Radiation from a D-dimensional collision of shock waves: Two dimensional reduction and Carter-Penrose diagram

TL;DR

This paper investigates the causal structure of a 2D reduced model of D-dimensional shock wave collisions, constructing Carter-Penrose diagrams to analyze singularities and causal boundaries for perturbative gravitational studies.

Contribution

It provides a method to construct Carter-Penrose diagrams for the 2D reduced background in shock wave collisions, applicable to all orders in perturbation theory.

Findings

Constructed conformal Carter-Penrose diagrams for the reduced model.

Identified singularities of source and Green's functions at each perturbative order.

Established a method for causal analysis in high-dimensional shock wave collisions.

Abstract

We analyse the causal structure of the two dimensional (2D) reduced background used in the perturbative treatment of a head-on collision of two $D$-dimensional Aichelburg-Sexl gravitational shock waves. After defining all causal boundaries, namely the future light-cone of the collision and the past light-cone of a future observer, we obtain characteristic coordinates using two independent methods. The first is a geometrical construction of the null rays which define the various light cones, using a parametric representation. The second is a transformation of the 2D reduced wave operator for the problem into a hyperbolic form. The characteristic coordinates are then compactified allowing us to represent all causal light rays in a conformal Carter-Penrose diagram. Our construction holds to all orders in perturbation theory. In particular, we can easily identify the singularities of the source functions and of the Green's functions appearing in the perturbative expansion, at each order, which is crucial for a successful numerical evaluation of any higher order corrections using this method.