Diffractive Nonlinear Geometrical Optics for Variational Wave Equations and the Einstein Equations

Transcription

1 Diffractive Nonlinear Geometrical Optics for Variational Wave Equations and the Einstein Equations Giuseppe Alì Istituto per le Applicazioni del Calcolo, Consilio Nazionale delle Ricerche, Napoli, and INFN-Gruppo c. Cosenza AND John K. Hunter University of California at Davis Abstract We derive an asymptotic solution of the vacuum Einstein equations that describes the propaation and diffraction of a localized, lare-amplitude, rapidly-varyin ravitational wave. We compare and contrast the resultin theory of stronly nonlinear eometrical optics for the Einstein equations with nonlinear eometrical optics theories for variational wave equations. c Wiley Periodicals, Inc. Introduction Geometrical optics and its eneralizations, such as the eometrical theory of diffraction, are a powerful approach to the study of wave propaation, for both linear and nonlinear waves. In this paper, we develop a theory of stronly nonlinear eometrical optics for ravitational wave solutions of the vacuum Einstein equations. Specifically, we derive asymptotic equations that describe the diffraction of localized, lare-amplitude, rapidly-varyin ravitational waves. These equations are a eneralization of the straihtforward non-diffractive, nonlinear eometrical optics equations for lare-amplitude ravitational waves derived in []. This stronly nonlinear theory differs fundamentally from the weakly nonlinear theory for small-amplitude ravitational waves obtained by Choquet-Bruhat [] and Isaacson [] because it captures the direct nonlinear self-interaction of the waves. Here, we use the term eometrical optics to refer to any asymptotic theory for the propaation of short-wavelenth, hih-frequency waves, irrespective of its area of physical application. Communications on Pure and Applied Mathematics, Vol., 35 () c Wiley Periodicals, Inc.

2 G. ALì AND J. K. HUNTER The Einstein equations may be derived from a variational principle, and when written with respect to a suitable aue they form a system of wave equations for the ravitational field. In order to explain the structure of nonlinear eometrical optics theories for ravitational waves and to motivate the form of our asymptotic expansion, it is useful to consider first such theories for a eneral class of variational wave equations. We describe straihtforward and diffractive eometrical optics theories for variational wave equations in Section. In the weakly nonlinear theory for waves with periodic waveforms, the scalar amplitude-waveform function a(θ, v) of the wave depends on a ray variable v and periodically on a fast phase variable θ. The wave-amplitude a satisfies the Hunter-Saxton equation (.), (.) { a v + ( a ) θ } + Na θ = { a θ a θ }, where the anular brackets denote an averae with respect to θ. The coefficient of the nonlinear terms in (.) may be interpreted as a derivative of the wave speed with respect to the wave amplitude. In addition, the wave-amplitude is coupled with a slowly-varyin mean field. The mean-field satisfies a system (.) of variational wave equations of the same form as the oriinal system with a source term proportional to the mean enery-density of the wave a θ. An application of this weakly-nonlinear expansion to the Einstein equations leads to the theory of Choquet-Bruhat [] and Isaacson []. In that case, the nonlinear coefficients correspondin to in the equations for the amplitudes of the ravitational waves are identically zero. This theory therefore describes the nonlinear interaction between a hih-frequency, oscillatory ravitational wave and a slowly-varyin mean ravitational field, but does not describe the direct nonlinear self-interaction of the ravitational wave itself. In Section.4, we make a distinction between enuinely nonlinear wavefields in variational wave equations, for which is never zero, and linearly deenerate wave-fields for which identically zero (see Definition.). All wave-fields in the Einstein equations are linearly deenerate. As observed in [3, 4, 6, 9, 9], for example, this fact reflects a fundamental deeneracy in the nonlinearity of the Einstein equations in comparison with eneral quasilinear wave equations. A property of the Einstein equations related to their linear deeneracy is that they possess an exact solution for non-distortin, lare-amplitude, plane waves, the Brinkmann solution [9,, 6, 7]. One can use this solution as the basis of a stronly nonlinear eometrical optics theory for lare-amplitude ravitational waves, which leads to a eneralization of the collidin plane wave equations []. We outline the resultin non-diffractive theory in Section.5. We derive our diffractive, stronly nonlinear eometrical optics solution of the vacuum Einstein equations in Section 3. This solution describes the propaation and diffraction of a thin, lare-amplitude ravitational wave, such as a pulse or sandwich wave. The simplest, and most basic, case is that of a plane-polarized

3 NONLINEAR GEOMETRICAL OPTICS 3 ravitational wave diffractin in a sinle direction. We summarize the resultin asymptotic equations here. We suppose that the polarization of the wave is alined with the diffraction direction. Then, with respect to a suitable coordinate system (u, v, y, z), the metric of the wave adopts the form (.) = e M( du εy dy ) dv + e U( e V dy + e V dz ) + O(ε ). Here, ε is a small parameter. The leadin-order metric component functions U, V, M and the first-order function Y depend upon a fast phase variable θ, an intermediate transverse variable η, and the slow ray variable v, where θ = u ε, η = y ε. The phase u is liht-like and the transverse coordinate y is space-like. This metric describes a ravitational wave whose wavefronts are close to the null-hypersurface u =. The wave is plane-polarized in the (y, z)-directions and diffracts in the y-direction. To write equations for the metric component functions in a concise form, we define a derivative D η and functions φ, ψ by D η = e ( ) (.3) U η + Y θ, (.4) φ = D η M e U Y θ, (.5) ψ = D η (U + V ). We note that εd η = e V (dy), where denotes the raisin operator from oneforms to vector fields. Then (U, V, M, Y ) are functions of (θ, η, v) which satisfy the followin system of PDEs: U θθ ( ) U (.6) θ + Vθ + Uθ M θ =, (.7) (φ + ψ) θ = ψ(u + V ) θ, U θv U θ U v = { e (U+V +M) D η φ + D η ψ } (.8) φ φψ ψ, V θv (U θ V v + U v V θ ) = { e (U+V +M) D η φ + } (.9) φ, M θv + (U θu v V θ V v ) = { e (U+V +M) D η ψ } (.) φ + ψ. This system is the main result of our analysis. It is an asymptotic reduction of the full vacuum Einstein equations to a ( + )-dimensional system of PDEs. The system provides a model nonlinear wave equation for eneral relativity, and should be useful, for example, in studyin the focusin of nonplanar ravitational waves and the effect of diffraction on the formation of sinularities. We plan to study these topics in future work.

