofboththeRWandTeukolskyformalisms,phrasedinourframework.InChapter4,thenecessaryandsucientconditionsfortheexistenceoftheIRG(inalargerclassofspacetimesthanweconsiderelsewhere)aredeterminedwiththeaidofourformoftheEinsteinequations.Chapter5thenusestheIRGmetricperturbationtoderivesomeimportantrelationshipsbetweenthecurvatureperturbationsrepresentedby0and4,whichareofimportancefortheinversionproblemdescribedinthischapter.Furthermore,thisapplicationshowcasessomeofourMaplepackage'smostusefulfeatures.InChapter6wethenpresentaverydierentapplicationofourframeworkinconjunctionwithmorestandardtechniquestoaddresstheissueofthenon-radiatedmultipoles. 22 PAGE 23 Inthischapterwedevelopthebasicformalismwewillbeworkingwithinfortheremainderofthiswork.WebeginwithadescriptionofthespincoecientformalismofNewmanandPenrose[ 22 ]andintroducethemodicationsofitduetoGeroch,HeldandPenrose[ 9 ].Withinthelatterformalism,wedevelopthepropertiesofthegeneralclassofspacetimeswithwhichwewillbeworking.Includedisadiscussionofgaugeandthegeneralframeworkofrelativisticperturbationtheory.Thechapterendswiththeintroductiontotheframeworkwewillexploitinsubsequentchapters. Webeginbyintroducinganorthogonaltetradofnullvectors,la;na;maandma,withlaandnabeingrealandmaandmabeingcomplexconjugates.Wewillimposearelativenormalization withallotherinnerproductsvanishing.Asanexampletokeepinmind,consideranorthonormaltetradonMinkowskispace,(ta;xa;ya;za),suchthattata=xaxa=yaya=zaza=1.Sincethevectorsareproperlynormalized,itiseasytoverifythat 23 PAGE 24 Forsimplicity,weintroducethefollowingnotationforourtetrad(borrowedfromChandrasekhar[ 29 ]):ea(i)=(la;na;ma;ma); 2{1 )nulltetrad(i)(j)=(i)(j)=0BBBBBBB@0100100000010010:1CCCCCCCA wherel(anb)=1 2(lanb+lbna). 24 PAGE 25 Itfollowsfromthedenitionthat Thereisatotaloftwelvespincomplexcoecients,individuallynamedasfollows and 2(manbralbmambramb);=1 2(mambrambmalbranb);=1 2(lanbralblambramb);=1 2(nambrambnalbranb):(2{7) Ourea(i)naturallydenefourindependent,non-commutingdirectionalderivativese(i)ea(i)@ @xa; @xa;=ma@ @xa;=na@ @xa;=ma@ @xa:(2{8) 25 PAGE 26 2(gacRbd+gbdRacgbcRadgadRbc)1 2(gacgbdgbcgad)R:(2{9) whereCabcd;Rabcd;RabandRdenotetheWeyltensor,Riemanntensor,RiccitensorandRicciscalar,respectively.SinceboththeRiccitensorandtheRicciscalarvanishintheabsenceofsources,theWeylandRiemanntensorsareidenticalinsource-freespacetimes.InthatsensetheWeyltensorrepresentsthepurelygravitationaldegreesoffreedom.TheRiemanntensoristhenexpressedpurelyintermsofthespincoecientsandtheirderivativesbycontractingallfourvectorindiceswithea(i)'sandmakinguseoftheRicciidentity, (rarbrbra)vk=Rabcdvd=Rabcdvd;(2{10) wherevdisanarbitraryvector.InfourdimensionstheRiemanntensorhastwentyindependentcomponentsandtheRiccitensorhasten,leavingtheWeyltensorwithtenindependentcomponents.IntheNPformalism,thistranslatesintovecomplexscalars: 2Cabcd(lanblcnd+lanbmcmd);3=Cabcdlanbmcnd;4=Cabcdnambncmd:(2{11) 26 PAGE 27 00=1 2R11;11=1 4(R12+R34);01=1 2R13;12=1 2R23;10=1 2R14;21=1 2R24;02=1 2R33;22=1 2R22;20=1 2R44;=1 24R:(2{12) TheeldequationsthenfollowfromEquations 2{9 and 2{10 .AfullsetofequationsfortheNPformalismiscomposedofthecommutators,theequationsinvolvingdependenceonmatter,andtheBianchiidentities.ThisisgiveninAppendixA. 9 ]introducedsomeconvenientmodicationsoftheNPformalism.Specically,theyidentiedthenotionsofspinandboostweightandmakeexplicituseofaninherentdiscretesymmetryoftheNPequations. IntheNPformalism,thereisanimplicitinvarianceunderacertaininterchangeofthebasisvectorswhichGHPhavebuiltonthroughtheintroductionoftheprime(0)operation,denedbyitsactiononthetetradvectors: (la)0=na;(ma)0=ma;(na)0=la;(ma)0=ma:(2{13) AglanceatEquations 2{6 and 2{7 suggeststheadoptionofachangeinnotation: andsimilarlyforthedirectionalderivativesofEquation 2{8 27 PAGE 28 2;~xa=cosxasinya;~ya=sinxa+cosya; ~la=rla;~na=r1na;~ma=eima;~ma=eima;(2{16) wherer=p 2{16 maybesummarizedby Aquantity,,isthensaidtobeoftypefp;qgif,underEquation 2{17 ,!pq.Alternatively[ 9 ],wemaysaythatpossessesspinweights=(pq)=2andboostweightb=(p+q)=2.ThepandqvaluesforthetetradvectorscanbereadofromEquation 2{17 .Theyallowonetodeterminethespinandboostweightsofthespin 30 ]or[ 29 ]. 28 PAGE 29 2{6 ,whilethespincoecientsinEquation 2{7 havenowelldenedspinorboostweightsince,underEquation 2{17 ,theypickuptermsinvolvingderivativesof.Whenactingonaquantityofwelldenedspinandboostweight,thedirectionalderivativesofEquation 2{8 bythemselvesalsofailtocreateanotherquantityofwelldenedweight.However,itispossibletocombinethespincoecientsinEquation 2{7 withtheactionofderivativeoperatorsinEquation 2{8 toconstructderivativeoperatorsthatdoproducenewquantitieswithwelldenedspinandboostweights.Withtakentobeoftypefp;qg,wecandenetheseoperatorsasfollows: whereandareIcelandiccharactersnamed\thorn"and\edth",respectively.Eachofthesederivativeshassomewelldenedtypefr;sginthesensethatwhentheyactonaquantityoftypefp;qg,aquantityoftypefr+p;s+qgisproduced.Thesenewderivativeoperatorsinherittheirtypefromtheircorrespondingtetradvectors: Itisquiteoftenusefultothinkof()and0(0)asspin(boost)weightraisingandloweringoperators,respectively.ThederivativesinEquation 2{18 canbecombinedtoformacovariantderivativeoperator: a=la0+nama0ma=ra1 2(p+q)nbralb+1 2(pq)mbramb:(2{20) Wenoteinpassingthatthisdenitiondenesthe\GHPconnection."OurprimaryuseforEquation 2{20 willbetoexpressthingsinGHPlanguageviathereplacementra!a.Withthesedenitions,allequationsintheNPformalismcanbetranslatedintoGHP 29 PAGE 30 2.3.1SpecializationtoPetrovTypeD whereCabcdistheWeyltensorintroducedinEquation 2{9 andthesquarebrackets[]denoteanti-symmetrization.Thevectorskadenetheso-calledprincipalnulldirectionsofthespace-time.Forsomespace-times,oneormoreoftheprincipalnullvectorscoincide.Thegeneralclassicationofspace-timesbasedonthenumberofuniqueprincipalnulldirectionsoftheWeyltensorwasgivenin1954byPetrov[ 31 ]andbearshisname.Itturnsoutthatalltheblackholesolutionsofastrophysicalinterest|includingSchwarzschild,KerrandKerr-Newman|areofPetrovtypeD,meaningtheypossesstwoprincipalnullvectors,eachwithdegeneracytwo.AccordingtotheGoldberg-Sachstheorem[ 32 ]anditscorollaries,foraspace-timeoftypeDwithlaandnaalignedalongtheprincipalnulldirectionsoftheWeyltensor,thefollowinghold(andreciprocally): Thisisequivalenttothestatementthatbothlaandnaarebothgeodesicandshear-free.Thus,intheNPandGHPformalisms,allblackholespace-timesareonequalfooting.IntheKerrspacetime,thecommonlyusedtetrad(alignedwiththeprincipalnulldirections) 30 PAGE 31 33 ],whichtakestheform 2(r2+a2cos2)r2+a2;;0;a; Clearly,Equations 2{22 helpsimplifytheGHPequationstremendously.However,typeDspacetimesaresospecialthattheirdescriptionintermsoftheGHPformalismisevenfurthersimplied.Suchsimplicationisdueinlargeparttotheexistenceofvariousobjectssatisfyingsuitablegeneralizations(andspecializations)ofKilling'sequation. 34 { 36 ],AB=(AB),isasolutionto whereAandA0arespinorindicesandtheparenthesesdenotesymmetrization.TherstconsequenceoftheexistenceofABisthatthequantity isaKillingvector|asatises Theproofofthisinspinorlanguagecanbefoundin[ 36 ],andtheGHPexpressioncanbeverieddirectlybymakingthereplacementra!aandutilizingtheexpressionsin 2{26 isalsoknownasthetwistorequation,whichprovidesadierentmeansofunderstandingitsrelevance. 31 PAGE 32 2{28 independently[ 36 ],soalltypeDspacetimespossesstwoindependentKillingvectors.ThesetwoKillingvectorseachgiverisetoaconstantofmotionalongageodesic.Inotherwords,ifuaistangenttoageodesic(ubrbua=0),thenauaisconservedalongua: wherethersttermvanishesasaconsequenceof(Killing's)Equation 2{28 andthesecondbecauseuaistangenttoageodesic. InadditiontotheexistenceoftwoKillingvectors,theKillingspinoralsogivesrisetotheconformalKillingtensor[ 35 37 ]: 2(22)1=3(l(anb)+m(amb));(2{30) whichalsoexistsineverytypeDbackground.TheconformalKillingtensorisalternativelydenedasasolutionto 3g(abrdPc)d:(2{31) ConformalKillingtensorsareusefulbecausetheygiverisetoconservedquantitiesalongnullgeodesics.Ifkaistangenttoanullgeodesic(kbrbka=0andkaka=0)thenthequantityPabkakbisconservedalongka: 3(kaka)kcrbPbc=0; PAGE 33 2{31 Incertaininstanceswecanextendthisideatoprovidearstintegralofthemotionfortimelikeandspacelikegeodesicsaswell.Suchanotioncanberealizedbydeningatensor,Kab=K(ab),thatsatises AquantitysatisfyingthisrelationiscalledaKilling-Staeckeltensor.NotethatbydenitionthemetricandsymmetricouterproductsofKillingvectorsbothsatisfyEquation 2{32 .WereservethenameKilling-Staeckeltensorforanobjectthatdoesnotreduceinthisway.ThisistobedistinguishedfromtheantisymmetricKilling-Yanotensorsatisfyingr(aYb)c=0; 38 39 ].BecausewewillnotmakeuseofKilling-Yanotensorshere,wewillfollowconventionallanguageandrefertotheKilling-StaeckeltensorassimplyaKillingtensor.Returningtothemainlineofdevelopment,giventheexistenceofaKillingtensor,wecanrecycletheargumentabove(nowusingEquation 2{32 insteadofEquation 2{31 )fortheconformalKillingtensortoshowthatthequantityKabuaubisconservedforanyuatangenttoageodesic,regardlessofwhetheritbetimelike,spacelikeornull.Thequestionthenarises:WhencanwendaKabthatsatisesEquation 2{32 ?Toanswerthisquestion,webeginbydecomposingtheKillingtensorintoitstrace-freepartanditstrace,accordingto 4Kgab;(2{33) withPabgab=0andK=Kabgab.Usingthisin(Killing's)Equation 2{32 anddividingtheresultingexpressionintotrace-freeandtracepartsgivestwoequations.Thetrace-freepartissimplyEquation 2{31 andsoPabistheconformalKillingtensor(asweanticipatedwith 33 PAGE 34 4rbK=0:(2{34) TheexistenceofaKsatisfyingthisconditionisbothnecessaryandsucientfortheexistenceoftheKillingtensor.Bymakingtheappropriatesubstitution(ra!a),usingEquation 2{30 andtakingcomponentswithrespecttothetetradvectors,weareledtothefollowing: ByapplyingallthecommutatorsinAppendixAtoKandmakinguseofEquation 2{35 ,wearriveataseriesofrelationswhichwecompactlywrite(followingChandrasekhar[ 29 ])as TheseintegrabilityconditionsarebothnecessaryandsucientfortheexistenceofaKsatisfyingEquation 2{34 andthusprovidenecessaryandsucientconditionsforexistenceoftheKillingtensorinatypeDbackground.Theyaresatisedforeverynon-acceleratingtypeDspacetime.Theserelationsaretheprimaryresultofthissection.ItisstraightforwardtoverifythatK=1 2(e2ic2=32+e2ic2=32),wheree2icisaphasefactorwhoseoriginswillbedescribedbelowinEquation 2{41 .ItfollowsthattheKillingtensormaybeexpressedas 8(eic1=32+eic1=32)2gab:(2{37) Historically,theKillingtensorwasdiscoveredbyCarter[ 40 41 ]whileconsideringtheseparationoftheHamilton-JacobiequationintheKerrbackground.TheconstantofmotionderivedfromtheKillingtensoristhusknownastheCarterconstant. Inanon-acceleratingspacetime,wherethefullKillingtensorisavailable,theKillingvectorinEquation 2{27 isrealuptoacomplexphase.IfwespecializetotheKerr 34 PAGE 35 WecanrewriteEquation 2{39 withthehelpofEquation 2{38 as whereCisa(possiblycomplex)functionannihilatedby.Thisisinfactnotaproof,butrathertherststepinone.AfullproofwouldconsistofshowingthatthisisconsistentwiththerestoftheGHPequationsandBianchiidentities.Thecoordinate-freeintegrationtechniqueintroducedinChapter5isideallysuitedforthis.FornowwetakeitasgiventhattheEquation 2{40 istrueineverytypeDbackground,forsomecomplex whichdenesthephasefactorintroducedinEquation 2{37 .ItturnsoutthatinalltypeDspacetimesnotpossessingNUTcharge,c=0.Moreimportantly,wenowhavetherelations 35 PAGE 36 2{27 isrealuptoacomplexphase(e2ic).Notealsothat0a=a.Whathappenedtotheother(linearlyindependent)Killingvector?Itisgivenby 81=32n[eic1=32eic1=32]2(0lbnb)[eic1=32+eic1=32]2(0mbmb)o: ProvingthatthisexpressionsatisesKilling'sequationingeneralisabitinvolved,andsincewe'llhavenodirectuseforEquation 2{43 insubsequentchapters,werefertheinterestedreaderelsewhere[ 36 ]fordetails.Onceagain,usingEquations 2{42 ,itisstraightforwardtoseethatEquation 2{43 isrealuptoaphase.UsingtheKinnersleytetradintheKerrspacetime,Equation 2{43 becomes Mb;(2{44) wheretaisthetimelikeKillingvectorandaistheaxialKillingvector.Becausebisproportionaltoa,itclearlyvanishesintheSchwarzschildspacetime.Thiscanalsobeenseenbynotingthat,intheSchwarzschildspacetime,=0=0andthuscomparisonsofEquations 2{27 and 2{43 revealthatthetwoKillingvectorsarenotlinearlyindependent[ 42 ].In[ 36 ]itisshownhowonecaninfersphericalsymmetryfromthisfact. 