Caustics and wave propagation in curved spacetimes

Abraham I. Harte1 and Theodore D. Drivas2

1Max-Planck-Institut fur Gravitationsphysik, Albert-Einstein-Institut,Am Muhlenberg 1, D-14476 Golm, Germany

2Department of Applied Mathematics and Statistics,The Johns Hopkins University, Baltimore, Maryland 21218, USA

(Dated: February 3, 2012)

We investigate the effects of light cone caustics on the propagation of linear scalar fields in genericfour-dimensional spacetimes. In particular, we analyze the singular structure of relevant Greenfunctions. As expected from general theorems, Green functions associated with wave equations areglobally singular along a large class of null geodesics. Despite this, the “nature” of the singularity ona given geodesic does not necessarily remain fixed. It can change character on encountering causticsof the light cone. These changes are studied by first deriving global Green functions for scalar fieldspropagating on smooth plane wave spacetimes. We then use Penrose limits to argue that thereis a sense in which the “leading order singular behavior” of a (typically unknown) Green functionassociated with a generic spacetime can always be understood using a (known) Green functionassociated with an appropriate plane wave spacetime. This correspondence is used to derive asimple rule describing how Green functions change their singular structure near some referencenull geodesic. Such changes depend only on the multiplicities of the conjugate points encounteredalong the reference geodesic. Using σ(p, p′) to denote a suitable generalization of Synge’s worldfunction, conjugate points with multiplicity 1 convert Green function singularities involving δ(σ)into singularities involving ±1/πσ (and vice-versa). Conjugate points with multiplicity 2 may beviewed as having the effect of two successive passes through conjugate points with multiplicity 1.Separately, we provide an extensive review of plane wave geometry that may be of independentinterest. Explicit forms for bitensors such as Synge’s function, the van Vleck determinant, and theparallel and Jacobi propagators are derived almost everywhere for all non-singular four-dimensionalplane waves. The asymptotic behaviors of various objects near caustics are also discussed.

I. INTRODUCTION

Disturbances in physical fields generically propagatethroughout the causal future of that initial disturbance.The character of this propagation is well-understood forpoints that are sufficiently close to said disturbance.More specifically, there exist fairly straightforward pro-cedures to construct Green functions1 for linear (or lin-earized) wave equations in regions which are sufficientlysmall that, roughly speaking, characteristic rays start-ing at one point do not cross each other at any otherpoint. What occurs outside of these regions – where thecharacteristics develop caustics – is considerably morecomplicated.

To be specific, consider linear hyperbolic equationswhose principal part is the d’Alembertian associated witha spacetime metric gab. Suppressing possible indices onthe field Φ and source ρ, let

LΦ = (gab∇a∇b + . . .)Φ = −4πρ, (1.1)

where ∇a is the natural derivative operator associated

1 The term “Green function” as used here coincides with thestandard definition of a fundamental solution. For the modelequation (1.1), a Green function is any solution to LG(p, p′) =−4πIδ(p, p′), where I denotes an appropriate identity operator(needed for non-scalar fields) and δ(p, p′) the Dirac distribu-tion. By contrast, some authors define Green functions some-what more restrictively. See, e.g., [1].

with gab. The omitted part of L in this equation may beany first order linear differential operator. Equations sat-isfied by Klein-Gordon fields, electromagnetic vector po-tentials, and linearized perturbations of Einstein’s equa-tion all fall into this class (for certain gauge choices).

Considerable insight into (1.1) may be obtained by con-structing an associated Green function. If the points pand p′ are sufficiently close, the retarded Green functionis known to have the form2 [2, 3] (again suppressing in-dices)

Gret(p, p′) = θ(p ≥ p′)

[U(p, p′)δ(σ(p, p′))

+ V(p, p′)Θ(−σ(p, p′))]

(1.2)

in four spacetime dimensions. Here, θ(p ≥ p′) is definedto equal unity if p is in the causal future of p′, and zerootherwise. Θ and δ are the one-dimensional Heavisideand Dirac distributions, respectively. σ(p, p′) = σ(p′, p)denotes Synge’s world function: a two-point scalar equalto one-half of the squared geodesic distance between itsarguments [2–4]. The bitensors U(p, p′) and V(p, p′) aremore complicated, although explicit procedures to com-pute them are known [2, 3].

The interpretation of (1.2) is very simple. It im-plies that disturbances at a point p′ are initially prop-agated “sharply” along future-directed null geodesics

2 Distributions like θ(p ≥ p′)δ(σ) appearing here are not a priori,well-defined. They are to be interpreted as, e.g., limε→0+ θ(p ≥p′)δ(σ + ε). See Sect. 4.1 of [2].

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[where σ(·, p′) = 0] with an influence proportional toU(·, p′). At least for fields where the differential oper-ator L has no first-order component, U(·, p′) is closelyrelated to the expansion of the congruence of future-directed null geodesics emanating from p′. As mighthave been expected, the importance of the δ(σ) term inthe Green function is therefore related to the focusingof null geodesics in the sense expected from geometricoptics. Apart from this, the second line of (1.2) indi-cates that there may also be contributions to field dis-turbances – known as a “tail” – that propagate alongall future-directed timelike geodesics emanating from p′

[where σ(·, p′) < 0].This description of wave propagation cannot usually be

applied throughout an entire spacetime. The Hadamardform (1.2) for the retarded Green function is guaranteedto be valid only in convex geodesic domains [2]. Indeed,the standard definition of σ breaks down when pairs ofpoints can be connected by more than one geodesic (orby none). It is the purpose of this paper to discuss globalproperties of Green functions in curved spacetimes. Inparticular, we focus on changes in Green functions arisingfrom the presence of caustics on light cones.

Some insight into global wave propagation may begained from general theorems on the propagation of sin-gularities in wave equations (see, e.g., Corollary 5 on p.121 of [5]). Roughly speaking, these state that singulari-ties are globally propagated along null geodesics. In par-ticular, the fact that Gret(·, p′) is initially singular alongall future-directed null geodesics emanating from p′ im-plies that it remains singular as these geodesics are ex-tended arbitrarily far into the future of p′ (even thoughEq. (1.2) does not necessarily hold in the distant future).Standard propagation-of-singularities theorems do not,however, describe the “character” of the singularity ona given null geodesic. Generically, the singular structureof a Green function can exhibit qualitative changes whenpassing through light cone caustics.

It is clear from (1.2) that retarded Green functionsinitially contain a term involving δ(σ). Recent computa-tions of Green functions for linear scalar fields in Nariai[6] and Schwarzschild [7] spacetimes have demonstratedthat there is a sense in which such terms are replaced bydifferent singular distributions after each encounter witha caustic of the light cone. Following a null geodesic for-wards in time from a source point p′, there appears tobe a sense in which the singular structure of Gret(·, p′)“oscillates” in the repeating 4-fold pattern [modulo anappropriate extension of the (positive) prefactor U(p, p′)appearing in (1.2)]

δ(σ)→ pv

(1

πσ

)→ −δ(σ)→ −pv

(1

πσ

)→ . . . , (1.3)

where “pv” denotes the Cauchy principal value.Ori has heuristically argued [8] that this phenomenon

should be generic for waves propagating through “astig-matic caustics” where light rays are focused in onlyone transverse direction. Furthermore, he claims that

the effect of “stronger” anastigmatic caustics that focusin both transverse directions simultaneously should beequivalent to two passes through astigmatic caustics. Ifall caustics are anastigmatic, this implies that the singu-lar structure should follow the repeating 2-fold pattern

δ(σ)→ −δ(σ)→ . . . (1.4)

Such patterns have indeed been observed in scalar Greenfunctions for both the Einstein static universe and theBertotti-Robinson spacetime [9].

Although there is a vast literature on wave propagationthrough caustics in various contexts [10–13], almost all ofthis has been carried out in the frequency domain. It isknown, for example, that there are cases where individualFourier modes of a field experience a phase change of π/2on passing through a caustic of the type associated withthe 4-fold pattern (1.3) [11]. One might therefore expectthere to be a sense in which four passes through astig-matic caustics return a Green function to its “originalform.” Stronger anastigmatic caustics associated withthe pattern (1.4) add a phase of 2(π/2) = π to eachFourier mode. Two passes through such caustics mighttherefore be expected to return a Green function to itsoriginal form.

Despite this type of argument, the simple “position-space” patterns (1.3) and (1.4) do not appear to havebeen systematically derived before except in a few specialcases. One problem is finding an appropriately-precisestatement of the result. The usual definition of σ breaksdown once caustics arise, so even the meanings of (1.3)and (1.4) are not immediately clear. Additionally, onecan only hope that there is a sense in which patterns likethese hold “near” null geodesics. Precisely formulating anotion of this type is not necessarily trivial.

The patterns (1.3) and (1.4) are also somewhat mys-terious from an intuitive perspective. How, for example,can a “sharp” distribution like δ(σ) instantaneously jumpinto the much more “spread out” pv(1/πσ)? One mayalso question how a causal Green function can extendinto regions where σ > 0. According to the typical def-inition of Synge’s function, this would appear to implythat disturbances in fields can propagate to points thatare causally disconnected from that disturbance.

We first investigate and resolve all of these issues infour-dimensional plane wave spacetimes. Retarded andadvanced Green functions associated with massless scalarfields propagating in all smooth plane wave spacetimesare derived explicitly. These geometries might be thoughtof as modelling strong gravitational waves emitted fromsome moderately distant astrophysical system. Certaincharacteristics of plane wave spacetimes are not, however,particularly realistic. More relevant are some of theirmathematical properties:

• Appropriately adjusting the amplitude and polar-ization profile of a plane wave geometry allowsthe construction of examples with any number andcombination of astigmatic and anastigmatic caus-tics. This is accomplished in a spacetime with

3

topology R4 and a metric whose coordinate com-ponents can be made globally smooth.

• Green functions associated with masslessminimally-coupled scalar fields or Maxwell fieldsare known to have nonzero tails V(p, p′) in almostevery four-dimensional spacetime. Essentially theonly nontrivial counterexamples are plane wavespacetimes [2, 14].

• Although generic plane wave spacetimes fail to be(globally) geodesically convex, there is a naturaldefinition for σ(p, p′) that holds almost everywhere.

• The geodesic structure of plane wave spacetimes isunderstood essentially in its entirety. This allowsσ(p, p′) and U(p, p′) to be computed in closed form.

Most importantly, we choose to work in plane wavespacetimes due to the existence of a procedure known asthe Penrose limit [15–17]. This provides a sense in whichthe geometry near any given null geodesic in an arbi-trary spacetime looks like the geometry of an appropri-ate plane wave spacetime. The Penrose limit preservesvarious properties of the original spacetime [17–19]; inparticular, the conjugate point structure of the chosen(or “reference”) geodesic.

We use Penrose limits to argue that most of the “lead-ing order” singular behavior of Green functions associ-ated with wave propagation in arbitrary spacetimes maybe understood using knowledge of Green functions in ap-propriate plane wave spacetimes. This singular structurenaturally splits into two components. One portion is as-sociated with the appearance of conjugate points on thereference geodesic with which the Penrose limit is per-formed. The effects of such singularities are, in a sense,determined quasi-locally. Their structure depends onlyon the multiplicities of the conjugate points appearingon the reference geodesic. In most cases, the resultingsingular structures follow either the 4- or 2-fold patterns(1.3) or (1.4) (although there exist fine-tuned examplesthat fall into neither category).

It is important to emphasize that despite this result,Green functions are not quasi-local objects. In general, itis not possible for all of a Green function’s singular struc-ture to be determined near some null geodesic using onlyknowledge of the geometry near that geodesic. Genuinelynonlocal effects must appear generically. These may bethought of as arising from certain pairs of null geodesicsthat start at one point, separate widely, and later inter-sect at a second point. We show that there is a sensein which all singularities associated with such phenom-ena are “confined.” Their effects do not propagate alongthe reference geodesic. Using the Penrose limit, singu-larities of this type correspond to isolated structures inplane wave Green functions that occur at locations whichcannot be predicted from any given set of initial data.Plane wave spacetimes are not globally hyperbolic, sotheir Green functions are, indeed, non-unique. The lackof uniqueness associated with plane wave Green functions

is, in fact, necessary if they are to model certain char-acteristics of generic Green functions with intrinsicallynonlocal components.

This paper is organized into three main parts. Sects.II and III define the plane wave geometry and provide anextensive discussion of its geometrical properties. Whilemuch of the material in these sections has been noted be-fore in various places [17, 20–25], some appears to be new(e.g., the behavior of various geometric objects near caus-tics). Only a few key results from Sect. III are neededto understand the majority of Sect. IV, where explicitglobal Green functions are constructed for all smoothfour-dimensional plane wave spacetimes. Sect. V finallyshows that knowledge of plane wave Green functions issufficient to understand the leading order singular struc-ture of Green functions associated with scalar fields in anarbitrary spacetime. Appendix A describes various prop-erties of two matrices central to describing the geometryof plane wave spacetimes. Appendix B establishes thatthe Green function obtained for plane wave spacetimes is– despite appearances – a well-defined distribution.

Notation

In this paper, abstract indices are represented usingletters taken from the beginning of the Latin alphabet:a, b, . . . Four-dimensional coordinate indices are repre-sented using Greek characters, while indices referringonly to the spatial coordinates x1, x2 or X1, X2 intro-duced below are denoted by i, j, . . . Where appropriate,units are used in which G = c = 1. Our sign conven-tions follow those of Wald [26]: The metric signatureis chosen to be (− + ++), the Riemann tensor is de-fined such that 2∇[a∇b]ωc = Rabc

dωd for any 1-formωa, and the Ricci tensor satisfies Rab = Racb

c. Space-time points on a manifold M are generally denoted byp, p′, etc. Coordinates associated with (say) p′ are them-selves primed. We often find occasion to abuse nota-tion in various ways that should be understandable fromcontext. For example, we often identify a function ofspacetime points with the equivalent function acting oncoordinates: e.g., f(p) = f(u, v, x1, x2) in a global chart(u, v, x1, x2) : M → R4. We also make extensive useof elementary vector and matrix notation to denote thespatial coordinate components of various tensors: e.g.,xᵀ = (x1 x2), (AB)ij = AikBkj , etc. The majority ofthis paper is concerned with plane wave spacetimes. InSect. V, more general spacetimes are considered as well.Quantities associated with these geometries are often dis-tinguished by the presence of a check mark. A non-planewave metric is often denoted by gab, for example.

4

II. PLANE WAVE SPACETIMES IN GENERAL

A. pp-waves

While definitions in the literature vary slightly, aspacetime (M, gab) is typically said to be a pp-wave ifit admits a null vector field `a that everywhere satisfies∇a`b = 0 (where ∇a is the Levi-Civita connection associ-ated with the spacetime metric gab). It is clear that anysuch `a must be Killing. Its integral curves form a nullgeodesic congruence that is non-expanding, shear-free,and twist-free. These curves are orthogonal to a familyof 2-surfaces that may be interpreted as constant-phase“wave surfaces.”

A large class of pp-wave metrics in four dimensions canbe written as [20, 28–30]

ds2 = −2dudv +H(u,x)du2 + |dx|2, (2.1)

where |dx|2 denotes (dx1)2 + (dx2)2. We assume thatthe coordinates (u, v,x) = (u, v, x1, x2) may take anyvalues in R4. The unconstrained function H(u,x) =H(u, x1, x2) fixes the waveform and its polarization.Note that ∂/∂v is both null and covariantly-constant.It may therefore be identified with `a:

`a =

(∂

∂v

)a. (2.2)

Use of the metric then shows that `a = −∇au. Thevector field `a is therefore a generator for hypersurfaces ofconstant u. 2-surfaces spanned by the spatial coordinatesx1 and x2 (at fixed u, v) are the aforementioned wavesurfaces.

All non-vanishing coordinate components of the Rie-mann tensor may be obtained from

Ruiuj = −1

2∂i∂jH(u,x), (2.3)

where ∂i := ∂/∂xi and i = 1, 2. The Ricci tensor canhave at most one nonzero component:

Ruu = −1

2∇2H(u,x). (2.4)

Here, ∇2 denotes the two-dimensional Euclidean Lapla-cian acting on the coordinates (x1, x2). It follows thatany pp-wave satisfying the vacuum Einstein equationRab = 0 has a waveform H(u,x) that is harmonic inthe spatial coordinates. The sum of any two harmonicfunctions is itself harmonic, so there is a sense in whichvacuum pp-wave metrics propagating in the same direc-tion remain vacuum under linear superposition.

Note that it follows from (2.1) and (2.4) that the Ricciscalar gabRab vanishes in all pp-wave spacetimes. Moregenerally, all locally-constructed curvature scalars van-ish in these geometries. Spacetimes with this property –of which the pp-waves are a special case – are known tobe members of the Kundt class [31] (The converse is not

true: there do exist Kundt metrics with non-vanishingcurvature scalars.). There is a sense in which such ge-ometries are the gravitational analogs of “null” electro-magnetic fields Fab satisfying FabF

ab = εabcdFabFcd = 0.Some of the simplest nontrivial examples of null elec-tromagnetic fields are plane waves propagating in flatspacetime. Similarly, some of the simplest nontrivial ge-ometries with vanishing curvature scalars are plane wavespacetimes. These are a subclass of pp-wave spacetimes.

B. Plane waves

Plane wave spacetimes are special pp-waves where thecurvature components Rµνλρ depend only on the u coor-dinate. This is interpreted as implying that the “waveamplitude” does not vary over any u = const. wavefront.It follows from (2.3) that the constant-curvature condi-tion arises if H(u,x) is at most quadratic in the spatialvariables x. A coordinate transformation may be used toeliminate any components of H independent of or linearin x1 and x2. The metric of a general plane wave space-time can therefore be written in the Brinkmann form

ds2 = −2dudv +Hij(u)xixjdu2 + |dx|2. (2.5)

Here, Hij(u) is an arbitrary symmetric 2× 2 matrix thatspecifies the wave’s amplitude and polarization profile.Except in Sect. V, the metric (2.5) is assumed to holdthroughout this paper. We shall restrict to non-singularplane waves where (u, v,x) ∈ R4 and Hij(u) is a collec-tion of smooth functions on R.