4 4 G. ALì AND J. K. HUNTER We discuss (.6) (.) in Section 4. The constraint equation (.6) is a nonlinear ODE in θ in which η, v appear as parameters. As shown in Proposition 4., this constraint is preserved by the remainin equations. Equation (.7) is a linear, nonhomoeneous ODE in θ for Y, whose coefficients depend on θ- and η-derivatives of (U, V, M). It may therefore be rearded as determinin Y in terms of (U, V, M). Equations (.8) (.) form a system of evolution equations for (U, V, M) which is coupled with Y. The main part of the system consists of a (+)-dimensional wave equation in (θ, η, v) for (U + V ), where (θ, v) are characteristic coordinates. When Y = and all functions are independent of η, equations (.6) (.) reduce to the collidin plane wave equations without the v-constraint equation (see Section.5). When all functions depend on (ξ, η) with ξ = θ λv for some constant λ, we et a system of PDEs in two variables that is studied further in []. This system describes space-times that are stationary with respect to an observer movin close to the speed of liht. We also derive asymptotic equations for the diffraction of ravitational waves with eneral polarizations in two transverse directions. These equations are iven by (3.) (3.4), (C.), and (C.3) (C.5), but they are much more complicated than the ones for plane-polarized waves written out above. Geometrical optics The Einstein equations do not form a hyperbolic system of PDEs because of their aue-covariance, but they are hyperbolic when written with respect to a suitable aue. In order to develop and interpret eometrical optics solutions for the Einstein equations, it is useful to bein by studyin eometrical optics solutions for hyperbolic systems of variational wave equations. We remark that there is an analoy between eometrical optics theories for hyperbolic systems of conservation laws [9] and variational wave equations. For example, the inviscid Burers equation [, ] is the analo of the Hunter-Saxton equation (.4), and the unsteady transonic small disturbance equation [8] is the analo of the two-dimensional Hunter-Saxton equation (.5). One can also derive lare-amplitude eometrical optics theories for linearly deenerate waves in hyperbolic conservation laws (see [33, 34], for example) that are analoous to the lare-amplitude theories described here for the Einstein equations. There are other nonlinear eometrical optics theories for dispersive waves, most notably Whitham s averaed Laranian method [37, 38] for lare-amplitude dispersive waves. These dispersive theories have a different character from the ones for nondispersive hyperbolic waves. Nonlinear dispersive waves have specific waveforms iven by travelin wave solutions in which the effects of dispersion and nonlinearity balance, whereas nondispersive hyperbolic wave equations and the Einstein equations have travelin wave solutions with arbitrary waveforms, which may distort as the waves propaate.

5 NONLINEAR GEOMETRICAL OPTICS 5 In straihtforward, non-diffractive theories of eometrical optics, waves are locally approximated by plane waves and propaate alon rays. In linear eometrical optics, the wave amplitude satisfies an ODE (the transport equation) alon a ray; in nonlinear eometrical optics, the amplitude-waveform function satisfies a nonlinear PDE in one space dimension (a eneralization of the transport equation) alon a ray. This difference arises because linear hyperbolic waves propaate without distortion, so one requires only an ODE alon each ray to determine the chane in the wave amplitude; by contrast, wave-steepenin and other effects distort the waveform of nonlinear hyperbolic waves, so one requires a PDE alon each ray to determine the chane in the wave amplitude and the waveform. Straihtforward eometrical optics, and a local plane-wave approximation, break down when the effects of wave-diffraction become important; for example, this occurs when a hih-frequency wave focuses at a caustic, or when a wave beam of lare (relative to its wavelenth), but finite, transverse extent spreads out. The effects of diffraction on a hih-frequency wave may be described by the inclusion of additional lenth-scales in the straihtforward eometrical optics asymptotic solution, and one then obtains asymptotic PDEs with a larer number of independent variables. Perhaps the most basic asymptotic solution that incorporates the effect of wave diffraction is the parabolic approximation described below for wave equations. In Section 3, we derive analoous asymptotic solutions of the Einstein equations that describe the diffraction of lare-amplitude ravitational waves.. The wave equation We bein by recallin eometrical optics theories for the linear wave equation (.) tt = (c ). Here, (t, x) is a scalar function, the wave speed c (t, x) is a iven smooth function, and x R d. Althouh well-known, these theories provide a useful backround for our analysis of nonlinear variational wave equations and the Einstein equations. We look for a short-wavelenth asymptotic solution = ε of (.), dependin on a small parameter ε, of the form ( ) u(t, x) ε (t, x) a, t, x as ε. The solution depends on a fast phase variable and slow space-time variables (t, x). ε θ = u ε Here, we use u/ε as a phase variable, rather than u/ε, for consistency with the diffractive expansion below.

6 6 G. ALì AND J. K. HUNTER One finds that the scalar-valued phase function u(t, x) satisfies the eikonal equation (.) u t = c u. The amplitude-waveform function a(θ, t, x) satisfies the equation (.3) a θv + Na θ =, where (.4) v = u t t c u is a derivative alon the rays associated with the phase u, and N(t, x) is iven by (.5) N = { utt (c u)}. Equation (.3) has solutions of the form a(θ, t, x) = A(t, x)f (θ), where A(t, x) is a wave-amplitude that satisfies an ODE alon a ray (the transport equation of linear eometrical optics) (.6) A v + N A =, and F(θ) is an arbitrary function that describes the waveform of the wave. For example, if F(θ) = e iθ, then the solution describes an oscillatory harmonic wave; if { θ n θ >, F(θ) = θ, then the solution describes a wavefront across which the normal derivative of of order n jumps; if F(θ) has compact support, then the solution describes a pulse; and if F(θ) has different limits as θ ±, then the solution describes a wave that carries a jump in. The term N A in the transport equation (.6) describes the effect of the ray eometry on the wave amplitude. The coefficient N becomes infinite when rays focus at a caustic. The straihtforward eometrical optics solution then becomes invalid, and diffractive effects must be taken into account [3]. There are many ways in which diffraction modifies straihtforward eometrical optics. Here, we consider one of the simplest diffractive expansions, iven by ( ) u(t, x) ε y(t, x) (t, x) a,, t, x as ε. ε ε This asymptotic solution depends upon an additional intermediate variable η = y ε, where y(t, x) is a scalar-valued transverse phase. This ansatz describes a hihfrequency wave whose wavefronts are close to u = constant that diffracts in the y-direction.

