On the compatibility of Lorentz metrics with linear connections on four-dimensional manifolds

Abstract

Abstract. This paper considers 4-dimensional manifolds upon which there is a Lorentz metric h and a symmetric connection Γ and which are originally assumed unrelated. It then derives sufficient conditions on h and Γ (expressed through the curvature tensor of Γ) for Γ to be the Levi-Civita connection of some (local) Lorentz metric g and calculates the relationship between g and h. Some examples are provided which help to assess the strength of the sufficient conditions derived.

Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds

“…To see this note that g ′ need not have Lorentz signature (−, +, +, +) but, if this is insisted upon, the curvature classification scheme, including the nature (timelike, spacelike or null) of F and * F in class B, r in class C and F in class D is the same whether taken for (the common curvature tensor) Riem with g or Riem with g ′ . In this sense it is a classification of Riem, independent of the metric generating Riem [6].…”

The principle of equivalence and projective structure in spacetimes

“…where, as mentioned earlier, Ru denotes the covector at p with components R ′ ab u b and similarly for Rz, Rx and Ry. Recalling that in this notation, ẽa are, respectively, the covectors with components u a , z a , x a and y a , one can substitute these expansions into (19) and (20) and equate coefficients of the e ′ a e ′ b in an obvious way. One finds that several of these vanish and certain others are forced to be equal.…”

“…To see this, consider the following example (see and cf. [19,20,21]). Let U be the open subset of R 4 given by x 0 > 0 and let ϕ : U → R be given by ϕ(x 0 , x 1 , x 2 , x 3 )=log x 0 .…”

“…Then, with a closed 1-form ψ defined on U by the components ψ a = (1 − e ϕ ) −1 ϕ a , define another connection ∇ ′ on U by (52). Then ∇ ′ is the desired connection on U because it is known that ∇ ′ is not a metric connection [19,20]]. Thus, knowledge of the total (unparametrised) geodesic structure of space-time does not determine whether the connection is metric or not.…”

The Reciprocal of the Fundamental Theorem of Riemannian Geometry

“…Finding the metric has been studied extensively [5,6,8,9,10,11,12,13,14,16,19,20,21,23,24,25,26,27,28]. These works provide a mixture of somewhat general algorithms and studies of existence and uniqueness.…”