[Paper Review] Quest for realistic non-singular black-hole geometries: Regular-center type

Hideki Maeda|arXiv (Cornell University)|Jul 10, 2021

Astrophysical Phenomena and Observations3 references4 citations

TL;DR

This paper proposes seven physical criteria to identify realistic non-singular black-hole models with a regular center, applying them to four spherically symmetric non-singular black holes and their rotating counterparts. It finds that only the Dymnikova and Fan-Wang models satisfy the dominant energy condition globally, with the rotating Dymnikova black hole being free from curvature singularities and closed timelike curves, making it the most physically viable candidate among the models studied.

ABSTRACT

We propose seven criteria to single out physically reasonable non-singular black-hole models and adopt them to four different spherically symmetric models with a regular center and their rotating counterparts. In general relativity, all such non-singular black holes are non-generic with a certain matter field including a class of nonlinear electromagnetic fields. According to a criterion that the effective energy-momentum tensor should satisfy all the standard energy conditions in asymptotically flat regions, the well-known Bardeen and Hayward black holes are discarded. In contrast, the Dymnikova and Fan-Wang black holes respect the dominant energy condition everywhere. Although the rotating Fan-Wang black hole contains a curvature singularity, the rotating Dymnikova black hole is free from scalar polynomial curvature singularities and closed timelike curves. In addition, the dominant energy condition is respected on and outside the event horizons in the latter case. The absence of parallelly propagated curvature singularities remains an open question.

Motivation & Objective

• To establish a set of physical criteria for identifying realistic non-singular black-hole geometries in general relativity.
• To assess the physical reasonableness of four spherically symmetric non-singular black-hole models and their rotating counterparts.
• To determine which models satisfy standard energy conditions, avoid curvature singularities, and preserve causality.
• To evaluate the behavior of geodesics and tidal forces near the center to test for pathological spacetime structures.
• To identify the most viable candidate for a physically realistic non-singular black hole based on multiple physical constraints.

Proposed method

• Define the effective energy-momentum tensor $\tilde{T}_{\mu\nu} = G_{\mu\nu}$ (with $c = 8\pi G = 1$) to analyze energy conditions without specifying a matter field.
• Apply standard energy conditions—Null, Weak, Dominant, and Strong Energy Conditions—to the effective energy-momentum tensor in asymptotically flat regions.
• Use the generalized Birkhoff’s theorem and nonlinear electromagnetic field solutions to explore the origin of non-singular black holes in modified gravity.
• Analyze curvature invariants and tidal forces via the Riemann tensor and Jacobi equations to detect singularities along geodesics.
• Investigate geodesic equations and photon sphere structure in rotating models to assess stability and causality.
• Evaluate the presence of closed timelike curves and scalar polynomial curvature singularities in rotating solutions.

Experimental results

Research questions

• RQ1 Which non-singular black-hole models satisfy the dominant energy condition everywhere, including at the center?
• RQ2 Do rotating counterparts of non-singular black holes avoid curvature singularities and closed timelike curves?
• RQ3 Can geodesics reach the center with finite affine parameter, and do tidal forces diverge there?
• RQ4 Which models respect the null and weak energy conditions in asymptotically flat regions?
• RQ5 Is the rotating Dymnikova black hole free from parallelly propagated curvature singularities?

Key findings

• The Bardeen and Hayward black holes violate the dominant energy condition in asymptotically flat regions and are therefore discarded as physically unreasonable.
• The Dymnikova and Fan-Wang black holes satisfy the dominant energy condition globally, with the Dymnikova model being particularly robust under rotation.
• The rotating Fan-Wang black hole contains a curvature singularity, disqualifying it as a physically viable model.
• The rotating Dymnikova black hole is free from scalar polynomial curvature singularities and closed timelike curves, satisfying the dominant energy condition on and outside the event horizon.
• Along radial geodesics with $E^2 = 1$, tidal forces and Jacobi fields diverge as $r \to 0$ only for $-2 < \alpha < 0$, indicating potential pathologies in certain mass functions.
• For $\alpha \geq 0$, curvature invariants and tidal forces remain finite at $r = 0$, suggesting a regular center for models with $M(r) \sim r^{3+\alpha}$ and $\alpha \geq 0$.