Einstein-Kropina Metrics and Their Application in Finsler Gravity
Abstract
We generalize the Einstein condition for Kropina metrics obtained in the positive definite setting by Zhang, Shen and others to all signatures. As examples of Einstein-Kropina metrics L=La,b, we construct new explicit positive definite ones and the first ones with Lorentzian signature. Next, we classify all Einstein-Kropina solutions to the vacuum equation of Finsler gravity by Pfeifer and Wohlfarth, in arbitrary dimension and including a possible cosmological constant Λ. For the Lorentzian and positive definite cases, the local picture is as follows. In dimension 4 or lower, only the trivial solution exists: a Minkowski or Euclidean space, essentially. In dimension 5 and higher, the solutions are those La,b for which the metric a is a product of the real line with a Ricci-flat metric (Lorentzian or Riemannian) and the vector field b is the unique unit vector on the first factor. As a very surprising rigidity phenomenon, all Einstein-Kropina solutions to the Λ-vacuum equation are Berwald and Ricci-flat, and the cosmological constant necessarily vanishes.
Versions
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BibTeX
@misc{heefer2026einsteinkropina,
title = {{Einstein-Kropina Metrics and Their Application in Finsler Gravity}},
author = {Sjors Heefer and Fidel F. Villaseñor and Andrea Fuster},
howpublished = {arXiv},
year = {2026},
month = jun,
doi = {10.48550/arxiv.2606.07121},
eprint = {2606.07121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.07121},
}
RIS
TY - UNPB
TI - Einstein-Kropina Metrics and Their Application in Finsler Gravity
AU - Sjors Heefer
AU - Fidel F. Villaseñor
AU - Andrea Fuster
PY - 2026
DA - 2026/06/05
PB - arXiv
DO - 10.48550/arxiv.2606.07121
UR - https://arxiv.org/abs/2606.07121
AB - We generalize the Einstein condition for Kropina metrics obtained in the positive definite setting by Zhang, Shen and others to all signatures. As examples of Einstein-Kropina metrics L=L_a,b, we construct new explicit positive definite ones and the first ones with Lorentzian signature. Next, we classify all Einstein-Kropina solutions to the vacuum equation of Finsler gravity by Pfeifer and Wohlfarth, in arbitrary dimension and including a possible cosmological constant Λ. For the Lorentzian and positive definite cases, the local picture is as follows. In dimension 4 or lower, only the trivial solution exists: a Minkowski or Euclidean space, essentially. In dimension 5 and higher, the solutions are those L_a,b for which the metric a is a product of the real line with a Ricci-flat metric (Lorentzian or Riemannian) and the vector field b is the unique unit vector on the first factor. As a very surprising rigidity phenomenon, all Einstein-Kropina solutions to the Λ-vacuum equation are Berwald and Ricci-flat, and the cosmological constant necessarily vanishes.
ER -