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

• URL: http://arxiv.org/abs/2402.04308v2
• Date: Thu, 4 Apr 2024 17:41:57 GMT
• Title: New Well-Posed Boundary Conditions for Semi-Classical Euclidean Gravity
• Authors: Xiaoyi Liu, Jorge E. Santos, Toby Wiseman,
• Abstract summary: Dirichlet conditions do not yield a well-posed elliptic system. Anderson and Dirichlet boundary conditions can be seen as the limits $p to 0$ and $infty$ of these. For $p 1/6$ there are unstable modes, even in the spherically symmetric and static sector.
• Score: 1.4630192509676045
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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