Notes on perturbation methods in general relativity

Transcription

1 Notes from phz 7608, Speial and General Relativity University of Florida, Spring 2005, Detweiler Notes on perturbation methods in general relativity These notes are not a substitute in any manner for lass letures. Please let me know if you find errors. Eah of setions V to XII disuss some aspet of perturbation theory, or derive some useful relationship. However, eah setion essentially stands alone; I might not atually disuss eah of these setions in lass. I. FIRST ORDER PERTURBATION ANALYSIS Perturbation analysis provides the framework for an understanding of the effets of a small mass moving through a bakground spaetime. The analysis begins with a bakground spaetime metri g whih is usually a vauum solution of the Einstein equations G g = 0. An objet of small mass µ then disturbs the geometry by an amount h = Oµ whih is governed by the perturbed Einstein equations with the stress-energy tensor δt = Oµ of the objet being the soure, E h = 8πδT + Oµ 2. 1 Here E h is the linear, seond order differential operator on symmetri, two-indexed tensors shematially defined by and G is the Einstein tensor of g, so that E h δg δg d h d, 2 2E h = 2 h + a b h 2 a h b + 2R a b d h d + g d h d 2 h 2R ah b + Rh g R d h d. 3 with h h g and a and R a b d being the derivative operator and Riemann tensor of g. In terms of h h 1 2 g h, this same equation is 2E h = 2 h 2 a hb + 2R a b d hd + g d hd 2R a h b + R h g R d hd 4 see MTW [13]. If h is a solution of Eq. 1 then it follows from Eq. 2 that g + h is an approximate solution of the Einstein equations with soure T + δt, G g + h = 8πT + δt + Oµ

2 The Bianhi identity implies that a E h = 0 6 for any symmetri tensor h ; this is disussed in setion VI. Thus, an integrility ondition for Eq. 1 is that the stress-energy tensor T be onserved in the bakground geometry g, a δt = Oµ 2. 7 Perturbation analysis at the seond order is no more diffiult formally than at the first. But the integrility ondition for the seond order equations is that T be onserved not in the bakground geometry, but in the first order perturbed geometry. Thus, before solving the seond order equations, it is neessary to hange the stress-energy tensor in a way whih is dependent upon the first order metri perturbations. This modifiation to T is said to result from the self-fore on the objet from its own gravitational field and inludes the dissipative effets of what is often referred to as radiation reation as well as other nonlinear aspets of general relativity. This modifiation to T is Oµ 2 beause T itself is Oµ. A desription of general, nth order perturbation analysis is given in setion VII. The proedure is similar to that just outlined. The stress-energy tensor must be onserved with the metri g n 1 in order to solve the nth order perturbed Einstein equation 60 for h n. In an implementation, the task then alternates between solving the equations of motion for the stress-energy tensor and solving the perturbed Einstein equation for the metri perturbation. Similar alternation of fous between the equations of motion and the field equations is present in post-newtonian analyses. For many interesting situations the objet is muh smaller than the length sale of the geometry through whih it moves. We expet, then, that the detailed struture of the soure should be unimportant in determining its subsequent motion. To fous on those details of the self-fore whih are independent of the objet s struture we first attempt to model the objet by an strat point partile with no spin angular momentum or internal struture. The stress-energy tensor of a point partile is δt = µ u a u b g δ 4 x a X a s ds 8 where X a s desribes the world line Γ of the partile in some oordinate system as a funtion of the proper time s along the world line. The naive replaement of a small objet by a delta-funtion distribution for the stressenergy tensor is satisfatory at first order in the perturbation analysis. The integrility ondition Eq. 7 requires the onservation of the perturbing stress-energy tensor. For a point partile this implies that the world line Γ of the partile is an approximate geodesi of the bakground metri g, with u a a u b = Oµ f setion VIII. The solution of Eq. 1 is formally straightforward, even for a distribution valued soure. This proedure has been used many times to study the emission of gravitational waves by a point mass orbiting a blak hole [1 3]. II. VECTOR AND TENSOR HARMONICS Many interesting problems involve perturbations of a spherially symmetri spaetime; these inlude perturbations of the Minkowskii metri, the Shwarzshild metri and also 2

