Explicit Fermi coordinates and tidal dynamics in de Sitter and Gödel spacetimes

Fermi coordinates are directly constructed in de Sitter and Gödel spacetimes and the corresponding exact coordinate transformations are given explicitly. The quasi-inertial Fermi coordinates are then employed to discuss the dynamics of a free test particle in these spacetimes and the results are compared to the corresponding generalized Jacobi equations that contain only the lowest-order tidal terms. The domain of validity of the generalized Jacobi equation is thus examined in these cases. Furthermore, the difficulty of constructing explicit Fermi coordinates in black hole spacetimes is demonstrated.

Figure 1

Schematic construction of Fermi coordinates at $P$.

Figure 2

Plot of $\dot{R}$ versus $HR$, $0\le HR\le \pi /2$, for $V_0=0.4$, $1/\sqrt{2}$ and 0.9. In each panel, the lower (upper) curve represents the motion according to the exact (approximate) equation of motion in Fermi coordinates. In the exact case, $\dot{R}$ vanishes at $HR=\pi /2$.

Figure 3

Plot of $\dot{\rho }$ versus $\Omega \rho$, $0\le \Omega \rho \le \sqrt{2}\ln (1+\sqrt{2})$, for $V_0=0.5$, $1/\sqrt{2}$ and 0.9. In the top and middle panels, the lower (upper) curves represent the motion according to the exact (approximate) equation of motion in Fermi coordinates. In the top panel, $\dot{\rho }$ monotonically increases in the approximate case. In the bottom panel, the inner curve corresponds to the exact equation of motion.