7 NONLINEAR GEOMETRICAL OPTICS 7 One finds that u satisfies the eikonal equation, as before, and y v =, meanin that y is constant alon the rays associated with u. Moreover, the amplitudewaveform function a(θ, η, t, x) satisfies the equation (.7) a θv + Na θ + Da ηη =, where the coefficient D(t, x) of the diffractive term is iven by (.8) D = y t c y. For harmonic solutions, we have a(θ, η, t, x) = A(η, t, x)e iθ, and equation (.7) reduces to a Schrödiner equation i {A v + N A} + D A ηη =. This parabolic approximation and its eneralizations are widely used in the study of wave propaation. In the simplest case of the diffraction of plane wave solutions of the twodimensional wave equation with wave speed c =, we may choose Equation (.7) is then tt = xx + yy, u = t x, y = y, v = t + x. (.9) a θv = a ηη. This equation describes waves that propaate in directions close to the positive x- direction, and is a wave equation in which θ and v are characteristic coordinates. We will see that equations with a similar structure to (.9) in their hihest-order derivatives arise from the Einstein equations.. A variational wave equation Next, we consider the followin nonlinear, scalar wave equation [] (.) tt (c () ) + c()c () =, where a prime denotes the derivative with respect to and we assume that the wave speed c R R + is a smooth, non-vanishin function. This equation is derived from the variational principle { δ t } c () dtdx =. The lobal existence of conservative weak solutions of (.) in one space-dimension is established in [8].

8 8 G. ALì AND J. K. HUNTER The structure of the nonlinear terms in (.) resembles that of the Einstein equations, althouh the effects of nonlinearity are qualitatively different because of the linear deeneracy of the Einstein equations. We look for an asymptotic solution of (.) of the form [6, 9, ] ( ) u(t, x) ε (t, x) (t, x) + ε a, t, x as ε. This solution represents a small-amplitude, hih-frequency perturbation of a slowlyvaryin field. The amplitude and the wavelenth of the perturbation are chosen to be of the same order of manitude because this leads to a balance between the effects of weak nonlinearity and the ray eometry. First, we suppose that the amplitude-waveform function a(θ, t, x) is a periodic function of the phase variable θ. We assume, without loss of enerality, that its mean with respect to θ is zero. We find that the phase u satisfies the eikonal equation (.) with c = c ( ). The mean-field satisfies the nonlinear wave equation (.) tt (c ) + c c + u c c a θ =, where the anular brackets denote an averae with respect to θ over a period, and c = c ( ). Equation (.) has the same form as the oriinal wave equation (.), with an additional source term proportional to the mean enery-density of the wave-field. The amplitude-waveform function a(θ, t, x) satisfies the periodic Hunter-Saxton equation, { ( ) } (.) a v + a + Na = θ θ { aθ a θ }, where v is the ray derivative defined in (.4), N is iven by (.5), and (.3) = u c c. The coefficient is proportional to the derivative of the wave-speed with respect to the wave amplitude, and it provides crucial information about the effect of nonlinearity on the waves; for a iven, we obtain a nonlinear PDE for the amplitudewaveform function only when. We may ensure that the hiher-order terms in this expansion for waves with periodic waveforms also depend periodically on θ. As a result, no secular terms arise in the expansion, and it is uniformly valid as ε for (t, x) = O() and θ = O(ε ). If we consider waves with localized waveforms, such as pulses or jumps, then we cannot in eneral eliminate secular terms in θ, but we may obtain an asymptotic solution that is valid near 3 the wave-front u = in which we nelect ε 3 Specifically, the expansion is valid when θ = O(), or u = O(ε ), and t = O().

9 NONLINEAR GEOMETRICAL OPTICS 9 mean-field effects. Thus, the backround field (t, x) is a solution of the oriinal wave equation, and a(θ, t, x) satisfies { ( ) } (.4) a v + a + Na = θ θ a θ. For results concernin the lobal existence of weak solutions of (.4), see [7] and references iven there. It is straihtforward to include diffractive effects in the expansion for localized waves. The asymptotic solution has the form ( ) u(t, x) ε (t, x) (t, x) + ε y(t, x) a,, t, x as ε. ε ε Here, is a solution of the oriinal wave equation, the phase u satisfies the eikonal equation at, the transverse function y is constant alon the rays associated with u, and a(θ, η, t, x) satisfies a ( + )-dimensional eneralization of the Hunter- Saxton equation (.4), (.5) { a v + where D is iven by (.8). ( ) } a + Na + θ θ Da ηη = a θ,.3 Hyperbolic systems of variational wave equations In this section, we consider a class of hyperbolic systems of nonlinear wave equations that are derived from a variational principle whose action is a quadratic function of the derivatives of the field with coefficients dependin on the field [] (.6) δ A αβ p pq () q dx =. x α x β Here, x = (x,..., x d ) R d+ are space-time variables, = (,..., m) R d+ R m are dependent variables, A αβ pq R m R are smooth coefficient functions, and we use the summation convention. We assume that A αβ pq = Aβα pq = Aαβ qp. The Euler-Larane equations associated with (.6) are (.7) G p [ ] =, where (.8) G p [ ] = x α { } A αβ q pq () x β A αβ qr p q r () x α x. β We assume that (.7) forms a hyperbolic system of PDEs. The scalar wave equation considered in the previous section is the simplest representative of this class of