2{3 .Bycontractingthelastlinewitheachofthetetradvectorsand 36 PAGE 37 2{32 ,weestablishthat$la=$na=$ma=$ma=0: Inthislight,itisreasonabletoexpectthatwecanconstructanoperator,V,relatedtotheKillingvectorthatcommuteswithallfouroftheGHPderivatives.Becauseofthefactthatspin-andboost-weightsenterexplicitlyintothecommutators(Equations A{1 { A{3 ),wewouldalsoexpectthatanysuchoperatorwouldcarryspin-andboost-weightdependence.Infact,suchanoperatorcanbeconstructed.Bytakingasouransatz:V=aa+pA+qB; 2{36 aresatised,whichimpliesaKillingtensorexists.Fornon-acceleratingspacetimeswethenhave 22p+ wherepandqrefertotheGHPtypeoftheobjectbeingactedon.ThisresulthasbeennotedbyJeryes[ 43 ],whoarrivedatitfromspinorconsiderations.IfwespecializetotheKerrspacetimeandtheKinnersleytetrad,itiseasytoseethatittakesthevalueM1=3@t+bM2=3(r2+a2cos2)1,wherebistheboost-weightofthequantitybeingactedon.DespitethisdierencebetweenthevectoraandtheoperatorV,wewillrefertotheminterchangeablyasaKillingvector.Similarly,wecanfollowthesameprocedurethatled 37 PAGE 38 2{45 toobtainasimilaroperatorassociatedwitha(Equation 2{43 ): 81=32n[eic1=32eic1=32]2(00)[eic1=32+eic1=32]2(00)+2(pq)01=32(e2ic1=321=32)2(p+q)01=32(e2ic1=32+1=32)1 2pe2ic1=32(e4ic21=321=322=322=32)1 2q1=32(e4ic21=321=322=322=32)o;(2{46) whichalsocommuteswithallfourGHPderivations. Onanalnote,weremarkthatinrecentworkBeyer[ 44 ]obtainedanoperatorrelatedtoKillingtensorthatcommuteswiththescalarwaveequation.Theoperatorhasthefeaturethatitisrstorderintime.InthiscontextitistemptingtoaskifthereexistsanoperatoranalogoustothosedenedfortheKillingvectorsthatcommuteswitheachoftheGHPderivatives.Theansweriscurrentlyunclearandsoweleaveitforfutureinvestigation. 2{36 inhand,wearenowinapositiontocompletelysimplifytheGHPequationsforthespecialcaseoftypeDbackgrounds.OurstartingpointistheGHPequationsandBianchiidentitiesadaptedtoaTypeDbackground: (2{48) (2{49) 38 PAGE 39 wherewehaveomittedthoseequationsthatcanbeobtaineddirectlybyutilizingtheoperationsofprimeandcomplexconjugation.Byapplyingthecommutatorsto2andmakinguseoftheequationsabove,welearnthat NotethattheprecedingequationsholdforalltypeDspacetimes.Nextwespecializetonon-acceleratingspacetimesbymakinguseofEquation 2{36 intheform0= 2{56 toobtain Nowwecomputethecommutator[;0]andusetheGHPequationsandtheappropriateversionofEquation 2{57 untilwearriveatanexpressioninwhichtheonlyderivativesare0and0.ThisexpressioncanthenbeusedwithEquations 2{51 and 2{36 tondthefollowingtworelations: 22 22 (2{60) andourtaskiscomplete.ItisworthpointingoutthatduetoEquations 2{36 ,theseexpressionsarenotunique.ThisisasignthatthereissomeredundancyintheGHPequations,whichistobeexpectedwhenweconsidersuchaspecialclassofspacetimes.Wealsopointoutthathavingexpressionsforeveryderivativeoneveryquantityofinterestissucient(butnotnecessary)tocompletelyintegratethebackgroundGHPequations.This 39 PAGE 40 33 ]andlaterbyHeld[ 45 ]fortheGHPequations.InChapter5,wewilldiscussthelatterofthesemethodsinmoredetail. 46 ],whosebasicresultswillbedevelopedhere.Beforeweaddresstherelativisticproblem,weverybrieyreviewrst-orderperturbationtheoryinaatspacetime.Inthatinstance,wethinkofthequantityofinterest,q=q(),asbeingparameterizedbysome,sothatq(0)correspondstotheunperturbedquantityandq(1)isthefullyperturbedquantitywhoserst-orderperturbationswewouldliketoconsider.Itfollowsfromwritingq()asaTaylorseriesinthattherst-orderperturbation,q,isgivenbyq=dq() 40 PAGE 41 Theimportantpointaboutthisprescriptionisthefactthatanotonlyfailstobeunique,butthereis,ingeneral,nopreferredchoiceforit.Achoiceofaismorecommonlyknownasachoiceofgauge.AccordingtoEquation 2{61 ,thedierencebetweenQcomputedwithaandaisgivenbyQQ=$Q; ~Q=Q$Q:(2{62) Notethatagaugetransformationinthissenserepresentsachangeinthewayweidentifypointsinthephysicalspacetimewithpointsinthebackground.Thisistobedistinguishedfromacoordinatetransformation,whichchangesthelabelingofcoordinatesinboththephysicalandbackgroundspacetimes. ThesignicanceofEquation 2{62 isthatunless$Q=0foreverya,thereissomeambiguityinidentifyingtheperturbation|wecan'tdierentiatebetweenthecontributions 41 PAGE 42 47 ].Adirectconsequenceofthisfactisthatthemetricperturbation,arguablythemostfundamentalquantitywedealwith,failstobegaugeinvariant.Fortunately,typeDspacetimescomeequippedwithtwogaugeinvariants,0and4,whichhavesimpleexpressionsintermsofthecomponentsofthemetricperturbation.Aswewillsee,appropriateuseofgaugefreedomsimpliesourcomputationstremendously. 20 ],Chrzanowski[ 18 ],andStewart[ 21 ],anditwillcertainlyplayacentralroleinanyfuturedescriptionofmetricperturbations.Thesecondingredient,theGHPformalismcomeswithseveraladvantages.Firstofall,theinherentcoordinateindependenceandnotationaleconomymakescalculationsingeneralspacetimestractable.Furthermore,byvirtueoftheGoldberg-Sachstheorem,wecandealwiththeentireclassoftypeDspacetimesatonce.Additionally,spin-andboost-weightsprovideusefulbookkeepingand,aswe'llsee,ausefulcontextforunderstandingtherolesthatvariousquantitiesplay.Lastbutnotleast,theuseofaspincoecientformalismhasprovedabsolutelycrucialforstudyingperturbationsofanythingotherthansphericallysymmetricspacetimes.WewillputtheseideastogethertocomputetheperturbedEinsteinequationsinamixedtetrad-tensorform.Thisistheheartofourwork. 42 PAGE 43 sothat,forexample,hll=hablalb.InorderforthistobevalidwithintheGHPformalism,eachcomponentofEquation 2{63 musthaveawell-denedspin-andboost-weight.Becausethebackgroundmetric(Equation 2{3 )isinvariantunderaspin-boost(Equation 2{17 )ithastypef0;0g,whichmustalsobethetypeofthemetricperturbation,hab.Thereforethetypeoftheindividualcomponentsofthemetricperturbationaredeterminedbytheirtetradindices: Allofthevectorsandtensorswewillconcernourselveswithcanbetreatedinthisway. ItisworthwhiletostophereandtakealookatwhatEquation 2{63 reallymeans.ComparingwithourtreatmentofSchwarzschild(Equation 1{4 ),wenotethatthescalarpartsofthemetricare\mixedup"inhll;hlnandhnn,allofwhichhavespinweightzerobutdierinboostweight.Similarly,thevectorpartsaregivenbyhlm;hnmandtheircomplexconjugatesandlikewisethetensorpiecesaregivenherebyhmm;hmmandhmm.However,theseidenticationsarecompletelyindependentofthebackgroundspacetime.Thus,inacertainsense,Equation 2{63 providesageneralizationoftheRWmode 43 PAGE 44 RecallourexpressionfortheperturbedEinsteinequations:Eab=1 2rcrchab1 2rarbhcc+rcr(ahb)c+1 2gab(rcrchddrcrdhcd): 2{63 ,wearriveattheperturbedEinsteinequationsinGHPform: 2cchab1 2abhcc+c(ahb)c+1 2gab(cchddcdhcd);(2{65) which(rightnow,atleast)don'tlookallthatdierent!ThetetradcomponentsofEquation 2{65 foranarbitraryalgebraicallyspecialbackgroundspacetimearegiveninAppendixB.Asidefromtheobviouscosmeticdierences,thereareseveralkeydistinctionsbetweenEquation 2{65 andthestandardformofmetricperturbationtheoryworthpointingout.Firstofall,ourformlacksthebackgroundEinsteinequationspresentinthestandardtreatment.TakingtheirplacearethebackgroundGHPequationsandBianchiidentities.Perhapsmoreimportantlyistheinherentcoordinateindependence.Coupledwiththeconceptsofspin-andboost-weight,thisallowsforacertainstructuralintuitionnotpresentincoordinatebasedtechniques.Thispointofviewwillbestressedthroughout. WritingEquation 2{65 isonething,butactuallycomputingitisanotherquestionentirely,whichwenowturnourattentionto. 44 PAGE 45 EverysessionbeginsbyinvokingGHPtools:> Tobeginwith,eachvariableisdirectlyspeciedbyitsusualname.ForexamplewouldbeenteredinMapleasconjugate(rho).Theprimedvariableshavea`1'appendedtotheend,sothat0wouldbeenteredasconjugate(rho1).TheWeylscalarsarerecognizedascapital'swiththeappropriatenumber,e.g.Psi2.Thederivatives,,0and0arerecognizedinMapleasth(),eth(),thp()andethp(),respectively.GHPtoolsrecognizesthetetradvectorsaslabelsindicatingthepositionoftheindexwiththeactualindexinparentheses.Forexamplelaandmcwouldbeinputaslup(a)andconjugate(mdn)(c).Finally,GHPtoolscontainsanarbitraryfunction,(inMaple:phi),thatisquiteusefulforgeneralcalculations.Amongst 48 49 ],calledGHPandGHPII.WestressthatGHPtoolsisnowayintendedtocompetewiththeseoranyotherMaplepackages. 