Quasi-Local Gravitational Thermodynamics

• Quasi-local gravitational thermodynamics is a framework that assigns thermodynamic variables like energy, temperature, and entropy to finite regions in spacetime.
• It utilizes Hamiltonian and Brown–York methods to define localized quantities and extends classical thermodynamic laws with generalized first law and Smarr relations.
• The approach applies across various gravitational theories, offering insights into phase transitions, holographic entanglement, and emergent dynamics in modified gravity.

Quasi-local gravitational thermodynamics is the framework that assigns thermodynamic variables—such as energy, temperature, entropy, pressure, and volume—not just globally (as for event horizons or spatial infinity), but to finite, bounded regions within a gravitating spacetime. Unlike global horizon mechanics, quasi-local approaches operate on finite boundaries (timelike, spacelike, or null), incorporate local geometric data, and are applicable beyond stationary black holes to dynamical, cosmological, and more general contexts. This framework unifies and extends classical black hole thermodynamics, incorporates generalized versions of the first law and Smarr relations, and underpins a range of modern approaches including holographic entanglement, the thermodynamic derivation of field equations, and novel notions of stability and phase structure in gravitational systems.

1. Hamiltonian and Brown–York Approaches to Quasi-local Thermodynamics

The Hamiltonian formulation and Brown–York construction provide the foundation for quasi-local thermodynamic quantities. For a D-dimensional spacetime, the Brown–York quasi-local energy is defined on a timelike hypersurface at $r=R$ using the extrinsic curvature $k$ and a reference value $k_0$: $$\varepsilon = -\frac{1}{8\pi}(k - k_0)\,, \qquad E(R) = \int_{S^{D-2}} \varepsilon\,d\Sigma = -\frac{B_{D-2}R^{D-2}}{8\pi}\left[\frac{D-2}{R}N(R) - \alpha_0(R)\right],$$ where $N(R)$ is the lapse, $B_{D-2}$ encodes the area of the $S^{D-2}$ sphere, and $\alpha_0(R)$ is chosen to ensure $E\to0$ as $M\to0$ (Fontana et al., 2018).

The local Tolman-redshifted temperature and quasi-local surface pressure at the boundary are

$$T_L(R) = \frac{\kappa}{2\pi N(R)},\qquad P(R) = -\frac{1}{(D-2)B_{D-2}R^{D-3}} \frac{\partial E}{\partial R}\Big|_{M,\Lambda},$$

with $\kappa$ the surface gravity at the horizon.

This framework is generalized by promoting the cosmological constant $\Lambda$ (or $A=|\Lambda|$) to a thermodynamic variable, yielding an extended phase space with conjugate pairs $(S, T_L)$, $(\mathcal V, -P)$, and $(A, \xi)$, subject to constraints ensuring homogeneity and consistent Legendre transforms. The corresponding quasilocal first law is

$dE(R) = T(R)\,dS - P(R)\,d\mathcal V + \xi\,dA,$

and the associated Smarr relation at finite $R$ becomes

$(D-2)E = (D-3)TS - 2P\mathcal V + (D-1)A\xi.$

At $R\to\infty$, one recovers standard global AdS black hole thermodynamics (Fontana et al., 2018).

2. Quasi-local First Laws, Equilibrium, and Thermodynamic Potentials

Quasi-local first laws extend the black hole thermodynamic identity to finite regions without requiring a Killing horizon. The local first law for arbitrary spacetimes in Einstein gravity, formulated on a stretched future light cone $\Sigma$ around a point $P$, is

$\Delta E = T\,\Delta S - W,$

where $\Delta E$ is the matter energy enclosed by $\Sigma$, $T$ is the Unruh temperature measured by accelerated observers, $\Delta S$ is the (reversible) change in gravitational entropy, and $W$ is the work done by pressure (Parikh et al., 2018). The entropy is given by the Bekenstein–Hawking formula $S = A / (4G\hbar)$ for Einstein gravity, and by the Wald entropy more generally.

