Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 10

The Einstein–Hilbert–Palatini formalism in pseudo-Finsler geometry

Pages: 3563 – 3631

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n10.a5

Authors

Miguel Ángel Javaloyes (Departamento de Matemáticas, Universidad de Murcia, Spain)

Miguel Sánchez (Departamento de Geometría y Topología, Facultad de Ciencias & IMAG (Centro de Excelencia María de Maeztu), Universidad de Granada, Spain)

Fidel F. Villaseñor (Departamento de Geometría y Topología, Facultad de Ciencias & IMAG (Centro de Excelencia María de Maeztu), Universidad de Granada, Spain)

Abstract

A systematic development of the so-called Palatini formalism is carried out for pseudo-Finsler metrics $L$ of any signature. Substituting in the classical Einstein–Hilbert–Palatini functional the scalar curvature by the Finslerian Ricci scalar constructed with an independent nonlinear connection $N$, the affine and metric equations for $(N,L)$ are obtained. In Lorentzian signature with vanishing mean Landsberg tensor Lani, both the Finslerian Hilbert metric equation and the classical Palatini conclusions are recovered by means of a combination of techniques involving the (Riemannian) maximum principle and an original argument about divisibility and fiberwise analyticity. Some of these findings are also extended to classical Riemannian solutions by using the eigenvalues of a Laplacian. When $\operatorname{Lan}_i \neq 0$, the Palatini conclusions fail necessarily, however, a good number of properties of the solutions remain. The framework and proofs are built up in detail.

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MAJ was partially supported by the projects PGC2018-097046-B-I00 and PID2021-124157NB-I00 funded by MCIN/ AEI /10.13039/501100011033/ FEDER “Una manera de hacer Europa” and Fundación Séneca project with reference 19901/GERM/15. This work is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. MS and FFV were partially supported by the project PID2020-116126GBI00 funded by MCIN/AEI/10.13039/501100011033, by the project PY20-01391 (PAIDI 2020) funded by Junta de Andalucía—FEDER and by the framework of IMAG-María de Maeztu grant CEX2020-001105-M funded by MCIN/AEI/10.13039/50110001103. FFV is partially supported also by an FPU grant (Formaci ón de Profesorado Universitario) from the Spanish Ministerio de Universidades.

Published 25 March 2024