Tails of Plane Wave Spacetimes: Wave-Wave Scattering in General Relativity

PHYSICAL REVIEW D 88, 084059 (2013) Tails of plane wave spacetimes: Wave-wave scattering in general relativity

Abraham I. Harte* Max-Planck-Institut fu¨r Gravitationsphysik, Albert-Einstein-Institut, Am Mu¨hlenberg 1, 14476 Golm, Germany (Received 25 September 2013; published 30 October 2013) One of the most important characteristics of light in flat spacetime is that it satisfies Huygens’ principle: Initial data for the vacuum Maxwell equations evolve sharply along null (and not timelike) geodesics. In flat spacetime, there are no tails which linger behind expanding wavefronts. Tails generically do exist, however, if the background spacetime is curved. The only nonflat vacuum geometries where electromagnetic fields satisfy Huygens’ principle are known to be those associated with gravitational plane waves. This paper investigates whether perturbations to the plane wave geometry itself also propagate without tails. First-order perturbations to all locally constructed curvature scalars are indeed found to satisfy Huygens’ principles. Despite this, gravitational tails do exist. Locally, they can only perturb one plane wave spacetime into another plane wave spacetime. A weak localized beam of gravitational radiation passing through an arbitrarily strong plane wave therefore leaves behind only a slight perturbation to the waveform of the background plane wave. The planar symmetry of that wave cannot be disturbed by any linear tail. These results are obtained by first deriving the retarded Green function for Lorenz-gauge metric perturbations and then analyzing its consequences for generic initial-value problems.

DOI: 10.1103/PhysRevD.88.084059 PACS numbers: 04.30.Nk, 04.30.w, 04.25.g

satisfying the vacuum Einstein equation R ¼ 0, I. INTRODUCTION ab Huygens’ principle holds if and only if the geometry In flat spacetime and in the absence of intervening is either flat or is associated with a plane-symmetric gravi- matter, electromagnetic fields do not develop tails. Light tational wave [1,2]. Slightly more can be said by noting propagates without dispersion, and bursts of radiation that Maxwell’s equations are conformally invariant. which are initially sharp remain sharp as they evolve. Electromagnetic waves therefore propagate tail-free on More precisely, the electromagnetic field FabðxÞ at an event all (not necessarily vacuum) metrics which are conformal x depends on current densities and initial data only on, and either to flat spacetime or to a plane wave. If any further not inside, the past null cone of x. This is referred to as possibilities exist, they are highly constrained by the Huygens’ principle. results reviewed in Refs. [1,3,4]. It is fundamental to our everyday lives that light and It is clear that plane wave spacetimes play a privileged sound as we tend to experience them satisfy at least ap- role in the theory of electromagnetic fields. This is not, proximate versions of Huygens’ principle. If this were not it should be emphasized, a peculiarity of Maxwell’s true, moving objects would appear to blur together, and equations. The massless Klein–Gordon equation also sat- spoken language would be impractical. There are, how- isfies Huygens’ principle in flat and plane wave spacetimes ever, many wave phenomena for which Huygens’ principle [5] but essentially no others [6]. Massless spin-1=2 does fail. A pebble falling into a pond, for example, fields propagate tail-free on plane wave backgrounds as produces water waves which persist at any fixed location well [3,7]. long after the initial wavefront has passed. Whether or not This paper explores whether perturbations to the plane Huygens’ principle is valid for any particular system may wave spacetimes themselves satisfy a version of Huygens’ be interpreted as a property of the initial-value problem principle. In physical terms, does a localized packet of associated with the particular differential equations used to gravitational waves passing through a background plane describe that system. From this point of view, only a wave leave behind a ‘‘wake?’’ The answer is that such a handful of reasonable wave equations satisfy Huygens’ wake does exist, although its form is remarkably simple. principle exactly. The vast majority of possibilities develop There are many motivations of studying plane wave tails. spacetimes. Their most obvious application is as exact In the context of general relativity, it is of particular models of gravitational radiation in general relativity. interest to understand how the curvature of spacetime is Separate from this, plane wave spacetimes also have a related to the presence of tails. The vacuum Maxwell number of unusual mathematical properties. Their rele- equations provide a particularly well-studied example. vance for (nongravitational) wave tails has already been Restricting to four-dimensional background spacetimes mentioned. Additionally, Penrose has shown [8] that plane waves arise as universal limits of all spacetimes in the *[email protected] vicinity of null geodesics. This might imply that certain

1550-7998=2013=88(8)=084059(13) 084059-1 Ó 2013 American Physical Society ABRAHAM I. HARTE PHYSICAL REVIEW D 88, 084059 (2013) aspects of ‘‘ultrarelativistic physics’’ in generic curved associated with the background. The waveform associated spacetimes can be understood using effective plane wave with that background may be perturbed, however. This is backgrounds. Ideas of this type have already been applied found to depend on only a single component of the initial to understand properties of classical [9] and quantum [10] metric perturbation. fields propagating on curved backgrounds. The Penrose Our results are derived by first considering the general limit and other features of plane wave spacetimes have problem of first-order perturbations on plane wave back- also produced applications in string theory and related grounds. We derive the retarded Green function for metric topics [11,12]. perturbations in Lorenz gauge and then use a Kirchhoff Regardless of motivation, we consider linear metric per- integral to write an arbitrary metric perturbation in terms of turbations onvacuum plane wave backgrounds in four space- Cauchy data and any sources which may be present. The time dimensions. The few existing results on Huygens’ tail of the resulting Green function is used to derive various principles in linearized general relativity cannot be used to properties satisfied by the ‘‘tail portion of the metric’’ in immediately answer whether gravitational tails exist on the Lorenz gauge. Lastly, an explicit gauge transformation plane wave backgrounds. Waylen [13] considered the line- is constructed which transforms the perturbed metric into arized Einstein equation on general backgrounds and in the the canonical form for a plane wave spacetime. ab Section II briefly reviews some properties of plane wave Lorenz gauge. He then asked whether the trace g ðxÞhabðxÞ spacetimes needed for the later development. Section III of the first-order metric perturbation hab at a point x depended on initial data lying inside the past null cone of then illustrates how simple test fields propagate on plane x. This was always found to occur unless the background wave backgrounds by deriving Green functions for scalar spacetime was either flat or a plane wave. Later, Noonan [14] and electromagnetic fields. While the main conclusions and Wu¨nsch [15] separately asked—again in Lorenz of this section have appeared elsewhere in various forms, they are rederived here more succinctly and to allow for a gauge—whether any part of habðxÞ depended on initial data inside the past null cone of x. They found that this more direct comparison to the gravitational case. The always occurred unless the background was flat. methods reviewed in Sec. III are applied in new ways in None of these results settles the issue. Even if a metric Sec. IV to derive a Green function for metric perturbations perturbation develops a tail in the Lorenz gauge, it might on plane wave backgrounds. This Green function has a tail which is interpreted in Sec. V. not do so in another gauge. This is precisely what occurs 3 1 for electromagnetic fields on plane wave backgrounds. In þ dimensions are assumed throughout this þ2 that case, the vector potential in the Lorenz gauge develops paper. The signature is taken to be , and the sign of the Riemann tensor is defined such that R d! ¼ a tail (as it does in all curved spacetimes [16]). Such tails abc d 2 ... can, however, be removed by appropriate gauge transfor- r½arb!c for any !a. Latin letters a; b; from the mations [17]. The electromagnetic field F ¼ 2r A beginning of the alphabet are used to denote abstract ab ½a b ... i satisfies Huygens’ principle even though A typically indices. The letters i; j; instead refer to coordinates x a ¼ 1 does not. (i , 2) which are transverse to the direction of propa- The problem of gauge is significantly more subtle for gation associated with the background plane wave. The coordinates xi are not to be confused with spacetime points metric perturbations. This is both because the class of denoted by x; y; ...(without indices). possible gauge transformations is larger than in electro- magnetism and also because it is less clear which observ- ables should be considered. The metric at a single point is II. PLANE WAVE SPACETIMES not very interesting, for example. One might instead con- Various aspects of plane wave spacetimes have been abcd sider perturbations to curvature scalars such as R Rabcd. reviewed in Refs. [9,18–20]. For our purposes, these are These scalars vanish on the background, so their first-order most naturally introduced as special cases of the plane- perturbations are gauge-invariant. It is shown here that all fronted waves with parallel rays (pp-waves). Although the such perturbations satisfy Huygens’ principles: Every lo- concept is more general, we focus only on pp-waves which cally constructed polynomial in the curvature and its de- are solutions to the vacuum Einstein equation Rab ¼ 0 in rivatives propagates tail-free on plane wave backgrounds. four spacetime dimensions. It is then sufficient to consider Despite this, there is a sense in which gravitational tails— a simply connected region M, which admits a vector field unlike their scalar or electromagnetic counterparts—do ‘a that is everywhere null, nonvanishing, and covariantly exist. Locally, these can only perturb one plane wave constant [18]: spacetime into another plane wave spacetime. Even initial ‘a‘ ¼r ‘a ¼ 0: (1) data with no symmetries whatsoever produce tails which a b 1 retain the full five-dimensional space of Killing fields The integral curves of ‘a are interpreted as the rays of the gravitational wave. These are free of expansion, shear, 1Some plane wave spacetimes admit a sixth Killing field. This and twist. There is a sense in which all such rays remain is not necessarily preserved by tails. parallel to one another and are orthogonal to a family

