in the Collisions of Gravitational Waves

It mainly focuses on two fundamental questions: First, what is the general structure of singularities and Killing-Cauchy horizons, which are formed in collisions of precisely plane-symmetric gravitational waves. Based on (non-rigorous) order-of-magnitude considerations, we discuss the collision outcome in two fundamentally different regimes of colliding wave parameters; these parameters are transverse sizes

CHAPTER2

INTRODUCTION AND SUMMARY

In section ill we study the propagation of a broad class of classical fields on a plane-symmetric spacetime with a Killing-Cauchy horizon of type I as in the above example. III shows that this behavior is quite general for plane-symmetric spacetimes with type I Killing-Cauchy horizons.

CLASSIFICATION

We will formulate the genericity condition on the data for { C7) in N during the proof. c). The initial data for cj> in N of the above class are generic if the solution is nonzero somewhere on the surface P (curve C) in S.

INSTABILITY OF HORIZONS OF TYPE II

Since Shas is past infinite null generators and S=H+(L), :E has no edge, meaning it is infinite in Killing ~; ~ directions.8 Consider generic, planar symmetric initial data for our scalar field .p on :E. By assumption (b) of the theorem, if Lie transmits the initial data, truncated in the manner of the previous paragraph, by killing.

CONCLUSIONS

The Killing-Cauchy horizon of type I Sin the Mink:owskian region of the plane sandwich-wave space-time described by the equations (l.1}- (1.5) with f 1=. The initial null boundary N consists of N2: works by the null surface { u =u (p 0)=1), which lies above { v =v (p 0 )); and from N1: the piece of the zero cone ( v =v (p 0)=const) lying above the surface { u =1) with the exception of a single generator of this cone lying in S N1 and N2 intersect at spacelik:e bisurface Zpa· The initial data for the plane-symmetric scalar field cp is zero on N, except on the dashed strip in N2, which lies between ZPo and the line (bisurface) v =v (p 1). If these data are generic, cp will be nonzero at some point q on the line C={X =0) lying in the Killing-Cauchy horizon S= { u = f 1.

FIGURE. 1.]

I FIGURE 2 I

CHAPTER3

INTRODUCTION AND OVERVIEW

This first paper in the series lays the groundwork for subsequent papers by (briefly) revising old results and presenting some new ones on the exact space-time collision of plane-waves and introducing the concept of a gravitational wave (GW) space-time --- of which Quasi-Plane Waves are a special case - and prove several theorems about GW spacetimes that imply some important properties of Quasi-Plane Waves. II we briefly review some global properties of exact solutions for so-called parallel-radius plane-front waves ("PP waves") and for plane waves. These theorems have important implications for quasi-plane waves (which are special cases of GW spacetime).

Since PP waves are always infinitely large in transverse extent, this result implies that near-plane waves (which have a finite transverse "size") must always exhibit diffraction. Since all PP waves are infinitely large in transverse extent, this theorem implies that near-plane waves must always exit.

EXACT PLANE WAVE AND PP-WAVE SPACETIMES: A REVIEW INTRODUCING OUR TERMINOLOGY AND NOTATION

These metric functions show, as we shall see, the effect of the plane sandwich wave (2.6) focusing on zero geodesic lines propagating in the u direction. A single null generator of a null cone CQ that runs parallel (and therefore does not intersect) with a plane wave is a single past infinite generator. Similar conclusions apply to null generators of achronic boundaries j+(p), where p eJ-(N) is a point sufficiently far from the wave (before the arrival of the wave).

III B) allows the zero generators of the achronal boundary j\zP) to intersect. It is interesting to reveal the depth of Tipler's theorem to note that for a single plane-wave spacetime the only conditions for the theorem that do not hold are strict plane symmetry and the existence of the partial Cauchy surface that satisfies.

GRAVITATIONAL-WAVE (GW) SPACETIMES

Since the proof of the theorem makes extensive use of the characteristic initial value formalism as developed by Penrose. We will now outline the construction of a local coordinate system and tetrad on M, which is particularly well suited to the discussion of the initial value problem. Then that portion of spacetime that lies to the future of the initial null boundary S=N1uN2uz is uniquely determined by the reduced initial data it induces on S.

Now the scalability in u, i.e. the freedom of coordinate transfonnations of the first kind, established by the arrangement that the wavefront N11 coincides with the zero surface (u =a. To find the spacetime that develops from these initial data, just put any of the quantities.

CHAPTER4

INTRODUCTION

The integration of this system of initial values ​​to obtain the metric coefficients in the interaction region bounded by N1uN2 is generally very difficult, indeed no general expression has been found for the solution in the generic case of plane wave collisions with non-parallel polarizations. This is in contrast to the direct method, where we integrate the initial data represented by the input plane waves and obtain a unique collimated plane wave spacetime. The reason for this behavior is that the same solution in the interaction region can be developed from several non-equivalent sets of initial data, while the result of the direct method of integration of given initial data is bounded to uniqueness by the well-known uniqueness results for hyperbolic systems.