It follows from (2.4) that the vacuum Einstein equationRab = 0 is satisfied if and only if Hij is trace-free: Tr H =0. The metric of any purely gravitational plane wave cantherefore be put into the form

ds2 = −2dudv +{h+(u)

[(x1)2 − (x2)2

]+ 2h×(u)x1x2

}du2 + |dx|2, (2.6)

where h+(u) and h×(u) are arbitrary functions repre-senting waveforms for the two polarization states of thegravitational wave. If these functions are proportional,the wave is said to be linearly polarized. A coordinaterotation can then be used to eliminate h×(u) in favor ofrescaling h+(u).

It is interesting to note that there is a sense in whichmetrics with the form (2.6) satisfy all generally covari-ant field equations that can be constructed purely fromthe metric and its derivatives [32]. Ricci-flat planewave spacetimes therefore provide a model for plane-symmetric gravitational radiation in general relativity aswell as many alternative theories of gravity.

In this paper, we do not restrict the discussion onlyto vacuum plane waves. One interesting class of non-vacuum plane waves are those that are conformally-flat.Such geometries must have Riemann tensors that are“pure trace.” It follows from inspection of (2.4) and (2.5)

5

that the metric of a conformally-flat plane wave can al-ways be put into the form

ds2 = −2dudv − h2(u)|x|2du2 + |dx|2 (2.7)

for some function h(u). There is at most one nonzeroRicci component in these coordinates:

Ruu = 2h2(u). (2.8)

If Einstein’s equation Rab − 12gabR = 8πTab is imposed,

the null energy condition holds for the stress-energy ten-sor Tab if only if h2(u) ≥ 0. This condition also impliesthe weak, dominant and strong energy conditions.

Conformally-flat plane wave spacetimes satisfyingh2(u) ≥ 0 may be interpreted as the gravitational fieldsassociated with plane electromagnetic waves in Einstein-Maxwell theory. In general, the stress-energy tensor ofan electromagnetic field Fab is

Tab =1

4π(FacFb

c − 1

4gabFcdF

cd). (2.9)

Inserting this into Einstein’s equation and using (2.8),it is easily observed that the plane wave geometry (2.7)may be associated with the electromagnetic plane wave

Fab = 2h(u)∇[au∇b]x1. (2.10)

This electromagnetic field is a solution to the vac-uum Maxwell equations. It also satisfies FabF

ab =εabcdF

abF cd = 0. The electric and magnetic fields seenby any observer are therefore equal in magnitude andorthogonal:

EaEa = BaB

a, EaBa = 0. (2.11)

Note, however, that (2.10) is but one possible Fab thatcould be associated with a given h(u). Other possibilitiesexist.

Although we make little use of it, it should be men-tioned that plane waves are often described in the lit-erature in terms of Rosen coordinates (U, V,X) insteadof the the Brinkmann coordinates (u, v,x) used in (2.5).The metric then takes the form

ds2 = −2dUdV +Hij(U)dXidXj . (2.12)

The 2 × 2 matrix Hij(U) appearing here depends non-algebraically and non-uniquely on Hij(u). Consider, inparticular, the transformation

u = U (2.13a)

v = V +1

2Eki(U)Ekj(U)XiXj (2.13b)

xi = Eij(U)Xj , (2.13c)

where the matrix Eij(U) is a nontrivial solution to thedifferential equation

E(U) = H(U)E(U). (2.14)

We also require that

EᵀE = (EᵀE)ᵀ. (2.15)

Applying (2.13) to the Brinkmann line element (2.5) withthese restrictions on E, one finds the Rosen line element(2.12) with

H(U) = Eᵀ(U)E(U). (2.16)

Rosen coordinates have the advantage of being moreclosely related than Brinkmann coordinates to intu-ition for gravitational waves built up from lineariz-ing Einstein’s equation about Minkowski spacetime intransverse-traceless gauge. Some properties of a planewave’s geodesics and symmetries are also more easily ex-pressed in terms of Rosen coordinates. Unfortunately,the metric (2.12) does not generally cover the entirespacetime. Rosen coordinates generically develop singu-larities that are not present in Brinkmann coordinates.There is also a considerable degree of “gauge freedom” inHij [i.e., there are many allowed solutions to (2.14) for agiven Hij ].

III. GEOMETRIC PROPERTIES OF PLANEWAVE SPACETIMES

Before discussing wave propagation in some back-ground spacetime, it is important to understand the ge-ometry of that background. This section discusses thesymmetries of plane wave spacetimes as well as theirgeodesic and causal structures. Bitensors such as Synge’sfunction, the van Vleck determinant, and the parallelpropagator are computed explicitly. Emphasis is placedon the focusing of geodesics and the asymptotic behav-ior of various bitensors near light cone caustics. Thesetopics are all important for the understanding of Greenfunctions associated with wave equations like (1.2).

A recurring object in the geometry of plane wave space-times is the matrix differential equation (2.14). This maybe viewed as a generalized oscillator equation where thewave profile H(U) acts like (the negative of) a “squaredfrequency matrix.” It is useful to describe all solutions bya linear combination of two particular solutions A(u, u′)and B(u, u′). We choose to define these matrices to besolutions of

∂2uA(u, u′) = H(u)A(u, u′) (3.1a)

∂2uB(u, u′) = H(u)B(u, u′) (3.1b)

satisfying the boundary conditions

[A] = [∂uB] = δ, [B] = [∂uA] = 0. (3.2)

Here, δ denotes the 2 × 2 identity matrix and [·] indi-cates the coincidence limit u→ u′. A(u, u′) and B(u, u′)are assumed to be matrices of functions that are smooththroughout R×R. Some of their properties are discussedin Appendix A.

6

We demonstrate below that A(u, u′) and B(u, u′) maybe used not only to describe the transformation betweenRosen and Brinkmann coordinates, but also to computethe spatial coordinate components of geodesics and Ja-cobi fields. Additionally, these matrices can be usedto identify conjugate points, explicitly compute variousbitensors, and construct Killing fields.

As an important example, consider a geodesic pass-ing through two points p and p′. It is shown in Sect.III C that these points are conjugate along the givengeodesic if and only if – abusing notation somewhat –their Brinkmann wavefront coordinates u = u(p) andu′ = u(p′) satisfy

det B(u, u′) = 0 (3.3)

and u 6= u′. The multiplicity of such a conjugate pair isequal to the nullity of the matrix B(u, u′). In the fourspacetime dimensions considered here, the multiplicitycannot exceed two. For any τ ∈ R, It is convenient todenote by Sτ the constant-u hyperplane

Sτ := {p ∈M : u(p) = τ}. (3.4)

If the pair (p, p′) are non-degenerate (multiplicity one)conjugate points, the set of all null geodesics emanat-ing from p′ and passing through Su is shown in Sect.III E to form a one-dimensional curve on that three-dimensional hyperplane. This represents astigmatic fo-cusing. Null geodesics emanating from a single point andpassing through a conjugate hyperplane with multiplic-ity two are momentarily focused back down to a singlepoint.

Note that (3.3) does not depend on any details of thegeodesic under consideration. It is purely a relation be-tween pairs of u coordinates. Geometrically, it may beinterpreted as distinguishing certain pairs (Su, Su′) of hy-perplanes. We call these “conjugate hyperplanes.” Theyplay a central role in the geometry of plane wave space-times.

It is convenient to denote the set of all solutions todet B(·, u′) = 0 by T (u′). The individual elements3 ofT (u′) are written as τn(u′) ∈ R \ {u′}:

T (u′) =⋃n

τn(u′). (3.5)

Here, the n are nonzero integers that order the elementsof T (u′) (if any). By convention, we set n > 0 if τn(u′) >

3 Conjugate points always occur discretely in plane wave space-times. In more general Lorentzian metrics, it is possible forthere to exist continuous sections of a geodesic that are con-jugate to one particular point on that geodesic [33, 34]. This is,however, only possible along spacelike geodesics. For plane wavespacetimes, the appearance of conjugate points along spacelikeand causal geodesics is governed by the same equation. T (u′) istherefore a countable set for every u′ ∈ R.

u′ and n < 0 otherwise. See Fig. 1 and further discussionin Sects. III B and III C below.

Given some preferred point p′ (or just its wavefrontcoordinate u′), the set T (u′) naturally divides the space-time into a collection of simply-connected sets Nn(u′).Define the region N0(u′) to be

N0(u′) := {p ∈M : u(p) ∈ (τ−1(u′), τ+1(u′))} (3.6)

if τ±1(u′) ∈ T (u′) both exist. If these elements do notexist, the appropriate endpoint(s) of the interval in thedefinition of N0(u′) is to be replaced by ±∞. If, say,τ+1(u′) and τ+2(u′) exist,

N1 := {p ∈M : u(p) ∈ (τ1, τ2)}. (3.7)

Similarly,

N−1 := {p ∈M : u(p) ∈ (τ−2, τ−1)} (3.8)

if τ−1(u′) and τ−2(u′) both exist. In general, Nn(u′) rep-resents the region between the nth hyperplane conjugateto Su′ and the “next one.” See Fig 1.

The region N0(u′) is a normal neighbhorhood of everypoint p′ for which u′ = u(p′). It is, however, shown inSect. III B below that all points in

N (u′) :=⋃n

Nn(u′) (3.9)

are connected to p′ by exactly one geodesic. In thissense, N (u′) is a “generalized normal neighborhood” ofp′. Note, however, that this set is not path-connectedunless T (u′) is the empty set (in which case N (u′) =N0(u′) = M). The geodesic lying in M that connectsa generic point in N (u′) with p′ need not lie entirely inN (u′). Note, however, that the set

M \ N (u′) =⋃n

Sτn(u′) (3.10)

has zero volume.

A. Symmetries

Plane wave spacetimes possess at least five linearlyindependent Killing vectors. One of these is clearly`a = (∂/∂v)a. The others have the form(

xiΞi(u)∂

∂v+ Ξi(u)

∂

∂xi

)a(3.11)

in terms of the Brinkmann coordinates (u, v,x). Here,Ξi(u) = Ξi(u) is any solution to

Ξ(u) = H(u)Ξ(u). (3.12)

This equation prescribes a total of four linearly indepen-dent Killing fields in addition to `a. Even more Killingfields may be found in certain special cases. Note that

7

(3.12) is very closely related to the modified oscillatorequation (2.14). In terms of the matrices A and B de-fined by (3.1) and (3.2), the general solution is

Ξ(u) = A(u, u′)Ξ(u′) + B(u, u′)Ξ(u′). (3.13)

The parameters u′, Ξ(u′), and Ξ(u′) appearing in thisequation may be varied arbitrarily.

Various types of non-Killing symmetries exist ingeneric plane wave spacetimes. For example, the vectorfield ζa := u`a is always a (proper) affine collineation.This means that ∇aLζgbc = 0. There is also a properhomothety ψa given by

ψa =

(2v

∂

∂v+ xi

∂

∂xi

)a. (3.14)

This satisfies Lψgab = 2gab. More extensive discussionsof the symmetries of plane wave spacetimes may be foundin [35, 36].

B. Geodesics

The geodesic structure of plane wave spacetimes is rel-atively straightforward to determine, yet still exhibits anumber of nontrivial features. First recall that the vectorfield `a = (∂/∂v)a is Killing. The quantity `az

a is there-fore conserved along any affinely-parameterized geodesicwith tangent za.

If `aza = 0 for a particular geodesic, that geodesic

remains in a single constant-u hypersurface. In theBrinkmann coordinates where (2.5) holds, the coordinatecomponents of such a geodesic satisfy

d

dszµ(s) = 0 (3.15)

for all µ = 1, . . . , 4. Any geodesic lying on a single wave-front therefore appears to be a (Euclidean) straight linein the coordinates (v,x). Furthermore, there exists ex-actly one geodesic connecting any pair of points with thesame u coordinates. The hypersurfaces Su are totallygeodesic hyperplanes.

Geodesics are more complicated when `aza 6= 0. In

these cases, the affine parameter of a geodesic may al-ways be rescaled such that `az

a = −1. Choosing theorigin of this parameter appropriately then allows it tobe identified with the coordinate u. Doing this, the spa-tial components zi(s) = zi(u) of a geodesic are easilyshown to satisfy

z(u) = H(u)z(u). (3.16)

This equation is identical to (3.12) and very similar to(2.14). As in (3.13), any possible z(u) can be written interms of the matrices A and B introduced above:

z(u) = A(u, u′)z(u′) + B(u, u′)z(u′). (3.17)

Τ2Hu ' LΤ1Hu ' LΤ-1Hu ' L u '

x1

u

N0Hu ' L N1Hu ' L

FIG. 1. This is a schematic illustration of geodesic focus-ing in a plane wave spacetime. Three conjugate hyperplanesassociated with the hypersurface Su′ are indicated togetherwith the projection of three geodesics onto the x1-u coordi-nate plane. The focusing here is assumed to act (at least)in the x1-direction. The regions N0(u′) and N1(u′) are alsoillustrated.

The v component of any geodesic is most easily foundusing the conservation law associated with the homothetyψa given by (3.14). In general, affine collineations – ofwhich homotheties (and Killing fields) are special cases– are associated with conserved quantities of the form

zaψa − 1

2szazbLψgab (3.18)

for any affinely-parameterized geodesic with tangentza(s) [37]. Using this and again identifying s with udemonstrates that

zv(u) = v′ + ε(u− u′)

+1

2

[z(u) · z(u)− z(u′) · z(u′)

], (3.19)

where the constant ε is given by

ε = −1

2zµ(u)zµ(u). (3.20)

All geodesics with initial coordinates (u′, v′, z′) and ini-tial spatial velocity z′ that are not confined to the hy-perplane Su′ have spatial coordinates which evolve via(3.17). The evolution of their v coordinates is easilyfound by combining (3.17) and (3.19).

Given any two distinct points on a particular geodesicwhere B−1(u, u′) exists, (3.17) may be used to solve forthe spatial velocity in terms of the starting and endingpoints x = z(u) and x′ = z(u′):

z′ = B−1(x−Ax′), (3.21a)

z = ∂uAx′ + ∂uBB−1(x−Ax′). (3.21b)

8

Using (3.19), the v coordinate of such a geodesic is givenby

zv(u) = v′ + ε(u− u′) +1

2

[xᵀ∂uAx′

+ (xᵀ∂uB− x′ᵀ)B−1(x−Ax′)]. (3.22)

The constant ε is unconstrained. It follows that twopoints p and p′ are connected by exactly one geodesicwhenever det B(u, u′) 6= 0. It was mentioned above thatpairs of points are also connected by exactly one geodesicwhen u = u′. Recalling (3.9) and the surrounding discus-sion, there therefore exists exactly one geodesic connect-ing any point p ∈ N (u′) to any point p′ with wavefrontcoordinate u′ = u(p′).

In all other cases, p and p′ lie on conjugate hyper-planes. The matrix rank of B(u, u′) is then strictly lessthan two. If a particular pair of hyperplanes is fixedtogether with spatial coordinates x′ on one of them, itfollows from (3.17) that the space of all possible x thatcan be reached by geodesics with initial spatial coordi-nates x′ has a dimension less than two. This implies thatalmost all points on conjugate hyperplanes are geodesi-cally disconnected (although they are always connectedby continuous non-geodesic curves).

Suppose, however, that two points lying on conjugatehyperplanes are known to be connected by one particulargeodesic. If the initial data for this geodesic is modifiedby adding to its initial spatial velocity any nontrivial vec-tor in the null space of B, it follows from (3.17) that thespatial endpoints of this new geodesic will be the same asthose of the original geodesic. Spacetime points that lieon conjugate hyperplanes and are connected by at leastone geodesic are therefore connected by an infinite num-ber of geodesics.

To summarize, all distinct pairs of points that do notlie on conjugate hyperplanes are connected by exactlyone geodesic. Almost all points lying on conjugate hy-perplanes fail to be connected by any geodesics. The re-mainder are connected by an infinite number of geodesics.More discussion may be found in Sect. III E below.

C. Conjugate points and Jacobi propagators

In general, plane waves satisfying standard energy con-ditions focus geodesics. It is evident from the above dis-cussion that, as claimed, pairs of points satisfying (3.3)must be conjugate along any geodesic connecting them.We now show that all conjugate points are of this form.Furthermore, we establish that the matrices A and Bcoincide with the spatial components of Jacobi propaga-tors Aaa′ and Baa′ (when the associated geodesic sat-isfies `az

a 6= 0 and the affine parameter is identifiedwith u). The full Jacobi propagators are known to beuseful for computing “generalized Killing fields” whichhave found application in understanding the motion ofextended matter distributions [38–43]. Although they

will not be used later, we provide explicit forms for thefull Jacobi propagators in terms of their spatial compo-nents A and B.

By definition, two points p and p′ lying on a particulargeodesic are said to be conjugate if there exists a nontriv-ial Jacobi field on that geodesic which vanishes at bothp and p′ (see, e.g., [26]). Consider an affinely parameter-ized geodesic z(s) as above. By definition, Jacobi fieldsξa(s) on this curve satisfy

D2ξa

ds2−Rbcdaξbzczd = 0, (3.23)

where D/ds denotes a covariant derivative in the direc-tion za. The Jacobi equation is linear, so its generalsolution has the form

ξa(s) = Aab′(s, s′)ξb

′(s′) +Bab′(s, s

′)D

ds′ξb′(s′), (3.24)

for some bitensors Aab′(s, s′) and Bab′(s, s

′) that dependonly on the spacetime metric and the chosen geodesic.These are called Jacobi propagators. They are solutionsto

0 =D2

ds2Aaa′ −RbcdaAba′ zczd, (3.25a)

=D2

ds2Baa′ −RbcdaBba′ zczd, (3.25b)

with the boundary conditions[DAaa′

ds

]= [Baa′ ] = 0, (3.26a)

[Aaa′ ] =

[DBaa′

ds

]= δaa′ . (3.26b)

If s and s′ are not too widely separated, Aaa′(s, s′) and

Baa′(s, s′) may be written explicitly in terms of Synge’s

function σ(p, p′) and its first two derivatives [38]. Ourmain interest lies in geometric properties of plane wavesoutside of the normal neighborhood where the traditionalderivation of such formulae breaks down. We thereforeconsign ourselves for now to somewhat less explicit com-ments on the Jacobi propagators that hold globally. Itis shown in Sect. III D below that Synge’s function may,in fact, be usefully extended beyond the normal neigh-borhood. It is, however, more convenient to express σ interms of the Jacobi propagators rather than writing theJacobi propagators in terms of σ.

