The validity of the universal relation between corrections to entropy and the extremality of Schwarzschild-de Sitter black holes under the GUP and EUP

Technical Summary: Validity of the Universal Relation under GUP and EUP for Schwarzschild-de Sitter Black Holes

Problem Statement
This paper investigates the validity of the Goon-Penco universal thermodynamic relation, which connects the derivative of the extremal mass bound to the derivative of entropy with respect to a perturbation parameter $\epsilon$. While previous studies (specifically Ref. [20]) confirmed that this relation holds for Schwarzschild-de Sitter (SdS) black holes in the Nariai limit under standard conditions, this work examines whether the relation remains valid when the system is subjected to corrections arising from the Generalized Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP). The central question is whether the additional terms introduced by these quantum gravity corrections can be reconciled with the universal relation through specific matching conditions, or if they fundamentally violate the extremality bounds.

Methodology
The authors employ a perturbative approach within the framework of the Nariai limit (where the event horizon and cosmological horizon coincide, $T \to 0$). The study proceeds in three main stages:

1. Baseline Review: The standard thermodynamic variables (mass $M$, entropy $S$, temperature $T$) for SdS black holes are reviewed under the action $I = \frac{1}{16\pi} \int d^4x \sqrt{-g} (R - 2(1+\epsilon)\Lambda)$, establishing the baseline Goon-Penco relation: $\frac{\partial M_{ext}}{\partial \epsilon} = \lim_{M \to M_{ext}} [-T (\frac{\partial S}{\partial \epsilon})_{M}]$.
2. GUP Analysis: The GUP is introduced as $\Delta x \Delta p \geq \frac{\hbar}{2}[1 + (\beta l_p / \hbar)^2 \Delta p^2]$. This leads to a corrected horizon radius $r'_H$ and a corresponding corrected entropy $S'$. The authors derive the modified mass, temperature, and the partial derivative of the perturbation parameter $\epsilon$ with respect to entropy, keeping mass fixed.
3. EUP Analysis: The EUP is introduced as $\Delta x \Delta p \geq \frac{\hbar}{2}(1 + \frac{\alpha}{L_*^2} \Delta x^2)$, where $L_*$ is a large fundamental distance scale. This results in a corrected horizon radius $r_H$ and modified entropy $S$. Similar to the GUP case, the authors derive the modified thermodynamic quantities and test the universal relation.

In both cases, the authors calculate the limit of the right-hand side of the Goon-Penco relation and compare it against the derivative of the extremal mass $\frac{\partial M_N}{\partial \epsilon}$. They attempt to find "matching conditions" (specific constraints on thermodynamic variables or parameters) that would force the two sides of the equation to be equal.

Key Contributions and Results

• GUP Violation: Under the GUP, the corrected entropy and mass introduce additional terms dependent on the dimensionless parameter $\beta$. When evaluating the limit $\lim_{M \to M_N} [-T' (\frac{\partial S}{\partial \epsilon})_M]$, the result contains extra terms involving $\beta$ that do not match the derivative of the extremal mass $\frac{\partial M_N}{\partial \epsilon}$. The authors demonstrate that the only way to satisfy the matching condition required for the universal relation is if $\beta l_p = 0$. Consequently, for any non-zero GUP correction, the universal relation is violated.
• EUP Violation: Under the EUP, the corrections depend on the parameter $\alpha/L_*^2$. The analysis yields a complex matching condition (Eq. 48) involving the perturbation parameter $\epsilon$, the cosmological radius $l$, and the EUP factor. The authors show that this equation has no solution for real physical quantities. Therefore, no matching condition can be imposed to restore the equality. The presence of the $\alpha$-term inherently breaks the universal relation.
• General Conclusion: The study concludes that the Goon-Penco universal relation is not robust against corrections from either GUP or EUP. The additional terms generated by these uncertainty principles cannot be canceled or matched by adjusting thermodynamic variables or imposing specific conditions, unless the influence of the GUP and EUP is entirely removed (i.e., $\beta=0$ and $\alpha=0$).

Significance and Claims
The paper argues that the corrected Heisenberg uncertainty principles (GUP and EUP) fundamentally violate the validity of the Goon-Penco extremality relation for Schwarzschild-de Sitter black holes. The authors claim that the "universal" nature of the relation is broken when quantum gravity effects are included in the form of these uncertainty principles.

The significance of this work lies in its challenge to the universality of the thermodynamic extremality relation in the context of quantum gravity. The authors suggest that if the GUP or EUP are valid descriptions of nature at certain scales, the specific thermodynamic link between entropy corrections and extremality bounds proposed by Goon and Penco does not hold for SdS black holes. The paper modestly proposes that these findings should prompt further exploration into whether similar violations occur for Reissner-Nordstrom-de Sitter and Kerr-de Sitter black holes under the same conditions, noting that additional terms involving charge or angular momentum are unlikely to be canceled by different parameter selections.