Kazakov–Solodukhin Black Hole: Quantum Gravity Insights

• The Kazakov–Solodukhin black hole is a quantum-corrected, spherically symmetric deformation of the Schwarzschild metric that replaces the singularity with a finite Planck-scale sphere.
• Its thermodynamics includes logarithmic entropy corrections, a finite mass remnant, and a Hawking–Page type phase transition, reflecting quantum gravitational effects.
• Observable features such as an enlarged shadow, narrowed photon/lensed rings, and altered polarization signatures distinguish it from classical GR black holes.

The Kazakov–Solodukhin black hole is a spherically symmetric, asymptotically flat solution that arises from quantum-gravitational corrections to the Schwarzschild geometry. Originally motivated by effective two-dimensional dilaton gravity treatments and embedding in deformed Hořava–Lifshitz gravity, it serves as a paradigmatic example of a "minimal deformation" of the Schwarzschild black hole, introducing Planck-scale modifications without radically altering the asymptotic or horizon structure. Its observational properties, thermodynamics, regularity, and quantum statistical behavior have been extensively analyzed, in both theoretical and phenomenological contexts.

1. Spacetime Structure and Quantum Deformations

The Kazakov–Solodukhin (KS) metric modifies the standard Schwarzschild line element through a deformation parameter $a$, often interpreted as encoding quantum gravitational corrections: $$ds^2 = -f^{\mathrm{(KS)}}(r)\,dt^2 + \frac{dr^2}{f^{\mathrm{(KS)}}(r)} + r^2 (d\theta^2 + \sin^2\theta\,d\phi^2)$$ where

$$f^{\mathrm{(KS)}}(r) = \frac{\sqrt{r^2 - a^2}}{r} - \frac{2M}{r}$$

For $a \to 0$, the Schwarzschild limit $f(r) = 1 - 2M/r$ is recovered. The parameter $a$ represents a minimal length, typically of order the Planck scale, and regularizes the classical singularity: the center $r=0$ is replaced by a sphere of radius $r=a$ with finite area, rather than a curvature singularity (Berry et al., 2021). The event horizon and photon sphere are shifted outward relative to Schwarzschild, and curvature invariants remain finite for sufficiently strong deformation $n \geq 5$ in generalizations (Berry et al., 2021).

2. Thermodynamic Properties and Phase Structure

The thermodynamics of the KS black hole displays rich features distinct from the classical case, especially when considered within a finite isothermal cavity to render the canonical ensemble well-defined in asymptotically flat space (Eune et al., 2012):

• Hawking and Tolman Temperatures: The local Tolman temperature at cavity radius $r$ is

$$T = \frac{T_H}{\sqrt{f(r)}},\quad T_H = \frac{-1 + 2\omega r_+^2}{8\pi r_+(1+\omega r_+^2)}$$

with $r_+$ the event horizon and $\omega$ a HL gravity coupling.

• Entropy:

$$S = \frac{A_H}{4} + \frac{\pi}{\omega} \ln\left(\frac{A_H}{4}\right) + S_0$$

exhibiting a logarithmic correction indicative of quantum effects and higher-derivative contributions.

• Internal Energy:

Derived by integrating the first law $dE = T\,dS$, leading to a nontrivial mass dependence and finite extremal remnant.

• Specific Heat and Stability:

Specific heat $C = \partial E / \partial T$ can be positive or negative; phase diagram analysis reveals small locally stable black holes (near extremality) and large stable black holes. For $\omega > \omega_{\text{cr}}$, there is a three-branch structure reminiscent of classical black hole phase transitions.

• Hawking–Page Type Transition and Tunneling:

A first-order phase transition occurs at a critical temperature $T_*$: above $T_*$, the large black hole phase is thermodynamically favored over hot flat space; below, the system may tunnel into flat space or the large black hole via quantum tunneling, especially from the unstable branch or extremal remnant (Eune et al., 2012).

