[Paper Review] On the analytic extension of regular rotating black holes

Tian Zhou, Leonardo Modesto|arXiv (Cornell University)|Mar 20, 2023

Astrophysical Phenomena and Observations4 citations

TL;DR

This paper investigates the analytic extension of regular rotating black holes (RRBHs) derived via the Newman-Janis algorithm, showing that most such spacetimes require extension to negative $ r $, where they become singular or non-analytic at the ring. To resolve this, the authors propose replacing the constant angular momentum $ a $ with a radial function $ a' = a r^4/(r^4 + L^4) $, which shrinks the ring to a point and restores regularity of curvature invariants, ensuring geodesic completeness and analyticity at $ r=0 $.

ABSTRACT

We hereby focus on the analytic geodesic extension of several regular rotating black holes (RRBHs) obtained throughout the Newman-Janis algorithm starting from some popular spherically symmetric regular black holes. It turns out that if the metric is not an even function of Boyer-Lindquist radial coordinate r, similarly to the Kerr spacetime, the metric has to be extended to negative values of r to ensure the analyticity of the geodesic equations (and in turn of the geodesics). Therefore, some of the extended RRBHs considered in this paper, such as the rotating Hayward black hole, are geodetically incomplete because they are singular somewhere for r < 0, and non-analytic at the ring located in $(r = 0, {θ= π/2})$. Conversely, other spacetimes, like nonlocal black holes, can be analytically extended to negative r. However, the real issue shows up at the ring, where, unfortunately, all the RRBHs studied in this paper fail to be regular. Indeed, at the ring, the Kretschmann invariant is finite but nonanalytic, while the higher derivative curvature invariants are divergent. In order to avoid such catastrophe, we propose a modification of the RRBHs in which the angular momentum is promoted to a function of the radial coordinate. According to our proposal, the angular momentum vanishes for $r ightarrow 0$ and the ring shrinks to a point. Therefore, the regularity properties of the regular spherically symmetric black holes are recovered for $r ightarrow 0$.

Motivation & Objective

• To assess the analyticity and geodesic completeness of regular rotating black holes (RRBHs) derived from spherically symmetric regular black holes via the Newman-Janis algorithm.
• To identify the conditions under which RRBHs remain regular and geodesically complete, particularly at the ring singularity.
• To resolve the issue of non-analyticity and curvature divergence at the ring by modifying the angular momentum to be a function of radial coordinate.
• To ensure compatibility with quantum gravity principles, such as the finite action principle involving higher-derivative curvature invariants.
• To propose a viable, physically consistent model of RRBHs that avoids singularities and maintains predictability in general relativity.

Proposed method

• Apply the Newman-Janis algorithm to spherically symmetric regular black holes (e.g., Hayward, Dymnikova) to generate rotating counterparts.
• Analyze the analyticity of geodesic equations by extending spacetime to negative $ r $, following the Kerr spacetime’s maximal extension.
• Evaluate curvature invariants—especially the Kretschmann scalar and higher-derivative invariants—near $ r=0 $ and at the ring ($ r=0, \theta=\pi/2 $).
• Propose a radial-dependent angular momentum function $ a' = a r^4/(r^4 + L^4) $, ensuring $ a' \to 0 $ as $ r \to 0 $, thus recovering spherical symmetry at the center.
• Assess the implications of this modification for geodesic completeness and analyticity, comparing with conformal gravity and non-analytic mass functions.
• Use the action natural selection principle to argue that only metrics with finite higher-derivative invariants contribute to the quantum path integral.

Experimental results

Research questions

• RQ1 Does the analytic extension of regular rotating black holes to negative $ r $ preserve the regularity of geodesics and curvature invariants?
• RQ2 Why do all standard RRBHs fail to be geodetically complete despite being regular at $ r=0 $, and what is the role of the ring singularity?
• RQ3 Can the angular momentum be made a function of $ r $ to eliminate non-analyticity and divergence of higher-derivative curvature invariants at the ring?
• RQ4 How does the proposed radial-dependent angular momentum $ a' $ restore regularity and analyticity in the spacetime near $ r=0 $?
• RQ5 What are the implications of this modification for quantum gravity, particularly in relation to the finite action principle and path integral selection of viable spacetimes?

Key findings

• The maximally extended rotating Hayward, renormalization group improved, and Dymnikova black holes are singular for $ r<0 $, violating analyticity of geodesic equations.
• All standard RRBHs exhibit non-analytic behavior and divergent higher-derivative curvature invariants at the ring ($ r=0, \theta=\pi/2 $), despite finite Kretschmann scalar.
• The proposed modification $ a' = a r^4/(r^4 + L^4) $ ensures $ a' \to 0 $ as $ r \to 0 $, shrinking the ring to a point and restoring regularity of curvature invariants.
• This modification restores analyticity and geodesic completeness, as the spacetime near $ r=0 $ becomes spherically symmetric and invariant under $ r \to -r $, satisfying conditions from prior work on spherically symmetric regular black holes.
• The resulting spacetime is compatible with the finite action principle in quantum gravity, as higher-derivative invariants remain finite.
• Alternative models, such as conformal gravity black holes or non-analytic mass functions, can also achieve geodesic completeness, though at the cost of losing analyticity or requiring non-standard dynamics.