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Wormholes originate from a special solution of the Einstein field equations (EFE), equations which are fundamental to general relativity. However, there seems to be a reason as to why these objects are fiction. I'm having a hard time assimilating this. How can the solution of a fundamental set of equations be off?

Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations (ME), the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress. BUT, I can't imagine a solution of ME that just can't be. Non-real solutions to ME usually appear when one deals with subtle conditions such as an infinite amount of charge, which is anyway unrealistic.

Noting that Einstein-Rosen bridges, a.k.a wormholes, are solutions of the vacuum EFE, I want to consider the vacuum solutions of ME. The vacuum ME can be solved by plane waves, which exist only in theory, but these solutions can be a good model to experimentally approximate "real" light from distant sources, or can be used as an essential tool in describing an electromagnetic spectrum via the Fourier transform. Nevertheless, wormholes have no correspondence in the real world, particularly because their existence requires the presence of exotic matter (negative mass/energy) and "bending" EFE, and so we don't know of anything that can be modeled as a wormhole to any approximation.

My question is: Are EFE likely incomplete, for they allow unreal solutions, or are there "unrealistic" solutions to ME in the sense described above?

On one hand, I can imagine EFE is missing a term that forbids negative mass/energy, similar to the time Maxwell found the missing displacement current in the electromagnetic equations that today bear his name. On the other hand, I am aware of vortex-like solutions to vacuum ME, but so far they seem to be invalid due to some sort of mistake in them.

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Suppose you saw an apple fall to the ground. You might observe that its velocity increases linearly with time. After studying for a while, you deduce the relevant equation of motion. Then, extrapolating back in time, you deduce that at some earlier moment the apple must have been moving in an upwards direction.

If the apple was thrown upwards by someone, then that deduction could be correct. But what if at some earlier time the apple was moving neither up nor down but hanging from the branch of a tree? In that case the deduction that it started off moving upwards would be wrong. The problem with it is not that upwards motion is impossible, but simply that that was not the motion that was happening because there was something else present: a tree.

The situation with many wormhole solutions is similar. Just because a manifold is a solution of the field equation, it does not follow that that manifold exists in nature. It would require the right initial conditions. The wormhole connected with the Schwarzschild solution (called Einstein-Rosen bridge) is an example. It is an allowed solution, just as apples can move upwards as well as downwards, but in order for this solution to be found in nature one would have to have the right initial conditions (like someone would have to throw the apple upwards). But the initial conditions require some sort of white hole-like structure. And how is that going to come about? There are many studies which show that it appears that there is no physical process, within classical physics at least, which can produce such a structure if one was not already there.

There are other solutions, for charged or rotating black holes, where the wormhole possibilities are more varied, but here too it is hard to find a classical process which could result in a wormhole, except possibly for some exotic process in the early universe which we don't know how to calculate. And whether the wormhole would be stable is a further important issue, because most wormholes are not stable.

Quantum physics is a good deal more subtle and it may allow processes which can produce wormholes. They would very likely be tiny and short-lived.

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