Hayward Regular Black Hole Overview

• Hayward regular black hole is a spacetime model that replaces Schwarzschild’s central singularity with a de Sitter-like core, ensuring finite curvature invariants.
• It employs a metric function f(r)=1 - (2mr²)/(r³+2ml²) that transitions from Schwarzschild behavior at large radii to de Sitter geometry near the center.
• The model serves as a testbed for analyzing geodesic structure, thermodynamic properties, and inner-horizon stability in various modified gravity and nonlinear electrodynamics frameworks.

Searching arXiv for recent and foundational papers on the Hayward regular black hole. arxiv_search({"query":"Hayward regular black hole metric thermodynamics geodesics regular black holes", "max_results": 10, "sort_by": "relevance"}) I found several relevant arXiv papers. I’ll use the most directly pertinent ones together with the supplied sources. The Hayward regular black hole is a static, spherically symmetric black-hole geometry in which the central curvature singularity of Schwarzschild is replaced by a regular de Sitter-like core. In its standard form, the line element is

$$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\theta^2+r^2\sin^2\theta\,d\phi^2,$$

with

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$

where $m$ is the mass parameter and $l$ is the Hayward regularization parameter. The geometry approaches Schwarzschild for large $r$, while near $r=0$ it approaches a de Sitter core; this is the basic mechanism by which the spacetime is regular rather than singular. The solution has become a standard testbed for questions involving geodesic structure, wave propagation, thermodynamics, rotating regular spacetimes, and the stability of inner horizons under perturbations (Abbas et al., 2014, Borissova et al., 18 Feb 2026).

1. Metric structure and parameterizations

In the notation used most often in the static literature, the Hayward metric function is

$f(r)=1-\frac{2mr^2}{r^3+2ml^2}.$

The parameter $l$ controls the departure from Schwarzschild and regularizes the central region. One formulation states that $l>0$ is related to the central energy density, with density scale $3/(8\pi l^2)$; another describes $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$0 as a Hayward length parameter, with $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$1 (Abbas et al., 2014, Iguchi, 15 Apr 2025).

At large radius,

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$2

so the spacetime is Schwarzschild-like. Near the origin,

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$3

so the core is de Sitter-like. In this sense, the Hayward solution is a regular black hole rather than a singular one (Abbas et al., 2014).

Different subliteratures use different parameters for the same regularizing role. In nonlinear-electrodynamics constructions, one finds

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$4

with magnetic charge $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$5 and normalized charge $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$6 (Paula et al., 2023). In rotating treatments on the equatorial plane, the effective mass function is often written as

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$7

where $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$8 is the deviation parameter; $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$9 recovers Kerr, and $m$0 together with $m$1 recovers Schwarzschild (Khan et al., 2021).

Context	Metric function or mass function	Regularization parameter
Static Hayward	$m$2	$m$3
NED-supported static form	$m$4	$m$5
Rotating equatorial form	$m$6	$m$7

These notational changes are model-dependent rather than purely cosmetic. In particular, the same regularizing effect is realized in the literature by different source models: nonlinear electrodynamics, effective magnetic charge, or modified gravity (Paula et al., 2023, Eichhorn et al., 1 Aug 2025).

2. Regularity, curvature, and horizon structure

The defining property of the Hayward spacetime is the finiteness of curvature invariants at the center. For the static metric, explicit invariants are

$m$8

$m$9

and

$l$0

These limits remain finite, so the center is nonsingular (Abbas et al., 2014).

A complementary curvature characterization uses the Weyl scalar $l$1 and Kretschmann scalar $l$2. In a dimensionless coordinate $l$3, one study gives

$l$4

$l$5

The key feature is that $l$6 is finite everywhere, including at $l$7, while $l$8 at the center. This vanishing of the Weyl curvature at the origin is tied to the de Sitter-like character of the core (Iguchi, 15 Apr 2025).