45 PAGE 46 Thefp;qgtypeofanyquantitymaybeobtainedbytheuseofthegetpqfunction,whichreturnspandq,inthatorder:> 1=0; =0; =0; 1=0; PAGE 47 TherealusefulnessofGHPtoolscomesnotfromitsbookkeepingabilities,butratheritsabilitytoperformsymboliccomputationswithintheGHPformalism.TheseabilitiesbeginwiththeDGHP()procedure,whichexpandsderivativesofobjectsoccurringinanexpressioninaccordancewiththerulesofderivations.Forexample> 1)> PAGE 48 1))> Todate,DGHP()canhandlepowersandlogarithms(theonlyfunctionsthisauthorhasencounteredintheGHPformalism),buttheprocedurecanbeeasilymodiedtoaccommodatejustaboutanyfunction.BuildingcomplicatedexpressionsinvolvinglinearcombinationsofderivativeandmultiplicativeoperatorsiseasilyachievedwiththehelpoftheGHPmult()procedure.TheseexpressionscanthenbeexpandedwithDGHP().Asanexample,considertheexpression()4:> Simplifyingsuchexpressionsis,inthecontextoftypeDspacetimewithoutacceleration,handledbythetdsimp()procedurethatsubstitutestheknownvaluesofthederivativesofthespincoecients(storedinthegloballyavailablelisttdspec;suchaprocedurecanbeeasilygeneralizedtoencompassanyspacetime,shouldtheneedarise)intoitsargument.Thusourpreviousexamplesimpliesconsiderably:> PAGE 49 Perhapsevenmoreusefulisthecomm()procedurewhichcommutesderivativesonanexpression.Ittakestwoarguments:therstisthetermwhosersttwoderivativeswillbecommutedandthesecondistheexpressionintowhichtheresultwillbesubstituted.Considerthefollowingexamples:> me1:=eth(th(ethp(ethp())))th(eth(ethp(ethp())))> 1ethp(ethp()) 1ethp(ethp())pq> me2:=th(th(thp()))th(thp(th()))> eth() 1th(ethp()) 1+ppth()1+3pp1pp1 1+3pq 1pq 1 3pq ComputingtheperturbedEinsteinequationsandWeylscalarsnecessarilyrequirestheabilitytocontractvariouscombinationsofthetetradvectors.Thisfunctionalityisprovidedbythetetcon()procedure,whichalsotakestwoarguments.Therstistheexpressionthatcontainstheuncontractedvectorsandthesecondisalistoftheindicestobecontractedover.Taketheexampleofcomputingthetraceofthemetric: 49 PAGE 50 Finally,GHPtoolsprovidessomefunctionalityfortranslatingexpressionsintoNPexpressionsthatcansubsequentlybeconvertedtoordinarycoordinateexpressions.ThisfunctionalityisprovidebytheaptlynamedprocedureGHP2NP(),whichtakesasitsinputaGHPexpression.Thefunctionalityprovidedbytheprocedureislimitedtoexpressionsinvolvingatmosttwoderivatives.Furthermore,thederivativesmustappearinaspeciedorderaccordingtothefollowingrules:(1)and0mustalwaysappeartotheleftofand0,(2)mustappeartotheleftof0and(3)mustappeartotheleftof0.Takethefollowingexample:> (hln)+ Inordertoaidintheconversionofsuchquantitiesintocoordinateexpressions,GHPtoolscontains,aslistsofarrays,somecommonlyusedtetradsintheKerrspacetime.Theyare:theKinnersleytetradwithindicesuptetupKanddowntetdnK,thesymmetrictetrad(tetupS,tetdnS)andthesymmetrictetradboostedbyafunctionB(t;r;;)andspunbyafunctionS(t;r;;)(tetupSB,tetdnSB).Thesearecalledsimplybyinvokingtheirnames: 50 PAGE 51 2Iasin()p 2(racos()I)p 2I(r2+a2)sin()p 2r22Mr+a2 2;0;1 2a(r22Mr+a2)sin()2 2Iasin()p 2(r+acos()I)p 2I(r2+a2)sin()p 51 PAGE 52 Asarstapplicationofourframework,wewillprovideamoredetaileddiscussionoftheRegge-WheelerandTeukolskyequations.ThisleadsquitenaturallytoadiscussionofthemetricperturbationgeneratedfromaHertzpotential,whichwillplayamajorroleinsubsequentchapters.Ourstartingpointisageneraldiscussionofparitythatdoesnotassumeeithersphericalsymmetryorangularseparationfromtheoutset. 50 ]. Ourrstassumptionisthatourspacetimemanifold,M,admitsaspacelike,closed2-surface,S,topologicallya2-sphere,withpositiveGaussiancurvatureandapositivedenitemetricgivenby wheremaandmaaretwomembersofanulltetrad.Forasphericallysymmetricbackgroundabisproportionaltothemetricofthe(round)2-sphereandmaandmacanbedirectlyassociatedwiththebackgroundmetric.Moregenerally,weallowforthe 52 PAGE 53 Forexampleconsidersomevector,va,denedinthespacetime:va=vlna+vnlavmmavmma: Theseareallthetoolsnecessaryforwhatfollows. WebeginbyconsideringtheprojectionofvectorsdenedinthespacetimeontoS.ToidentifytheoddandevenparitypieceswestartbydecomposingageneralvectoronS (3{4) 53 PAGE 54 Symmetric,trace-freetwo-indexedtensorsonSalsohaveasimpleparitydecomposition.Itiseasytorecognizethe(two)componentsofsuchtensorsasspin-weight2scalars.Thatis,thecomponentsareoftypefb2;b2g.WeconsidertheparitydecompositiononSbycreatingthetensorfromavectoronS,a,withboost-weightbandspin-weight0: whichcaninturnbefurtherdecomposedintoitsevenandoddparitypiecesbyapplyingEquation 3{4 toyield whichprovidesuswithameansofidentifyingtheevenandoddbitsofsymmetrictrace-freetensorsonS.Thisresultgeneralizesquiteeasilyton-indexedsymmetrictrace-freetensors(withcomponentsofspin-weightnandboost-weightb)onS: 8 ]fordetails)(i.e.,the\i"comesalongfortheride). 54 PAGE 55 Inpractice,wearegenerallygivensomespin-andboost-weightedscalar,(and/oritscomplexconjugate),andwemerelywanttoidentifytheeven-andodd-paritypieceswithoutexplicitlydecomposingitaccordingtoEquation 3{7 .InthiscaseEquation 3{7 allowsustodosobysimplywriting Inthecontextofaspacetimewherelaandnaarexedbyconsiderationsotherthanbeingorthogonaltomaandma(e.g.PetrovtypeD,wherewewouldlikethemalignedwiththeprincipalnulldirections),butmaandmafailtoformaclosed2-surface(theKerrspacetimeprovidesonesuchexample;thiscanbeseenbynotingthatand0don'tcommute),thequestionarisesofwhetherornotsomethinglikeEquation 3{8 isstillusefultoconsider.Itappearsso.Insuchacasethedecompositiontheorems(therstlinesofEquations 3{5 and 3{4 )failtobetrue,butthisisn'taseriousissue.Becauseabandabstillallowustodecomposetensorsintotheir\proper"and\pseudo"pieces,inplaceofEquation 3{7 wehave where\even"and\odd"arewritteninquotestoemphasizethefactthattheyreallyrefertorealandimaginaryinthiscontextandthebarovertauindicatestheproperspin-andboost-weight.Clearly,Equation 3{9 ,lackstheadvantagepresentinEquation 3{7 ofbeingabletoputalloftheangulardependenceintoand0andregardtheentiretensorasarisingfromthetworealscalarsevenandodd.Neverthelessitprovidesausefuldecompositionofspin-andboost-weightedscalars,withoutseparationofvariables,that 55 PAGE 56 3{8 inarbitrarybackgrounds.Furthermoreinthelimitthatmaandmabecomesurface-forming(e.g.thea!0limitoftheKerrspacetime),Equation 3{9 becomesEquation 3{7 .Thisisoneavenueforunderstandingwhyparityplayssuchanimportantroleintheperturbationtheoryofsphericallysymmetricbackgrounds.Inthecontextofnulltetradformalismswecanseetheseeminglyunmotivatedactofperformingparitydecomposition,whichdoesnotgeneralizewell,asarisingfromthequitenatural(andperhapsmorefundamental)actofseparatingquantitiesintotheirrealandimaginaryparts,whichisentirelygeneral.Inthislight,itmakessensethatourattentionwouldbefocusedonparitybecausetherstperturbativeanalysistookplaceinthesphericallysymmetricSchwarzschildbackgroundinwhichonecannotdierentiatebetweenthetwodecompositionsbutparityhassignicancethere.Regardless,theonlyusewemakeoftheseresults,exceptforsomeremarksinChapter5,isbelowinthecaseoftheSchwarzschildbackgroundwherethepointismoot. 24 ]whoobtained sin2hh=0;(3{10) 56 PAGE 57 sin@(sinht+@ht)=0; sin@(sinhr+@hr)=0; asthemode-independentexpressionoftheRWgauge.NowwecantransformthisdescriptionintoGHPlanguage.Itisarelativelystraightforwardprocessnowtowritethetetradcomponentsofthemetricperturbation(hll,hln,etc.)intermsofthecoordinatecomponentsofthemetricperturbation(htt,hrr,etc.)andinverttherelations.Withthisknowledgeinhand,itbecomesevidentthatEquations 3{10 and 3{11 aresimplycombinationsofhmm=0andhmm=0: 3{12 and 3{13 arecombinationsof0hlm+hlm=0andhnm+0hnm=0; Inthislanguage,itisnaturaltogeneralizetheseconditionstomoregeneraltypeDspacetimesonthebasisofspin-weightconsiderations.ThespiritoftheRWgaugesuggeststhatwekeeptherequirementthatnospin2componentsenterthemetricperturbation.Therequirementonthespin1componentsiseasilygeneralizablebyputtinginpiecesproportionaltoand0whichbothvanishintheSchwarzschildbackground. 57 PAGE 58 whereaandbare(generallycomplex)constantsthatmustbedeterminedbysomeothermeans.NotethattheformofEquations 3{14 isrestrictedbyrequiringthegaugerestrictionstobeinvariantunderbothprimeandcomplexconjugation.ThefullutilityofthegeneralizedRWgaugeremainstobeexplored,butitisclearthatanysimplicationitbringswillapplyuniformlytoalltypeDspacetimes. withallotherbackgroundquantitiesvanishing,sothesituationisimmediatelysimplied.Nextweproceedwiththeparitydecompositionbywritingthecomponentsofthemetricperturbationas,forexample,hlm=hevenlm+ihoddlm,hlm=hevenlmihoddlm,etc.Notetherelativeminussignsbetweentheodd-paritybitsandtheircomplexconjugates.Fromhereonwewillspecializetoodd-parityandthusdropthe\odd"labelsandfactorsofisincenoconfusioncanarise.Withthisspecialization,ourgaugeconditionsnowread: 58 PAGE 59 (00)hlm+()hnm=0 (3{17) (00)hlm+()hnm=0; wherethesecondrelationfollowsfromcomplexconjugationoftherst(orintegrationofEmm),andwehavesetthe\integrationconstant"tozeroforconvenience(itwouldcancelbelow).WenowturnourattentiontoElm.BysuccessiveapplicationsofEquation 3{17 wecaneliminatealltermsinvolvinghnm,arrivingat 2n(020+402042)hlm22hnmo(3{19) Takingtheprimeofthis(whichintroducesanoverallminussignbecauseoftheparitydecomposition)leadstoasimilarexpressionforEnm.Nextwetakethe(sourcefree)combination (020)0Elm+(2)Enm=0:(3{20) Wecanremovefromthisexpressionallreferencestohlmand0hnmusingthegaugeconditionsinEquations 3{16 ,which,aftersomeseriouscommutingleadstothequite 59 PAGE 60 ThisistheRegge-Wheelerequation.Wecancleanitupabitbyrecognizingtheobjectbeingactedonas2_odd2=hnm00hlm,theodd-paritypieceoftheperturbationof2.FurthermoretheoperatorinEquation 3{21 isthewaveoperator,2,intheSchwarzschildbackgrounduptoafactorof1=2.Makingtheseidentications,wenowhavefortheRegge-Wheelerequation: (2+82)2=32_odd2=0:(3{22) Asimilarequationfor_odd2=Im(_2)waspreviouslyderivedbyPrice[ 51 ](whoseonlyrelationtothepresentauthorisEquation 3{22 ),whoshowedthat(moduloangulardependence),Im(_2)isthetimederivativeoftheRegge-Wheelervariable.Moreover,withoutreferencetoIm(_2)Jezierski[ 52 ]arrivedatanequationforodd-parityperturbationsthatisessentiallyidenticaltoEquation 3{22 ,thoughphrasedinmorestandardlanguage.Additionally,ananalysisbyNolan[ 53 ]wholookedattheperturbedWeylscalarsintermsofgaugeinvariantsofthemetricperturbationshowedexplicitlytherelationbetweenIm_2andthegaugeinvariantquantityassociatedwiththeRWvariable.Furthermore,Nolanpointsoutthatbecause2isrealinthebackground,theperturbationofitsimaginarypartis,whenwerestrictourattentiontoodd-parity,gaugeinvariantinthesensediscussedinChapter2.Perhapsmoresurprisingly,NolanfurtherassertsthatthisistrueoftheperturbationsofalltheWeylscalars,whichemphasizesthefactthatodd-parityperturbationsofsphericallysymmetricspacetimesareobtainablebyvirtuallyanymeans. OnethingthatsetsourtreatmentofRWapartfromothersisoursparinguseofsphericalsymmetry.TheonlyplacewemakeexplicituseofitisinEquations 3{15 ,whichdenesthebackgroundGHPquantities.Thiscertainlysimpliesthesubsequentcalculationsconsiderably,butfailstofullyexploitthebackgroundsymmetry.In 60 PAGE 61 TheZerilliequation[ 7 ]describingeven-parityperturbationsoftheSchwarzschildspacetimehassofareludedadirectdescriptionintermsofgaugeinvariantperturbationsoftheWeylscalars.However,theinformationcontainedwithintheZerilliequationcanbeobtainedthroughthemetricperturbationthatfollowsfromtheTeukolskyequation,whichisthefocusoftheremainderofthischapter. 10 { 12 ]camedirectlyfromconsideringperturbationsoftheWeylscalars.We,however,areinterestedinobtainingitdirectlyfromtheEinsteinequation.UsingTeukolsky'sexpressionsforthesourcesof0and4,wecanobtainthisdirectly.ThesourcesoftheTeukolskyequationaregivenby whereT0andT4arethesourcesfor0and4,respectively.MakingthereplacementTab=1 8Eabintheequationsaboveleads(afterproperlyrearrangingthederivativeswith 61 PAGE 62 [(4)(00)(40)(00)32]0=4T0; [(0400)()(040)()32]4=4T4; where,intermsofthecomponentsofthemetricperturbation 2((0)(0)hll+()()hmm[()(20)+(0)(2)]h(lm)); 2((0)(0)hnn+(00)(00)hmm[(00)(02)+(0)(020)]h(nm)); andwheretheparentheses,(),aroundthetetradindicesdenotesymmetrization.Itisbothinterestingandimportanttonotethat,intheKerrspacetime,thecoordinatedescriptionofEquation 3{26 doesnotleadtotheseparableequationdiscussedinChapter1(Equation 1{17 ).Toobtainaseparableequation,anextrafactorof4=32mustbebroughtin,resultinginthefollowingexpression: [(00)(+3)(0)(+3)32]4=324=44=32T4:(3{29) Belowwewillseethesameexpressionarisingfromverydierentconsiderations. 