The Einstein-Hilbert-Palatini formalism in Pseudo-Finsler Geometry

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 $\mathrm{N}$, the affine and metric equations for $(\mathrm{N},L)$ are obtained. In Lorentzian signature with vanishing mean Landsberg tensor Lan${}_i$, 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 (positive definite) Riemannian metrics by using the eigenvalues of the Laplacian. When Lan${}_i\ne 0$, the Palatini conclusions fail necessarily, however, a good number of properties of the solutions remain. The framework and proofs are built up in detail.

Minor modifications of style and updated reference. To appear in Adv. Theor. Math. Phys. 50 pages

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Cited by in corpus (6)

• An anisotropic gravity theory
• Cosmological Landsberg Finsler spacetimes
• On the significance of the stress-energy tensor in Finsler spacetimes
• From kinetic gases to an exponentially expanding universe - The Finsler-Friedmann equation
• General off-diagonal integrability of metric and nonmetric geometric flow and Finsler-Lagrange-Hamilton modified Einstein equations
• Characteristic tensors for almost Finsler manifolds