Holographic Thermodynamic Pressure

• The paper outlines how bulk pressure emerges by promoting the cosmological constant to a thermodynamic variable, recasting black hole mass as enthalpy.
• It details the holographic duality where varying pressure corresponds to shifting parameters in the dual gauge theory, influencing the AdS radius and degrees of freedom.
• The study compares bulk and boundary definitions of pressure using quasi-local methods and analytic techniques to elucidate distinct phase behaviors in gravitational systems.

The holographic definition of thermodynamic pressure is a set of related but distinct prescriptions for how the notion of pressure and its thermodynamic conjugate volume are formulated, interpreted, and computed in gravitational systems with a holographic (gauge/gravity dual) description. In the context of gravitational thermodynamics—especially for black holes—pressure arises not as a primitive boundary observable but as an emergent variable, typically associated with the bulk cosmological constant via the extended thermodynamic first law. Its holographic meaning and operational status can differ drastically depending on the theoretical framework, the focus on bulk or boundary thermodynamics, and the physical ensemble under consideration.

1. Bulk Gravitational Definition: Cosmological Constant as Pressure

The foundational insight is that, in certain gravitational theories, the negative bulk cosmological constant $\Lambda$ can be promoted to a thermodynamic variable, identified with pressure via the relation

$p = -\frac{\Lambda}{8\pi G}$

where $G$ is Newton's constant, and units $c = \hbar = k_B = 1$ are typically adopted. In $D$ spacetime dimensions, writing $\Lambda = -\frac{(D-1)(D-2)}{2\ell^2}$ yields

$p = \frac{(D-1)(D-2)}{16\pi G \ell^2}$

This "bulk pressure" enters the black-hole first law, where the mass parameter $M$ is recast from internal energy to enthalpy, and its conjugate "thermodynamic volume" $V$ is defined by

$V = \left(\frac{\partial M}{\partial p}\right)_S$

This extended first law reads

$$dM = TdS + Vdp + \sum_i \Phi_i dq_i + \sum_j \Omega_j dJ_j$$

for temperature $T$, entropy $S$, electromagnetic potentials $\Phi_i$, electric charges $q_i$, angular velocities $\Omega_j$, and angular momenta $J_j$ (Johnson, 2014, Visser, 2021, Nie et al., 2015). In static AdS–Reissner–Nordström black holes, $V$ typically coincides (up to factors) with the geometric volume enclosed by the horizon.

2. Holographic Dual: Field-Theory Interpretation and RG Flows

Within the AdS/CFT or general gauge/gravity duality, the bulk cosmological constant sets the radius $\ell$ of the AdS space and thereby determines the dual field theory's number of degrees of freedom $N$ (e.g., via the central charge or rank of the gauge group). Varying $p$ in the bulk, therefore, corresponds not to a thermodynamic transformation within a single boundary theory but to navigating the space of theories parameterized by $N$. In this dictionary:

• $M \leftrightarrow U$, energy of the boundary theory (for fixed $\Lambda$).
• Allowing $\Lambda$ to vary: $M=H=U+pV$, with the bulk Euclidean action computing the Gibbs free energy $G = U - TS + pV$.
• The dual field theory does not possess $p$ as a thermodynamic variable; rather, $p$ encodes a "coupling-space" coordinate.
• Engine cycles in this picture are interpreted as sequences of renormalization group flows, corresponding to traversing closed orbits in $(T, S, p)$ space, with RG trajectories returning to equivalent endpoints in the field-theory space (Johnson, 2014, Visser, 2021).

The bulk $V$ is not simply the spatial volume of the boundary theory and is not directly observable as a thermodynamic variable on the boundary. Instead, it is conjectured to represent a "chemical-potential-like" quantity, possibly quantifying changes in the landscape of degrees of freedom across dualities.

3. Quasi-local and Boundary Constructions: Brown–York Prescription

Quasi-local thermodynamics, via the Brown–York formalism, offers a geometric route to defining pressure and energy for gravitational systems in bounded domains. The Brown–York stress tensor,

$$\tau_{ij} = \frac{1}{8\pi G} \left( K_{ij} - K \gamma_{ij} \right)$$

with $K_{ij}$ the extrinsic curvature of the boundary and $\gamma_{ij}$ its induced metric, underpins this construction (Gu et al., 2010, Borsboom et al., 4 Feb 2026). The boundary pressure $p$ is identified as the trace over spatial directions:

$p = e^\phi \frac{1}{2} \gamma^{ab} \tau_{ab}$

for lapse function $\phi$. In the static spherically symmetric case, for a boundary at radius $r_B$, the (background-subtracted) surface pressure is

$$P(r_B, r_h) = \frac{1}{8\pi G} \left[ \frac{d-3}{r_B h(r_B)} + \frac{N'(r_B)}{h(r_B) N(r_B)} \right]$$

This pressure is conjugate to variations in the (hyper-)area $A$ of the boundary, giving rise to a quasi-local first law of the form

$\delta E = T\delta S - P \delta A$

where $E$ is the Brown–York quasi-local energy (Savvidou et al., 2013, Borsboom et al., 4 Feb 2026, Gu et al., 2010).