20 ]usedspinormethods.Shortlyafterthat,Chrzanowski[ 54 ]obtainedessentiallythesameresultusingfactorizedGreen'sfunctions.Sometimelater,Stewart[ 21 ]enteredthegameandprovidedanewderivationrootedinspinormethods.Eventually,Wald[ 55 ]introduceda 62 PAGE 63 Wald'smethodiscenteredaroundthenotionofadjoints.Considersomelineardierentialoperator,L,thattakesn-indextensoreldsintom-indextensorelds.Itsadjoint,Ly,whichtakesm-indextensoreldsinton-indextensoreldsisdenedby forsometensoreldsa1:::amandb1:::bnandsomevectoreldsa.IfLy=L,thenLisself-adjoint.Animportantpropertyofadjointsisthatfortwolinearoperators,LandM,(LM)y=MyLy.NowletE=E(hab)denotedthelinearEinsteinoperator,StheoperatorthatgiveseitheroftheTeukolskyequationsfromE(Equation 3{23 or 3{24 ),O=O(0or4)thesource-freeTeukolskyoperator(Equation 3{25 or 3{29 )andT=T(hab)theoperatorthatactsonthemetricperturbationtogive0or4(Equation 3{23 or 3{24 ).ThentheTeukolskyequationscanbewrittenconciselyas Itfollowsbytakingtheadjointthat wherewehaveusedthefactthattheperturbedEinsteinequationsareself-adjoint.Thus,ifsatisesOy=0,thenSyisasolutiontotheperturbedEinsteinequations!ThisremarkablysimpleandelegantresultholdsforanysystemhavingtheformofEquation 3{31 ,wheneverEisself-adjoint. InordertoapplythisresulttotheTeukolskyequationwenotethatscalarsareallself-adjointandtheadjointsoftheGHPderivativesaregivenby 63 PAGE 64 SupposenowthatwehaveasolutiontotheTeukolskyequationfor0,sothatOisgivenbytheleftsideofEquation 3{25 andSisgivenbytherightsideofEquation 3{23 (withTabreplacedwithEab).Wald'smethodthentellsusthatifOy=0,thenhab=SyisasolutiontotheperturbedEinsteinequations.UsingEquations 3{33 wecancomputeSy: wherewe'veaddedthecomplexconjugate(c.c.)tomakethemetricperturbationrealandremainstobespecied.UsingEquations 3{33 ,itisclearthattheadjointofEquation 3{25 is [(00)(+3)(0)(+3)32]=0;(3{36) whichispreciselytheequationsatisedby4=324(c.f.Equation 3{29 ),previouslyobtainedthroughseparabilityconsiderationsintheKerrspacetime.However,obtainingEquation 3{36 requirednoreferencetoseparationofvariablesinaparticularspacetimeandthusappliestoalltypeDspacetimes.Itisimportanttonotethatalthoughsatisesthesameequationas4=324,itisnottheperturbationof4forthemetricitgenerates(Equation 3{35 ).InChapter5wewillexplore'sconnectionto4morecarefully. ThoughthederivationofEquation 3{35 wasquitesimple,itfailstoyieldanyinformationaboutthegaugeinwhichthemetricperturbationexists.Inthisparticularinstance,itisfairlystraightforwardtoverifythatthemetricperturbationwe'vebeenled 64 PAGE 65 whichisknownintheliteratureastheingoingradiationgauge(IRG),anunfortunatenamebecauseingoingradiationiscarriedbylaandEquation 3{37 tellsusthatthemetricperturbationiscompletelyorthogonaltola.Thusthereisonlyoutgoingradiationintheingoingradiationgauge!ObtainingthegaugeconditionsinEquations 3{37 and 3{38 ismorenaturalintheapproachesofCohenandKegeles[ 20 ]andStewart[ 21 ].Onestartlingaspectofthegaugeconditionsisthatthereareveofthem.Thisbeingthecase,wemustbeconcernedaboutthecircumstancesunderwhichthemetricperturbationintheIRGiswell-dened.Thisisthesubjectofthenextchapter. OurderivationbeganwiththeTeukolskyequationfor0.HadweinsteadstartedwiththeTeukolskyequationfor4=324,wewouldbeledtoametricperturbationintermsofaHertzpotential,0,thatsatisestheTeukolskyequationfor0.TheresultingmetricperturbationandgaugeconditionsarethensimplytheGHPprimeofEquations 3{35 3{37 and 3{38 ,respectively.Inthiscase,themetricperturbationexistsintheso-calledoutgoingradiationgauge(ORG).Fortheremainderofthiswork,wewillfocusourattentionontheIRGmetricperturbation,butalltheresultsholdfortheORGperturbationaswell. OnanalnoteweremarkthattheTeukolskyequationfor0(Equation 3{25 )actuallyexistsinthemoregeneraltypeIIspacetimes,withoutitscompanionfor4.Inthiscase,Wald'smethodalsoleadstometricperturbation(intheIRG;noORGexistshere),withapotential,,satisfyingtheadjointofEquation 3{25 ,which,inthisinstance,isnottheequationfortheperturbationof4. 65 PAGE 66 Inthepreviouschapter,itwasseenthattheperturbationsoftheWeylscalarsleadquitenaturallytometricperturbationsintheradiationgauges,(seeminglyover-)speciedbyveconditions.Inthischapterwewillexploretheprecisecircumstancesunderwhichonecanimposeallveoftheseconditions.ThiswillrequireustoexaminetheperturbedEinsteintensor,whichpresentstheneedtointegratesomeofthecomponents.Forthis,wewillappealtoacoordinate-freeintegrationtechniquebasedontheGHPformalism,duetoHeld[ 45 56 ].Thegeneralityofthesemethodsallowustoprovetheresultforamuchbroaderclassofspacetimesthanwehaveencounteredsofar,namely,PetrovtypeII,whichwewillseeisthelargestclassofspacetimeforwhichtheradiationgaugesaredened.Webeginwithamorethoroughdiscussionoftheradiationgaugesandtheirorigin.Mostofthischapteristakenfrompublishedwork[ 57 ]. 58 ](forperturbationsofPetrovtypeIIspacetimes)andChrzanowski[ 54 ](whoconsideredperturbationsofPetrovtypeDspacetimes),buttheworkthatcomesclosesttoourcontributionindescribingtheiroriginisthatofStewart[ 21 ],againforthemoregeneralcaseoftypeIIspacetimes. IntypeIIbackgroundspacetimes,theIRGisdenedbytheconditions 66 PAGE 67 IntypeIIbackgroundspacetimes,onlyoneortheotheroftheseoptionsexists(IRGorORG),whereasinPetrovtypeDbackgroundspacetimes,thereisthepossibilityofdeningbothgauges.FortheremainderofthisworkwefocusontheIRG.ResultsfortheORGcanbeobtainedbymakingthereplacementla$na. Equations 4{1 translateintoalgebraicconditionsonthecomponentsofthemetricperturbation.Wewillrefertothefourconditionsin( 4{1 )asthelhgaugeconditions. TheconditioninEquation 4{2 willbereferredtoasthetraceconditionandcanbeexpressedintermsofthecomponentsofthemetricperturbationashlnhmm=0;which,whenEquation 4{5 isimposed,simplyreads BecausetheIRGconstitutesatotalofveconditionsonthemetricperturbation,insteadofthefouronemightexpectforagaugecondition,itisnecessarytoensurethattheextraconditiondoesnotinterferewithanyphysicaldegreeoffreedomintheproblem, 59 ],thoughtheyarenottheconditionsoriginallyintroducedforgravitationwiththatname[ 60 ].Itmaywellbethatthisdescriptionissuitablemoregenerally,althoughpresumablywithoutthespecicgeometricalinterpretationof[ 59 ]. 67 PAGE 68 B{1 ofAppendixB,inwhicheverytermwouldberemovedbyEquations 4{5 and 4{6 ,renderingEquation B{1 inoperablewheneverithasanon-zerosource.InthenextsectionwewilllooktotheperturbedEinsteinequationstodeterminethecircumstancesunderwhichwecansafelyimposeallveoftheconditionsthatconstitutetheIRG. ItisusefultonotethesimilaritybetweenthefullIRG,( 4{1 ),andthemorecommonlyknowntransversetraceless(TT)gaugedenedby which,ataglance,alsoappearstobeover-specied.Infact,theTTgaugeexistsforanyvacuumperturbationofanarbitrary,globallyhyperbolic,vacuumsolution[ 61 ],becauseimposingthedierentialpartofthegaugedoesnotexhaustalloftheavailablegaugefreedom.Interestinglyenough,Stewart'sanalysisintermsofHertzpotentials[ 21 ]beginsbyconsideringametricperturbationintheTTgauge.However,inordertoconstructthecurvedspaceanalogueofaHertzpotential,heiscompelledtoperformatransformationthatdestroysEquation 4{7 andinsteadyieldsametricperturbationintheIRG. Consideragaugetransformationonthemetricperturbationgeneratedbyagaugevector,a.Tocreateatransformedmetricinthelhgauge,thegaugeconditionsin 21 ]ortheelectromagneticexampleinChapter1foramoredetailedexplanation. 68 PAGE 69 4{5 require wherethesemicolondenotesthecovariantderivative.Intermsofcomponentsthisreads 2l=hll;0l+n+(+0)m+(+0)m=hln;(+)m+(+0)l=hlm;(+)m+(0+0)l=hlm:(4{9) Similarly,forthetraceconditioninEquation 4{6 tobesatisedbythegaugetransformedmetric,werequire Anyextragaugetransformationthatsatisesla(a;b)=0|solvesthehomogeneousformofEquation 4{9 |preservesthefourlhgaugeconditionsinEquations 4{5 .Thisiswhatismeantbyresidualgaugefreedom.Wewillexplicitlyusethisresidualgaugefreedomtoimposethelhandtraceconditionssimultaneously,thusestablishingtheIRG.Wewillndthatsomegaugefreedomstillremains,asexplainedinSection 4.3 Now,weturnourattentiontothegeneralcaseoftypeIIbackgroundspacetimes. 45 46 ]. 69 PAGE 70 [;~0]x=0;[;~]x=0;[;~0]x=0;(4{11) where[a;b]denotesthecommutatorbetweenaandb.TheexplicitformoftheoperatorsisgiveninAppendixC.Thenextstep,theheartofHeld'smethod,istoexploittheGHPequation=2,anditscomplexconjugate,=2,toexpresseverythingasapolynomialintermsofand,withcoecientsthatareannihilatedby.Held'smethodisthenbroughttocompletionbychoosingfourindependentquantitiestouseascoordinates[ 56 62 ].Inthiswork,wewillnottakethisextrastep.FortypeIIspacetimes(andtheacceleratingC-metrics),thisstephasnotbeencarriedout,whileforallremainingtypeDspacetimes,ithasbeencarriedthroughtocompletion[ 45 46 ]. InaspacetimemoregeneralthantypeII,thereisnopossibilityofhavingarepeatedPND.WhenarepeatedPNDexists,wecanappealtotheGoldberg-Sachstheorem[ 32 ]andset==0=1=0inEquationsB-1{B-7.FollowingHeld'spartialintegrationofPetrovtypeIIbackgrounds[ 56 ],wealsoperformanullrotation(keepinglaxed,butchangingna)toset=0.Asaconsequence,itfollowsfromtheGHPequationsthat0=0.NowweareinapositiontoaddressthequestionofwhenthefullIRGcanbeimposed.FirstweapplythelhgaugeconditionsinEquations 4{5 toEquationsB-1{B-7.WhilemostoftheperturbedEinsteinequationsdependonseveralcomponentsofthemetricperturbation,afterimposingEquations 4{5 ,theexpressionforElldependsonlyonhmmandthell-componentoftheperturbedEinsteintensorsimplybecomes 70 PAGE 71 4{12 andourargumentwouldnothold.WeimmediatelyseethatTll=0isnecessarytosatisfythetraceconditioninEquation 4{6 .NextweturnourattentiontothequestionofwhethertheconditionEll=0,issucienttoimposeEquation 4{6 usingresidualgaugefreedom. InordertoaddressthisquestionwewillintegrateEll=0andtheresidualgaugevector,givenbythehomogeneousformofEquations 4{9 .FullintegrationofthehomogeneousformofEquations 4{9 iscarriedoutinAppendixC,butwewillworkthroughtheintegrationofEll=0heretoillustratethemethod.WebeginbyrewritingEquation 4{12 ,withthehelpof=2anditscomplexconjugate,as: 3 Integratingoncegives andanotherintegrationleadsto +1 2b(+):(4{15) However,hmmis,bydenition,arealquantity,soweaddthecomplexconjugateandusebtorepresentarealquantityinthesecondterm.Thenalresultisthat +b(+):(4{16) 71 PAGE 72 4{9 ,ascarriedoutinAppendixC,leadstothefollowingsolutionforthecomponentsoftheresidualgaugevector: 21 ~0l+1 2(2+2)l;m=1 m~l;m=1 where2isrelatedtothebackgroundcurvaturevia2=23.InordertousethisresidualgaugefreedomtoimposethefullIRG,wereturntothegaugetransformationforhmm(Equation 4{10 )whichbecomes,aftersomemanipulation(usingEquations C{6 { C{9 andEquation C{13 ), h~m+~0li+(+)[1 2(~0~+~~000)l+n]:(4{18) Inthisformitisclearthatwecanimposethetracecondition(Equation 4{6 )ofthefullIRGifwechooseourgaugevectorsothat ~0m+~0l=a;1 2(~0~+~~000)l+n=b:(4{19) WehavenowshownbyconstructionthattheconditionTll=0isbothnecessaryandsucientforimposingthefullIRGinatypeIIbackground.Weturnnexttodiscussingthecompleteextentoftheresidualgaugefreedominmoredetail. 