10 G. ALì AND J. K. HUNTER equations. For discussions of recent local existence results for quasilinear systems of wave equations and the Einstein equations, see [5, 5]. The weakly nonlinear eometrical optics solution of (.7) has the form [6, 9, ] ( ) u(x) (.9) ε (x) (x) + ε a, x R(x) as ε, ε where R d+ R m is a backround field, u R d+ R is a phase function, a R R d+ R is an amplitude-waveform function, and R R d+ R m is a vector field. The phase u satisfies the eikonal equation (.) det [ u x αu x β A αβ pq ( ) ] =. We introduce the sinular m m matrix C with components (.) C pq = u x αu x β A αβ pq ( ). Then the vector R in (.9), with components R p, is a null-vector of C, so that C pq R q =. Here, and in (.9), we assume that we are dealin with a simple characteristic, meanin that the null-space of C is one-dimensional. We write the scalar-valued amplitude-waveform function a(θ, x) as a function of the fast phase variable θ = u/ε and the slow variables x. If a(θ, x) is a periodic function of θ with zero mean, then the mean-field (x) satisfies the equation [ ] (.) G p = H p a θ, where the anular brackets denote an averae with respect to θ, and Aqr αβ (.3) H p = u x αu x β ( ) R q R r. p Equation (.) has the same form as the oriinal equation (.7) for with an additional source term proportional to aθ. The H p are interaction coefficients that describe the effect of the wave on the mean-field. The equation for a is (.) with (.4) v = u x β A αβ pq ( ) R p R q x α, = u x αu x β A αβ qr p ( ) R p R q R r, N = { ux x α β A αβ pq ( ) R p R q} Aqr αβ u x α ( ) r p x R p R q. β For localized waveforms, the mean-field interactions may be nelected, and we can obtain diffractive versions of this expansion as before. These equations eneralize easily to the case when (.7) has a multiple characteristic of constant multiplicity n. In that case, one obtains a mean-field equation whose source term is a sum of averaes of products of θ-derivatives of the

11 NONLINEAR GEOMETRICAL OPTICS amplitude-waveform functions, and an n n system of Hunter-Saxton equations for the amplitude-waveform functions. In particular, if the vectors {R,..., R n } form a basis of the null-space of the matrix C defined in (.), then the coefficients i jk of the nonlinear terms in the system of equations for the amplitude-waveform functions are iven by (.5) i jk = u x αu x β A αβ qr p.4 Linearly deenerate wave equations ( ) R p i R q j Rr k, i, j, k n. The coefficient in (.4) may be interpreted as a derivative of a wave speed with respect to the wave amplitude, as is explained further in [3]. Motivated by the correspondin definition for hyperbolic conservation laws introduced by Lax [8], we make the followin definition. Definition.. A simple characteristic of a hyperbolic system of variational equations (.7) (.8) is enuinely nonlinear (respectively, linearly deenerate) if, for every R m and every non-zero du R d+ belonin to the solution branch of (.) associated with the characteristic, the quantity defined in (.4) is non-zero (respectively, zero). A multiple characteristic of constant multiplicity n is linearly deenerate if all coefficients i jk defined in (.5) are zero. We say that a system is enuinely nonlinear (respectively, linearly deenerate) if all of its characteristics are enuinely nonlinear (respectively, linearly deenerate). Thus, for linearly deenerate wave equations, the amplitude-waveform function in the weakly nonlinear theory correspondin to the ansatz (.9) satisfies a linear PDE. Quadratically nonlinear interactions with a mean-field may still occur, however: it follows from (.3) (.4) that = H p R p, so we may have = but H p. Even if a characteristic of a nonlinear wave equation is not linearly deenerate, a loss of enuine nonlinearity may occur in which vanishes at a particular point R m and direction du R d+. For example, from (.3), the scalar wave equation (.) is enuinely nonlinear if c () for all R, has a loss of enuine nonlinearity if c () = for some R, and is linearly deenerate if c is constant, when it reduces to the linear wave equation. Systems of wave equations may be linearly deenerate without reducin to linear PDEs. We emphasize that these definitions of enuine nonlinearity and linear deeneracy for wave equations are analoous to but different from the definitions for hyperbolic conservation laws. If a enuinely nonlinear variational wave equation for were rewritten as a first-order hyperbolic system for and F =, it would be classified as a linearly deenerate first-order hyperbolic system since the wave speeds of the oriinal wave equation are independent of the derivative. For further discussion of linearly deenerate hyperbolic systems see [6].

12 G. ALì AND J. K. HUNTER Our interest here in linearly deenerate variational systems of wave equations is that they may possess lare-amplitude, non-distortin, plane wave solutions. In that case, weakly nonlinear eometrical optics does not capture the nonlinear selfinteraction of a wave, and one requires a stronly nonlinear theory. Non-distortin plane wave solutions exist only if the equations satisfy an additional deeneracy condition, namely that the coefficients H p in (.3) vanish on the plane wave solutions. We study these questions for eneral variational systems in [3]. Here, we discuss the correspondin theory for the Einstein equations..5 The Einstein equations The Einstein equations do not fall exactly into the class of variational wave equations defined in Section.3 because of their aue-covariance. They can be derived from a variational principle of the form (.6), obtained after an interation by parts in the Einstein-Hilbert action, but the resultin Euler-Larane equations are not hyperbolic. Nevertheless as is well-known they become hyperbolic when written with respect to suitable coordinates, such as wave (or harmonic) coordinates. The equation for the metric, with components µν, then adopts a similar form to (.7) (.8), namely x α { αβ µν x β } H µν () (, ) =, where H µν is a quadratic form in the metric derivatives with coefficients dependin on. (See [3] for an explicit expression.) The weakly nonlinear expansion described in Section.3 for variational systems of wave equations corresponds to the expansion of Choquet-Bruhat [] and Isaacson [] for the Einstein equations. The Einstein equations have multiple characteristics, so one obtains a system of equations for the amplitude-waveform functions, but all of the nonlinear coefficients (.5) in these equations are zero. Thus, in this sense, the Einstein equations form a linearly deenerate system of variational wave equations. Physically, the linear deeneracy is a consequence of the transverse nature of ravitational waves. The velocity of a ravitational wave that propaates in the x-direction is determined by the (t, x)-metric components, but (in a suitable transverse aue) the wave carries variations in the (y, z)-metric components. Thus, the wave amplitude does not affect the wave velocity. The linear deeneracy of the Einstein equations is related to the polarized null condition of Choquet-Bruhat [3, 4] and the weak null condition of Lindblad and Rodnianski [9]. This condition is what permits the existence of lobal smooth small-amplitude perturbations of Minkowski space-time [5, 4, 3], for example, and similar results would not be true for ( + 3)-dimensional variational wave equations in which. See also [4] for related lobal existence results for other quasilinear wave equations.

