  Static spherically symmetric uncoupled scalar space-times have no event
horizon and a divergent Kretschmann singularity at the origin of the
coordinates. The singularity is always present so that non-static solutions
have been sought to see if the singularities can develop from an initially
singular free space-time. In flat space-time the Klein-Gordon equation
$\Box\ph=0$ has the non-static spherically symmetric solution $\ph=\si(v)/r$,
where $\si(v)$ is a once differentiable function of the null coordinate $v$. In
particular the function $\si(v)$ can be taken to be initially zero and then
grow, thus producing a singularity in the scalar field. A similar situation
occurs when the scalar field is coupled to gravity via Einstein's equations;
the solution also develops a divergent Kretschmann invariant singularity, but
it has no overall energy. To overcome this Bekenstein's theorems are applied to
give two corresponding conformally coupled solutions. One of these has positive
ADM mass and has the properties: i) it develops a Kretschmann invariant
singularity, ii)it has no event horizon, iii)it has a well-defined source,
iv)it has well-defined junction condition to Minkowski space-time, v)it is
asymptotically flat with positive overall energy. This paper presents this
solution and several other non-static scalar solutions. The properties of these
solutions which are studied are limited to the following three: i)whether the
solution can be joined to Minkowski space-time, ii)whether the solution is
asymptotically flat, iii)and if so what the solutions' Bondi and ADM masses
are.