4{19 involvethreerealdegreesoffreedom(aiscomplex),itturnsoutthatonlytworealdegreesofgaugefreedomarerequiredtofullyremoveanysolutionofEquation 4{13 forthetracehmm.Toseethisweintroducethefollowingidentity: =(+)1 1 72 PAGE 73 4{16 as 2(a+a) +[1 2(aa)+b](+):(4{21) Inasimilarfashion,werewriteEquation 4{18 as 2(~0m+~m)+~0li +h1 2(~0m~m)1 2(~0~+~~000)l+ni(+);(4{22) inwhicheachcoecientinbigsquarebracketsispurelyreal.Now,supposewehaveaparticularsolutionforEll=0(i.e.,a,aandbarexed)andourtaskistosolveforthecomponentsofthegaugevectorwhichremovesthissolution.BycomparingEquations 4{21 and 4{22 weseethat,foranygivenmandm,wecanxl(uptoasolutionof~0l=0)via ~0l=1 2(a+a)1 2(~0m+~m);(4{23) andwecanxnbysetting 2(aa)+b+1 2(~0~+~~000)l1 2(~0m~m);(4{24) tocompletelyeliminatethenonzerohmm,thusimposingthefullIRGwhilestillleavingtwocompletelyunconstraineddegreesofgaugefreedom,mandm.OnceintheIRG,Equations 4{23 and 4{24 ,witha,aandbsettozeroandmandmarbitrary,givetheremainingcomponentsofagaugevectorpreservingtheIRG.ItiscurrentlyunclearhowtotakeadvantageofthisremaininggaugefreedomtosimplifytheanalysisofperturbationsinthefullIRG. 33 ].Whilethe 73 PAGE 74 45 ].ThecompleteintegrationrequiresthateachoflaandnabealignedwithoneofthetwoPNDs.InthatcasewecanexploitthefullpoweroftheGoldberg-Sachstheoremanditscorollariestoset=0==0=0=1=3=4=0,whilemaintaining6=0and06=0.Inthissectionwerepeatthepreviouscalculationwiththisdierentchoiceoftetrad. TheresultofintegratingEll=0isthesameasinthecaseofatypeIIbackground,giveninEquation 4{16 .Theresidualgaugevector,however,nowhasthefollowing,morecomplex,form(detailsoftheintegrationaregiveninAppendixC): 2l+1 2l+l+1 2l1 2+h 21 ~0l[(~+)+(~0+)]l+m m1 2m1 m1 l1 +l(~0+)l;(4{25) 74 PAGE 75 h~m+~0l+Bi+(+)h1 2(~0~+~~000)l+nAi;(4{26) wherewehaveintroduced(notethatBispurelyimaginary) 2f2~+~()+gl+c.c.;B=1 4f4~+~()+52glc.c.;(4{27) withc.c.indicatingthecomplexconjugate.Integrationofthebackgroundswhere6=0and6=0usingtheHeldtechniquehasnotmadeitswayintotheliteratureandisbeyondthescopeofthepresentwork.Asaresult,derivativesofandappearinEquations 4{27 butdonoharmtoourargument.Choosinganygaugevectorthatsatises ~0m+~0lB=a;1 2(~0~+~~000)l+nA=b;(4{28) willservetoimposethetraceconditioninthefullIRG.OnceagainwehaveestablishedthatTll=0isbothanecessaryandsucientconditionfortheexistenceofthefullIRG.Notethatbysetting==0(i.e.,ignoringtheC-metrics)inthebackground,A=B=0,andtheresultisvirtuallyidenticaltoEquations 4{18 and 4{19 .Thereisonesimplicationinthatnow0=0[ 46 ].ThefullextentoftheremainingresidualgaugefreedominEquations 4{28 canbedemonstratedalongthesamelinesasusedinSection 4.3 .AsforthecaseofatypeIIbackground,itresideschieyinthefreelyspeciablemandm. 75 PAGE 76 ForperturbationswithTll=0,ourcharacterizationoftheresidualgaugefreedomissucientlycompletethatwecanexplicitlydemonstratetherequiredgaugechoicetoremoveanynon-zerosolutionforthetraceobtainedviaEll=0.Thus,intypeIIspacetimes,radiationgaugescanbeestablishedbyagenuinegaugechoice,evenifonlyafterasolutionofEll=0ischosen. TherearesubtledierencesbetweenthegeneraltypeIIcaseandthemorerestrictedtypeDcase,astherearealsointheconstructionofHertzpotentialsforthetwocases.Stewart[ 21 ]writesoutthetypeIIcaseratherfullyforanIRG.Inthiscase,theperturbationin0istetradandgaugeinvariant,whilethepotentialsatisestheadjoint(inthesensedetailedbyWald[ 55 ])ofthes=+2Teukolskyequation.Remarkably,inthetypeDcase,thisadjointisactuallythes=2Teukolskyequation,alsosatisedbythegaugeandtetradinvariantperturbationin4.InthetypeIIcase,theadjointequationisthesameasintypeD,but4isnolongertetradinvariant.ComparedtothetypeDresult,theexpressionfor4givenbyStewarthasmanyextratermsdependingon0and0,sopresumablyitdoesnotsatisfythesameequationasthepotential.Asaconsequence,metricreconstructionwouldberestrictedtobeingbuiltaroundtheperturbationfor0(c.f.thecommentsattheendofChapter3). 76 PAGE 77 63 ]hasarguedthatmassandangularmomentumperturbationsarenotgivenbyanysolutiontotheTeukolskyequations,andStewart[ 21 ]hasshownthatthesecannotberepresentedinaradiationgaugeintermsofapotential.Whatwehavedoneisidentifythegaugefreedomwhichremainsinthefullysatisedradiationgauges,neitherinterferingwiththeradiationgaugeprescriptionnorrulingoutthepossibilityofmassandangularmomentumperturbations.ByrealizingtheexplicitconstructionoftheradiationgaugesfortypeIIbackgroundspacetimesandbyidentifyingtheremaininggaugefreedomwhichtheyallow,wehave,inasense,completedataskinitiallyembarkeduponbyStewart[ 21 ],thoughinthedierentcontextofHertzpotentials. 77 PAGE 78 Havingestablishedtheconditionsfortheexistenceoftheradiationgauges,wewillusethecorrespondingmetricperturbationstoestablishsomeusefulrelationshipsbetweentheperturbedWeylscalarsknowngenerally(andquiteloosely)astheTeukolsky-Starobinskyidentities.BecauseHertzpotentialsaresolutionstotheTeukolskyequation,theseidentitieshaveimmediaterelevanceformetricreconstructionintheIRG,bothinthetime-domainapproachofLoustoandWhiting[ 25 ]andthefrequencydomainapproachofOri[ 23 ]. TheoriginalanalysisofTeukolsky[ 11 12 ]wasbasedontheasymptoticformofthesolutionsoftheseparatedangularandradialfunctionsintheKerrspacetimeaswellasatheoremduetoStarobinskyandChurilov[ 64 ].OnlylaterdidChandrasekharprovideafullanalysis,whichisnicelysummarizedinhisbook[ 29 ].Ouranalysis,however,willbeentirelysymbolic,involvingonlyGHPquantities.Thisapproachhastheadvantagenotonlyofapplyingtoalargerclassofspacetimes,butdisplayingthestructureinherentintheidentitiesinamuchmoreobviousway.AsimilaranalysisofsomeoftheidentitieswewilldiscusswaspreviouslyundertakenintheNPformalismbyTorresdelCastillo[ 65 ]andlatertranslatedintoGHPbyOrtigoza[ 66 ].TheseprioranalysesmadeuseofthemostgeneraltypeDspacetimeandtranslatedbackandforthbetweencoordinate-basedandcoordinate-freeexpressions.Incontrast,ourapproachwillnotmakeanyreferencetothechoiceofcoordinatesoratetrad(otherthanrequiringittobealignedwiththeprincipalnulldirections).Becauseofthis,ourapproachwillshowcaseoneofGHPtools'greateststrengths{theabilitytocommuteseveralderivativeswithrelativeease. Ourstartingpointisthe(source-free)IRGmetricperturbationgivenbyhab=flalb()(+3)l(amb)[(+)(+3)+(+0)(+3)]+mamb()(+3)g+c.c.: 3{35 ) 78 PAGE 79 3{35 ,theactualperturbedWeylscalarsfollowdirectlyfromEquations 3{27 and 3{28 24; 2n0434=32[1=32(000+022)]o: Theterminsquarebrackets[]inEquation 5{2 isactuallyjusttheoperatorformofthe(generallycomplex)Killingvector(actingon,whichhastypef4;0g)discussedinChapter2.WecanfurthercombinetherelationsinEquations 5{1 and 5{2 toeliminateanyreferencetothepotentials.TherststepistoactonEquation 5{2 with4=32,whichgivesus 2n44=320434Vo:(5{3) Commutingtheeightderivativesontherstterm(usingGHPtools,ofcourse)yieldstheusefulidentity whichwewillhaveoccasiontoexploitagain.CommutingthederivativesinthesecondtermofEquation 5{3 posesnoproblembecauseVcommuteswitheverything.NowitisasimplemattertoidentifytheresultingexpressionwiththetermsinEquations 5{1 and 5{2 toarriveatthefollowing 2inseveralearlierpapers.Stewart[ 21 ]andChrzanowski[ 18 ]havethesefactorscorrect,thelatterwithdierentsignconventions. 79 PAGE 80 65 ]andOrtigoza[ 66 ],whereexplicitcoordinateexpressionswereused,Equations 5{5 and 5{6 bothappeartobetrue.Thisfactappearstobecoincidentalsinceitisunclearhowitfollowsingeneralfromthefundamentalequationsofperturbationtheory.Theremainderoftheidentitieswewillpresenthavenotappearedintheliteratureinthisformandwecanonlyclaimtheyholdfornon-acceleratingtypeDspacetimes. Beforewecontinue,we'lltakealookatthecontentofEquations 5{5 and 5{6 inthecontextoftheKerrspacetime.Ifwewrite0R+2(r)S+2(;)and4=324R2(r)S2(;)andunderstandthetimedependenceofeachtobegivenbyei!t,thenEquation 5{5 tellsus:(1)theresultoffourradialderivativesonR+2isproportionaltoR2and(2)theresultoffourangularderivativesonS2isproportionaltoS+2.ThesameistrueofEquation 5{6 withthe+'sand'sswapped.NotethatEquations 5{1 and 5{2 (andtheirprimesintheORG)sayessentiallythesamethingwiththesubtledierencethattheangularandradialfunctionsarenotobviouslysolutionstothesameperturbation.NosuchambiguityarisesinEquations 5{5 and 5{6 Remarkably,wecanactuallytakethingsastepfurtherandarriveatexpressionsfor0and4independently.Webeginbyacting044=32onEquation 5{5 : Byrecallingthathasthesametypeas4=324(2carriesnoweight),wecansimplytaketheprimeandconjugateofEquation 5{4 ,anduseittocommutethederivativeson 80 PAGE 81 wherewemadeuseofEquation 5{6 inthesecondlineandcommutedeverythingthroughVinthethirdline.ThesecondterminEquation 5{7 becomes 3044=32V0=3V044=320=3V(044=324+3V4)=3V044=324+9VV4; wherewemadeuseofthecomplexconjugateofEquation 5{6 .Combiningtheseresultsgivesus wherewetooktheprimeoftherstequationtogetthesecondone\forfree."ThesearethesecondformoftheTeukolsky-Starobinskyidentities.Wenoteinpassingthatinthecontextoftheseparatedsolutionsof0and4=324,therelationsaboveallowforthedeterminationofthemagnitudeoftheproportionalityconstantrelatingR+2andR2[ 29 ]. Surprisingly,thisisn'ttheendofthestory.RecallthatinatypeDspacetimewealsohaveatourdisposaltheoutgoingradiationgaugewhere 2n40+34=32V0o; 2040; 81 PAGE 82 5{1 and 5{2 .NotethatwhereassatisestheTeukolskyequationfor4=324,0satisestheadjointequation|theTeukolskyequationfor0.FromthecomplexconjugateoftheprecedingequationsandtheirIRGcounterparts,wegetthefollowing: therstformoftheTeukolsky-Starobinskyrelationshipsforpotentials.NotethedierencebetweentheaboveandEquations 5{5 and 5{6 ,particularlythemissingfactorsof4=32andthefactthatVappears.AswithEquations 5{5 and 5{6 ,wecanobtainrelationsforeachpotentialindividuallybyacting044=32onEquation 5{14 andfurtherexploiting(theprimedconjugateof)Equation 5{4 .Theresultisthat Wecansummarizethislastidentitybywriting [044=324044=324+9VV4=32]f4=324;g=0; [44=320444=3204+9VV24=3]f4=320;0g=0: BardeenhasrecentlypointedoutanissueinthestandardtreatmentoftheTeukolsky-Starobinskyidentities[ 67 ].Inparticular,hendsthat,intheSchwarzschildbackground,thereisahithertounnoticedrelativesigndierencebetweentheodd-andeven-parityinthetermproportionalto@t(alternatively!whentimeseparationisperformed),whichbycontinuitypresumablypersistsintheKerrbackground.Bardeenarguesusingstandardtechniquesthatdon'tmakeclearthedierencebetweenthe'sandtheircomplexconjugatesontheright-hand-sidesofEquations 5{5 and 5{6 .However,recallingourdiscussionofparityinChapter3,aglanceattheseequationsrevealsthat 82 PAGE 83 83 PAGE 84 Inthischapterwewilladdresstheissueofthenon-radiatedmultipolesalludedtoinChapter1.TheissueisthatthemetricconstructedfromaHertzpotentialisincompleteinthesensethatitsmultipoledecompositionnecessarilybeginsat`=2becausetheangulardependenceofthepotentialisthatofaspin-weight2angularfunction.Toseethisexplicitly,wefocusourattentionontheIRGmetricperturbation(Equation 3{35 )intheSchwarzschildspacetime,wherethepotential,,canbedecomposedintosomeradialfunction,R(r),withexponentialtimedependence,ei!t,andaspin-weight2sphericalharmonic,2Y`m(;)(seeAppendixD,fordetailsaboutthespin-weightedsphericalharmonics).Ignoringtheradialandtimedependence,weseethatthecomponentsofthemetricperturbationhaveangulardependencegivenby