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It is well known that when gravitational plane waves propagating on an otherwise flat background collide, they produce spacetime singularities. In this paper we consider the problem of whether (or under what conditions) singularities can be produced by the collision of gravitational waves with finite but very large transverse sizes. On the basis of (nonrigorous) order-of-magnitude considerations, we discuss the outcome of the collision in two fundamentally different regimes for the parameters of the colliding waves; these parameters are the transverse sizes ( L T ) i , typical amplitudes h i , typical reduced wavelengths λ / i ≡ λ i /2π, thickneses a i , and focal lengths f i ∼λ / i 2 / a i h i 2 ( i =1,2) of the waves 1 and 2. For the first parameter regime where ( L T ) 1 ≫( L T ) 2 and h 1 ≫ h 2 , we conjecture the following. (i) If ( L T ) 2 ≪√λ / 2 f 1 ( h 1 / h 2 ) 1 / 4 , the almost-plane wave 2 will be focused by will be focused by wave 1 down to a finite, minimum size, then diffract and disperse Fig. 1(a). (ii) If ( L T ) 2 ≫√λ / 2 f 1 ( h 1 / h 2 ) 1 / 4 (and if wave 1 is sufficiently anastigmatic), wave 2 will be focused by wave 1 so strongly that it forms a singularity surrounded by a horizon, and the end result is a black hole flying away from wave 1 Fig. 1(b). For the second parameter regime where ( L T ) 1 ∼( L T ) 2 ≡ L T and h 1 ∼ h 2 , we conjecture that if L T ≫√ f 1 f 2 ≡ f , a horizon forms around the two colliding waves shortly before their collision, and the collision produces a black hole that is at rest with respect to the reference frame in which f 1 ∼ f 2 ∼ f (Fig. 2). As a first step in proving this conjecture, we give a rigorous analysis of the second regime in the case L T ≫ f , for the special situation of colliding parallel-polarized (almost-plane) gravitational waves which are exactly plane-symmetric across a region of transverse size ≫f , but which fall off in an arbitrary way at larger transverse distances. Our rigorous analysis shows that this collision is guaranteed to produce a spacetime singularity with the same local structure as in an exact plane-wave collision, but it does not prove that the singularity is surrounded by a horizon.
Physical Review D – American Physical Society (APS)
Published: Sep 15, 1988
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