This formalism recovers the standard global first law for stationary horizons and generalizes directly to higher-curvature theories, directly connecting local geometric quantities (area change, surface gravity) to the thermodynamic structure.

In a different geometric approach, minimizing the mean extrinsic curvature of the bounding surface ("screen") yields a variational principle for quasilocal equilibrium. The quasilocal Helmholtz free energy is identified as

$$F = \frac{1}{16\pi}\oint_\mathbb{S} \sqrt{k^2 - \ell^2}\, d\mathbb{S} ,$$

where $k, \ell$ are traces of extrinsic curvatures. Equilibrium corresponds to the screen sitting at a generalized apparent horizon ($\sqrt{k^2 - \ell^2}=0$), and the free energy minimum coincides with hydrodynamic equilibrium for spherically symmetric spacetimes (Uzun et al., 2015).

3. Extended Phase Space: Additional Thermodynamic Variables

Recent developments recognize the cosmological constant $\Lambda$, the gravitational constant $G$, and other couplings as dynamical variables within the quasilocal thermodynamic phase space. In the Hamiltonian and ADT (Abbott–Deser–Tekin) formalism, both $\Lambda$ and $G^{-1}$ are promoted to conserved charges with conjugate potentials (Kim et al., 10 Sep 2025): $$\delta M = T\,\delta S + \Phi\,\delta(G^{-1}) - V\,\delta P,$$ where $\Phi$ is the potential conjugate to $1/G$ and $V$ is the thermodynamic volume conjugate to $P = -\Lambda/8\pi G$. This formalism is fully consistent with extended first laws and Smarr relations derived via Euler scaling in arbitrary dimensions.

The inclusion of such charges enhances the flexibility and scope of quasi-local thermodynamics, allowing the exploration of novel phase behaviors, generalized susceptibilities, and sector splitting in gravitational decoupling scenarios (e.g., in Lovelock-Unique-Vacuum theory or MGD) (Estrada, 2021, Estrada et al., 2020). Here, the total energy, pressure, and temperature can be decomposed additively into standard and quasi sectors, both sharing the same entropy and volume but admitting distinct thermodynamic features and phase transitions.

4. Quasi-local Thermodynamics in Various Gravitational Theories

Quasi-local constructs extend beyond General Relativity to scalar–tensor theories and higher-curvature gravities. For scalar–tensor gravity in both Einstein and Jordan frames, the Brown–York tensor and associated quasi-local parameters (energy, entropy, temperature, pressure) have natural conformal relations, leaving the Gibbs first law invariant under frame transformations: $\delta E = T\,\delta S + p\,\delta A + \cdots,$ with $S_J = S_E$, $T_J = T_E/\sqrt{\phi}$, and $E_J = \sqrt{\phi} E_E$, maintaining the form of the first law for Killing and isolated horizons (Bhattacharya et al., 2023).

In Lovelock–Unique–Vacuum theories, the gravitational decoupling method generates new stationary black hole solutions whose quasi-local energy explicitly splits into seed and "quasi" sector contributions, and whose modified first law maintains internal consistency without the need for background subtraction or counterterms (Estrada, 2021). The thermodynamic variables (entropy, temperature, specific heat) are calculable at the quasi-local level and can exhibit novel horizon structures (such as multiple inner horizons) and stability behavior.

5. Thermodynamics of Causal Diamonds and Local Screens

The thermodynamics of causal diamonds—intersections of the future of one point and the past of another—generalizes the notion of quasi-local equilibrium beyond stationary black holes to maximally symmetric and dynamical backgrounds. Each diamond admits a unique conformal Killing flow, and the integrated Noether charge identities yield Smarr-type relations involving area, volume, and $\Lambda$: $(d-2)\kappa\,A = (d-1)k\,V + 2V_\zeta\Lambda,$ and a first law

$$\delta H_\zeta^{\rm matter} = \frac{1}{8\pi G}\left[ -\kappa\,\delta A + \kappa\,k\,\delta V - V_\zeta\,\delta\Lambda \right].$$

Assigning area-proportional entropy and negative temperature to the diamond establishes a thermodynamically self-consistent quasi-local mechanics, capable of unifying static patch de Sitter, Rindler, and AdS wedge thermodynamics (Jacobson et al., 2018).