andsimilarlyforhlmandhmm.Becausethespin-weightedsphericalharmonicsareundenedforjsj>`,theaboveexpressionsmakeitclearthatthemetricperturbationinthisgaugehasno`=0;1piecesandthereforeprovidesanincompletedescriptionofthephysicalspacetime.Bycontinuity,thesituationpersistsintheKerrspacetime.Howincompleteisthisdescription? Forthemajorityofthiswork,wehavefocusedourattentionongravitationalradiationintypeDspacetimes.Thisinformationiscontainedintheperturbationofeither0or4,aresultestablishedbyWald[ 16 ].Inparticular,Waldwasabletoshowthatwell-behavedperturbationsof0and4determineeachotherandfurthermorethateitheronecharacterizestheentireperturbationofthespacetimeupto\trivial"perturbationsinmassandangularmomentum.With0and4determinedbytheHertzpotential 84 PAGE 85 5{1 and 5{2 )thisbegsthequestionofwhyweshouldconcernourselveswithsuchtrivialities. Theansweris,inpart,thatthesetrivialperturbationsrepresentthelargestcontributiontotheself-force,asshownbyDetweilerandPoisson[ 17 ].Althoughitisunclearifsuchcontributionspersistinallgaugeinvariantquantitiesofinterest,suchascertaincharacterizationsoftheorbitalmotionoftheparticle[ 68 ],thereisinfactamorecompellingreasontobeconcernedwiththenon-radiatedmultipoles.Inrecentwork,Keidl,FriedmanandWiseman[ 69 ]havelookedattheproblemofcomputingtheself-forceinaradiationgaugeinthecontextofastaticparticleintheSchwarzschildspacetime.Intheircalculation,theyfoundtheperturbationsofmassandangularmomentumarisingintheconstructionofaHertzpotential.Thus,althoughtheHertzpotentialcannotbeusedtodeterminetheseperturbations,itmuststill\know"aboutthemandtheymustbedeterminedbysomeothermeans. Inthischapterwewillpresentageneralprescriptionforcomputingthenon-radiatedmultipoles.Morespecically,wewillconsidertheproblemofcomputingtheshiftsinmassandangularmomentumduetoapointsourceinacircular(geodesic),equatorialorbitaroundablackhole.SpecicallyweareafterexpressionsforManda,theshiftsinmassandangularmomentum,intermsoftheorbitingparticle'smass,,andorbitalparameters.Theideaisquitesimple:matchaninteriorspacetime,(gab;M),toanexteriorspacetime,(g+ab;M+),dieringonlyinmassandangularmomentum,onahypersurface(ofcodimension1),p,containingtheperturbation.Thebasicconditionsforagoodmatchingare(1)thatthemetriciscontinuousacrosspand(2)therstderivativesofthemetricarecontinuousexceptwherethesourceisinnite.TheseconditionsarecompatiblewithIsrael'squitegeneraljunctionconditions[ 70 ]. Beforewecandoanymatching,wemustrstdeterminethegeometryofp.Insphericallysymmetricspacetimes,theobviouschoiceisthesimplest|the(round)2-sphere,aswe'llseebelowinourcalculationinSchwarzschild.FortheKerrspacetime, 85 PAGE 86 Oncewe'veagreedonap,fulllingourrstmatchingconditionrequiresustosimplyequatethecomponentsofthemetric(onp).Inotherwords, [gab]g+abjpgabjp=0;(6{4) wherejpindicatestherestrictiontop.Theonly(slight)complicationthatariseshereisensuringthatthereisenoughfreedominthemetricperturbationtoperformthematching.Thiswillgenerallyrequireperformingagaugetransformationontheinteriorandexteriorspacetimes.Althoughthisintroducessomegaugedependenceintotheproblem,theendresult-Mora-isinfactgaugeinvariant,aswewillseebelow. Imposingthesecondconditionisabitmoreinvolvedbecauseofthepresenceofthesource.Bychoosingagoodmatchingsurface,p,wecaneectively\smearout"theangulardependenceofthesource.If,forexample,pisa2-sphere,wecanusethecompletenessrelationstowritetheangulardeltafunctionaccordingto Similarrelationsholdforcompletesetsoffunctionsondierentclosed2-surfaces.Thesourcenowconsistssolelyofaradialdeltafunction.Tohandlethis,weimposetheperturbedEinsteinequationsas,forexample, lim!0Zr0+r0Eabdr=Zr0+r08Tabdr!;(6{6) whereEabdenotestheperturbedEinsteintensorandTabdenotesthestress-energytensorofthesourceandr0isthelocationofpasseenfrombothsides.ForadeltafunctionsourcedueaparticleofmassinacircularequatorialorbitoftheKerrspacetime, g(rr0)(cos)(t);(6{7) 86 PAGE 87 d;0;0;d d)isthefour-velocityoftheparticleparameterizedbypropertime(),r0istheradiusoftheorbitand=d dt.Forcircularequatorialgeodesics d=(r20+a2)T d=~La~E+aT with where~E=E=and~L=L=aretheenergyandangularmomentumperunitmass,respectively.WecanrecoverthecorrespondingresultfortheSchwarzschildspacetimebysimplytakinga!0.BecausetheintegrationinEquation 6{6 ispurelyradial,itisclearthattheonlytermsthatactuallyparticipateintheintegralontheleftsidearethoseinvolvingtworadialderivatives.ThisiswhereourformoftheperturbedEinsteinequationscomesin.WhileitisgenerallyquitetediousandimpracticaltocomputetheperturbedEinsteintensorforabackgroundmoregeneralthanSchwarzschildandpickoutthetermsinvolvingtwoderivatives,itisaquitetrivialtaskfortheEinsteinequationsinGHPform.Allweneedtodoispickoutthepiecesinvolvingtwoofand0(amindlesstaskwiththeaidofGHPtools),pluginourfavoritetetradandvola!Notethattheseconditionsonthesecondderivativesaregenerallyinvariantwithrespecttochoiceoftetrad.Becauseofthis,wewillwritethejumpconditionsoutinthesymmetrictetrad,whichisobtainablefromtheKinnersleytetradbyasimplespin-boost(Equation 2{16 ) 87 PAGE 88 2~2;0;a 2~2;0;a Withthistetradchoice,theradialjumpconditionsare: wheretheomittedequationsfollowbytakingtheprimeand/orcomplexconjugateofthoselisted(thefactorsofand~2remainunchanged;afeatureofthesymmetrictetrad),anditisunderstoodthatequalityonlyholdsinthesenseofEquation 6{6 .AtaglanceEquations 6{14 { 6{18 mayappearinconsistent,withthesameleft-hand-sidebeingequatedtodierentright-hand-sides.Infact,thecirculargeodesicnatureofuaensuresthatthisisnotthecase. Whatwehavenotyetaddressedisthequestionofwhat,precisely,wemeanbymassandangularmomentum.SuitabledenitionsarisefromtheHamiltoniantreatmentofGeneralRelativityinitiatedbyArnowitt,DeserandMisner[ 71 ].ThegeneralideaisthatbecauseMinkowskispaceprovidesanunambiguousnotionofenergyandangularmomentumthroughtimetranslationsandrotations,respectively,wecanadaptthesenotionstocurvedspacesifthemetricbecomesMinkowskianatspacelikeinnity.ThustheADMdenitionsrequireustorestrictourattentiontoasymptoticallyatspacetimes, 88 PAGE 89 72 ],butforourpurposesitwillsucetosimplyconsidertheasymptoticfalloofthecomponentsofthemetric.Moreprecisely,forasetofcoordinates(x;y;z)inametric,gab,andr=p TheseconditionsaresatisedbytheSchwarzschildandKerrspacetimeswewishtoconsider,butwemustbecarefultochooseanappropriategaugeforthemetricperturbationtoensurethatEquations 6{19 aresatised.Assuminganasymptoticallyatspacetime,theADMmassisdenedby 16limS!1IS(DbabDa)radS;(6{20) wherethesymbolsneedabitofexplanation:wedenotethehypersurfaceofconstanttbytanditsboundarybyS.Thethree-metricontisab.Thenab=ab0ab,with0abbeingthemetricofatspacetime(inthesamecoordinatesasab)and=ab(0)ab.Additionally,Daisthecovariantderivativecompatiblewith0ab,raistheunitnormaltoS,anddSisthesurfaceelementonS.Foranarbitrarymetricperturbation,hab,thisevaluatesto 16limr!1Z20Z02rsinhrrdd;(6{21) 89 PAGE 90 8limS!1IS(KabKab)arbdS;(6{22) wherewehaveintroducedtheextrinsiccurvature,Kab,oftandtherotationalKillingvectora.ForagenericmetricperturbationoftheKerrspacetime,wehave 8limr!1Z20Z0rsinht1 2r2sin@rhtdd:(6{23) Thoughthesedenitionsprovidethemostgeneralprescriptionforcomputingthemassandangularmomentum,forstationaryandaxiallysymmetricspacetimes(thosecontainingbothtimelikeandaxialKillingvectors),theKomarformulae[ 73 ]evaluatedatinnityallowustocomputethevalueoftheperturbations 2Tgab)natbp 2Tgab)nabp whereisspacelikehypersurfacethatextendstoinnity,naistheunitnormaltoit,taandaarethetimelikeandaxialKillingvectorsandp 6{7 { 6{12 ,theKomarformulaegive(fortheKerrspacetime) 90 PAGE 91 wheref=12M=r.AccordingtoBirkho'stheorem,theonlystatic,sphericallysymmetricsolutiontotheEinsteinequationsistheSchwarzschildsolution.Thus,weareassuredfromtheoutsetthatperturbingthemasswillsimplyleadustoanotherSchwarzschildspacetimewithamassM+M.Thenonzerocomponentsofthecorrespondingmetricperturbationaregivenby rhrr=2M rf2;(6{29) whichiseasilyobtainedbylinearizingamassperturbationofEquation 6{28 .Inordertocharacterizemassperturbationsmoregenerally,wewillintroducemorefreedomby 91 PAGE 92 wherewe'vetakenacuefromRegge&Wheeleranddecomposedthegaugevectorintosphericalharmonics.Notetheabsenceofandcomponentsinourgaugevector.Wehavedeliberatelyomittedthesecomponentsonthegroundsthattheyinterferewiththeforminvarianceofthemetric.InordertodeterminethefunctionsP(t;r)andQ(t;r)aswellastheappropriate`andm,we'lllookattheircontributiontothemetricperturbation.Ourgaugetransformation,ab=$gab,hastheform where\sym"meanssymmetricandwe'vedroppedthefunctionaldependenciesforsimplicity.First,we'llfurtherspecializethegaugetransformationbyinsistingonpreservingtheformofEquation 6{28 .Aconsequenceofthisisthathtr=(f@rP(t;r)+f1@tQ(t;r))=0: 92 PAGE 93 6{19 requireQ(r)=O1 r; r2f; WecangiveaninterpretationtobyconsideringEquation 6{33 withM=Q=0,inwhichcaseitisclearthatisjustarescalingofthetimecoordinate. Inordertoperformthematching,weneedtoadaptourgenericperturbationtotheinteriorandexteriorspacetimesandchooseaparticulargaugetoperformthematching.Wewillbeginwiththedescriptionofthemetricontheinterior,gab.HereM=0,sotheperturbationispuregauge.Furthermore,ontheinteriorthereisnoneedtoimposeasymptoticatness.Instead,wewillchooseQ(r)sothattheinteriormetricisregularonthehorizonandleavetheformofP(t)untouched.Asuitablechoiceis r02M!i;(6{37) wherer=r0isthelocationofpandisaconstantinsertedfordimensionalreasonsandi>0.Thevaluesofandwillbedeterminedfromthejumpconditions.Proceedingto 93 PAGE 94 r2M!j;(6{38) where,inanticipationofthematching,we'vechosenthesamedimensionalconstant,,thatweusedinthedescriptionoftheinteriorspacetimeandj2.Withbothmetricsspeciedwenowturnourattentiontomatchingthespacetimes. Becausebothbackgroundmetricsarethesame,itwillsucetomatchtheperturbationsonly.Byimposing[hab]=0,wearriveatthreeuniqueconditions: r0+f0[]Y00+M r20[Q]Y00=0; dr#Y00M[Q]Y00r0M=0; [Q]=0; whereweusedf0=f(r0).OurchoicesforQ+andQ( 6{38 6{37 )ensurethatthethirdconditionissatised.WecansolveEquations 6{39 and 6{40 togetequationsfor[]andM: []="dQ dr#=(i+j) dr#Y00=(i+j)Y00; wherewe'vemadeuseofEquations 6{38 and 6{37 .Nextwewillusethejumpconditionstosolvefor. Applicationofthejumpconditions(Equations 6{14 { 6{18 )issimpliedbythefactthatourmetricperturbationispurespin-0.Thusweonlyneedconsiderthejumpconditionsforthespin-0componentsofthemetricperturbation(hll,hln,hnnandhmm).ForsimplicitywewillworkwithEquation 6{15 ,thoughitcanbedirectlyveriedthatthe 94 PAGE 95 6{7 ,wehaveforthetetradcomponentsoftherelevantobjects: rr2M r02M!iY00 rr02M r2M!jY00 16~2 r20f0(rr0)(cos)(t); withalltheMdependencereplacedaccordingtoEquation 6{43 .ImposingEquation 6{6 thenleadsto"@hmm r20f0(cos)(t); r20f01X`=0`Xm=`Y`m(=2;t)Y`m(;);(6{47) wherewe'vedecomposedtheangulardeltafunctionsaccordingtoEquation 6{5 .WecaneliminatethesumontherightsideofEquation 6{48 bymultiplyingbothsidesbyY00(;),integratingoverthesphereandexploitingtheorthogonalityofthesphericalharmonics.Theresultisthat i+j;(6{48) wherewe'veusedY00(;)=Y00(;)=(4)1=2.Finally,wehave []=(4)1=2~E r02M Theseequationscompleteourconstructionofthematchedspacetime.Notethattheaboveonlyrestrictsthedierencebetweenontheinteriorandexterior.IfwerecallEquation 6{41 ,weseethatthesameisgenerallytrueofQ(r)aswellifwedroptherequirementsofregularityintheinteriorandasymptoticatnessintheexterior. 