Virtual work, thermodynamic structure of the spacetime, and black hole criticality

TL;DR

The paper develops a generalized Euclidean gravity framework by incorporating 'virtual geometries' that need not satisfy Einstein equations, enabling direct analysis of black hole thermodynamics and criticality through a Landau-Ginzburg potential. It defines a virtual thermodynamic potential ${\mathcal{G}}(T,\Phi,x_h)$ from the regularized Euclidean action on virtual geometries and shows its derivative with respect to the horizon radius encodes a virtual work term, yielding a modified first law that becomes the standard one on-shell. Specializing to an exact asymptotically flat hairy KK black hole with a dilaton potential, the authors identify a finite-temperature critical point and an inverted swallowtail, and provide a Landau-Ginzburg description near criticality. The approach offers a unified, model-dependent method to analyze black hole phase structure in flat space and can be extended to other gravity theories with scalar potentials or higher-derivative corrections.

Abstract

We propose a new way to relate the black hole thermodynamics and geometry by generalizing the Euclidean formalism to include "virtual geometries", which do not necessarily satisfy Einstein equations. This provides a physically well motivated route to study black hole criticality and obtain the Landau Ginzburg potential. We compute the "virtual thermodynamic potential" and show that it satisfies a modified quantum statistical relation that is compatible with the first law of black hole thermodynamics supplemented with an extra term, interpreted as virtual work in previous literature. The novelty is that, within our formalism, we can explicitly compute this term as the first derivative of the virtual thermodynamic potential with respect to the horizon radius that is considered as the order parameter. Imposing the physical condition that the first derivative vanishes is at the basis of the matching between the first law of black hole thermodynamics and (one of the) Einstein equations evaluated at the horizon. Interestingly, imposing the physical conditions that the second and third derivatives vanish, we can concretely study the criticality and existence of swallow tails. As a specific example, we apply this formalism to an exact four dimensional asymptotically flat hairy black hole, namely the generalized Kaluza Klein (KK) black hole when the dilaton potential is included, and show that it is thermodynamically stable and has a non-trivial critical behaviour corresponding to an inverted swallowtail.

Figures (4)

• Figure 1: Top panel: $\mathcal{G}$ vs $T$ displaying a standard swallowtail structure. Bottom panel: the corresponding Landau-Ginzburg potential, $\Psi$ vs $\hat{\rho}$, for three representative isotherms. For the leftmost isotherm (the red vertical line in the first plot), the globally stable phase corresponds to the branch of small black hole which corresponds to the global minimum of the uppermost curve in the bottom panel. For the rightmost isotherm in top panel, the branch of large black hole phase is now the globally stable one.
• Figure 2: Top panel: $\mathcal{G} = \mathcal{G}(T)$ for the value $\Phi = 0.70388$, for which the inverted swallowtail structure occurs. The cross, dot, and circle markers indicate small, intermediate, and large black holes, respectively, which intersect the isotherms $T \approx 0.0601$, $T \approx 0.0602$, and $T\approx 0.0603$. Bottom panel: $\Psi = \Psi(\hat{\rho})$ for $\hat{\psi} \approx 0.00057$ ($\Phi = 0.70388$), for the same three isotherms from the top panel.
• Figure 3: Left panel: 3D plot of $\mathcal{G}$ vs $T$ and $\Phi$. Right panel: Constant-$\Phi$ curves. Near the critical point, the free energy surface develops a characteristic swallowtail structure, but in this case it appears inverted, with the 'tail' pointing downward.
• Figure 4: $\mathcal{G}(T)$ for decreasing conjugate potentials $\Phi=\tfrac{1}{\sqrt{2}}\approx0.70711$, $0.70710$, $0.705$, and $\Phi_c\approx0.70348$. In particular, the top-right panel zooms into the physically relevant region with $T_1<T^*$, where the thermodynamically favored black hole branch dominates over flat space.