[Paper Review] Cardy-Verlinde Formula and asymptotically flat rotating Charged black holes

Jiliang Jing|ArXiv.org|Feb 8, 2002

Black Holes and Theoretical Physics10 references3 citations

TL;DR

This paper generalizes the Cardy-Verlinde formula to asymptotically flat rotating charged black holes in Einstein-Maxwell theory and low-energy string effective field theories, including Kerr-Newman, EMDA, Kaluza-Klein, and Sen black holes. It shows that while the Casimir energy definition is similar to AdS/dS counterparts for Kerr-Newman, the formula takes a distinct form; for EMDA, Kaluza-Klein, and Sen black holes, the formula reduces to the Kerr black hole form with modified coefficients, and entropy bounds are tighter than Bekenstein's for Kaluza-Klein and Sen cases.

ABSTRACT

The Cardy-Verlinde formula is generalized to the asymptotically flat rotating charged black holes in the Einstein-Maxwell theory and low-energy effective field theory describing string by using some typical spacetimes, such as the Kerr-Newman, Einstein-Maxwell-dilaton-axion, Kaluza-Klein, and Sen black holes. For the Kerr-Newman black hole, the definition of the Casimir energy takes the same form as that of the Kerr-Newman-AdS$_4$ and Kerr-Newman-dS$_4$ black holes, while the Cardy-Verlinde formula possesses different from since the Casimir energy does not appear in the extensive energy. The Einstein-Maxwell-dilaton-axion, Kaluza-Klein, and Sen black holes have special property: The definition of the Casimir energy for these black holes is similar to that of the Kerr-Newman black hole, but the Cardy-Verlinde formula takes the same form as that of the Kerr black hole. Furthermore, we also study the entropy bounds for the systems in which the matters surrounds these black holes. We find that the bound for the case of the Kerr-Newman black hole is related to its charge, and the bound for the cases of the EMDA, Kaluza-Klein, and Sen black holes can be expressed as a unified form. A surprising result is that the entropy bounds for the Kaluza-Klein and Sen black holes are tighter than the Bekenstein one.

Motivation & Objective

• To extend the Cardy-Verlinde formula to asymptotically flat rotating charged black holes, which remain unexplored despite its success in AdS/dS spacetimes.
• To investigate whether the entropy of conformal field theories (CFTs) on the boundary of these black holes matches their Bekenstein-Hawking entropy.
• To derive and analyze entropy bounds for systems where matter surrounds these black holes, especially comparing them to the Bekenstein bound.
• To determine whether the Cardy-Verlinde formula maintains its form or requires modification for non-AdS black holes with charge and rotation.
• To explore the role of Casimir energy and extensive energy in the generalized formula across different black hole types.

Proposed method

• Uses the Cardy-Verlinde formula in the form $ S = \frac{2\pi r_{+}}{n} \sqrt{E_c \cdot 2(E - E_Q)} $ for Kerr-Newman black holes, with $ E_Q $ as the electric potential energy.
• Applies the unified Cardy-Verlinde form $ S = \frac{2\pi r_{+}}{\sqrt{a_1b_1}} \sqrt{E_c \cdot 2E} $ for EMDA, Kaluza-Klein, and Sen black holes, with $ \sqrt{a_1b_1} $ derived from geometric and thermodynamic parameters.
• Calculates the Casimir energy $ E_c $ via thermodynamic and geometric relations, using the horizon radius $ r_+ $, mass $ M $, angular momentum $ a $, and charge $ Q $.
• Employs the covariant phase method and Carlip’s boundary conditions to construct the Virasoro algebra and central charge at the horizon, linking CFT entropy to Bekenstein-Hawking entropy.
• Derives entropy bounds by defining $ S_B = 2\pi R E \frac{n}{\sqrt{a_1b_1}} $, comparing $ \frac{n}{\sqrt{a_1b_1}} $ to 1 to assess tightness relative to Bekenstein’s bound.
• Analyzes specific black hole solutions: Kerr-Newman, EMDA, Kaluza-Klein, and Sen, using their known metric and thermodynamic properties.

Experimental results

Research questions

• RQ1 Does the Cardy-Verlinde formula hold for asymptotically flat rotating charged black holes such as Kerr-Newman and EMDA?
• RQ2 How does the definition of Casimir energy and extensive energy differ in flat spacetime compared to AdS/dS black holes?
• RQ3 Can the entropy bound for matter surrounding these black holes be tighter than the Bekenstein bound, and under what conditions?
• RQ4 Why do EMDA, Kaluza-Klein, and Sen black holes exhibit the same Cardy-Verlinde form as the Kerr black hole despite different charges and structures?
• RQ5 Is the agreement between CFT entropy and Bekenstein-Hawking entropy preserved in flat spacetime black holes via the Virasoro algebra construction?

Key findings

• For the Kerr-Newman black hole, the Casimir energy is defined similarly to Kerr-Newman-AdS/dS cases, but the extensive energy excludes the electric potential energy, leading to a modified Cardy-Verlinde formula: $ S = \frac{2\pi r_{+}}{n} \sqrt{E_c \cdot 2(E - E_Q)} $.
• For EMDA, Kaluza-Klein, and Sen black holes, the Casimir energy definition is similar to Kerr-Newman, but the Cardy-Verlinde formula reduces to the form $ S = \frac{2\pi r_{+}}{\sqrt{a_1b_1}} \sqrt{E_c \cdot 2E} $, identical to the Kerr black hole formula.
• The coefficient $ \sqrt{a_1b_1} $ is $ n $ for EMDA, $ \frac{n(1 - v^2/2)}{\sqrt{1 - v^2}} $ for Kaluza-Klein, and $ \frac{n(1 + \cosh\alpha\cosh\beta)}{\cosh\alpha + \cosh\beta} $ for Sen black holes.
• The entropy of the CFT on the horizon precisely matches the Bekenstein-Hawking entropy for all four black hole types, confirming consistency via CFT methods.
• The entropy bound for the Kaluza-Klein and Sen black holes is tighter than Bekenstein’s bound because $ \frac{n}{\sqrt{a_1b_1}} \leq 1 $, with equality only in special limits.
• For the Kerr-Newman black hole, the entropy bound $ S \leq 2\pi R(E - \frac{Q^2}{2r_+}) $ is tightened by the electric charge and reduces to Bekenstein’s bound when $ Q = 0 $.