084059-2 TAILS OF PLANE WAVE SPACETIMES: WAVE-WAVE ... PHYSICAL REVIEW D 88, 084059 (2013) of planar wavefronts. It follows from the simple- Killing vector fields. Gravitational plane waves may there- connectedness of M and r½a‘b ¼ 0 that a scalar field u fore be defined as those pp-waves which admit at least five exists such that linearly independent Killing fields [22]. This can be shown to imply that for a plane wave spacetime, Hðu; xiÞ can be at ‘a ¼rau: (2) most quadratic in the two transverse coordinates xi.A u describes the phase of the gravitational wave. coordinate transformation may always be used to eliminate i Hypersurfaces of constant u are null and are referred to all terms in Hðu; x Þ which are either independent of or i as wave surfaces. linear in x . Plane wave spacetimes may therefore be 2 2 Any vacuum spacetime which admits a constant null described by a symmetric matrix HijðuÞ satisfying a vector field ‘ also admits a nonvanishing and covariantly k i j Hðu; x Þ¼HijðuÞx x : (8) constant null 2-form fab ¼ f½ab [18]. fab and its dual cd 2 fab ¼ ab fcd= (which is also null and constant) Einstein’s equation (6) then reduces to the trivial algebraic describe the plane of polarization associated with the constraint that Hij be trace-free. This leaves two free gravitational wave. This is always transverse to the direc- functions of u corresponding to waveforms2 associated b b 0 tion of propagation in the sense that fab‘ ¼ fab‘ ¼ . with the two polarization states of the gravitational wave. An explicit metric for pp-wave spacetimes may be It can be convenient to replace the real matrix HijðuÞ by constructed using a phase coordinate u, which satisfies a single complex function H ðuÞ via Eq. (2), an affine parameter v defined by H ¼ H11 þ iH12: (9) @ ‘a ¼ ; (3) @v Similarly complexifying the transverse coordinates using i 1 1 and two more scalar fields x (i ¼ , 2) related to fab via ¼ pffiffiffi ðx1 þ ix2Þ; (10) 2 1 2 fab ¼ ‘½arbx ;fab ¼‘½arbx : (4) the pp-wave line element (5) with a plane-wave–type These choices result in the so-called Brinkmann metric profile (8) is then equivalent to 2 k 2 1 2 2 2 ds ¼2dudv þ Hðu; x Þdu þðdx Þ þðdx Þ (5) ds2 ¼ 2ðdudv þ<½H ðuÞ2du2 þ ddÞ: (11) for general pp-wave spacetimes. The vacuum Einstein Here, overbars denote complex conjugates, and < returns equation constrains the wave profile H to satisfy the real component of its argument. Both the real and r2Hðu; xiÞ¼0; (6) complex representations of plane wave spacetimes are used interchangeably below. 2 2 2 r ¼ 1 þ 2 H where @x @x is usual Laplace operator in two The complex waveform constitutes a nearly gauge- dimensions. H is otherwise arbitrary. invariant way to describe a vacuum plane wave spacetime. The Brinkmann coordinates ðu; v; xiÞ have a number To see this, first note that vector fields ‘a satisfying Eq. (1) of useful properties. Unlike coordinate systems which are unique up to an overall constant. Phases u satisfying are more closely related to typical perturbative discussions Eq. (2) may therefore be altered only by constant affine of gravitational radiation in transverse-traceless gauge, transformations. and can vary by rotations in the Brinkmann coordinates do not develop coordinate singu- complex plane but also by more subtle u-dependent ‘‘trans- larities. Brinkmann coordinates are also harmonic and lations’’ which affect v as well. All coordinate transforma- place the metric into the Kerr–Schild form tions that preserve the form of the Brinkmann metric (11) are known [18]. Applying them results in each plane wave g ¼ þ H‘ ‘ : (7) ab ab a b spacetime being describable by a 3-parameter family of b b Here, ab is flat, and ‘a ¼ gab‘ ¼ ab‘ . The vector field possible waveforms: ‘a is null with respect to both and g . Viewing H‘ ‘ ab ab a b H 2H ic as an arbitrarily large perturbation on the ‘‘background’’ ðuÞ!a ðau bÞe : (12) ab, Eq. (6) implies that this perturbation satisfies the Here, a 0, b, and c are arbitrary real constants. These linearized as well as the fully nonlinear Einstein equations cannot change the overall ‘‘shape’’ of a nonconstant wave- (which is true for all Kerr–Schild metrics [21]). Indeed, form. Note, however, that the overall magnitude of a pp-waves obey exact linear superposition and other properties typically associated with waves in linear theories. 2 This linearity is generally not apparent in other coordinate This notion of waveform is different from the one typically used in perturbative descriptions of gravitational radiation. Hij systems. d directly describes the local curvature Rabc [see Eq. (16)]. The In general, pp-waves admit only one Killing vector. waveforms associated with perturbation theory in the transverse- Recall, however, that electromagnetic plane waves in flat traceless gauge must instead be differentiated twice in order to d spacetime are preserved by a five-dimensional space of recover Rabc .

