Affine sphere spacetimes which satisfy the relativity principle

Abstract

In the context of Lorentz-Finsler spacetime theories the relativity principle holds at a spacetime point if the indicatrix (observer space) is homogeneous. We point out that in four spacetime dimensions there are just three kinematical models which respect an exact form of the relativity principle and for which all observers agree on the spacetime volume. They have necessarily affine sphere indicatrices. For them every observer which looks at a flash of light emitted by a point would observe, respectively, an expanding (a) sphere, (b) tetrahedron, or (c) cone, with barycenter at the point. The first model corresponds to Lorentzian relativity, the second one has been studied by several authors though the relationship with affine spheres passed unnoticed, and the last one has not been previously recognized and it is studied here in some detail. The symmetry groups are O + (3, 1), R 3 , O + (2, 1) × R, respectively. In the second part, devoted to the general relativistic theory, we show that the field equations can be obtained by gauging the Finsler Lagrangian symmetry while avoiding direct use of Finslerian curvatures. We construct some notable affine sphere spacetimes which in the appropriate velocity limit return the Schwarzschild, Kerr-Schild, Kerr-de Sitter, Kerr-Newman, Taub, and FLRW spacetimes, respectively. CONTENTS

Affine Sphere Relativity

“…For space reasons the discussion of the relativity principle and a study of some models satisfying it will be given in a different work [69].…”

Riemann–Finsler geometry and Lorentz-violating scalar fields

“…We can therefore extend the interpretation of Eqs. (17) and (18) to d ≥ n−2 when desired. Note that from this perspective the Finsler structure F (d) 1 with d > n can be viewed as a natural generalization of the Randers structure, in which the 1-form is replaced by a symmetric (d − n + 2)-form.…”

“…Next, to gain insight about the various Riemann-Finsler spaces governed by (k (d) ) j 1 ... j d−n+2 , we perform some explicit calculations for the Finsler structure (18). The expressions for key properties below are derived at first order in k (d) j 1 ... j l .…”

Gravity theory through affine spheres

“…In [2] we have argued that the relativity principle holds at an event x if a linear subgroups of the endomorphisms of T x M acts transitively on the indicatrix I x . In other words, the velocity space is homogeneous and so for an observer it becomes impossible to infer its position on velocity space from local measurements.…”

“…In this work we are going to present some recent results on anisotropic gravity theories obtained in [1,2]. Although, ultimately the theory is of Lorentz-Finsler type we wish to introduce it from a different angle, emphasizing the role of affine differential geometry in its development.…”