Hawking-Page phase transitions of black holes in the Hamiltonian formalism

This paper applies the Hamiltonian formalism to analyze Hawking-Page phase transitions in BTZ, Reissner-Nordström, and Kerr-Newman black holes, demonstrating that the system's Hamiltonian corresponds to its thermodynamic free energy and revealing how electric charge and rotation enable the coexistence of black hole and thermal soliton states in off-shell configurations.

The Gist

Imagine you are a cosmic chef trying to understand the behavior of black holes. In the universe, black holes aren't just static, dark pits; they are hot, energetic objects that can "boil," "freeze," and even change their state of matter, much like water turning into ice or steam.

This paper is about a specific recipe for cooking these black holes: The Hawking-Page Phase Transition. It's the moment a black hole decides to either exist as a giant, hot object or dissolve into a sea of thermal radiation (like steam).

Here is the story of the paper, broken down into simple concepts and everyday analogies.

1. The New Tool: The "Hamiltonian" Kitchen Scale

Traditionally, physicists calculate the energy of a black hole using a complex set of rules called "Action" (think of this as a long, tedious recipe book).

In this paper, the authors (Tran Ngoc Thien and Vo Quoc Phong) decide to use a different tool: The Hamiltonian Formalism.

• The Analogy: Imagine you want to know the weight of a cake. You could weigh every single ingredient separately and add them up (the "Action" method). Or, you could just put the whole cake on a scale (the "Hamiltonian" method).
• The Discovery: The authors found that for black holes, the "Hamiltonian" (the total energy of the system) is exactly the same as the Free Energy (the thermodynamic potential that decides if the black hole is stable).
• Why it matters: It's a shortcut! They proved you can get the same accurate results about black hole behavior without doing the heavy lifting of the traditional method.

2. The Three Test Subjects

To test their new "scale," they looked at three different types of black holes, like testing a new thermometer on water, oil, and mercury:

• The BTZ Black Hole: A simple, 3-dimensional black hole. It's the "training wheels" version.
• The Reissner-Nordstrom (RN) Black Hole: A 4-dimensional black hole that has an electric charge (like a static shock).
• The Kerr-Newman (KN) Black Hole: The most complex one. It has an electric charge AND it is spinning (like a top).

3. The Two Ways to Look: "On-Shell" vs. "Off-Shell"

This is the most crucial part of the paper. The authors looked at these black holes in two different "modes":

Mode A: On-Shell (The "Perfect" Reality)

• The Analogy: Imagine a perfectly smooth, round marble rolling down a hill. It follows the laws of physics exactly without any wobbles.
• What happens: In this perfect world, the transition from "Radiation" to "Black Hole" happens instantly.
• The Result: It's a First-Order Phase Transition. Think of it like water boiling at exactly 100°C. One second it's liquid, the next it's gas. There is no middle ground. The black hole appears suddenly once the temperature gets high enough.

Mode B: Off-Shell (The "Realistic" Reality with Fluctuations)

• The Analogy: Now imagine that marble is actually a wobbly, squishy gelatin cube. It jiggles, fluctuates, and isn't perfectly smooth. It represents a black hole that is allowed to have small, random fluctuations (like quantum jitter).
• What happens: When you allow these wobbles, the transition changes. It doesn't happen instantly anymore.
• The Result: It becomes a Second-Order Phase Transition. Think of it like a magnet losing its magnetism as it heats up. The change is gradual. The "Black Hole" state and the "Radiation" state can coexist for a while, slowly swapping places as the temperature changes.

4. The Twist: Charge and Spin Change the Rules

The paper found some fascinating differences when they added Electric Charge and Spin:

• The Electric Charge (RN Black Hole):

  • The Analogy: Imagine the black hole is wearing a heavy backpack (the charge). It needs a minimum amount of "muscle" (mass) just to stand up.
  • The Result: In the "wobbly" (off-shell) world, the charge creates a minimum mass threshold. The black hole and the radiation can coexist, but the black hole can't exist if it's too light. The transition is smooth and continuous.