084059-3 ABRAHAM I. HARTE PHYSICAL REVIEW D 88, 084059 (2013) particular H is not invariant. There is no intrinsic notion where overdots denote derivatives with respect to u. for the amplitude of a gravitational plane wave unless Solutions to this equation directly determine almost all additional structure is introduced (such as a preferred interesting properties of plane wave spacetimes. They observer). describe not only the behaviors of individual geodesics Without any such structure, it is difficult to find any local but also geodesic deviation, parallel transport, distances scalar invariants in plane wave spacetimes. All polyno- between finitely separated points, and so on. mials in the curvature and its derivatives vanish, for Relations of this type have previously been formulated example. A nontrivial invariant may nevertheless be con- using the real form H structed from a particular using [18,23] j x€i ¼ Hijx (19) ½@ ln H ðuÞ2 IðuÞ¼ u : (13) of Eq. (18)[9]. This equation is linear, so its solutions jH ð Þj u depend linearly on initial conditions. Specifying these This is unique up to affine transformations of its argument. conditions at a phase u0, there exist 2 2 real matrices It is not, however, a reasonable measure of amplitude; Aij and Bij such that I ¼ 0 when H is a nonzero constant even though this ð Þ¼ ð 0Þ jð 0Þþ ð 0Þ _ jð 0Þ case is physically nontrivial. xi u Aij u; u x u Bij u; u x u : (20) Local physics in a curved spacetime is largely deter- The two propagators here satisfy mined by the Riemann tensor R d. For plane waves, the abc 2 0 0 only independent coordinate components of the curvature @uAijðu; u Þ¼HikðuÞAkjðu; u Þ (21a) 2 0 0 are determined by @uBijðu; u Þ¼HikðuÞBkjðu; u Þ (21b)

Ruiuj ¼Hij: (14) together with

It can be convenient to reexpress this equation in terms of a ½Aij¼½@uBij¼ij; ½@uAij¼½Bij¼0: (22) complex null tetrad ð‘a;na;ma; m aÞ. Let ‘a be defined by Eq. (3) and Brackets are used in these last equations to denote the coincidence limit lim u!u0 . Knowledge of the waveform @ 2 @ @ H na ¼ þ<ðH Þ ;ma ¼ : (15) Hij (or equivalently ) is sufficient to compute Aij and @u @v @ Bij everywhere. Detailed properties of these matrices are All scalar products of these vector fields vanish except for discussed in Refs. [9,20]. a a Equation (19) is analogous to the equations which ‘ na ¼1 and m m a ¼ 1. The associated Weyl scalars vanish except for describe Newtonian masses coupled by springs with time-varying stiffnesses. That Hij is trace-free implies a b c d H 4 ¼ Cabcdn m n m ¼ : (16) that some of these stiffnesses must be negative. In cases where H is periodic, Eq. (19) is a form of Hill’s equation. Derivatives of the tetrad are ij This arises in many areas of physics and engineering and is b b b b ran ¼ 2‘a<ðH m Þ; ram ¼ H ‘a‘ ; (17) known to have a rich phenomenology [24]. No particular form for Hij is assumed here, however (other than a from which it follows that all divergences and sufficient degree of regularity). d’Alembertians of the tetrad components must vanish It is intuitively clear that solutions to oscillatorlike equa- a 0 h a b a 0 (e.g., ran ¼ and n ¼r rbn ¼ ). tions tend to eventually pass through zero. Fixing some u0, there generically exist some values of U u0 such that 0 A. Geodesics and associated bitensors det BijðU; u Þ¼0. This means that in Eq. (20), the initial i 0 Geodesics in plane wave spacetimes fall into one of two transverse velocity x_ ðu Þ of a geodesic may be varied in the 0 classes. The simpler (and less interesting) case comprises null space of BijðU; u Þ without affecting the final trans- those geodesics which remain confined to a single hyper- verse position xiðUÞ. The initial and final v coordinates of surface of constant u. The only causal geodesics with this such a family of geodesics may also be arranged to coin- property are the null integral curves of ‘a, the rays of the cide. Two points in a plane wave spacetime can therefore gravitational wave. More interesting are the geodesics be connected by an infinite number of geodesics when a det 0 which pass through wave surfaces. Recalling that ‘ is Bij ¼ . This clearly signals the boundary of a normal Killing, the u coordinate may be used as an affine parame- neighborhood. It is also closely related to the largest ter for all such geodesics. All timelike and almost all null domains on which initial data may be used to uniquely geodesics are in this class. Their transverse coordinates , determine the future (since plane wave spacetimes are not satisfy globally hyperbolic [25]). For simplicity, this paper € restricts attention to regions which are sufficiently small ¼ H ; (18) 0 0 that det Bijðu; u Þ 0 for all relevant phases u u .Any

084059-4 TAILS OF PLANE WAVE SPACETIMES: WAVE-WAVE ... PHYSICAL REVIEW D 88, 084059 (2013) pair of points is then connected by exactly one geodesic, III. SCALAR AND ELECTROMAGNETIC FIELDS there are no conjugate points, and initial-value problems Before considering metric perturbations propagating on associated with standard equations are well-posed. a plane wave background, it is instructive to first review the Various geometric objects important for the construction behavior of scalar and electromagnetic fields in this con- of Green functions may be found explicitly in terms of text. As outlined above, we restrict attention only to A and B (see, e.g., Refs. [9,10]). Synge’s function ij ij regions which are sufficiently small that all initial value ðx; x0Þ¼ðx0;xÞ, which returns one-half of the squared 0 problems are well-posed and exactly one geodesic passes geodesic distance between the points x and x ,is through any two distinct points. 1 Consider a massless scalar field c propagating on a ¼ ð 0Þ½2ð 0Þþð BB1Þ i j 2 u u v v @u ijx x plane wave background. Supposing the existence of a 1 0i 0j 1 0i j charge density , this satisfies þðB AÞijx x 2ðB Þijx x : (23) 1 hc ¼rar c ¼4: (28) Here ðB Þij denotes the matrix inverse of Bij. By assump- a tion, ðu u0Þ½B1ðu; u0Þ exists in all regions which are ij It is convenient to introduce a retarded Green function of interest here. ðx; x0Þ is positive, negative, or zero if its Gðx; x0Þ. Let arguments are spacelike-, timelike-, or null-separated. Another important bitensor is the ‘‘scalarized’’ van hGðx; x0Þ¼4ðx; x0Þ; (29) Vleck determinant ðx; x0Þ¼ðx0;xÞ. In plane wave spacetimes, and demand that Gðx; x0Þ vanish when x is not in the causal 0 ðu u0Þ2 future of x . Given an appropriate spacelike hypersurface ðx; x0Þ¼ : (24) on which to impose initial data, arbitrary solutions to det B ðu; u0Þ ij Eq. (28) in the future of may be written in terms of the Holding x0 fixed, this is related to the expansion of the Kirchhoff integral congruence of geodesics passing through x0 [26]. Z depends only on the phase coordinates of its arguments c ðxÞ¼ ðx0ÞGðx;x0ÞdV0 in plane wave spacetimes. DþðÞ 1 Z The last important bitensor needed here is the parallel 0 c0 0 0 c0 0 þ ½c ðx Þr Gðx;x ÞGðx;x Þr c ðx ÞdS 0 : a 0 a 0 4 c propagator g a0 ðx; x Þ¼ga0 ðx ;xÞ. Contracting this 0 with any vector at x returns that same vector parallel- (30) propagated to x along the unique geodesic which passes 0 a þ through both x and x . Coordinate components of g a0 have Here D ðÞ denotes the future domain of dependence of been computed in Ref. [9]. Reexpressed in terms of the null . Tails exist when perturbations to the source or initial tetrad (15), data can travel along timelike as well as null geodesics. a a a It follows from Eq. (30) that this occurs when the support g 0 ¼‘ ½n 0 þ ‘ 0 2<ðm 0 Þ n ‘ 0 a a a a a of G includes regions where its arguments are timelike- a þ 2<½m ðm a0 ‘a0 Þ: (25) separated. If no such regions exist, G is said to be c 0 tail-free and satisfies Huygens’ principle. The complex function ðx; x Þ appearing here may be It has long been known that the retarded Green function found by first considering a geodesic yðÞ which passes 0 associated with Eq. (28) in a plane wave background is through two given points x and x . Using Eq. (20) with tail-free [5]. In full, it is the boundary conditions yiðuÞ¼xi and yiðu0Þ¼x0i, the 0 1=2 0 0 transverse components of such a geodesic satisfy Gðx; x Þ¼ ðx; x Þððx; x ÞÞ: (31) B 0 B1 0 j j 0 0k yiðÞ¼½ ð; u Þ ðu; u Þij½x A kðu; u Þx 0 0 ðÞ¼ðÞðx>xÞ, and ðx>xÞ is a distribution 0 0j 0 þ Aijð; u Þx : (26) which is equal to 1 if x is in the future of x and vanishes otherwise. denotes Synge’s world function and the van Then is given by Vleck determinant. These two-point scalars are explicitly 1 given by Eqs. (23) and (24) in plane wave spacetimes. ¼ pffiffiffi f½y_ 1ðuÞy_ 1ðu0Þ þ i½y_ 2ðuÞy_ 2ðu0Þg: (27) 2 Noting that flat ¼ 1 everywhere in flat spacetime, the form of Eq. (31) is identical to that of the retarded Green This is independent of v, v0 and linear in xi, x0i (or ¼ ð Þ function Gflat flat associated with a flat back- equivalently , ). It also antisymmetric in its arguments: ground. Both solutions are concentrated only on light 0 0 a ðx; x Þ¼ðx ;xÞ. The bitensors , , g a0 are all trivial cones (where ¼ 0 or flat ¼ 0). The only qualitative to compute once Aij and Bij are known. These latter difference between the flat and plane wave Green functions matrices in turn depend only on the waveform Hij. is that is not constant in the plane wave case. This is

