Invariant Approach to the Geometry of Spaces in General Relativity

Abstract

A procedure is described for obtaining a complete, invariant classification of the local, analytic geometries and matter fields in general relativity by a finite number of algebraic steps. The approach is based on an extension of the classification scheme to include differential invariants of all orders and to provide maximally determined standard frames of vectors at each point. It is further shown that the resultant invariant functions can be replaced, in a finite number of algebraic steps, by special invariant functions which, while still uniquely representative of the geometry, can be assigned arbitrarily to produce all possible local, analytic solutions to the Einstein equations, in this representation. It is suggested that this type and special function scheme, obtainable from ideal geometric measurements in a finite number of steps, could be useful in general relativity. Unfortunately, due to the extensive algebra involved, this scheme has not yet been explicitly calculated, even for empty spaces.

A scalar invariant and the local geometry of a class of static spacetimes

“…This result, as suggested by Brans [3] and further developed by Karlhede [4], may be used to provide a coordinate independent characterization of gravitational fields in general relativity. Based on this idea, Karlhede et al [5] have investigated the simplest scalar invariant, I ≡ R µνρσ;δ R µνρσ;δ , constructed from the first covariant derivative of the curvature tensor, and found that the behavior of this locally measurable scalar reveals the effect of passage through the event horizon of the Schwarzschild spacetime -a static solution to the vacuum Einstein equation, R µν = 0.…”

“…Thus the local geometry of STE spacetimes with I B = 0 is of considerable interest. In general, for the class of 4-dimensional STE spacetimes, we find that the vanishing of I B implies that (3) Σ is locally foliated by 2-dimensional surfaces, (2) S, of constant curvature. From the resulting structure of (3) Σ, we show that any nonflat static solution to R µν = Λg µν must be either Kottler-type or Nariai-type [subsection 3.4] if I B = 0.…”

Intrinsic geometry of a null hypersurface

“…Two examples of such work are given by references [2,4]. We also note that geometrical structures, in Cartan's sense, induced on null hypersurfaces in n-dimensional space-times have been investigated in reference [3].…”

Tilted cosmological models of Bianchi type V

“…(However it is in practice often convenient to use the twice contracted Bianchi identities since they give simple relations, but they may of course be derived from the other equations). Hence the system (13,14) is the complete system. It is a set of first order ordinary differential equations, where the independent variable is ρ, and algebraic constraints.…”

“…To find the solutions we use a method for construction of solutions to Einstein's equations, [13][14][15][16], based on the invariant classification scheme by Cartan and Karlhede [17,18]. The method is shortly described in section II.…”