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Spherically-symmetric solutions in general relativity using a tetrad-based approach

Published version
Peer-reviewed

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Authors

Lasenby, AN 
Kim, Do Young 
Hobson, Michael 

Abstract

We present a tetrad-based method for solving the Einstein field equations for spherically-symmetric systems and compare it with the widely-used Lemaître– Tolman–Bondi (LTB) model. In particular, we focus on the issues of gauge ambiguity and the use of comoving versus ‘physical’ coordinate systems. We also clarify the correspondences between the two approaches, and illustrate their differences by applying them to the classic examples of the Schwarzschild and Friedmann–Lemaître– Robertson–Walker spacetimes. We demonstrate that the tetrad-based method does not suffer from the gauge freedoms inherent to the LTB model, naturally accommodates non-uniform pressure and has a more transparent physical interpretation. We further apply our tetrad-based method to a generalised form of ‘Swiss cheese’ model, which consists of an interior spherical region surrounded by a spherical shell of vacuum that is embedded in an exterior background universe. In general, we allow the fluid in the interior and exterior regions to support pressure, and do not demand that the interior region be compensated. We pay particular attention to the form of the solution in the intervening vacuum region and illustrate the validity of Birkhoff’s theorem at both the metric and tetrad level. We then reconsider critically the original theoretical arguments underlying the so-called Rh = ct cosmological model, which has recently received considerable attention. These considerations in turn illustrate the interesting behaviour of a number of ‘horizons’ in general cosmological models.

Description

Keywords

Lemaitre-Tolman-Bondi model, Swiss cheese model, Spherically symmetric solutions, Inhomogeneities, R-h = ct model

Journal Title

General Relativity and Gravitation

Conference Name

Journal ISSN

0001-7701
1572-9532

Volume Title

50

Publisher

Springer Nature
Sponsorship
Science and Technology Facilities Council (ST/M001172/1)