13 NONLINEAR GEOMETRICAL OPTICS 3 The Einstein equations possess non-distortin, plane wave solutions for lareamplitude ravitational waves, and these solutions form the basis of a stronly nonlinear eometrical optics theory for lare-amplitude ravitational waves. It does not, however, appear possible to obtain a self-consistent theory for oscillatory lare-amplitude waves, since the mean enery-momentum associated with an extended wave-packet would enerate a very stron backround curvature of space-time. This restriction is related to the fact that mean-field interactions with oscillatory waves already occur at leadin order in the small-amplitude theory. We therefore consider localized waves. An asymptotic theory for the propaation of localized, lare-amplitude, rapidly varyin ravitational waves into slowly varyin space-times was developed in []. In this theory, the metric of a plane-polarized wave may be written with respect to a suitable coordinate system (u, v, y, z) as (.6) = e M du dv + e U+V dy + e U V dz + O(ε ). Here, the leadin-order metric component functions U, V, M depend on θ, v, where the fast phase variable θ is iven by θ = u ε. The metric components may also depend on y, z, but these variables occur as parameters in the asymptotic equations. Then (U, V, M) are functions of (θ, v) which satisfy the followin PDEs: (.7) (.8) (.9) (.3) U θθ ( ) U θ + Vθ + Uθ M θ =, U θv U θ U v =, V θv (U θ V v + U v V θ ) =, M θv + (U θu v V θ V v ) =. Equations (.8) (.3) are wave equations for (U, V, M) in characteristic coordinates (θ, v), and (.7) is a constraint which is preserved by (.8) (.3). These equations correspond to a well-known exact solution of the Einstein equations, the collidin plane wave solution [7, 3, 35, 36], without the usual constraint equation in v, U vv ( ) U v + Vv + Uv M v =. This constraint need not be satisfied if the slowly-varyin space-time into which the wave propaates is not that of a counter-propaatin ravitational wave. If it is not satisfied, then the resultin metric is an asymptotic solution of the Einstein equations but not an exact solution. The asymptotic equations we derive in this paper are a diffractive eneralization of (.7) (.3).

14 4 G. ALì AND J. K. HUNTER 3 The asymptotic expansion In this section, we outline our asymptotic expansion of the Einstein equations, and specialize it to the case of plane-polarized ravitational waves that diffract in a sinle direction. Our oal is to explain the structure of the expansion and the resultin perturbation equations. The detailed alebra is summarized in the appendices. 3. The eneral expansion Let be a Lorentzian metric. We denote the covariant components of with respect to a local coordinate system x α by αβ. The connection coefficients Ɣ λ αβ and the covariant components R αβ of the Ricci curvature tensor associated with are defined by Ɣ λ αβ = ( βµ λµ x + αµ α x ) αβ (3.), β x µ R αβ = Ɣλ αβ Ɣλ βλ (3.) + Ɣ λ αβɣ µ x λ x α λµ Ɣ µ αλɣ λ βµ. Here, and below, Greek indices α, β, λ, µ run over the values,,, 3. The vacuum Einstein equations may be written as (3.3) R αβ =. We look for asymptotic solutions of (3.3) with metrics of the form ( u(x α ) =, ya (x α ) ) (3.4), x α ; ε, ε ε (θ, η a, x α ; ε) = (θ, η a, x α ) + ε (θ, η a, x α ) + ε (θ, η a, x α ) + O(ε 3 ). Here, ε is a small parameter, u is a phase, y a with a =, 3 are transverse variables, and θ = u ε, ηa = ya ε are stretched variables. The ansatz in (3.4) corresponds to a metric that varies rapidly and stronly in the u-direction, with less rapid variations in the y a -directions, and slow variations in x α. We remark that the form of this ansatz is relative to a class of local coordinate systems, since an ε-dependent chane of coordinates can alter the way in which the metric depends on ε. It would be desirable to ive a eometrically intrinsic characterization of such an ansatz and to carry out the expansion in a lobal, coordinate-invariant way, but we do not attempt to do so here. We use (3.4) in the Einstein equations (3.3), expandin derivatives of a function f with respect to x µ as (3.5) x µ f ( u ε, ya ε, x α ) = ε f,θu µ + ε f,ā yµ a + f,µ,

15 NONLINEAR GEOMETRICAL OPTICS 5 and then treatin (θ, η a, x α ) as independent variables. In (3.5) and below, we use the shorthand notation f,θ = f, f,ā = f θ η a, f,µ = f (3.6) θ,x x µ, θ,η a η a,x u µ = u x, µ ya µ = ya x. µ We will omit commas from derivatives when this does not lead to any confusion. Usin the expansions of the connection coefficients and the Ricci curvature components iven in Appendix A, we find that to et a non-trivial asymptotic solution of (3.3) at the leadin orders, the phase u must be null to leadin order, and the transverse variables y a must be constant alon the rays associated with u to leadin order. Moreover, the leadin-order metric must have the form of the collidin plane wave metric. To carry out the expansion in detail, we use the aue-covariance of the Einstein equations to make a choice of coordinates that is adapted to the metric in (3.4). We assume that u is approximately null up to the order ε and that y, y 3 are constant alon the rays associated with u up to the order ε. Furthermore, we assume that we can extend (u, y, y 3 ) to a local coordinate system (x, x, x, x 3 ) = (u, v, y, y 3 ). Then, as shown in Appendix B, we can use appropriate aue transformations, which involve a near identity transformation of the phase and the transverse variables, to write a eneral metric (3.4) as (3.7) = dx dx + ab dx a dx b } +ε { a dx dx a + ab dx a dx b + ε i j dx i dx j + O(ε 3 ). In (3.7) and below, indices i, j, k,... take on the values,, 3, while indices a, b, c,... take on the values, 3. We raise and lower indices usin the leadin order metric components; for example, we write ( αβ) = ( αβ ), h α = αβ h β. We also show in Appendix B that we have one additional aue freedom in (3.7), involvin a nonlinear transformation of the phase, which we can use to set either = or one of the three components, 3, equal to zero. We will exploit this aue freedom later, since it is convenient in formulatin a variational principle for the asymptotic equations (see Appendix D).