084059-5 ABRAHAM I. HARTE PHYSICAL REVIEW D 88, 084059 (2013) essentially a geometric optics effect related to the focusing Ref. [9] that !1and ð=jjÞ1=2 is finite for almost of null geodesics by the spacetime curvature. all approaches to such a hypersurface. The tail appearing in In generic (not necessarily plane wave) spacetimes, the Eq. (36) therefore tends to zero almost everywhere in this first term in Eq. (31) remains as is. It describes the propa- limit. gation of disturbances along null geodesics. A second term Of more direct physical interest is the behavior of an may also arise, however: electromagnetic field Fab. This is known to propagate without tails in plane wave backgrounds [17]. Unlike in G ¼ 1=2 ðÞþV ðÞ: (32) the scalar case, gauge is an important ingredient here. V is relevant for timelike-separated points and is therefore Green functions for vector potentials Aa in commonly referred to as the tail of G.IfV 0, disturbances propa- used gauges do develop tails in plane wave backgrounds. gate in timelike as well as null directions. The effects of These can nevertheless be removed by appropriate gauge localized perturbations may then persist long after they are transformations Aa ! Aa þra and are therefore physi- first observed. Mathematically, it is straightforward to cally irrelevant. show that tails do not occur for massless scalar fields on Consider a vector potential Aa in the Lorenz gauge. plane wave backgrounds. Substituting Eq. (32) into Introducing an electromagnetic current Ja, Maxwell’s Eq. (29) shows that in any spacetime, equations are equivalent to h 4 a 0 hV ¼ 0 (33) Aa ¼ Ja; r Aa ¼ (37) everywhere, and in vacuum spacetimes. The Hadamard ansatz for an associated retarded Green function is 1 1 a 1=2 raV þ ðh 2ÞV ¼ h (34) a0 a0 a0 2 2 Ga ¼ Ua ðÞþVa ðÞ: (38) when ¼ 0. The self-adjointness of Eq. (28) also implies This must be a solution to that Vðx; x0Þ¼Vðx0;xÞ. Fixing x0 and noting that h a0 4 a0 aðx; x0Þ¼raðx; x0Þ is tangent to the geodesic connect- Ga ¼ ga (39) 0 ing x to x, Eq. (34) acts like an ordinary differential in a vacuum spacetime. As in the scalar case, the first equation along each null geodesic which passes through 0 (‘‘direct’’) term in G a has the same form in any space- x0. Integrating it provides characteristic initial data that can a time [26]: be used to solve Eq. (33) and obtain V everywhere. In a a0 1=2 a0 plane wave background, recall from Eq. (24) that Ua ¼ ga : (40) 1=2ðx; x0Þ is a scalar field depending only on the phase a0 coordinates u, u0 of its arguments. Any scalar of this type ga is the parallel propagator, and is again the van Vleck 1 2 a0 has vanishing d’Alembertian, so h = ¼ 0. The initial determinant. The appearance of ga here may be inter- data for V therefore vanishes, and the unique solution to preted as a reason why polarizations associated with elec- Eq. (33)isV ¼ 0. This recovers Eq. (31). tromagnetic waves are parallel-propagated in the limit of a0 Tails typically do appear if the field equation (28)is geometric optics. In plane wave backgrounds, ga is modified in some way. Consider the addition of a mass explicitly given by Eq. (25). term The self-adjointness of the wave equation implies that 2 the tail is symmetric in its arguments: ðh Þc ¼4: (35) 0 0 Vaa0 ðx; x Þ¼Va0aðx; x Þ: (41) Solutions to this equation develop tails even in flat spacetime. Remarkably, the form of the tail in plane wave back- Substituting Eqs. (38) into (39) also shows that grounds is exactly the same as in the flat case. Using J1 to 0 hV a ¼ 0 (42) denote a Bessel function of the first kind, the associated a retarded Green function is everywhere, and " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! # 2 1 1 J1ð 2 Þ 0 0 0 1=2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi br V a þ ðh 2ÞV a ¼ hU a (43) G ¼ ðÞ ðÞ : (36) b a 2 a 2 a 22 when ¼ 0. Equation (43) provides characteristic initial Initially sharp perturbations to a massive scalar field do not data for Eq. (42). Unlike in the massless scalar case, this appear sharp to timelike observers. Consider holding the data is not trivial. It may be found by first noting that the second argument of G ðx; x0Þ fixed and extending x as far van Vleck determinant satisfies the transport equation [26] as possible into the future before encountering a u ¼ U det 0 0 1 wave surface where BijðU; u Þ¼ . Excluding degen- a 1=2 h 4 1=2 0 0 ra þ ð Þ ¼ : (44) erate cases where BijðU; u Þ vanishes entirely, it is shown in 2

084059-6 TAILS OF PLANE WAVE SPACETIMES: WAVE-WAVE ... PHYSICAL REVIEW D 88, 084059 (2013)

a0 1=2 a0 Factoring this out of the tail by defining Wa ¼ Va , the initial data is chosen appropriately and the current is Eq. (43) reduces to everywhere conserved. 1 It is evident from Eqs. (50) and (51) that perturbations in b a0 a0 a0 A generically propagate along timelike as well as null rbWa þ Wa ¼ hga : (45) a 2 geodesics. The same cannot be said for perturbations to The main advantage of Eq. (45) over Eq. (43) is that the Fab. This can be shown by applying an exterior derivative to Eq. (51) in order to obtain a Kirchhoff formula for Fab. left-hand side of Eq. (45) is a total derivative. Consider a 0 0 0 0 a a null geodesic yðuÞ such that yðu Þ¼x . Then, The relevant propagator is clearly Gab ¼r½aGb , which has no tail: D 0 1 0 0 a 0 h a 0 ½ð u ÞWa ðyðÞ;xÞ ¼ ga ðyðÞ;xÞ (46) a0 0 d 2 r½aVb ðÞ¼ : (52) on a plane wave background. Initial data for the tail is Fab therefore satisfies Huygens’ principle on plane wave entirely determined by the degree to which parallel- backgrounds, as claimed. transported vectors satisfy the vector wave equation. Directly computing the right-hand side of Eq. (46) using Eq. (25) and properties of the null tetrad ð‘a;na;ma; m aÞ IV. GRAVITATIONAL PERTURBATIONS described in Sec. II, At least within normal neighborhoods, both scalar and 0 0 electromagnetic test fields on plane wave backgrounds hg a ¼r2ðÞ‘ ‘a : (47) a a behave essentially as they do in flat spacetime: Pulses of Recalling that is linear in the transverse coordinates xi, radiation which are initially sharp remain sharp as they x0i and is independent of v, v0, r2ðÞ can depend only on propagate. We now consider perturbations to the plane 0 a0 wave metric itself. the phase coordinates u, u . The initial data for Va is therefore Metric perturbations on plane wave spacetimes have previously been considered in several contexts. The theory a0 0 1=2 0 0 a0 Va ðx; x Þ¼ ðx; x ÞZðu; u Þ‘a‘ ; (48) of colliding plane waves is particularly well-developed, containing many exact solutions to the fully nonlinear where Einstein equation [27]. Yurtsever has also discussed collid-

