New Well-Posed Boundary Conditions for Semi-Classical Euclidean Gravity

TL;DR

The paper introduces a one-parameter family of generalized conformal boundary conditions for four-dimensional Euclidean gravity in a finite cavity, unifying Anderson and Dirichlet by fixing the conformal boundary data and a Weyl-weighted extrinsic-curvature combination. It develops a Euclidean thermodynamic framework with a first-law formulation and analyzes stability across hot space and Schwarzschild infillings, finding Euclidean stability for $p>1/6$ but Euclidean negative modes for $p<1/6$, and showing a York–Hawking–Page–type transition independent of $p$. In the Lorentzian sector, the authors derive a dynamical boundary (brane) equation and show spherical instabilities align with $p<1/6$ but, for large black holes and $p>1/6$, reveal additional boundary-driven instabilities that do not manifest in the Euclidean analysis, especially for non-spherical perturbations, suggesting a potential lack of smoothness in the Euclidean path integral. Overall, the work exposes a deep tension between stable Euclidean thermodynamics and unstable Lorentzian dynamics under these boundary conditions, raising questions about the role of horizon smoothness and the appropriate ensemble in quantum gravity with boundaries.

Abstract

We consider four-dimensional Euclidean gravity in a finite cavity. Dirichlet conditions do not yield a well-posed elliptic system, and Anderson has suggested boundary conditions that do. Here we point out that there exists a one-parameter family of boundary conditions, parameterized by a constant $p$, where a suitably Weyl rescaled boundary metric is fixed, and all give a well-posed elliptic system. Anderson and Dirichlet boundary conditions can be seen as the limits $p \to 0$ and $\infty$ of these. Focussing on static Euclidean solutions, we derive a thermodynamic first law. Restricting to a spherical spatial boundary, the infillings are flat space or the Schwarzschild solution, and have similar thermodynamics to the Dirichlet case. We consider smooth Euclidean fluctuations about the flat space saddle; for $p > 1/6$ the spectrum of the Lichnerowicz operator is stable -- its eigenvalues have positive real part. Thus we may regard large $p$ as a regularization of the ill-posed Dirichlet boundary conditions. However for $p < 1/6$ there are unstable modes, even in the spherically symmetric and static sector. We then turn to Lorentzian signature. For $p < 1/6$ we may understand this spherical Euclidean instability as being paired with a Lorentzian instability associated with the dynamics of the boundary itself. However, a mystery emerges when we consider perturbations that break spherical symmetry. Here we find a plethora of dynamically unstable modes even for $p > 1/6$, contrasting starkly with the Euclidean stability we found. Thus we seemingly obtain a system with stable thermodynamics, but unstable dynamics, calling into question the standard assumption of smoothness that we have implemented when discussing the Euclidean theory.

Figures (22)

• Figure 1: $p$ as a function of $\tilde{\lambda}$ for the static spherical modes $n=\ell_S=0$. The green dot marks $p=\frac{1}{6}$, where $\tilde{\lambda}$ is precisely $0$. The red dashed curve gives the asymptotic behavior of $p(\tilde{\lambda})$ as $\tilde{\lambda}\to-\infty$.
• Figure 2: $p$ as a function of $\tilde{\lambda}$ for the static non-spherical modes with $\ell_S=2$ (left panel) and $\ell_S=4$ (right panel). The green dots mark $\left(0,\frac{1}{12}(2-\ell_S-2\ell_S^2)\right)$.
• Figure 3: Contour plot of $\mathrm{Re}\,p$ in the complex $\tilde{\lambda}$ plane for $\ell_S=2$ (left panel) and $\ell_S=4$ (right panel). The red dashed curves are where $\mathrm{Im}\,p=0$ that corresponds to a physical unstable mode.
• Figure 4: $\mathrm{Re}\,p$ as a function of $\varpi$ at $\tilde{\lambda}=0$ for different $\ell_S$'s. The dashed lines give the asymptotic behaviors. The black solid line is $\mathrm{Re}\,p=0$.
• Figure 5: $\mathrm{Re}\,p$ as a function of $\mathrm{Im}\,\tilde{\lambda}$ at $\varpi=0$ for different $\ell_S$'s. The dashed lines give the asymptotic behaviors. The black solid line is $\mathrm{Re}\,p=0$.