3 metris for non-rotating neutron stars. We an take advantage of a spherially symmetri bakground by deomposing various salar, vetor and tensor fields in terms of salar, vetor and tensor spherial harmonis. For the angular omponents of vetors and tensors, we find it onvenient to follow Thorne s desription of the pure-spin vetor and tensor harmonis [4], whih are losely related to the harmoni deomposition used by Regge and Wheeler [1]. For example, the spin-1 vetor harmonis generated by the spherial harmoni funtion Y lm are the even parity and the odd parity where is the metri of a onstant t,r two-sphere, and is the Levi-Civita tensor on the same two-sphere. Here also Y Elm a = rσ a b b Y lm 9 Y Blm a = rɛ a b b Y lm, 10 σ g + u a u b n a n b 11 ɛ ɛ tr 12 ɛ trθφ = r 2 sin θ, 13 ɛ θφ = r 2 sin θ, 14 and u a and n a are the unit normals of surfaes of onstant t and onstant r, respetively. In the usual Shwarzshild oordinates the omponents of σ A B and ɛ A B are and σ A B = δ θ Aδ B θ + δ φ A δb φ 15 ɛ A B = δ θ Aδ B φ sin θ 1 + δ φ A δb θ sin θ. 16 In the same oordinates the omponents of u a and n a are u a = 1 2M/r 1/2 δ t a 17 n a = 1 2M/r 1/2 δ r a. 18 The terms even and odd parity are often replaed by eletri and magneti, or by polar and axial. We generalize this approah: For a vetor field ξ a, the parts σ a b ξ b whih are tangent to a two-sphere may be desribed by two potentials ξ ev and ξ od via σ a b ξ b = rσ a b b ξ ev rɛ a b b ξ od. 19 The potentials ξ ev and ξ od are generally funtions of all of the spaetime oordinates and are guaranteed to exist by the invertibility of the two dimensional Laplaian on a two-sphere. The fators of r are inluded for onveniene. 3

4 The notation for a ovariant vetor field is ondensed by defining even and odd parity vetors assoiated with the potential ξ ev and with the potential ξ od ξ ev a rσ a b b ξ ev 20 ξ od a rɛ a b b ξ od. 21 The four independent omponents of a ovariant vetor in a spherially symmetri geometry may be written as a sum of the form ξ a dx a = ξ t dt + ξ r dr + ξa ev + ξa od dx A 22 in terms of the four funtions ξ t, ξ r, ξ ev and ξ od. The apital index A is used here just as a reminder that the vetor to whih it is attahed is tangent to the two-sphere. The A index should otherwise be onsidered an ordinary spaetime index in the ovariant spirit of Eq. 19-Eq. 21. Similarly for a symmetri tensor field h, the parts whih are tangent to a two-sphere σ a σ b d h d may be desribed by the trae with respet to σ and by two potentials h ev and h od via σ a σ b d h d = 1 2 htr σ + r 2 2σ a σ b d σ d e e h ev r 2 σ σ d σ d e e h ev 2r 2 ɛ a σ b d σd e e h od 23 The potentials h ev and h od are generally funtions of all of the spaetime oordinates and are guaranteed to exist by theorems involving solutions of ellipti equations on a two-sphere. The fators of r 2 are inluded for onveniene. The notation for a ovariant tensor field is ondensed by defining trae-free tensors tangent to a two-sphere and assoiated with the potential h ev and with the potential h od h ev r 2 2σ a σ b d σ d e e h ev r 2 σ σ d σ d e e h ev 24 h od 2r 2 ɛ a σ b d σd e e h od. 25 The ten independent omponents of a symmetri ovariant tensor h in a spherially symmetri geometry may be written as a sum of the form h dx a dx b = h tt dt 2 + 2h tr dt dr + 2 h ev ta + h od ta dt dx A + h rr dr h ev ra + h od ra dr dx A + h tr σ AB + h ev AB + hab od dx A dx B 26 in terms of the ten funtions h tt, h tr, h ev t, h od t, h rr, h ev r, h od r, h tr, h ev and h od, As with the vetor field, the apital indies A and B are used here just as a reminder that the vetor or tensor to whih they are attahed is tangent to the two-sphere. Otherwise, they should be onsidered ordinary spaetime indies in the ovariant spirit of Eq. 23-Eq. 25. The desriptions of vetor and tensor potentials in Eqs. 19 and 23 on a two-sphere ould have been written with a derivative operator involving the usual angular oordinates. 4

