Self Tuning Scalar Fields in Spherically Symmetric Spacetimes
  We search for self tuning solutions to the Einstein-scalar field equations
for the simplest class of `Fab-Four' models with constant potentials. We first
review the conditions under which self tuning occurs in a cosmological
spacetime, and by introducing a small modification to the original theory -
introducing the second and third Galileon terms - show how one can obtain de
Sitter states where the expansion rate is independent of the vacuum energy. We
then consider whether the same self tuning mechanism can persist in a
spherically symmetric inhomogeneous spacetime. We show that there are no
asymptotically flat solutions to the field equations in which the vacuum energy
is screened, other than the trivial one (Minkowski space). We then consider the
possibility of constructing Schwarzschild de Sitter spacetimes for the modified
Fab Four plus Galileon theory. We argue that the only model that can
successfully screen the vacuum energy in both an FLRW and Schwarzschild de
Sitter spacetime is one containing `John' $\sim G^{\mu}{}_{\nu}
\partial_{\mu}\phi\partial^{\nu}\phi$ and a canonical kinetic term $\sim
\partial_{\alpha}\phi \partial^{\alpha}\phi$. This behaviour was first observed
in (Babichev&Charmousis,2013). The screening mechanism, which requires
redundancy of the scalar field equation in the `vacuum', fails for the `Paul'
term in an inhomogeneous spacetime.