16 6 G. ALì AND J. K. HUNTER We use (3.7) in (3.) (3.) and expand the result with respect to ε. We find that (3.8) R αβ = ε 4 4 R αβ + ε 3 3 R αβ + ε R αβ +O(ε ), where explicit expressions for the nonzero components of the Ricci tensor up to the order ε are summarized in Appendix C. Usin (3.8) in (3.3) and equatin coefficients of ε 4, ε 3 and ε to zero, we et (3.9) (3.) (3.) 4 R αβ =, 3 R αβ =, R αβ =. These perturbation equations lead to a closed set of equations for the leadin-order and first-order components of the metric, as we now explain. The only component of (3.9) that is not identically satisfied is (3.) 4 R =. From (3.) and (C.), it follows that αβ satisfies,θ ab ab,θ 4 ac bc,θ bd ad,θ =. ( ab ab,θ ),θ + This equation is a constraint on the leadin order metric which involves only θ- derivatives, and has the same form as the constraint equation for a sinle ravitational plane wave. The nonzero components of the Ricci tensor at the next order in ε are The condition (3.3) 3 R, 3 R a. 3 R a=, yields equations for a and their derivatives with respect to θ, with nonhomoeneous terms dependin on θ- and η-derivatives of the leadin order metric. Thus, (3.3) provides two equations relatin the first-order perturbation in the metric to the leadin order metric. The equation 3 R = is a sinle equation that is homoeneous in the first-order components ab, as can be seen from (C.). A nonzero solution of this equation corresponds physically to a free, small-amplitude ravitational wave of strenth of the order ε propaatin in the space-time of the lare-amplitude ravitational wave. Retainin a nonzero

17 NONLINEAR GEOMETRICAL OPTICS 7 solution of this homoenous equation would not chane our final equations for the lare-amplitude wave, and for simplicity we assume that ab =. This assumption is also consistent with the hiher-order perturbation equations. At the next order in ε, the nonzero components of the Ricci curvature are R, R, R a, R ab. The perturbation equations for R and R a in (3.) ive non-homoenous equations for the second-order metric components ab, a, with source terms dependin on the lower-order components of the metric. These equations are satisfied by a suitable choice of the second-order metric components, and they are decoupled from the equations for the leadin-order and first-order metric components. We therefore do not consider them further here. The remainin perturbation equations are (3.4) R =, R ab=. By use of the aue freedom mentioned at the beinnin of this section, we may set =. In that case, (3.4) provides a set of four equations relatin the leadin-order and first-order metric components, and their derivatives. We remark that, in eneral, one cannot eliminate secular terms from the asymptotic solution that are unbounded as the fast phase variable θ tends to infinity. As a result, the validity of the asymptotic equations is restricted to a thin layer of thickness of the order ε about the hypersurface u =, where θ = O(). A lobal asymptotic solution can be obtained by matchin the inner solution inside this layer with appropriate outer solutions, such as slowly varyin space-times on either side of the wave. In this paper, however, we focus on the construction of asymptotic solutions for localized ravitational waves, and do not consider any matchin problems. Summarizin these results, we find that after a suitable aue transformation, an asymptotic solution of the Einstein equations (3.3) of the form (3.4) may be written as (3.5) = dx dx + ab dx a dx b + ε a dx dx a + O(ε ), where u = x, y a = x a and the six metric components, ab, a

18 8 G. ALì AND J. K. HUNTER satisfy a system (3.) (3.4) of seven equations. Equation (3.) is an ODE in θ, and, in the case of plane-polarized waves, we show that it is a aue-type constraint that is preserved by the remainin equations (3.3) (3.4). The explicit form of the equations follows from the expressions in (C.), (C.3) (C.5) for the correspondin components in the expansion of of the Ricci tensor that appear in (3.) (3.4). In eneral, these equations are very complicated, but they simplify considerably in special cases. 3. Plane-polarized ravitational waves In this section, we specialize our asymptotic solution to the case of a planepolarized ravitational wave that diffracts in a sinle direction, and write out the resultin equations explicitly. We choose coordinates (3.6) (x, x, x, x 3 ) = (u, v, y, z) in which the metric has the form (3.5). We suppose that the metric depends on the variables θ, η, v where the phase variable θ and the transverse variable η are defined by θ = u ε, η = y ε, and is independent of the second transverse variable ζ = z/ε. This means that the wave diffracts only in the y-direction. We could also allow the metric to depend on z, which would appear in the final equations as a parameter. We consider a leadin-order metric that has the form of the collidin plane wave metric for a plane-polarized wave, polarized in the (y, z)-directions. We have seen that we need only retain the hiher order components a,. From (C.3), one component of (3.3) is homoeneous in 3, since there is no dependence on ζ, and we may assume that 3 =. We therefore take the special form of the metric in (3.7) iven by (3.7) = e M dudv + e U+V dy + e U V dz +εy e M dydv + ε T e M dv + O(ε ), where the functions (U, V, M, Y, T ) depend on (θ, η, v). We recall that we have the aue-freedom to set either M, Y, or T equal to zero. In writin the equations, we choose to set T =. We then et the metric (.). The asymptotic metric (3.7) must satisfy equations (3.) (3.4). First, usin (3.7) in (3.) and simplifyin the result, we obtain, after some alebra, the θ- constraint equation (.6).

19 NONLINEAR GEOMETRICAL OPTICS 9 Next, usin (3.7) and (C.3), we find that the only nontrivial component of (3.3) is (3.8) 3 R =. After the introduction of D η, φ, ψ defined in (.3) (.5) and some alebra, we find that (3.8) may be written as (.7). Finally, usin (3.7) and (C.4) (C.5), we find that the only nontrivial components of (3.4) are (3.9) R =, R =, R 33=. After some alebra, we may write these equations as (.8) (.). We have also checked the results by use of MAPLE. 3.3 The variational principle Equations (3.) (3.4) can be derived from a variational principle, which is obtained by expandin the variational principle for the Einstein equations. In order to formulate a variational principle, it is necessary to retain the component T =, which was set equal to zero in our previous choice of coordinates. Variations with respect to this component yield the aue-type constraint (3.), and it may be set to zero after takin variations with respect to it. The eneral form of the asymptotic variational principle is iven in Appendix D. Here, we specialize it to the case of a plane-polarized ravitational wave considered in Section 3.. Usin the metric (3.7) in (D.3) we find, after some alebra, that the variational principle (D.4) for (.6) (.) is δs () =, S () = L ( ) dθ dv dη, where the Laranian L ( ) may be written as (3.) L ( ) = e U{ M θv + 4U θv V θ V v 3U θ U v T θθ + T θ (M θ + U θ ) + T (M θθ + U θθ 3 U θ V θ ) +e (U+V +M) [ D η φ D η ψ + 3 φ + φψ + ψ ] }. Variations of S () with respect to the second-order metric component T lead to the θ-constraint equation (.6). Variations with respect to the first-order metric component Y, which appears in φ, ψ, and D η, lead to equation (.7). Variations with respect to U, V, M lead to the evolution equations (.8) (.), after we set T =.