5 However, this would loud the ovariant nature of the deomposition whih is learly revealed ove. The desription of the vetor and tensor omponents in terms of potentials takes advantage of the the natural symmetry of the bakground geometry. For example, if a potential is a funtion of r and t times a Y lm then the resulting vetor or tensor field is the same funtion times the vetor or tensor spherial harmoni with the same l, m pair. Expressions suh as the perturbed Einstein tensor take a partiularly simple form when written in terms of the potentials in plae of the omponents. III. INTEGRATION OF THE SCALAR WAVE EQUATION IN THE SCHWARZ- SCHILD GEOMETRY The salar field resulting from a harge q moving in a irular orbit of the Shwarzshild geometry provides an elementary example whih ontains many of the same interesting features of the metri perturbations. The wave equation for the salar field is 2 ψ = 4πρ, 27 where the salar field soure ρ represents a point harge q moving through spaetime along a worldline desribed by oordinates z a τ. This soure is ρx = q g 1/2 δ 4 x a z a τdτ = q g 1/2 dt/dτ 1 δ 3 x i z i t, 28 with τ being the proper time along the worldline. For a irular orbit at radius R, expanding ρ in terms of spherial harmoni omponents provides ρ = q g 1/2 δr Rδθ π/2δφ Ωtδt tτdτ where and = r 2 qδr Rδθ π/2δφ Ωtdt/dτ = q lm 4πR δr Reiωmt Y lm θ, φ, 29 lm q lm = 4πq R ω m mω, 30 Y lm π/2, 0, 31 dt/dτ dt dτ = 1 1 3M/R, 32 whih follows from the knowledge of the irular geodesis of the Shwarzshild geometry. Also, deomposing ψ provides ψ = l,m ψ lm r e iωmt Y lm θ, φ, 33 r 5

6 and the lm omponent of the salar wave equation beomes [ ] d 2 ψ lm /r 2r M dψ lm /r ω 2 r 2 ll ψ dr 2 rr 2M dr r 2M 2 lm /r = q lm rr 2M R 2M δr R. 34 This may also be written in terms of the tortoise oordinate r = r + 2M lnr/2m 1 and dr /dr = r/r 2M 35 as d 2 ψ lm dr 2 + [ ω 2 1 2M r ll M/r r 2 ] ψ lm = 4π1 3M/R 1/2 qy lm π/2, 0δr R.36 We know that Y l, m = 1 m Y l,m, 37 and the reality of ρ and of the final solution for ψt, r, θ, φ requires similar expressions for q l, m and ψ l, m. The boundary onditions of interest require only downgoing waves at the event horizon and we let one homogeneous solution of Eq. 34 be ψ H lm = e iωr, r 2M. 38 A seond homogeneous solution of Eq. 34 with only outgoing waves at infinity is ψ lm = e iωr r. 39 We an numerially integrate Eq. 34 from very near the event horizon out to the radius of the orbit R to give ψ H, and we an similarly integrate Eq. 34 from near infinity in to R to give ψ. The retarded field is then given as a multiple of ψ H for r < R and as a multiple of ψ for r > R, { ψ ret lm = A ψlm H, r < R B ψlm, r > R. 40 The oeffiients A and B are determined both by properly mathing the disontinuity in dψ lm / dr at R with the δ-funtion soure and also by requiring that ψ lm be ontinuous at R. The disontinuity in dψ lm / dr at R is B dψ lm dr A dψh lm dr R = q lm R 2M, 41 and the ontinuity then yields A ψlm H dψlm ψlm dψlm H = ψ q lm lm dr dr R 2M, 42 and B ψlm H dψlm ψlm dψlm H = ψ H q lm lm dr dr R 2M. 43 6