Assuming the existence of a dual field theory on the boundary, these geometric quantities are mapped to the physical pressure and volume of the dual thermodynamic system. The "holographic" volume is then interpreted as the area of the boundary and provides a natural system-size variable, enabling extensivity and large-system limits.

4. Alternative Notions: Thermodynamic Holography and Analytic Continuation

A more formal but broadly applicable notion of "holographic thermodynamic pressure" leverages the analyticity of the partition function $Z(\beta, V)$ as a function of system volume $V$ (Wei et al., 2014). For systems whose Hamiltonian depends analytically on $V$, the partition function is analytic in a complex neighborhood, and the standard Cauchy formula yields a holographic representation for pressure:

$$p(V_0) = \frac{1}{\beta} \frac{1}{2\pi i} \oint_{\partial D} \frac{Z(\beta,V)}{Z(\beta,V_0)(V-V_0)^2} dV$$

Thus, the pressure at point $V_0$ in parameter space is determined by boundary data—i.e., the values of $Z$ on $\partial D$ in the complex $V$-plane. This is a direct thermodynamic analog of the holographic principle, providing a way to reconstruct volumetric derivatives from codimension-1 data (Wei et al., 2014).

5. Comparative Perspectives: Bulk versus Boundary Pressure

There exist operationally distinct but mathematically precise definitions of "pressure" in holographic settings, notably:

Definition	Pressure Variable	Conjugate Volume
Bulk/Extended (p$_\text{bulk}$)	$p = -\Lambda/(8\pi G)$	$V = (\partial M/\partial p)_S$
Boundary stress-tensor (p$_\text{bdry}$)	$p_{\rm fluid}$ from $T_{ab}$	Actual spatial boundary volume

In extended black-hole thermodynamics ("black-hole chemistry"), $p_\text{bulk}$ is associated with varying the cosmological constant and identifies the gravitational mass as enthalpy, leading to $p$–$V$ critical phenomena and van der Waals–like behavior (Nie et al., 2015). In contrast, the boundary stress-tensor pressure $p_\text{bdry}$ arises from the holographically renormalized energy-momentum tensor and encodes the physical fluid pressure of the boundary field theory (Nie et al., 2015).

Varying $p_\text{bulk}$ corresponds to changing the underlying dual CFT (modifying $N$ or central charge), not to a process within a fixed field theory. $p_\text{bdry}$, by contrast, is the true thermodynamic pressure within a single theory. The two definitions lead to sharply distinct thermodynamic phase diagrams and critical structure (Nie et al., 2015).

6. Extensions, Subtleties, and Open Problems

• The precise identification of the field-theory dual of the bulk thermodynamic volume $V$ remains unknown; it cannot be the literal boundary spatial volume, since that is fixed once $N$ is set (Johnson, 2014, Visser, 2021).
• In large-$N$ gauge theories, the Euler relation includes energy, entropy, and a chemical potential for the number of colors $N$, but not "volume" or "pressure" conjugate to bulk $p$ and $V_\text{th}$; instead, the bulk Smarr relation packages the boundary chemical potential and pressure terms into a single conjugate (Visser, 2021).
• In quasi-local constructions, all variables—including pressure—are encoded in the boundary geometry and extrinsic curvature, demonstrating classical gravitational holography independent of quantum considerations (Savvidou et al., 2013).
• For general gravitational systems, the extended first law may include not only $p$–$V$ terms but also variations in $G$, electric charge, angular momentum, and their duals, reflecting the complex structure of the phase space and the scaling symmetries of gravity (Visser, 2021).
• The correct physical interpretation of engine cycles in the context of holography is that they describe RG "circuits" in coupling space rather than conventional thermodynamic cycles in a single statistical system (Johnson, 2014).

7. Physical Implications and Examples

• In holographic heat-engine constructions, classic thermodynamic cycles (Carnot, Stirling, etc.) can be implemented in black-hole (AdS) spacetimes, with efficiency calculable in terms of $p$ and $V$ (Johnson, 2014).
• In boundary quantum field theory, pressure defined via the stress tensor governs the conventional response to volume changes, while the extended bulk pressure is invisible as an independent operator (Nie et al., 2015). The two notions may not coincide numerically and, under variation, explore different sectors of the state space.
• For self-gravitating equilibrium systems, the entire thermodynamics (including pressure) is encoded by the data on the boundary two-surface: its induced metric and extrinsic curvature. This classical result is a manifestation of the holographic structure of general relativity (Savvidou et al., 2013).
• In Gauss–Bonnet gravity, both "bulk" and "boundary" definitions coexist but furnish qualitatively distinct $P$–$T$ phase diagrams, emphasizing the necessity of specifying the thermodynamic ensemble and the intended physical meaning when discussing "pressure" in holographic contexts (Nie et al., 2015).

Overall, the holographic definition of thermodynamic pressure is both structurally robust in the bulk gravitational theory and conceptually nontrivial in its field-theory duals. It establishes a firm framework for extended gravitational thermodynamics, but its precise operational meaning on the boundary remains the subject of ongoing research and interpretation.