Black holes at a finite distance: Quasi-local restricted phase space formalism

This paper extends the restricted phase space formalism to quasi-local regimes with static observers at finite distances, demonstrating that RN black holes in this setting exhibit thermodynamic behaviors and phase transitions strikingly similar to asymptotic RN-AdS black holes, including Hawking-Page-like transitions in the neutral limit, provided an extra pair of thermodynamic variables (pressure and boundary area) is included.

The Gist

Imagine you are trying to understand the weather inside a storm. If you stand on the ground far away, you see the storm as a whole, a giant swirling mass. But if you climb a ladder and stand just a few feet away from the eye of the storm, the wind feels different, the pressure changes, and the rules of how the storm behaves might look completely different.

This paper is about doing exactly that with black holes.

The Big Idea: Changing Your Viewpoint

For decades, physicists have studied black holes as if they were observing them from "infinity"—a theoretical spot so far away that the black hole's gravity has no effect on the observer's measurements. This is like looking at a storm from a satellite.

The authors, Bai-Hao Huang and Liu Zhao, decided to ask: What if we move the observer closer? What if we place a thermometer and a pressure gauge at a specific, finite distance from the black hole, like standing on a ladder near the storm?

They took a specific mathematical toolkit called the Restricted Phase Space (RPS) formalism—which is like a very strict, perfect rulebook for black hole thermodynamics that works well for distant observers—and adapted it for these "close-up" observers. They call this the Quasi-Local approach.

The New Rules of the Game

When you move the observer closer, the physics changes in a surprising way. The authors found that to make the math work correctly (specifically to keep the "Euler relation," a fundamental rule of thermodynamics, valid), they had to add two new variables to the black hole's recipe:

1. Surface Pressure ($P$): Just like air pressure pushes on a balloon, the space around the black hole exerts a pressure on the observer's location.
2. Surface Area ($A$): The size of the "window" or sphere where the observer is standing.

In the old "distant" view, these two didn't matter. In the "close-up" view, they are essential ingredients, just like temperature and entropy.

The Surprise: A Black Hole That Behaves Like a Different Black Hole

The most exciting discovery in the paper is how the black hole behaves when viewed up close.

• The Old View (Distant): If you look at a standard, electrically charged black hole (called an RN black hole) from far away, it is boring. It's thermodynamically unstable and doesn't undergo any "phase transitions." It's like a cup of coffee that just sits there cooling down; it never suddenly boils or freezes.
• The New View (Close-Up): When the authors moved the observer closer, that same boring black hole suddenly became exciting. It started behaving exactly like a charged black hole in an Anti-de Sitter (AdS) universe (a universe with a negative cosmological constant, which is a very different theoretical setting).

The Analogy:
Imagine a calm lake (the distant view). Nothing happens; it's just water. But if you dive just a few feet below the surface (the quasi-local view), you discover currents, whirlpools, and turbulence that you couldn't see from the boat. The lake hasn't changed, but your experience of it has completely transformed.

What Happens in This New View?

The authors analyzed these "close-up" black holes and found:

1. Phase Transitions: Just like water turning into ice or steam, these black holes can undergo sudden changes in state. They found that if you keep the electric charge constant, the black hole can jump between different temperature states. This is something that never happens when you look at them from far away.
2. The "Swallowtail" Shape: When they graphed the energy of the black hole, the curve looked like a swallow's tail. In physics, this specific shape is a signature that a first-order phase transition is happening (a sudden, dramatic change, like water boiling).
3. The Neutral Limit (Schwarzschild Black Holes): Even for a black hole with no electric charge (just pure gravity), the close-up view revealed something special. In the distant view, these black holes are thermodynamically unstable and don't undergo phase transitions. But up close, they show signs of Hawking-Page transitions. This is a famous phenomenon where a black hole can spontaneously turn into a cloud of hot radiation (thermal gas) and vice versa. The authors showed this happens for any black hole if you are close enough, not just the special ones in AdS space.

Why Does This Matter?

The paper concludes with a profound realization: Different observers see different realities.

In physics, we often assume that a black hole has one set of properties. This paper proves that its thermodynamic personality—whether it's stable, whether it changes phase, whether it has pressure—depends entirely on where you are standing.

• From far away: The black hole is a simple, thermodynamically unstable object without phase transitions.
• From close up: The black hole is a complex, dynamic system with pressure, area, and the ability to undergo dramatic phase changes.

The authors didn't propose any new technology or medical application. Instead, they refined our theoretical understanding, showing that the "rules" of black hole thermodynamics are not absolute; they are relative to the observer's distance, much like how the weather feels different depending on whether you are on a mountain peak or in a valley.

Technical Summary

Technical Summary: Quasi-local Restricted Phase Space Formalism for Black Holes

Problem Statement
Black hole thermodynamics (BHT) has traditionally been formulated with observers implicitly placed at asymptotic infinity, utilizing ADM energy and Hawking temperature. While the Restricted Phase Space (RPS) formalism successfully restores Euler homogeneity and standard thermodynamic definitions (such as subsystems and stability criteria) for asymptotic observers, it does not account for observers located at a finite radial distance. In realistic observational scenarios, such as potential verification within the solar system, observers are finite-distance. Existing quasi-local descriptions (e.g., Brown-York) successfully define local energy and pressure but fail to satisfy Euler homogeneity, a crucial feature for extensive thermodynamics. This paper addresses the gap by extending the RPS formalism to the quasi-local regime to determine if a consistent, Euler-homogeneous thermodynamic framework exists for finite-distance observers and how this alters the thermodynamic behavior of black holes.