1 Z u ing waves which are planar only over large but finite 0 0 2 ðu u ÞZðu; u Þ¼ dr ðÞjðyðÞ;x0Þ: (49) regions [28,29]. 2 0 u These results do not allow for localized perturbations The reciprocity relation (41) implies that Zðu; u0Þ¼ and are therefore inadequate to discuss the presence of Zðu0;uÞ. tails. Linear perturbations of plane wave spacetimes which Equation (48) arose as the solution to Eq. (43). As such, are not plane-symmetric have been derived before [30,31], it is a priori valid only when its arguments are null- although not in a form which appears to be useful for our separated. Note, however, that Eq. (48) is a solution to purposes. The approach taken here is to instead describe the bulk wave equation (42) everywhere. No additional perturbations of plane wave spacetimes in terms of a Green work is required to extend the tail into the interiors of null function associated with the linearized Einstein equation. cones. The complete retarded Green function for Lorenz- As is usual in general relativistic perturbation theory, we gauge vector potentials in plane wave backgrounds is consider a 1-parameter family of metrics, 2 a0 1=2 a0 a0 g^ ðÞ¼g þ h þ Oð Þ; (53) Ga ¼ ½ga ðÞZ‘a‘ ðÞ: (50) ab ab ab A general vector potential in Lorenz gauge may be which tend smoothly to a vacuum plane wave background ! 0þ derived from this Green function using the Kirchhoff gab as . Only first-order perturbations are consid- integral ered here. Carets are used to denote quantities associated with the full (background þ perturbation) spacetime. Z a0 0 All indices are raised and lowered using the background Aa ¼ Gb Ja0 dV DþðÞ metric gab. 1 Z It is convenient to work in the Lorenz gauge. This is c0 a0 a0 c0 þ ð 0 r r 0 Þ 0 4 Aa Ga Ga Aa dSc : (51) defined by demanding that the trace-reversed metric per- TR cd 2 turbation hab ¼ hab gabg hcd= be divergence-free: h This produces solutions to the wave equation Aa ¼ b TR 0 r hab ¼ : (54) 4Ja in terms of the current Ja and initial data prescribed on . It does not enforce the gauge condition, however. Einstein’s equation linearized about gab is then Electromagnetic fields F ¼ 2r A determined by ab ½a b h TR 2 c d TR 16 Eq. (51) are true solutions to Maxwell’s equations only if hab þ Ra b hcd ¼ Tab; (55)

084059-7 ABRAHAM I. HARTE PHYSICAL REVIEW D 88, 084059 (2013) where Tab denotes the perturbed stress-energy tensor. to the tail V associated with massless scalar fields via ab a0b0 a0b0 0 All derivative operators are those associated with the g Vab ¼ g V [26]. But V ¼ in plane wave back- background. grounds, so General solutions to Eq. (55) may be constructed using ab a0b0 0 Green functions. Consider a two-point distribution g Vaba0b0 ¼ Vaba0b0 g ¼ : (63) ab 0 G a0b0 ðx; x Þ, which is separately symmetric in the indices ab 0 0 g hab therefore propagates only along null geodesics in ab and a b and which satisfies the Lorenz gauge. This is the sense in which Waylen [13] h a0b0 2 c d a0b0 4 ða0 b0Þ has claimed that metric perturbations on plane wave back- Gab þ Ra b Gcd ¼ ga gb : (56) grounds satisfy Huygens’ principle. Other components of The Hadamard ansatz for a retarded solution to this hab do not behave so simply. To see this, it is sufficient to equation is a0b0 note that regularity of Vab in Eq. (60) implies that the a0b0 a0b0 a0b0 tail has the coincidence limit [26] Gab ¼ Uab ðÞþVab ðÞ: (57) In every spacetime, the first term here is known to be given ½Vabc0d0 ¼Rac0bd0 : (64) by [26] The right-hand side of this equation does not vanish, so 0 0 a0b0 ða0 b0Þ1=2 V a b cannot vanish either. Uab ¼ ga gb : (58) ab As in the electromagnetic case above, it is convenient to The polarization tensor of a gravitational wave is therefore factor the van Vleck determinant out of the tail by defining parallel-propagated in the limit of geometric optics. Its a0b0 1=2 a0b0 ‘‘amplitude’’ also varies according to the usual focusing Wab ¼ Vab : (65) factor 1=2. a0b0 Equation (60) then reduces to The gravitational tail Vab appearing in Eq. (57) does D 0 0 1 0 0 0 0 not have a universal form. It satisfies ½ð u0ÞW a b ¼ hðg ða g b ÞÞþRc d g ða g b Þ d ab 2 a b a b c d hV a0b0 þ 2R c dV a0b0 ¼ 0 (59) ab a b cd (66) everywhere with characteristic initial data determined by along null geodesics passing through x0. The characteristic 1 cr V a0b0 þ ðh 2ÞV a0b0 initial data needed to obtain the tail follows from this c ab 2 ab equation by using Eq. (25) and contracting with all pos-

1 0 0 0 sible tetrad components. The resulting calculation is tedi- ¼ hU a b þ R a bU a b 2 ab c d ab (60) ous but straightforward. It results in an expression for V a0b0 which is valid when ¼ 0. Whatever is not taken for all null-separated points x, x0. The self-adjointness of ab into account by this data may be parametrized by the Eq. (55) implies the reciprocity relation a0b0 a0b0 addition of Lab for some smooth Lab . The result 0 0 Vaba0b0 ðx; x Þ¼Va0b0abðx ;xÞ: (61) of these computations is that a0b0 1=2 a0b0 4 a0 b0 As in the scalar and electromagnetic cases discussed Vab ¼

(62) Z u ðu u0ÞYðu; u0Þ¼ dðr2ð2Þ2H Þ: (68) Only perturbations which satisfy the gauge condition (54) u0 are true solutions to the linearized Einstein equation. As a a The only portion of the gravitational Green function consequence, r T ¼ 0 and the initial data appearing in 0 0 ab which is left to be determined is L a b . Substituting Eq. (62) must satisfy the gauge condition in an appropriate ab Eq. (67) into Eq. (59) while using Eqs. (49) and (68) shows sense. 0 0 that One component of V a b may be derived immediately. ab 1 Taking the trace of Eq. (55) and noting the appropriate a0b0 2 2 a0 b0 0 0 L ¼ ðjYj þ 8Z Þ‘ ‘ ‘ ‘ : (69) a b ab 4 a b boundary conditions, the gravitational tail Vab is related