First note that for geodesics confined to a singleconstant-u hypersurface, the Jacobi propagators are sim-ply

Aµµ′ = δµµ′ , Bµµ′ = (s− s′)δµµ′ (3.27)

in the Brinkmann coordinates where (2.5) holds. It istherefore clear that there can be no conjugate pointsalong any geodesic satisfying `az

a = 0.Now consider a geodesic where `az

a 6= 0, and againidentify its affine parameter with u. It is apparent from

9

inspection of (3.23) that za(u) and (u−u′)za(u) are bothJacobi fields. This means that

Aaa′(u, u′)za

′(u′) = za(u), (3.28a)

Baa′(u, u′)za

′(u′) = (u− u′)za(u). (3.28b)

Also note that the restriction of any affine collineation(such as a Killing vector) to a particular geodesic is a Ja-cobi field on that geodesic. This means, for example, thatthe covariantly-constant null vector `a associated with ageneric plane wave spacetime can be used to generateJacobi fields. So can (u− u′)`a. Hence,

Aaa′(u, u′)`a

′(z(u′)) = `a(z(u)), (3.29a)

Baa′(u, u′)`a

′(z(u′)) = (u− u′)`a(z(u)). (3.29b)

Furthermore, application of (2.3) and (3.25) showsthat

0 = ∂2u(`aA

aa′) = ∂2

u(`aBaa′), (3.30a)

= ∂2u(zaA

aa′) = ∂2

u(zaBaa′). (3.30b)

Using the coincidence limits (3.26) of Aaa′ and Baa′ , theappropriate solutions to these differential equations areseen to be

`aAaa′(u, u

′) = `a′ , (3.31a)

`aBaa′(u, u

′) = (u− u′)`a′ , (3.31b)

zaAaa′(u, u

′) = za′ , (3.31c)

zaBaa′(u, u

′) = (u− u′)za′ . (3.31d)

Also note that the components Aii′ and Bii′ inBrinkmann coordinates coincide with the matrices A andB defined by (3.1) and (3.2) above.

If two points z(u) and z(u′) (with u 6= u′) are conjugateon a particular geodesic, there must exist nonzero vectorsλa′

at z(u′) such that

Baa′(u, u′)λa

′= 0. (3.32)

Contracting this with `a and za while using (3.31) showsthat

`a′λa′ = za′λ

a′ = 0. (3.33)

Applying (3.31) again then shows that (3.32) can alwaysbe replaced by the weaker condition

Biµ′(u, u′)λµ

′= 0. (3.34)

Using (3.29) and (3.33) further demonstrates that dis-tinct points z(u) and z(u′) in plane wave spacetimes areconjugate if and only if u and u′ satisfy (3.3). As claimedat the beginning of Sect. III, all conjugate points may beidentified by computing the zeros of det B.

For completeness, we now write down all coordinatecomponents of the Jacobi propagators using the eigen-vector equations (3.28), (3.29), and (3.31). Applying therelations involving `a,

Aµv′ = δµv , Bµv′ = (u− u′)δµv , (3.35a)

Auµ′ = δu′

µ′ , Buµ′ = (u− u′)δu′

µ′ . (3.35b)

Using (3.19),

Aiu′ =(z−Az′

)i, (3.36a)

Avi′ =(zᵀA− z′

)i′, (3.36b)

Avu′ =1

2

[(xᵀHx− x′ᵀH′x′)

+ |z′|2 + |z|2 − 2zᵀAz′], (3.36c)

and

Biu′ =[(u− u′)z−Bz′

]i, (3.37a)

Bvi′ =[zᵀB− (u− u′)z′

]i′, (3.37b)

Bvu′ =1

2(u− u′)

[(xᵀHx− x′ᵀH′x′)

+ |z′|2 + |z|2 − 2zᵀBz′]. (3.37c)

Although the Jacobi propagators are defined along aparticular geodesic, they are easily reinterpreted as biten-sors on spacetime for all pairs of points p and p′ that donot lie on conjugate hyperplanes. This may be done ex-plicitly by using (3.21) to replace z and z′ with x, x′, A,and B. It is then straightforward to build vector fields onspacetime equal to Jacobi fields along all geodesics ema-nating from some preferred origin. Fixing that origin, theresulting fields form a 20-dimensional vector space. Theycan be interpreted as “generalized affine collineations”associated with the chosen origin [40]. A certain ten-dimensional subset generalize the Killing fields (and in-clude any real Killing fields that may exist).

D. Bitensors

Two-point tensors (or bitensors) are useful for, amongother things, the evaluation of Green functions in curvedspacetimes. Foremost among these is Synge’s world func-tion σ(p, p′) = σ(p′, p), which is equal to one-half of thesquared geodesic distance between its arguments. This istypically defined only in those regions where both pointscan be connected by exactly one geodesic. More gen-erally, it is possible to define a closely related two-pointscalar σz(s, s

′) associated with a particular geodesic z(s):

σz(s, s′) :=

1

2(s′ − s)

∫ s′

s

gab(z(t))za(t)zb(t)dt. (3.38)

If the geodesic in this equation is the only geodesic con-necting two points z(s) and z(s′), the ordinary worldfunction is related via

σ(z(s), z(s′)) = σz(s, s′). (3.39)

Sect. III B has established that distinct points in planewave spacetimes are connected by exactly one geodesicas long as they do not lie on conjugate hyperplanes. Eq.(3.39) may therefore be used to define Synge’s functionfor all pairs of points that do not lie on conjugate hyper-planes.

10

The definition (3.38) for σz is straightforward to evalu-ate explicitly in the general plane wave metric (2.5) alongany geodesic satisfying `az

a = −1 (with, once again, sidentified with u). Integrating by parts and using (3.16)gives

σz(u, u′) =

1

2(u− u′) (z · z− 2zv)|uu′ . (3.40)

Removing the z appearing here using (3.21) and substi-tuting the result into (3.39),

σ(p, p′) =1

2(u− u′)

[− 2(v − v′) + xᵀ∂uAx′

+ (xᵀ∂uB− x′ᵀ)B−1(x−Ax′)]. (3.41)

∂uA may be eliminated from this equation using (A2)and the symmetry of ∂uBB−1 established in AppendixA:

σ(p, p′) =1

2(u− u′)

[− 2(v − v′) + xᵀ∂uBB−1x

+ x′ᵀB−1Ax′ − 2x′ᵀB−1x]. (3.42)

Both of these relations are valid as long as u /∈ T (u′)and u 6= u′. If u = u′, Synge’s function reduces to theEuclidean expression

σ(p, p′) =1

2|x− x′|2. (3.43)

Note that unless x and x′ are chosen in a very particularway, σ diverges as u → τn(u′) ∈ T (u′). This is a man-ifestation of the aforementioned fact that most pairs ofpoints on conjugate hyperplanes cannot be connected byany geodesics.

The (scalarized) van Vleck determinant ∆(p, p′) is of-ten defined4 terms of σ(p, p′) via

∆(p, p′) = −det[−∇µ∇µ′σ(p, p′)]√−g(p)

√−g(p′)

, (3.44)

where g(p) := det gµν(p). This is a biscalar with coinci-dence limit [∆] = 1. It is closely related to the expansionof the congruence of null geodesics emanating from p′ [3].Inserting (2.5) and (3.42) into (3.44) shows that

∆(p, p′) = det(∂i∂i′σ) =(u− u′)2

det B(u, u′)(3.45)

4 It is also common to define the van Vleck determinant as thesolution to a certain “transport equation” along the geodesicconnecting its arguments [2, 3]. Unfortunately, ∆(p, p′) be-comes unbounded near conjugate hyperplanes. It is not clearhow to unambiguously extend the solution of a differential equa-tion through these singularities, so we adopt the definition (3.44)instead.

wherever σ is defined. It is evident from (3.3) that ∆(·, p′)is unbounded near any element of T (u′). We have as-sumed that B is everywhere finite, so ∆(·, p′) can neverpass through zero. It may switch signs, however. It isshown in Sect. III E that this occurs only when passingthrough conjugate hyperplanes with multiplicity 1.

The parallel propagator gaa′(p, p′) is another impor-

tant bitensor. Although this is not required to constructthe scalar Green functions discussed in most of this pa-per, it does appear in Green functions associated withelectromagnetic fields and metric perturbations [3]. Wetherefore include it for completeness. gaa′(p, p

′) satisfies

D

dsgaa′(z(s), z(s

′)) = 0 (3.46)

along any geodesic z(s) connecting its two arguments.It also has the coincidence limit [gaa′ ] = δaa′ . As thename implies, gaa′(p, p

′) parallel transports vectors fromp′ to p (or covectors from p to p′) when there is a uniquegeodesic connecting these points.

If p and p′ are connected by a single geodesic withtangent za(u), it is clear that `a and za are both left-and right-eigenvectors of gaa′(p, p

′):

`agaa′ = `a′ , gaa′`

a′ = `a, (3.47a)

zagaa′ = za′ , gaa′ z

a′ = za. (3.47b)

Applying the first of these equations demonstrates that

guµ′ = δu′

µ′ , gµv′ = δµv . (3.48)

A direct calculation using (3.46) also shows that

gii′ = δii′ . (3.49)

The remaining components of the parallel propagator are

giu′ = (z− z′)i, gvi′ = (z− z′)i′ , (3.50a)

gvu′ =1

2

[(xᵀHx− x′ᵀH′x′) + |z− z′|2

]. (3.50b)

Applying (3.21) removes any reference to geodesic veloc-ities in these equations:

giu′ = gvi =[(∂uB− δ)B−1(x−Ax′)

+ ∂uAx′]i, (3.51a)

gvu′ =1

2

[(xᵀHx− x′ᵀH′x′) + |giu′ |2

]. (3.51b)

E. Effects of caustics

Essentially all difficulties related to deriving Greenfunctions in plane wave spacetimes arise from the poorbehavior of various bitensors near conjugate hyper-planes. This is, in turn, a consequence of geodesic non-uniqueness in these regions. We now discuss the struc-ture of geodesics and the asymptotic forms of σ(p, p′) and∆(p, p′) near conjugate hyperplanes. Both of these topicsdepend on knowledge of the spatial Jacobi propagatorsA(u, u′) and B(u, u′) near conjugate hyperplanes.

11

1. A and B near conjugate hyperplanes

It is simplest to discuss the behavior of the spatial Ja-cobi propagators near conjugate hyperplanes associatedwith degenerate (multiplicity 2) conjugate points. Con-sider a particular τn(u′) ∈ T (u′) where

Bn(u′) := B(τn(u′), u′) = 0. (3.52)

It follows from (A2) that An(u′) := A(τn(u′), u′) and

∂uBn(u′) := ∂uB(u, u′)|u=τn(u′) are both invertible (even

though Bn is not). Furthermore,

∂uBn = (A−1n )ᵀ 6= 0. (3.53)

Using these expressions to expand B(u, u′) when u isnear τn,

B(u, u′) ∼ (u− τn)(A−1n )ᵀ. (3.54)

The “∼” symbol is used here to denote a relation thatholds asymptotically as u → τn. To leading order, thedeterminant of B(u, u′) in this limit is

det B(u, u′) ∼ (u− τn)2

det An

. (3.55)

Its inverse is clearly

B−1(u, u′) ∼ (u− τn)−1Aᵀn. (3.56)

Now consider a different τn ∈ T (u′) associated with a

non-degenerate (multiplicity 1) conjugate point. Bn isthen nonzero and has matrix rank 1. Using the matrixdeterminant lemma [44] together with (A2), the Jacobipropagators at (τn(u′), u′) are easily seen to satisfy

det An det ∂uBn = 1 + Tr (∂uAᵀ

nBn). (3.57)

We shall only consider cases where

Tr (∂uAᵀ

nBn) 6= −1. (3.58)

This is a technical condition which we use to ensure thatAn and ∂uBn are both invertible. There do exist planewaves where (3.58) is violated, but these examples mustbe very finely-tuned.

Applying the matrix determinant lemma gives an ap-proximation for the determinant of B(u, u′) when u isnear τn. To lowest nontrivial order,

det B(u, u′) ∼ (u− τn)Tr [Bn(∂uBn)−1]

× det ∂uBn. (3.59)

The inverse of B(u, u′) in the limit u → τn follows fromthe Sherman-Morrison formula [44]:

B−1(u, u′) ∼ (u− τn)−1(∂uBn)−1

×(δ − Bn(∂uBn)−1

Tr [Bn(∂uBn)−1]

). (3.60)

Assumption (3.58) implies that ∂uBn has matrix rank

2. Bn has rank 1 for the non-degenerate case consid-

ered here, so Bn(∂uBn)−1 must also have rank 1. It

is shown in Appendix A that Bn(∂uBn)−1 is also sym-metric. As a result, it possesses one non-zero eigenvalueand a non-vanishing trace. The matrix in parentheses in(3.60) is therefore a two-dimensional projection operator.It is symmetric with eigenvalues 0 and 1. There thereforeexists a unit vector qn satisfying(

δ − Bn(∂uBn)−1

Tr [Bn(∂uBn)−1]

)qn = qn. (3.61)

This is unique up to a sign. In terms of qn,

B−1(u, u′) ∼ (u− τn)−1(∂uBn)−1(qn ⊗ qn). (3.62)

Eq. (3.61) may be used to see that qn is in the (left-)

null space of Bn:

qᵀnBn = 0. (3.63)

Using (A2), the u-derivative of B on a non-degenerateconjugate hyperplane is given by

∂uBn = (A−1n )ᵀ(δ + ∂uA

ᵀ

nBn). (3.64)

This may be substituted into (3.59) and (3.60) in orderto provide explicit approximations for the inverse anddeterminant of B(u, u′) near a non-degenerate conjugate

hyperplane in terms of An, ∂uAn, and Bn. Eqs. (3.55)and (3.56) serve the same purpose for degenerate conju-gate points.

B may be interpreted as a “focusing matrix.” Neardegenerate conjugate points, (3.56) implies that B−1 di-verges in both spatial directions. In the case of a non-degenerate conjugate point, (3.62) illustrates how B−1

diverges in only “one direction.” This distinction isclosely related to the behavior of geodesics near conju-gate points with different multiplicities.

2. Geodesics on conjugate hyperplanes

It is shown in Sect. III B that pairs of points onconjugate hyperplanes are connected either by an infi-nite number of geodesics or by none. This is a conse-quence of the fact that all geodesics – timelike, space-like, or null – emanating from a single point and reach-ing a conjugate hyperplane are focused down to a one-or two-dimensional region on that three-dimensional sur-face. The null geodesics are focused to either a point ora line. We now demonstrate that the latter case occurson non-degenerate conjugate hyperplanes, and is oftenreferred to as astigmatic focusing. Scenarios where allnull geodesics are momentarily brought to a single pointoccur only on degenerate conjugate hyperplanes. This isanastigmatic focusing.

12

Consider a point p′ and a hyperplane Sτn(u′) that isconjugate to Su′ . Suppose that the conjugate points asso-ciated with this pair are degenerate, so Bn = B(τn, u

′) =0. It then follows from (3.17) that all geodesics (of anytype) emanating from a particular point p′ with spatialcoordinates x′ are focused to

x = An(u′)x′ (3.65)

as they pass through Sτn(u′). Use of (3.17), (3.19), and(3.53) then shows that the v coordinates of all geodesicsare given by

v′ + ε(τn − u′) +1

2x′ᵀAᵀ

n∂uAnx′ (3.66)

on Sτn . The only free parameter here is ε, which is de-fined by (3.20). The set of all geodesics emanating fromp′ are therefore focused down to a line on Sτn . Nullgeodesics (where ε = 0) are focused down to a point. In-deed, almost the entire null cone of a point p′ is focusedto a point on every degenerate hyperplane conjugate toSu′ . The lone exception is the null geodesic generated by`a. This lies entirely in Su′ .

The situation is slightly more complicated for non-degenerate conjugate points. In these cases, Bn can bewritten as the outer product of two non-zero vectors:

Bn = pn ⊗ mn. (3.67)

There is no loss of generality in supposing that |pn|2 = 1.Substitution of (3.67) into (3.17) shows that the spatialcomponents of all geodesics starting at x′ lie on the line

x = Anx′ + tpn (3.68)

as they pass through Sτn . The parameter t = mn · z′ ap-pearing in this equation can be any real number. Com-bining (3.67) with (3.63) shows that

pn · qn = 0, (3.69)

where qn is defined by (3.61). In this sense, a plane wavemay be thought of as focusing geodesics in the directionqn.

Eqs. (3.61) and (3.69) imply that pn is an eigenvector

of Bn(∂uBn)−1. In particular,

Bᵀnpn = Tr [Bn(∂uBn)−1](∂uBn)ᵀpn. (3.70)

Using this together with (3.17), (3.19), (3.68), and (A2),the v coordinates of geodesics starting at a single pointand passing through a non-degenerate conjugate hyper-plane Sτn satisfy

v′ + ε(τn − u′) +1

2x′ᵀAᵀ

n∂uAnx′ + tpᵀn∂uAnx′

+1

2t2Tr [Bn(∂uBn)−1]−1. (3.71)

There are two free parameters here: ε and t. For nullgeodesics, ε vanishes. The intersection of Sτn with thelight cone of a point p′ is therefore a one-dimensionalcurve. In particular, it is a parabola in the coordinates(v,x).

