[Paper Review] Toward an invariant definition of repulsive gravity

Orlando Luongo, Hernando Quevedo|arXiv (Cornell University)|May 25, 2010

Cosmology and Gravitation Theories4 citations

TL;DR

This paper proposes an invariant definition of repulsive gravity using curvature eigenvalues from the Riemann tensor, overcoming coordinate-dependent methods like effective potential analysis. It identifies regions of repulsion as intervals between the first spatial extremum of an eigenvalue and the singularity, yielding physically plausible radii for Reissner-Nordström and Kerr naked singularities.

ABSTRACT

A remarkable property of naked singularities in general relativity is their repulsive nature. The effects generated by repulsive gravity are usually investigated by analyzing the trajectories of test particles which move in the effective potential of a naked singularity. This method is, however, coordinate and observer dependent. We propose to use the properties of the Riemann tensor in order to establish in an invariant manner the regions where repulsive gravity plays a dominant role. In particular, we show that in the case of the Kerr-Newman singularity and its special subcases the method delivers plausible results.

Motivation & Objective

• To overcome the coordinate and observer dependence of traditional methods for identifying repulsive gravity in naked singularities.
• To establish a geometric, invariant criterion for repulsive gravity using the Riemann curvature tensor’s eigenvalues.
• To validate the method on known naked singularities (Reissner-Nordström, Kerr, Kerr-Newman) with known physical behavior.
• To provide a framework applicable to singularities without black hole counterparts, such as those with mass quadrupole moments.
• To use the concept of repulsion regions as a tool for matching interior and exterior solutions in general relativity.

Proposed method

• Uses the SO(3,C) representation of the Riemann curvature tensor to decompose it into Weyl, trace-free Ricci, and scalar parts.
• Analyzes the complex eigenvalues of the curvature matrix R = W + E + S, which are invariant under coordinate transformations.
• Defines a region of repulsion as the spacetime interval between the first spatial extremum (dλₙ/dxⁱ = 0) and the singularity, ensuring invariance.
• Applies the method to the Kerr-Newman metric, deriving the radius of repulsion as the largest root of a quartic equation on the symmetry axis.
• Compares results with test particle dynamics and known instability radii to validate physical consistency.
• Uses curvature eigenvalues instead of second-order invariants (e.g., Kretschmann scalar), which fail to detect sign changes in effective mass.

Experimental results

Research questions

• RQ1 Can repulsive gravity be defined in a coordinate- and observer-independent manner using curvature invariants?
• RQ2 Do curvature eigenvalues reliably identify regions where gravitational repulsion dominates in naked singularities?
• RQ3 How do the eigenvalue extremum structure and sign changes relate to known regions of instability in test particle motion?
• RQ4 Can the method distinguish between black hole and naked singularity spacetimes based on curvature eigenvalue behavior?
• RQ5 Is the concept of a 'region of repulsion' useful for matching interior and exterior solutions in general relativity?

Key findings

• The curvature eigenvalue λ = M/r³ for the Schwarzschild metric changes sign under M → −M, indicating a physically meaningful distinction between attractive and repulsive singularities.
• For the Reissner-Nordström naked singularity, the radius of repulsion is R_rep^RN = 2Q²/M, which lies within the unstable circular orbit region.
• For the Kerr naked singularity, R_rep^K = (1+√2)a cosθ, showing repulsion is absent on the equatorial plane (θ = π/2) and dominant elsewhere.
• On the symmetry axis, the Kerr-Newman radius of repulsion is the largest root of the quartic equation Mr⁴ − 2Q²r³ − 6Ma²r² + 2a²Q²r + Ma⁴ = 0.
• The method consistently identifies repulsion regions in singularities with mass quadrupole moments, even without black hole counterparts.
• The invariant extremum condition ∂λₙ/∂xⁱ = 0 provides a geometric, coordinate-independent definition of the boundary of the repulsion region.