The principle of equivalence and projective structure in spacetimes

Abstract

This paper discusses the extent to which one can determine the spacetime metric from a knowledge of a certain subset of the (unparametrised) geodesics of its Levi-Civita connection, that is, from the experimental evidence of the equivalence principle. It is shown that, if the space-time concerned is known to be vacuum, then the Levi-Civita connection is uniquely determined and its associated metric is uniquely determined up to a choice of units of measurement, by the specification of these geodesics. It is further demonstrated that if two space-times share the same unparametrised geodesics and only one is assumed vacuum then their Levi-Civita connections are again equal (and so the other metric is also a vacuum metric) and the first result above is recovered.

Projective structure and holonomy in four-dimensional Lorentz manifolds

“…If B m is spanned by the bivector F then, at m, R abcd = αF ab F cd (3) for 0 = α ∈ R and R a[bcd] = 0 implies that F a[b F cd] = 0 from which it may be checked that F is necessarily simple.…”

“…Then ∇ and ∇ ′ (or g and g ′ , or (M, g) and (M, g ′ )) are said to be projectively related (on M )). [In fact, it is sufficient that, for each m ∈ M , ∇ and ∇ ′ share a non-empty subset of unparametrised g-timelike geodesics whose directions at m span a non-empty open subset in the usual topology on the collection of 1-dimensional subspaces (directions) of T m M [3].] Although, in general, a projectively related pair ∇ and ∇ ′ may still be expected to differ, it turns out that in many interesting situations they are necessarily equal.…”

“…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.…”

“…Now apply the Ricci identity to a and use (15) to get (a ab;cd − a ab;dc =)a ae R e bcd + a be R e acd = g ac λ bd + g bc λ ad − g ad λ bc − g bd λ ac (18) where λ ab = λ a;b (= λ ba ). This leads to the following lemma which is (mostly) a special case of a more detailed result in [3,23] and for which a definition is required. Suppose m ∈ M , that F ∈ Λ m M and that the curvature tensor Riem of (M, g) satisfies R ab cd F cd = αF ab (α ∈ R) at m, so that F may be regarded as a (real) eigenvector of the map f in (1) (recalling the liberties taken in identifying bivectors described in section 1).…”

“…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.…”