3. Bitensors near conjugate hyperplanes

The bitensors discussed in Sect. III D are not definedif their arguments lie on conjugate hyperplanes. Despitethis, expansions for B(u, u′) obtained above may be usedto understand how σ(p, p′) and ∆(p, p′) behave near con-jugate hyperplanes.

First consider σ(p, p′) if the u coordinate of p is close(but not equal) to some τn(u′) ∈ T (u′) associated withdegenerate conjugate points. In this region, the un-bounded matrix B−1(u, u′) almost always dominates in(3.41). Using that equation together with (3.53) and(3.56),

σ(p, p′) ∼ −1

2

(τn − u′

τn − u

)|x− Anx′|2. (3.72)

As before, An := A(τn, u′). It follows that σ diverges as

one approaches a degenerate conjugate hyperplane ex-cept if the approach is at the special spatial coordinatesx = Anx′. Recalling (3.65), these are the coordinates towhich all geodesics emanating from p′ are focused to onSτn .

The behavior of the van Vleck determinant near adegenerate conjugate hyperplane is easily found using(3.45) and (3.55):

∆(p, p′) ∼(τn − u′

τn − u

)2

det An. (3.73)

det An cannot vanish, so ∆ always diverges like (τn−u)−2

as u → τn. Also note that the van Vleck determinantretains its sign before and after the singularity.

Similar equations may be derived if τn is associatedwith a non-degenerate conjugate point. First note thatin this case, (A2), (3.63), and the symmetry of B(∂uB)−1

imply that

(qn ⊗ qn)An = (qn ⊗ qn)[(∂uBn)−1]ᵀ, (3.74)

where qn is defined by (3.61). Using this identity to-gether with (3.41) and (3.62) establishes that

σ(p, p′) ∼ −1

2

(τn − u′

τn − u

)[qn · (x− Anx′)]2. (3.75)

It is clear from this equation and (3.69) that σ divergesas u → τn unless x satisfies (3.68). This is analogous towhat occured in the case of degenerate point: σ divergesas one attempts to approach pairs of points that are notconnected by any geodesics.

The behavior of the van Vleck determinant near a non-degenerate conjugate point is easily determined using(3.59) and (3.45):

∆(p, p′) ∼ − (τn − u)−1(τn − u′)2

Tr[Bn(∂uBn)−1] det(∂uBn). (3.76)

Note that this diverges more slowly as u→ τn than in thecase of degenerate conjugate hyperplanes (as expected

13

due to the weaker focusing). It is also evident that ∆switches sign after passing through a non-degenerate con-jugate hyperplane.

F. Causal structure

Plane wave spacetimes are not globally hyperbolic [45].This is easily confirmed by considering two points p andp′ that are conjugate along some causal geodesic withinitial spatial velocity z′. As argued in Sect. III B, suchpoints are connected by an infinite number of geodesics.Indeed, they are connected by an infinite number ofcausal geodesics. p and p′ are conjugate on all of them.

This may be seen by considering a causal geodesic con-necting two points p and p′. Suppose that u(p) = τn(u′).If the affine parameter of the connecting geodesic is iden-tified with u, consider a new geodesic (with the sameaffine parameter) where the initial data is shifted suchthat

z′ → z′ + tλ, (3.77)

ε→ ε+t

2

(z′ · λ− z · (∂uBnλ)

τn − u′

). (3.78)

Here, λ is any vector satisfying Bnλ = 0 and t ∈ R. εdenotes the constant defined by (3.20). It is easily veri-fied that the resulting geodesic still passes through p andp′. Indeed, varying t produces a 1-parameter family ofgeodesics passing through these points. It is clear that tmay be increased without bound in at least one directionwhile retaining the causal nature of the geodesics (im-plied by ε ≥ 0). There therefore exist causal geodesicsconnecting p and p′ with arbitrarily large initial veloci-ties.

Recall that B(u, u′) has maximal rank when u /∈ T (u′).It then follows from (3.17) and the unboundedness of|z′| that there exist causal geodesics connecting p and p′

that reach arbitrarily large values of |x| between thesepoints. If p is in the future of p′, (causal past of p) ∩(causal future of p′) is therefore unbounded. Global hy-perbolicity requires that all such sets be compact, soplane waves with conjugate points cannot be globally hy-perbolic.

One consequence of this is that causally-connectedpoints can fail to be connected by any causal geodesics.Certain pairs of points connected by spacelike geodesics(and not by any other types of geodesic) may also be con-nected by accelerated curves that are everywhere causal.Avez and Seifert have shown that this cannot happen inglobally hyperbolic spacetimes [46, 47].

To see that this does indeed occur in plane wave space-times, consider two points p and p′ that do not lie onconjugate hyperplanes. Suppose that u > u′, and thatthere exists exactly one τn(u′) ∈ T (u′) that lies betweenu and u′. The discussion in Sect. III B implies that p andp′ are connected by a unique geodesic. That geodesic is

spacelike whenever

σ(p, p′) > 0. (3.79)

Choose a third point p′′, where u′′ 6= τn lies betweenu and u′. Consider a curve constructed by stitching to-gether the unique geodesic connecting p′ to p′′ with theunique geodesic connecting p′′ to p. We now show thatit is possible to choose p′′ such that, despite (3.79), bothof these geodesics are causal:

σ(p′, p′′) ≤ 0, σ(p′′, p) ≤ 0. (3.80)

Suppose for definiteness that u′′ = τn − ε for someε > 0. If the x′′ are not spatial coordinates to whichgeodesics starting at x′ must focus to as u → τn, theexpansions (3.72) and (3.75) show that σ(p′, p′′) can bemade arbitrarily negative by choosing ε to be sufficientlysmall. Essentially any geodesic from p′ to p′′ can there-fore be made timelike by placing u′′ sufficiently close to(but less than) τn. One then needs to choose v′′ and x′′

such that σ(p′′, p) ≤ 0. v′′ is entirely free, while x′′ is onlyconstrained not to equal (3.65) or (3.68) (with x→ x′′).These parameters can always be adjusted to ensure thatthe geodesic from p′′ to p is causal. It follows that all pairsof points separated by exactly one conjugate hyperplaneare causally connected. This is true despite that somesuch pairs are not connected by any causal geodesics.

This argument may be extended to points separated bymultiple conjugate hyperplanes using curves constructedby stitching together increasing numbers of geodesic seg-ments. The result is the same: Any two points separatedby at least one conjugate hyperplane are in causal con-tact. More discussion of causality in plane wave – andmore generally, pp-wave – spacetimes may be found in[48].

G. Examples

We now illustrate the concepts just discussed by con-sidering three examples of plane wave spacetimes.

1. Symmetric electromagnetic plane wave

The simplest nontrivial plane wave spacetime is asymmetric conformally-flat geometry whose associatedstress-energy tensor satisfies the weak energy condition.Following the discussion in Sect. II B, the profile H(u) ofsuch a wave is given by H(u) = −h2δ for some constanth > 0. Rescaling the u and v coordinates, there is no lossof generality in setting h = 1:

H = −δ. (3.81)

Hence,

ds2 = −2dudv − |x|2du2 + |dx|2. (3.82)

14

Recalling (2.10), this metric may be interpreted as thegeometry associated with the electromagnetic field

Fab = 2∇[au∇b]x1. (3.83)

A timelike geodesic observer at the spatial origin x = 0and with the unit 4-velocity

za =1√2

(∂

∂u+

∂

∂v

)a(3.84)

would view Fab as being composed of constant electricand magnetic fields in the x1-x2 plane:

Ea := Fabzb = − 1√

2(dx1)a, (3.85a)

Ba := −1

2εabcdzbFcd = − 1√

2

(∂

∂x2

)a. (3.85b)

Regardless of interpretation, it follows from (3.1) and(3.2) that the spatial components of the Jacobi propaga-tors are

A(u, u′) = δ cos(u− u′), (3.86a)

B(u, u′) = δ sin(u− u′). (3.86b)

It is evident from (3.3) that the conjugate hyperplanesare equally spaced and occur at the u coordinates

τn(u′) = u′ + nπ, (3.87)

where n is any nonzero integer. In these spacetimes,there are an infinite number of conjugate points alongany (inextendible) geodesic satisfying `az

a 6= 0. All ofthese conjugate points have multiplicity 2. Regardless ofinitial velocity, all geodesics with initial spatial coordi-nates z(u′) on Su′ have spatial coordinates (−1)nz(u′)on Sτn(u′).

The van Vleck determinant is easily computed using(3.45) and (3.86):

∆(p, p′) =

[(u− u′)

sin(u− u′)

]2

. (3.88)

As expected from the discussion in Sect. III E, ∆(p, p′) ispositive everywhere it is defined and diverges like (τn −u)−2 if u→ τn(u′).

For reference, Synge’s function may be computed using(3.41) and (3.86):

σ =1

2(u− u′)

[− 2(v − v′) + cot(u− u′)

×(|x|2 + |x′|2 − 2x · x′ sec(u− u′)

) ]. (3.89)

In terms of the Rosen coordinates (U, V,X) discussedat the end of Sect. II B, the metric of a homogeneousconformally-flat plane wave may be written in the form(2.12) with, e.g.,

H(U) = Aᵀ(U,U ′)A(U,U ′) = δ cos2(U − U ′). (3.90)

Here, U ′ is interpreted as an arbitrary parameter. It isevident that (no matter the choice of U ′), there exist val-ues of U where H(U) = 0. The Rosen metric is singularat these points even though the Brinkmann metric (3.82)is everywhere well-defined.

2. Symmetric gravitational plane wave

The simplest example of a plane wave admitting non-degenerate conjugate points is the linearly polarized sym-metric gravitational wave described by

H(u) =

(1 00 −1

). (3.91)

It is easily verified that

A(u, u′) =

(cosh(u− u′) 0

0 cos(u− u′)

), (3.92)

and

B(u, u′) =

(sinh(u− u′) 0

0 sin(u− u′)

). (3.93)

The conjugate hyperplanes are again given by (3.87). Un-like in the previous example, however, the associated con-jugate points all have multiplicity 1: Focusing occurs onlyin the x2-direction.

The van Vleck determinant for this spacetime is

∆(p, p′) =(u− u′)2

sin(u− u′) sinh(u− u′). (3.94)

Following the general trends derived in Sect. III E, thisdiverges like (τn − u)−1 if u → τn(u′); a slower growththan occurs in the degenerate example (3.88). It is alsoclear that ∆ switches sign on passing through each con-jugate hyperplane.

A plane wave spacetime with H(u) given by (3.91) maybe written in Rosen coordinates (2.12) using, e.g.,

H(U) =

(cosh2(U − U ′) 0

0 cos2(U − U ′)

). (3.95)

Once again, the Rosen coordinates become singular whilethe Brinkmann coordinates do not.

3. A more realistic example

Although very simple, neither (3.90) nor (3.95) lookvery much like “ordinary” oscillating waves. Considerinstead a linearly polarized vacuum plane wave with theprofile

H(u) =h

2

(1 00 −1

)cosu. (3.96)

15

Here, 0 < h < 1 is a constant. In this case, A(u, u′) andB(u, u′) are appropriate linear combinations of Mathieufunctions.

Properties of these functions are not particularly intu-itive, so it is instructive to consider perturbative solutionswhen h� 1. One such solution of (2.14) is

E(U) = δ − h

2

(1 00 −1

)cosU +O(h2). (3.97)

Substituting this into (2.12) and (2.16), the metric maybe written in Rosen coordinates as

ds2 = −2dUdV + |dX|2

− h[(dX1)2 − (dX2)2] cosU +O(h2). (3.98)

This is the form of a polarized monochromatic gravita-tional plane wave as one would expect from linearizedgeneral relativity in transverse-traceless gauge.

Continuing to assume that h is small, the spatial Jacobipropagators are approximately given by

A(u, u′) = δ − h

2

(1 00 −1

)×[(cosu− cosu′) + (u− u′) sinu′

]+O(h2). (3.99)

and

B(u, u′) = (u− u′)δ + h

(1 00 −1

)[(sinu− sinu′)

− 1

2(u− u′)(cosu+ cosu′)

]+O(h2). (3.100)

Hence,

det B(u, u′) = (u− u′)2 +O(h2) (3.101)

and

∆(p, p′) = 1 +O(h2). (3.102)

It follows that there are no conjugate points in this ap-proximation.

Exact solutions for A and B in terms of Mathieu func-tions display much more interesting behavior. Conjugatepoints occur generically. Indeed, det B(u, u′) is an ap-proximately sinusoidal function of u:

det B(u, u′) ≈ (const.)× [1− cos ν(h)(u− u′)]. (3.103)

This heuristic approximation rapidly improves as h →0. See Fig. 2 for a case where it starts to break down(h = 2/3). The period of the oscillations in det B isdetermined by the Mathieu characteristic exponent ν(h),and is always greater than 2π. It is roughly given by

2π

ν(h)≈ 2.82π

h(3.104)

if h is not too large (the relative error in this estimatefor the period is approximately 10% if h = 2/3). As

2 Π 4 Π 6 Π 8 Π

u

5

10

15

det BHu, 0L

FIG. 2. det B(u, 0) = u2/∆ for a plane wave spacetime withH(u) given by (3.96) and h = 2/3. The zeroes correspondto locations of conjugate hyperplanes. They occur in closely-spaced pairs separated by approximately 3.8π.

illustrated in Fig. 2, conjugate points generically (butnot universally) occur in closely-spaced pairs separatedby roughly 2π/ν(h). Such points have multiplicity 1. It ispossible for there to exist conjugate points of multiplicity2 – which do not occur in pairs – although this requiresfinely-tuned values of h.

IV. GREEN FUNCTIONS IN PLANE WAVESPACETIMES

Consider a massless scalar field Φ propagating (with-out gravitational backreaction) in a plane wave space-time. Allowing for a scalar charge density ρ, such a fieldsatisfies the wave equation

−4πρ = gab∇a∇bΦ= [−2∂u∂v −Hij(u)xixj∂2

v +∇2]Φ. (4.1)

Significant insight into the solutions of this equation maybe obtained by computing an associated Green functionG(p, p′). Green functions are defined here to be any dis-tributional solutions to the wave equation with (zeroth-order) sources localized to a single spacetime point:

∇a∇aG(p, p′) = −4πδ(p, p′). (4.2)

There are, of course, many solutions to this equation.Any one of the them may be used to obtain some solution

Φρ(p) :=

∫ρ(p′)G(p, p′)dV ′ (4.3)

to (4.1). More general solutions can be built by addingto Φρ an appropriate homogeneous solution Φ0 satisfying∇a∇aΦ0 = 0. Alternatively, appropriate Green functionstogether with initial data may be used to convert thewave equation into a Kirchhoff-type integral equation (atleast in small regions) [2, 3].

16

If p and p′ are sufficiently close, one particular solutionto (4.2) is the “retarded5 solution” (1.2) [2, 3]. The biten-sors U(p, p′) and V(p, p′) appearing in that formula areknown for the four-dimensional plane wave spacetimesconsidered here [2, 14]. In such cases, the “tail” V(p, p′)vanishes and the “direct” portion is determined by thevan Vleck determinant described in Sect. III D:

U(p, p′) =√

∆(p, p′). (4.4)

It follows that

Gret(p, p′) = θ(p ≥ p′)

√∆(p, p′)δ

(σ(p, p′)

). (4.5)

The advanced solution Gadv(p, p′) is the same with theobvious replacement θ(p ≥ p′) → θ(p′ ≥ p). We mainlyfocus on the “symmetric Green function”

GS(p, p′) :=1

2[Gret(p, p

′) +Gadv(p, p′)], (4.6)

from which the advanced and retarded solutions are eas-ily extracted. If p and p′ are sufficiently close, it is clearfrom (4.5) that

GS(p, p′) =1

2

√∆(p, p′)δ

(σ(p, p′)

). (4.7)

At least for short distances, (4.5) implies that distur-bances in Φ are propagated only on – and not inside – thelight cones of those disturbances. Signals from sourcesthat turn on and off sharply are themselves sharp. This“Huygens’ principle” is a very special property of mass-less scalar fields in four-dimensional plane wave space-times. In almost all other cases, retarded Green func-tions have support inside the light cone [i.e., V(p, p′) 6= 0in (1.2)] [2, 14]. Even for massless scalar fields in planewave spacetimes, Huygens’ principle is not necessarilyvalid globally. It is shown below that the appropriateextension of (4.5) fails to be everywhere sharp if thereexist non-degenerate conjugate hyperplanes.

It is evident from the discussion in Sect. III E thatthe form (4.7) for the symmetric Green function becomesproblematic if p and p′ are too widely separated. Ifa plane wave spacetime admits conjugate hyperplanes,there exist pairs of points for which the bitensors σ(p, p′)and ∆(p, p′) appearing in that formula fail to be defined.One can therefore expect (4.5) to be valid only for p ina neighborhood of p′ that does not intersect any hyper-planes conjugate to Su′ . The largest such neigborhood isthe set N0(u′) defined by (3.6). Eqs. (4.5) and (4.7) areindeed valid solutions to (4.2) throughout

{p, p′ ∈M : p ∈ N0(u(p′))}. (4.8)

5 Notions of “advanced” and “retarded” used here are quasi-local.They refer only to the causal properties of a solution when itsarguments are sufficiently close together. No claims are maderegarding the behavior of, e.g., Gret at infinity [where expressionslike (1.2) are not valid].

