[Paper Review] Thermodynamics of Kerr Newman de Sitter Black Hole and dS/CFT Correspondence

M. Dehghani, Hadi Khajehazad|arXiv (Cornell University)|Sep 24, 2002

Astrophysical Phenomena and Observations7 citations

TL;DR

This paper investigates the quasilocal thermodynamics of four-dimensional Kerr-Newman-de Sitter black holes using intrinsic boundary counterterms in both canonical and grand-canonical ensembles. It derives temperature, conserved quantities, and stability conditions via Hessian determinant analysis, and analytically confirms that temperature equals surface gravity times Tolman redshift factor in the non-rotating limit.

ABSTRACT

We consider the quasilocal thermodynamics of asymptotic de Sitter rotating charged black holes in both the canonical and the grand-canonical ensembles. Using the minimal number of intrinsic boundary counterterms, we carry out an analysis of the quasilocal thermodynamics of four-dimensional Kerr-Newman-de Sitter black hole for virtually all possible values of the mass, rotation and charge parameters that leave the quasilocal boundary inside the cosmological event horizon. Specifically, we compute the temperature and the conserved quantities of the system. We also perform a quasilocal stability analysis by computing the determinant of Hessian matrix of the energy as a function of its thermodynamic variables in both the canonical and the grand-canonical ensembles and obtain a complete set of phase diagrams. Finally, we investigate the non-rotating case analytically, and find that the temperature is equal to the product of the surface gravity (divided by $2\\pi$) and the Tolman redshift factor.

Motivation & Objective

• To analyze the quasilocal thermodynamics of asymptotically de Sitter rotating and charged black holes within a well-defined boundary framework.
• To compute temperature and conserved quantities (mass, angular momentum, charge) for the Kerr-Newman-de Sitter black hole across physically allowed parameter regimes.
• To perform a quasilocal stability analysis using the Hessian determinant of the energy in both canonical and grand-canonical ensembles.
• To investigate the non-rotating limit analytically and verify consistency with known thermodynamic relations.
• To contribute to the dS/CFT correspondence by providing a thermodynamically consistent description of de Sitter black holes with intrinsic boundary counterterms.

Proposed method

• Applies minimal intrinsic boundary counterterms to regularize the gravitational action and extract finite conserved charges on a boundary inside the cosmological horizon.
• Computes the temperature via the surface gravity and Tolman redshift factor, particularly verifying this relation analytically in the non-rotating case.
• Derives the energy as a function of thermodynamic variables (mass, angular momentum, charge) and evaluates the Hessian determinant to assess thermodynamic stability.
• Performs analysis in both canonical (fixed angular momentum and charge) and grand-canonical (fixed chemical potentials) ensembles to obtain complete phase diagrams.
• Uses the first law of thermodynamics in the quasilocal formalism to relate variations in energy to changes in extensive variables.
• Validates results by comparing the non-rotating limit with known expressions involving surface gravity and Tolman redshift factor.

Experimental results

Research questions

• RQ1 How do intrinsic boundary counterterms enable a consistent computation of thermodynamic quantities for Kerr-Newman-de Sitter black holes?
• RQ2 What is the behavior of temperature and conserved charges across the full range of allowed mass, rotation, and charge parameters within the cosmological horizon?
• RQ3 How does the Hessian determinant of the energy signal thermodynamic stability in both canonical and grand-canonical ensembles?
• RQ4 Does the temperature in the non-rotating limit satisfy the relation T = (κ / 2π) × z, where z is the Tolman redshift factor?
• RQ5 How do the resulting phase diagrams reflect the thermodynamic structure of the system in de Sitter spacetime?

Key findings

• The temperature of the Kerr-Newman-de Sitter black hole is consistently computed using surface gravity and Tolman redshift factor, with full consistency confirmed in the non-rotating limit.
• The Hessian determinant of the energy provides a complete stability analysis, enabling the construction of phase diagrams in both canonical and grand-canonical ensembles.
• Conserved quantities such as mass, angular momentum, and charge are derived using minimal boundary counterterms, ensuring finiteness and physical consistency.
• The non-rotating case yields an exact analytical verification that temperature equals (κ / 2π) times the Tolman redshift factor.
• The analysis is valid for all physically allowed values of mass, rotation, and charge parameters that keep the quasilocal boundary inside the cosmological horizon.
• The framework supports the dS/CFT correspondence by providing a well-defined, finite thermodynamic description of de Sitter black holes with intrinsic boundary terms.