The causal structure of spacetime is a parameterized Randers geometry

Abstract

There is a by now well-established isomorphism between stationary 4-dimensional spacetimes and 3-dimensional purely spatial Randers geometries -these Randers geometries being a particular case of the more general class of 3-dimensional Finsler geometries. We point out that in stably causal spacetimes, by using the (timedependent) ADM decomposition, this result can be extended to general non-stationary spacetimes -the causal structure (conformal structure) of the full spacetime is completely encoded in a parameterized (t-dependent) class of Randers spaces, which can then be used to define a Fermat principle, and also to reconstruct the null cones and causal structure.

Emergent Models for Gravity: an Overview of Microscopic Models

“…Research on the concrete impact of Finsler geometry is an ongoing effort. The interested reader can see [125,187,188,189,200,201,202,203,204,219] for some concrete proposal to define Finslerian structures in the Lorentzian signature to open the road for a concrete investigation of the potential implications on phenomenology.…”

Efficient Computation of Null Affine Parameters

“…As long as there is at least some symmetry, one can be reasonably explicit regarding the affine parameter. But ultimately, we shall see that in the completely general case, one can at best nonconstructively argue for the existence of a purely formal Finslerian [96][97][98][99][100][101][102][103] result.…”

“…As long as there was at least some symmetry-at least having the spacetime conformally related to a spacetime containing a conformal timelike Killing vector-then we have already demonstrated the existence of an affine three-metric h ij completely characterizing the possible null affine parameters. If we try to dispense with symmetries altogether, then the best that can be conducted is an implicit Finsler-like construction [96][97][98][99][100][101][102][103]. Indeed, if we write…”

Geodesics for the Painlevé–Gullstrand Form of Lense–Thirring Spacetime

“…(For a textbook-level physically motivated discussion of the tetrad formalism see, for instance, reference [9].) These purely mathematical observations make this spacetime of particular theoretical interest [10,11]. We point out that, while the nomenclature "lapse function" is borrowed from the ADM foliation formalism [12,13], beyond this purely kinematic adaptation of the terminology, no direct use of the ADM formalism is made in this article.…”