Our strategy for constructing a global Green functionGS(·, p′) first demands that (4.7) hold throughout the“zeroth normal neighborhood” N0(u′). Sect. IV A thenderives similar formulae in all of the remaining Nn(u′)[where σ(·, p′) and ∆(·, p′) remain well-defined]. The re-sult involves two free parameters for each n, and is avalid solution to (4.2) throughout the generalized normalneighborhood N (u′). Sect. IV B demonstrates how toextend this solution through the conjugate hyperplanesthat separate the disjoint components Nn(u′) of N (u′).Enforcing the wave equation on conjugate hyperplanesrelates the various free parameters to each other in a sim-ple way. This fixes the singularity structure of GS(·, p′)along almost all null geodesics passing through p′.

It is important to emphasize that this construction pro-duces only one of many possible Green functions. Weessentially state that a solution to (4.2) is known in someregion, and extend this using “initial data” on the bound-ary of that region. Here, the relevant boundaries are hy-persurfaces of constant u. Even in flat spacetime, initialdata imposed in this way does not yield a unique solutionto a wave equation. Unlike in flat spacetime, however,plane wave geometries do not admit appropriate Cauchysurfaces that can be used instead. As explained in Sect.III F, plane waves are not globally hyperbolic. That theGreen function we construct fails to be unique is actuallyshown to be a benefit in Sect. V, where the leading ordersingularity structure of Green functions in generic space-times is shown to be determined by plane wave Greenfunctions.

A. Green functions in the generalized normalneighborhood

Outside of the region (4.8), GS(p, p′) must be a solutionto the homogeneous wave equation

∇a∇aGS(p, p′) = 0. (4.9)

There are, of course, many solutions to this equation. Al-though the Hadamard form (4.7) breaks down outside of(4.8), one might still consider a “Hadamard-like” ansatz

GS(p, p′) =1

2

√|∆(p, p′)|gn

(σ(p, p′)

)(4.10)

for all p ∈ Nn(u′) and p′ ∈ M . Here, gn(σ) is some as-yet undetermined distribution (for n 6= 0). Recall thatthe bitensors σ(p, p′) and ∆(p, p′) appearing in (4.10) arewell-defined in the region where that equation is valid.Also note that, as shown in Sect. III D, ∆(p, p′) is fi-nite and nonzero everywhere it is defined. This biscalarmay be negative, however, which necessitates the abso-lute value appearing in (4.10).

Substituting (4.10) into (4.9) yields the ordinary dif-ferential equation

σd2gndσ2

+ 2dgndσ

= 0. (4.11)

17

The general distributional solution of this is

gn(σ) = αnδ(σ) + pv (βn/σ) + γn, (4.12)

where αn, βn, and γn are arbitrary constants and “pv”denotes the Cauchy principal value. The term involvingγn is not interesting, so we discard it at this point6.

It follows that for any p ∈ Nn(u′) (with n possiblyvanishing),

GS(p, p′) =1

2

√|∆(p, p′)|

[αnδ

(σ(p, p′)

)+ pv

(βn/σ(p, p′)

)]. (4.13)

Comparison with (4.7) shows that

α0 = 1, β0 = 0. (4.14)

Eq. (4.13) provides a class of possible forms forGS(p, p′) for all p in the generalized normal neighborhoodN (u′). The coefficients αn, βn are undetermined at thispoint (for n 6= 0), which reflects the fact that the waveequation (4.2) has been not been solved everywhere. Inparticular, it has not been solved on the boundaries ∂Nnseparating the disconnected components of N . Demand-ing that the wave equation be solved everywhere pro-vides algebraic matching conditions that relate (αn, βn)to (αn+1, βn+1) or (αn−1, βn−1). Using (4.14) as “ini-tial data,” these matching conditions provide a uniqueprescription for these parameters everywhere.

B. Green functions on conjugate hyperplanes

Demanding that the wave equation (4.2) be satisfiedon a boundary ∂Nn first requires defining what couldpossibly be meant by GS(p, p′) in such regions. Roughlyspeaking, one would like to define objects that behavelike, e.g., √

|∆|δ(σ)Θ(u− τn), (4.15a)

pv

(√|∆|σ

)Θ(u− τn). (4.15b)

∆(p, p′) and σ(p, p′) are both ill-behaved as u→ τn, so itis not obvious that there is any way to define distributionsof this type. Nevertheless, appropriate distributions maybe guessed that are well-defined everywhere and “looklike” (4.15) for all u away from conjugate hyperplanes(where the meaning of those expressions is unambigu-ous).

6 The term involving γn adds to the Green function somethingwhich depends only on u and u′. All distributions in these vari-ables are solutions to the homogeneous equation (4.9), and maytherefore be freely added or removed from a particular Greenfunction.

To be more precise, we must now treat Green func-tions “properly” as distributions; i.e., they are linearfunctionals acting on an appropriate space of test func-tions7. Specifically, GS(p, p′) takes as input a sourcepoint p′ ∈ M as well as a test function ϕ(p) : M → Rthat is in the space C∞0 (M) of smooth scalar functionswith compact support:

GS : C∞0 (M)×M → R. (4.16)

Given any test function ϕ ∈ C∞0 (M), the action of theGreen function at p′ is denoted by

〈GS(p, p′), ϕ(p)〉. (4.17)

It is also common to write this as∫GS(p, p′)ϕ(p)dV, (4.18)

which is the notation we have already been using.Now, differential equations like (4.2) are really a type

of shorthand notation. For every ϕ ∈ C∞0 (M) and everyp′ ∈M ,

〈GS(p, p′),∇a∇aϕ(p)〉 = −4πϕ(p′). (4.19)

Arguments given above already imply that this equationis satisfied if the support of ϕ lies entirely in N (u′).

We now proceed by providing an ansatz for〈GS(p, p′), ϕ(p)〉 that applies for test functions with sup-ports that do not lie entirely in N (u′). By linearity,it suffices to consider test functions ϕn (with n 6= 0)whose supports intersect at most one conjugate hyper-plane; specifically Sτn(u′). Denote by Tn(u′) a connectedopen neighborhood of the hyperplane Sτn(u′) that doesnot intersect any other conjugate hyperplanes (or, fortechnical reasons, Su′). One could choose, for example,

T2 = N1 ∪N2 ∪ Sτ2 (4.20)

if τ2 exists. Regardless, use the notation ϕn to denotetest functions with the property

ϕn ∈ C∞0(Tn(u′)

). (4.21)

The action of GS(·, p′) on a general test function may beobtained by summing its action on various ϕn and on atest function with support only in N .

We now introduce two new functionals8 G]n±(p, p′) and

G[n±(p, p′) that act on arbitrary ϕn satisfying (4.21).

7 Not every linear functional on an appropriate space of test func-tions is a distribution. There must additionally be a certain sensein which the functional is continuous with respect to sequencesof test functions. Equivalently, it must be possible to boundthe action of a distribution on an arbitrary test function usingcertain semi-norm estimates. See, e.g., Appendix B or [49].

8 The ] notation on G]n±

was chosen to remind the reader thatthis distribution is related to the “sharp” propagation of signalsassociated with δ-functions. The “1/σ-like” behavior of G[

n±is,

by comparison, rather “flat.”

18

These are required to have the forms

G]n± →√|∆|δ(σ)Θ

(± (u− τn)

), (4.22a)

G[n± → pv

(√|∆|σ

)Θ(± (u− τn)

). (4.22b)

if the support of ϕn does not intersect Sτn . For moregeneral ϕn, we provide the explicit definitions

〈G]n± , ϕn〉 := ± limε→0+

∫ ±∞τn±ε

du

∫R2

d2x

×

( √|∆|

|u− u′|

)ϕn(u, v′ + χ,x), (4.23)

and

〈G[n± , ϕn〉 := ∓ limε→0+

∫ ±∞τn±ε

du

∫R2

d2x

∫ ∞0

dΣ

× Σ−1

(√|∆|

u− u′

)[ϕn(u, v′ + χ+ Σ,x)

− ϕn(u, v′ + χ− Σ,x)]. (4.24)

Here, the function χ(u, u′; x,x′) is chosen to be the valueof v − v′ which ensures that p is connected to p′ via anull geodesic:

σ(u, v′ + χ(u, u′; x,x′),x;u′, v′,x′

)= 0. (4.25)

Referring to (3.42), χ(u, u′; x,x′) is given by

χ =1

2

[xᵀ∂uBB−1x + x′ᵀB−1Ax′

− 2x′ᵀB−1x]

(4.26)

in terms of the matrices A(u, u′) and B(u, u′) defined by(3.1) and (3.2). It is shown in Appendix B that (4.23)and (4.24) are well-defined distributions: All integralsconverge and appropriate semi-norm estimates may bederived.

Given (4.13) and (4.22), it is now natural to proposethe ansatz

〈GS, ϕn〉 =1

2

(αn−1〈G]n− , ϕn〉+ αn〈G]n+ , ϕn〉

+ βn−1〈G[n− , ϕn〉+ βn〈G[n+ , ϕn〉)

(4.27)

if n > 0. The same expression holds with the replace-ments

(αn−1, βn−1, αn, βn)→ (αn, βn, αn+1, βn+1) (4.28)

if n < 0. It is clear from (4.13) and (4.22) thatGS satisfies the wave equation (4.19) if ϕn has nosupport on Sτn(u′). For more general test functions,〈GS,∇a∇aϕn〉 6= 0 unless the αn and βn are related in

a particular way. We now compute 〈G]n± ,∇a∇aϕn〉 and

〈G[n± ,∇a∇aϕn〉.

Letting

ϕ := ϕn(u, v′ + χ(u, u′; x,x′),x), (4.29)

ϕ′ := ∂vϕn(u, v′ + χ(u, u′; x,x′),x), (4.30)

note that√|∆|

|u− u′|∇a∇aϕn(u, v′ + χ,x) = −2∂u

( √|∆|

|u− u′|ϕ′

)

+∇2

( √|∆|

|u− u′|ϕ

)− 2∂i

( √|∆|

|u− u′|ϕ′∂iχ

). (4.31)

This is easily verified using direct computation togetherwith (3.1), (3.45), (4.26), (A4) and the symmetry of thematrices ∂uAA−1 and ∂uBB−1 established in AppendixA. Substitution into (4.23) shows that

〈G]n± ,∇a∇aϕn〉 = ±2 lim

u→τ±n

∫R2

d2x

( √|∆|

|u− u′|

)× ∂vϕn(u, v′ + χ(u, u′; x,x′),x). (4.32)

Using (4.24), the 1/σ-type portions of the Green func-tion are seen to produce the following errors in the waveequation:

〈G[n± ,∇a∇aϕn〉 = ∓2 lim

u→τ±n

∫R2

d2x

∫ ∞0

dΣ

× Σ−1

( √|∆|

|u− u′|

)[∂vϕn(u, v′ + χ+ Σ,x)

− ∂vϕn(u, v′ + χ− Σ,x)]. (4.33)

Evaluating 〈GS,∇a∇aϕn〉 requires simplifying these twoexpressions and then applying (4.27). The result dependson the multiplicity of the conjugate points associatedwith τn, and requires expansions for the Jacobi propa-gators and bitensors derived in Sect. III E.

1. Degenerate conjugate points

First consider the case where the conjugate hyper-planes associated with some τn(u′) ∈ T (u′) are related todegenerate conjugate points. Applying (3.42) and (3.72)to (4.26) then shows that for all p ∈ Tn(u′),

χ(u, u′; x,x′) = −1

2

|x− Anx′|2

τn − u+ χn(u, u′; x,x′). (4.34)

Here, An(u′) = A(τn(u′), u′) and χn(u, u′; x,x′) is afunction that is well-behaved in all of its arguments.

Substituting (4.34) into (4.32) and using (3.73) to-gether with the variable transformation

x→ x :=x− Anx′√|τn − u|

(4.35)

19

yields

〈G]n± ,∇a∇aϕn〉 = ±2

√|det An|

∫R2

d2x

∂vϕn(τn, v′ ± 1

2|x|2 + χn, Anx′). (4.36)

Transforming into polar coordinates in the usual way, theintegral on the right-hand side of this equation may beevaluated explicitly:

〈G]n± ,∇a∇aϕn〉 = −4π

√|det An|

× ϕn(τn, v′ + χn(τn, u

′; Anx′,x′), Anx′). (4.37)

The discussion in Sect. III E may be used to show thatthe argument of ϕn appearing here is the point to whichall null geodesics starting at p′ focus to on Sτn(u′).

Using similar arguments together with (4.33), the waveoperator acting on the 1/σ part of the Green functionproduces

〈G[n± ,∇a∇aϕn〉 = 4π

√|det An|

∫ ∞0

dΣ

× Σ−1[ϕn(τn, v

′ + χn + Σ, Anx′)

− ϕn(τn, v′ + χn − Σ, Anx′)

]. (4.38)

This depends on ϕn at all points on Sτn(u′) that are con-nected to p′ by geodesics of any type. It is not propor-

tional to 〈G]n± ,∇a∇aϕn〉 as given by (4.37).

Substituting (4.27), (4.37), and (4.38) into (4.19)shows that the wave equation can be satisfied on a de-generate conjugate hyperplane Sτn(u′) if and only if

αn = −αn−1, βn = −βn−1 (4.39)

when n > 0 or

αn = −αn+1, βn = −βn+1 (4.40)

when n < 0. If these relations are satisfied for a par-ticular n, ∇a∇aGS(p, p′) = 0 throughout Tn(u′). Theyimply that Green functions tend to switch sign on passingthrough degenerate conjugate hyperplanes.

2. Simple conjugate points

The non-degenerate case is treated similarly. Choose aparticular τn(u′) ∈ T (u′) associated with non-degenerateconjugate points. Eq. (3.75) then implies that if u issufficiently close to τn(u′), χ(u, u′; x,x′) has the form

χ(u, u′; x,x′) = −1

2

[qn · (x− Anx′)]2

τn − u+ χn(u, u′; x,x′), (4.41)

where χn(u, u′; x,x′) is smooth. Recall that the unit vec-tor qn is associated with τn(u′) as a solution to the eigen-vector problem (3.61).

The form of χ near a simple conjugate hyperplane sug-gests the coordinate transformation x→ x, where

x1 :=qn · (x− Anx′)√

|τn − u|, (4.42a)

x2 := pn · (x− Anx′). (4.42b)

Here, pn is a unit vector satisfying pn · qn = 0. Thesigns of pn and qn are to be chosen such that the orderedpair of coordinates (x1, x2) have the same orientation as(x1, x2). Using (3.76) and (4.32),

〈G]n± ,∇a∇aϕn〉 =

±2√|Tr[Bn(∂uBn)−1] det(∂uBn)|

×∫R2

d2x∂vϕn(τn, v′ ± 1

2(x1)2 + χn, Anx′ + pnx

2).

(4.43)

Similar simplifications of (4.33) result in

〈G[n± ,∇a∇aϕn〉 =

2π√|Tr[Bn(∂uBn)−1] det(∂uBn)|

×∫R2

d2x∂vϕ(τn, v′ ∓ 1

2(x1)2 + χn, Anx′ + pnx

2).

(4.44)

Substituting these two equations into (4.27), one seesthat GS(p, p′) satisfies the wave equation throughoutTn(u′) when

αn = −πβn−1, βn =αn−1

π, (4.45)

if n > 0 or

αn = πβn+1, βn = −αn+1

π(4.46)

if n < 0. Unlike in the degenerate case, these relationsshow that the qualitative character of plane wave Greenfunctions changes on passing through simple conjugatehyperplanes. It switches from having a δ(σ)-type singu-larity to a 1/σ-type singularity (or vice-versa).

C. Global solutions

We now have a recipe for constructing global Greenfunctions associated with the scalar wave equation (4.1).In particular, GS(p, p′) has the form (4.27), where the dis-

tributions G]n±(p, p′) and G[n±(p, p′) are defined by (4.23)and (4.24). Retarded and advanced Green functionsGret(p, p

′) and Gadv(p, p′) are extracted from GS(p, p′) inthe obvious way. All coefficients αn and βn appearing in

front of G]n±(p, p′) and G[n±(p, p′) in (4.27) are determinedonly by the geometry and the choice of source point p′.α0 and β0 are given by (4.14). These effectively serve asinitial conditions for the matching equations (4.39)-(4.40)

20

and (4.45)-(4.46) that determine αn and βn when n 6= 0.As mentioned above, Gret(p, p

′) and Gadv(p, p′) are notthe only Green functions that “quasi-locally” appear tobe “retarded” or “advanced.” This lack of uniqueness isdiscussed more fully at the end of this section.

Restricting to their action on test functions with sup-ports that do not intersect any conjugate hyperplanes,GS(p, p′), Gret(p, p

′), and Gadv(p, p′) may be summarizedusing only a few simple rules. Recalling the definition ofthe “nth normal neighborhood” Nn(u′) provided in Sect.III, the generic form of GS(p, p′) when acting on testfunctions with support only in Nn(u′) is given by (4.10).Similarly, the retarded Green function may be written as

Gret(p, p′) = Θ(u− u′)

√|∆|gn(σ) (4.47)

when acting on test functions with support entirely con-tained in Nn(u′). The distributions gn(σ) appearing inthese equations are what we refer to as a Green function’s“singular structure.” They are given by either ±δ(σ)or ±pv(1/πσ). Which of these possibilities is appropri-ate depends only on the pattern of multiplicities of theconjugate hyperplanes lying between p and p′. ConsiderGS(·, p′), Gret(·, p′), or Gadv(·, p′) along some curve em-anating from p′. If this curve intersects a hyperplaneSτn(u′) conjugate to Su′ , the singular structure changesaccording to the rules:

• When the conjugate pair (Sτn(u′), Su′) has multi-plicity 1,

± δ(σ)→ ±pv

(1

πσ

), (4.48)

and

± pv

(1

πσ

)→ ∓δ(σ). (4.49)

if traversing Sτn(u′) in a direction of increasing u.Signs on the right-hand sides of both of these re-placement rules are reversed if traversing Sτn(u′) ina direction of decreasing u.

