[Paper Review] The Planck length as a duality of the Cosmological Constant: S-dS and S-AdS thermodynamics from a single expression

Ivan Arraut|arXiv (Cornell University)|May 31, 2012

Noncommutative and Quantum Gravity Theories20 references3 citations

TL;DR

This paper proposes a unified Generalized Uncertainty Principle (GUP) incorporating both the Planck length ($l_{\text{pl}}$) as an ultraviolet (UV) cutoff and the cosmological constant scale ($r_\Lambda = 1/\sqrt{\Lambda}$) as an infrared (IR) cutoff, demonstrating that this single GUP expression accurately reproduces the thermodynamics of both Schwarzschild-de Sitter (S-dS) and Schwarzschild-anti-de Sitter (S-AdS) black holes without requiring analytic continuation of parameters. The key result is that the temperature of the S-dS black hole, when measured by a 'natural' static observer at $l_0 = (\frac{3}{2}r_s r_\Lambda^2)^{1/3}$, matches the S-AdS thermodynamics under the same GUP framework.

ABSTRACT

In this paper we suggest that the Planck length $l_{pl}$ and the Cosmological Constant scale $r_Λ=\frac{1}{\sqrtΛ}$ could in principle be dual each other if we take seriously the so-called q-Bargmann Fock space representation as has been previously suggested by Kempf and others and if additionally we introduce $l_{pl}$ as an ultraviolet cut-off and $r_Λ=\frac{1}{\sqrtΛ}$ as an infrared one. As a consequence, it is possible to demonstrate that a Generalized Uncertainty Principle (GUP) given by $ΔX ΔP\geq \frac{\hbar}{2}+\frac{l_{pl}^2}{2\hbar}(ΔP)^2+\frac{\hbar}{2r_Λ^2}(ΔX)^2$, can reproduce appropriately the thermodynamic behavior for both, the Schwarzschild Anti de-Sitter (S-AdS) and the Schwarzschild de-Sitter (S-dS) space without making any analytic extension for the coefficient (parameter) related to the minimum uncertainty in momentum (already suggested in the literature). This is possible if the Black Hole temperature is described with respect to the "natural" Static Observer for the S-dS case located at a distance $l_0=(3/2r_s r_Λ^2)^{1/3}$.

Motivation & Objective

• To resolve inconsistencies in existing GUP approaches that require analytic continuation of IR parameters for S-dS black hole thermodynamics.
• To unify the thermodynamic description of S-dS and S-AdS black holes using a single GUP expression with both Planck length and cosmological constant scales as fundamental cutoffs.
• To demonstrate that the GUP with UV and IR cutoffs naturally reproduces the minimum temperature in S-AdS and the correct temperature profile in S-dS without ad hoc assumptions.
• To clarify the role of the 'natural' static observer in S-dS spacetime and its impact on surface gravity and temperature corrections.

Proposed method

• Derive a generalized uncertainty principle (GUP) of the form $\Delta X\Delta P \geq \frac{\hbar}{2} + \frac{l_{\text{pl}}^2}{2\hbar}(\Delta P)^2 + \frac{\hbar}{2r_\Lambda^2}(\Delta X)^2$, incorporating both Planck length ($l_{\text{pl}}$) as UV cutoff and $r_\Lambda = 1/\sqrt{\Lambda}$ as IR cutoff.
• Apply the GUP to black hole thermodynamics by identifying $\Delta X \approx r_+$ (event horizon) and $\Delta P \approx \kappa$ (surface gravity), using the q-Bargmann Fock space formalism to justify the duality between UV and IR scales.
• Reinterpret the S-dS black hole temperature by measuring it with respect to the 'natural' static observer located at $l_0 = (\frac{3}{2}r_s r_\Lambda^2)^{1/3}$, which modifies the normalization of the Killing vector and corrects the surface gravity.
• Expand the surface gravity expressions for both S-dS and S-AdS cases in the limit $M \ll M_{\text{max}}$ and $M \ll M_{\text{crit}}$, respectively, to compare first-order corrections from the GUP.
• Match the S-dS temperature near extremality ($r_{BH} \to r_\Lambda$) with the S-AdS temperature by solving for the horizon radius $r_+ = C r_\Lambda$ and deriving the constraint $C = 1 + \frac{2}{3}\epsilon - \frac{2}{\sqrt{3}}\sqrt{\epsilon}$, showing agreement in the limit $\epsilon \to 0$.
• Use the four solutions of the GUP equation to distinguish between negative and positive heat capacity regions, confirming consistency with known thermodynamic behaviors in both S-dS and S-AdS spacetimes.

