SV-Regular Black Holes

• SV-regular black holes are non-singular, static, spherically symmetric solutions that replace the central singularity with a de Sitter core using a radial substitution.
• They employ a metric deformation (r → √(r² + a²)) that ensures all curvature invariants remain finite while recovering Schwarzschild–AdS behavior at large radii.
• Their unique thermodynamic properties, including modified Hawking temperature and entropy with van-der-Waals like phase transitions, offer new insights into merger dynamics and gravitational-wave bounds.

SV-regular black holes, or singularity-free black holes constructed via the Simpson–Visser (SV) regularization scheme, represent a class of non-singular, static and spherically symmetric solutions to Einstein's equations. These spacetimes admit a regular center in place of a curvature singularity, while retaining the essential causal feature of an event horizon. SV-regular black holes extend the paradigm of regular black holes—initiated by Bardeen and Hayward—by applying a universal regularization technique that deforms the Schwarzschild or Schwarzschild–AdS metrics through a simple radial substitution, yielding models free of central blow-up and amenable to thermodynamic and dynamical analysis (Neves, 2017, Kumar et al., 26 Nov 2025).

1. Metric Structure and Regularization Prescription

The general SV-regular black hole metric in an AdS background is given by

$$ds^2 = -f(r,a)\,dt^2 + \frac{dr^2}{f(r,a)} + (r^2 + a^2)\,d\Omega^2,$$

with the lapse function

$$f(r,a) = 1 - \frac{2M}{\sqrt{r^2 + a^2}} + \frac{r^2 + a^2}{L^2},$$

where $M$ is the ADM mass, $L$ the AdS curvature radius, and $a > 0$ is the SV-regularization parameter (Kumar et al., 26 Nov 2025). Setting $a=0$ recovers the conventional Schwarzschild–(AdS) solution, which is singular at $r=0$. The SV regularization ($r \rightarrow \sqrt{r^2+a^2}$) ensures that for any $a>0$, the geometry is regular everywhere on $r \in (-\infty,+\infty)$.

For large $r$, the metric asymptotes to Schwarzschild–AdS, reproducing the correct ADM mass: $$f(r,a) \to 1 - \frac{2M}{r} + \frac{r^2}{L^2}, \quad \text{as} \ r \to \infty.$$ For $r \to 0$, the lapse expands as $f(r,a) \approx 1 - 2M/a + a^2/L^2$, manifesting finiteness at the center and realizing a de Sitter-like core.

2. Regularity, Curvature Scalars, and Horizon Structure

Regularity requires all curvature invariants to remain finite for all $r$. The Kretschmann scalar,

$K(r) = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma},$

admits a closed analytic expression in terms of derivatives of $f(r,a)$ (Neves, 2017). For $a>0$, near $r=0$: $$f(r,a) \approx 1 - \frac{2M}{a} + \frac{a^2}{L^2}, \quad f'(r,a) \propto r/a^3,$$ and thus $K(r)$ and all Ricci invariants are finite, avoiding the Schwarzschild singularity.

Horizons are located by real, positive roots of $f(r,a) = 0$. Depending on $M$, $a$, and $L$, the model admits up to two horizons: an outer event horizon $r_+$ and, for certain parameters, an inner Cauchy horizon $r_-$. In the extremal case ($M=M_\text{ext}$), the two roots coalesce; for sub-extremal parameters, horizons are absent and the solution describes a regular non-black-hole object (Kumar et al., 26 Nov 2025, Neves, 2017).

3. Thermodynamics: Temperature, Entropy, and Phase Structure

SV-regular black holes exhibit modifications to standard black hole thermodynamics:

• Hawking temperature: The surface gravity yields

$$T(r_h,a,L) = \frac{1}{4\pi} \left[ \frac{r_h}{r_h^2+a^2} + \frac{3r_h}{L^2} \right]$$

where $r_h$ is the event horizon radius. For $a \to 0$, this recovers the Schwarzschild–AdS temperature (Kumar et al., 26 Nov 2025).

• Entropy: Derived from the first law $dM = T dS$, the entropy is

$$S(r_h,a) = \pi \left[ r_h \sqrt{r_h^2 + a^2} - a^2 \ln\left(\frac{\sqrt{r_h^2 + a^2} - r_h}{a}\right) \right],$$

reverting to the Bekenstein–Hawking area law $S = \pi r_h^2$ as $a \to 0$. The SV correction introduces a logarithmic term proportional to $a^2$ (Kumar et al., 26 Nov 2025).

