  In a series of recent papers we developed a formulation of general relativity
in which spacetime and the dynamics of matter evolve with a Poincar\'e
invariant parameter $\tau$. In this paper, we apply the formalism to derive the
metric induced by a `static' event evolving uniformly along its $t$-axis at the
spatial origin ${\mathbf x} = 0$. The metric is shown to vary with $t$ and
$\tau$, as well as spatial distance $r$, taking its maximum value for a test
particle at the retarded time $\tau = t - r/c$. In the resulting picture, an
event localized in space and time produces a metric field similarly localized,
where both evolve in $\tau$. We first derive this metric as a solution to the
wave equation in linearized field theory, and discuss its limitations by
studying the geodesic motion it produces for an evolving event. By then
examining this solution in the 4+1 formalism, which poses an initial value
problem for the metric under $\tau$-evolution, we clarify these limitations and
indicate how they may be overcome in a solution to the full nonlinear field
equations.

General Physics