Relativistic gravity gradiometry

In general relativity, relativistic gravity gradiometry involves the measurement of the relativistic tidal matrix, which is theoretically obtained from the projection of the Riemann curvature tensor onto the orthonormal tetrad frame of an observer. The observer’s 4-velocity vector defines its local temporal axis and its local spatial frame is defined by a set of three orthonormal nonrotating gyro directions. The general tidal matrix for the timelike geodesics of Kerr spacetime has been calculated by Marck [Proc. R. Soc. A 385, 431 (1983)]. We are interested in the measured components of the curvature tensor along the inclined “circular” geodesic orbit of a test mass about a slowly rotating astronomical object of mass $M$ and angular momentum $J$. Therefore, we specialize Marck’s results to such a “circular” orbit that is tilted with respect to the equatorial plane of the Kerr source. To linear order in $J$, we recover the gravitomagnetic beating phenomenon [B. Mashhoon and D. S. Theiss, Phys. Rev. Lett. 49, 1542 (1982)], where the beat frequency is the frequency of geodetic precession. The beat effect shows up as a special long-period gravitomagnetic part of the relativistic tidal matrix; moreover, the effect’s short-term manifestations are contained in certain post-Newtonian secular terms. The physical interpretation of this effect is briefly discussed.

Figure 1

This schematic plot depicts the construction of a natural spatial triad that is parallel propagated along a stable circular geodesic orbit in the equatorial $(x,y)$ plane of the exterior Kerr spacetime. The result coincides in this case with the general frame constructed by Marck for an arbitrary geodesic world line in the exterior Kerr spacetime. That is, Marck’s construction can be viewed as a simple generalization of the method illustrated in this figure.

Figure 2

Schematic depiction of the tilted circular orbit followed by a test mass $m$ around a Schwarzschild source of mass $M\gg m$. The inclination of the orbit is given by the angle $\alpha$. Moreover, $\eta$ is the angular position of $m$ in the orbital plane measured from the line of the ascending node. At $\tau =0$, $m$ has spherical polar coordinates $(r_0,{\theta }_0,{\varphi }_0)$, where $\cos {\theta }_0=\sin \alpha \sin {\eta }_0$ and ${\varphi }_0=\arctan (\cos \alpha \tan {\eta }_0)+{\phi }_0-{\eta }_0$.