Geodesic motion in Bogoslovsky-Finsler spacetimes

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 $\mathrm{ISIM}(2)$. 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 with a homogeneous $pp$-wave solution of the Einstein-Maxwell equations. We conclude by extending the theory to the Finsler-Friedmann-Lemaître model.

Figure 1

Consistently with Eq. (4.13), 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 Eq. (4.22), in a $b$-dependent term, which is linear in retarded time, $u$. Experiments indicate that the anisotropy, and hence $b$, is very small. When $b\to 1$, the trajectory approaches the massless one (in heavy black), consistently with Eq. (4.22).

Figure 2

The conformal factor [Eq. (8.21b)] of the Friedmann-Lemaître model [Eq. (8.1)], expressed as a function of the conformal time $\eta$, obtained by numerical integration of Eq. (8.21).

Figure 3

(a) $\sigma (\eta )$ and (b) $w(\eta )$ in Eq. (9.10) plotted for blue $\mathbf{b}=\mathbf{0}$ and for red $\mathbf{b}=\mathbf{0.5}$.

Figure 4

For blue $\mathbf{b}=\mathbf{0}$, all trajectories follow straight lines and have identical evolution. For red $\mathbf{b}=\mathbf{0.5}$, $z(\eta )$ becomes different from the transverse trajectories [$x(\eta )$, $y(\eta )$] consistently with Eq. (9.13), as shown in Fig. 5.

Figure 5

For red $b>0$, the motion in the $x\text{−}z$ plane is not more along a straight line (as it is for blue $\mathbf{b}=\mathbf{0}$). The Hubble friction slows down the $z$ motion for blue $\mathbf{b}=\mathbf{0}$, but not when red $b>0$.

Figure 6

For the Friedmann-Lemaître model for blue $\mathbf{b}=\mathbf{0}$, the 3D trajectory is a straight line. For the Bogoslovsky-Finsler-F modification for red $\mathbf{b}=\mathbf{0.5}$, however, while the projection to the $x-y$ plane is still along a straight line, the $z$ component becomes curved, consistently with Fig. 5.