[Paper Review] Toward a Gravitation Theory in Berwald--Finsler Space

Xin Li, Zhe Chang|ArXiv.org|Nov 13, 2007

Advanced Differential Geometry Research1 references6 citations

TL;DR

This paper proposes a gravitational theory in Berwald-Finsler space using the Chern connection and Bianchi identities, leading to a nonsymmetric geometric field equation that implies spontaneous local Lorentz invariance violation. Nontrivial solutions are derived for Randers-type Berwald spaces, including a 5D cosmological model with a constant vector field, demonstrating the viability of Finsler geometry as a framework for Lorentz-violating gravity beyond general relativity.

ABSTRACT

Finsler geometry is a natural and fundamental generalization of Riemann geometry. The Finsler structure depends on both coordinates and velocities. It is defined as a function on tangent bundle of a manifold. We use the Bianchi identities satisfied by Chern curvature to set up a gravitation theory in Berwald-Finsler space. The geometric part of the gravitational field equation is nonsymmetric in general. This indicates that the local Lorentz invariance is violated. Nontrivial solutions of the gravitational field equation are presented.

Motivation & Objective

• To develop a gravitational field theory in Berwald-Finsler geometry as a generalization of general relativity.
• To investigate the implications of local Lorentz invariance violation in Finsler spacetime.
• To construct nontrivial solutions of the gravitational field equation in Berwald-Finsler spaces, particularly in Randers-type geometries.
• To explore the role of the Chern connection and curvature in formulating a consistent gravity theory beyond Riemannian geometry.

Proposed method

• Utilizes the torsion-free Chern connection on Finsler manifolds to define a canonical linear connection compatible with the Finsler metric.
• Applies the first and second Bianchi identities of the Chern curvature to derive the field equations in Berwald-Finsler space.
• Derives the gravitational field equation in the form $ R_{jl} - \frac{1}{2}g_{jl}S + \left\{ \frac{1}{2}B^{k}_{kjl} + B^{k}_{jlk} \right\} = 8\pi G T_{jl} $, where the geometric part is generally nonsymmetric.
• Imposes the Berwald condition $ P^i_{jkl} = 0 $, reducing the curvature to the $ hh $-part and simplifying the field equation.
• Constructs explicit solutions in 4D and 5D Randers spaces of Berwald type, using the Robertson-Walker metric and a constant 1-form $ \tilde{b}_i $.
• Verifies that in Berwald spaces, the geodesic spray coefficients and Chern connection reduce to the Levi-Civita connection of the underlying Riemannian metric.

Experimental results

Research questions

• RQ1 How can a consistent gravitational field theory be formulated in Berwald-Finsler geometry using the Chern connection and curvature?
• RQ2 What are the implications of the nonsymmetric geometric part in the field equation for local Lorentz invariance?
• RQ3 Can nontrivial solutions of the gravitational field equation be constructed in Berwald-Finsler spaces, particularly in cosmological settings?
• RQ4 How do the field equations reduce in Randers spaces of Berwald type, and what is the role of the vector field $ \tilde{b}_i $ in generating solutions?

Key findings

• The geometric part of the gravitational field equation in Berwald-Finsler space is generally nonsymmetric, indicating spontaneous violation of local Lorentz invariance.
• The field equation reduces to $ R^{i}_{jl} - \frac{1}{2}g^{i}_{l}S + \left\{ \frac{1}{2}B^{k}_{kjl} + B^{k}_{jlk} \right\} = 8\pi G T_{jl} $, with the energy-momentum tensor $ T_{jl} $ not necessarily symmetric.
• In Randers spaces of Berwald type, the geodesic spray coefficients and Chern connection reduce to the Levi-Civita connection of the underlying Riemannian metric $ \tilde{a}_{ij} $, simplifying dynamics.
• A nontrivial 5D solution is found with $ \tilde{a}_{ij} = \text{diag}(1, -a^2(t)/(1-kr^2), -a^2(t)r^2, -a^2(t)r^2\sin^2\theta, 0) $ and $ \tilde{b}_i = \{0,0,0,0,c\} $, where $ c $ is a constant.
• The condition $ \tilde{b}_{i|j} = 0 $ holds in Berwald-type Randers spaces, ensuring compatibility with the Riemannian Christoffel symbols.
• The field equation admits solutions in 4D with the Robertson-Walker metric under the constraint $ \dot{a}^2 + k = 0 $, indicating a consistent cosmological model in this framework.