From what I understand there is no physical evidence for wormholes, but are their certain solutions to the einstein field equations that allow for them?
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Yes, there's a long history of them, beginning with the original Einstein-Rosen bridge and the maximally extended Schwarzschild metric (white hole / black hole). Problems with this original "wormhole" is that not only does not fit physical reality in that, black holes do not have an infinite existence into the past as a white hole (they emerge from collapsing stars), but trying to traverse such a "wormhole" is impossible since it pinches off before even light can traverse it.
Jump ahead several decades where the first traversable wormhole metric is actually constructed by Ellis (1973). The only way his wormholes are traversable (i.e., don't pinch off) is he proposed a scalar field that exhibits the properties of a phantom field (i.e., a negative kinetic term), which is usually seen as unphysical, except in a few out-there dark energy models.
Now jump to the late 1980s where the more canonical traversable wormhole by Morris and Thorne, with corrections by Matt Visser (1989). These wormholes stay open by requiring negative energy densities. How much? On par with a Jupiter mass for a meter radius wormhole ($-1.898 \times 10^{27}\mathrm{kg}$). Such astronomical energy densities remain about the same to this day (see Krasnikov 2003).
So yes, one can find wormhole "solutions" by running through the Einstein Field Equations. However, every such solution ends up requiring vast amounts of negative energy, which probably doesn't exist, so these solutions are pedagogical: fun ways to learn General Relativity.
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$\begingroup$ You actually run the Einstein field equations forward. The EFEs put the stress-energy tensor in terms of the metric, not the other way around. Determining the exact metric created by a given stress-energy is actually impossible because many different metrics correspond to the same stress-energy - Minkowski (flat) space and Schwarzschild and Kerr black holes are all vacuum solutions with $T^{\alpha\beta}=0$. The EFEs don't tell you spacetime geometry from stress-energy, they tell you stress-energy from spacetime geometry. $\endgroup$ Commented Aug 3 at 17:44
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1$\begingroup$ @controlgroup Thanks for pointing that out. I removed the offending phrases. $\endgroup$– HokonCommented Aug 3 at 17:47
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There are solutions to the Einstein Field equations that describe a wormhole (i.e. a black hole connected to a white hole). But there is no evidence that these objects actually exist.
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$\begingroup$ (+1) Could you describe how the geodesic path looks like when traveling through a wormhole solution in EFE, please? Jumping through a wormhole is defined as "teleporting" from one point in this 4D volume and suddenly appearing at another point in the same 4D volume, correct? But it seems to me that the geodesic equation implies velocity and position continuity, so could you elaborate how would "jumping through a worm hole" look as a continuous path on geodesics, please? $\endgroup$– JamesCommented Aug 3 at 23:04
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$\begingroup$ Huh? @James what exactly are you asking for? $\endgroup$– MikeCommented Aug 9 at 16:03
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$\begingroup$ @Mike I meant, on the ER-bridge metric, for instance, two black holes are baked into the metric, and then the topology of space is "folded" so that one hole connects to the other hole, correct? This requires an object at coordinate, e.g. (1,3,5,7) to suddenly connect to another location e.g. (24,26,28,30). But test objects always move following the geodesic equation, right? Then, there must be a continuous velocity/position path $ds$ of the test object described by the geodesic equation describing this sudden movement from (1,3,5,7) to a far off point (24,26,28,30). What is this description? $\endgroup$– JamesCommented Aug 9 at 23:27
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$\begingroup$ @Mike or I suppose one may be able to just define explicitly which points are neighbors to which other points... On a Pacman screen, one could define that the left of coordinate [0,0] is [100,0], and that is it, all the mechanics should work as before... The geodesic equation that prescribes moving forward from (1,3,5,7) will simply write the new location as (24,26,28,30) according to some table of connected points we feed in. $\endgroup$– JamesCommented Aug 10 at 0:25
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$\begingroup$ @James well I'm not sure of the answer is this case. $\endgroup$– MikeCommented Aug 15 at 18:03