[Paper Review] Microcanonical Action and the Entropy of Hawking Radiation

Juan F. Pedraza, Andrew Svesko+2 more|arXiv (Cornell University)|Nov 12, 2021

Black Holes and Theoretical Physics71 references4 citations

TL;DR

This paper derives the island formula for Hawking radiation entropy in Jackiw-Teitelboim (JT) gravity using the microcanonical ensemble, showing that the generalized entropy is minimized via the on-shell microcanonical action. The key result is that the microcanonical action equals minus the Wald entropy, and islands emerge from maximizing entropy at fixed energy, reproducing the unitary Page curve without the replica trick.

ABSTRACT

The island formula -- an extremization prescription for generalized entropy -- is known to result in a unitary Page curve for the entropy of Hawking radiation. This semi-classical entropy formula has been derived for Jackiw-Teitelboim (JT) gravity coupled to conformal matter using the "replica trick" to evaluate the Euclidean path integral. Alternatively, for eternal Anti-de Sitter black holes, we derive the extremization of generalized entropy from minimizing the microcanonical action of an entanglement wedge. The on-shell action is minus the entropy and arises in the saddle-point approximation of the (nonreplicated) microcanonical path integral. When the black hole is coupled to a bath, islands emerge from maximizing the entropy at fixed energy, consistent with the island formula. Our method applies to JT gravity as well as other two-dimensional dilaton gravity theories.

Motivation & Objective

• To derive the island formula for Hawking radiation entropy in JT gravity using the microcanonical ensemble instead of the replica trick.
• To show that the on-shell microcanonical action is equal to minus the Wald entropy, linking action minimization to entropy extremization.
• To demonstrate how islands emerge in eternal AdS2 black holes coupled to a bath through maximization of entropy at fixed energy.
• To generalize the derivation to two-dimensional dilaton gravity theories beyond JT gravity.

Proposed method

• Uses the microcanonical path integral to compute the gravitational partition function without replication, avoiding the replica trick.
• Applies the saddle-point approximation to the nonreplicated microcanonical path integral, where the on-shell action determines the entropy.
• Derives the on-shell microcanonical action as minus the Wald entropy via Noether charge and symplectic current formalism.
• Identifies the energy $ E_0 $ as the variation of the on-shell action, linking it to the Hamiltonian generating time evolution.
• Shows equivalence between the microcanonical action and the Gibbons-Hawking-York (GHY) boundary term in the limit approaching the bifurcation surface.
• Uses Cartan's magic formula and symplectic geometry to relate variations of the action to conserved charges and energy.

Experimental results

Research questions

• RQ1 Can the island formula for Hawking radiation entropy be derived without the replica trick in JT gravity?
• RQ2 How does the microcanonical action relate to the Wald entropy in two-dimensional dilaton gravity?
• RQ3 What is the role of the on-shell microcanonical action in determining the entropy of entanglement wedges?
• RQ4 How do islands emerge in eternal AdS2 black holes coupled to a bath via entropy maximization at fixed energy?
• RQ5 Is the microcanonical action formulation equivalent to the standard GHY boundary term in the limit of approaching the bifurcation surface?

Key findings

• The on-shell microcanonical action is equal to minus the Wald entropy, establishing $ I^{ ext{mc}}_{ ext{E}} = -S_{ ext{W}} $.
• The microcanonical action is minimized at fixed energy $ E_0 $, which corresponds to maximizing entropy, consistent with the island formula.
• Islands emerge naturally from maximizing entropy at fixed energy in eternal AdS2 black holes coupled to a bath.
• The derivation avoids the replica trick and instead uses the standard thermal partition function in the microcanonical ensemble.
• The equivalence between the microcanonical action and the GHY boundary term is proven in the limit approaching the bifurcation surface, confirming consistency with known results.
• The method applies to general two-dimensional dilaton gravity theories, not just JT gravity.