New Well-Posed Boundary Conditions for Semi-Classical Euclidean Gravity
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929303209377792 |
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| author | Liu, Xiaoyi Santos, Jorge E. Wiseman, Toby |
| author_facet | Liu, Xiaoyi Santos, Jorge E. Wiseman, Toby |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04308 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New Well-Posed Boundary Conditions for Semi-Classical Euclidean Gravity Liu, Xiaoyi Santos, Jorge E. Wiseman, Toby High Energy Physics - Theory General Relativity and Quantum Cosmology Quantum Physics 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. |
| title | New Well-Posed Boundary Conditions for Semi-Classical Euclidean Gravity |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology Quantum Physics |
| url | https://arxiv.org/abs/2402.04308 |