Quantum corrections can be incorporated by including the matter entanglement entropy, giving rise to stationarity conditions for the generalized entropy $S_{\rm gen}$—the sum of area/4G and matter entropy—under fixed volume and $\Lambda$. This "entanglement equilibrium" directly yields the semiclassical Einstein equation in the small diamond limit.

6. Entanglement, Integrated Energy Conditions, and Holographic Extensions

Quasi-local gravitational thermodynamics interfaces with quantum information via the concept of entanglement density: the second variation of von Neumann entropy under infinitesimal deformations of the region. Strong subadditivity enforces positivity of this density, which in holographic duals equates to an integrated null energy condition on the bulk extremal surface: $$n_\pm = \frac{1}{8G_N}\int_{\Gamma_\Sigma} \sqrt{\gamma} N_\pm^\mu N_\pm^\nu E_{\mu\nu} \geq 0,$$ with $E_{\mu\nu}$ the Einstein tensor plus $\Lambda$ term (Bhattacharya et al., 2014). This structure provides a generalized "second law" for arbitrary extremal surfaces, rather than just horizons, and is foundational for modern approaches to emergent gravity and holography.

In 2D nearly de Sitter gravity (Jackiw–Teitelboim), the quasi-local action and microcanonical entropy are shown to be equivalent, providing a direct derivation of the island rule for fine-grained radiation entropy—thus embedding the statistical structure of quantum gravity in a quasi-local thermodynamic setup (Svesko et al., 2022).

7. Phase Structure, Stability, and Physical Implications

The quasilocal formalism reveals rich phase structures absent in purely global descriptions. For SAdS black holes, the heat capacity at fixed boundary diverges at specific values of $r_+$, indicating a local transition between small/large black hole phases analogous to the Hawking–Page transition, but now for a system inside a finite-radius cavity (Fontana et al., 2018). Additional compressibilities and susceptibilities can be defined due to the multiplicity of thermodynamic variables at the boundary.

In gravitational decoupling scenarios, the sum of quasi and standard sector temperature contributions can drive second-order phase transitions, with quasi-sector instabilities manifest even when the standard sector remains regular (Estrada et al., 2020). In Lovelock–Unique–Vacuum black holes with decoupling sources, novel inner horizon structures and guaranteed positive specific heat differentiate the quasilocal predictions from canonical approaches (Estrada, 2021).

Quasi-local approaches also clarify equilibrium conditions: for spherically symmetric spacetimes, minimization of the mean extrinsic curvature (i.e., of the quasilocal free energy) precisely locates the screen at an apparent horizon, and this stationarity coincides with hydrodynamic equilibrium (Uzun et al., 2015). A corresponding quasi-local virial relation further links the proper mass-energy content to binding energy in a general-relativistic setting.

---

Key References:

• (Fontana et al., 2018): Hamiltonian quasilocal thermodynamics for SAdS black holes
• (Parikh et al., 2018): Local first law for arbitrary causal regions
• (Kim et al., 10 Sep 2025): $G$ as a quasi-local conserved thermodynamic charge
• (Jacobson et al., 2018): Smarr/first laws and negative temperature for causal diamonds
• (Estrada, 2021, Estrada et al., 2020): Gravitational decoupling and quasilocal sector splitting
• (Uzun et al., 2015): Thermodynamic equilibrium via extrinsic geometry
• (Bhattacharya et al., 2014): Entanglement density and local second laws
• (Bhattacharya et al., 2023): Brown–York thermodynamics in scalar–tensor theory
• (Svesko et al., 2022): Islands and microcanonical entropy in 2D gravity

Quasi-local gravitational thermodynamics thus provides a nonperturbative, coordinate-invariant foundation for the statistical mechanics of gravitating systems, unifying geometric, field-theoretic, and quantum-information perspectives across an array of gravitational settings.