[Paper Review] Deformations of JT Gravity and Phase Transitions

Edward Witten|arXiv (Cornell University)|Jun 5, 2020

Black Holes and Theoretical Physics27 references38 citations

TL;DR

The paper analyzes classical black hole solutions in generalized 2D dilaton gravity with a dilaton potential W(phi), showing how energy, entropy, and temperature relate, and revealing possible first-order phase transitions analogous to Hawking-Page transitions.

ABSTRACT

We re-examine the black hole solutions in classical theories of dilaton gravity in two dimensions. We consider an arbitrary dilaton potential such that there are black hole solutions asymptotic at infinity to the nearly $\mathrm{AdS}_2$ solutions of JT gravity, and such that the black hole energy and entropy are bounded below. We show that if there is a black hole solution with negative specific heat at some temperature $T$, then at the same temperature there is a black hole solution with lower free energy and positive specific heat. As the temperature is increased from 0 to infinity, the black hole energy and entropy increase monotonically but not necessarily continuously; there can be first order phase transitions, similar to the Hawking-Page transition. These theories can also have solutions corresponding to closed universes.

Motivation & Objective

• Motivate and study a class of two-dimensional dilaton gravity models that deform JT gravity while preserving asymptotic AdS2 behavior.
• Derive the classical black hole solutions and their thermodynamics within the generalized action I_b = -1/2 ∫ d^2x sqrt(g) (phi R + W(phi)).
• Investigate thermodynamic stability and show how negative specific heat black holes imply the existence of a lower free energy, positive specific heat counterpart at the same temperature.
• Explore phase structure as temperature varies, including possible first-order transitions and Hawking-Page-like behavior, as well as ground-state changes at zero temperature.

Proposed method

• Utilize the general dilaton gravity action with W(phi) and derive black hole solutions in Euclidean signature.
• Impose phi=horizon and derive A(r) via A'(r)=W(r) with phi=r, leading to A(r)=∫ W(r) dr.
• Compute thermodynamic quantities by evaluating the bulk and GHY surface action to obtain E=b/2 and S=2π phi_h + S0.
• Analyze stability by examining dE/dT and the sign of W'(phi_h) to determine canonical ensemble stability.
• Study phase transitions by comparing free energies of multiple black hole branches at fixed temperature and by identifying conditions under which transitions minimize F.

Experimental results

Research questions

• RQ1 Under what conditions does a black hole solution exist for a given horizon value phi_h?
• RQ2 How does the presence of negative specific heat at a given temperature relate to the existence of a lower free energy, positive specific heat solution at the same temperature?
• RQ3 What is the nature of phase transitions as temperature increases, and when do first-order (Hawking-Page-like) transitions occur?
• RQ4 How do generalized potentials W(phi) influence the ground state and possible closed-universe solutions?

Key findings

• For a black hole with phi_h where W(phi_h) > 0, the temperature is T = W(phi_h)/(4π).
• The energy difference between two horizons satisfies ΔE = (1/2) ∫_{phi1}^{phi2} dφ W(φ).
• If a black hole has negative specific heat at some T, there exists another black hole at the same T with lower free energy and positive specific heat.
• As temperature increases, energy and entropy increase monotonically (though not necessarily continuously), allowing first-order phase transitions akin to Hawking-Page.
• Ground states can change discontinuously at zero temperature when varying W(φ).
• The framework accommodates both black hole solutions and closed-universe (de Sitter-like) Euclidean solutions in this class of models.