For the solutions constructed in this paper, the metric in the interaction region of the colliding plane wave space times is obtained from the insides of the static, axisymmetric "distorted black holes" (Weyl) solutions that have an inside. The construction by which we build our colliding plane wave space times is described in detail in the next section (Sec. II) for the Schwarzschild metric, along with a discussion of the properties of the resulting colliding plane wave solution.

THE SOLUTION OBTAINED FROM THE SCHWARZSCHILD METRIC We first write the Schwarzschild metric inside the horizon (i.e ., for r <2M) as

Note that in the best-known plane wave collision solution5•2•3 there are also corresponding null surfaces. With any of the above boundary arguments, the field equations hold on the two planes { u =v =0}. This restriction on the general solution (3.9) is necessary for the existence of the inner region r <2M, because the expression.

we can proceed to the construction of the corresponding family of colliding spacetime plane waves. For each choice of parameters ( dk }), the above construction yields a unique plane wave collision solution.

EXAMPLES

The curvature singularity, which, in the generic case, constitutes an achronal future c -boundary in the interaction region of the solution (3.24), is located at. The reasons for this are the u ~ v symmetry of the metric (3.18) in the interaction region, and the u ~ v symmetry of the Penrose prescription to extend the metric beyond the interaction region. iii). The zero levels N1={u=O) and N2={v=O) are the previous wavefronts of the incoming plane waves 1 and 2.

The geometry of the interaction region I is uniquely determined by the solution of the initial value problem above. The region J in Schwarzschild spacetime, to which the interaction region of the plane wave collision solution (2.4), is locally isometric. This region J is shown shaded in this figure, which is drawn in the plane { t=const), { =O, 1t).

CHAPTERS

THE FIELD EQUATIONS FOR COLLIDING PARALLEL-POLARIZED GRAVITATIONAL PLANE WAVES AND THEIR SOLUTION

We will use this coordinate freedom below when we discuss the initial value problem for the field equations.] Therefore, the coordinate system (u ,v ,x ,y ) is the direct analog of the Rosen-type coordinates associated with each of the incident colliding plane waves. This measurement freedom also shows itself in the choice of initial data on the characteristic initial surface { u =0} U { v =0}: The choice of initial data { M (u =0, v ), M (u, v ). =0)} for the metric function M is completely arbitrary, since,. Our choice of gauge, Eq. 2.8), implies that the metric in region II (where u ~0, . v :sO ) describing the geometry of the incoming colliding wave propagating in the v direction (right in Fig. 1) is given by.

2.9) and (2.10)], in the more precise mathematical description of the initial value problem, the initial data is completely determined by only the two freely specifiable functions V 1(u ), and V 2(v. In the linearized regime ( when V 1 , V 2«1), the functions V 1 and V 2 correspond to the time-dependent physical amplitudes of the incoming, colliding gravitational waves [cf.

The Field equations and their solution in the (a,[3) coordinates

In other words, the singularities of the coordinate system (a,l3,x ,y) consist of the singularities of the (u ,v ,x ,y) coordinates (when there are any), and the singularity along the initial characteristic surface. Therefore, the field equations in the new coordinate system [Eqs. 2.44)] will involve only two unknown variables instead of the three functions M , V , and U involved in Eq. We now proceed with the mathematical analysis of the initial value problem defined by Eq.

We are now in a position to write down the complete formulation in the (a,j3,.x,y) coordinate system of the metric and field equations in the interaction region of a colliding parallel polarized plane wave space-time. This completes the formulation of the initial value problem for the function V (a.,p) or, equivalently, for the function V (r ,s) [cf.

A: Inhomogeneous Kasner Singularity Before proceeding to a full mathematical analysis of the asymptotic structure. and where y is Euler's constant.19 From Eq. 3.2) it immediately follows, using the field equation (2.44b), that the asymptotic structure of the function Q ( a,l3) near a.=O is determined by. Thus, in the next paragraph, we will analyze the asymptotic behavior of the solution (2.60) near the singularity a.=O and obtain explicit forms for the functions £(13) and o(l3), expressed in the ten initial data (2.49) for the incoming waves. Then, in the continuation of this section, we will use the analysis carried out so far to investigate the asymptotic structure of the space-time metric (2.43) in the vicinity of the singular surface a.=O.

Consider the space-time metric (2.43) in the vicinity of such a world line as the observer approaches the singularity a=O at a fixed spatial coordinate 13. 3.4) and (3.7), the asymptotic behavior of the metric along the observer's world line as a~O can is expressed as. On the right-hand side of Eq. 3.14), all quantities that depend on 13 must be considered constants when the metric g (l3) is interpreted as the asymptotic limit of the metric (2.43 ); this asymptotic metric describes a region of spacetime which is arbitrarily large in the Killing x, y directions, but which extends (in general) very little [over an interval of 13 small enough that the variation in £(13) is negligible] in the 13 direction and which covers an area (0,11) in the coordinate a.