Horizons are determined by $l$9. In the $r$0 parametrization, there is a critical mass

$r$1

If $r$2, the metric has no zero; if $r$3, it has one degenerate zero at $r$4; if $r$5, it has two horizons, an outer event horizon $r$6 and an inner horizon $r$7 (Abbas et al., 2014). Equivalent statements appear in the $r$8 notation: there are no horizons for $r$9, one degenerate horizon for $r=0$0, and two horizons for $r=0$1 (Hu et al., 2021).

This already excludes a common misconception: regularity does not imply the absence of horizons. The standard Hayward geometry generically has an event horizon and an inner Cauchy horizon. In deformed settings the horizon structure can be richer. For example, in the Hayward black hole surrounded by quintessence, the lapse function

$r=0$2

can yield two or three horizons, including a quintessence horizon $r=0$3 (Pedraza et al., 2021).

3. Geodesic structure, wave propagation, and tidal effects

The geodesic equations admit a standard first-integral form. In the equatorial plane, with conserved energy $r=0$4 and angular momentum $r=0$5,

$r=0$6

and the radial equation can be written as

$r=0$7

with $r=0$8 for null geodesics and $r=0$9 for timelike geodesics. The corresponding effective potential is

$f(r)=1-\frac{2mr^2}{r^3+2ml^2}.$0

For massive particles, minima of $f(r)=1-\frac{2mr^2}{r^3+2ml^2}.$1 indicate stable circular orbits (Abbas et al., 2014).

Detailed numerical analyses classify the resulting orbit families. One study finds four time-like orbit types—planetary or bound orbits, circular orbits, escape orbits, and absorbing orbits—and three null-geodesic types—unstable circular orbits, escape orbits, and absorbing orbits (Hu et al., 2021). Another phase-plane analysis of timelike motion identifies unstable circular orbits, separate stable orbits, stable hyperbolic orbits, and elliptical orbits; for $f(r)=1-\frac{2mr^2}{r^3+2ml^2}.$2, particles with $f(r)=1-\frac{2mr^2}{r^3+2ml^2}.$3 plunge into the black hole, while the innermost stable circular orbit occurs at $f(r)=1-\frac{2mr^2}{r^3+2ml^2}.$4 for $f(r)=1-\frac{2mr^2}{r^3+2ml^2}.$5 (Hu et al., 2019). For $f(r)=1-\frac{2mr^2}{r^3+2ml^2}.$6, a different study finds a critical angular momentum $f(r)=1-\frac{2mr^2}{r^3+2ml^2}.$7: escape orbits exist only for $f(r)=1-\frac{2mr^2}{r^3+2ml^2}.$8, and the precession direction changes with $f(r)=1-\frac{2mr^2}{r^3+2ml^2}.$9, being clockwise for $l$0 and counterclockwise for $l$1 (Hu et al., 2021).

Painlevé–Gullstrand coordinates make the interior behavior especially transparent. For radial infall in the Hayward metric,

$l$2

At the outer horizon, $l$3; the speed reaches a maximum at

$l$4

then decreases, and at the center $l$5. This contrasts with Schwarzschild and Reissner–Nordström behavior near the origin and reflects the repulsive effect of the regular core (Perez-Roman et al., 2018).

Wave propagation exhibits the same regular-core imprint. For a neutral massless scalar field in the static Hayward background, the radial equation becomes

$l$6

with

$l$7

Low-frequency absorption satisfies

$l$8

while at high frequency the cross section oscillates around the geometric capture cross section $l$9 and is described by a sinc approximation involving the Lyapunov exponent of the unstable null circular orbit. The same study shows that Hayward black holes can mimic some absorption and scattering results of Reissner–Nordström black holes for suitable charge choices, for example similar absorption cross sections for $l>0$0 and $l>0$1 (Paula et al., 2023).

