Dear MBN, if you allow the future metric to be sourced by an arbitrary $T_{\mu\nu}$ that doesn't have to be derived from any actual type of matter or fields, then your spliced spacetime may be constructed. However, it's simply because Einstein's equations become tautologies if you allow $T_{\mu\nu}$ to be anything. For any geometry (metric), you may find a $T_{\mu\nu}$ profile such that Einstein's equations are satisfied - just calculate the "right" stress-energy tensor from Einstein's equations.
Just take the Minkowski space for $t<0$ and extend the metric tensor components to arbitrary functions of $t$ that are infinitely differentiable, even at $t=0$, but that are non-constant for positive $t$. For example, you may take such functions to be
$$g_{\mu\nu}(t) = g_{\mu\nu}(t<0) + C_{\mu\nu} \exp(-1/t^2)$$
for positive $t$.
Now, calculate the curvature and Einstein's tensor out of this arbitrary metric, and you will know what $T_{\mu\nu}$ should be declared to be the source of this gravitational field. However, you will never derive such a "suddenly turning on" matter source out of any well-defined equations. It's because you would face a similar problem e.g. for the electromagnetic field that could source the strange spliced gravitational field.
However, the electromagnetic source also can't be turned on "suddenly", unless it's sourced by an electric charge distribution $j_\mu$ that also has to be turned on suddenly. So something has to be "externally inserted" to any physical system to achieve the change of the behavior in $t=0$.
What's more important is that you can't get your spliced spacetime out of the vacuum Einstein's equations, i.e. those with $T_{\mu\nu}=0$. Einstein's equations are second-order partial differential equations for the metric components $g_{\mu\nu}(t,x,y,z)$. Throw away all the indices and useless dimensions and solve a morally similar problem, the equation
$$d^2 x/dt^2 = 0$$
Clearly, this equation has a unique solution for given initial conditions. Well, in this particular form, the solutions have to be linear functions, but more generally, the solution is uniquely determined by the initial conditions. The same can be proved for the partial differential equations from a broad class - including Einstein's equations.
To summarize, you can say that the change of the behavior can only be achieved if some external sources of the fields are manually added at $t=0$. Of course, if you gradually turn them on, the response will be gradually increasing, too. But there is nothing mysterious about it. If you suddenly "buy" some matter from another spacetime and insert it to Einstein's equations starting from $t=0$, e.g. by an infinitely smooth function, the metric starts to get curved in a similar way after $t=0$, too.
Because the stress-energy tensor has to have a vanishing covariant divergence (because the Einstein's tensor has the same property identically), it will constrain the kinds of stress-energy tensors that may be turned in this way. It's likely that the required stress-energy tensor - e.g. one calculate from the metric I wrote in a displayed equation above - always violates some (or all) energy conditions right after $t=0$. In particular, the components of the required $T_{\mu\nu}$ will be "mostly spacelike" which is forbidden by the null or dominant energy conditions.
