Rotating regular black holes (1302.6075v2)

Abstract: The formation of spacetime singularities is a quite common phenomenon in General Relativity and it is regulated by specific theorems. It is widely believed that spacetime singularities do not exist in Nature, but that they represent a limitation of the classical theory. While we do not yet have any solid theory of quantum gravity, toy models of black hole solutions without singularities have been proposed. So far, there are only non-rotating regular black holes in the literature. These metrics can be hardly tested by astrophysical observations, as the black hole spin plays a fundamental role in any astrophysical process. In this letter, we apply the Newman-Janis algorithm to the Hayward and to the Bardeen black hole metrics. In both cases, we obtain a family of rotating solutions. Every solution corresponds to a different matter configuration. Each family has one solution with special properties, which can be written in Kerr-like form in Boyer-Lindquist coordinates. These special solutions are of Petrov type D, they are singularity free, but they violate the weak energy condition for a non-vanishing spin and their curvature invariants have different values at $r=0$ depending on the way one approaches the origin. We propose a natural prescription to have rotating solutions with a minimal violation of the weak energy condition and without the questionable property of the curvature invariants at the origin.

• The paper extends non-singular black hole solutions by applying the Newman-Janis algorithm to introduce rotation.
• It examines Hayward and Bardeen metrics to derive type-I and type-II rotating solutions that avoid singularities yet violate the weak energy condition.
• The study proposes a spin-dependent mass function modification to address curvature anomalies and align with energy condition requirements.

A Critical Evaluation of "Rotating Regular Black Holes"

The paper "Rotating regular black holes" by Cosimo Bambi and Leonardo Modesto undertakes an investigation into the extension of compact object solutions in general relativity by examining rotating, non-singular black holes. The primary motivation lies in addressing the theoretical limitations of classical general relativity, which suggests the existence of spacetime singularities. Such singularities pose significant challenges to the predictability of physical laws and highlight potential areas where quantum effects could provide resolution. This paper serves as an intermediary exploration in the absence of a fully developed theory of quantum gravity by utilizing solutions that might capture essential quantum gravitational properties.

Methodological Framework

Central to the paper is the application of the Newman-Janis algorithm to derive rotating solutions from existing non-rotating metrics, particularly the Hayward and Bardeen metrics. The Newman-Janis transformation, initially designed to model the Kerr solution, provides a systematic approach to introduce angular momentum to spherically symmetric spacetimes. The algorithm involves a series of coordinate transformations followed by complexification aimed at generating a metric akin to the desired rotating solution.

For the Hayward metric, the selected complexifications result in solutions categorized as type-I and type-II. Type-I solutions, deemed as a trivial extension, replicate the Kerr form in Boyer-Lindquist coordinates with a mass function derived from the non-rotating solution. In contrast, type-II solutions introduce additional theoretical complexity without retaining the simplified form but offer a broader class of metrics reflecting varied physical conditions.

The Bardeen metric undergoes a similar analysis, with resultant manifestations of type-I and type-II solutions preserving the structure of the non-rotating Hayward solutions.

Analytical Outcomes

The derived rotating solutions exhibit certain crucial properties:

• Singularity-Free Structure: Both Hayward-like and Bardeen-like rotating solutions avert spacetime singularities. This contributes to the discussion on alternative gravitational theories and regular black hole metrics without the causal pathologies associated with singularities.
• Violation of the Weak Energy Condition: Despite maintaining freedom from singularities, the paper highlights that the weak energy condition is violated in the presence of rotation. This noncompliance introduces significant considerations in how one interprets these solutions in a physical context, questioning their empirical validity and capability to model astrophysical phenomena accurately.
• Behavior at the Origin: A peculiar observation is that curvature invariants have different values at the origin, depending on the approach path, which the authors term as the "de Sitter belt." This peculiar fitting in a Newtonian framework raises additional inquiries into the completeness of these metrics.

Theoretical and Practical Implications

The construction of rotating regular black hole solutions is theoretically significant as it bridges classical black hole physics and potential quantum deviations. In practice, the ability to derive rotating metrics opens avenues for observational tests in astrophysical settings, where spin plays a crucial role in phenomena such as accretion disk dynamics and high-energy emissions.

Moreover, the further exploration into modifying the Newman-Janis approach by introducing a spin-dependent mass function is an innovative conceptual leap. This proposal aims to rectify both the weak energy condition violation and curvature behavior anomalies, albeit necessitating further paper to ascertain its efficacy.

Speculations and Future Direction

While the results elucidate certain aspects of non-singular black holes, the paper suggests several speculative pathways for further research:

1. Examination of alternative complexification schemes within the Newman-Janis algorithm to extend the regular metrics' applicability to more physically viable settings.
2. Consideration of alternative variational frameworks or modified theories of gravity, incorporating these solutions to yield a coherent, singularity-free theory that respects classical energy conditions.
3. Development of observational strategies capable of distinguishing these regular solutions from standard Kerr black holes, potentially offering novel insights into the nature of spacetime around rotating compact objects.

In conclusion, Bambi and Modesto's investigation of rotating regular black holes provides foundational insights and implores continued dialogue bridging theoretical constructs to observational pragmatics in contemporary gravitational physics.