Geodesic motion in Bogoslovsky-Finsler Spacetimes

TL;DR

The paper analyzes the free motion of massive particles in Bogoslovsky-Finsler deformations of plane gravitational waves and of Friedmann-Lemaître cosmologies, revealing that transverse dynamics remain identical to the background while the longitudinal coordinate acquires a $b$-dependent correction. By deriving the Finsler geodesics in Baldwin-Jeffery-Rosen coordinates and examining the underlying Carroll/VSR-type symmetries, the authors show that the partially broken Carroll symmetry persists and leads to integrable dynamics, with the deformed U-V boost blending with dilations to produce conserved quantities. The work includes an Einstein-Maxwell pp-wave example and extends the construction to a BF-Friedmann-Lemaître cosmology, where masses can become time-dependent via a conformal factor $f(\\eta)$. These results establish the viability and symmetry structure of Bogoslovsky-Finsler spacetimes in GR contexts and hint at potential observational consequences in anisotropic early-universe scenarios.

Abstract

We study the free motion of a massive particle moving in the background of a Finslerian deformation of a plane gravitational wave in Einstein's General Relativity. The deformation is a curved version of a one-parameter family of Relativistic Finsler structures introduced by Bogoslovsky, which are invariant under a certain deformation of Cohen and Glashow's Very Special Relativity group ISIM(2). The partially broken Carroll Symmetry we derive using Baldwin-Jeffery-Rosen coordinates allows us to integrate the geodesics equations. The transverse coordinates of timelike Finsler-geodesics are identical to those of the underlying plane gravitational wave for any value of the Bogoslovsky-Finsler parameter $b$. We then replace the underlying plane gravitational wave by a homogenous pp-wave solution of the Einstein-Maxwell equations. We conclude by extending the theory to the Finsler-Friedmann-Lemaitre model.

Figures (6)

• Figure 1: Consistently with \ref{['transmot']}, the Bogoslovsky-Finsler geodesics project to the same curve in 2D transverse space for all values of the parameter $b$ while their $v$ coordinates differ, according to \ref{['BFvgeos']}, in a $b$-dependent term, which is linear in retarded time, $u$. Experiments indicate that the anisotropy and hence $b$ is very small. When $b\to1$ the trajectory approaches to the massless one (in heavy black), consistently with \ref{['BFvgeos']}.
• Figure 2: The conformal factor \ref{['f(eta)']} of the Friedmann-Lemaı̂tre model \ref{['FLmetric']}, expressed as function of the conformal time, $\eta$, obtained by numerical integration of \ref{['fetaeta']}.
• Figure 3: (a) $\sigma(\eta)$ and (b) $w(\eta)$ in \ref{['wrep']} plotted for $\bf b=0$ and for $\bf b=0.5$.
• Figure 4: For ${\bf b=0}$ all trajectories follow straight lines and have identical evolution. For ${\bf b=0.5}$$z(\eta)$ become different from the transverse trajectories ($x(\eta), y(\eta)$) consistently with \ref{['xyzeta']}, as shown in Fig.\ref{['xzplaneb5']}.
• Figure 5: For $\textcolor{red}{{\bf b}>0}$ the motion in the $x-z$ plane is not more along a straight line (as it is for ${\bf b}=0$). The Hubble friction slows down the $z$ motion for ${\bf b}=0$ but not when ${\bf b}>0$.