• When the conjugate pair (Sτn(u′), Su′) has multi-plicity 2, the singular structure switches sign:

±δ(σ)→ ∓δ(σ), (4.50)

±pv

(1

πσ

)→ ∓pv

(1

πσ

). (4.51)

This is equivalent to the effect of two passesthrough conjugate hyperplanes with multiplicity 1(where these passes either both increase u or bothdecrease u).

The simplest nontrivial application of these rules oc-curs in plane waves where all conjugate points are de-generate. In these cases, one finds from (4.39) and (4.40)

that αn = (−1)n and βn = 0. The symmetric Greenfunction is therefore given by

GS(p, p′) =1

2(−1)n

√|∆|δ(σ) (4.52)

when p ∈ Nn(u′). This implies that the retarded Greenfunction is

Gret(p, p′) = Θ(u− u′)(−1)n

√|∆|δ(σ) (4.53)

in the same region. Throughout the generalized normalneighborhood N (u′), the supports of GS(·, p′), Gret(·, p′),and Gadv(·, p′) are confined to the σ(·, p′) = 0 light conesof p′. The singular structure of these Green functionsdoes, however, change sign on each pass through a con-jugate hyperplane. It follows the 2-fold pattern (1.4).

All conjugate points associated with conformally-flatplane waves are degenerate, so their Green functions arealways of the form (4.52). In the symmetric case wherethe wave amplitude h(u) in (2.7) remains constant, it isshown in Sect. III G that there exist an infinite numberof degenerate conjugate hyperplanes (for any u′) at loca-tions given by (3.87). Using (3.88), the retarded Greenfunction for such a spacetime is

Gret(p, p′) = Θ(u− u′)

[(u− u′)

sin(u− u′)

]δ(σ) (4.54)

if h = 1 and u−u′ 6= nπ (for all nonzero integers n). Eq.(3.89) provides an explicit coordinate expression for σ inthis case.

If all conjugate points in a particular plane wave space-time are non-degenerate, the scalar Green function hasthe repeating 4-fold singularity structure (1.3) ratherthan the 2-fold structure (1.4) found in the purely de-generate case. Applying (4.45) and (4.46) to (4.13) and(4.14) for some p ∈ Nn(u′),

Gret(p, p′) = Θ(u− u′)

√|∆|

×

{(−1)

n2 δ(σ) if n even,

(−1)n−12 pv(1/πσ) if n odd.

(4.55)

This is, in a sense, the generic form for retarded Greenfunctions in plane wave spacetimes. It is not correct ifthere exist degenerate conjugate hyperplanes9, althoughsuch structures tend to be “fragile.” Consider, for ex-ample, a plane wave that initially possesses a degener-ate conjugate hyperplane. If such a spacetime is per-turbed by slightly changing Hij(u), the original degener-ate hyperplane tends – but is not guaranteed – to split

9 Degenerate and non-degenerate conjugate hyperplanes may ex-ist in the same spacetime. Examples of this may be found byfine-tuning the parameter h appearing in (3.96). The singularstructure of the Green function in such cases deviates from thesimple patterns (1.3) and (1.4). Rules (4.48)-(4.51) must thenbe applied on a case-by-case basis.

21

into two closely-spaced non-degenerate conjugate hyper-planes. Passing through one non-degenerate hyperplanemight therefore be viewed as physically equivalent toquickly passing through two non-degenerate conjugatehyperplanes. Indeed, we have found that two passesthrough non-degenerate hyperplanes has the same effecton a scalar Green function as one pass through a degen-erate conjugate hyperplane.

As a simple example of the 4-fold singularity structureexhibited in (4.55), consider a symmetric gravitationalplane wave where the wave profile Hij(u) is given by(3.91). Such spacetimes have an infinite number of non-degenerate conjugate hyperplanes at the locations (3.87).The retarded Green function in this case is explicitly

Gret(p, p′) =

((u− u′)Θ(u− u′)√

| sin(u− u′)| sinh(u− u′)

)

×

{(−1)

n2 δ(σ) if n even,

(−1)n−12 pv(1/πσ) if n odd.

(4.56)

n is given by

n = b(u− u′)/πc , (4.57)

where b·c denotes the floor function. A coordinate ex-pression for the σ appearing here may be found by sub-stituting (3.92) and (3.93) into (3.41).

Before moving on, recall that two questions are posedin the introduction regarding the qualitative way inwhich a Green function may change its singular struc-ture. First, how can a distribution like δ(σ) “smoothlytransition” into something like pv(1/σ)? In plane waveGreen functions, this change occurs on u = const. hyper-planes. Furthermore, the discussion in Sect. III E impliesthat σ →∞ on almost all (but not quite all) approachesto such surfaces. This means that like δ(σ), pv(1/σ) van-ishes almost everywhere when approaching a conjugatehyperplane where it might transition into a δ-function.

It is also asked in the introduction how it is possible fora retarded Green function to involve a term proportionalto pv(1/σ) when σ(p, p′) > 0 traditionally implies thatthe points p and p′ are causally disconnected. In planewave spacetimes, σ(p, p′) > 0 implies that the (only)geodesic connecting p and p′ is spacelike. Despite this,the discussion of Sect. III F implies that such points arestill in causal contact. They may be always be connectedby non-geodesic curves that are everywhere causal.

Non-uniqueness of plane wave Green functions

Recall that we have constructed a “retarded Greenfunction” Gret(p, p

′) by demanding that it solve (4.2) ev-erywhere and that it be equal to (4.5) for all p ∈ N0(u′).Other distributions also satisfy these constraints. Onemay consider, e.g.,

Gret(p, p′) + Γ(p, p′), (4.58)

where Γ(p, p′) is some solution to ∇a∇aΓ(p, p′) = 0 thatvanishes when u < τ1(u′). Any object of this form mayreasonably be interpreted as a “retarded Green function.”

Nontrivial distributions Γ(p, p′) always exist. Con-sider, for example, anything with depends purely on uand u′ and that vanishes when u < τ1(u′). As anotherchoice, suppose that Γ(p, p′) is, for fixed p′, concentratedon a constant-u hypersurface St(u′) [where t(u′) ≥ τ1(u′)].Then, e.g.,

Γ(p, p′) = δ(t(u′)− u)γ(x,x′), (4.59)

for any γ(x,x′) that is harmonic at least in its first ar-gument: ∇2γ(x,x′) = 0. Many other possibilities alsoexist.

V. GREEN FUNCTIONS IN GENERALSPACETIMES

Up to this point, we have focused on the propagationof (test) scalar fields Φ on plane wave backgrounds. Asoutlined in the introduction, plane wave spacetimes havea number of mathematically attractive features. Theyare not, however, physically realistic on large scales.Plane wave geometries are not asymptotically flat, noreven globally hyperbolic. Despite this, one might hopethat there is a sense in which our results remain “essen-tially correct” for physically realistic plane waves wherethe metric is adequately approximated by (2.5) only insome finite region10. We now argue for a significantlystronger result: The singular behavior of Green functionsin generic spacetimes is, to leading order, equivalent tothe singular behavior of Green functions in appropriateplane wave spacetimes. This is similar to a statementproposed in [23].

The correspondence with plane wave spacetimes is mo-tivated by two observations. First, general theorems re-garding the propagation of singularities imply that thesingular supports of generic Green functions lie on nullgeodesics. Second, there is a sense in which the geome-try “near” a null geodesic in any spacetime is equivalent– via what is known as a Penrose limit [15–17] – to thegeometry of an appropriate plane wave spacetime. Fur-thermore, one might expect that the behavior of a genericGreen function near its singular support (i.e., near a nullgeodesic) could be at least partially understood usingthe geometry of that region. It would then follow that“most of” the singular structure of a Green function ina spacetime (M, gab) near an affinely-parameterized nullgeodesic z(u) might be found by computing the Green

10 A somewhat more realistic model of a simple gravitational wave isa pp-wave where the profile function H(u,x) appearing in (2.1) isquadratic in x in some finite region and subquadratic as |x| → ∞.Geometries of this type are discussed in, e.g., [27, 30]. Theircausal properties do not display the pathologies of “pure” planewave spacetimes.

22

function associated with a plane wave spacetime (M, gab)obtained from (M, gab) and z(u) using a Penrose limit.

In order to demonstrate this, we first review the def-inition of the Penrose limit. An appropriate notion ofa Green function’s “leading order singular behavior” isthen defined. Near a given null geodesic, it is argued thatthis object is reproduced by a Green function associatedwith an appropriate plane wave spacetime. The resultsof Sect. IV are then applied to determine the singularstructure of Green functions for scalar fields propagatingin arbitrary four-dimensional spacetimes.

A. Penrose limits

As formulated in [16], the Penrose limit takes as inputa null geodesic z(u) in a spacetime (M, gab), and usesthis to construct a null generalization of a Fermi normalcoordinate system11 (u, v, x). Assume for simplicity thatthe reference geodesic z(u) is defined for all u ∈ R andthat gab is smooth along this curve. Next, construct atetrad {ea±(u), eai (u)} on z(u) that is parallel-propagatedalong z(u) with respect to gab. Let the first element ofthis tetrad be the null tangent to the reference geodesic:

ea+(u) =dz(u)

du. (5.1)

Let the second element ea−(u) of the tetrad also be null,

and suppose that it satisfies gabea+e

b− = −1. The final

two elements eai (u) of the tetrad are to be spacelike andorthonormal. They are orthogonal to the two null vectorsea+(u) and ea−(u):

gabea+e

bi = gabe

a−e

bi = 0. (5.2)

Given some point p ∈ M sufficiently near the referencegeodesic z(u), identify a u coordinate associated with pby solving the equation

ea−(u)∇aσ(z(u), p) = 0. (5.3)

Here, σ(p, p′) denotes Synge’s function in the spacetime(M, gab). Once u = u(p) has been fixed using (5.3), theremaining three coordinates of p are determined by defin-ing the tetrad components of the “separation vector”−∇aσ(z(u(p)), p) to be the coordinates v(p) and x(p):

− ∇aσ = vea− + xieai . (5.4)

Inverting this relation,

v(p) := ea+(u(p))∇aσ(z(u(p)), p), (5.5a)

xi(p) := −δijeaj (u(p))∇aσ(z(u(p)), p). (5.5b)

11 A check mark is omitted on the symbol u because this coordinateis not rescaled in (5.6) below.

Together, these equations and (5.3) define a Fermi-likecoordinate system (u, v, x) near the reference geodesic(i.e., for v and x sufficiently small that Synge’s functionremains single-valued).

The Penrose limit involves a 1-parameter family oftransformations on the components gµν of the metric inthe coordinates (u, v, x). Consider, in particular, the sub-stitutions

u→ u, v → v := λ−2v, x→ x := λ−1x (5.6)

for any λ > 0. In the limit λ → 0, this transformation“zooms up” on the reference geodesic z(u). All compo-nents gµν of the metric in the coordinate system (u, v,x)vanish as λ → 0. Expanding the line element in powersof λ, the first non-vanishing term is [16]

ds2 = λ2[− 2dudv − R+i+j(u)xixjdu2 + |dx|2

]+O(λ3). (5.7)

Here, R+i+j(u) denotes the appropriate tetrad compo-nents of the Riemann tensor on the reference geodesic:

R+i+j(u) := Rabcd(z(u))ea+(u)ebi (u)ec+(u)edj (u). (5.8)

Comparing (2.5) and (5.7), it is clear that

gµν := limλ→0

λ−2gµν (5.9)

is – regardless of the original geometry – the metric ofa plane wave spacetime in Brinkmann coordinates withthe profile

Hij(u) = −R+i+j(u). (5.10)

In this sense, the geometry near any12 inextendible nullgeodesic is equivalent to that of an appropriate planewave spacetime.

For any choice of reference geodesic, the Penrose limitpreserves various properties of the original spacetime(M, gab) in the associated plane wave spacetime (M, gab)[17–19]. For example, conformally-flat spacetimes arealways mapped to conformally-flat plane waves. Vac-uum (Ricci-flat) spacetimes are mapped to vacuum planewaves. Additionally, the number of linearly independentKilling fields cannot decrease after taking a Penrose limit.

For every u ∈ R, the Penrose limit maps the pointz(u) ∈ M on the reference geodesic into a point withBrinkmann coordinates (u, 0, 0, 0) in the associated planewave spacetime. This means that the reference curve –

12 If the reference geodesic intersects a singularity and thereforecannot be extended to infinitely large values of its affine param-eter, one can still perform a Penrose limit. The only differenceis that the resulting plane wave spacetime is slightly differentfrom the type described in Sects. II and III. In such cases, thecoordinate u would no longer take all values in R and the waveprofile Hij(u) may be unbounded for finite u.

23

which is a null geodesic with respect to gab – is mappedinto a curve that is a null geodesic with respect to gab.The conjugate point structure of the reference geodesic isidentical in both the original and plane wave spacetimes.

Another important property of the Penrose limit isits effect on a geodesic in the original spacetime whichintersects the reference geodesic at, say, z(u0). It isstraightforward to show that all such curves (which arenot infinitesimal deformations of the reference geodesic)are mapped to the u = u0 hyperplane Su0 . They re-main geodesics with respect to the plane wave metric gab,and are therefore straight lines in the coordinates (v,x).These straight lines always pass through v = x = 0. Anynull or timelike geodesics (and some spacelike geodesics)which intersect z(u) are mapped to a curve which is bothnull and geodesic with respect to gab. The only suchcurve confined to Su0

has the Brinkmann coordinatesu = u0 and x = 0. It is generated by `a = (∂/∂v)a.

B. Singular structure of generic scalar Greenfunctions

We now consider a Green function G(p, p′) associatedwith the scalar wave equation

LΦ = −4πρ (5.11)

in an arbitrary spacetime (M, gab). As in (1.1), the prin-cipal part of the linear differential operator L is to begiven by gab∇a∇b. Unlike in the plane wave field equa-tion (4.1), we allow for additional terms involving at mostone derivative (so fields with, e.g., mass or non-minimalcoupling to the curvature may be considered). Any Greenfunction associated with (5.11) is required to satisfy

〈G(p, p′), L†ϕ(p)〉 = −4πϕ(p′) (5.12)

for all test functions ϕ ∈ C∞0 (M). Here, L† denotes theadjoint of L.

Penrose limits can be thought of as zooming in on aparticular null geodesic z(u). It follows that a plane waveGreen function G(p, p′) could only be expected to de-scribe the action of a generic Green function G(p, p′) nearz(u). A mathematically precise statement of this formis that we would like to consider the action of G(p, p′)on test functions ϕ(u, v,x) ∈ C∞0 (R4) that are fixed inthe scaled coordinates (u, v,x) related to the Fermi-likecoordinates (u, v, x) via (5.6). Given some ϕ, define a1-parameter family of test functions ϕλ such that

ϕλ(u, v, x) := ϕ(u, λ−2v, λ−1x) (5.13)

for all λ > 0. Test functions of this type always remainnear z(u) when λ is sufficiently small.

The action of any smooth function – call it V(p) – ona test function ϕλ of the form (5.13) is given by

〈V, ϕλ〉 =

∫dudvd2x V(u, v, x)ϕλ(u, v, x)

= λ4

∫dudvd2x V(u, λ2v, λx)ϕ(u, v,x). (5.14)

This clearly scales like λ4 as λ→ 0. One would thereforeexpect any tail terms in G(p, p′) to scale like λ4 whenacting on test functions ϕλ.

Portions of 〈G(p, p′), ϕλ(p)〉 depending on parts ofG(p, p′) which are singular on z(u) decrease more slowlythan λ4 in the Penrose limit λ → 0. Consider, for ex-ample, Synge’s function σ(p, p′) for pairs of points thatare sufficiently close that the standard definition of thisobject remains well-defined. Then the definition (3.38)and the Penrose limit metric gab given by (5.9) suggestthat

σ ∼ λ−2σ. (5.15)

Hence,

δ(σ) ∼ λ−2δ(σ), pv

(1

σ

)∼ λ−2pv

(1

σ

). (5.16)

These are the most singular terms that one would expectto find in G(p, p′). It is therefore reasonable to expectthat

〈G, ϕλ〉 ∼ λ2 (5.17)

as λ→ 0.We now use this heuristic argument as motivation to

define the “leading order singular portion” of a genericscalar Green function G(p, p′). For any p′ = z(u′) ∈ Mlying on the reference geodesic and any test function ϕλthat is, as describe above, fixed in the scaled coordinates(u, v,x), define a linear operator G(p, p′) by

〈G(p, p′), ϕ(p)〉 := limλ→0

λ−2〈G(p, p′), ϕλ(p)〉. (5.18)

The Hadamard form (4.5) and the estimates (5.16) guar-antee that this limit exists at least for test functionswhose supports lies sufficiently close to p′. We assume,however, that the limit exists for all test functions of theform (5.13).

The notation in (5.18) suggests that G(p, p′) is a Greenfunction in an appropriate plane wave spacetime. To es-tablish that this is indeed the case, consider families oftest functions generated by gab∇a∇bϕ, where gab is theinverse of the Penrose limit metric (5.9) and ∇a is the as-sociated covariant derivative operator. Substitution into(5.18) and use of (5.12) shows that

〈G, gab∇a∇bϕ〉 = limλ→0〈G, L†ϕλ〉 = −4πϕ(p′) (5.19)

for all ϕ ∈ C∞0 (R4). Comparison with (4.19) shows thatthe operator G(p, p′) is a Green function for a scalar fieldpropagating on a plane wave spacetime with metric (5.9).It follows that appropriate components of generic Greenfunctions behave like plane wave Green functions nearnull geodesics. Properties of plane wave Green functionsderived in Sect. IV may therefore be used to understandsome aspects of scalar Green functions in more generalspacetimes.