084059-8 TAILS OF PLANE WAVE SPACETIMES: WAVE-WAVE ... PHYSICAL REVIEW D 88, 084059 (2013) Note that the scalar coefficient in this equation depends only for some scalar field H. Stress-energy conservation and on u and u0. Together, Eqs. (49), (57), (58), and (67)–(69) the Lorenz gauge condition require completely describe the retarded Green function for L T T 0 Lorenz-gauge metric perturbations on vacuum plane ‘ ¼ @v ¼ @vH ¼ : (74) wave backgrounds. This appears to be the first closed- Perturbations in this class therefore share some degree of form example of a Green function for Einstein’s equation plane symmetry with the background spacetime. linearized off of a nonflat vacuum background (see, how- More than this, the perturbed metric is exactly that of a ever, Refs. [32,33] for discussions of Green functions on (not necessarily vacuum) pp-wave spacetime with the pro- Friedmann–Robertson–Walker backgrounds). i j k file HijðuÞx x þ Hðu; x Þ. Although this result has been The scalar, electromagnetic, and gravitational Green obtained using the linearized Einstein equation, it is functions derived here all have the remarkable property actually an exact solution to the fully nonlinear Einstein that their Hadamard series terminate at finite order. In equation. That the linearized solution is also an exact generic spacetimes, tails of Green functions can be solution is a consequence of the fact that the perturbation obtained systematically using the series expansion (73) is in extended Kerr–Schild form [21]. X1 0 0 n 0 V ðx; x Þ¼ Vn ðx; x Þ ðx; x Þ: (70) n¼0 V. INTERPRETING THE TAIL This is called a Hadamard series and is motivated by the It was shown in Sec. III that tails associated with observation that initial data equations such as Eq. (60) are Lorenz-gauge vector potentials are pure gauge in plane ordinary differential equations which apply when ¼ 0 wave backgrounds. It is interesting to ask whether a similar (see, e.g., Refs. [34]). Terms which are higher order in situation arises for the tail associated with the gravitational a0b0 effectively expand the solution ‘‘into the light cones.’’ Green function Gab derived in Sec. IV. In the electro- Conditions may be placed on the coefficients Vn such magnetic case, all physical information is contained in the that these functions are uniquely determined by ordinary Faraday tensor Fab. A ‘‘relevant’’ electromagnetic tail differential equations for all n. The results of Sec. III imply therefore exists if and only if initial data for Fab propagates that all Hadamard coefficients vanish for massless along timelike curves. It follows from Eq. (52) that this scalar fields propagating on plane wave backgrounds. For does not occur, so any tails associated with the vector aa0 potential are unphysical. Lorenz-gauge vector potentials, Vn vanishes for all n 1. The tail associated with Lorenz-gauge metric per- Unfortunately, there is no direct analog of Fab which can turbations admits nonvanishing Hadamard coefficients describe the geometry of spacetime in a way that is both only when n ¼ 0 or n ¼ 1. In all of these cases, complete and gauge-invariant. The perturbed metric Hadamard expansions result in finite series. More pre- changes drastically in different gauges and is therefore difficult to interpret directly. The perturbed Riemann ten- cisely, the Hadamard coefficients Vn vanish for all n ðtensor rank of the relevant fieldÞ. It is not clear if this sor is less dependent on gauge but not completely inde- pattern continues for higher-rank fields propagating on pendent of it. Nevertheless, the curvature may be used to plane wave backgrounds. construct various quantities which are gauge-invariant. One possible route is to compute perturbations to all scalars A. Plane-fronted perturbations 0 0 abcd a bcdf a b R Rabcd; r R raRbcdf; ... (75) The tail associated with Gab is discussed in detail in Sec. V. One interesting class of perturbations may, how- formed from local polynomials in the Riemann tensor and ever, be understood immediately. Consider sources with its derivatives (together with g and ). Every scalar the form T ¼ T‘ ‘ and initial data satisfying ab abcd ab a b of this type is known to vanish in the background space- h / ‘ ‘ ;dSrch / ‘ ‘ (71) time. Their perturbations are therefore gauge-invariant. ab a b c ab a b Local knowledge of all curvature scalars places strong on a spacelike hypersurface . Equations (25), (57), (58), constraints on the local geometry [35]. Here, all tail-related (62), (65), (67), and (69) imply that the tail cannot affect perturbations to the curvature scalars may be shown to any metric perturbation to the future of . The relevant vanish. This is a gauge-invariant statement. It does not, portion of the Green function is simply however, exhaust all physically relevant information which may be extracted from hab. a0b0 1=2 Gab ‘a0 ‘b0 ¼ ðÞ‘a‘b; (72) Geometries with vanishing curvature scalars are known to comprise a certain subset of the Kundt spacetimes [36]. from which it follows that A metric gâb is said to be of the Kundt type if it admits a null vector field ‘â which is geodesic, expansion-free, hab ¼ H‘a‘b (73) shear-free, and twist-free. All pp-waves (and therefore

084059-9 ABRAHAM I. HARTE PHYSICAL REVIEW D 88, 084059 (2013) plane waves) are in this class. Our strategy to understand on both u and xi and is at most linear in the latter coor- a0b0 the tail portion of Gab is to first identify a perturbation dinates. hnn is at most linear in v (with a coefficient i ‘â of the background vector field ‘a which establishes the depending only on u) and at most quadratic in x . perturbed spacetime to locally be in the Kundt class. Together, these observations strongly constrain the geo- Indeed, we obtain a nontrivial null vector field for which metric character of the perturbed spacetime. a ^ â 2 Recall that the vector field ‘ ¼ @=@v is both null and the first derivative vanishes entirely: rb‘ ¼ Oð Þ. This implies that through first order in the small parameter , covariantly constant with respect to the background metric g . ‘a is also null with respect to the perturbed metric g^ . tail effects can only perturb a background plane wave into a ab ab ^ a 0 pp-wave. We construct an explicit gauge transformation It is easily verified, however, that rb‘ in general. Nevertheless, the rescaled vector field which transforms the perturbed metric gâb into the canonical Brinkmann form (5). The perturbed wave profile may â then be read off directly from the metric components. It is ‘ ¼ð1 þ Þ‘a (76) found to be at most quadratic in the transverse coordinates, ^ so the perturbed spacetime is locally a plane wave and not a is both null and constant with respect to gab [through OðÞ] more general type of pp-wave. if the real scalar field ðuÞ satisfies Before carrying out this procedure, it is first important to 1 note that no universal properties can be expected to hold _ ¼ @ h : (77) for the perturbed metric as a whole and for completely 2 v nn arbitrary initial data. Only the ‘‘tail components’’ of metric â perturbations might be expected to have a simple descrip- That such an ‘ exists is highly nontrivial. It implies that ^ tion for all possible choices of initial data. This concept gab must describe a pp-wave through OðÞ. The lack of must be made precise. In general, it is difficult to disen- uniqueness associated with corresponds to the usual â tangle a meaningful tail from the remainder of a generic freedom to rescale ‘ by a multiplicative constant. metric perturbation. Naively, the Kirchhoff integral (62) Any pp-wave spacetime must admit not only a constant could be used to isolate only those portions of habðxÞ which null vector field but also a constant null 2-form. In the involve data in the interior of the past light cone of x. This background spacetime, examples of such 2-forms are the [ f and f of Eq. (4). The complex 2-form ‘ m ¼ could be called the ‘‘tail’’ habðxÞ of the full metric pertur- ab ab pffiffiffi ½a b 2 bation at a particular point x. It is, however, impossible to ðfab ifabÞ= and its conjugate are null and constant make geometrically meaningful statements knowing only a as well. Preferred perturbations m^ a to the background [ a metric at a single point. Derivatives of hab must also be tetrad component m may therefore be found by demand- ^½a b computed at x. These have Kirchhoff representations of ing that ‘ m^ be null and constant with respect to gâb their own, which involve data that is no longer guaranteed (together with the usual requirements that m^ a be null, â a b to be confined only to the interior of the past light cone of orthogonal to ‘ , and that gâbm^ m^ ¼ 1). This results in x. Derivatives of objects which appear to be ‘‘pure tails’’ are not pure tails themselves. This is because initial data 1 1 ^ a ¼ 1 þ a a a which lies sufficiently close to—but not on—the past light m i 2 hmm m 2 hmmm @‘ : cone of x can intersect the past light cone of a neighboring (78) point x þ dx. Although our assumptions can be weakened, this prob- ^ a â lem is avoided here by considering only those cases m is considerably less unique than ‘ . It depends on the 3 ^ a where it does not arise. First suppose for simplicity that choice of an arbitrary real function ðu; ; Þ. m also depends on ðuÞ, which is real and satisfies Tab ¼ 0. Fixing a particular point x, let all nontrivial data on the initial hypersurface vanish where that hypersur- _ face meets the past light cone of x. Also demand that ¼=ð@hmnÞ: (79) similar conditions exist for all points in an open neighborhood of x. This corresponds to considering general Here = denotes the imaginary component of its argument. vacuum perturbations observed only in regions which are Different solutions for correspond to the usual freedom ^ a timelike- or spacelike- (but not null-)separated to all to vary m by a constant complex phase. â nontrivial initial data. Given particular choices for ‘ and m^ a within the class nâ Under these restrictions, various properties of hab may just described, a unique real vector field may be ^ â ^b be deduced directly from inspection of Eqs. (62) and (67). constructed which is null, satisfies gab‘ n ¼1, and is ab ^ a ^ a The first of these is that g hab ¼ 0. Using the notation orthogonal to m and m : a b hn‘ ¼ habn ‘ , it follows that hn‘ ¼ hmm . Additionally, h ¼ h ¼ 0. The components h , h , and h 3 ^ a ‘‘ m‘ mm mm ‘n Only @ is relevant for m . appears undifferentiated in the depend only on the phase coordinate u. hmn may depend coordinate transformation (86).