20 G. ALì AND J. K. HUNTER 4 Properties of the equations In this section, we study some properties of the equations for plane-polarized waves that are derived in the previous section. We write out the structure of the hihest-order derivatives that appear in the equations, and use this to formulate a reasonable IBVP for them. We also show that the θ-constraint equation is preserved by the evolution in v, and that the linearized equations are consistent with the linearized equations for ravitational waves in the parabolic approximation. 4. Structure of equations and an IBVP In this subsection, we consider the structure of equations (.6) (.) in more detail. The first equation, (.6), is an ODE with respect to θ relatin (U, V, M). As we show in the next section, this equation is a aue-type constraint which holds for all v if it holds for v =, say. We may therefore nelect this equation provided that the initial data at v = is compatible with it. The remainin equations (.7) (.) form a system of equations for U, V, M, Y. In order to exhibit their structure, we rewrite them in a way that shows explicitly how the hihest, second-order, derivatives appear. Usin (.3) (.5), we may rewrite equation (.7) as { Y θθ {(V + M) θ Y } θ + U θ U θ V θ } (4.) V θ Y = (U + V + M) θη + M η U θ (U + V ) η V θ. If U, V, M are assumed known, then this equation is a linear ODE in θ for Y. We may rewrite equations (.8) (.) as (4.) (4.3) (4.4) (U + V + M) θv (U + V ) θ (U + V ) v = { e (U+V +M) } φ φψ, (U + V ) θv U θ (U + V ) v U v (U + V ) θ, = { e (U+V +M) Dη (U + V ) φψ ψ } U θv U θ U v = +M){ e (U+V Dη (U + V + M) D ( ) η e U Y θ φ φψ ψ }. Examinin the terms that involve second order derivatives, we see that these equations consist of a ( + )-dimensional wave equation in (θ, η, v) for (U + V ), and two ( + )-dimensional wave equations in (θ, v), for (U + V + M) and U,

21 NONLINEAR GEOMETRICAL OPTICS in which (θ, v) are characteristic coordinates. An additional second-order derivative term, proportional to D η Y θ, appears in the equation for U. The function Y is also coupled with the evolution equations throuh the dependence of the transverse derivative D η, iven in (.3), on Y. In order to specify a unique solution of these equations, we expect that we need to supplement the ODE (4.) with data for Y and Y θ on θ =, say, and the evolution equations (4.) (4.4) with characteristic initial data for (U, V, M) on θ = and v =. Thus, a reasonable IBVP for (4.) (4.4) in the reion θ >, < η <, and v > is (U, V, M) = (U, V, M ) on v =, (U, V, M) = (U, V, M ) on θ =, (Y, Y θ ) = (Y, Y ) on θ =. Here, (U, V, M ) are iven functions of (θ, η) that satisfy the constraint U θθ ( U θ + V θ) + Uθ M θ =, and (U, V, M, Y, Y ) are iven compatible functions of (η, v). This data may be interpreted as initial data for the state of the wave on the hypersurface v =, and boundary data on the leadin wavefront θ =. 4. The constraint equations In this section, we show that the constraint equation (.6) is preserved. Proposition 4.. Suppose that (U, V, M, Y ) are smooth functions that satisfy (.7) (.). Let (4.5) F = U θθ Then (4.6) F v = U v F. ( ) U θ + Vθ + Uθ M θ. Proof. We write the evolution equations (.8) (.) as (4.7) (4.8) (4.9) U θv U θ U v = e (U+V +M) A, V θv (U θ V v + U v V θ ) = e (U+V +M) B, M θv + (U θu v V v V θ ) = e (U+V +M) C,

22 G. ALì AND J. K. HUNTER where, usin the notation defined in (.3) (.5), (4.) (4.) (4.) A = D η φ + D η ψ φ φψ ψ, B = D η φ + φ, C = D η ψ φ + ψ. Differentiatin (4.5) with respect to v, and usin (4.7) (4.9) and (4.5) to replace U θv, V θv, M θv and U θθ in the result, we find that (4.3) F v = U v F + e (U+V +M) D, where (4.4) D = A θ (U + V ) θ A V θ B + U θ C. Differentiatin equation (4.) for A with respect to θ, introducin the commutator [ θ, D η ] of θ and D η, and usin equations (.5) and (.7), we compute that (4.5) A θ = θ { D η (φ + ψ) φ φψ ψ } = D η (φ + ψ) θ + [ θ, D η ] (φ + ψ) (φ + ψ)φθ (φ + ψ)ψ θ = D η {ψ(u + V ) θ } (φ + ψ)φ θ (φ + ψ)ψ θ + [ θ, D η ] (φ + ψ) = (U + V ) θ D η ψ + ψ D η (U + V ) θ (φ + ψ)φ θ (φ + ψ)ψ θ + [ θ, D η ] (φ + ψ) = (U + V ) θ D η ψ + ψ θ D η (U + V ) ψ [ θ, D η ] (U + V ) (φ + ψ)φ θ (φ + ψ)ψ θ + [ θ, D η ] (φ + ψ) = (U + V ) θ D η ψ + ψψ θ (φ + ψ)φ θ (φ + ψ)ψ θ + [ θ, D η ] (φ + ψ) ψ [ θ, D η ] (U + V ) = (U + V ) θ D η ψ (φ + ψ)(φ + ψ) θ + [ θ, D η ] (φ + ψ) ψ [ θ, D η ] (U + V ) = (U + V ) θ { Dη ψ ψ(φ + ψ) } + [ θ, D η ] (φ + ψ) ψ [ θ, D η ] (U + V ). From the definition of D η in (.3), it follows that [ θ, D η ] = Uθ D η + e U Y θ θ.

