Thermodynamics of thin-shell wormholes

Abstract: We develop a unified thermodynamic description of dynamic thin-shell wormholes, starting from the transparent (vacuum bulk) case and then relaxing the transparency condition to include bulk matter crossing the throat. For isolated shells, we show that the entropy is conserved under transparent evolution, $TdS=0$, which holds for arbitrary dynamical motion of the throat within the assumptions of the thin-shell formalism. When bulk matter is present, the generalised first law becomes $T\dot{S}=AΦ$, where $Φ$ is the net energy flux; entropy increases (decreases) when matter flows into (out of) the shell. Explicit expressions for the flux are provided for null dust and massless scalar fields, and quantum pair production (Hawking-like emission) is also discussed. A formulation of the generalised second law is presented, and it is consistent with standard thermodynamic expectations under conditions where heat flows from hotter to colder regions. As a concrete astrophysical application, we study accretion of null dust: the critical accretion rate above which the throat becomes dynamically unstable depends sensitively on the spacetime geometry and the shell equation of state, highlighting the environmental dependence of these configurations. Fluctuations in the energy flux induce a stochastic force on the throat dynamics, leading to a Langevin description with an associated fluctuation-dissipation relation. This results in diffusion of the throat radius and additional entropy production. The framework applies to a broad class of spherically symmetric thin-shell constructions within general relativity. Our results provide evidence that thin-shell wormholes admit a consistent thermodynamic description, quantify their sensitivity to accretion processes, and suggest possible avenues for probing exotic compact objects with future gravitational wave observations.

• The paper introduces a unified thermodynamic framework that generalizes static results to dynamic thin-shell wormholes via explicit energy and entropy relations.
• The analysis employs the Israel-Lanczos junction conditions to derive surface energy density and pressure, enabling stability and critical accretion rate computations.
• The study incorporates stochastic force modeling and quantum corrections, providing practical insights on wormhole evolution and potential observational signatures.

Thermodynamics of Thin-Shell Wormholes: Unified Formalism and Implications

Overview

The paper "Thermodynamics of thin-shell wormholes" (2606.26370) provides a rigorous and comprehensive treatment of the thermodynamic properties of dynamic thin-shell wormholes within general relativity. The authors introduce a unified framework that generalizes previous static and equilibrium results to arbitrary dynamical motion of the wormhole throat, systematically addressing both transparent scenarios (vacuum bulk) and situations where bulk matter fluxes cross the throat. The formalism captures adiabatic evolution, energy and entropy exchange, and stochastic effects, with explicit expressions for the first and second laws, evaporation, accretion, and critical stability thresholds. This expands the theoretical toolkit for investigating exotic compact objects, their thermodynamics, and prospective observational consequences.

Thin-Shell Formalism

The construction utilizes two spherically symmetric spacetimes, excising inner regions and identifying their boundaries to create a geodesically complete manifold with a dynamic throat. The Israel-Lanczos junction conditions determine the surface stress-energy tensor through the jump in extrinsic curvature. The formalism accommodates both symmetric and asymmetric bulk metrics, yielding explicit expressions for surface energy density and tangential pressure as functions of the embedding and throat motion (see Eqs. 14–15). The conservation equation holds in the absence of net bulk matter flux, independent of the shell equation of state, underpinning the subsequent thermodynamic analysis.

Thermodynamic Laws in Transparent Wormholes

First Law, Adiabatic Invariance, and Dynamic Temperature

For transparent evolution (vacuum bulks), the shell obeys $dU + PdA = 0$, which translates via the Gibbs relation to $TdS = 0$—demonstrating strict adiabatic invariance of entropy across arbitrary dynamical trajectories. Shell temperature, defined through an effective surface gravity (symmetric and averaged for asymmetric cases), generalizes Unruh-Hawking relations and adapts to non-static geometries. Shell entropy remains constant, but temperature varies with throat dynamics, implying an adiabatic, non-equilibrium evolution.

