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

TL;DR

The paper develops a unified, analytic framework to compute the relativistic tidal tensor in stationary axisymmetric spacetimes by leveraging ZAMO frames and standard frame transformations, avoiding the traditional, case-by-case tetrad construction. It provides complete analytic (in terms of elementary functions) expressions for the local tidal tensor in spherical spacetimes and demonstrates the method on RN, Kerr, and KN spacetimes, plus a rotating wormhole, enabling systematic Hills-mass calculations for both black holes and naked singularities. Key outcomes include finite Hills masses for Kerr-Newman naked singularities, infinite Hills mass for Kerr-like naked singularities, and detailed IBSO/IBCO-based tidal analyses that reveal how charge, spin, and inclination shape tidal forces. The results offer a practical route to constrain strong-field gravity via tidal observations (e.g., TDEs and photon-ring in EHT images) and come with computational tools (symbolic notebook and C++/Eigen workflow) to extend tidal studies to broader metrics.

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.

Figures (8)

• Figure 1: Tidal eigenvalues on IBCO in RN spacetime, with $\lambda_{1}$ (green dashed), $\lambda_{2}$ (orange dash-dotted), $\lambda_{3}$ (purple dotted), and the sum $\sum \lambda_{i}$ (black solid).
• Figure 2: Evolution of tidal eigenvalues along prograde IBSO geodesics in Kerr BH, with radial distance (green dashed), polar angle (orange dotted), and tidal eigenvalues (purple solid). Grey dashed lines indicate theoretical eigenvalues as the particle crosses the equatorial plane.
• Figure 3: Evolution of tidal eigenvalues along prograde IBSO geodesics in KN BH, with radial distance (green dashed), polar angle (orange dotted), and tidal eigenvalues (purple solid).
• Figure 4: Heatmaps of radii of closest approach (IBSOs for BH; zero angular momentum orbits for NS) in Kerr-Newman spacetime. Black dashed line separates BH regime from NS regime ($a^2 + Q^2 = 1$). Different colour maps are used for the two regimes.
• Figure 5: Heatmaps of the strongest stretching tidal eigenvalue in Kerr-Newman spacetime. Black dashed line separates BH regime from NS regime ($a^2 + Q^2 = 1$). Different colour maps are used for the two regimes. All eigenvalues are evaluated as the particle crosses the equatorial plane at the closest approach (IBSOs for BH; zero angular momentum orbits for NS).