Deformed AdS-Schwarzschild Black Holes

• Deformed AdS–Schwarzschild black holes are solutions to Einstein’s equations with a negative cosmological constant that incorporate deformation parameters α and β altering the standard geometry.
• The deformation introduces rich thermodynamic behavior with van der Waals-like phase transitions, characterized by modified Hawking temperature, swallowtail Gibbs free energy, and multiple stability regions.
• This framework provides actionable insights for testing modified gravity theories and understanding holographic dual field theory implications in quantum-corrected black hole systems.

A deformed AdS–Schwarzschild black hole is a solution to Einstein's equations with negative cosmological constant that incorporates systematic modifications—driven either by additional matter fields, gravitational decoupling, quantum corrections, or higher-derivative terms—relative to the standard AdS–Schwarzschild geometry. These deformations introduce new control parameters (notably, a deformation parameter α and often a regularizing scale β), which adjust the geometric, thermodynamic, and holographically dual field theory properties of the black hole. The resulting solutions serve as fertile ground for exploring nontrivial phase structures, transport coefficients, and connections to the AdS/CFT correspondence, as well as providing templates for precision tests of gravitational theories beyond General Relativity.

1. Geometric Structure and Definition of the Deformation

The general metric for a deformed AdS–Schwarzschild black hole is characterized by

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

with a deformed lapse function

$$F(r) = 1 - \frac{2M}{r} + \frac{r^2}{l^2} + \alpha\frac{\beta^2 + 3r^2 + 3\beta r}{3 r (\beta + r)^3},$$

where $M$ is the ADM mass, $l$ the AdS radius (with cosmological constant $\Lambda = -3/l^2$), and the deformation is controlled by the parameter $\alpha$; $\beta$ is a regularizing scale ensuring smoothness at $r=0$ (Khosravipoor et al., 2023, Panigrahi et al., 20 Aug 2025). When $\alpha=0$, the standard AdS–Schwarzschild solution is recovered; for $\beta=0$, the correction reduces to a $1/r^2$ term, resembling a charged (Reissner–Nordström–AdS) black hole with $Q^2 \sim \alpha$.

This minimal geometric deformation arises naturally via the gravitational decoupling (GD) method, which introduces an auxiliary energy-momentum source with energy density $\mathcal{E}(r) = \alpha/[\kappa (\beta + r)^4]$ that satisfies the weak energy condition (Khosravipoor et al., 2023). Smoothness of the solution and horizon regularity place upper bounds on $\alpha$.

2. Extended Thermodynamics and Phase Structure

In extended black hole thermodynamics, the cosmological constant is identified with thermodynamic pressure $P = 3/(8\pi l^2)$ and its conjugate with thermodynamic volume $V = (4\pi/3) r_h^3$ where $r_h$ is the horizon radius. The Hawking temperature is given by

$$T_H = \frac{1}{4\pi r_h} \left[1 + r_h^2 \left(\frac{3}{l^2} - \frac{\alpha}{(\beta + r_h)^4}\right)\right]$$

and the black hole entropy remains the standard area law $S = \pi r_h^2$ (Khosravipoor et al., 2023).

The deformation parameter $\alpha$ strongly influences the $T_H(r_h)$ profile. For intermediate $\alpha$ (with nonzero $\beta$), the $T_H(r_h)$ curve exhibits two inflection points, leading to multiple black hole branches distinguished by different heat capacities. In this regime the $P{-}v$ (or $P{-}r_h$) diagram displays a van der Waals–like “swallowtail” structure and first-order phase transition, characterized by a discontinuity in volume at constant pressure. The Gibbs free energy $G = M - T_H S$ develops a characteristic swallowtail in this parameter range, reflecting phase coexistence and metastability (Panigrahi et al., 20 Aug 2025). For $\alpha$ outside the critical window these features disappear, and only Hawking–Page–like transitions remain.

Critical exponents at the van der Waals–like coexistence point are $\alpha=0$, $\beta=1/2$, $\gamma=1$, $\delta=3$—consistent with mean-field theory (Panigrahi et al., 20 Aug 2025). The system also undergoes the classic Hawking–Page transition (from thermal AdS to large black hole) at a critical temperature $T_{\mathrm{HP}}$ whose location increases with $\alpha$.

Table: Thermodynamic Effects of Geometric Deformation Parameters

Parameter	Physical Role	Effect on Thermodynamics
$\alpha$	Geometric Deformation	Induces vdW transition, raises $T_{\mathrm{HP}}$
$\beta$	Regularization scale (avoids singularities)	Alters location of inflection and critical points
$l$	AdS radius / Pressure	Governs scale of $T, P, V$

For $\beta=0$ the solution behaves like a charged AdS black hole, with $\alpha$ mapping to $Q^2$.

