Identifying Berwald Finsler Geometries

Abstract

Berwald geometries are Finsler geometries close to (pseudo)-Riemannian geometries. We establish a simple first order partial differential equation as necessary and sufficient condition, which a given Finsler Lagrangian has to satisfy to be of Berwald type. Applied to $(\alpha ,\beta )$-Finsler spaces, respectively $(A,B)$-Finsler spacetimes, this reduces to a necessary and sufficient condition for the Levi-Civita covariant derivative of the defining $1$-form. We illustrate our results with novel examples of $(\alpha ,\beta )$-Berwald geometries which represent Finslerian versions of Kundt (constant scalar invariant) spacetimes. The results generalize earlier findings by Tavakol and van den Bergh, as well as the Berwald conditions for Randers and m-Kropina resp. very special/general relativity geometries.

arXiv comment17 pages, results on $(\alpha ,\beta )$-Finsler geometries extended, explicit examples added, updated to journal version

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Cited by in corpus (7)

• Randers pp-waves
• 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
• Relativistic deformed kinematics: from flat to curved spacetimes
• Cosmological Landsberg Finsler spacetimes
• Modelling Cosmic Springs with Finsler and Generalised Finsler Geometries