Methodology
The authors extend the RPS formalism to spherically symmetric solutions in Einstein-Maxwell theory (Reissner-Nordström [RN] and RN-AdS black holes) with static observers located at a finite radius $R$.

1. Action and Boundary Terms: The on-shell Euclidean action is defined over a finite spacetime region $M$ bounded by a hypersurface $\partial M$ at $r=R$. A background counter-term (pure AdS or Minkowski) is subtracted to ensure finite results.
2. Quasi-local Variables: Using the Brown-York formalism, the authors extract the quasi-local energy density $\varepsilon$ and surface stress tensor $s_{ab}$ from the boundary stress tensor. These yield the quasi-local energy $E$ (interpreted as internal energy) and surface pressure $P$.
3. Thermodynamic Conjugates: The formalism introduces a new conjugate pair $(\hat{P}, \hat{A})$, where $\hat{P} = PG$ and $\hat{A} = A/G = 4\pi R^2/G$. The temperature is the Tolman temperature $T_R = T_H / \sqrt{f(R)}$, while the entropy $S$ remains the Bekenstein-Hawking entropy. A chemical potential $\mu_R$ is defined via the Euclidean action and the rescaled pressure-area term.
4. Verification: The authors explicitly calculate these variables for RN and RN-AdS black holes to verify the validity of the First Law ($dE = T_R dS - \hat{P} d\hat{A} + \hat{\Phi}_R d\hat{Q} + \mu_R dN$) and the Euler relation ($E = T_R S - \hat{P} \hat{A} + \hat{\Phi}_R \hat{Q} + \mu_R N$).
5. Stability and Phase Transitions: Stability is analyzed via the Hessian of the internal energy. The paper investigates phase transitions by examining the monotonicity of the $T_R-S$ relation under isocharge and isovoltage conditions, comparing the finite-distance results against the asymptotic limit ($R \to \infty$).

Key Contributions

• Formal Extension: The paper successfully constructs a "quasi-local RPS formalism" that maintains full Euler homogeneity for black hole thermodynamics when observers are at a finite distance.
• Thermodynamic Consistency: It proves that the First Law and Euler relation hold for RN and RN-AdS black holes in this regime, provided the extra conjugate pair $(\hat{P}, \hat{A})$ is included.
• Qualitative Shift in RN Behavior: The study reveals that while RN-AdS black holes show only quantitative shifts in the quasi-local regime, pure RN black holes undergo a qualitative change. In the asymptotic description, RN black holes are thermodynamically unstable with no phase transitions. In the quasi-local description, they exhibit stable phases and first-order isocharge temperature-entropy ($T_R-S$) phase transitions, behaving similarly to RN-AdS black holes in the asymptotic regime.
• Hawking-Page-like Transitions: In the neutral (Schwarzschild) limit, the quasi-local description predicts Hawking-Page-like transitions (transitions between black holes and thermal gases) for finite $R$, a phenomenon absent in the asymptotic description of asymptotically flat Schwarzschild black holes.

Results

• Isocharge Transitions: For RN black holes with fixed charge and finite $R$, the $T_R-S$ relation becomes non-monotonic, allowing for phase transitions. The critical parameters ($S_c, \hat{Q}_c, T_c$) are derived, and the Helmholtz free energy exhibits a "swallow tail" structure, indicating first-order transitions that become second-order at the critical point.
• Isovoltage Stability: In contrast to the isocharge case, isovoltage processes (fixed electric potential $\hat{\Phi}_R$) do not exhibit phase transitions; the system possesses a single stable phase for $S > S_{min}$.
• Schwarzschild Limit: For neutral black holes, the Euclidean action $I_E$ acquires zeros at finite $R$ (specifically at $r_h = 8R/9$), signaling a phase transition between the black hole state and a thermal gas state, analogous to the Hawking-Page transition in AdS space.
• Asymptotic Recovery: As $R \to \infty$, the surface pressure $\hat{P} \to 0$, the conjugate pair decouples, and the formalism naturally reduces to the standard asymptotic RPS formalism, recovering the known instability and lack of phase transitions for asymptotically flat RN black holes.

Significance
The paper claims that its primary significance lies in demonstrating that the thermodynamic behavior of black holes is observer-dependent in a fundamental way. It provides an explicit example where the same physical system (an RN black hole) exhibits stable thermodynamic phases and phase transitions when observed from a finite distance, whereas it appears unstable and transition-free when observed from infinity. This supports the broader consensus in relativistic physics that different observers perceive different phenomena for the same underlying process. Furthermore, the work establishes that the RPS formalism is robust enough to be extended beyond asymptotic infinity, offering a consistent framework for quasi-local thermodynamics with full Euler homogeneity. The authors note that this is an initial step, with future work needed to address non-static observers, non-spherically symmetric black holes, and other gravity models.