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SUMMARY:New Well-Posed Boundary Conditions for Semi-Classical Euclidean Gr
 avity - Jorge Santos\, University of Cambridge
DTSTART:20240606T120000Z
DTEND:20240606T130000Z
UID:TALK215746@talks.cam.ac.uk
CONTACT:Jackson Fliss
DESCRIPTION:We consider four-dimensional Euclidean gravity in a finite cav
 ity. 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\, an
 d all give a well-posed elliptic system. Anderson and Dirichlet boundary c
 onditions can be seen as the limits p → 0 and ∞ of these. Focussing on
  static Euclidean solutions\, we derive a thermodynamic first law. Restric
 ting to a spherical spatial boundary\, the infillings are flat space or th
 e Schwarzschild solution and have similar thermodynamics to the Dirichlet 
 case. We consider smooth Euclidean fluctuations about the flat space saddl
 e\; for p > 1/6 the spectrum of the Lichnerowicz operator is stable – it
 s eigenvalues have a positive real part. Thus we may regard large p as a r
 egularisation of the ill-posed Dirichlet boundary conditions. However\, fo
 r 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 u
 nderstand this spherical Euclidean instability as being paired with a Lore
 ntzian instability associated with the dynamics of the boundary itself. Ho
 wever\, a mystery emerges when we consider perturbations that break spheri
 cal symmetry. Here we find a plethora of dynamically unstable modes even f
 or p > 1/6\, contrasting starkly with the Euclidean stability we found. Th
 us we seemingly obtain a system with stable thermodynamics\, but unstable 
 dynamics\, calling into question the standard assumption of smoothness tha
 t we have implemented when discussing the Euclidean theory.
LOCATION:Potter Room (B1.19)
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