I know there are solutions to Einstein's field equations that give a wormhole geometry. But they are time independent. They are static. Is there a process where empty flat spacetime can evolve into a wormhole by an appropriate flow of matter and energy and negative energy?
If so, it would change the topology of spacetime. Does General Relativity permit this? How would a hole in spacetime form? What determines where the other mouth of the wormhole would be located?
The current state of knowledge is basically that we don't know.
The topology change is contentious - it is unclear if it is permitted, but there are also disagreements about why and how. The standard approach is to shout "quantum gravity!" and escape in the confusion.
One classical argument for why making a wormhole would be problematic is the topology censorship theorem. It states:
Every causal curve extending from past null infinity to future null infinity can be continuously deformed to a curve near infinity.
Roughly speaking, this says that an observer, whose trip begins and ends near infinity, and who thus remains outside all black holes, is unable to probe any nontrivial topological structures.
Now, there are issues with the theorem (since it assumes the null energy condition that wormholes and quantum fields often break, and some topological assumptions). But it seems to be a good reason to suspect wormhole formation or existence is not allowed unless it gets hidden by a topologically spherical event horizon. Except that general relativity on its own seems to be too much of a local theory to be really good defence against non-trivial topology.
There are quantum gravity papers arguing that a cosmic string breaking by tunnelling can produce traversable wormholes and gleefully break the (classical) topological censorship. Here the wormholes show up at the ends of the string, initially next to each other.
Making a solution of the formation of a wormhole isn't too hard (I'm lying, it is hard), but whether or not that's a reasonable solution is a different matter.
The simplest case is to just consider the case of the collapse of a wormhole in reverse. Take a spacetime that is originally just two copies of $\mathbb{R}^3$, and at some point, remove a point from both copies. That point will then grow in size, to an open ball, on which you can identify the boundaries (this will only work for a ball of radius $> 0$, so the initial singularity is indeed a singularity).
This is entirely artificial, but it does illustrate some of the problems. While there is nothing fundamentally wrong with this solution, you can just ask why the singularity would develop in such a way that the edges would identify, instead of simply remaining a singularity, or even appearing at all.
By the various theorems relating to spacetime topology, any such change in topology will be in some way unpalatable. It will either involve closed timelike curves or singularities in some way. In particular, by Geroch's theorem, it cannot be globally hyperbolic. If your spacetime isn't globally hyperbolic, there is also no guarantee of the uniqueness of development. There exists a theorem saying that any spacetime developping closed timelike curve also has a possible development without them, and I suspect the same may be true for wormholes.
A possibility that goes back to the very origin of wormholes in the 1950's is the microscopic structure of spacetime. Some theories have it that quantum gravity, along with having a sum over every possible metric (in the path integral formalism), there may also be a sum over topologies, and spacetime may simply evolve wormholes at that scale naturally. In other words, we may get something like
\begin{equation} Z = \sum_{M \in \mathrm{Top}} \int \mathcal{D}g \exp[i \left(\int_M (R_g + L_M) d\mu[g]\right)] \end{equation}
Whether that is accurate, and whether this allows for the production of traversable macroscopic wormholes is another matter.
Unfortunately the answer is that we don't know, and most likely a classical theory like General Relativity that concerns itself mostly with local properties of geometry is inherently unfit to aboard the problem of global topology
The clues we get from General Relativity is that some static adiabatic (i.e. reversible) solutions with spacetimes topologically inequivalent to Minkowski do exist, but they require unphysical negative energy.
If we relax the reversibility requirement on trajectories, we might consider static Kerr geometries as unidirectional wormholes, which can exist without any negative energy required
Intuitively, a topology change of spacetime cannot (shouldn't?) occur without at least a transient singularity occurring somewhere in the spacetime, but no one has ever obtained the equivalent of a maximally extended spacetime from a finite-time collapse (see this answer)
