Characterizing repulsive gravity with curvature eigenvalues

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 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 repulsion 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.

Figure 1

Behavior of the curvature eigenvalues ${\lambda }_1$ (dashed curve) and ${\lambda }_3$ (solid curve) in terms of the radial coordinate $r$ for the values $M=1$ and $Q^2=2$ that correspond to a naked singularity.

Figure 2

Behavior of the curvature eigenvalues ${\lambda }_3$ in terms of the radial coordinate $r$ for $M=1$ and $Q=\sqrt{0.9}$ (red dotted curve), $Q=1$ (blue solid curve), $Q=\sqrt{1.1}$ (black dashed curve) and $Q=\sqrt{1.2}$ (green dash-dotted curve).

Figure 3

Region determined by the repulsion radius $r_{\text{rep}}^K=(1+\sqrt{2})a\cos \theta$ in a Kerr spacetime with $a=1$. Only the projection of the repulsion spheres on a plane orthogonal to the equatorial plane is here illustrated.

Figure 4

The zeros of the polynomial (35) determine the repulsion radius of a Kerr-Newman naked singularity with $M=1$, $Q=\sqrt{2}$, and $a=\sqrt{10}$, for different values of the azimuthal angle: $\theta =0$ (solid curve), $\theta =\frac{\pi }{6}$ (long-dashed curve), $\theta =\frac{\pi }{3}$ (dash-dotted curve), $\theta =\frac{\pi }{2}$ (dotted curve).

Figure 5

Location of the repulsion regions of a Kerr-Newman extreme black hole with $M=1$, $Q=\frac{1}{2}$ and $a=\sqrt{\frac{3}{4}}$. The repulsion region closer to the central singularity exists only for $\theta \ne 0$, and corresponds to a second root of the polynomial (35) which exists only outside the equator.