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

TL;DR

This paper presents a unified analytical method to compute the relativistic tidal tensor in stationary axisymmetric spacetimes, enabling better analysis of tidal forces in strong gravity regimes and extending the concept of Hills mass to naked singularities.

Contribution

It introduces a general, streamlined approach for deriving the relativistic tidal tensor applicable to various stationary, axisymmetric spacetimes, simplifying previous case-by-case methods.

Findings

Applied method to Schwarzschild, Reissner-Nordstrom, Kerr, Kerr-Newman, and wormhole metrics.

Extended Hills mass concept to naked singularity spacetimes.

Demonstrated utility in analyzing tidal forces in strong-field gravity.

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.