On the non metrizability of Berwald Finsler spacetimes

Abstract

We investigate whether Szabo's metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern-Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi-Civita connection of some Riemannian metric. We show for the first time that this result does not extend to Finsler spacetimes. More precisely, we find a large class of Berwald spacetimes for which the Ricci tensor of the affine connection is not symmetric. The fundamental difference from positive definite Finsler spaces that makes such an asymmetry possible, is the fact that generally, Finsler spacetimes satisfy certain smoothness properties only on a proper conic subset of the slit tangent bundle. Indeed, we prove that when the Finsler Lagrangian is smooth on the entire slit tangent bundle, the Ricci tensor must necessarily be symmetric. For large classes of Finsler spacetimes, however, the Berwald property does not imply that the affine structure is equivalent to the affine structure of a pseudo-Riemannian metric. Instead, the affine structure is that of metric-affine geometry with vanishing torsion.

arXiv comment12 pages, contribution to the Special Issue "Finsler Modification of Classical General Relativity" in the Journal Universe

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• Searches for beyond-Riemann gravity
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• Randers pp-waves
• Identifying Berwald Finsler Geometries
• Finsler gravitational waves of $(\alpha ,\beta )$-type and their observational signature
• Four-dimensional SO(3)-spherically symmetric Berwald Finsler spaces
• On the metrizability of $m$-Kropina spaces with closed null 1-form
• Geometry of weighted Lorentz-Finsler manifolds II: A splitting theorem
• Cosmological Landsberg Finsler spacetimes
• On the significance of the stress-energy tensor in Finsler spacetimes
• Birkhoff Theorem for Berwald Finsler spacetimes
• Berwald $m$-Kropina Spaces of Arbitrary Signature: Metrizability and Ricci-Flatness
• Metrizability of SO(3)-invariant connections: Riemann versus Finsler
• Concavity of spacetimes