Local spacetime effects on gyroscope systems

We give a precise theoretical description of initially aligned sets of orthogonal gyroscopes which are transported along different paths from some initial point to the same final point in spacetime. These gyroscope systems can be used to synchronize separated observers’ spatial frames by free fall along timelike geodesics. We find that initially aligned gyroscope systems, or spatial frames, lose their synchronization due to the curvature of spacetime and their relative motion. On the basis of our results we propose a simple experiment that enables observers to determine locally whether their spacetime is described by a rotating Kerr or a nonrotating Schwarzschild metric.

Figure 1

Parallel transport of a vector along two different paths.

Figure 2

Motion of initial gyroscope axes $e$ at $\gamma (0,0)$ to final gyroscope axes $f^{(1)}$ and $f^{(2)}$ at $\gamma (t,s)$ along different paths: ${\text{}}^{(1)}$ via $\gamma (t,0)$ and ${\text{}}^{(2)}$ via $\gamma (0,s)$. Finite Lorentz boosts changing orthogonality to $\dot{\gamma }$ into orthogonality to ${\gamma }^{\prime }$ are indicated by $\curvearrowright$, the reverse change by $\curvearrowleft$.