  A five-dimensional solution to Einstein's equations coupled to a scalar field
has been proposed as a partial solution to the cosmological constant problem:
the effect of arbitrary vacuum energy (tension) of a 3-brane is cancelled;
however, the scalar field becomes singular at some finite proper distance in
the extra dimension. We show that in the original model with a vanishing bulk
potential for the scalar, the solution is a saddle point which is unstable to
expansion or contraction of the brane world. We construct exact time-dependent
solutions which generalize the static solution, and demonstrate that they do
not conserve energy on the brane; thus they do not have an effective 4-D field
theoretic description. When a bulk scalar field potential is added, the
boundary conditions on the brane cannot be trivially satisfied, raising hope
that the self-tuning mechanism may still give some insight into the
cosmological constant problem in this case.