3. Holographic Dictionary and Boundary CFT Interpretation

Through the AdS/CFT correspondence, the extended black hole thermodynamics maps onto the dual large-$N$ CFT as follows (Panigrahi et al., 20 Aug 2025):

• The AdS curvature radius $l$ sets the central charge $C = l^2/(4 G_N)$.
• The black hole energy, entropy, and temperature in the bulk rescale to $E = M/\omega$, $S = 4\pi C x^2$, $T = T_H/\omega$ with $x = r_h/l$, where the conformal factor $\omega$ matches the boundary CFT radius.
• The boundary CFT, considered in various ensembles (fixed volume $V$, central charge $C$, chemical potential $\mu$, or boundary pressure $p$), displays phase transition structures directly correlated with bulk behavior.

In the fixed $(\mathcal{V},C)$ ensemble, the free energy $F=E-TS$ displays the same van der Waals-like swallowtail as the bulk, and the Hawking–Page transition maps to a confinement–deconfinement transition in the dual theory. The critical line and branching of the phase diagram remain controlled by $\alpha$ and $\beta$, with universal exponents persisting on both sides of the duality.

4. Stability, Criticality, and Extended/Generalized Phenomena

The heat capacity $C_P$ and response functions identify stable/unstable black hole phases: $C_P < 0$ indicates instability (small black holes), $C_P > 0$ corresponds to stable (large) black holes. Divergences of $C_P$ correlate with the inflection points and thus with the locus of first-order phase transitions.

In the presence of suitable $\alpha$, the system admits two distinct critical points $T_{c1}, T_{c2}$, yielding an intermediate regime where the phase transition is realized; outside this range, monotonic thermodynamic response and the absence of a first-order transition prevail (Panigrahi et al., 20 Aug 2025). These structures echo those found in charged or rotating AdS black holes, but arise here exclusively from minimal geometric deformation.

The thermodynamic first law retains its standard extended form $dM = T_H\,dS + V\,dP$ with the mass $M$ interpreted as bulk enthalpy, matching the interpretation in terms of CFT energy. The Smarr relation is modified only by the bulk–boundary scaling.

5. Influence on Holographic Phase Transitions and Critical Exponents

The GD deformation parameter $\alpha$ is pivotal in driving the system between Hawking–Page–only and van der Waals–type phase structure. As α increases (β fixed), the critical region opens up and the phase coexistence is more pronounced, pushing the deconfinement transition in the dual CFT to higher temperatures and larger horizon radii. Phase diagrams constructed for $(T_H, S)$, $(P, V)$, $(F, T_H)$, and their CFT analogues display clear swallowtail and branching behavior over the appropriate parameter intervals.

Critical exponents are

$$\alpha = 0, \quad \beta = 1/2, \quad \gamma = 1, \quad \delta = 3,$$

indicating classic mean-field universality, even as the details of the phase diagram are controlled by the GD deformation.

6. Physical Implications and Connections to Modified Gravity

Deformed AdS–Schwarzschild black holes, realized via gravitational decoupling and similar minimal deformations, serve as archetypal examples in the paper of quantum-corrected and matter-corrected black holes. The fully analytic, parameter-controllable solutions allow for explicit tracking of the onset and nature of criticality, facilitating direct holographic mapping to dual large-$N$ gauge theories with variable central charge and volume.

These deformations also provide insights into how black hole thermodynamics is altered by nontrivial bulk stress-energy sources, as well as guiding the interpretation of astrophysical observables (e.g., black hole shadows, ringdown dynamics) in the context of non-Einsteinian gravity.

7. Mathematical Formulæ and Summary Structure

• Metric Function:

$$F(r) = 1 - \frac{2M}{r} + \frac{r^2}{l^2} + \alpha \frac{\beta^2 + 3r^2 + 3\beta r}{3 r (\beta + r)^3}$$

• Temperature:

$$T_H = \frac{1}{4\pi r_h} \left[1 + r_h^2\left(\frac{3}{l^2} - \frac{\alpha}{(\beta + r_h)^4}\right)\right]$$

• Equation of State (specific volume $v=2r_h$):

$$P = \frac{T_H}{v} - \frac{1}{2\pi v^2} + \frac{\alpha}{8\pi (\beta + v/2)^4}$$

• First Law in the bulk:

$dM = T_H dS + V dP$

• CFT Mapping:

$$E = \frac{2C x}{R} \left[1 + x^2 + \frac{\widetilde{\alpha}(b^2 + 3x^2 + 3bx)}{3x(b + x)^3}\right]$$

Here $x = r_h/l$, $\widetilde{\alpha} = \alpha/l^2$, $b = \beta/l$, and $R$ is the rescaled CFT radius.

This construction provides a minimal, highly tractable, and analytically explicit framework for exploring rich black hole phase structures and critical phenomena, both in bulk and in holographically dual theories, controlled by a single geometric deformation parameter (Panigrahi et al., 20 Aug 2025, Khosravipoor et al., 2023).