Gravitational Observatories

Understand

We consider four-dimensional general relativity with vanishing cosmological constant defined on a manifold with a boundary.

• In Lorentzian signature, the timelike boundary is of the form $\boldsymbol{\sigma} \times \mathbb{R}$, with $\boldsymbol{\sigma}$ a spatial two-manifold that we take to be either flat or $S^2$.
• In Euclidean signature, we take the boundary to be $S^2\times S^1$.
• We consider conformal boundary conditions, whereby the conformal class of the induced metric and trace $K$ of the extrinsic curvature are fixed at the timelike boundary.