Notes on non-singular models of black holes

Abstract: We discuss static spherically symmetric metrics which represent non-singular black holes in four- and higher-dimensional spacetime. We impose a set of restrictions, such as a regularity of the metric at the center $r=0$ and Schwarzschild asymptotic behavior at large $r$. We assume that the metric besides mass $M$ contains an additional parameter $\ell$, which determines the scale where modification of the solution of the Einstein equations becomes significant. We require that the modified metric obeys the limiting curvature condition, that is its curvature is uniformly restricted by the value $\sim \ell^{-2}$. We also make a "more technical" assumption that the metric coefficients are rational functions of $r$. In particular, the invariant $(\nabla r)^2$ has the form $P_n(r)/\tilde{P}_n(r)$, where $P_n$ and $\tilde{P}_n$ are polynomials of the order of $n$. We discuss first the case of four dimensions. We show that when $n\le 2$ such a metric cannot describe a non-singular black hole. For $n=3$ we find a suitable metric, which besides $M$ and $\ell$ contains a dimensionless numerical parameter. When this parameter vanishes the obtained metric coincides with Hayward's one. The characteristic property of such spacetimes is $-\xi^2=(\nabla r)^2$, where $\xi^2$ is a time-like at infinity Killing vector. We describe a possible generalization of a non-singular black-hole metric to the case when this equality is violated. We also obtain a metric for a charged non-singular black hole obeying similar restrictions as the neutral one, and construct higher dimensional models of neutral and charged black holes.

Non-Singular Models of Black Holes

The paper "Notes on non-singular models of black holes" by Valeri P. Frolov provides a thorough investigation into the concept of non-singular black holes within the framework of modified general relativity. General Relativity (GR), while extremely successful in describing gravitational phenomena at large scales, is known to encounter issues such as singularities where physical quantities become infinite. This paper aims to extend traditional black hole models by incorporating a non-singular approach.

Overview of the Models

The paper is centered around static spherically symmetric spacetime metrics that describe non-singular black holes in both four and higher-dimensional spaces. These models are constrained by specific requirements, such as regularity at the center ($r=0$) and asymptotic Schwarzschild behavior at large distances. The introduction of an additional parameter $\ell$ serves as a scale for when modifications to the solutions of the Einstein equations become significant.

A key aspect of these models is the implementation of a "limiting curvature condition," meaning that the curvature is uniformly bounded by some value $\sim \ell^{-2}$. This condition is posited to avoid the singularities that plague classical solutions like the Schwarzschild metric. Frolov also assumes "more technical" conditions, requiring the metric coefficients to be rational functions of $r$, which streamlines the mathematical complexity of the solutions while maintaining physical plausibility.

Main Findings

1. Four-Dimensional Non-Singular Solutions: In four-dimensional spacetime, it is shown that when using rational polynomial functions of degree $n\leq2$, it is impossible to formulate a non-singular black hole. However, for $n=3$, a suitable metric form emerges. This model aligns with Hayward's existing metric when a certain dimensionless parameter vanishes.

2. Generalization to Higher Dimensions: The analysis is extended to higher-dimensional metrics, which also respects similar constraints as the four-dimensional case, providing a consistent framework for a broader range of black hole solutions.

3. Charged Non-Singular Black Holes: The paper includes a model for charged black holes that meets the set regularity and limiting curvature conditions similarly to their neutral counterparts.

Implications and Future Prospects

These non-singular black hole models offer significant theoretical implications. They present a potential framework for resolving singularities inherent in classical GR models, which has been a longstanding issue in theoretical physics. The approach suggests that in regions of high curvature, gravity may be modified in a way that prevents singularities, offering a pathway towards an ultraviolet-complete theory of gravity.

Practically, although these models remain largely theoretical, they raise interesting possibilities for astrophysical black holes and the ultimate fate of such objects as they emit Hawking radiation and potentially evaporate over cosmological timescales. The resolution of singularities also intersects with the information loss paradox by allowing for the prospect of information recovery post evaporation.

In concluding, the paper provides a pivotal step toward rethinking the internal structure of black holes. While no explicit claims are made about a groundbreaking nature for these models, they nonetheless provide a fertile ground for further exploration in theoretical and possibly quantum gravity, proposing modifications that could align with quantum principles. Future studies may focus on the empirical verification of these models and their application in other contexts, such as cosmology and high-energy physics.