The presence of a minimal mass remnant $M_{\min} = 1/\sqrt{2\omega}$ (extremal limit) and the possibility of tunneling decay into vacuum distinguish the KS black hole thermodynamics from classical Schwarzschild evaporation scenarios.

3. Geodesic Structure, Regularity, and Force Properties

The KS black hole replaces the curvature singularity at $r=0$ with a Planck-scale 2-sphere, allowing all geodesics to exist only for $r > a$ (Nozari et al., 2020, Berry et al., 2021). The effective potential for null and timelike geodesics is modified: $$V_{\text{eff}}(r) = f(r)\,\left(\frac{L^2}{r^2} + \epsilon\right)$$ with $\epsilon=0$ (null) or $-1$ (timelike). The innermost accessible region is no longer $r \to 0$ but $r=a$, where metric, connection, and curvature invariants can be rendered finite for appropriate choice of deformation parameter (Berry et al., 2021). The force derived from the effective potential reveals that the quantum term acts repulsively in the interior, providing a possible driver for local acceleration—a scenario in which quantum effects, rather than quintessence/dark energy (which is found to be attractive on these scales), account for repulsive cosmic acceleration (Nozari et al., 2020).

4. Optical Signatures: Shadows, Photon Rings, and Gravitational Lensing

Shadow Geometry and Observational Constraints

Quantum corrections in the KS metric systematically enlarge the event horizon, photon sphere, and critical impact parameter $b_p$ relative to Schwarzschild: $$r_h^{\mathrm{(KS)}} = \sqrt{4M^2 + a^2},\quad r_p^{\mathrm{(KS)}} = \sqrt{\frac{3}{2}\left[a^2 + \sqrt{2a^2 + 9} + 3\right]}$$ As a result, the shadow (defined by $b_p$) increases in size for spherical symmetry and face-on viewing angles (Gong et al., 28 Aug 2025). The shadow radius in the thin accretion disk model is determined purely by the geometry and is insensitive to the accretion emission model; EHT data constrains the deformation parameter to $a/M \lesssim 0.23$ (1$\sigma$), $a/M \lesssim 0.94$ (2$\sigma$) (Gong et al., 28 Aug 2025).

Photon and Lensed Ring Structure

Photon and lensed rings in the KS black hole are narrower than in Schwarzschild or magnetic (Ghosh–Kumar, GK) black holes. This narrowing results from the outward-shifted, shallower photon sphere and affects the demagnification and integrated intensity of multi-orbit photon emissions seen in optically thin disk models (Gong et al., 28 Aug 2025). Direct emission dominates the observed brightness, with photon/lensed ring contributions suppressed.

Gravitational Lensing and Deflection

At large distances (weak deflection limit), the deflection angle receives an enhancement quadratic in $a$: $\beta \approx \frac{4M}{b} + \frac{3\pi a^2}{8b^2}$ In the presence of a plasma medium, additional terms appear, increasing the deflection angle further through the plasma's refractive index. These corrections may be accessible to precision lensing experiments (Javed et al., 2022).

5. Accretion, Emission, and Polarization Phenomena

Spherical Accretion

KS black holes under spherical accretion exhibit a larger shadow (as $a$ increases) and decreased integrated intensity compared to Schwarzschild. Radial 4-velocity and proper energy density of infalling matter remain finite at $r = a$, in contrast to the divergence at $r \to 0$ in uncorrected solutions, due to the regularized central region (Nozari et al., 2020).

Thin Disk and Optical Appearance

For optically thin, geometrically thin disks, direct emission constitutes the main contributor to observed intensity. Photon and lensed rings, being narrower, have limited impact on total brightness. The observed integrated intensity across the shadow is decreased as $a$ increases, and rings appear compressed in $b$-parameter space (Gong et al., 28 Aug 2025, Huang et al., 2023).

