On the Generation of Equivalent &amp;lsquo;Black Hole&amp;rsquo; Metrics: A Review

Stephen J Crothers

doi:10.3844/ajssp.2015.28.44

Abstract

Various line-elements purporting different types of black hole universes have been advanced by cosmologists but a means by which the required infinite set of equivalent metrics can be generated has evaded them. Without such a means the theory of black holes is not only incomplete but also ill-posed. Notwithstanding, the mathematical form by which the infinite set of equivalent metrics is generated was first revealed in 2005, from other quarters and it has in turn revealed significant properties of black hole universes which cosmology has not realised. The general metric ground-form from which the infinite set of equivalent 'black hole' metrics can be generated is presented herein and its consequences explored.

Highlights

The simplest black hole solution to Einstein’s field equations is the ‘Schwarzschild solution’
Purported black hole metrics are not restricted to spherical symmetry or the absence of matter, the latter being, according to Einstein, everything except his gravitational field (Einstein, 1916)
A means for generating an infinite set of equivalent solutions for black hole universes must not be restricted to spherical symmetry in the absence of matter; the restriction applied by (Fromholz et al, 2013)

Summary

Introduction

The simplest black hole solution to Einstein’s field equations is the ‘Schwarzschild solution’. Expressions (18) constitute the metric ground-form for generating the infinite set of equivalent solutions for Schwarzschild spacetime. Equations (5) and (25) reveal the nature of this parameter -it is both the radius and the inverse square root of the Gaussian curvature of the spherically symmetric surface in the spatial section of metric (5) for Minkowski spacetime. The metric ground-form to generate an infinite set of equivalent solutions for Schwarzschild spacetime in isotropic coordinates is (Crothers, 2006):. Gaussian curvature for the surface in the spatial section of the Kerr-Newman ground-form (32) has been obtained (Crothers, 2014b), which reduces to the Schwarzschild form Note that if q = 0 (42) reduces to the invariant Gaussian curvature of the surface in the spatial section of the isotropic Schwarzschild ground-form.

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Conclusion