Characterizing repulsive gravity with curvature eigenvalues

TL;DR

This paper introduces an invariant method based on curvature eigenvalues to identify regions of repulsive gravity around compact objects, providing a coordinate-independent way to define a repulsion radius and analyze its dependence on physical parameters.

Contribution

It proposes a novel, invariant approach using curvature eigenvalues to characterize repulsive gravity regions, improving upon coordinate-dependent methods.

Findings

Repulsive regions can exist around charged and rotating black holes.

The size and shape of repulsion spheres depend on mass, charge, and angular momentum.

Positive mass can produce repulsive gravity when combined with charge or spin.

Abstract

Repulsive gravity has been investigated in several scenarios near compact objects by using different intuitive approaches. Here, we propose an invariant method to characterize regions of repulsive gravity, associated to black holes and naked singularities. Our method is based upon the behavior of the curvature tensor eigenvalues, and leads to an invariant definition of a \emph{repulsion radius}. The repulsion radius determines a physical region, which can be interpreted as a repulsion sphere, where the effects due to repulsive gravity naturally arise. Further, we show that the use of effective masses to characterize repulsive regions can lead to coordinate-dependent results whereas, in our approach, repulsion emerges as a consequence of the spacetime geometry in a completely invariant way. Our definition is tested in the spacetime of an electrically charged Kerr naked singularity and in all its limiting cases. We show that a positive mass can generate repulsive gravity if it is equipped with an electric charge or an angular momentum. We obtain reasonable results for the spacetime regions contained inside the repulsion sphere whose size and shape depend on the value of the mass, charge and angular momentum. Consequently, we define repulsive gravity as a classical relativistic effect by using the geometry of spacetime only.