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On the Non Metrizability of Berwald Finsler Spacetimes

by 1,†, 1,†, 2,*,† and 3,†
Department of Mathematics and Computer Science, Eindhoven University of Technology, Groene Loper 5, 5612AZ Eindhoven, The Netherlands
Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia
Faculty of Mathematics and Computer Science, Transilvania University, Iuliu Maniu Str. 50, 500091 Brasov, Romania
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Universe 2020, 6(5), 64;
Received: 28 February 2020 / Revised: 24 April 2020 / Accepted: 27 April 2020 / Published: 1 May 2020
(This article belongs to the Special Issue Finsler Modification of Classical General Relativity)
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 general 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 a metric-affine geometry with vanishing torsion. View Full-Text
Keywords: Finsler geometry; Berwald spaces; Berwald spacetimes; Szabo’s theorem; metrizability Finsler geometry; Berwald spaces; Berwald spacetimes; Szabo’s theorem; metrizability
MDPI and ACS Style

Fuster, A.; Heefer, S.; Pfeifer, C.; Voicu, N. On the Non Metrizability of Berwald Finsler Spacetimes. Universe 2020, 6, 64.

AMA Style

Fuster A, Heefer S, Pfeifer C, Voicu N. On the Non Metrizability of Berwald Finsler Spacetimes. Universe. 2020; 6(5):64.

Chicago/Turabian Style

Fuster, Andrea, Sjors Heefer, Christian Pfeifer, and Nicoleta Voicu. 2020. "On the Non Metrizability of Berwald Finsler Spacetimes" Universe 6, no. 5: 64.

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