Abstract
We revisit the physical arguments that led to the definition of the stress–energy tensor T in the Lorentz–Finsler setting (M,L) starting with classical relativity. Both the standard heuristic approach using fluids and the Lagrangian one are taken into account. In particular, we argue that the Finslerian breaking of Lorentz symmetry makes T an anisotropic 2-tensor (i.e., a tensor for each L-timelike direction), in contrast with the energy-momentum vectors defined on M. Such a tensor is compared with different ones obtained by using a Lagrangian approach. The notion of divergence is revised from a geometric viewpoint, and, then, the conservation laws of T for each observer field are revisited. We introduce a natural anisotropic Lie bracket derivation, which leads to a divergence obtained from the volume element and the non-linear connection associated with L alone. The computation of this divergence selects the Chern anisotropic connection, thus giving a geometric interpretation to previous choices in the literature.
Highlights
This article has a double aim in Lorentz–Finsler geometry
We introduce new notions of the Lie bracket and the derivative associated with a nonlinear connection and applicable to anisotropic tensors fields, which appear naturally in Finsler geometry
Finslerian modifications of General Relativity aim to find a tensor T collecting the possible anisotropies in the distribution of energy, momentum, and stress, which will serve as a source for the ( Lorentz–Finsler) geometry of the spacetime [1,2,3,4,5]
Summary
This article has a double aim in Lorentz–Finsler geometry. The first one is to revisit the physical grounds of the stress–energy tensor T, Section 3. The second one is to revise geometrically the notion of divergence, Section 4, yielding consequences about the conservation of T, Section 5 With this aim, we introduce new notions of the Lie bracket and the derivative associated with a nonlinear connection and applicable to anisotropic tensors fields, which appear naturally in Finsler geometry. Section 2.3), and (b) there is a significant variety of possible extensions of the relativistic kinematic objects to the Finsler case, at least from the geometric viewpoint (see Appendix A) In the Finslerian case, the Lie derivative and bracket do not make sense for arbitrary anisotropic vector fields This difficulty was circumvented by Rund [18], who redefined div( Z ) in such a way that a type of divergence theorem held. We show that a combination of Rund’s and Minguzzi’s methods to compute the boundary terms allows one to obtain appropriate decay rates (namely, the properly Finslerian hypothesis (49)), which ensure the conservation
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