24

To be more precise, fix a background spacetime(M, gab) in which a scalar field Φ propagates accordingto (5.11). Fix a point p′ ∈ M corresponding to the lo-cation of some small disturbance in Φ. The effect ofsuch a disturbance may now be followed in a neighbor-hood of some affinely-parameterized null geodesic z(u)which passes through p′ = z(u′). Perform a Penroselimit using z(u) and the metric gab. Such a limit re-quires a choice of tetrad {ea±(u), eai (u)} along the refer-ence geodesic. This is to be constructed using the pre-scription described in Sect. V A. Defining a Fermi-like co-ordinate system (u, v, x) and performing the scaling (5.6)produces [via (5.9)] a plane wave metric in Brinkmanncoordinates with wave profile (5.10).

The (say) retarded Green function Gret(p, p′) associ-

ated with (5.12) is related to the retarded plane waveGreen function Gret(p, p

′) constructed in Sect. IV C. Forany test function ϕ ∈ C∞0 (R4), (5.13) may be rewrittenas

limλ→0

λ−2〈Gret, ϕλ〉 = 〈Gret, ϕ〉+ 〈Γ, ϕ〉. (5.20)

Here, Γ(p, p′) is an appropriate solution to the homoge-neous wave equation

〈Γ, gab∇a∇bϕ〉 = 0. (5.21)

Γ(·, p′) vanishes in the Penrose limit of any normal neigh-borhood of p′ [as computed in the original spacetime(M, gab)], but need not vanish globally. We return tothis point shortly.

For the moment, consider only the first term on theright-hand side of (5.20). It is clear from the discussion inSect. IV C that, fixing p′, Gret(·, p′) is proportional eitherto δ(σ) or pv(1/σ). It can switch between these twopossibilities and also switch sign. If there is nothing inΓ(·, p′) that is singular on curves where σ(·, p′) = 0, suchterms provide a precise sense in which generic retardedGreen functions Gret(p, p

′) have singularities that “looklike” either δ(σ) or pv(1/σ) near null geodesics [whereσ(·, p′) = 0]. It is simple to determine which of theseforms is appropriate by considering the points conjugateto p′ along z(u). These may be found by first using (5.10)to construct H(u) from Rabc

d(z(u)). H(u) can then beused to compute the matrix B(u, u′) defined by (3.1) and(3.2). A point z(τn) is conjugate to z(u′) = p′ if and onlyif det B(τn, u

′) = 0. The multiplicity of such a conjugatepair is equal to the nullity of B(τn, u

′).The (discrete set of) points conjugate to z(u′) on z(u)

generically divide the reference geodesic into a number ofregions corresponding to the Nn(u′) of Sect. III. Rules(4.48)-(4.51) may be used to find the leading order singu-lar structure of Gret(·, p′) in each of these regions usingonly the multiplicities of intervening conjugate points.

This argument takes into account only the contributionto Gret(p, p

′) from the first term on the right-hand sideof (5.20). In general, the second term on the right-handside of this equation may also be important. It is re-quired if, as described in the introduction, null geodesics

emanating from p′ later intersect z(u) at a point thatis not conjugate to p′. Such phenomena introduce newsingularities in a neighborhood of the reference geodesicwhose locations cannot be predicted using only the lim-ited geometric information preserved in the Penrose limit.Penrose limits map any null geodesic in the full space-time which intersects the reference geodesic at, say, z(u0)into a null geodesic in the plane wave spacetime confinedto the hyperplane Su0

. One might therefore expect totake into account the singularities transported by suchgeodesics using an appropriate Γ(p, p′) in (5.20) that issingular when u = u0, v ∈ R, and x = 0. It does not,however, appear clear precisely what form such solutionsshould take. We merely note that they can exist andthat their singular support extends to |v| → ∞. Suchsingularities appear quite different from those associatedwith Gret(p, p

′). They are, in a sense, “frozen” at spe-cific points on the reference geodesic (as viewed in thefull background spacetime).

Examples

We now discuss some consequences of the above re-sults. General statements are made regarding Greenfunctions associated with conformally-flat spacetimesand an important class of vacuum spacetimes. Some spe-cific examples are also mentioned briefly.

The simplest general statement following from the ar-gument of Sect. V B concerns scalar Green functionsin spacetimes whose metrics are conformally-flat. Asnoted above, all Penrose limits of conformally-flat space-times are conformally-flat plane waves. Furthermore, allconformally-flat plane waves have metrics gab with theform (2.7). It is evident from this together with (2.5)and (3.1)-(3.3) that all conjugate points in such space-times have multiplicity 2. The plane wave Green func-tion Gret(p, p

′) is therefore given by (4.53). Via (5.20),a similar structure also appears in the Green functionGret(p, p

′) associated with the full spacetime (M, gab).This provides the sense in which the 2-fold singular struc-ture (1.4) suggested by Ori [8] is present in retardedscalar Green functions associated with all conformally-flat spacetimes.

For null geodesics passing through a vacuum (Ricci-flat) region of some spacetime (M, gab), the associatedPenrose limit is a vacuum plane wave with the metric(2.6). The wave profile in such a case satisfies

Tr H(u) = δijR+i+j(z(u)) = 0. (5.22)

If

R+i+j(z(u)) = h(u)Hij (5.23)

for some constant matrix H, the resulting plane wave issaid to be linearly polarized. An appropriate rotationof the spatial components eai (u) of the tetrad used to

25

perform the Penrose limit can then be used to set

H = ±(

1 00 −1

). (5.24)

It follows that the h(u) appearing in (5.23) can be iden-tified (up to a sign) with the h+(u) in (2.6). If h(u) iseither entirely non-negative or entirely non-positive, itis clear from (3.1)-(3.3) that any conjugate points whichmay exist must have multiplicity 1. Gret(p, p

′) then hasthe form (4.55). It follows from (5.20) that in the vac-uum case where the Riemann tensor along the referencegeodesic has the form (5.23) and the h(u) appearing inthat equation does not pass through zero (but may some-times equal zero), Gret(p, p

′) contains the repeating 4-foldpattern of singular structures (1.3).

Many of the explicit computations of four-dimensionalGreen functions found in the literature fall into oneof the two classes of spacetimes just described. Inthe conformally-flat case, scalar Green functions havebeen computed in both the Einstein static universe andBertotti-Robinson spacetimes [9]. As expected from ourargument using Penrose limits, the singular structures ofretarded Green functions associated with both of thesespacetimes have been found to have a 2-fold pattern ofsingular structures with the form (1.4).

Another important example in the literature isthe retarded scalar Green function associated withSchwarzschild spacetime. All Penrose limits ofSchwarzschild13 have the form (5.23) with h(u) ≥ 0[17, 24]. We therefore predict that retarded Green func-tions in Schwarzschild possess the 4-fold singular struc-ture (1.3). This is indeed what was observed in the ex-plicit computations carried out in [7].

More generally, we may consider scalar Green func-tions associated with all Kerr spacetimes. Penrose limitsof Kerr (and all other Petrov type D spacetimes) are dis-cussed in [24]. Once again, these limits result in waveprofiles with the form (5.23). For a reference geodesicwith specific angular momentum about the z-axis lz andCarter constant q, it is shown in [24] that

h(u) =3M [(a− lz)2 + q]

[r2(u) + a2 cos2 θ(u)]5/2. (5.25)

Here, M and aM are the mass and angular momentumassociated with the Kerr background. r(u) and θ(u) arethe Boyer-Lindquist coordinates of the reference geodesicat the affine time u. It is clear that h(u) cannot changesign, so we predict that retarded scalar Green functionsin Kerr spacetime display the 4-fold singular structure(1.3).

13 Some null geodesics of Schwarzschild intersect the central sin-gularity. The associated Penrose limits are then singular planewaves. Before this point, however, all discussion above remainsvalid.

C. Tensor Green functions

Thus far, we have considered only Green functionsassociated with the propagation of scalar fields. It isstraightforward to partially extend our results to also al-low for fields with nonzero tensor rank. In particular,we now show that the leading order singular structure oftensor Green functions in arbitrary spacetimes can be un-derstood using appropriate plane wave Green functions.No attempt is made, however, to also derive the form ofthose plane wave Green functions as we have done in thescalar case.

As an example, consider a field Aa propagating on aspacetime (M, gab) and satisfying

LAa = −4πρa. (5.26)

Here L is any second-order linear differential opera-tor whose principal part is equal to the d’Alembertiangab∇a∇b. The wave equation (5.26) is naturally associ-

ated with Green functions Gaa′(p, p′) satisfying

〈Gaa′(p, p′), L†ϕa(p)〉 = −4πϕa

′(p′) (5.27)

for all smooth vector fields ϕa(p) with compact support.Now choose a point p′ ∈ M and consider the behavior

of Gaa′(·, p′) near a null geodesic z(u) passing through

p′ = z(u′). As above, we choose a tetrad and construct aFermi-like coordinate system (u, v, x) in a neighborhoodof z(u). It is also useful to consider the scaled coordinates(u, v,x) defined by (5.6).

We now seek an analog of (5.18). This requires choos-ing an appropriate family of test functions similar to(5.13). Given an arbitrary test function ϕµ(u, v,x) inthe plane wave spacetime which results from the Pen-rose limit of (M, gab) and z(u), reverse the coordinatetransformation (5.6) to obtain

ϕuλ(u, v, x) := ϕu(u, λ−2v, λ−1x) (5.28a)

ϕvλ(u, v, x) := λ2ϕv(u, λ−2v, λ−1x) (5.28b)

ϕiλ(u, v, x) := λϕi(u, λ−2v, λ−1x) (5.28c)

in the unscaled coordinates (u, v, x). Similarly, choosesome covector vµ′ which remains fixed (for all λ > 0) inthe scaled coordinates (u, v,x). Then define

〈Gµµ′, ϕµvµ′〉 := lim

λ→0λ−2〈Gµµ

′, ϕµλvµ′〉. (5.29)

Considering test functions of the form gνρ∇ν∇ρϕµ,where gµν denotes the Penrose limit metric (5.9), it isstraightforward to show using (5.27) that

〈Gµµ′, vµ′g

νρ∇ν∇ρϕµ〉 = −4πϕµ′(p′)vµ′ . (5.30)

It follows that the operator Gµµ′(p, p′) is, as the nota-

tion suggests, a Green function associated with the planewave spacetime obtained by taking a Penrose limit with(M, gab) and z(u).

26

This argument carries through essentially withoutchange for Green functions associated with all higher-rank tensor fields. We have thus established that there isa sense in which the leading order singular behavior of alltensor wave equations can be understood by consideringappropriate plane wave Green functions.

VI. DISCUSSION

This paper discusses the transport of disturbances inscalar fields propagating on curved spacetimes. In partic-ular, we study how light cone caustics affect the characterof singularities appearing in the relevant Green functions.This problem is addressed in two steps. First, explicitGreen functions are obtained for massless scalar fieldspropagating on all non-singular four-dimensional planewave spacetimes. We then show in Sect. V that Penroselimits provide a sense in which certain aspects of thesesolutions are universal: The leading order singular struc-ture of scalar Green functions associated with essentiallyall four-dimensional spacetimes can be described by ap-propriate plane wave Green functions.

The plane wave Green functions we obtain are sum-marized in Sect. IV C. They are globally defined andfully explicit [up to the calculation of the 2 × 2 matri-ces A and B defined by (3.1) and (3.2)]. Almost every-where, plane wave Green functions are found to have a“Hadamard-like” component. Using σ and ∆ to denoteSynge’s function and the van Vleck determinant (whichare well-defined almost everywhere in plane wave space-times), we find that there exists a retarded Green func-

tion that switches between the forms ±√|∆|δ(σ) and

±√|∆|pv(1/πσ).

As described in Sect. V B, there is a sense inwhich (say) retarded Green functions Gret(·, p′) satisfying(5.12) in generic spacetimes contain similar Hadamard-like terms near any future-directed null geodesic ema-nating from p′. Fixing some point p on such a nullgeodesic (that is not conjugate to p′), precisely whichHadamard form is appropriate depends only on the pat-tern of multiplicities of all points conjugate to p′ that liebetween p′ and p. Following rules (4.48)-(4.51), cross-ing a non-degenerate (multiplicity 1) conjugate point is

found to change a Green function involving√|∆|δ(σ)

into one involving√|∆|pv(1/πσ). Conversely, non-

degenerate conjugate points transform Green functionsproportional to

√|∆|pv(1/πσ) into ones proportional to

−√|∆|δ(σ). One pass through a conjugate point with

multiplicity two is seen to have the same effect as twopasses through conjugate points with multiplicity one.This merely reverses signs:

√|∆|δ(σ) → −

√|∆|δ(σ) or√

|∆|pv(1/πσ)→ −√|∆|pv(1/πσ).

In this way, we have derived and made significantlymore precise Ori’s comments [8] regarding changes in thesingularity structure of Green functions due to light conecaustics. The result is a simple universal rule that is –

unlike most results regarding caustics – naturally statedin terms of distributions on the spacetime manifold (asopposed to statements on the tangent bundle involvingFourier transforms).

It is interesting to note that the object σ appearing inthe leading order singular structure of a generic Greenfunction Gret is not the world function σ associated withthe spacetime (M, gab). σ is typically ill-defined whenits arguments are widely separated. σ is, instead, theworld function of an appropriate plane wave spacetimeobtained via a Penrose limit. This is well-defined al-most everywhere. Similar comments also apply to thevan Vleck determinant ∆, which effectively measures the“strength” of the leading order singular terms appearingin a generic retarded Green function. Explicit forms forboth σ and ∆ are easily computed in arbitrary spacetimesusing the results of Sects. III D and V A.

The rules we derive for changes in a Green function’ssingular structure have a simple heuristic interpretation.One might think of degenerate conjugate points as eventswhere bundles of light rays are perfectly focused in everydirection. Sharp solutions involving δ(σ) might there-fore be expected to remain sharp after passing througha degenerate conjugate point. Similarly, more diffuse so-lutions like pv(1/σ) might be expect to remain diffusein such an encounter. Conjugate points with multiplic-ity 1 are different. They focus null geodesics in only onetransverse direction. It is therefore reasonable to expectsharp solutions like δ(σ) to be “blurred out” by suchstructures. Somewhat less intuitive is that the nature ofthis blurring is always such that another pass througha non-degenerate conjugate point “resharpens” the fieldback into a form involving δ(σ).

An important special case of this work concerns the be-havior of retarded Green functions associated with scalarfields in the Kerr spacetime. All conjugate points ap-pearing on null geodesics of Kerr are non-degenerate.Scalar Green functions in Kerr therefore change singular-ity structure according to the 4-fold pattern (1.3). Thisresult includes as a special case the 4-fold behavior ob-served by Dolan and Ottewill [7] in Schwarzschild Greenfunctions.

The problem of wave propagation in curved spacetimeis a very general one with many applications. Our resultsmay therefore be useful in a number of fields. One pos-sible application concerns the computation of self-forces:What is the force exerted by a small charge on itself in acurved spacetime? One may assume that the total scalarfield is the retarded solution and find the force that thisexerts on a given charge distribution. In generic space-times, the result depends on the charge’s past history atleast via the tail term V(p, p′) appearing in (1.2). It has,however, been less clear precisely how light cone caus-tics in the distant past contribute to an object’s self-field(see, e.g., [6, 7, 50] for some related discussion). Moregenerally, it is important to understand “how much” of acharge’s past history influences its current self-field. Ourresults may be useful in answering this question.

27

Other applications could exist even in systems wherethe spacetime curvature is negligible. Wave equations oncurved spacetimes are mathematically equivalent to var-ious physically different problems in flat spacetime. Forexample, one might use the same equations to describethe propagation of acoustic waves in a moving fluid orelectromagnetic waves in certain classes of permeable ma-terials [51, 52]. It would be interesting to translate theresults of this paper to more readily apply to problemssuch as these. The physical meaning of the Penrose limitwould be particularly interesting to understand in someof these “analog gravity” systems.

There are two additional ways in which this work couldbe extended. Most obviously, it would be extremely use-ful to generalize our results to apply to wave equationsinvolving tensor fields with nonzero rank. The singularitystructure of disturbances in, e.g., electromagnetic fieldsand metric perturbations could then be understood in arelatively simple way. We carry out one portion of thistask in Sect. V C, where Penrose limits are used to relategeneric tensor Green functions to appropriate plane waveGreen functions. Although there does not appear to beany significant obstacle to doing so, we have not madeany attempt to compute plane wave Green functions forhigher-rank tensor fields in this paper. A complete dis-cussion of the leading order singularity structure of tensorGreen functions must therefore wait for later work.

It might also be interesting to extend our work tohigher numbers of dimensions. In the four-dimensionalcase considered here, the rules describing how Greenfunctions transition between different singular structuressuggest that passing through a conjugate point with mul-tiplicity 2 is equivalent to two passes through conjugatepoints with multiplicity 1. While it appears intuitivelyplausible that such a rule generalizes for larger multiplic-ities, it would be interesting to verify this directly. Is onepass through a conjugate point with multiplicity n ≥ 1in a spacetime with dimension d ≥ n + 2 equivalent ton passes through conjugate points with multiplicity 1?Questions related to higher dimensions are perhaps notonly of mathematical interest. Higher-dimensional planewave spacetimes and Penrose limits have found extensiveuse in string theory and related subjects [17, 53–55].

Appendix A: Properties of A(u, u′) and B(u, u′)

The 2 × 2 matrices A(u, u′) and B(u, u′) defined by(3.1) and (3.2) play a central role in the geometry ofplane wave spacetimes. This appendix briefly reviewssome of their most important properties.

First note that any two solutions C(u) and D(u) tothe modified oscillator equation (2.14) satisfy

∂u(Cᵀ∂uD− ∂uCᵀD) = 0. (A1)

This is a simple application of Abel’s identity. Usingthe boundary conditions (3.2) with the identifications

C(u)→ A(u, u′) and D(u)→ B(u, u′), it follows that

Aᵀ∂uB− ∂uAᵀB = δ. (A2)

In general, A and B are not symmetric matrices. Sim-ple combinations of them are, however, symmetric. Ap-plying (A1) with C,D → A demonstrates that Aᵀ∂uAis one such example:

Aᵀ∂uA = (Aᵀ∂uA)ᵀ. (A3)

Right-multiplying this equation with A−1 and left-multiplying with (A−1)ᵀ shows that ∂uAA−1 and itsinverse are also symmetric (wherever they exist). All ofthese comments continue to hold with the replacementA→ B.