Experimental results

Research questions

• RQ1 Can a single GUP expression with both Planck length and cosmological constant scales as cutoffs reproduce the thermodynamics of both S-dS and S-AdS black holes without analytic continuation?
• RQ2 How does the choice of observer—specifically the 'natural' static observer in S-dS spacetime—affect the surface gravity and temperature derived from the GUP?
• RQ3 What is the physical significance of the geometric mean scale $l_0 = (l_{\text{pl}} r_\Lambda)^{1/2}$ in the context of GUP and black hole thermodynamics?
• RQ4 Why do the S-dS and S-AdS black hole temperatures become indistinguishable near $r_{BH} \to r_\Lambda$ and $r_+ \to r_\Lambda$, respectively, under the same GUP framework?
• RQ5 How does the GUP framework naturally account for both the minimum temperature in S-AdS and the non-monotonic temperature behavior in S-dS without additional assumptions?

Key findings

• The GUP expression $\Delta X\Delta P \geq \frac{\hbar}{2} + \frac{l_{\text{pl}}^2}{2\hbar}(\Delta P)^2 + \frac{\hbar}{2r_\Lambda^2}(\Delta X)^2$ successfully reproduces the thermodynamics of both S-dS and S-AdS black holes without requiring analytic continuation of the IR parameter.
• The S-dS black hole temperature, when measured by the 'natural' static observer at $l_0 = (\frac{3}{2}r_s r_\Lambda^2)^{1/3}$, yields a corrected surface gravity $\kappa_{\text{dS}} \approx \frac{1}{4GM} + \frac{9^{1/3}}{8}(r_{\Lambda}^2 GM)^{-1/3}$, which accounts for the normalization of the Killing vector and ensures consistent negative heat capacity.
• For the S-AdS case, the same GUP expression reproduces the critical mass $M_{\text{crit}} = \frac{2}{3}\frac{m_{\text{pl}}^2}{m_\Lambda}$ at which the heat capacity changes sign, with the temperature given by $\kappa_{\text{AdS}} \approx \frac{1}{4GM} + \frac{GM}{r_\Lambda^2}$ in the $M \ll M_{\text{crit}}$ regime.
• Near the cosmological horizon ($r_{BH} \to r_\Lambda$), the S-dS temperature $\kappa_{\text{dS}} \approx \frac{1}{r_\Lambda}(1 + \frac{2}{3}\epsilon)$ matches the S-AdS temperature $\kappa_{\text{AdS}} \approx \frac{1}{r_\Lambda}(\frac{1}{2C} + \frac{C}{2})$ when $C = 1 + \frac{2}{3}\epsilon - \frac{2}{\sqrt{3}}\sqrt{\epsilon}$, showing agreement in the extremal limit $\epsilon \to 0$.
• The four solutions of the GUP equation correctly describe both the negative heat capacity region (with minimum and maximum temperatures) and the positive heat capacity region for S-AdS, confirming consistency with known thermodynamic behavior.
• The geometric mean scale $l_0 = (l_{\text{pl}} r_\Lambda)^{1/2}$ corresponds to the extremal condition of the GUP, where the uncertainty product reaches its minimum, and serves as a unifying UV-IR duality scale.