Finsler Geometry, Spacetime & Gravity -- From Metrizability of Berwald Spaces to Exact Vacuum Solutions in Finsler Gravity

Abstract

This PhD dissertation covers a range of topics in Finsler geometry and Finsler gravity, most notably: (i) the characterization of Berwald spaces, (ii) pseudo-Riemann (non-)metrizability of Berwald spaces, (iii) $(α,β)$-metrics, (iv) exact solutions to Pfeifer and Wohlfarth's vacuum field equation in Finsler gravity, and (v) Finsler gravitational waves and their observational signature. An extended abstract can be found in the dissertation itself.

Figures (4)

• Figure 1: The figures show, for several representative values of $\rho$, a $3$D projection of the light cone and the signature of the fundamental tensor of the (even dimensional) modified Randers metric $F = \text{sgn}(A)\alpha + |\beta|$, in coordinates such that at $x\in M$ one has $A = -(y^0)^2+(y^1)^2+\dots +(y^2)^{n-1}$ and $\beta = \rho \,y^0$ (in the timelike case) or $\beta = \rho(y^0+y^1)$ (in the null case) or $\beta = \rho \,y^1$ (in the spacelike case). Such coordinates can always be chosen. Green regions correspond to Lorentzian signature and red regions to non-Lorentzian signature. Figures \ref{['fig:L1']} - \ref{['fig:S3']} show the physically reasonable scenarios, where $|b|^2>-1$. In that case, two cones can be observed. The inner cone is the true light cone of $F$ (i.e. the set $F=0$), and the outer cone is the light cone of $a_{ij}$ (i.e. the set $A=0$). The only region with non-Lorentzian signature (disregarding irregularities) is precisely the gap in between the two cones, including their boundaries. If on the other hand $|b|^2=-1$ (\ref{['fig:T5']}) then the light 'cone' of $F$ is the line $y^1=y^2=y^3=0$. And if $|b|^2<-1$ (\ref{['fig:T4']}) then the light 'cone' of $F$ consists only of the origin. Therefore we deem the latter two cases not physically interesting.
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Theorems & Definitions (164)

• Definition 1.2.1
• Definition 1.2.2
• Definition 1.2.3
• Theorem 1.2.4: Euler's Theorem Bao
• Definition 1.4.1
• Definition 1.4.2
• Proposition 2.2.1
• proof
• Proposition 2.2.2
• Theorem 2.3.1: Fundamental lemma of Finsler geometry