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

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• Jorge Santos, University of Cambridge
• Thursday 06 June 2024, 13:00-14:00
• Potter Room (B1.19).

If you have a question about this talk, please contact Jackson Fliss.

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 a one-parameter family of boundary conditions exists, parameterised 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 → 0 and ∞ 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 a positive real part. Thus we may regard large p as a regularisation 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.

This talk is part of the Quantum Fields and Strings Seminars series.

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