SV-Regularization in Black Hole Models

• SV-Regularization parameter is a scalar that smooths black hole singularities by modifying the radial coordinate using √(r² + a²).
• It influences horizon structure and thermodynamic properties, introducing corrections in temperature, entropy, and phase transitions.
• This tunable parameter enables controlled transitions from singular black holes to traversable wormholes, bridging gravitational physics and regularization techniques.

The SV-regularization parameter is a fundamental scalar quantity that governs the behavior of regularization schemes originating in gravitational physics and, by analogy, plays crucial roles in other scientific domains where singularity-avoidance or controlled penalization is required. In the context of the Simpson–Visser regularization—central to recent developments in regular black hole models—the SV-regularization parameter (typically denoted $a$) controls the degree to which central singularities are smoothed, thus altering both geometric and thermodynamic properties of the resulting spacetime. This parameter serves as a tunable measure of how strongly the regularization modifies the underlying physical or mathematical object, and systematically interpolates between the singular (or unregularized) and fully regularized regimes.

1. Definition and Geometric Role of the SV-Regularization Parameter

The SV-regularization parameter $a > 0$ enters the spacetime metric of a regularized black hole by deforming the standard Schwarzschild radial coordinate, replacing $r$ with $\sqrt{r^2 + a^2}$. Concretely, the SV–AdS metric reads

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

where the lapse function is

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

Here, $M$ is the ADM mass and $\ell$ is the AdS radius. The parameter $a$ regularizes the central curvature singularity, introducing, at $r=0$, a smooth “throat” of radius $a$. When $a$ exceeds a threshold, the geometry is further modified into a traversable wormhole topology (Kumar et al., 26 Nov 2025).

2. Influence on Horizon Structure and Solution Space

The presence and value of the SV-regularization parameter directly control the black hole’s horizon configuration. Horizons are determined by solving $f(r_h, a) = 0$, yielding real roots only when

$\frac{(\ell^2 - u)^2}{3u} > a^2,$

where $u$ is a function of $M$ and $\ell$. In the asymptotically flat limit $(\ell \to \infty)$, the existence of regular BHs is guaranteed for $a < 2M$, reproducing the SV-Schwarzschild bound. This parameter thus demarcates the regime where a regular black hole solution exists versus when a non-singular, non-black-hole (e.g., wormhole) spacetime results (Kumar et al., 26 Nov 2025).

3. Impact on Thermodynamic Quantities and Laws

The SV-regularization parameter modifies all key thermodynamic quantities:

• Temperature: The Hawking temperature for the outer horizon is

$$T = \frac{1}{4\pi}\left[ \frac{r_h}{r_h^2 + a^2} + \frac{3 r_h}{\ell^2} \right].$$

• Entropy: Integrating the first law yields

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

with nontrivial logarithmic corrections introduced by $a$. In the limit $a \rightarrow 0$, the standard area law $S = A/4$ is recovered (Kumar et al., 26 Nov 2025).

• Free Energy: The Helmholtz free energy incorporates $a$ through both the mass function and entropy, leading to corrections affecting phase stability.

These explicit dependences introduce new physical features, including the appearance of extremal (zero-temperature) regular black holes and the removal of the Hawking–Page transition in AdS, replaced by an extremal background as $T \to 0$ (Kumar et al., 26 Nov 2025).

4. Phase Transitions and Critical Behavior

The SV parameter induces novel phase behavior in black hole thermodynamics. For any $a > 0$, the temperature function $T(r_h)$ develops local maxima and minima, producing multiple solution branches (stable small and large black holes, unstable intermediates). The free energy as a function of temperature exhibits a swallowtail structure characteristic of first-order transitions, with the form and disappearance of the swallowtail governed by the critical value $a_c$. Beyond $a_c$, only a single black hole branch remains and the transition becomes second-order (Kumar et al., 26 Nov 2025).

No Hawking–Page transition is present for SV-black holes; even for vanishing temperature, the spacetime retains a regular, extremal black hole configuration instead of reverting to pure thermal AdS. The SV-regularization parameter hence controls not only regularity but also the qualitative nature of the system’s thermodynamic and phase properties.

5. Merger Physics and Entropy/Mass Bounds

The SV-regularization parameter exerts a critical influence on the entropy and mass bounds arising from the generalized second law in black hole mergers. For a merger of two equal-mass SV–AdS black holes, the final entropy $S_f$ must satisfy

$S_f \geq 2 S_i,$

with $S$ given above. The corresponding mass bound on the post-merger remnant $M_f(a)$ is highly nontrivial: as $a$ increases, the maximum allowed $M_f$ initially rises above the standard Schwarzschild–AdS value, peaks at a critical $a_*$, then decreases sharply for larger $a$. This non-monotonic dependence is absent in the unregularized case and is regulated directly by the logarithmic term in the SV-entropy formula (Kumar et al., 26 Nov 2025).

Quantitative summary of merger bound behavior:

Parameter regime	Mass bound behavior	Qualitative result
$a \ll M$	Increases with $a$	Bound exceeds standard case
$a \approx a_*$	Peaks	Maximal post-merger $M_f$
$a > a_*$	Decreases sharply	Upper bound drops below standard case

6. Physical Interpretation and Regularity Significance

The SV-regularization parameter quantifies the “strength” of central smoothing: $a = 0$ returns the singular Schwarzschild (or Schwarzschild–AdS) black hole, $0 < a < 2M$ yields a regular black hole with a finite minimal area at $r=0$, and $a > 2M$ gives rise to a traversable wormhole. The parameter thus interpolates between distinct geometric and physical regimes, providing a controlled lever to paper deviations from classical singular solutions. All thermodynamic expressions, phase structures, and merger bounds reduce smoothly to their standard general relativistic forms as $a \rightarrow 0$, confirming the robustness of the SV parameterization (Kumar et al., 26 Nov 2025).

7. Broader Context and Extensions

While the SV-regularization parameter arises in the context of regular black hole geometries, the mechanism of introducing a dimensionful smoothing scale to regulate pathological behavior has direct analogs in other fields, including statistical regularization (e.g., Tikhonov and ridge-based estimators), variational optimization (cubic regularization in nonconvex minimization), and machine learning (hyperparameter selection in SVMs and deep networks). In all these contexts, the regularization parameter orchestrates a balance between fidelity to data or underlying geometry and the suppression of instability or singular behavior, confirming its centrality in contemporary mathematical physics and applied computation.