084059-10 TAILS OF PLANE WAVE SPACETIMES: WAVE-WAVE ... PHYSICAL REVIEW D 88, 084059 (2013) 1 _ 1 â ¼½1 þ ð Þ a þ a ¼ þ½ =ð Þ ð þ Þ n hmm n 2 hnn‘ @ hmn i @hmn 2 @u hmm hmm : 2 a <½ð@ þ hmnÞm : (80) (88) ð â â ^ a ^ aÞ can exist only if the right-hand side of this last equation is Together, the vector fields ‘ ; n ; m ; m form a null tetrad in the perturbed spacetime with very similar prop- independent of both and . As a consequence, is erties to the null tetrad ð‘a;na;ma; m aÞ defined by Eqs. (3) constrained by the two integrability conditions: and (15) in the background spacetime. The perturbed tetrad 1 ¼ <ð Þ is not unique. It depends on a real function ðu; ; Þ as @@ 2 @uhmm @hmn ; (89) well as two real numbers corresponding to initial conditions for the differential equations, Eqs. (77) and (79). 2 1 â ^ ^b @ ¼ @ h @ h : (90) Given that ‘ is constant, the 1-form gab‘ must be 2 u mm mn closed. It follows that there exists a scalar field u^ such that The right-hand sides of both of these relations depend only ^b gâb‘ ¼rau:^ (81) on u,so must be quadratic in , . For any real function 0ðuÞ, the most general is By analogy with Eq. (2), this defines a phase coordinate u^ ^ 1 for the perturbed spacetime. Similarly, scalar fields v^ and ¼ þ< 2½ _ ð j Þ þ 0 hmn ¼0 2 @vhnn may be introduced such that 1 2 @ @ þ ð@ h 2@ h Þ : (91) ‘â ¼ ; m^ a ¼ : (82) 2 u mm mn @v^ @^ The Lorenz gauge condition (54) has been used here in the Coordinates with these properties fix seven out of ten form metric components. The remainder follow from demand- 1 ing that <ð Þ¼ ð Þ @hmn 2 @uhmm @vhnn : (92) @ @ nâ ¼ þ h (83) @u^ @v^ It is clear from Eq. (91) that the subquadratic components of may be varied arbitrarily by appropriate choices of ^ ^ for some real hðu;^ ; Þ [cf. Eq. (15)]. The perturbed metric and 0. then reduces to the explicit Brinkmann form This freedom can be used to simplify the wave profile h. Applying the various coordinate transformations and c2 2 ^ ^ 2 ds ¼ 2ðdud^ v^ þ hdu^ þ ddÞþOð Þ (84) comparing to Eq. (83), for a pp-wave with profile 2h. _ ^2 ^ ^ h ¼< ½ð1 þ 2ði ÞÞH H H h m m Coordinate (or gauge) transformations ðu; v; ; Þ! ^ ^ 1 ðu;^ v;^ ; Þ with these properties may be constructed 2 H ^ þ ð ^ Þþ 2 hnn v@vhnn @u : (93) explicitly and used to write h in terms of hab. Noting Eqs. (76) and (81), the perturbed and background phase This has been written in terms of the perturbed Brinkmann coordinates are related by ^ ^ coordinates ðu;^ v;^ ; Þ. It satisfies @v^ h ¼ 0, as required for _ u^ðuÞ¼u þ ðuÞ; ðuÞ¼ðuÞh ðuÞ: (85) a general pp-wave profile. Furthermore, h is at most qua- mm dratic in the transverse coordinates ^, ^. This means that Different choices for and correspond to the fact that a the perturbed spacetime is actually a plane wave and not constant affine transformation applied to any valid phase another type of pp-wave. coordinate produces another valid phase coordinate. gâb satisfies the linearized Einstein equation by The remaining coordinate transformations which imply construction, so Eqs. (82) and (83) are @ ^@^h ¼ 0: (94) v^ðu; v; ; Þ¼½1 ðuÞv þ ðu; ; Þ; (86) Further simplifications arise from appropriate choices for ^ 1 0 and . Letting ðu; ; Þ¼ 1 þ ðiðuÞþ h ðuÞÞ 2 mm 1 _ ¼ ð Þj 1 0 2 hnn v@vhnn ¼0 (95) þ ð Þ þ ð Þ 2 hmm u u ; (87) eliminates all terms in h which are independent of the where ðuÞ is complex and satisfies transverse coordinates. Different solutions to Eq. (95)

084059-11 ABRAHAM I. HARTE PHYSICAL REVIEW D 88, 084059 (2013) correspond to definitions for v^ coordinates which differ by additive constants. All terms linear in ^ and its conjugate which appear in h may be eliminated by demanding that satisfy 1 € H ¼ þ @uhmn @hnn : (96) 2 ¼0 In the absence of metric perturbations, this reduces to the background geodesic equation (18). Different solutions for affect the definitions for both v^ and ^ in ways which may be identified with motions generated by Killing fields. FIG. 1. An initial value problem for vacuum perturbations on a Applying Eqs. (94)–(96) to Eq. (93) reduces the wave plane wave spacetime. Nontrivial initial data are imposed on profile to the canonical form [cf. Eq. (11)] only in small regions around the two points i1, i2. The unshaded ^ ^ ^ ^2 regions are locally determined by the background plane wave. hðu;^ ; Þ¼<½H ðu^Þ ; (97) The three shaded regions are locally plane waves, each of which may have different waveforms. The (finite-width) boundaries with the perturbed waveform between each of the plane wave regions can be arbitrarily ^ _ complicated. Also illustrated is a timelike worldline representing H ¼½1 þ 2ði ÞH H the path of a local observer.

