The relativistic tidal tensor: general solutions for stationary axisymmetric spacetimes and the Hills mass of naked singularities

Abstract

The tidal forces experienced on an orbit contain, in principle, information about the underlying spacetime an object is moving through. Astronomical observations often probe the properties of tidal forces in the relativistic regime, and could thus in principle be leveraged to examine the properties of strong-field gravity, provided that a general procedure for computing the relativistic tidal tensor is known. Existing techniques for deriving the tidal tensor rely on cumbersome, case-by-case methods. This paper introduces a unified analytical approach to deriving the tidal accelerations experienced by a test particle in any stationary, axisymmetric spacetime. This technique uses standard relativistic frame transformations and is built around the zero angular momentum observer frame. The method's utility is demonstrated in the four traditional black hole metrics: Schwarzschild, Reissner-Nordstrom, Kerr, and Kerr-Newman, as well as a particular wormhole metric. As an example of a possible astronomical application of this work, we discuss the concept of the Hills mass, the maximum mass at which a black hole can disrupt a star, and extend its definition to various naked singularity metrics.