Geodesic Completeness of Regular Black Holes

To solve the singularity issue in Einstein’s general relativity(GR), many regular black hole (RBH) solutions have been proposed. We here focus on the singularity problem for RBHs. The curvature singularity does not exist in RBHs but the geodetical completion, the usual definition of regular spacetimes, may not hold. Thus, we have to investigate the geodesic completeness of RBH solutions. The Hayward metric has a de-Sitter core so 𝑟 = 0 is no longer a curvature singularity. We compute the proper time to reach 𝑟 = 0 for radial geodesics and found that particles with enough energy can reach 𝑟 = 0 in finite proper time. Therefore, we are forced to extend the Hayward metric to 𝑟 < 0, until arriving at the Schwarzchild-like singularity 𝑟 = −𝐿. And it is easy to show that geodesics reach the singularity in finite proper time so the maximally extended Hayward spacetime is geodesically incomplete. We then study the generalized Hayward metric and found that the geodesic completeness depends on the metric function. Similarly, other black holes with de-Sitter cores, such as the Nicolini black hole, Lee-Wick black hole, Bardeen’s black hole, and the black hole from loop quantum gravity are also geodesically complete. But Visser’s black hole with a Minkowski core is not geodesically complete since the metric is ill-defined in the limit 𝑟 → 0−, even though the curvature is regular at 𝑟 = 0. For the renormalization-group-inspired black holes, the geodesic completeness is up to the parameter 𝛾. Nevertheless, even though many regular black holes are geodesically complete, the spacetimes are not stable because of the multi-horizon structure.

Then, we show that conformal symmetry is a great idea to solve the singularity problem. In this way, the spacetime metric cannot be the solution of GR but a conformally rescaled metric. The reselecting of the conformal gauge can make singular spacetime to be geodesically complete. Conformal transformation does not change the causal structure so there is no new horizon caused, which makes spacetime unstable. We demonstrate that the conformal rescaled spherically symmetric metric, such as the Schwarzchild metric, can be geodetically complete for massless, massive, and conformal-coupled particles. Afterward, we show that we always can choose a conformal gauge to ensure the spacetime completion of massless, massive, and conformally coupled particles for general metrics with some causal conditions.

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