[Paper Review] Wormhole Thermodynamics at Apparent Horizons

Mubasher Jamil, M. Ali Akbar|ArXiv.org|Nov 13, 2009

Hydraulic Fracturing and Reservoir Analysis3 citations

TL;DR

This paper demonstrates that the Einstein field equations for an evolving Lorentzian wormhole with shape function $b(r) = r_0^2/r$ can be rewritten as a first law of thermodynamics, $dE = TdS + WdV$, at its apparent horizons. By associating temperature $T = \kappa/2\pi$, entropy $S = A/4G$, and work density $W = (\rho - p)/2$ with the horizons, the authors show that the dynamics of the wormhole’s inner and outer apparent horizons—where the inner expands and the outer contracts—satisfy thermodynamic identities analogous to those in FRW cosmology.

ABSTRACT

In this paper, we discuss the thermodynamic properties of the evolving Lorentzian wormhole. For the shape function $b(r) = r_{0}^2/r$, it is shown that the wormhole spacetime admits two apparent horizons, the inner and the outer one. The inner horizon expands while the outer contracts with the passage of time. Corresponding to these horizons, we have three types of wormholes, regular, extreme and the naked wormholes. Moreover, it is shown that the Einstein field equations can be rewritten as a first law of thermodynamics $dE=TdS+WdV$, at the apparent horizons of the wormhole, where $E=ρV$, $T = κ/2π$, $S=A/4G$, $W=(ρ-P)/2$ and $V = {4/3}π ilde{r}_{A+}^3$ are the total matter energy, horizon temperature, wormhole entropy, work density and the volume of the wormhole respectively.

Motivation & Objective

• To extend the thermodynamic interpretation of gravity—previously established for black holes and FRW cosmologies—to evolving Lorentzian wormholes.
• To investigate whether the Einstein field equations of wormhole spacetimes can be expressed as a first law of thermodynamics at apparent horizons.
• To analyze the dynamical behavior of inner and outer apparent horizons in a time-evolving wormhole geometry with a specific shape function.
• To explore the thermodynamic consistency of wormhole spacetimes, particularly in the context of energy, entropy, and work density at horizons.
• To lay the foundation for extending this thermodynamic framework to extended theories of gravity in the future.

Proposed method

• Assumes a time-dependent wormhole metric with $\Phi(t,r) = 0$ and shape function $b(r) = r_0^2/r$, leading to a spacetime with two apparent horizons.
• Derives the Friedman-like equations from the Einstein field equations under spherical symmetry and a perfect fluid energy-momentum tensor.
• Identifies the apparent horizons $\tilde{r}_{A+}$ (outer) and $\tilde{r}_{A-}$ (inner) by solving $h^{ab}\partial_a \tilde{r} \partial_b \tilde{r} = 0$.
• Computes the surface gravity $\kappa$ at the outer horizon $\tilde{r}_{A+}$, leading to temperature $T = \kappa/2\pi$ and entropy $S = A/4G$.
• Re-expresses the Friedman-like equation (8) at the apparent horizon by multiplying with a time-dependent factor to isolate $TdS$.
• Derives $dE = TdS + WdV$ by combining the differential of matter energy $E = \rho V$ with the thermodynamic form, identifying $W = (\rho - p)/2$ as work density.

Experimental results

Research questions

• RQ1 Can the Einstein field equations of an evolving Lorentzian wormhole be expressed as a first law of thermodynamics at its apparent horizons?
• RQ2 How do the inner and outer apparent horizons of the wormhole evolve over time, and what are their thermodynamic properties?
• RQ3 What is the role of the shape function $b(r) = r_0^2/r$ in enabling two distinct apparent horizons and their thermodynamic behavior?
• RQ4 Does the thermodynamic identity $dE = TdS + WdV$ hold for wormholes in the same way as it does for FRW cosmologies?
• RQ5 What are the implications of this thermodynamic formulation for extreme or naked wormhole configurations?

Key findings

• The wormhole with shape function $b(r) = r_0^2/r$ admits two apparent horizons: an inner horizon that expands and an outer horizon that contracts over time.
• At the outer apparent horizon $\tilde{r}_{A+}$, the Einstein field equations reduce to the first law of thermodynamics: $dE = TdS + WdV$, where $T = \kappa/2\pi$, $S = A/4G$, $E = \rho V$, and $W = (\rho - p)/2$.
• The surface gravity $\kappa$ at $\tilde{r}_{A+}$ is given by $\kappa = -\frac{1}{\tilde{r}_{A+}}\left(1 - \frac{\dot{\tilde{r}}_{A+}}{2H\tilde{r}_{A+}}\right)\left(1 - \frac{2a^2 r_0^2}{\tilde{r}_{A+}^2}\right)$, which reduces to the FRW form when $r_0 \to 0$.
• The entropy $S$ is proportional to the area of the apparent horizon: $S = \frac{A}{4G} = \frac{\pi \tilde{r}_{A+}^2}{G}$, consistent with black hole thermodynamics.
• The thermodynamic identity also holds at the inner horizon $\tilde{r}_{A-}$, indicating that both horizons support a thermal interpretation.
• In the extreme case where $p = -\rho$, the first law reduces to the standard form $dE = TdS - pdV$, analogous to cosmological thermodynamics.