Semiclassical Charged Black Holes

• Semiclassical charged black holes are gravitational objects where classical metrics are enhanced by quantum corrections, affecting horizon structure and thermodynamics.
• Quantum corrections introduce logarithmic entropy terms and modify quasinormal mode spectra, altering both evaporation rates and oscillation frequencies.
• Extensions via quantum gravity and nonlinear electrodynamics yield regularized horizons, observable quantum hair, and potential remnant structures as dark matter candidates.

A semiclassical charged black hole is a gravitational object whose spacetime geometry and observable features arise from the interplay between classical solutions of the Einstein-Maxwell-dilaton system and leading quantum corrections to both thermodynamic and dynamical properties. Such corrections account for effects like quantum entropy contributions, vacuum polarization, quantum-corrected quasinormal modes, regularization of horizon divergences, and modified evaporation rates. These effects are systematically derived within the frameworks of quantum field theory in curved spacetime, effective actions from string theory, or direct quantization schemes, and are especially prominent for charged black holes due to the rich structure of their horizons, coupling to gauge fields, and nontrivial thermodynamic behavior.

1. Classical Backgrounds and the Semiclassical Limit

The prototypical classical charged black hole is the Reissner–Nordström solution or, in string-theoretic contexts, the Gibbons–Maeda–Garfinkle–Horowitz–Strominger (GMGHS) black hole. In such spacetimes, the line element is typically specified by mass $M$, charge $Q$, and possibly additional fields (e.g., dilaton $\phi$):

$$ds^2 = -f(r) dt^2 + f(r)^{-1} dr^2 + R^2(r) d\Omega^2,$$

where $f(r)$ and $R(r)$ encode the gravitational and gauge structure.

The standard semiclassical regime is defined by the dominance of the classical geometry with quantum corrections computed perturbatively in $\hbar$ or via the effective action. The black hole entropy at this level is given by the Bekenstein–Hawking area law ($S_0 = A/4\hbar$), with $A$ the event horizon area.

For the GMGHS charged black hole, which generalizes the Reissner–Nordström geometry via the inclusion of a dilaton field and arises from the low-energy effective action of heterotic string theory, the extremal and non-extremal regimes, as well as the presence of a nontrivial electric potential, lead to thermodynamic and scattering properties distinct from those of electrically neutral black holes (Larranaga, 2010).

2. Quantum Corrections to Black Hole Entropy

Semiclassical charged black holes admit quantum corrections to their entropy beyond the area law. These corrections may be derived using the quantum tunneling approach, path-integral expansions, or thermodynamic integrability constraints:

• Quantum Tunneling Expansion: The black hole action for a radiating particle is expanded as

$$I(r, t) = I_0(r,t) + \hbar I_1(r,t) + \hbar^2 I_2(r,t) + \ldots$$

with higher-order terms capturing quantum effects.

• First Law and Exactness: The first law,

$dM = T dS + \Phi dQ,$

when written in differential form for the entropy $S(M, Q)$, requires

$dS = \frac{1}{T} dM - \frac{\Phi}{T} dQ.$

The exactness of $dS$ (as an exact differential) imposes the integrability condition,

$$\frac{\partial (1/T)}{\partial Q} = \frac{\partial (-\Phi/T)}{\partial M},$$

ensuring that $S(M, Q)$ is path-independent. Quantum corrections modify $T$ and can be resummed in a series,

$S(M, Q) = S_0 + S_1 + S_2 + \cdots,$

where $S_1 \propto \ln(A)$ is the leading logarithmic correction and higher $S_j$ scale with powers of inverse area.

• String-inspired Results: In the GMGHS black hole, the corrected entropy reads

$$S(M,Q) = \frac{A}{4\hbar} + \pi\beta_1 \ln|A| + \sum_{j>1} \frac{\pi\beta_j \hbar^{j-1}}{1-j} \left(\frac{A}{4\pi}\right)^{1-j},$$

where $\beta_j$ are dimensionless constants and $A$ depends on both $r_H$ and $Q$ (Larranaga, 2010).

Leading logarithmic corrections are a universal feature, observed across different quantum gravity approaches for charged black holes.

3. Quantum Backreaction and Quasinormal Mode Spectrum

Quantum vacuum polarization—primarily via massive quantum fields (scalar, vector, spinor)—induces corrections to the classical Einstein-Maxwell equations:

$$G_{\mu\nu} = 8\pi (T_{\mu\nu}^{\text{classical}} + \langle T_{\mu\nu} \rangle_{\text{ren}}).$$

• The quantum stress tensor $\langle T_{\mu\nu}\rangle_{\text{ren}}$ is calculated using the Schwinger–DeWitt heat kernel expansion; the leading effect for massive fields is proportional to $1/m^2$ where $m$ is the field mass (Piedra et al., 2010).
• The resulting geometry exhibits a shift in the horizon position and metric coefficients. For instance, the radial function $1/B(r)$ is modified from its Reissner–Nordström form by quantum corrections, while the function $A(r)$ acquires an exponential of integrated stress-energy differences.
• The QNM spectrum for a test scalar (or field) in the quantum-corrected geometry is governed by an effective potential $V(r) = V^c(r) + (\epsilon/\pi) U(r)$, with $V^c$ the classical Regge–Wheeler potential and $U(r)$ encoding corrections. The WKB quantization formula,

$$i \frac{\omega^2 - V_0}{\sqrt{-2 V_0''}} = n + \frac{1}{2} + \text{higher WKB terms},$$

shows that quantum corrections shift the real part of $\omega$ upward and the imaginary part (damping) downward: oscillations persist longer and at higher frequencies relative to the classical case. This effect, found for physically motivated parameters, results in black holes that act as “better oscillators.” Enhanced quality factors $Q \propto |\text{Re}\,\omega|/|\text{Im}\,\omega|$ may have observational consequences (Piedra et al., 2010).

