Relativistic tidal divergences in circular orbits and the dynamics of light rings

TL;DR

The paper investigates relativistic tidal forces on observers in circular geodesics around static, spherically symmetric ultracompact objects and shows that tidal stresses diverge as orbits approach null circular geodesics (light rings), with the outer ring at $r_+=3M$ in Schwarzschild exteriors and an interior inner ring $r_-$ whose position depends on the interior model. It derives the tidal equations by projecting the geodesic deviation equation onto a circular-orbit tetrad using spin-connection formalism, yielding explicit components that diverge near light rings. Through isotropic constant-density and Florides anisotropic star models, the work demonstrates that tidal divergences arise near both inner and outer light rings, with model-dependent interior behavior (e.g., center regularity in Florides but divergence near $r_-$). The results suggest light rings act as dynamical regulators of ultracompact horizonless objects, potentially constraining nonlinear instabilities and informing the physics of black hole mimickers, with future work extending to rotation and alternative theories of gravity.

Abstract

Tidal forces acting on orbiting bodies arise from inhomogeneities in the gravitational field, generating stresses that can deform or even disrupt these objects. In this work, we analyze relativistic tidal forces associated with ultracompact objects described by static and spherically symmetric spacetimes, focusing on observers in circular geodesic motion. We show that, in contrast to the case of radial geodesics, tidal forces diverge as the orbit approaches null circular geodesics. As illustrative examples, we study two uniform-density stellar models: one isotropic and another supported purely by tangential stresses. We conjecture that the divergence of tidal forces near light rings may play a role in the nonlinear stability of ultracompact, horizonless objects.

Figures (5)

• Figure 1: Illustrative picture of ultacompact objects. The marginally stable circular orbit, at $r = 6M$, is indicated by the dashed blue circle. The outer unstable light ring, at $r = 3M$, appears in green. For objects without a horizon, there may be an inner stable light ring, represented in orange, whose position depends on the star's radius. In this example, we have an uniform density isotropic star of radius $R = 2.4M$.
• Figure 2: Specific energy for circular orbits around isotropic stars and for the Florides model for anisotropic objects. The curves are associated with different values of compactness $R/M$.
• Figure 3: Specific angular momentum for circular orbits around isotropic stars and for the Florides model for anisotropic objects. The curves are associated with different values of compactness $R/M$.
• Figure 4: Radial and angular components of the tidal forces acting on a body in circular orbit inside an isotropic star. The curves are classified according to the star's compactness, $R/M$.
• Figure 5: Radial and angular components of the tidal forces acting on a body in circular orbit inside an anisotropic star based on Florides' solution. The curves are classified according to the star's compactness, $R/M$.