• Helmholtz Free Energy: In canonical ensemble,

$F = M(r_h,a) - T(r_h,a)\,S(r_h,a),$

leads to a modified phase structure, displaying van-der-Waals–like behavior: for $a=0$, there is a standard Hawking–Page transition; for $a>0$, black holes exist at all temperatures down to extremality ($T \to 0$ at $r_h = a$), with a first-order small-large black hole phase transition up to a critical $a_c$ (Kumar et al., 26 Nov 2025).

4. Binary Mergers and Second Law Constraints

SV-regular black holes show distinctive behavior in binary merger scenarios:

• The second law requires that the entropy of the final (post-merger) black hole, $S_f$, satisfies $S_f \geq 2 S_i$, where $S_i$ is the initial entropy of each equal-mass SV-black hole (Kumar et al., 26 Nov 2025).
• Expressing $M_f = M(r_h^f, a)$ with $S_f$ as above, the allowed minimum final mass $M_f(a)$ is non-monotonic: For small $a$, $M_f(a)$ increases above the classical Schwarzschild–AdS value, peaking at some $a \approx a_{\text{max}}$, then declines for larger $a$ (flat-space limit $L \to \infty$ preserves this trend).
• This suggests that SV-regularization introduces an extra core entropy that affects gravitational-wave bounds and merger remnant mass predictions.

5. Effective Matter Content and Energy Conditions

The Einstein tensor for SV-regular black holes corresponds to a stress-energy tensor with anisotropic fluid components: $$T^\mu{}_\nu = \mathrm{diag}(-\rho(r), p_r(r), p_t(r), p_t(r)),$$ where, for suitable $m(r)$,

$$\rho(r) = \frac{m'(r)}{4\pi r^2}, \quad p_r(r) = -\frac{m'(r)}{4\pi r^2}, \quad p_t(r) = -\frac{m''(r)}{8\pi r}.$$

Regularity ($m \sim r^3$ near $r=0$) ensures all components remain finite at the origin (Neves, 2017).

For Bardeen-type and Hayward-type regular black holes, the weak energy condition can be satisfied globally if suitable Lagrangians—such as those from nonlinear electrodynamics—are employed. The strong energy condition is violated near the de Sitter core, permitting evasion of classical singularity theorems.

6. Global Structure, Comparison, and Physical Observables

The global (Penrose) structure of SV-regular black holes echoes that of Reissner–Nordström but replaces the inner timelike singularity with a regular de Sitter core (Neves, 2017). The event horizon persists, and, for appropriate parameters, the spacetime also admits a Cauchy horizon.

Physically, the parameters $M$ and $a$ (or scale parameters $g$, $\ell$ in alternative models) specify the asymptotic mass and the core regularization scale—which can be interpreted as arising from quantum gravity effects or nonlinear electromagnetic charges. At large radius, SV-regular black holes are observationally indistinguishable from Schwarzschild–AdS, but corrections become manifest in strong field phenomena such as quasinormal mode spectra, black hole shadow size, accretion disk emission profiles, and post-merger mass bounds in gravitational-wave events (Kumar et al., 26 Nov 2025).

The typical instability of the inner (Cauchy) horizon to mass inflation persists, though in certain nonlinear parameter regimes stability can be achieved (Neves, 2017). SV-regularization realizes explicit, causal, and thermodynamically consistent models that contest the inevitability of classical curvature singularities.

7. Explicit Examples and Relations to Other Regular Black Holes

Early models—Bardeen and Hayward—can be cast within the SV framework. For example, the Bardeen mass function

$m(r) = \frac{M r^3}{(r^2 + g^2)^{3/2}},$

and the Hayward mass function

$m(r) = \frac{M r^3}{r^3 + \ell^3},$

both yield $f(r)$ which, for $r \to 0$, approximates a de Sitter core (Neves, 2017). SV-regular black holes generalize these to arbitrary backgrounds (including AdS) and provide a systematic regularization via $r \rightarrow \sqrt{r^2 + a^2}$, unifying various constructions and extending their applicability to broader gravitational and thermodynamic analyses.

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Key References:

• "Well-behaved relativity: regular black holes" (Neves, 2017)
• "Simpson-Visser-AdS Black Holes: Thermodynamics and Binary Merger" (Kumar et al., 26 Nov 2025)