On limits of spacetimes-a coordinate-free approach

Abstract

A coordinate-free approach to limits of spacetimes is developed. The limits of the Schwarzschild metric as the mass parameter tends to 0 or ∞ are studied, extending previous results. Besides the known Petrov type D and 0 limits, three vacuum plane-wave solutions of Petrov type N are found to be limits of the Schwarzschild spacetime.

Selected Solutions of Einstein’s Field Equations: Their Role in General Relativity and Astrophysics

“…The two metrics g αβ and g ′ αβ are then equivalent if and only if there exist coordinate and Lorentz transformations under which one whole set of frame components goes into the other. In a practical algorithm given by Karlhede [51], recently summarized and used in [52], the number of derivations required is reduced.…”

The Levi-Civita space–time as a limiting case of the γ space–time

“…Since the limit m → ∞ taken on the γ metric in the form (2.1) diverges, we use the Cartan scalar approach to obtain a finite limit [7,8].…”

“…Later, [7] re-obtained these limits by using the Cartan scalar technique, and extended the results presenting new limits of the Schwarzschild metric and developing an approach to find all limits of a given spacetime (see also [8,15,16])…”

Inhomogeneous cosmological models with homogeneous inner hypersurface geometry

“…Limits of space-times are considered in [12] using a coordinate-free approach, cf. also the references cited in [12].…”

“…also the references cited in [12]. In [7, 13 -16] a topology in the space of Lie algebras was considered which shall be put in relation to the limit of space-times in sct.…”