Repulsive gravity in regular black holes

Abstract: We evaluate the effects of repulsive gravity using first order geometric invariants, \textit{i.e.}, the Ricci scalar and the eigenvalues of the Riemann curvature tensor, for three regular black holes, namely the Bardeen, Hayward, and Dymnikova spacetimes. To examine the repulsive effects, we calculate their respective onsets and regions of repulsive gravity. Afterwards, we compare the repulsive regions obtained from these metrics among themselves and then with the predictions got from the Reissner-Nordstr\"{o}m and Schwarzschild-de Sitter. We find that the Dymnikova spacetime does not exhibit regions in which gravity changes its sign. A notable characteristic, observed in all these metrics, is that the repulsive regions appear to be unaffected by the mass that generates the regular black hole. This property emerges due to the invariants employed in our analysis, which do not change sign through \emph{linear combinations} of the mass and the free coefficients of the metrics. As a result, gravity can change sign independently of the specific values acquired by the mass. This conclusion suggests a potential \emph{incompleteness} of regular solutions, particularly in terms of their repulsive effects. To further highlight this finding, we numerically compute, for the Reissner-Nordstr\"{o}m and Schwarzschild-de Sitter solutions, the values of mass, $M$, that emulate the repulsive effects found in the Bardeen and Hayward spacetimes. These selected values of $M$ provide evidence that regular black holes do not incorporate repulsive effects by means of the masses used to generate the solutions themselves. Implications and physical consequences of these results are then discussed in detail.