Projective equivalence of Einstein spaces in general relativity

Abstract

There has been some recent interest in the relation between two spacetimes which have the same geodesic paths, that is, spacetimes which are projectively equivalent (sometimes called geodesically equivalent). This paper presents a short and accessible proof of the theorem that if two spacetimes have the same geodesic paths and one of them is an Einstein space then (either each is of constant curvature or) their Lévi-Civitá connections are identical. It also clarifies the relationship between their associated metrics. The results are extended to include the signatures (+ + ++) and (− − ++), and some examples and discussion are given in the case of dimension n > 4. Some remarks are also made which show how these results may be useful in the study of projective symmetry.

Projective structure and holonomy in four-dimensional Lorentz manifolds

“…), g ′ = cg on M (0 = c ∈ R) (and so (M, g ′ ) is also vacuum). For this case g ′ has the same signature as g (up to an overall minus sign [5]). This result is important for describing the power of the equivalence principle in general relativity theory.…”

“…A particularly important case of such a study arises where the original pair (M, g) is a space-time which is also an Einstein space so that the Ricci and metric tensors are related by Ricc = R 4 g. This problem has been discussed in several places (see the bibliography in [6]). The particular case which is, perhaps, of most importance in general relativity arises when the Ricci scalar vanishes and then (M, g) is a vacuum (Ricciflat ) space-time and this is discussed in [1,3,5,8]. It turns out that if (M, g) is a space-time which is an Einstein space and if g ′ is another metric on M projectively related to g, either (M, g) and (M, g ′ ) are each of constant curvature, or the Levi-Civita connections ∇ and ∇ ′ of g and g ′ , respectively, are equal.…”

“…There has been some recent interest in the study of projective relatedness of (metric) connections and, in particular, within Einstein's theory [1,2,3,4,5,6,7,8,9]. Thus, roughly speaking, one assumes that one has two Lorentz metrics on a given space-time M whose Levi-Civita connections give rise to the same set of geodesic paths (unparametrised geodesics) on the space-time manifold and then tries to find the relationship between these metrics and connections.…”

Projective structure and holonomy in general relativity

“…A solution of the problem in the general case seems to be rather difficult and the idea is to consider certain special cases in order to make the problem more tractable. For the important situation when (M, g) is a vacuum space-time a very strong result is available and has been discovered, to a large extent independently, in [2,3,4,5]. This result says that if (M, g) is vacuum and not flat and if a space-time (M, g ) with Levi-Civita connection ∇ is projectively related to it then ∇ = ∇ (and so (M, g ) is also vacuum) and, with exclusion of the case when (M, g) is a pp-wave, g = cg for some constant c [3,4].…”

“…However, the problem can, to some extent, be simplified by adopting the Sinjukov transformation ( [15], see also [4,5,8,9]). This technique involves introducing another non-degenerate second order symmetric tensor a and another 1-form λ to replace g and ψ and which are defined in terms of them by…”