7 IV. PERTURBATIONS OF THE SCHWARZSCHILD GEOMETRY The Regge-Wheeler equation for odd parity, l 2, soure-free metri perturbations of the Shwarzshild geometry is [ d 2 ψ lm + ω 2 1 2M ] ll + 1 6M/r ψ dr 2 r r 2 lm = The even parity perturbations with l 2 are desribed in a similar manner by the Zerilli equation whih is algebraially more omplex than Eq. 44, but struturally very similar. In fat, any homogeneous solution of the Zerilli equation may be written as a fairly simple linear ombination of a solution of the Regge-Wheeler equation and its derivative. And the magnitudes of the refletion and transmission amplitudes of these rather different equations are, in fat idential. A. Free osillations of a blak hole The free osillations of a blak hole are variously desribed as the quasi-normal modes or the ring-down of the hole. These osillations are mathematially desribed as a homogeneous solution of, say, the Regge-Wheeler equation whih has only outgoing radiation in the limit that r and simultaneously only downgoing radiation in the limit that r 2M. An equivalent desription of a free osillation is in terms of the sattering amplitude. Fous on a gravitational wave, of a speifi frequeny, being sent in from infinity toward the blak hole with a unit amplitude. Some of the wave is sorbed by the blak hole and some is refleted. Thus { e iωr + B ref e iωr, r + ψ lm = 45 A s e iωr, r. The omplex amplitudes A s and B ref are omplex funtions of the omplex frequeny. A frequeny of a free osillation is then haraterized as being a pole in A s that harateristi frequeny has a solution to the wave equation with purely outgoing waves at infinity and downgoing waves at the event horizon. The imaginary part of the eigenfrequeny gives the e-folding time of the damping, and the real part gives the frequeny of the osillations. The frequenies of the even and the odd parity free osillations of a blak hole are idential, and were first alulated bak in the 70 s [5, 6]. Exerise 1: Show that the separated equation for ψr, in the usual Shwarzshild oordinates, is the result given in Eq. 36 Exerise 2: Consider the toy problem, whih uses Prie s potential, where the separated equation for ψr is d 2 ψ lm + [ ω 2 V r dr 2 ] ψ lm = 0 46 over the range < r < +, and where V r = O for r < 3M ll + 1 = for r r 2 3M. 47 7

8 Reall that quasi-normal modes are those solutions to the vauum wave equation whih have outgoing waves at large r, i.e. ψ lm r exp iωr as r +, and downgoing waves at the event horizon, i.e. ψ lm r expiωr as r. Finding the quasinormal modes is, thus, an eigenvalue problem for the omplex frequenies ω whose wave funtions Ψ satisfy appropriate boundary onditions. Find all of the eigen-frequenies of these quasi-normal modes for l = 0, 1 and 2. Hint: Use ψ lm r expiωr for r < 3M and ψ lm r a spherial Bessel funtion for r > 3M. Sale ψ lm so that it is ontinuous at r = 3M and then find ω suh that the dψ lm /dr is also ontinuous at r = 3M. Exerise 3: With Prie s potential rather than the atual potential of Eq. 36, let a point soure with a salar harge q be in a irular orbit at radius r = R with orbital frequeny Ω = M/R 3 1/2. Calulate the amplitude of the salar radiation that goes out at infinity and that goes down the blak hole for all values of m 2 l and l = 0 or 1. For simpliity, assume that the lm omponent of the wave equation takes the form d 2 ψ lm dr 2 + [ ω 2 V r ] ψ lm = q lm δr R. 48 To make this problem a bit less ompliated, you may assume that the radius of the orbit of the salar harge is at R = 3M. V. GAUGE ISSUES A. Gauge transformations In perturbation analyses of general relativity [7 9], one onsiders the differene in the atual metri g at of an interesting, perturbed spaetime and the strat metri g 0 of some given, bakground spaetime. The differene h = g at g 0 49 is assumed to be infinitesimal, say Oh. Typially, one determines a set of linear equations whih govern h by expanding the Einstein equations through Oh. The results are often used to resolve interesting issues onerning the stility of the bakground, or the propagation and emission of gravitational waves by a perturbing soure. However, Eq. 49 is ambiguous: The metris g at and g0 are given on different manifolds. For a given event on one manifold at whih orresponding event on the other manifold is the subtration to be performed? Usually a oordinate system ommon to both spaetimes indues an impliit mapping between the manifolds and defines the subtration. Yet, the presene of the perturbation allows ambiguity. An infinitesimal oordinate transformation of the perturbed spaetime x a = x a + ξ a, where ξ a = Oh, 50 not only hanges the omponents of a tensor at Oh, in the usual way, but also hanges the mapping between the two manifolds in Eq. 49. After the transformation Eq. 50, h new = gd 0 + h old x x d d x a x g 0 b + ξ g0. 51 x 8

