Friends, it is a well-known fact that solutions to general relativity which describe traversable wormholes require the existence of exotic matter, i.e., matter which violates the null energy condition (NEC). The latter states that for any null vector k^a, the energy-momentum tensor Tab satisfies Tab k^a k^b ≥ 0. The violation of the NEC is a consequence of the “flaring-out condition” which follows from the existence of a wormhole throat where its radius is minimal, and where, Gab k^a k^b < 0 for some null vector k^a. The Einstein equations Gab = 8πGTab then imply that also Tab k^a k^b < 0. This was first shown for static, spherically symmetric wormholes (non-rotating) and later generalized to non-static, non-symmetric wormholes (rotating). However, this result crucially depends on the gravitational field equations, and indeed in modified gravity theories the requirement of exotic matter can be minimized or even completely avoided. In these theories the NEC, being a condition on the matter energy-momentum tensor, does not translate to a geometric constraint in the same way as in general relativity, and instead leads to possibly weaker conditions on the allowed geometries.
Manuel Hohmann in his paper, aimed to show that wormhole solutions without exotic matter also exist in multimetric gravity theories (recently modified gravity theory).
“In our paper we studied wormholes in the context of a recently discussed class of multimetric gravity theories containing N ≥ 2 metric tensors and a corresponding number of standard model copies ϕ.”— wrote authors of the study.
He presented a static, spherically symmetric, traversable wormhole solution to multimetric gravity which is sustained by only non-exotic matter, i.e., matter which satisfies all energy conditions. The possibility of this solution arises from the fact that under certain conditions the multimetric gravitational field equations reduce to the Einstein equations, but with a negative effective gravitational constant. They also calculated Arnowitt-Deser-Misner mass of their wormhole solution and showed that the Arnowitt-Deser-Misner mass of this wormhole vanishes, so that it appears massless to observers in the asymptotically flat spacetime.
“Given a wormhole solution which is sustained by only non-exotic matter, one may speculate on the possibility to create and maintain such a wormhole. Unfortunately already a simple thought experiment shows that the construction of a multimetric wormhole is beyond any practical feasibility, and unlikely to happen naturally.“— told Hohmann, lead author of the study
The main problem arises from the fact that in order to work with the simplified field equations, they needed to construct a solution in which all metrics, and therefore the energy-momentum tensors for all standard model copies (refer note below for details), are equal. They further needed to assumed that different standard model copies repel each other, so that the effective gravitational constant in the field equations, Gab = 8π × Geff × Tab, becomes negative. It thus follows that the assembly of a wormhole sustained by equal amounts of all N matter types is obstructed by their mutual repulsion, and thus unlikely to occur naturally.
Also for a civilization constituted by one of the standard model copies the construction of a multimetric wormhole is practically impossible, since the only way to interact with and assemble the other N − 1 matter types, which appear dark to this civilization, is by means of gravity. It would rather need the cooperation of N civilizations, each constituted by a different matter type, coordinating their efforts using gravitational communication, to effectively manipulate the sustaining matter of the wormhole.
Finally, even if these civilizations should manage to assemble a wormhole, it is very likely to be unstable against fluctuations, since any shift of the different matter types with respect to each other would cause them to further separate and disrupt the wormhole due to their mutual repulsion.
They concluded that even though non-exotic wormhole solutions to multimetric gravity theories exist, their construction and maintenance is far from being feasible even by the joint effort of several advanced civilizations. Similar considerations may apply to other spacetime geometries which require exotic matter in general relativity, such as warp drives: even if one finds a warp drive spacetime which does not require exotic matter in multimetric gravity, it will probably suffer from the same drawbacks which obstruct its practical realization.
Note: Standard model copies is the solution to the hierarchy problem, according to which the universe contains multiple sectors, each of which is governed by its own version of the Standard Model with its own Higgs vacuum expectation value.
Reference: Manuel Hohmann, “Traversable wormholes without exotic matter in multimetric repulsive gravity”, Phys. Rev. D 89, 087503 – Published 9 April 2014. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.89.087503
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