23 NONLINEAR GEOMETRICAL OPTICS 3 Hence, usin (.5) and (.7), we et [ θ, D η ] (φ + ψ) = Uθ D η (φ + ψ) + e U Y θ (φ + ψ) θ = U θ D η (φ + ψ) + ψe U Y θ (U + V ) θ, [ θ, D η ] (U + V ) = Uθ D η (U + V ) + e U Y θ (U + V ) θ = U θ ψ + e U Y θ (U + V ) θ. Usin these equations in (4.5), and simplifyin the result, we find that A θ = (U + V ) θ ( Dη ψ φψ ψ ) + U θ ( Dη φ + D η ψ ψ ). Finally, usin this equation and (4.) (4.) in (4.4), and simplifyin the result, we find that D =. It follows from (4.3) that F satisfies (4.6). 4.3 Linearization We consider the small-amplitude limit of (.6) (.) in which U, V, M, Y. From the constraint equation (.6), we have U = O(V ), so U is of hiher-order in a linearized approximation and can be nelected, while V describes the waveprofile. (See in [3] for further discussion.) From (.3) (.4), we also have in this approximation that Linearization of (.7) yields D η = η, φ = M η Y θ, ψ = V η. (φ + ψ) θ =, while linearization the evolution equations (.8) (.) ives = (φ + ψ) η, V θv = φ η, M θv = ψ η. Nelectin functions of interation for simplicity, we find that these equations are satisfied if M = V and Y =. In that case, V satisfies an equation of the form (.9): V θv = V ηη. The correspondin linearized metric is iven by = ( V )dudv + ( + V )dy + ( V )dz.

24 4 G. ALì AND J. K. HUNTER Linearization of the Einstein equations, with a suitable choice of aue, leads to a set of linear wave equations for the metric components. One can verify that this linearization of our asymptotic solution arees with what is obtained by an application of the parabolic approximation described in Section. to the linearized Einstein equations for plane-polarized waves. Appendix A: Expansion of connection coefficients and Ricci tensor In this appendix, we write out the expansions as ε of the connection coefficients and the Ricci curvature components associated with a metric whose components have the form αβ = αβ ( u(x) ε, ya (x), x; ε ε ), αβ ( θ, η a, x; ε ) = αβ ( θ, η a, x ) + ε αβ ( θ, η a, x ) + ε αβ ( θ, η a, x ) + O(ε 3 ). The contravariant metric components αβ satisfy αµ µβ = δ α β. Expandin this equation in a power series in ε and solvin for αβ, we et (A.) αβ = αβ ε αβ ε ( αβ µν αµ βν ) + O(ε 3 ), = h αβ + ε h αβ + ε h αβ + O(ε 3 ), where αβ is the inverse of αβ. Here and below, we use the leadin order metric components to raise indices, so that αβ = αµ βν µν, αβ = αµ βν µν. We note that, with this convention, the first-order term in the expansion of the contravariant metric components αβ with respect to ε is h αβ = αβ, not αβ. We use the notation for derivatives in (3.6). The expansion of the connection coefficients (or Christoffel symbols) iven in (3.) is Ɣ λ αβ = ε Ɣ λ αβ + ε Ɣ λ αβ+ Ɣ λ αβ + O(ε),

25 NONLINEAR GEOMETRICAL OPTICS 5 where Ɣ λ αβ = Ɣ λ αβ = + Ɣ λ αβ = ( ) βµ,θ λµ u α + αµ,θ u β αβ,θ u µ ( ) βµ,ā λµ yα a + αµ,ā yβ a αβ,ā yµ a ( ) βµ,θ λµ u α + αµ,θ u β αβ,θ u µ ( ) βµ,θ λµ u α + αµ,θ u β αβ,θ u µ, ( ) βµ,α λµ + αµ,β αβ,µ ( ) βµ,ā λµ yα a + αµ,ā yβ a αβ,ā yµ a ( ) βµ,ā λµ yα a + αµ,ā yβ a αβ,ā yµ a ( ) βµ,θ λµ u α + αµ,θ u β αβ,θ u µ ( ) βµ,θ λµ u α + αµ,θ u β αβ,θ u µ ( ) βµ,θ h λµ u α + αµ,θ u β αβ,θ u µ., where The expansion of the Ricci tensor is R αβ = ε 4 4 R αβ + ε 3 3 R αβ + ε R αβ +O(ε ), 4 R αβ = Ɣ µ αβ,θu µ Ɣ µ βµ,θu α + Ɣ µ αβ Ɣ ν µν Ɣ µ αν Ɣ ν βµ, 3 R αβ = Ɣ µ αβ,θu µ Ɣ µ βµ,θu α + Ɣ µ αβ,ā y a µ Ɣ µ βµ,ā y a α + Ɣ µ αβ Ɣ ν µν+ Ɣ µ αβ Ɣ ν µν Ɣ µ αν Ɣ ν βµ Ɣ µ αν Ɣ ν βµ, R αβ = Ɣ µ αβ,θu µ Ɣ µ βµ,θu α + Ɣ µ αβ,ā y a µ Ɣ µ βµ,ā y a α + Ɣ µ αβ,µ Ɣ µ βµ,α+ Ɣ µ αβ Ɣ ν µν+ Ɣ µ αβ Ɣ ν µν Ɣ µ αν Ɣ ν βµ Ɣ µ αν Ɣ ν βµ+ Ɣ µ αβ Ɣ ν µν Ɣ µ αν Ɣ ν βµ.

26 6 G. ALì AND J. K. HUNTER Appendix B: Gaue transformations We consider a Lorentzian metric and local coordinates x α in which = αβ dx α dx β. We denote the contravariant form of by = αβ α β. We suppose that the metric depends upon a small parameter ε, and has the asymptotic expansion (3.4) as ε where u, y a, a =, 3, are independent functions. Furthermore, we assume that (B.) (B.) (du, du) = O(ε ), (du, dy a ) = O(ε), a =, 3. The first condition states that du is an approximate null form up to the order ε. The second condition states that y a is constant alon the rays associated with u up to the order ε. In this appendix, we show that under these assumptions, after a near-identity transformation of the phase u and the transverse variables y a, we can choose local coordinates x α in which u = x, y a = x a and the metric adopts the form in (3.7): (B.3) = dx dx + ab dx a dx b } +ε { a dx dx a + ab dx a dx b + ε i j dx i dx j + O(ε 3 ). As before, indices α, β,... take on the values,,, 3; indices i, j, k,... take on the values,, 3; and indices a, b, c,... take on the values, 3. We also show that by a nonlinear transformation of the phase u, we can impose any one of the followin four conditions: =, =, 3 =, =. To bein with, iven a metric of the form (3.4), we choose a local coordinate system x α in which (B.4) x = u, x a = y a. Then (B.) (B.) are equivalent to (B.5) These conditions imply that (B.6) Moreover, we must have = =, =, (B.7) det ab >, a =. a =.