On the non metrizability of Berwald Finsler spacetimes

Andrea Fuster, Sjors Heefer, Christian Pfeifer (Corresponding author), Nicoleta Voicu

Research output: Contribution to journalArticleAcademicpeer-review

2 Citations (Scopus)
23 Downloads (Pure)


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.
Original languageEnglish
Article number64
Number of pages12
Issue number5
Publication statusPublished - 1 May 2020

Bibliographical note

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


  • differential geometry
  • general relativtiy
  • mathematics
  • finsler geometry
  • finsler graviy
  • Finsler geometry
  • Berwald spaces
  • Metrizability
  • Szabo's theorem
  • Berwald spacetimes

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