95 PAGE 96 6{23 ,itisclearthatourmetricperturbationwillacquireanhtcomponent.RealizingthisasaperturbationtowardstheKerrspacetime,wewillwriteitas r;(6{51) whichisjustthelinearizationabouta=J=Mofthecorrespondingcomponentofthe(background)Kerrmetric.Becauseofthis,therewillbenonzerocontributionstohlm,hnmandtheircomplexconjugateswhichmeansthatwemustnowtakeparityintoconsideration.Tothatendwewillintroduceagaugevectorwithcomponents (6{52) (6{53) 2(+0)+S(t;r)i sin wherewe'vedenedY+`m=1 2(+0)Y`m=1 2(1Y`m+1Y`m)andY`m=i 6{51 has.Forananswer,welooktothesourceterms.AquickcomputationrevealsthatTlm=Tnm=Tlm=Tnm,fromwhichitfollows 96 PAGE 97 wherethe\-"onhab,referringtotheinteriorspacetime,istobedistinguishedfromthe\-"onY`m,whichreferstoacombinationofspin-weight1sphericalharmonics.Inthissituation,wemustmodifyourrequirementofforminvariance(whichisalreadybrokenbytheperturbation)totherequirementthatonlyhtremainsnonzero,whichpreservestheminimumfreedomtomatchtotheexterior.Firstwesetht=0,whichimpliesY`m=0or1Y`m=1Y`m.Thiscanonlyholdifm=0,whichmeanstheperturbationisaxiallysymmetric.Movingon,weturnourattentiontoeliminatingh.ThisentailscosY+`0sin@Y+`0=0; 21Y`0+1Y`0sin: 97 PAGE 98 4sin:(6{57) Finally,itiseasytosethr@rS2r1S=0; r; withallothercomponentsvanishing. Continuityofthemetricperturbation([hab]=0)requires wherewe'veusedtheequalityof1Y10toexpandY+10.Asbefore,theradialjumpconditionswilldetermine.Inthiscasewe'llusetheodd-parity(imaginary)partofEquation 6{17 .Therelevanttetradcomponentsaregivenby: 16~2 r30f1=20(rr0)(cos)(t): 98 PAGE 99 f1=201Y10=i16~L r30f1=20(cos)(t)=i16~L r30f1=201Xl=0`Xm=`1Y`m(=2;t)1Y`m(;): Equation 6{57 weobtain r30p ItthenfollowsfromEquation 6{60 that whichispreciselywhatistobeexpected|alloftheangularmomentuminthe(otherwisenon-rotating)spacetimecomesfromtheangularmomentumoftheparticle.OnceagainwecanverifydirectlyfromEquation 6{23 thatwehavecorrectlyidentiedtheangularmomentumofthespacetime. Incontrasttothesituationwithmassandangularmomentumperturbations,whereitwaseasytowritedownthegeneralformoftheperturbations,wehavenogeneralformforthemetricperturbation.Withoutpriorknowledgeoftheperturbation,wemustresorttosolvingtheEinsteinequationstodeterminetheperturbation.ThishasbeencarriedoutbybothZerilli[ 7 ]andDetweiler&Poisson[ 17 ].Theresultisametricperturbationthatcan 99 PAGE 100 Mr20sinsin(t)(rr0);hrr=2~E Mr20 Mr20coscos(t)(rr0);hr=~E Mr20sinsin(t)(rr0): 7 ]thatthegaugetransformationleadingtothisdescriptioncanbeinterpretedasatransformationfromanon-inertialframetetheredtothecentralblackholetothecenterofmassreferenceframe. Theprimaryissuewithtreatingthenon-radiatedmultipolesinthecontextofmatchedspacetimesisthechoiceofthematchingsurface,p.Mostofourdiscussionwillbefocusedonthisissue. 16 ],describedearlier,assuringusthatinnitesimalmassperturbationsoftheKerrsolutionleadtootherKerrsolutions(withinnitesimallydierentmasses,ofcourse)becausesuchperturbationsdonotcontributetheperturbationsof0or4(whichwewillverifyshortly).Thuswehavethe 100 PAGE 101 r2+a2cos2; r2+a2cos2; r2+a2cos2: BecausethecalculationsintheKerrspacetimearesignicantlymorecomplicated,wewilltakeashortcuttodeterminingtheangulardependenceoftheperturbationbylookingatthetetradcomponentsofthemetricperturbation,aresultwhichwewillinanycaseuseshortly.Inthesymmetrictetrad(Equations 6{13 )wehave withallothercomponentsvanishing.Becausebothhllandhnnarespin-weight0,theyhaveanaturaldecompositioninto`=0,m=0scalar(ordinary)sphericalharmonics.Furthermore,utilizingEquation 3{27 weseethat andsimilarlyfor4.Therefore,accordingtoWald'stheorem,weareensuredthatEquations 6{70 and 6{71 areaperturbationtowardsanotherKerrsolution. Withtheangulardependencedetermined,weareledtoconsideragaugevectoroftheform 101 PAGE 102 whilestationarityagainnecessitates Nextweturnourattentiontothematchingproblem. Inordertoclarifytheissuesinvolvedinthematchingproblem,we'lltakealookatthematchingconditionsthemselves.Supposewe'vechosensomep,buthaveyettospecifyitexplicitly.Thatis,wehavenotyetwritten(orimposed)r=something.Thefullsetofmatchingconditionsnowtaketheform d#2arM=0; dr#rM d#(a2cos2+r2(r2+a2)+2amr)=0; where=r22Mr+a2and~2=r2+a2cos2asbeforeandwehaveimposedtheconditioninEquation 6{79 intheothers.NotethatthisreducestotheSchwarzschildresultinEquations 6{39 { 6{41 bytakinga!0andsettingr=r0.Thissetofequations 102 PAGE 103 dr#; []=(r2+a2)2 dr#; []= (r2+a2)2[]; d#=a2sin2 whichisagaineasilyseentoreducetotheSchwarzschildresultintheappropriatelimit.Fromtheseequationswecanseeclearlytheissuesinvolvedinchoosingamatchingsurface.First,becausetheleftsidesofEquations 6{81 { 6{84 areallconstant,thismustbereectedintherightsidesaswell,whichcurrentlyexhibitdependenceonbothrand.Presumably,somechoiceofr=r()willenforcethis,thoughitiscurrentlyunclearwhatthatchoicemightbe.Notethatbecauseofthis,r=constantsurfacesdonotappeartobegoodformatching. WhatwehaveencounteredappearstobeaninstanceofalongstandingproblemwithmatchingtheKerrsolutiontoasource[ 74 75 ].Namely,thereisnoknownmattersolutionthatcorrectlyreproducesthemultipolestructureofthefullKerrgeometry.Inourproblem,we'retryingtoforcetheissuebyspecifyingboththemetricandthesource.Ontheotherhand,becausewe'renotmatchingtheentiresource,whichincludesquadrupoleandhighermoments,butonlythenon-radiatedmultipolesthatmerelytakeusfromoneKerrsolutiontothenext,itisnotclearthatthematching(inthisinstance)shouldfail.Thoughweareunabletoperformthematchinghere,wemaintainthatnothingforbidsit. Mostauthorsfacedwiththisissueturntothe\slowrotation"approximationandkeeponlytermslinearina.InthisapproximationtheKerrmetriccanbeviewedastherstorderperturbationoftheSchwarzschildsolutiontotheKerrsolution.Thatis,thebackgroundisgivenbySchwarzschildplusatermidenticaltothatinEquation 6{59 .It 103 PAGE 104 6{81 { 6{84 andallowforamatchingonr=constantsurfaces(whichareround2-spheresinthiscase).BecausethisapproachfailstoshednewlightonthesituationinthefullKerrspacetime,wewillnotfollowithere.Instead,wewillfocusonEquations 6{66 6{69 ,whichweknowtobecorrect. Let'sreviewthesituation.WehaveestablishedthatthemetricperturbationinEquations 6{66 { 6{69 isaperturbationtowardsanotherKerrsolutionwithdieringmass.Furthermore,wepreviouslyestablishedthatM=~E(Equation 6{26 ).Theproblemisthatwearecurrentlyunabletoperformthematching.Inpractice,therelevantportionofthespacetimeistheexteriorwheregravitationalradiationandthenon-radiatedmultipolesareobservedfarawayfromthesource.Becauseofthis,wecontendthatconsiderationsfromtheKomarformulaandWald'stheoremtogetherprovidethecorrectperturbationintheexteriorspacetime,independentlyofanymatchingconsiderations.ThusourresultislikelyusefulintheEMRIproblemeventhoughwelackthemetricperturbationeverywhereinthespacetime.Moreover,theperturbationisstillsimpletointerpretandasymptoticallyat,soitisamenabletosomeanalysis. Thisbeingthecase,weremarkthatmassperturbationsoftheKerrbackgroundremainconnedtothes=0sectoroftheperturbation.Itislikelythatthisistrueingeneral(atleastintypeD),butageneralproofofthisremainselusive.Furthermore,contrarytowhatonemightexpectintheKerrspacetime,themassperturbationdoesnotmixsphericalharmonic`-modes,butispurely`=0.Wenowturnourattentiontoangularmomentumperturbations. 104 PAGE 105 (r2+a2cos2)2; Thecorrespondingtetradcomponents(inthesymmetrictetrad)aregivenby ~2; ~2; wherewehaveomittedthecomplexconjugates.Thoughitisnotimmediatelyobvious,thisperturbationmakesnocontributionto0or4,ensuringthatthisisavalidangularmomentumperturbation. Inlightofrelativelystraightforwardresultsformassperturbations,thenontrivialformofEquations 6{90 { 6{94 comesasasurprise.Unlikemassperturbations,angularmomentumperturbationsarenotconnedtoasinglessector,whereasonemightexpectthemtobeexclusivelys=1,asintuitionfromworkingintheSchwarzschildbackgroundwouldleadustobelieve.Notethatalthoughtheperturbationappearsinthes=2sectorofthemetric,thevanishingofthes=2componentsoftheWeylcurvaturekeep 105 PAGE 106 Anothersurprisingfeatureisthefactthattheperturbationiscomplexandthusexhibitsbothtypesof\parity".Althoughthestaticnatureoftheperturbationguaranteesspin-weightedsphericalharmonicangulardependence,wemustbecarefulnottospeakofparityintheSchwarzschildsense,butrathertherealandimaginarypartsoftheperturbation.Inanycasetheimplicationsofthisfactarepresentlyunclearandremaintobedeterminedinfuturework. Firsto,onemayspeculatethatourrequirementofforminvarianceisperhapstoostricttoallowforapropermatching.Thisdoesnotappeartobethecase.AresultofCarter[ 76 ]impliesthat,duetostationarityandaxialsymmetry,theKerrmetric(inBoyer-Lindquistcoordinates)haspreciselytheminimumnumberofnonzerocomponents.Havingestablishedindependentlythatthemassandangularmomentumperturbationspreservethesespropertiesofthebackground,Carter'sresultsuggeststhattheproblemlieselsewhere. Thisleadsustoconsiderwhethertheintroductionofaninnitesimallythinshellofmatter(whichiseectivelywhatpis),necessarilyintroducesnon-Kerrperturbations.Ashell(ofsomecurrentlyunspeciedshape)wouldpresumablybeadierentiallyrotatingobject.Itisunclearwhetherthisdisruptsthestationarityoraxialsymmetryoftheexteriorspacetimebytheintroductionofperturbationsthatwehaveneglected 106 PAGE 107 16 ]actuallyspeciestwoothertypesofperturbationsthat0and4cannotaccountfor:perturbationstowardstheacceleratingC-metricsandperturbationstowardtheNUTsolution.IntheworkofKeidl,et.al.[ 69 ],wheretheyconcernedthemselveswithastaticparticleintheSchwarzschildgeometry,itwasfoundthatthespacetimeontheinteriordiersfromthatontheexteriorbyaperturbationtowardstheC-metrics.ThismakesphysicalsensebecauseastaticparticleisnotonageodesicoftheSchwarzschildspacetimeandthusrequiresaccelerationtokeepitinplace.Thoughwehavenoobviousphysicalreasontoexpecttheseperturbationsforcircular,equatorialorbitsoftheKerrgeometryandevidencefromtheSchwarzschildcalculationsuggeststheyshouldnotcontribute,wehavenotyetprovenaresulteitherway. Finally,onequestionthatwehaveoverlookedentirelyisthequestionofthestabilityofathinshell.IntheSchwarzschildbackground,thisproblemhasbeensolvedbyBrady,LoukoandPoisson[ 77 ],whoshowedthatathinshellisstableandsatisesthedominantenergyconditionalmostallthewayuptothelocationofthecircularphotonorbit(locatedatr=3M).TherearenosuchresultstoreportonfortheKerrspacetime.TheclosestthingtoastepinthisdirectionistheworkofMusgraveandLake[ 78 ],whoconsiderthematchingoftwoKerrspacetimeswithdierentvaluesofmassandangularmomentum.Unfortunately,theseauthorswereforcedtoresorttotheslowrotationapproximationdiscussedearlier.Strictlyspeaking,withoutknowledgeoftheexistenceofastableshellofmattersucientlyclosetotheblackhole,wearelefttoquestionthevalidityofourprocedure.Thisisaproblemweleaveforfuturework. 