• The Spin (Kerr-Newman Black Hole):

  • The Analogy: Now the black hole is not just wearing a backpack; it's also spinning like a figure skater.
  • The Result: The spin is so powerful that it cancels out the weird "wobbles" of the off-shell world.
  • The Surprise: Even in the "wobbly" off-shell mode, the spinning black hole behaves as if it were perfect. The distinction between the "Black Hole" and the "Radiation" blurs completely. They coexist throughout the entire process. The spin essentially smooths out the rough edges of the transition, making it a continuous, gentle shift rather than a sudden jump.

Summary: What Did They Learn?

1. The Shortcut Works: You can use the Hamiltonian (the "scale") to study black holes just as well as the traditional "recipe book" (Action), and it saves a lot of math time.
2. Reality is Wobbly: When you look at black holes realistically (allowing for quantum fluctuations/off-shell), the sudden "snap" of a phase transition turns into a smooth, gradual slide.
3. Spin is a Stabilizer: If a black hole is spinning, it makes the transition between states even smoother, allowing the black hole and the radiation to coexist peacefully for longer.

The Big Picture:
This paper helps us understand that the universe isn't just a series of sudden, sharp jumps. When we look closer (using the off-shell method), the changes are often gradual and fluid. Furthermore, the properties of the black hole itself (like how fast it spins) dictate how smoothly these cosmic changes happen. This brings us one step closer to understanding how gravity and quantum mechanics fit together.

Technical Summary

1. Problem Statement

The Hawking-Page (HP) phase transition is a critical phenomenon in black hole thermodynamics describing the transition between a stable black hole and thermal radiation (soliton state) in Anti-de Sitter (AdS) spacetime. Traditionally, this transition is studied using the Euclidean path integral approach (action principle), where the free energy is derived from the on-shell action.

The paper addresses the following gaps:

• Methodological Alternative: There is a need to verify thermodynamic quantities and phase transitions using the Hamiltonian formalism to ensure consistency with the action principle and to provide a canonical framework for the quantization of the gravitational field.
• Off-Shell Dynamics: Most existing studies focus on the on-shell case (where fields satisfy equations of motion). The behavior of the HP transition in the off-shell configuration (where fields do not strictly satisfy equations of motion, allowing for fluctuations and conical singularities) remains less explored, particularly for charged and rotating black holes.
• Complex Black Holes: The influence of electric charge (Reissner-Nordström) and rotation (Kerr-Newman) on the nature (order) of the phase transition in both on-shell and off-shell regimes requires further investigation.

2. Methodology

The authors employ the Hamiltonian formalism of general relativity to derive the thermodynamic free energy of black holes.

• Canonical Formulation:
  • The gravitational action is transformed via a Legendre transformation from the Lagrangian density ($L$) to the Hamiltonian density ($H$).
  • The conjugate momentum tensor $p_{\mu\nu}$ is defined relative to the metric tensor $g_{\mu\nu}$.
  • The total Hamiltonian $H$ is constructed as the sum of a bulk term ($H_M$) and a boundary term ($H_{\partial M}$), derived from the Einstein-Hilbert action plus higher-derivative corrections (for BTZ) or electromagnetic terms (for RN and KN).
• Identification of Free Energy:
  • The central hypothesis tested is that the Hamiltonian of the black hole system corresponds directly to its thermodynamic free energy ($F = H$).
• On-Shell vs. Off-Shell Analysis:
  • On-Shell: The metric satisfies the Einstein field equations. The transition is determined by equating the Hamiltonian of the black hole ($H_{bh}$) with that of the thermal soliton ($H_{soliton}$).
  • Off-Shell: The authors introduce conical singularities (deficit angles $2\pi(1-\alpha)$) at the horizon or origin to represent fluctuations. The Lagrangian is modified to include singular terms ($\delta_\Sigma$). The Hamiltonian is calculated by integrating over these singular surfaces, allowing the study of the transition as a continuous function of mass ($M$) rather than a discrete jump.
• Models Studied:
  1. BTZ Black Hole: 3D AdS black hole (with higher-derivative corrections).
  2. Reissner-Nordström (RN) Black Hole: 4D charged, non-rotating AdS black hole.
  3. Kerr-Newman (KN) Black Hole: 4D charged, rotating AdS black hole.

