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The Casimir effect is real, and it has been measured (it’s actually a nuisance for building nano-scale machines, but that’s a different story). Negative energy is a reality in our Universe.
And where there’s negative energy, there’s the potential for building stable, traversable wormholes. There’s just one problem: We’ll need to solve the biggest outstanding problem in physics to have a hope of realizing this potential.
Physicists are confident that the ultimate answer to building wormholes lies in the unknown territory of quantum gravity, the marriage of quantum mechanics and general relativity. GR tells us that wormholes may exist, but only if the right conditions are allowed (i.e., negative energy). And quantum mechanics—as expressed through quantum fields—tells us how to make negative energy. But we’re not sure how those two puzzle pieces fit together. We have no theory of quantum gravity.
For example, it’s not clear if the negative energy found in situations like the Casimir effect is the right kind of negative energy. It’s negative relative to the rest of the Universe, which may be enough to create and stabilize a wormhole, but maybe not. We might need negative energy in the absolute sense, which could be just as fantastical as negative matter.
The negative energy found in the Casimir effect is also incredibly weak and small-scale. Sure, you can point to the microscopic separation between two parallel metal plates and confidently say that negative energy exists there, but we don’t know how to scale that effect up into a macroscopic object.
We might be able to build wormholes with more exotic structures. For example, cosmic strings are the theoretical fractures left in spacetime from when the four forces of nature split off from each other in the very early universe. It might be possible to thread these cosmic strings through the open throat of a wormhole, “anchoring” the ends like the cables holding up a suspension bridge, thereby stabilizing the wormhole for transit. But while most cosmologists are confident that cosmic strings exist, no such strings have been found.
Theoretical physicists have also discovered that some theories of modified gravity, originally designed to explain the phenomenon of dark energy, may allow the presence of stable wormholes without any exotic forms of matter or energy. But those theories of modified gravity also predict that the speed of gravity is slower than the speed of light, which is difficult to reconcile with the 2017 observation of gravitational waves from a kilonova (a merger of two neutron stars), which showed that gravity and light travel at nearly the same speed.
String theory hopes to become a solution to the problem of quantum gravity by replacing the point-like particles in physics with extended objects, known variously as strings and branes. And indeed, some theorists have discovered that string theory may allow the existence of stable wormholes. Alas, string theory is not complete and so far has failed to provide a solvable theory of physics.
Investigations into the nature of quantum fields near the event horizons around black holes have found that it might—might—be possible to build a stable wormhole by contorting its shape. But those wormholes must be incredibly tiny, no bigger than about 10^-35 meters across, which is… less than useful. And those same mathematics rely on a host of simplifying assumptions about the nature of quantum gravity that may not hold up.
This is where modern wormhole research sits. Physicists are fascinated by wormholes because they provide a powerful laboratory for studying quantum gravity. Also, they're really cool. So while I don’t recommend planning a trip to the Andromeda galaxy quite yet, I can’t quite rule it out as a possibility, either.