Using these observations together with (A2) furthershows that almost everywhere,

BAᵀ = (∂uBB−1 − ∂uAA−1)−1. (A4)

The right-hand side of this equation is symmetric, soBAᵀ must be symmetric as well.

A similar argument shows that the symmetry condition(2.15) on the matrix E(U) involved in the transformationbetween Rosen and Brinkmann coordinates is automat-ically satisfied for all U if it satisfied for any U . It alsofollows that EE−1 is symmetric wherever it exists.

A and B can be computed by directly solving the dif-ferential equations (3.1) with the boundary conditions(3.2). Alternatively, both matrices may be found by in-tegrating any single E(u) satisfying (2.14) and (2.15).Use of (2.16) and (A1) with C→ E and D→ B demon-strates that

B(u, u′) =

∫ u

u′du′′E(u)H−1(u′′)Eᵀ(u′) (A5)

as long as H−1(·) is defined throughout the interval(u, u′). The same assumption also leads to a formulafor A:

A(u, u′) = δ +

∫ u

u′du′′E(u)H−1(u′′)

× [E(u′′)− E(u′)]ᵀ. (A6)

Similar equations may be derived for u and u′ arbitrarilyseparated by considering a number of different E matri-ces which are invertible in a suitable set of overlappingintervals. One interesting consequence of (A5) is that

B(u, u′) = −Bᵀ(u′, u). (A7)

Appendix B: Distributional character of G]n±

and G[n±

It is not a priori clear that the functionals G]n±(p, p′)

and G[n±(p, p′) used in the ansatz (4.27) for the scalarGreen function are well-defined. From their definitions

28

(4.23) and (4.24), these objects take as input an “obser-vation point” p′ = (u′, v′,x′) ∈ R4 and any test functionϕn(p) = ϕn(u, v,x) that is smooth and with compactsupport in the set Tn(p′) defined in Sect. IV B.

By definition, a linear form 〈G]n±(p, p′), ϕn(p)〉 is a dis-tribution if and only if, for every fixed p′ and every com-pact subset ω ⊂ Tn(u′), there exist semi-norm estimatesof the form

|〈G]n± , ϕω〉| ≤ Cω∑|α|≤Nω

sup |∂αϕω| (B1)

for all ϕω with suppϕω ⊂ ω [49]. α denotes a multi-index and |α| its order. The non-negative numbers Cωand Nω depend only on the region ω, and not on anydetails of the test function. Estimates of this type arestraightforward if ω does not pass through the hyper-plane Sτn(u′) conjugate to Su′ . More generally, one mustconsider separately the possible multiplicities the conju-gate pair (Sτn(u′), Su′).

It is simplest to derive estimates like (B1) by choosingfinite (nonzero) numbers uω, xω, and vω such that thecompact rectangular region defined by

|u− τn(u′)| < uω, |v − v′| < vω, (B2a)

|x− Anx′| < xω, (B2b)

entirely encloses ω ⊂ Tn(u′). As in Sect. III E, An(u′) :=A(τn(u′), u′

). It is also convenient to introduce the ∞-

norm ‖ · ‖ω in ω:

‖ · ‖ω := supω| · |. (B3)

1. Degenerate conjugate points

Suppose that τn(u′) is associated with conjugate pointsof multiplicity 2. Choose an arbitrary compact regionω ⊂ Tn(u′) and finite numbers uω, xω, and vω that definea rectangular region (B2) which encloses ω.

Consider the definition (4.23) for 〈G]n± , ϕω〉 whensuppϕω ⊂ ω. This consists of an integral over the co-ordinates u and x. It is clear that the integrand vanisheswhenever |x − Anx′| > xω. If, however, ϕω has sup-port sufficiently close to Sτn , a sharper bound may be

placed on the maximum value of |x − Anx′| that needsto be considered by using vω. This is because the func-tion χ(u, u′; x,x′) defined by (4.26) has the form (4.34) inthis region. It follows that the integrand vanishes when∣∣∣∣∣χn − 1

2

|x− Anx′|2

τn − u

∣∣∣∣∣ > vω. (B4)

This is implied by the stronger condition

|x− Anx′|2 > 2|τn − u| (vω + ‖χn‖ω) . (B5)

It follows that the spatial support of the integrand in(4.23) shrinks at least as fast as

√|τn − u| as u→ τn.

Without loss of generality, we may choose vω to besufficiently large that, e.g.,

vω > ‖χn‖ω. (B6)

The spatial integral in (4.23) may therefore be limited to

|x| < 2√vω, (B7)

where x is defined by (4.35). Using x as an integrationvariable, (4.23) changes to

〈G]n± , ϕω〉 := ± limε→0+

∫ τn±uω

τn±εdu

∫|x|<2

√vω

d2x(√|(τn − u)2∆||u− u′|

)ϕω(u, v′ + χ,x). (B8)

It follows from (3.73) that the integrand in this equationis everywhere bounded. Hence,

|〈G]n± , ϕω〉| ≤ 4πuωvω

∥∥∥∥∥√

(τn − u)2|∆||u− u′|

∥∥∥∥∥ω

× sup |ϕω| <∞. (B9)

Although ∆ is, in general, unbounded in ω, this argu-

ment shows that 〈G]n± , ϕω〉 is always finite. All integralsin its definition converge. Furthermore, they convergeabsolutely. The order of integration does not matter in

(4.23). This establishes that the G]n± are well-defined lin-ear functionals. They are also distributions. Eq. (B9)provides a semi-norm estimate of the form (B1) withNω = 0 and

Cω = 4πuωvω

∥∥∥∥∥√

(τn − u)2|∆||u− u′|

∥∥∥∥∥ω

. (B10)

Nω and Cω depend, as required, only on ω (and not onthe test function ϕω).

Establishing similar bounds for the functionals G[n± ismore complicated. Although it is again useful to changethe integration variable x to x in the definition (4.24),this leaves a result where neither the integrand nor theintegration volume are bounded.

The first simplification arises by using (4.34) and (B6)to deduce that the integrand in (4.24) vanishes when,e.g.,

|x| > 0, Σ >1

2|x|2 + 2vω, (B11)

or

|x| > 2√vω, Σ <

1

2|x|2 − 2vω. (B12)

Additionally, it is clear from (4.35) and (B2) that theintegrand also vanishes when

|x| > xω√|τn − u|

. (B13)

29

We shall assume that xω has been chosen to be suffi-ciently large that

xω >√

6uωvω, (B14)

so

xω√|τn − u|

>√

6vω. (B15)

The factor of 6 here is a fairly arbitrary number. Anyother choice greater than 4 could also be used.

These considerations suggest that (4.24) should besplit into two parts:

〈G[n± , ϕω〉 = ∓ limε→0+

∫ τn±uω

τn±εdu

(√(τn − u)2|∆|u− u′

)[∫|x|<√

6vω

d2x

∫ 5vω

0

dΣ

∫ 1

−1

dν ∂vϕω(u, v′ + χ+ νΣ,x)

∓∫ |x|<xω/

√|τn−u|

|x|>√

6vω

d2x

∫ 12 |x|

2+2vω

12 |x|2−2vω

dΣ

(ϕω(u, v′ + χ∓ Σ,x)

Σ

)], (B16)

Both integrands here are manifestly bounded. The inte-gral on the top line is also carried out over a finite volume.It is therefore trivial to provide a bound for it. The inte-gral on the lower line must be bounded somewhat morecarefully. The final result is that

|〈G[n± , ϕω〉| ≤ 4πvω

∥∥∥∥∥√

(u− τn)2|∆|u− u′

∥∥∥∥∥ω

× (15uωvω sup |∂vϕω|+ ζω sup |ϕω|), (B17)

where

ζω := 2

∫ uω

0

du

∫ xω/√

4uvω

√3/2

dr r ln

(r2 + 1

r2 − 1

). (B18)

Noting that

0 < r ln

(r2 + 1

r2 − 1

)< 2 (B19)

in the relevant range, a straightforward integration showsthat ζω < ∞. It follows that the linear functional〈G[n± , ϕω〉 is always finite. It is bounded by the semi-

norm estimate (B17), so G[n± is a distribution.

2. Non-degenerate conjugate points

Suppose now that τn(u′) is associated with conjugatepoints of multiplicity 1 and choose a compact region ω ⊂Tn(u′) in the same manner as in the previous case. Using(4.41), the integrand of (4.23) is easily seen to vanishwhen ∣∣∣∣∣χn − 1

2

[qn · (x− Anx′)]2

τn − u

∣∣∣∣∣ > vω. (B20)

Introducing the variables (x1, x2) defined by (4.42), thisinequality is implied by the stronger condition

(x1)2 > 2(vω + ‖χn‖ω). (B21)

Also note that the integrand of (4.23) vanishes when

|x2| > xω. (B22)

We assume that the parameters uω, vω, and xω arechosen such that (B6) holds. It then follows that (4.23)can be rewritten as

〈G]n± , ϕω〉 := ± limε→0+

∫ τn±uω

τn±εdu

∫|x1|<2

√vω

dx1

∫|x2|<xω

dx2

(√|(τn − u)∆||u− u′|

)ϕω(u, v′ + χ,x).

(B23)

It follows from (3.76) that the integrand in this equationis everywhere bounded. This means that

|〈G]n± , ϕω〉| ≤ 8uωxω√vω

∥∥∥∥∥√|(τn − u)∆||u− u′|

∥∥∥∥∥ω

× sup |ϕω| <∞. (B24)

Again, we see that all integrals in the definition of G]n±converge. Eq. (B24) provides a semi-norm estimate ofthe form (B1), so this operator is, as claimed, a distribu-tion near non-degenerate conjugate hyperplanes.

We now establish similar bounds for the functionalsG[n± . Using (4.41) and (B6) we find that the integrandin (4.24) vanishes when

|x1| > 0, Σ >1

2|x1|2 + 2vω, (B25)

30

or

|x1| > 2√vω, Σ <

1

2|x1|2 − 2vω. (B26)

This integrand also vanishes when

|x1| > xω√|τn − u|

or |x2| > xω. (B27)

We assume again that the (B14) holds, so (B15) is true.Eq. (4.24) can then be rewritten as

〈G[n± , ϕω〉 = ∓ limε→0+

∫ τn±uω

τn±εdu

(√|(τn − u)∆|u− u′

)∫|x2|<xω

dx2

[∫|x1|<

√6vω

dx1

∫ 5vω

0

dΣ

∫ 1

−1

dν ∂vϕω(u, v′ + χ+ νΣ,x)

∓∫ |x1|<xω/

√|τn−u|

|x1|>√

6vω

dx1

∫ 12 |x

1|2+2vω

12 |x1|2−2vω

dΣ

(ϕω(u, v′ + χ∓ Σ,x)

Σ

)].

(B28)

This implies the bound

|〈G[n± , ϕω〉| ≤ 8xω√vω

∥∥∥∥∥√|(u− τn)∆|u− u′

∥∥∥∥∥ω

× (5√

6uωvω sup |∂vϕω|+ Υω sup |ϕω|), (B29)

where

Υω :=

∫ uω

0

du

∫ xω/√

4uvω

√3/2

dr ln

(r2 + 1

r2 − 1

). (B30)

The integrand in this equation is bounded from aboveby 2 (and from below by 0), so Υω < ∞. It followsthat the linear functional G[n± is a distribution near non-degenerate conjugate hyperplanes.

Together, the results of this appendix establish that for

any nonzero integer n such that τn(u′) ∈ T (u′), G]n±(p, p′)

and G[n±(p, p′) are well-defined distributions throughoutTn(u′).

ACKNOWLEDGMENTS

This work was supported in part by NSF grant PHY08-54807 to the University of Chicago. The authors thankBarry Wardell, Aaryn Tonita, Marc Casals, and RobertWald for useful discussions and comments.

[1] C. Bar, N. Ginoux, and F. Pfaffle, Wave Equationson Lorentzian Manifolds and Quantization (EuropeanMathematical Society, Zurich, 2007)

[2] F. G. Friedlander, The Wave Equation on a CurvedSpace-time (Cambridge University Press, Cambridge,1975)

[3] E. Poisson, Living Rev. Relativity 7, 6 (2004)[4] J. L. Synge, Relativity: The General Theory, (North-

Holland, Amsterdam, 1960)[5] C. Bar and K. Fredenhagen, Quantum Field Theory in

Curved Spacetime (Springer-Verlag, Berlin, 2009)[6] M. Casals, S. Dolan, A. C. Ottewill, and B. Wardell,

Phys. Rev. D 79, 124043 (2009)[7] S. Dolan and A. C. Ottewill, Phys. Rev. D 84, 104002

(2011)[8] A. Ori, unpublished report

http://physics.technion.ac.il/ amos/acoustic.pdf (2009)[9] M. Casals, private communication (2012)

[10] V. I. Arnold, “Singularities of Caustics and Wave Fronts”

[11] Yu. A. Kravtsov, Yu. I. Orlov, and M. G. Edelev, Caus-tics, Catastrophes and Wave Fields (Springer-Verlag,Berlin, 1999), 2nd ed.

[12] W. Hasse, M. Kriele, and V. Perlick, Classical and Quan-tum Gravity 13, 1161 (1996)

[13] J. Ehlers, Ann. Phys. (Leipzig) 9, 307 (2000)[14] P. Gunther, Huygens’ Principle and Hyperbolic Equations

(Academic Press, London, 1988)[15] R. Penrose, p. 271 in Differential Geometry and relativity

(Reidel, Dordrecht, 1976)[16] M. Blau, D. Frank, and S. Weiss, Classical and Quantum

Gravity 23, 3993 (2006)[17] M. Blau, lecture notes

http://www.blau.itp.unibe.ch/lecturesPP.pdf (2006)[18] R. Geroch, Commun. Math. Phys. 13, 180 (1969)[19] M. Blau, J. Figueroa-O’Farrill, and G. Papadopoulos,

Classical Quantum Gravity 19, 4753 (2002)[20] J. Ehlers and W. Kundt in Gravitation: An Introduction

to Current Research, edited by L. Witten (Wiley, NewYork, 1962)

31

[21] P. E. Ehrlich and G. Emch, Rev. Math. Phys. 4, 163(1992)

[22] P. E. Ehrlich and G. Emch in Proc. Symposia in PureMathematics, Amer. Math. Soc. 54 part 2, 203 (1993)

[23] T. J. Hollowood and G. M. Shore J. High Energy Phys.12, 091 (2008)

[24] T. J. Hollowood, G. M. Shore, and R. J. Stanley J. HighEnergy Phys. 08, 089 (2009)

[25] J. K. Beem, P. E. Ehrlich, and K. L. Easley, GlobalLorentzian Geometry (Dekker, New York, 1996), 2nd ed.

[26] R. M. Wald, General Relativity (University of ChicagoPress, Chicago, 1984)

[27] A. M. Candela, J. L. Flores, and M. Sanchez, Gen. Rel-ativ. Gravit. 35, 631 (2003)

[28] J. B. Griffiths, Colliding Plane Waves in General Rela-tivity (Clarendon Press, Oxford, 1991)

[29] G. S. Hall, Symmetries and Curvature Structure in Gen-eral Relativity (World Scientific, Singapore, 2004)

[30] J. L. Flores and M. Sanchez, Classical Quantum Gravity20, 2275 (2003)

[31] V. Pravda, A. Pravdova, A. Coley, and R. Milson, Clas-sical Quantum Gravity 19, 6213 (2002)

[32] C. G. Torre, Classical Quantum Gravity 38, 653 (2006)[33] A. D. Helfer, Pacific J. Math 164, 321 (1994)[34] P. Piccione and D. V. Tausk, Commun. Anal. Geom. 11,

33 (2003)[35] R. Maartens and S. D. Maharaj, Classical Quantum

Gravity 8, 503 (1991)[36] A. J. Keane and B. O. J. Tupper, Classical Quantum

Gravity 27, 245011 (2010)[37] G. H. Katzin and J. Levine, J. Math. Phys. 22, 1878

(1981)

[38] W. G. Dixon, Proc. R. Soc. A 314, 499 (1970)[39] A. I. Harte, Classical Quantum Gravity 25, 235020

(2008)[40] A. I. Harte, Classical Quantum Gravity 25, 205008

(2008)[41] A. I. Harte, Classical Quantum Gravity 26, 155015

(2009)[42] A. I. Harte, arXiv:1103.0543[43] W. G. Dixon, Phil. Trans. Roy. Soc. A 277, 59 (1974)[44] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B.

P. Flannery, Numerical Recipes in C: The Art of Sci-entific Computing (Cambridge University Press, Cam-bridge, 1992)

[45] R. Penrose, Rev. Mod. Phys. 37, 215 (1965)[46] H.-J. Seifert, Z. Naturforsch. 22, 1356 (1967)[47] A. Avez, Ann. Inst. Fourier 13, 105 (1963)[48] V. E. Hubeny and M. Rangamani, J. High Energy Phys.

12, 043 (2002)[49] G. Friedlander and M. Joshi, Introduction to the The-

ory of Distributions (Cambridge University Press, Cam-bridge, 1998), 2nd ed.

[50] T. D. Drivas and S. E. Gralla, Classical Quantum Gravity28, 145025 (2011)

[51] C. Barcelo, S. Liberati, and M. Visser, Living Rev. Rel-ativity 14, 3 (2011)

[52] D. Christodoulou, The formation of shocks in 3-dimensional fluids (European Mathematical Society,Zurich, 2007)

[53] D. Amati and C. Klimcık, Phys. Lett. B 219, 443 (1989)[54] G. T. Horowitz and A. R. Steif, Phys. Rev. Lett. 64, 260

(1990)[55] D. Berenstein, J. M. Maldacena, and H. Nastase, J. High

Energy Phys. 04, 013 (2002)