Polarization Signatures

The influence of quantum corrections on polarization images is pronounced: increasing $a$ leads to a reduction in polarization intensity at fixed radius and an expansion of the region where polarization is detectable (Guo et al., 21 May 2024). The relationship is monotonic—the greater the quantum deformation, the more pronounced the "anti-polarization" effect. These effects persist regardless of observer inclination, magnetic field configuration, or emitting fluid velocity, providing a testable signature for quantum gravitational corrections (Guo et al., 21 May 2024).

6. Quantum Tunneling, Information, and Statistical Correspondence

The quantum-corrected KS metric modifies the Hawking radiation process:

• The tunneling amplitude, calculated via the Parikh–Wilczek method for the deformed metric, leads to a nonthermal spectrum with enhanced correlations between emitted quanta, thus partially resolving the information loss problem (Hajebrahimi et al., 2020).
• The deformation parameter acts analogously to an "imaginary" electric charge term ($Q \equiv i(\sqrt{2}/2)a$), mimicking features of Reissner–Nordström black holes such as the presence of inner and outer horizons (Hajebrahimi et al., 2020, Nozari et al., 2020).
• The end state of evaporation is a finite mass remnant, which can decay into vacuum via quantum tunneling, in contrast to continuous evaporation in classical scenarios (Eune et al., 2012).

Quantum statistical mechanics links the KS black hole to non-singlet matrix quantum mechanics, with microstate counting in matrix models reproducing black hole entropy and elucidating the emergence of the remnant and the evaporation dynamics (Betzios et al., 2022, Ahmadain et al., 2022). These approaches reveal that the KS black hole serves as an avatar of a two-cut (or plateau) phase in large-$N$ 2D Yang–Mills theory, realized at the Douglas–Kazakov transition (Gorsky et al., 2016).

7. Comparative and Observational Distinction

Compared with other minimal deformations—such as the Ghosh–Kumar (GK) black hole, which is magnetically charged—the KS black hole uniquely combines increased shadow size, lower intensity, and narrower photon/lensed rings, whereas the GK has a contracted shadow and higher intensity (Gong et al., 28 Aug 2025). These distinctions are robust against varying accretion and emission models under the assumptions of spherical symmetry and non-rotating configurations.

Contemporary EHT observations set empirical constraints on the allowed parameter space for quantum deformations. Additional observational channels—including gravitational lensing, ringdown echoes, and high-resolution polarimetry—may further distinguish KS black holes from both Schwarzschild and other regular/magnetically charged solutions.

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Key Mathematical Relations (Kazakov–Solodukhin, $M=1$ units):

Property	Schwarzschild	Kazakov–Solodukhin	Ghosh–Kumar
Event horizon $r_h$	$2M$	$\sqrt{4M^2 + a^2}$	$\sqrt{4M^2 - a^2}$
Photon sphere $r_p$	$3M$	$\sqrt{\frac{3}{2}(a^2 + \sqrt{2a^2+9} + 3)}$	$({\text{solution of cubic}})$
Critical impact $b_p$	$3\sqrt{3}$	(function of $a$ above)	(contracted w.r.t. Schwartz)
Shadow (thin disk) radius	$b_p$	$b_p$	$b_p$

Optical and Thermodynamic Distinctions:

• Shadow size (KS): increasing with $a$
• Integrated flux (KS): decreasing with $a$ for fixed accretion
• Photon/lensed ring widths (KS): decrease with $a$; direct emission dominates
• Polarization intensity (KS): monotonic decrease with $a$; polarization region expands
• Thermodynamic stability: small and large stable black holes, Hawking–Page transition, nonvanishing remnant mass

Implications: The Kazakov–Solodukhin black hole, through its deformation parameter $a$, encodes quantum gravitational effects in a manner directly accessible to both theoretical and observational investigation, providing constraints and potential observable deviations from classical GR black hole predictions in the shadow size, intensity profile, photon ring structure, polarization signatures, and evaporation dynamics.