9 The ξ in the last term aounts for the Oh hange in the event of the bakground used in the subtration. After an expansion, this provides a new desription of h h new = h old = h old gb 0 ξ x ξ d a g0 b x g0 b ξ x L ξ g 0 = h old 2 a ξ b 52 through Oh; the symbol L represents the Lie derivative and a is the ovariant derivative ompatible with g 0. The ation of suh an infinitesimal oordinate transformation is alled a gauge transformation and does not hange the atual perturbed manifold, but it does hange the oordinate desription of the perturbed manifold. A similar irumstane holds with general oordinate transformations. A hange in oordinate system reates a hange in desription. But, general ovariane ditates that atual physial measurements must be desrible in a manner whih is invariant under a hange in oordinates. Thus, one usually desribes physially interesting quantities stritly in terms of geometrial salars whih, by nature, are oordinate independent. In a perturbation analysis any physially interesting result ought to be desrible in a manner whih is gauge invariant. B. Gauge invariant quantities Gauge invariant quantities appear to fall into a few different ategories. The hange in any geometrial quantity under a gauge transformation is determined by the Lie derivative of that same quantity on the bakground manifold. This is demonstrated for the gauge transformation of a metri perturbation in Eq. 52. Thus, if a geometrial quantity vanishes in the bakground, but not in the perturbed metri, then it will be gauge invariant. Examples inlude the Newman-Penrose salars Ψ 0 and Ψ 4 whih vanish for the Kerr metri. In the perturbed Kerr metri Ψ 0 and Ψ 4 are non zero, gauge invariant and the basis for perturbation analyses of rotating blak holes. A seond example has the bakground metri being a vauum solution of the Einstein equations, so its Rii tensor R vanishes. The Rii tensor of a perturbation of this metri is then unhanged by a gauge transformation. This is diretly demonstrated in IX. Some quantities whih are assoiated with a symmetry of the perturbed geometry are gauge invariant. For example a geodesi of a perturbed Shwarzshild metri, where the perturbation is axisymmetri with Killing field k a, has a onstant of motion k a u b g 0 + h whih is gauge invariant. Another symmetry example involves the Shwarzshild geometry with an arbitrary perturbation. It is a fat that a gauge transformation an always be found, suh that the resulting h has the omponents h θθ, h θφ and h φφ all equal to zero. In this gauge, the surfaes of onstant r and t are geometrial two-spheres, even while the manifold as whole has no symmetry. The area of eah two-sphere an be used to define a radial salar field R whih is onstant on eah of these two-spheres. This salar field on the perturbed Shwarzshild manifold is independent of gauge. However, its oordinate desription in terms of the usual t, r, θ, φ oordinates does hange under a gauge transformation. Quantities whih are arefully desribed by a physial measurement are gauge invariant. For example, the aeleration of a world line ould be measured with masses and springs 9

10 by an observer moving along a world line in a perturbed geometry. The magnitude of the aeleration is a salar and is gauge invariant. If the world line has zero aeleration, then it is a geodesi. Therefore, a geodesi of a perturbed metri remains a geodesi under a gauge transformation even while its oordinate desription hanges by Oh. The mass and angular momentum are other gauge invariant quantities whih might be measured by distant observers in an asymptotially flat spaetime. A small mass orbiting a larger blak hole perturbs the blak hole metri and emits gravitational waves. The gravitational waveform measured at a large distane is also gauge invariant. VI. PERTURBED BIANCHI IDENTITY The Bianhi identity is Contration on and b implies that R dea b + e R da b + d R ea b = b R dea b = 0 54 for a vauum solution of the Einstein equations. This result is used often in the derivations of identities involving E h. The definition of the operator E for a vauum spaetime is so that 2E h = 2 h + a b h 2 a h b + 2R a b d h d + g d h d 2 h, 55 2 a E h = a h + a a b h a a h b a b h a + 2 a R a b d h d + 2R a b d a h d + b d h d b h. = a a h b a a h b a b h a + b a h a + R a b d h ad + 2R a b d a h d = The seond equality follows after use of the Rii identity to interhange the order of derivatives on the first, seond and last terms as well as repeated uses of R = 0 and Eq. 54 for vauum spaetimes. The final result follows after use of the Rii identity on the first two terms and on the seond two terms of the seond equality, and the appliation of symmetries of the Riemann tensor on the remainder. If h is not C 3 then a E h = 0 in a distributional sense. To show this, hoose an arbitrary, smooth test vetor field λ a with ompat support. Consider the integral of λ b a E h over a suffiiently large region. Integrate by parts one and disard the surfae term. Next use Eq. 77 and disard the surfae terms to obtain an integral of h E λ. This integral is zero from Eq. 68. These steps also provide an alternative derivation of Eq. 56 in the event that h is C 3, as well. 10

