On the metrizability of m -Kropina spaces with closed null one-form

authored by: Sjors Heefer, Christian Pfeifer, Jorn Van Voorthuizen, Andrea Fuster
Abstract: We investigate the local metrizability of Finsler spaces with m-Kropina metric F = α1+mβ-m, where β is a closed null one-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric α and one-form β have a very specific form in certain coordinates. In particular, when the signature of α is Lorentzian, α belongs to a certain subclass of the Kundt class and β generates the corresponding null congruence, and this generalizes in a natural way to arbitrary signature. We use this result to prove that the affine connection on such an m-Kropina space is locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor constructed from the affine connection is symmetric. In particular, we construct all counterexamples of this type to Szabo's metrization theorem, which has only been proven for positive definite Finsler metrics that are regular on all of the slit tangent bundle.
Organisation(s): QuantumFrontiers
External Organisation(s): Eindhoven University of Technology (TU/e)
University of Bremen
Type: Article
Journal: Journal of mathematical physics
Volume: 64
ISSN: 0022-2488
Publication date: 01.02.2023
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Statistical and Nonlinear Physics, Mathematical Physics
Electronic version(s): https://doi.org/10.1063/5.0130523 (Access: Open)

BibTeX

@article{88e7368725b14b48869a4d1176767682,
title = "On the metrizability of m -Kropina spaces with closed null one-form",
abstract = "We investigate the local metrizability of Finsler spaces with m-Kropina metric F = α1+mβ-m, where β is a closed null one-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric α and one-form β have a very specific form in certain coordinates. In particular, when the signature of α is Lorentzian, α belongs to a certain subclass of the Kundt class and β generates the corresponding null congruence, and this generalizes in a natural way to arbitrary signature. We use this result to prove that the affine connection on such an m-Kropina space is locally metrizable by a (pseudo-)Riemannian metric if and only if the Ricci tensor constructed from the affine connection is symmetric. In particular, we construct all counterexamples of this type to Szabo's metrization theorem, which has only been proven for positive definite Finsler metrics that are regular on all of the slit tangent bundle.",
author = "Sjors Heefer and Christian Pfeifer and {Van Voorthuizen}, Jorn and Andrea Fuster",
note = "Funding Information: C.P. was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Project No. 420243324 and acknowledges support from cluster of excellence Quantum Frontiers funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany{\textquoteright}s Excellence Strategy—EXC-2123 QuantumFrontiers—390837967. All of us would like to acknowledge networking support provided by the COST Action CA18108, supported by COST (European Cooperation in Science and Technology). ",
year = "2023",
month = feb,
day = "1",
doi = "10.1063/5.0130523",
language = "English",
volume = "64",
journal = "Journal of mathematical physics",
issn = "0022-2488",
publisher = "American Institute of Physics",
number = "2",
}