1 2 2 þ ð@ h þ @ h 2@ @ h Þ: (98) 2 nn u mm u mn future light cones of the nontrivial initial data and may all Through OðÞ, the tail of any linear perturbation to a plane have different waveforms. wave spacetime with waveform H is another plane wave Except for two brief events associated with the wave- ^ spacetime with waveform H . This is a complete descrip- fronts expanding outward from i1 and i2, the timelike tion for the local geometry and is unique up to the observer illustrated in Fig. 1 can use purely local measure- 3-parameter family of rescalings described by Eq. (12). ments to conclude that the surrounding geometry is always The integration constants associated with solving a plane wave. The instantaneous waveform of that plane Eqs. (77), (79), and (85) for , , and represent infini- wave can also be determined using local measurements. tesimal versions of these rescalings and may therefore be Such an observer could not, however, conclusively deter- ignored. mine whether or not the waveform observed immediately after a particular wavefront is really ‘‘different’’ from the Using the Kirchhoff representation (62) for hab, the perturbed waveform can be written as an integral over waveform observed before that wavefront. Such a deter- initial data specified on a spacelike hypersurface . mination requires the use of experiments which sample the a0b0 H^ geometry over finite distances. Using the explicit form (67) for the tail Vab , may ^ These results are but one application of the Green func- be shown to depend only on g‘‘j ¼ h‘‘j and the first a0b0 derivative of this quantity normal to . In this sense, plane tion Gab derived in Sec. IV. Another possible applica- wave tails arise from only a single component of the initial tion concerns the effect of a compact object’s ‘‘own metric perturbation. gravitational field’’ on its motion through a gravitational plane wave. Indeed, the gravitational self-force is most easily expressed in terms of the tail of an appropriate VI. DISCUSSION Green function [26,37–39]. A ‘‘self-torque’’ [38] could At least within normal neighborhoods, we have shown be also be computed in this way. More generally, it might that linear tails in vacuum general relativity cannot perturb be possible to use Penrose limits in order to understand a gravitational plane wave into anything other than another some aspects of the self-force problem on generic back- plane wave. The scattering of two gravitational waves— grounds by considering equivalent problems on plane wave one an initially localized burst and the other a much backgrounds. This is especially likely to be relevant for the stronger plane wave—can leave behind only a very par- ultrarelativistic self-force problem [40]. ticular type of tail. This situation is illustrated in Fig. 1 for Another potential application for the plane wave Green a0b0 the particular case where initial metric perturbations exist function Gab involves the long-time behavior of Green only in the vicinities of two distinct points i1 and i2. functions associated with nonplane wave backgrounds. Regions which are null-separated from the initial data Within a normal neighborhood, the most singular portion near i1 and i2 do not have any universal form. of such a Green function always has the form a0b0 Everywhere else in the future of is locally a plane Uab ðÞ. The structure of this term can change over wave. Different plane wave regions are separated by the large distances. Penrose limits were used in Ref. [9]to

084059-12 TAILS OF PLANE WAVE SPACETIMES: WAVE-WAVE ... PHYSICAL REVIEW D 88, 084059 (2013) derive such changes for scalar Green functions G by This pattern is universal. An analogous result for linearized reducing the problem to an equivalent one in an appropri- metric perturbations could likely be established if the ate plane wave background. This showed that every plane wave Green function of Sec. IV could be extended encounter with a nondegenerate conjugate point leads to beyond the normal neighborhood. changes in the ‘‘leading-order singularity structure’’ of G with the form ACKNOWLEDGMENTS jj1=2 The author thanks Seth Hopper and Brien Nolan for jj1=2ðÞ!pv !jj1=2ðÞ!... (99) useful discussions.

[1] P. Gu¨nther, Huygen’s Principle and Hyperbolic [20] A. I. Harte, Classical Quantum Gravity 30, 075011 (2013). Differential Equations (Academic Press, San Diego, [21] B. C. Xanthopoulos, J. Math. Phys. (N.Y.) 19, 1607 1988). (1978). [2] P. Gu¨nther and V. Wu¨nsch, Mathematische Nachrichten [22] H. Bondi and I. Robinson, Proc. R. Soc. A 251, 519 63, 97 (1974). (1959). [3] M. Belger, R. Schimming, and V. Wu¨nsch, Z. Anal. [23] A. Coley, D. McNutt, and R. Milson, Classical Quantum Anwend. 16, 9 (1997). Gravity 29, 235023 (2012). [4] S. R. Czapor and R. G. McLenaghan, Acta Phys. Pol. B [24] W. Magnus and S. Winkler, Hill’s Equation (Dover, New Proc. Suppl. 1, 77 (2008). York, 1979). [5] P. Gu¨nther, Arch. Ration. Mech. Anal. 18, 103 (1965). [25] R. Penrose, Rev. Mod. Phys. 37, 215 (1965). [6] R. G. McLenaghan, Math. Proc. Cambridge Philos. Soc. [26] E. Poisson, A. Pound, and I. Vega, Living Rev. Relativity 65, 139 (1969). 14, 7 (2011). [7] V. Wu¨nsch, Beitra¨ge zur Analysis 13, 147 (1979). [27] J. B. Grifﬁths, Colliding Plane Waves in General [8] R. Penrose, Differential Geometry and Relativity (Reidel, Relativity (Clarendon, Oxford, 1991). Dordrecht, 1976), p. 271. [28] U. Yurtsever, Phys. Rev. D 38, 1731 (1988). [9] A. I. Harte and T. D. Drivas, Phys. Rev. D 85, 124039 [29] U. Yurtsever, Phys. Rev. D 40, 329 (1989). (2012). [30] B. C. Xanthopoulos, J. Math. Phys. (N.Y.) 30, 2626 [10] T. J. Hollowood and G. M. Shore, J. High Energy Phys. 12 (1989). (2008) 091. [31] B. C. Xanthopoulos, J. Math. Phys. (N.Y.) 32, 1866 [11] A. A. Tseytlin, Classical Quantum Gravity 12, 2365 (1991). (1995). [32] P. C. Waylen, Proc. R. Soc. A 362, 233 (1978). [12] D. Sadri and M. M. Sheikh-Jabbari, Rev. Mod. Phys. 76, [33] R. R. Caldwell, Phys. Rev. D 48, 4688 (1993). 853 (2004). [34] F. G. Friedlander, The Wave Equation on a Curved [13] P.C. Waylen, Proc. R. Soc. A 321, 397 (1971). Spacetime (Cambridge University Press, Cambridge, [14] T. W. Noonan, Astrophys. J. 343, 849 (1989). England, 1975). [15] V. Wu¨nsch, Gen. Relativ. Gravit. 22, 843 (1990). [35] A. Coley, S. Hervik, and N. Pelavas, Classical Quantum [16] T. W. Noonan, Astrophys. J. 341, 786 (1989). Gravity 26, 025013 (2009). [17] H. P. Ku¨nzle, Math. Proc. Cambridge Philos. Soc. 64, 779 [36] V. Pravda, A. Pravdova´, A. Coley, and R. Milson, (1968). Classical Quantum Gravity 19, 6213 (2002). [18] J. Ehlers and W. Kundt, Gravitation: An Introduction to [37] A. Pound, Phys. Rev. D 81, 024023 (2010). Current Research, edited by L. Witten (Wiley, New York, [38] A. I. Harte, Classical Quantum Gravity 29, 055012 (2012). 1962). [39] S. E. Gralla and R. M. Wald, Classical Quantum Gravity [19] J. K. Beem, P.E. Ehrlich, and K. L. Easley, Global 25, 205009 (2008). Lorentzian Geometry (Dekker, New York, 1996), 2nd ed. [40] C. R. Galley and R. A. Porto, arXiv:1302.4486.

084059-13