11 VII. FORMAL nth ORDER PERTURBATION ANALYSIS In general perturbation analysis, let the g of Eq. 3 be an exat solution to the vauum Einstein equations, g 0, and iteratively define where Assume that we are given g n 1 g n and T n = g n 1 + h n 57 h n = Oµ n. 58 = Oµ, with G n 1 8πT n = Oµ n. 59 If h n is a solution of Eq. 59 from E h n = G n 1 8πT n + Oµ n+1, 60 then it follows from the definition of the operator E h in Eq. 2 that G n n 8πT = Oµ n+1, 61 and h n is an Oµ n improvement to the approximate solution to the Einstein equations. The Bianhi identity implies that a E h = 0 62 for any symmetri C 3 tensor field h, as shown in VI. It is also shown that if h is not C 3 then Eq. 62 holds in a distributional sense. Thus an integrility ondition of Eq. 60 is that Note, however, that a G n 1 8πT n a G n 1 = a n 1 G n 1 + Γ a a 8πT n = Oµ n G n 1 b 8πT n 8πT n b Γ G n 1a 8πT na, 64 where a n 1 is the derivative operator of gn 1, and Γ a b is the onnetion relating the derivative operators a and a n 1. The Bianhi identity implies that a n 1G n 1 = 0, 65 and the terms in Eq. 64 involving Γ a b are order µn+1 beause of Eq. 59 and the fat that Γ a b = Oµ. Thus, the approximate vanishing of the right hand side of Eq. 64 is the integrility ondition for Eq. 60, a n 1T n = Oµ n In other words, before Eq. 60 an be solved for h n, it is neessary that the perturbing stress tensor be adjusted to be onserved with the metri g n 1 and to satisfy Eq

12 VIII. b T = 0 IMPLIES THE GEODESIC EQUATION FOR A POINT MASS. We follow an example in referene [10]. In Eq. 8, δ 4 x a X a s/ g is a salar field, and the fator u b may be defined as a vetor field by extension, in any smooth manner, away from the world line. Then, [ g a + u u a b T = µg a + u b u a u b u a δ 4 x a X a s g + u a b u b g δ 4 x a X a s ] ds b u a u b = µ δ 4 x a X a s ds 67 g where the seond equality follows from properties of the projetion operator g a + u u a. If b T = 0, then it neessarily follows that the oeffiient of the delta funtion must be zero for all proper times. A onsequene is that u b b u a = 0, the geodesi equation. A more formal proof of this result is in Poisson s review of the self-fore [11], p 89. IX. GAUGE INVARIANCE OF E h For a bakground geometry whih is a vauum solution of the Einstein equations, an infinitesimal gauge transformation, x a new = x a + ξ a, with ξ a = Oµ hanges the metri perturbation, h new = h 2 a ξ b + Oµ 2. But the operator E h is invariant under suh a oordinate transformation, E ξ = This result follows immediately from the fat that the hange in the perturbation of the Einstein tensor E under a gauge transformation is the Lie derivative of the bakground Einstein tensor L ξ G. For a vauum bakground spaetime, this is zero. Equation 68 also follows from diret substitution into 2E h = 2 h + a b h 2 a h b + 2R a b d h d + g d h d 2 h 69 with h = 2 a ξ b. It is easiest to onsider the fator of g separately, fator of g = d ξ d + d d ξ 2 a a b ξ b = 2 d ξ d 2 a a b ξ b = The seond equality follows after use of the Rii identity on the first two indies of the seond term, use of R = 0 for a vauum spaetime and a releling of the indies. The final result follows after use of the Rii identity on the seond term of the seond equality and use of R = 0 for a vauum spaetime. With h = 2 a ξ b, the remainder of E 2 ξ is remainder = a ξ b + b ξ a + 2 a b ξ a b ξ a ξ b b a ξ b ξ a + 2R a b d ξ d + 2R a b d d ξ

