  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. The problem of linearised gravity is analysed using the
Kodama-Ishibashi formalism. It is shown that for a round metric on $S^2$ with
constant $K$, there are modes that grow exponentially in time. We discuss a
method to control the growing modes by varying $K$. The growing modes are
absent for a conformally flat induced metric on the timelike boundary. We
provide evidence that the Dirichlet problem for a spherical boundary does not
suffer from non-uniqueness issues at the linearised level. We consider the
extension of black hole thermodynamics to the case of conformal boundary
conditions, and show that the form of the Bekenstein-Hawking entropy is
retained.