In the charged Hayward spacetime,

$l>0$2

tidal forces remain finite at $l>0$3 and can vanish and change sign. The geodesic-deviation equations reduce to

$l>0$4

Unlike Schwarzschild, the radial tidal force may vanish outside the event horizon, while the angular tidal force can vanish between the event and Cauchy horizons (Junior et al., 2020).

4. Thermodynamics and entropy

Thermodynamic analyses of the Hayward family follow the same parameter dependence that governs the geometry. In a normalization where $l>0$5, one study gives the Hawking temperature of the static Hayward black hole as

$l>0$6

For the extremal case $l>0$7, $l>0$8; for $l>0$9, $3/(8\pi l^2)$0. The same work takes the entropy to be $3/(8\pi l^2)$1 and gives the heat capacity

$3/(8\pi l^2)$2

which it reports as nonnegative (Halilsoy et al., 2013).

For the rotating regular Hayward black hole generated by the Newman–Janis algorithm, the horizon condition is

$3/(8\pi l^2)$3

Using surface gravity,

$3/(8\pi l^2)$4

the Hawking temperature becomes

$3/(8\pi l^2)$5

Increasing $3/(8\pi l^2)$6 lowers $3/(8\pi l^2)$7, while increasing $3/(8\pi l^2)$8 raises it. In the same framework, a generalized-uncertainty-principle correction gives

$3/(8\pi l^2)$9

so the quantum-corrected temperature is reduced when $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$00. The entropy analysis uses

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$01

with a leading logarithmic correction

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$02

These results are interpreted there as supporting a remnant-like late stage of evaporation (Ali et al., 2022).

In anti-de Sitter spacetime, the regular Hayward black hole supports a van der Waals–like critical structure. In the $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$03, $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$04 Hayward-AdS sector,

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$05

with temperature

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$06

The entropy and thermodynamic volume are modified to

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$07

and the magnetic potential $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$08 is used as an order parameter through

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$09

A Landau analysis yields the mean-field exponents $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$10, $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$11, $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$12, and $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$13 (Kumara et al., 2020).

A distinct entropy concept is the Weyl-curvature-based gravitational entropy. Using

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$14

one finds for the Hayward black hole

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$15

This vanishes at the center and at $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$16, increases monotonically for $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$17, and approaches $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$18 asymptotically. On the horizons, the outer-horizon entropy decreases monotonically with the Hayward parameter, the inner-horizon entropy has a maximum at intermediate values, and the extremal horizon entropy is zero. The same analysis also raises a technical controversy: the entropy-density prescription $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$19 is described as not mathematically rigorous, and with the direct definition $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$20 the density becomes imaginary between the horizons and negative in a region near the inner horizon (Iguchi, 15 Apr 2025).

5. Rotation, energetic processes, and Kerr-like extensions

Rotation is introduced in one line of work through the Newman–Janis algorithm. Starting from the static metric in advanced null coordinates,

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$21

and performing the complex shift

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$22

one obtains a rotating regular Hayward geometry with

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$23

Setting $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$24 recovers Kerr; setting $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$25, $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$26, and $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$27 in the thermodynamic formulas recovers Schwarzschild (Ali et al., 2022).

In alternative rotating parameterizations, the deviation from Kerr is encoded by $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$28. On the equatorial plane,

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$29

and the horizon equation $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$30 admits two roots for subextremal spin, one degenerate root at extremality, and no horizon above a critical spin. For fixed $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$31, the extremal spin $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$32 decreases and the degenerate horizon radius $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$33 increases as $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$34 increases (Amir et al., 2015).

The equatorial geodesics of the rotating regular Hayward black hole support the same Kerr-like notions of innermost stable circular orbit, photon orbit, ergoregion, and Penrose process, but with quantitatively shifted radii and efficiencies. One analysis reports that both $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$35 and $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$36 enlarge the ergoregion, decrease the ISCO radius slightly, and increase the Penrose efficiency; for $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$37 and $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$38, the maximal efficiency reaches the Kerr extremal value of about $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$39, while for nonzero $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$40 it can exceed the Kerr value (Khan et al., 2021).

