Hayward Regular Black-Hole Model

The Hayward model replaces the Schwarzschild singularity by a de Sitter-like core through a compact mass function. Hayward’s original 2006 construction, complete short article, pp. 031103-1–031103-4 was dynamical and organized around trapping horizons and evaporation. The familiar static line element is exceptionally useful for testing horizon merger and regular-core extensions; Ansoldi 2008, §§2.2–2.6 places it among the broader regular-centre families. Without a specified action and fields it is not a prediction of a unique covariant theory.

Hayward geometry and identity

Property	Statement
Object class	Effective model
Theory	Phenomenological metric, with several inequivalent proposed completions
Dimension and symmetry	Four-dimensional, static, spherical baseline
Parameters	ADM mass $M$ and short-distance length $\ell$
Causal regimes	Two horizons, a degenerate horizon, or a horizonless regular geometry
Evidence	Mathematical model; no evidence for astrophysical realization

With $m=GM>0$, $\ell>0$, and $r$ the areal radius, the standard static form is

$$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2d\Omega_2^2, \qquad f(r)=1-\frac{2m r^2}{r^3+2m\ell^2}.$$

Equivalently, $f=1-2\mu(r)/r$ with $\mu(r)=m r^3/(r^3+2m\ell^2)$. The metric does not specify whether $\ell$ is fixed, runs with $m$, or arises from matter, higher curvature, or quantum geometry. Those choices change the physical model.

Horizon threshold and local regularity

Positive horizon radii solve

$$P(r)=r^3-2mr^2+2m\ell^2=0.$$

At a degenerate horizon, $P=P'=0$. The nonzero double root satisfies $r_{\mathrm{ext}}=4m/3$, which gives

$$m_{\mathrm{ext}}=\frac{3\sqrt3}{4}\ell, \qquad r_{\mathrm{ext}}=\sqrt3\,\ell.$$

For $m>m_{\mathrm{ext}}$ there are inner and outer horizons; equality is extremal; below it the geometry is horizonless, all within the $\ell>0$ regular family. The separate $\ell=0$ boundary is Schwarzschild and restores its singular center. This provides a controlled branch check rather than an inference from a plot.

Near $r=0$,

$$f(r)=1-\frac{r^2}{\ell^2}+O(r^5),$$

so the center approaches a static patch of de Sitter space with $\Lambda_{\mathrm{eff}}=3/\ell^2$. Polynomial curvature scalars remain finite. At fixed $r>0$, $\ell\to0$ yields $f\to1-2m/r$, recovering Schwarzschild. These are two independent checks: one probes the core and one the exterior.

Curvature regularity is not the same as geodesic completeness. One must specify the manifold through $r=0$, verify the differentiability of the extension, and solve causal geodesics. Nor does a regular core establish stability of the inner horizon, which can be sensitive to the mass-inflation blueshift mechanism of Poisson and Israel 1990, §§II–IV, pp. 1797–1807 and to backreaction; Frolov 2016, §§I–II and V emphasizes the additional assumptions hidden in nonsingular black-hole models. Specifying an Exact Spacetime Solution distinguishes the geometry, fields, domain, and field-equation claims.

Effective stress tensor and completions

If the metric is inserted into the Einstein tensor, one may define an anisotropic effective source. That bookkeeping device displays which energy conditions the geometry would require in Einstein gravity; it does not identify microscopic matter. The central equation of state is vacuum-like, and the strong energy condition must fail near the regular core. Other energy-condition statements depend on the chosen static or dynamical version.

Hayward’s original construction was motivated by a nonsingular dynamical black hole with trapping horizons and evaporation. Freezing it into a static metric discards part of that causal problem. Conversely, proposing a nonlinear-electrodynamics or modified-gravity action requires checking all field equations, coupling normalizations, regular horizon data, and the $\ell\to0$ branch; Maeda 2022, §§2.1, 3.1, and 4 shows how restrictive realistic regular-centre criteria can be. Constructing Alternative-Theory Solution Branches explains how to carry out that calculation.

The strongest established result is geometric: the displayed family gives an analytically transparent, finite-curvature interpolation with a computable horizon threshold and Schwarzschild exterior limit. No general theorem shows that realistic collapse produces it.

The horizon mechanics also reveals the extremal boundary. With the Killing field normalized by $g_{tt}\to-1$ at infinity, $\kappa_h=f'(r_h)/2$ vanishes at the double root. Away from extremality, a quantum-field temperature $T_h=|\kappa_h|/(2\pi)$ can be assigned to an individual stationary horizon. Turning that relation into a global evaporation law requires a quantum state, greybody factors, backreaction, and a rule for evolving $m$ and possibly $\ell$; the static metric supplies none of those ingredients.

Observable use and principal obstruction

Geodesics, shadows, accretion spectra, ringdown, and evaporation can be computed after declaring how matter and perturbations couple. A fit of $\ell/m$ is a constraint on that complete forward model. It is not a measurement of a universal “quantum length.” Plasma, spin, inclination, emissivity, waveform flexibility, and the inner boundary prescription can mimic geometric deviations. The inference logic belongs to Null Tests and Alternative-Model Tests.

The principal obstruction is completion-dependence: different actions can share a similar background metric but have different perturbative degrees of freedom, stability, and formation. As of the present theoretical record, no empirical observation selects the Hayward model over Kerr or over other regular metrics.

This leaves a strict hierarchy of status. The center is curvature-regular for the stated mass function. Geodesic completeness belongs to a chosen global extension. Energy conditions belong to the declared effective or fundamental stress tensor. Stability belongs to the coupled perturbation equations. Claims about a nonsingular evaporation endpoint require all four plus a controlled semiclassical evolution, and therefore exceed what the static ansatz establishes.

References

• Ansoldi, Stefano. “Spherical Black Holes with Regular Center: A Review of Existing Models Including a Recent Realization with Gaussian Sources.” (2008). arXiv:0802.0330.
• Frolov, Valeri P. “Notes on Nonsingular Models of Black Holes.” Physical Review D 94 (2016), 104056. doi:10.1103/PhysRevD.94.104056.
• Hayward, Sean A. “Formation and Evaporation of Nonsingular Black Holes.” Physical Review Letters 96 (2006), 031103. doi:10.1103/PhysRevLett.96.031103.
• Maeda, Hideki. “Quest for Realistic Non-Singular Black-Hole Geometries: Regular-Center Type.” Journal of High Energy Physics 11 (2022), 108. doi:10.1007/JHEP11(2022)108.
• Poisson, Eric, and Werner Israel. “Internal Structure of Black Holes.” Physical Review D 41 (1990), 1796–1809. doi:10.1103/PhysRevD.41.1796.