4. Thermodynamic and Radiative Properties in Lower-Dimensional Theories

In three dimensions, charged dilaton black holes in non-asymptotically flat spacetimes (e.g., with cosmological constant $\Lambda$) provide testbeds for semiclassical analysis with exact solvability.

• The metric function $f(r)$ admits factorization (e.g., $f(r) = 8\Lambda (r - r_+)(r - r_-)$), which enables exact solutions of the wave equation for test fields in terms of hypergeometric functions.
• Radiation Spectrum Dependence on Charge and $\Lambda$: The Hawking temperature obtained from the high-frequency limit of the radiation spectrum yields the uncharged result, $T = \Lambda / \pi$. Inclusion of charge is only properly encoded in the low-frequency regime, with the temperature depending on both horizon radii $r_+$ and $r_-$:

$$T_{II} = \frac{\Lambda}{\pi}\left( 1 - \frac{r_-}{r_+} \right).$$

Thus, the correct charged Hawking temperature is only accessible through a low-frequency analysis; the field’s long-wavelength components probe the global (non-asymptotically flat) geometric features (Sakalli, 2012).

5. Semiclassical Horizon Fluctuations, Matter Observables, and Hair

Treating the black hole horizon itself as a quantum variable—rather than an infinitely sharp classical surface—regularizes classic divergences and reveals new physics:

• Horizon Fluctuations: The black hole wavefunction may be modeled as a Gaussian centered at the classical horizon location with width $\sim 1/\sqrt{S_{BH}}$.
• Observable Computation Prescription: Observables $O(R)$ dependent on the horizon location $R$ are averaged over these fluctuations:

$$\langle O \rangle = \int dR\, |\psi_{BH}(R)|^2\, O(R).$$

This procedure ensures that classically divergent matter densities at the horizon become finite, scaling as inverse powers of $1/S_{BH}$.

• Hair and Global Charges: Quantum smearing of the horizon allows for small, nonzero expectation values of classically forbidden operators—encoded at order $1/S_{BH}$. As a result, forbidden “hair” or global charge can leak into observable quantities, and information about the interior is not absolutely hidden (Brustein et al., 2013).
• Resolution of Paradoxes: This formalism addresses the infinite redshift, information loss, and trans-Planckian issues endemic to the classical description, indicating that semiclassical corrections play a pivotal role in rendering black hole physics physically consistent and unitary.

6. Quantum Gravity Extensions, Remnants, and Nonlinear Electrodynamics

Incorporating quantum gravitational corrections and nonlinear field sources yields new structures and possible endpoints for black hole evaporation:

• Palatini Quadratic Gravity: By adopting the Palatini (first-order) formalism in gravity with quadratic curvature terms, one obtains second-order field equations free from metric ghost instabilities (Lobo et al., 2013). The resulting solutions for charged black holes are free from curvature singularities, exhibit a minimum radius (interpreted as a wormhole throat), and, when regularity conditions are imposed, have a discrete mass spectrum:

$$M \simeq 1.23605 \left(\frac{N_q}{N_q^c}\right)^{3/2} m_P, \quad N_q^c \approx 16.$$

These can act as stable solitonic (geon) remnants, which are astrophysically relevant as dark matter candidates or relics of primordial black holes.

• Quantum Oppenheimer–Snyder Models: Quantum-corrected exterior geometries (APS metrics) can be matched to dust interiors via thin shells with charge, and charge is implemented via nonlinear electrodynamics with power-law field invariants (e.g., $L \propto (-F)^s$). The equilibrium dynamics of the shell lead to stability criteria, regularized horizons, and potentially nonsingular global spacetime structure (Mazharimousavi, 12 Feb 2025).

7. Quantum Gravity and the S-Matrix for Collapse and Evaporation

The semiclassical S-matrix approach enables a consistent computation of gravitational transition probabilities, including for charged shells, by carefully accounting for metric backreaction:

• By integrating the semiclassical action over complex contours, the tunneling exponent for collapse-evaporation processes can be robustly associated with the Bekenstein–Hawking entropy, independent of coordinate choices, with the probability for a process given by $\exp(-S_{BH})$ (Bezrukov et al., 2015).

This approach is applicable to both neutral and charged systems (with proper considerations for Cauchy horizon instabilities) and is suggestive of a deep connection between semiclassical probabilities and the informational content of black holes.

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These results collectively delineate the current understanding of semiclassical charged black holes. They connect microscopic field-theoretic corrections, quantum regularization of classic pathologies, observable implications for thermodynamics and ringdown signals, and the extended solution space allowed by quantum gravitational theories. The interplay between classical geometry and semiclassical quantum effects is essential for a consistent and predictive description of charged black holes in a regime relevant for quantum gravity and high-energy astrophysics.