Colliding plane waves in general relativity by J.B. Griffiths

By J.B. Griffiths

The collision and non-linear interplay of aircraft waves in Einstein's common conception of relativity has obtained massive recognition lately. at first, it was once largely inspiration that such collisions inevitable produce curvature singularities. extra lately, although, an incredibly wealthy constitution of such space-times has been came across. This quantity offers a unified and accomplished survey to the present learn during this subject with a purpose to be compatible for graduate scholars and learn employees whose learn lies quite often relativity. the 1st 8 chapters current the historical past to the topic, introduce the sphere equations, and contain a dialogue of a few qualitative facets in their answer. an in depth account is integrated of the Kahn-Penrose answer because it shows the overall personality of so much colliding airplane wave options. The latter 1/2 the booklet is dedicated to a listing of extra precise suggestions describing the collision of either gravitational and electromagnetic airplane waves. This features a dialogue of the importance of recognized suggestions and a precis of themes of present learn curiosity. for this reason, the ebook will serve either as a useful study reference and in addition because the capacity to educate and learn this lively zone of study normally relativity.

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However, the metric functions U , V , W and M must take different forms in the four regions. It is now necessary to consider the conditions that should be imposed on these functions at the boundaries of the four regions. 1) imply that V , W and M are continuous and that U is smooth across these null boundaries. 24), U = − log(f (u) + g(v)). 2) It is therefore necessary that f (u) and g(v) are at least C 1 . 20) to describe a collision of plane waves, it must be assumed that U , V , W and M are functions of u and v in region IV, are functions of u only in region II, are functions of v only in region III, and are constants in region I.

Assume initially that the congruence starts in a vacuum region of space-time with Φ00 and Ψ0 both zero, and that the geodesics are parallel having zero contraction, twist and shear, so that ρ and σ are both zero. 1) are identically satisfied. Now assume that this congruence enters a region containing matter, for which Φ00 is non-zero. 1a) implies that ρ must become increasingly positive. This indicates that the congruence must start to contract. Eventually this congruence will focus. These properties have been described in detail by Penrose (1966).

1) imply that V , W and M are continuous and that U is smooth across these null boundaries. 24), U = − log(f (u) + g(v)). 2) It is therefore necessary that f (u) and g(v) are at least C 1 . 20) to describe a collision of plane waves, it must be assumed that U , V , W and M are functions of u and v in region IV, are functions of u only in region II, are functions of v only in region III, and are constants in region I. Concentrating initially on the metric function e−U = f + g, it is appropriate to choose f= g= 1 2 1 2 for u ≤ 0, for v ≤ 0, f (0) = 0 g (0) = 0.

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