GUP-Corrected Tunneling in Black Hole Physics

• The paper demonstrates that GUP corrections modify the tunneling probability by deforming the canonical commutation relations, leading to non-thermal spectra and lower Hawking temperatures.
• It employs deformed semiclassical methods, such as the WKB and Hamilton–Jacobi approaches, to capture quantum gravity effects and the parameter dependence influencing black hole evaporation.
• GUP corrections extend beyond black hole physics to quantum mechanics and cosmology, affecting tunneling in nonrelativistic barriers and early universe vacuum transitions.

The Generalized Uncertainty Principle (GUP) corrected tunneling probability is a quantum gravity-motivated modification of semiclassical tunneling rates in black hole evaporation and quantum mechanics, reflecting the existence of a minimal measurable length. These corrections arise from deforming the canonical commutation relations, which alters the underlying wave equations and, consequently, the imaginary part of the action that governs tunneling amplitudes. Such effects lead to non-thermal spectra, modified Hawking temperatures, reduced black hole evaporation rates, and, in some scenarios, the formation of remnants.

1. Generalized Uncertainty Principle Foundations

The canonical Heisenberg commutation relations, $[x_i, p_j] = i \hbar\, \delta_{ij}$, are generalized in the GUP framework to include higher-order momentum terms: $$[x_i, p_j] = i \hbar\, (1 + \alpha\, p^2)\, \delta_{ij}$$ or, more generally, for models with both linear and quadratic terms,

$$\Delta x\, \Delta p \geq \frac{\hbar}{2}\left(1 - \beta\, \Delta p + \gamma\, (\Delta p)^2\right)$$

where $\alpha$, $\beta$, and $\gamma$ are deformation parameters, typically $\sim \ell_p^2$ for Planck length $\ell_p$. This modification leads to a minimal observable length and impacts all quantum systems at sufficiently high energies or small scales.

In quantum tunneling, the GUP modifies the effective momentum operator. For instance, in position representation,

$p \to p_0(1 + \alpha p_0^2)$

where $p_0 = -i\hbar \partial_x$, and the field equations (e.g., Klein–Gordon or Dirac) inherit these deformations (Övgün, 2015, Blado et al., 2015).

2. GUP Modification of Tunneling Probability: Formalism and Methodologies

The central methodology adapts the standard WKB or Hamilton–Jacobi approach for tunneling, now with GUP-deformed operators. For a generic black hole background, the scalar field or Dirac equation is rewritten, introducing $\alpha$-dependent corrections. The radial equation for the forbidden region yields a deformed under-barrier momentum $p_{\rm GUP}(x)$: $$|p_{\rm GUP}(x)| = \frac{1}{\sqrt{\beta}} \arctan[\sqrt{\beta} p(x)] \approx p(x) - \frac{\beta}{3} p(x)^3 + \mathcal{O}(\beta^2)$$ for a small deformation parameter $\beta$, with $p(x)$ the semiclassical momentum (Blado et al., 2015).

The WKB exponent becomes, to first order,

$$S_{\rm GUP} = \int dx\, |p_{\rm GUP}(x)| = S - \frac{\beta}{3} \int dx\, [p(x)]^3$$

and the tunneling probability

$$T_{\rm GUP} \approx \exp\left( -\frac{2}{\hbar} S_{\rm GUP} \right ) = T\, \left[ 1 + \frac{2\beta}{3\hbar} \int dx\, [p(x)]^3 \right ]$$

with $T$ the undeformed result (Blado et al., 2015).

In black hole backgrounds, the GUP enters via a deformation of the conserved energy and the momentum under the horizon. For spherically symmetric spacetimes, the radial momentum typically takes the deformed form

$p_r \rightarrow p_r (1 + \alpha p_r^2)$

which, in the context of Hamilton–Jacobi tunneling, directly affects the imaginary part of the radial action $\Im S$ (Övgün, 2015, Silva et al., 2011). The form and order of the correction may depend on the specific realization of the GUP (linear, quadratic, or both).

3. GUP-Corrected Tunneling in Black Hole Spacetimes

GUP corrections have been systematically derived in diverse black hole backgrounds and for various particle species:

• Schwarzschild black hole with electromagnetic immersion: The radial part of the action for entangled scalar or fermion particles yields a GUP-corrected imaginary part (Övgün, 2015):

$$\Im W_{\pm} = \pm \frac{\pi E M (1+a)^2}{2 a \sqrt{1-2\alpha m^2}}$$

leading to a modified Hawking temperature

$$T_{\rm GUP} = \frac{a}{2\pi M (1+a)^2} \sqrt{1-2\alpha m^2}$$

The GUP correction is linear in $\alpha$ and depends on particle mass $m$, lowering both the tunneling probability and the temperature. This holds identically for scalar and fermion cases (Övgün, 2015).

• Black holes in quintessence backgrounds: By deforming the effective black hole mass, explicit closed-form expressions for the GUP-corrected tunneling probability for both massless and massive particles are derived. For instance, in the purely GUP case (Sen, 2024):

$$\Gamma = \exp \left\{ -2\pi \left[ (2M - 4\beta)(\Omega - m) - (\Omega^2 - m^2) - 8\beta \ln \left( \frac{|\Omega - M|}{|m - M|} \right) \right] \right\}$$

where $\beta$ is the GUP deformation parameter and the corrections encapsulate both energy and mass dependence.

• General black hole geometry and remnant formation: Across all backgrounds, GUP corrections generically tend to lower the Hawking temperature, delay evaporation, and can result in a finite remnant mass, as in $T_{\rm GUP}$ tending to zero for sufficiently small black holes (Javed et al., 2018, Rizwan et al., 2018, Bhandari et al., 2024).