107 PAGE 108 Firstandforemost,bytakingadvantageoftheGHPformalism,ourframeworkemphasizesandexploitsthosefeaturescommontoallblackholespacetimes|theirnullstructureasmanifestedintheirPetrovtype|which,sinceTeukolsky'sderivationoftheequationthatbearshisname,hasbeentheonlyprovenroadtoprogressinthisdicultsubject.SuchfeatureshavemadeanappearancethroughthesimplicationinthebackgroundGHPequationsdiscussedinChapter1.Thesehaveleadtousefulsimplicationsthroughout.Besidesthesefeatures,thebuilt-inconceptsofspin-andboost-weighthaveallowedussomeintuitiveinsightintothenatureofthefundamentalquantities,withoutresortingtoseparationofvariables. ThecreationofGHPtoolsistheonlyreasonanyofthisworkwasfeasibleintherstplace.Coordinate-independencecomesatthepriceofhavingtoperformmanynontrivialsymboliccomputations.GHPtoolshasnotonlyallowedustoperformsuchcomputations,butalsotopresenttheminafullysimpliedway,bringingsomeclarityeventopreviouslyknownresults.ThisisperhapsmostevidentinourtreatmentoftheTeukolsky-Starobinskyidentities,wheretheuseofGHPtoolsmaskedallofthehorrendouscomputationalcomplexityinvolvedintheirderivation,byprovidingsimpleandconciseresultsintheend. Furthermore,thecoordinate-freenatureofourframeworkhasfurtherallowedustoworkingreatgenerality.Thiswasseeninourtreatmentofthecommutingoperatorsof 108 PAGE 109 Finally,ourtreatmentofthenon-radiatedmultipolesdemonstratesthepowerofourframeworkwhencombinedwithexistingtechniques.OurresultsintheKerrspacetimerepresenttherstattemptattreatingthispartoftheperturbation.Thoughwewereunabletoobtainthedescriptionintermsofamatchedspacetime,weneverthelessprovidedaperturbationsuitableforuseinmetricreconstruction. Perhapsmostpressingisthegeneralizationofourresultforthenon-radiatedmultipolesintheKerrspacetimetoencompassmoregeneralorbits.Inparticular,orbitsnotlyingintheequatorialplaneareofparticularinterest.Suchorbitsnecessarilycontaino-axisangularmomentum,whichinturnarewidelythoughttoberelatedtoCarter'sconstant(associatedwiththeKillingtensor).ForsuchorbitstheKomarformulaefailtocompletelycharacterizetheseo-axisangularmomentumcomponents,soitisclearthatwemustlookelsewhereforasolution.OnepotentialavenueforprogressistheEinsteinequationsthemselves.Aswenotedinthepreviouschapter,massandangularmomentumperturbationsarebothstationaryperturbationswithangulardependencecharacterizedbythespin-weightedsphericalharmonics.ThesimplicationsthisbringsforworkingwiththeEinsteinequationsisimmenseandmayprovetomaketheproblemtractable,withoutrecoursetopurelynumericalmethods.Inanycase,itseemsclearthatourframework, 109 PAGE 110 AnotheravenueworthpursuingisthecommutingoperatorassociatedwiththeKillingtensorduetoBeyer[ 44 ](cf.Chapter1).RecallthatBeyer'soperatorcommuteswiththescalarwaveequationinKerr.ItisverytemptingtothinkthatsuchanoperatorwouldexistfortheTeukolskyequationaswell.TheGHPformalism,andGHPtools(ofcourse),providetheidealenvironmentinwhichtostudysuchquestions.Furthermore,inthecontextofworkperformedbyJeryes[ 79 ]concerningtheimplicationsoftheexistenceoftheKillingspinor(whichincludesadiscussionoftheTeukolsky-Starobinksyidentities),itisnaturaltothinkthatsuchanoperatormayinfactshedsomenewlightontheTeukolsky-StarobinskyidentitiesintheformpresentedinChapter5.Additionally,theexistenceofageneralizationofBeyer'soperatorcarrieswithitthepossibilityofnewdecompositionoffunctionsintheKerrspacetime|justastheexistenceoftheKillingvectors@ @tand@ @leadtoseparationintandaccordingtoei!tandeim(respectively),theeigenfunctionsofageneralizedBeyeroperatormayprovideanewseparationofvariablesintheKerrspacetime.Thisiscertainlyapossibilityworthpursuing. Finally,bothGHPtoolsandourformoftheperturbedEinsteinequationsareentirelygeneralandreadyforusebyresearchersinterestedinmoregeneral(orevenmorespecialized)backgroundsthanPetrovtypeD.Inparticular,theclassoftypeIIspacetimesseemsalikelycandidateforfurtheranalysis,especiallywiththeaidoftheintegrationtechniqueofHeld.Wehaveonlybeguntoscratchthesurfaceofthewidevarietyofproblemsthesetoolscanhelpsolve. 110 PAGE 111 Inthisappendix,wegivetheGHPcommutators,eldequationsandBianchiidentities,aswellasthederivativesofthetetradvectors.Thefullsetofequationsisobtainedbyapplyingtothoselistedprime,complexconjugationorboth.Whenactingonaquantityoftypefp;qg,thecommutatorsare: [;0]=(0)+(0)0p(00+2+11)q(00+2+11); [;]=+000p(00+1)q(00+01); [;0]=(00)+()0+p(00+211)q(00+211): TheGHPequationsare: TheBianchiidentitiesaregivenby: 111 PAGE 112 Finally,thederivativesofthetetradvectorsaregivenby alb=la(mb+mb)na(mb+mb)+ma(mb+mb)+ma(mb+mb) (A{14) amb=la(0lb+nb)na(0lb+nb)+ma(0lb+nb)+ma(0lb+nb) (A{15) 112 PAGE 113 InthisappendixwewritethecomponentsoftheperturbedEinsteintensorforanarbitraryalgebraicallyspecial(PetrovtypeII)background.WehaveassumedthePNDisalignedwithlaandmadeuseoftheGoldberg-Sachstheorem.NotethattheequationsforElm,EnmandEmmarecomplex,soElm=Elmandsoon: 113 PAGE 114 2f0(00)+0(00)+(20)0+(020)0+200ghll+1 2f()+()ghnn+1 2f(0+0+)(0)(0+30+30)+2(+0)+(2)0+(020)0(+)(0+0)22ghln+1 2f(020)(00)+(0+0+0)0(00)(20)0(2)0+0(0)ghlm+1 2f(020)(0)+(0+0+0)0(00)(2)0(2)0+0(0)ghlm+1 2f(2)(0)+(0+)(+)2(00)2ghnm+1 2f(2)()+(0+)(+)2(0)2ghnm+1 2f(0)(00)+(0)0ghmm+1 2f()(0)+(0)0ghmm+1 2f(0+0)(+0)+(000+)(2+2)+(020)+(2)0+(3020)+0(32)20+20+20()ghmm; 2f(00)(0)+(20)0()0+(0+0)+0(00+)+3+0ghll+1 2f(+)(+0)(3+0)20ghln+1 2f(0+0)(2)+(0+2020)40+22+(0+)(20)(0+02)0(40)ghlm+1 2f(2)0(+24)20(0)ghlm+1 2f(2)+2()ghnm PAGE 115 2f()(00+)+2ghmm+1 2f(+)(+0)+20(2)(0)+2ghmm; 2f(00)0+00+00ghll+1 2f(+)(0)(020+)+0()ghnn+1 2f((00+0)(0+0)(030+)0+(+0)020320ghln+f0(020)0(02)+1 24ghlm+1 2f(0(020)+0(2+20)+0(040+2)+20(00)+200ghlm+1 2f0(020)+0(2+2)2(0)ghnm+1 2f(00)(+2)+(020)+20()222+(03)+0(20+40)(02)ghnm+1 2f(00)000ghmm+1 2f(00)(+0)+2000(2+2)+0(0)0ghmm+1 2f(0+00)(00+)+2(020)(00)0+200+(0)0+003ghmm; PAGE 116 2f0(000)+200+0(0)0(0)+200ghll+1 2f()+2ghnn+1 2f(0+00)(+)0(+)+(0+00)2+(0)(0)+0(20)0(2+0)2(0)+200+0ghln+1 2f(020)(02)+(0+2020)+2(0)0020020()3ghlm+1 2f(020)(2)+(0+202)+2(00)00020020()3ghlm+1 2f(2)(020)+0(22)2+40ghnm+1 2f(2)(20)+0(22)2+40ghnm+1 2f(0)0(00)(2)0ghmm+1 2f()0(0)(2)0ghmm+1 2f20(00)()0(00+0)0(+)(020)0+(0+2)0(0)+(+)0()22ghmm: 116 PAGE 117 WeprovidedetailsoftheintegrationthatleadtoEquation 4{17 and 4{25 .Asitturnsout,thetypeIIcalculationisactuallymuchsimplerthanthethetypeDcalculationbecauseitusesatetradinwhich=0=0.ThereforewewillworkoutthetypeDcalculationindetailandthetypeIIresultmostlyfollowsbysettingcertainquantitiestozero,asindicatedbelow. Wewillneedsomeresults(andtheircomplexconjugates)fromtheintegrationofthetypeDbackground: ~0= 22(~+1 2)2++2+1 21 2+1 21 (~+); 2=3: Asnotedinthetext,6=0leadstotheacceleratingC-metrics,whichweincludeforfullgenerality.HenceforththecorrespondingquantitiesintypeIIspacetimescanbeobtainedbysetting==)0and)2 117 PAGE 118 22(2+); 2=23; theequationfor~0notfollowingfromthelimitingprocessmentionedabove.Notethatthequantity~0isneverusedinanyoftheintegrationsweperforminthetypeIIbackgroundspacetime.Wewillalsoneedthedenitionsofthenewoperators: ~0=0~~0+(p )+1 2(p2 ~= ; ~0=0 wherepandqlabeltheGHPtypeofthequantitybeingactedon.Additionally,inSections 4.2 and 4.4 wemakeuseofthecommutator [~;~0]=00 1 221 +~ 221 +~0 o;(C{13) whichisvalidintypeDand(with=0)typeIIspacetimes. Wenowbeginwith whichintegratestriviallytogive Withthisinformationinhand,wecannowintegratetheequationgoverningm: (+)m+(+0)l=0:(C{16) 118 PAGE 119 C{11 withp=1leadsto 1 (m)+0l+~l l=0;(C{17) which,aftersubstitutingEquation C{3 ,thecomplexconjugateofEquation C{4 andEquation C{15 alongwithsomerearranging,yields Integrationthengivesus l1 andthesolutionformthenfollowsfromcomplexconjugation +l(~0+)l:(C{20) Finally,weareinapositiontodealwithn,bywriting intermsofHeld'soperators(Equations C{1 C{3 and C{4 )as l1 22 SubstitutingEquations C{3 C{4 C{5 C{15 C{19 and C{20 ,rearrangingtermsandlettingthedustsettleleadsto 2l2+1 2l2l1 +l(2+2)[2(~+)+2(~0+)]l[(~+)+(~0+)]l+2m1 +2m1 119 PAGE 120 2l+1 2l+l+1 2l1 2+h 21 ~0l[(~+)+(~0+)]l+m m1 2m1 m1 andourtaskiscomplete. 120 PAGE 121 Inthisappendix,wepresentthebasicsofthetheoryofspin-weightedsphericalharmonics[ 15 80 ].ThesefunctionshaveanaturalplaceintheGHPformalismandprovideasimplealternativetothemorecomplicatedtensorsphericalharmonics.Thediscussioninthissectiontakesplaceontheround2-sphere.Inthatcase,theactionofonsomequantity,,ofspin-weightsisgivenby @+icsc@ @#(sin)s;(D{1) andtheactionof0is @icsc@ @#(sin)s:(D{2) Thespin-weightedsphericalharmonics,sY`m(;),arethendenedintermsoftheordinarysphericalharmonicsby (`+s)!sY`m(;)0s`;q (`s)!(1)s(0)sY`m(;)`s0;(D{3) butareundenedforjsj>`.ThebasicpropertiesofthesY`mareeasilyseentobe Foreachvalueofs,thespin-weightedsphericalharmonicsarecomplete: 121 PAGE 122 122 PAGE 175 [1] Y.Mino,M.Sasaki,andT.Tanaka,Phys.Rev.D55,3457(1997). 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[80] R.PenroseandW.Rindler,SpinorsandSpacetimeVolume1,CambridgeUniversityPress,NeyYork,1984. 178 PAGE 179 Larrywasbornin1978,inElPaso,Texas.Heistheeldestchildof(theelder)LarryPriceandPamelaVilla.Atlastcount,hehasapproximately6siblings.Fromtheagesofaboutvetotwelve,heattendedafunnysortofschoolwherethestudentswereallforcedtodressthesameandgatheronFridaystolistentoamaninadressreadfromabigbook.Hewastreatedwellthere,buthisentryintotheTexaspublicschoolsysteminthefthgradeprovedtobeagoodmove.Inmiddleschool,Larryrealizedheunderstoodalgebramuchbetterthanhisteacher(whohappenedtoalsobetheschool'sbasketballcoach),apointthathemadeclearinclassateveryopportunity.Itgoeswithoutsayingthathisinitialdesiretopubliclyhumiliatejockssubsequentlygrewintoamuchdeeperinterestinmathematicsandphysics.Theseinterestswerefurtheredinhighschool,whereLarryexploredotherareasaswell.Amongtheseisthetheater.FewpeopleareawarethatLarryhasperformedinleadingrolesinseveralmusicals,aswellasanoperetta.Upongraduatinghighschoolin1997,LarrydecidedthatitwouldbebesttogetasfarawayfromElPasoashecould.Tothisend,heattendedasmallliberalartsschoolnamedReedCollegeinPortland,Oregon,wherehespentsomeofthebestyearsofhislife.ReedprovidedavaluableopportunityforLarrytofurtherpursuethesciencesandreadsomereallygreatbooksatthesametime.Italsogavehimtheopportunitytointeractwithmanyinterestingpeoplefromwidelydierentbackgrounds.ItwastherethatLarrycameincontactwithNickWheeler,atrulyuniqueindividualwhoremainsatrustedmentor.Alas,allgoodthingsmustcometoanend,andsoLarrygraduatedfromReedwithaB.A.inphysicsin2001.Withhispathuncertainatthetime,LarrydecidedtostayinPortlandforthefollowingyear.ThereLarrytriedhishandasacomputationalchemistforSchrodinger,Inc.Thepeopletherewerefantasticandthepaychecksweren'tbad,butheneedmore 179 PAGE 180 180 |