Finsler geometric extension of Einstein gravity

We construct gravitational dynamics for Finsler spacetimes in terms of an action integral on the unit tangent bundle. These spacetimes are generalizations of Lorentzian metric manifolds which satisfy necessary causality properties. A coupling procedure for matter fields to Finsler gravity completes our new theory that consistently becomes equivalent to Einstein gravity in the limit of metric geometry. We provide a precise geometric definition of observers and their measurements and show that the transformations, by means of which different observers communicate, form a groupoid that generalizes the usual Lorentz group. Moreover, we discuss the implementation of Finsler spacetime symmetries. We use our results to analyze a particular spacetime model that leads to Finsler geometric refinements of the linearized Schwarzschild solution.

Figure 1

Geometric structures implemented in every tangent space $T_{x}M$ by definition 1 of Finsler spacetimes. The solid lines show the guaranteed cone of timelike vectors with the shells of unit timelike vectors and null boundary. The dotted lines indicate a potentially more complex null structure.

Figure 2

Tangent bundle geometry: decomposition of $T_{(x,y)}TM$ into horizontal and vertical parts.

Figure 3

Transformation between two observer frames: the frame $\{e_{\mu }\}$ in $H_{(x,y)}TM$ is first parallely transported to $\{{\widehat{e}}_{\mu }\}$ in $H_{(x,z)}TM$ and second, Lorentz transformed into the final frame $\{f_{\mu }\}$.

Figure 4

Numerical fly-by solutions of the geodesic equations for linearized Schwarzschild geometry (dashed line) and the bimetric Finsler refinement (solid line) with $a_1=0$, $a_3\simeq 0.156$, and $a_4=0.1$. The mass is centered at the origin and has Schwarzschild radius $r_0=0.1$. The initial conditions are $r(0)=0.5$, $\dot{r}(0)=0.02$, $\varphi (0)=0$, $\dot{\varphi }(0)=1.1$, and $t(0)=0$ for both curves, and $\dot{t}(0)$ is calculated from the respective unit normalization condition $F(x,\dot{x})=1$.