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

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 R h = ct cosmological model, which has recently B Do Young Kim

Perturbations and the Future Conformal Boundary

“…This can be done using a tetrad approach to GR (see e.g. [19]), but we are going to indicate the needed relationship schematically here, using some notation from 'Gauge Theory Gravity' (GTG -see [20]), which is particularly convenient for conformal metrics. The notation is for a vector-valued function of vectors, h, which is essentially the square root of the metric, and arises as the local gauge field corresponding to gauging translations.…”

The origin of rest-mass energy

“…The timedependent gravitational horizon is not necessarily a null surface, but is sometimes confused with one. Some [53][54][55][56] have suggested that objects beyond R h (t 0 ) ≡ c/H 0 are observable today (at time t 0 ), which is not correct [57][58][59]. Almost certainly some of this discourse is due to a confusion between coordinate and proper speeds in general relativity.…”

An alternative approach to modelling a cosmic void and its effect on the cosmic microwave background

“…As a consequence, we have for some time adopted a different, tetrad-based method for solving the Einstein field equations for spherically-symmetric systems (Lasenby et al 1998;Nandra et al 2012aNandra et al ,b, 2013Kim et al 2018). Aside from straightforwardly accommodating pressure (which we will not consider here), the method has no gauge ambiguities in non-vacuum regions and is expressed in terms of a non-comoving radial coordinate that results in a Eulerian picture of the fluid evolution with a clear physical interpretation.…”