The Bañados–Silk–West mechanism also survives regularization. For two particles of equal rest mass coming from rest at infinity, the center-of-mass energy is computed explicitly in the rotating Hayward background. In the extremal case, if one particle carries the critical angular momentum $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$41, then $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$42 near the degenerate horizon; in nonextremal cases the center-of-mass energy remains finite, although the upper bound increases with $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$43 (Amir et al., 2015).

The near-horizon regularization also modifies binary black-hole energetics. In head-on collisions of two rotating Hayward black holes, the spin-interaction potential depends on the alignment of the spins, being repulsive for parallel alignment and attractive for anti-parallel alignment, and the thermodynamic upper bound on gravitational radiation increases with the Hayward parameter $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$44. Comparison with GW150914 and GW151226 is used in that work to limit the size of the deviation parameter (Gwak, 2017).

6. Interior dynamics, stability, and theoretical realizations

The static regularity of the Hayward metric does not by itself settle the fate of the inner horizon under dynamical perturbations. In one fully nonlinear scalar-collapse study, weak scalar perturbations leave the inner horizon at a finite radius, whereas strong scalar perturbations drive the inner horizon to zero volume and produce a spacelike singularity. Near the threshold between these regimes, the inner-horizon radius obeys the scaling law

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$45

This work states that sufficiently strong scalar collapse effectively converts the Hayward interior into a Schwarzschild-like geometry (Shao et al., 28 Nov 2025).

A different backreaction analysis finds a softer instability pattern. For the Hayward mass profile

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$46

the interior evolution equation is nonlinear in the shell mass parameter, and this prevents the exponential growth characteristic of standard mass inflation. In that framework, the interior mass parameter $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$47 approaches the finite value

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$48

while the quasi-local Misner–Sharp mass diverges only as a power law and the Kretschmann scalar has a polynomial divergence at the Cauchy horizon. The resulting singularity is described there as weak, with a possible $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$49 extension beyond the inner horizon (Bonanno et al., 2020).

Taken together, these results distinguish two issues that are often conflated. Regularity of the static center is one property; dynamical stability of the Cauchy horizon is another. This suggests that the Hayward geometry should not be identified simply with “absence of singular behavior” once strong perturbations and full backreaction are included.

Theoretical embeddings of the Hayward metric have diversified considerably. In nonlinear-electrodynamics models, the metric arises from Einstein gravity minimally coupled to a magnetic monopole sector, with Lagrangians that do not reduce to Maxwell form in the weak-field limit (Halilsoy et al., 2013, Paula et al., 2023). In pure-gravity constructions, the metric appears as a vacuum solution of four-dimensional non-polynomial quasi-topological gravities. There, the static ansatz

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$50

is governed by an algebraic master equation

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$51

and choosing

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$52

reproduces exactly

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$53

This realizes the Hayward black hole as a vacuum solution of pure gravity in four dimensions (Borissova et al., 18 Feb 2026).

An even more recent modified-gravity realization derives a Hayward-type metric from a vector-tensor action with two vector fields carrying primary hair. Regularity requires $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$54, yielding

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$55

and the original Hayward metric is recovered by the choice $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$56. In that framework, choosing

$f(r)=1-\frac{2mr^2}{r^3+2ml^2},$57

produces a regular extremal black hole for all masses, with a degenerate horizon at $f(r)=1-\frac{2mr^2}{r^3+2ml^2},$58 and vanishing surface gravity, explicitly aimed at avoiding mass-inflation instability (Eichhorn et al., 1 Aug 2025).

These action-based embeddings show that the Hayward regular black hole is no longer treated only as a phenomenological line element. It now appears in nonlinear-electrodynamics, higher-curvature, and vector-hair constructions, while the open question of inner-horizon stability remains an active point of comparison across models (Kumar et al., 2020, Eichhorn et al., 1 Aug 2025).