Notably, the effective temperature is a geometric quantity linked to local acceleration, not necessarily associated with thermal emission. Genuine Hawking-like radiation requires specific causal structures not always present in wormhole throats, but the formalism remains consistent with the thermal interpretation in appropriate limits.

Bekenstein Bound and Quantum Corrections

The Bekenstein bound is shown to constrain the shell entropy according to instantaneous throat radius and energy, maintaining validity throughout dynamic evolution under quasi-equilibrium assumptions. Quantum corrections to entropy manifest as logarithmic terms, breaking strict adiabatic invariance on long timescales and allowing for slow entropy evolution with progressing throat dynamics. The emission of quantum particles (Hawking-like radiation) is also addressed phenomenologically, demonstrating consistency with the generalized second law: shell entropy decrease is compensated by the entropy carried away by emitted radiation.

Matter Fluxes Across the Throat: Relaxing Transparency

Generalized First and Second Laws

When bulk matter (null dust, scalar fields, pair production) crosses the throat, the surface Bianchi identity acquires a source term representing net energy influx. The generalized first law becomes $TdS = A\Phi$, directly linking entropy variation to energy flux per unit area. Entropy increases (decreases) when energy enters (leaves) the shell, and the formalism accommodates both classical and quantum sources.

Explicit flux expressions are derived:

• For null dust, influx is always positive, enhancing shell entropy and temperature.
• For scalar fields, flux can be positive or negative, allowing for both accretion and evaporative cooling.
• Quantum emission (modeled as Hawking-like radiation) induces energy loss and entropy decrease in the shell, again offset by the entropy of emitted quanta.

The generalized second law (GSL) is scrutinized: total entropy, including both shell and bulk contributions, is shown to obey standard thermodynamic constraints when heat flows from hotter to colder regions.

Accretion, Critical Rates, and Instability

Detailed analysis of null dust accretion reveals a critical rate beyond which the wormhole throat becomes dynamically unstable, with thresholds sensitively dependent on spacetime geometry and shell equation of state. This demonstrates strong environmental dependence and delineates regimes of stable versus unstable throat evolution. Entropy production rates are computed explicitly for slow accretion, and isothermal approximations are discussed.

Stochastic Dynamics and Fluctuation-Dissipation

Fluctuations in bulk energy flux induce stochastic forces on the throat, leading to a Langevin description with a fluctuation-dissipation relation. This framework captures the entropy increase due to random throat motion and provides a bridge to stochastic thermodynamics. The damping coefficient, derived from the mean flux and equation of state, governs stability: positive damping results in equilibrium, negative damping in instability. Microscopic entropy production is explicitly related to stochastic noise, and equipartition is recovered in equilibrium. Applications include Planck-scale throats and the derivation of probability distributions and escape times for stochastic throat dynamics, as well as potential gravitational wave emission signatures.

Practical and Theoretical Implications

The results establish that thin-shell wormholes admit a consistent thermodynamic description, valid for a broad class of spherically symmetric configurations. The framework quantifies environmental sensitivity, accretion-driven instability, and the subtle interplay between classical, quantum, and stochastic processes. The thermodynamic treatment—especially the formulation of generalized laws and entropy evolution—lays the foundation for probing nontrivial compact objects (including gravastars, regular black holes, and boson stars) and assessing their observability via gravitational wave instrumentation.

Future developments are anticipated in:

• Coupled dynamics with bulk field equations beyond prescribed fluxes
• Extension to non-spherical perturbations and identification of universal thermodynamic signatures
• Microscopic modeling of noise via open quantum systems
• Connections to information theory and wormhole evaporation remnants
• Computationally modeling stochastic GW signals for advanced detectors

Conclusion

This paper establishes a unified and technically robust thermodynamic framework for dynamic thin-shell wormholes, generalizing classical adiabatic invariance to non-equilibrium entropy production, environmental sensitivity, and stochastic dynamics under matter fluxes. The results rigorously quantify the conditions for stability and thermalization and provide explicit numerical estimates for entropy, critical accretion rates, and stochastic effects. The formalism offers new perspectives for theoretical investigation and potential observational strategies for exotic compact objects in gravitational physics.