3. Key Contributions

1. Validation of Hamiltonian as Free Energy: The paper rigorously demonstrates that for BTZ, RN, and KN black holes in AdS space, the Hamiltonian derived via canonical formalism is identical to the free energy obtained via the Euclidean action method. This provides an independent consistency check without additional computational cost.
2. Off-Shell Extension: The authors successfully extend the HP phase transition analysis to the off-shell regime for charged and rotating black holes, a domain previously less explored in this specific formalism.
3. Characterization of Phase Transition Orders:
   • The study distinguishes between first-order (discontinuous, instantaneous jump) and second-order (continuous) phase transitions based on the on-shell vs. off-shell configurations.
   • It reveals how electric charge and rotation fundamentally alter the thermodynamic landscape, specifically regarding the existence of minimum mass thresholds and the coexistence of states.

4. Key Results

A. Banados-Teitelboim-Zanelli (BTZ) Black Hole

• On-Shell: The transition occurs instantaneously at a critical temperature $T_c = 1/(2\pi l)$. The free energy curves intersect at a single point, indicating a first-order phase transition.
• Off-Shell: By introducing a deficit angle, the free energy of the soliton and black hole states are connected continuously at $M=0$. The transition is continuous (second-order).

B. Reissner-Nordström (RN) Black Hole (Charged)

• On-Shell: The presence of electric charge $Q$ imposes a minimum mass threshold for the existence of an event horizon. The transition remains first-order, occurring instantaneously when $H_{bh} = H_{soliton}$.
• Off-Shell: The electric charge creates a lower bound for mass and an upper bound for the soliton state. Crucially, the free energy curves of the soliton and black hole intersect and overlap, allowing for the coexistence of both states. The transition is continuous (second-order), redistributing statistical weights gradually as temperature changes.

C. Kerr-Newman (KN) Black Hole (Charged & Rotating)

• On-Shell: Similar to RN, charge and rotation impose a mass threshold. The transition is first-order.
• Off-Shell: The rotation parameter $a$ plays a unique role. It cancels out arbitrary surface contributions in the Hamiltonian calculation.
  • The free energy of the soliton state becomes identical to that of the black hole state.
  • The rotation removes the upper energy bound of the soliton, allowing the coexistence of soliton and black hole states throughout the entire thermodynamic process.
  • The transition is continuous (second-order), with the dominant state shifting gradually based on temperature.

5. Significance and Outlook

• Theoretical Consistency: The work confirms that the Hamiltonian formalism is a robust alternative to the Euclidean action method for deriving black hole thermodynamics, reinforcing the link between canonical gravity and thermodynamics.
• Quantum Gravity Insights: By identifying the Hamiltonian with free energy and analyzing off-shell fluctuations (conical singularities), the paper suggests a pathway toward the quantization of the gravitational field. The continuous nature of off-shell transitions implies a smoother statistical redistribution between vacuum (soliton) and black hole states, which is relevant for understanding black hole evaporation and information paradoxes.
• Role of Charge and Rotation: The study highlights that while charge introduces mass thresholds, rotation (in the off-shell case) can fundamentally alter the topology of the phase space, enabling the simultaneous coexistence of distinct thermodynamic phases.

In conclusion, the paper establishes that while Hawking-Page transitions appear as first-order jumps in the classical (on-shell) limit, they manifest as continuous, second-order transitions in the off-shell regime, with charge and rotation playing pivotal roles in defining the stability and coexistence of black hole and soliton states.