In general relativity, it doesn't really make sense to talk about a hypersurface of constant time as if that had some intrinsic physical meaning. Coordinates such as a time coordinate are arbitrary in GR. So to state your question correctly, what you really want to talk about is a spacelike surface or Cauchy surface. Let's call this surface S.

The answer to your question is no. The field equations imply that the stress-energy has zero divergence, which is a local statement of conservation of energy-momentum. We don't have a global Gauss's law in a curved spacetime, but local conservation is enough to rule out your scenario. For example, suppose that a hydrogen atom pops into existence at some point in spacetime, as in the old steady-state cosmological models. For a sufficiently small neighborhood of this point, curvature is negligible, and we can adopt Minkowski coordinates in which the atom is at rest. In these coordinates, $\partial_\kappa T^{\kappa t}=\partial T^{tt}/\partial t\ne 0$.

There are independent physical reasons why such a scenario is problematic. We expect the matter fields to obey some wave equation, but if the wave and all its derivatoves are zero on a Cauchy surface, then it would seem to violate causality if the wave were later to be nonzero. Also, it will be impossible to avoid a violation of Lorentz invariance, since there is no preferred frame of reference in which a newly created particle should be at rest.