4. Parameter Dependence and Particle Species

GUP-induced corrections are strongly model- and species-dependent:

• For Dirac and Klein–Gordon equations, the corrections at leading order in $\alpha$ (or $\beta$) yield identical forms for the tunneling probability and temperature for both scalar and fermion emission in minimally coupled scenarios (Övgün, 2015, Gecim et al., 2017).
• In non-minimally coupled, rotating, or charged backgrounds, the corrections display more intricate dependence on the particle mass and angular momentum, and potentially on other quantum numbers (e.g., total angular momentum in 2+1 black holes) (Gecim et al., 2017, Gecim et al., 2017, Singh et al., 2019).
• The temperature reduction is generally encoded as

$$T_H^{\rm GUP} = T_H^{(0)}\, [1 - \alpha\, f(m, J, \dots)]$$

where $f(m, J, ...)$ is a model-specific linear or quadratic function of particle quantum numbers.

5. Universal Properties and Physical Implications

• Thermal Deviations and Correlations: GUP corrections generally render the emission spectrum non-strictly thermal:

$$\Gamma \sim \exp\left( -\frac{\omega}{T_H^{\rm GUP}} + \lambda \omega^2 + \cdots \right )$$

The explicit appearance of energy-dependent terms produces correlations between emission probabilities for different quanta, a necessary condition for preservation of information and unitarity (0804.4221, Bhandari et al., 2024).

• Remnants and Stopping Evaporation: For sufficiently large deformation parameters or as the mass approaches the Planck scale, $T_H^{\rm GUP}$ vanishes before the black hole completely evaporates. This mechanism produces stable remnants in all detailed studies (e.g., Schwarzschild, Reissner–Nordström, BTZ, and hair black holes) (Rizwan et al., 2018, Haldar et al., 2017, Sen, 2024, Bhandari et al., 2024).
• Universality and Spectrum Quantization: While area quantization and the quantum of area conserve their universal character even under GUP and non-commutative corrections, the level spacing becomes non-uniform, increasing as evaporation progresses (Silva et al., 2011, Haldar et al., 2017).

6. Quantum Mechanical and Cosmological Applications

GUP-corrected tunneling probabilities have been applied beyond black hole evaporation:

• Nonrelativistic Quantum Barriers: In simple potential barrier problems (rectangular, alpha decay, quantum cosmogenesis), the GUP correction tends to increase tunneling rates, in contrast to suppression typically found in gravitational tunneling (Blado et al., 2015, 0901.1768).
• Quantum Cosmology: The GUP modifies Wheeler–DeWitt vacuum transitions, generally enhancing tunneling probabilities in the “ultraviolet” (small scale factor) regime, while suppressing them for large scale factor transitions, with implications for early universe nucleation (Garcia-Compean et al., 2022).

7. Limiting Cases, Consistency, and Model Dependence

• Reduction to Classical Results: In all models, the GUP parameter $\alpha \to 0$ or $\beta \to 0$ recovers the standard Hawking–Parikh–Wilczek tunneling outcomes.
• Order and Structure of Corrections: Leading-order quadratic GUPs can yield vanishing corrections at $O(\beta)$ in some backgrounds (e.g., Schwarzschild, see (Silva et al., 2011)), and only higher order or mixed (linear + quadratic) deformations yield modifications to the exponential part of $\Gamma$.
• Model Dependence: The realization of the GUP (quadratic, linear + quadratic, higher order), as well as background geometry and particle coupling, can qualitatively change the nature (sign, magnitude, scaling) of the correction.

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References:

• "Entangled Particles Tunneling From a Schwarzschild Black Hole immersed in an Electromagnetic Universe with GUP" (Övgün, 2015)
• "Quantum tunneling driven by quintessence and the role of GUP" (Sen, 2024)
• "Effects of the Generalized Uncertainty Principle on Quantum Tunneling" (Blado et al., 2015)
• "The black-hole area spectrum in the tunneling formalism with GUP" (Silva et al., 2011)
• "Quantum Gravity Effect on the Tunneling Particles from 2+1 dimensional New-type Black Hole" (Gecim et al., 2017)
• "Effect of GUP on Hawking radiation of BTZ black hole" (Singh et al., 2019)
• "Tunnelling mechanism in non-commutative space with generalized uncertainty principle and Bohr-like black hole" (Haldar et al., 2017)
• "Quantum Tunneling from the Charged Non-Rotating BTZ Black Hole with GUP" (Sadeghi et al., 2016)
• "The GUP effect on Hawking Radiation of the 2+1 dimensional Black Hole" (Gecim et al., 2017)
• "Charged fermions tunneling from stationary axially symmetric black holes with generalized uncertainty principle" (Rizwan et al., 2018)
• "Massive Vector Particles Tunneling From Noncommutative Charged Black Holes and its GUP-corrected Thermodynamics" (Övgün et al., 2015)
• "Phenomenological Implications of the Generalized Uncertainty Principle" (0901.1768)
• "Quantum Gravity and Recovery of Information in Black Hole Evaporation" (0804.4221)
• "Minimal length effects in black hole thermodynamics from tunneling formalism" (Gangopadhyay, 2014)
• "Lorentzian Vacuum Transitions with a Generalized Uncertainty Principle" (Garcia-Compean et al., 2022)
• "Quantum Gravity Corrections to Hawking Radiation via GUP" (Bhandari et al., 2024)
• "Hawking radiation from cubic and quartic black holes via tunneling of GUP corrected scalar and fermion particles" (Javed et al., 2018)