Semi-classical spacetime thermodynamics (2509.05052v1)

Abstract: We derive the semi-classical gravitational dynamics from thermodynamics of local stretched light cones in 2-dimensional dilaton gravity, explicitly treating the backreaction of quantum matter through the conformal anomaly's effect on the generalized entropy. We also sketch the extension of this analysis to the conformal anomaly in 4-dimensional semi-classical gravity. In direct connection to this problem, we also tackle the appropriate definition of Wald entropy in thermodynamic derivation of equations of motion for classical scalar-tensor theories. For the class of Brans-Dicke theories, including 2-dimensional dilaton gravity, we show that the equations of motion follow from the dynamical Wald entropy associated with local causal horizons.

• The paper derives semi-classical gravitational dynamics from thermodynamic principles using stretched light cones and the Clausius relation.
• The paper establishes the role of augmented Wald entropy in resolving non-minimal couplings and recovering Einstein’s and modified gravity equations in 2D scenarios.
• The paper identifies limitations in 4D, where anomaly-induced curvature terms expose discrepancies in effective semi-classical equations, urging further study.

Semi-classical Spacetime Thermodynamics: A Technical Analysis

Introduction and Motivation

The paper "Semi-classical spacetime thermodynamics" (2509.05052) presents a rigorous derivation of semi-classical gravitational dynamics from local thermodynamic principles, focusing on stretched light cones in 2D dilaton gravity and extending the analysis to 4D semi-classical gravity. The authors explicitly incorporate quantum matter backreaction via the conformal anomaly's contribution to generalized entropy, and address the correct definition of Wald entropy in scalar-tensor theories, including Brans-Dicke and $f(R)$ gravity. The work situates itself within the paradigm initiated by Jacobson, where gravitational field equations emerge from local thermodynamic equilibrium conditions, and advances this program by treating quantum effects and non-minimal couplings in a unified framework.

Thermodynamic Derivation of Gravitational Dynamics

Stretched Light Cones and Local Thermodynamics

The construction of stretched light cones (SLCs) provides a local, spherically symmetric causal surface suitable for assigning entropy and temperature via the Unruh effect. The metric expansion in Riemann normal coordinates around a point $P$ allows the definition of a timelike hyperboloid $\Sigma$ just outside the null cone, with acceleration $a = 1/\alpha$ and Unruh temperature $T_U = \hbar a / 2\pi$. The entropy associated with $\Sigma$ is computed via the Wald prescription, and the Clausius relation $\Delta S_{\text{rev}} = \Delta Q / T_U$ is imposed for reversible processes, with irreversible entropy production subtracted.

Recovery of Einstein and Modified Gravity Equations

For general relativity, the reversible change in Wald entropy and the Clausius entropy flux from matter yield the Einstein equations at a point $P$:

$$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu}$$

The method generalizes to Lagrangians constructed from the metric and Riemann tensor, with Wald entropy replaced by the Noether charge formula. For $f(R)$ gravity, the entropy functional and equilibrium condition recover the full equations of motion, including higher-derivative terms.

Scalar-Tensor and Non-minimal Coupling

Standard Wald entropy fails to capture contributions from non-minimally coupled scalars and does not yield the scalar field equation. The authors employ an augmented Wald entropy, motivated by recent dynamical entropy prescriptions, which includes additional terms proportional to the boost generator and a symmetric tensor $M^{\lambda\mu}$ derived from the Lagrangian. This modification is essential for Brans-Dicke and scalar-tensor theories, and the entropy functional is shown to be equivalent to the dynamical Wald entropy for non-stationary horizons.

Semi-classical Gravity and Generalized Entropy

2D Dilaton Gravity and the Conformal Anomaly

In 2D, the backreaction of quantum matter is fully encoded by the conformal anomaly and the Polyakov action. The generalized entropy $S_{\text{gen}}$ combines gravitational and quantum matter contributions and is manifestly finite and cutoff-independent when the running of $G$ is accounted for. The authors demonstrate that the augmented Wald entropy for SLCs coincides with $S_{\text{gen}}$ in 2D JT gravity, and the equilibrium condition $\Delta S_{\text{gen}} = 0$ yields the full non-linear semi-classical equations, including explicit backreaction terms.

The equations of motion derived from the thermodynamic condition match those from the Polyakov effective action:

• Metric variation yields the semi-classical Einstein equations with anomaly-induced stress-energy.
• Dilaton and anomaly field variations produce the correct scalar equations, reproducing the trace anomaly.

This result is robust for large central charge $c$ and is consistent with previous analyses of black hole entropy in JT gravity.

Extension to 4D Semi-classical Gravity

In 4D, the conformal anomaly includes curvature-squared terms and cannot be derived from a local, diffeomorphism-invariant action. The authors construct a local scalar-tensor effective action that reproduces the anomaly (excluding the $R^2$ term) and derive the corresponding augmented Wald entropy. The thermodynamic derivation recovers the effective equations of motion, but does not fully capture the semi-classical backreaction, as direct calculations of the renormalized stress-energy tensor reveal discrepancies. The generalized entropy in 4D is not explicitly known, and its identification remains an open problem.

Implications and Future Directions

Theoretical and Practical Consequences

• Generalized Entropy as a Unifying Principle: The equivalence between augmented Wald entropy and generalized entropy in 2D provides a concrete realization of the thermodynamic origin of semi-classical gravity, including quantum backreaction.
• Dynamical Wald Entropy: The dynamical prescription resolves ambiguities in entropy definitions for non-stationary horizons and is essential for scalar-tensor and semi-classical theories.
• Limitations in Higher Dimensions: The inability to fully encode the $R^2$ anomaly and the mismatch in effective dynamics highlight the need for a more complete understanding of generalized entropy in 4D.

Numerical and Conceptual Results

• Explicit Recovery of Semi-classical Equations: In 2D, the method yields the full non-linear semi-classical equations, including anomaly-induced terms, with no divergence in entropy.
• Ambiguity Resolution: The dynamical entropy prescription is shown to be invariant under total divergence shifts in the Lagrangian, providing a unique entropy functional for local causal horizons.

Prospects for AI and Quantum Gravity

• Statistical Interpretation: The connection between dynamical Wald entropy and generalized entropy suggests a possible statistical mechanical underpinning, potentially accessible via von Neumann algebra techniques.
• Extension to Arbitrary Subregions: Recent work on generalized entropy in gravitational settings may allow the extension of these results to arbitrary subregions and more general quantum field theories.
• Semi-classical Gravity in AI Models: The formalism may inform the development of AI models for quantum gravity, particularly in the simulation of backreaction and entropy production in semi-classical spacetimes.

Conclusion

The paper provides a technically rigorous framework for deriving semi-classical gravitational dynamics from local thermodynamic principles, with explicit treatment of quantum backreaction via the conformal anomaly and generalized entropy. The approach is successful in 2D, where the augmented Wald entropy coincides with the generalized entropy and yields the full semi-classical equations. In 4D, the method recovers the effective anomaly-induced dynamics but does not fully capture the semi-classical backreaction, reflecting the limitations of current entropy functionals. The dynamical Wald entropy prescription resolves ambiguities in non-stationary settings and offers a promising direction for future research, both in gravitational theory and in the statistical interpretation of entropy in quantum gravity. Further work is required to identify the full generalized entropy in higher dimensions and to understand its implications for the microscopic structure of spacetime.