13 The analysis of this expression is lengthy but not diffiult. It begins with using the Rii identity upon the seond and third indies of the first, seond, fourth and sixth terms and upon the first and seond indies of the fifth and seventh terms. The resulting terms with three derivatives may be paired up in a way to use the Rii identity again and to redue the entire expression to one involving only single derivatives. This also requires appliation of Eq. 54. That the entire expression is zero, then follows from the symmetries of the Riemann tensor. X. GREEN S THEOREM FOR E Assume that the bakground geometry is a vauum spaetime; i.e. R = 0 = R. The operator E h in Eq. 3, with an arbitrary tensor k, satisfies the identity where 2k E h = F k, h k, h, 72 F k, h k h 1 2 k h 2k b 1 2 gb k a h 1 2 g h 73 or, in terms of h h 1 2 g h, and or F k, h k h 2 k b a h 74 k, h k h 1 2 k h 2 a k a 1 2 ga k b h b 1 2 g bh 2k R d a b h d. 75 k, h k h 2 a ka b hb 2 k R a b d hd. 76 Note that the inner produt, k, h = h, k is symmetri under the interhange of h and k. It follows that k E h h E k = 1 2 [F k, h F h, k]. 77 Whih is a tensor version of Green s theorem for the differential operator E h. Exerise 4: Assume that the bakground geometry is a vauum solution of the Einstein equations. From Eqs. 3 and 72, derive Eqs. 73 and 76. Hint: Contrat Eq. 3 with an arbitrary symmetri tensor k, and move k inside a in eah term by differentiating by parts. The boundary terms onstitute F k, h. 13

14 XI. GAUSS LAW Consider a ompat region of an n-dimensional manifold, whih has a boundary defined by some salar field t =onst. Let the unit normal to the boundary be n a = ɛn a t = ɛ a t/ɛg b b t t 1/2, 78 where ɛ is ±1 depending upon whether n a is spaelike or timelike, respetively. It is useful to note a variety of forms of Gauss law: b A b g d n x = x ga b d n x = b b t A b g d n 1 x nb A b = g d n 1 x = n b A b γ d n 1 x 79 N where the metri on the boundary is γ = g ɛn a n b, and we use g det = γ det N XII. SINGULAR GAUGE TRANSFORMATIONS Let ξ a be a, possibly distribution valued, vetor field. And let h = 2 a ξ b, as for a gauge transformation. Also, let k be a smooth test tensor with ompat support. Then k E h g d 4 x = h E k g d 4 x = 2 a ξ b E k g d 4 x, 81 from Eq. 77, after dropping the divergene term. An integration by parts and appliation of the perturbed Bianhi identity 56 yields k E h g d 4 x = 2 ξ b a [E k] g d 4 x = Thus, we demonstrate that given a solution to the inhomogeneous, perturbed Einstein equations Eq. 1, even a distributional gauge transformation leaves a distributional valued metri perturbation that ontinues to satisfy the perturbed Einstein equations. [1] T. Regge and J. A. Wheeler, Phys. Rev. 108, [2] M. Davis, R. Ruffini, W. H. Press, and R. Prie, Phys. Rev. Lett. 27, [3] F. J. Zerilli, Phys. Rev. D 2, [4] K. S. Thorne, Rev. Mod. Phys. 52, [5] S. Chandrasekhar and S. Detweiler, Pro. R. So. London 344, [6] S. Detweiler, Astrophys. J. 239,

15 [7] R. Sahs, in Relativity, Groups and Topology, edited by B. DeWitt and C. DeWitt Gordon and Breah, New York, [8] J. M. Stewart and M. Walker, Pro. R. So. London 341, [9] J. M. Bardeen, Phys. Rev. D 22, [10] A. P. Lightman, W. H. Press, R. H. Prie, and S. A. Teukolsky, Problem Book in Relativity and Gravitation Prineton University Press, Prineton, [11] E. Poisson, Living Rev. Relativity 7, , [12] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation Freeman, San Fransiso, [13] Eq of MTW [12] is atually based upon δr = 0, however for a perturbation of a non vauum spaetime, it seems preferle to base the analysis on δ2r g R = 16πδT. Note that if δ2t g T = 0, then 2δT = h T + g h d T d. This explains the last two terms in Eq. 4 whih are not in Eq of MTW 15