SPECIAL ISSUE — Selected Papers of III Amazonian Symposium on Physics; Guest Editors: Carlos A. R. Herdeiro, Emanuele Berti, Vitor Cardoso, Luis C. B. Crispino, Leonardo Gualtieri and Ulrich SperhakeNo Access

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

Flávio S. Coelho

Departamento de Física da Universidade de Aveiro, Campus de Santiago, 3810-183 Aveiro, Portugal

Center for Research and Development in Mathematics and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal

E-mail Address: [email protected]

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and

Marco O. P. Sampaio

http://orcid.org/0000-0001-9493-2770

Departamento de Física da Universidade de Aveiro, Campus de Santiago, 3810-183 Aveiro, Portugal

Center for Research and Development in Mathematics and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal

E-mail Address: [email protected]

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https://doi.org/10.1142/S0218271816410108Cited by:0

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Abstract

We analyze 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.

Keywords:

PACS: 04.50.−h, 04.20.Cv, 04.30.Db, 04.50.Gh

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Published: 9 May 2016

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Vol. 25, No. 09
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Received 28 March 2016

Accepted 3 April 2016
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Radiation from a D-dimensional collision of shock waves: Two-dimensional reduction and Carter–Penrose diagram

Abstract

References

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