There are now positive energy soliton solutions to the warp drive[1] Still not exactly practical, but one less fundamental problem to solve. [1] https://arxiv.org/abs/2201.00652
Hyper-Fast Positive Energy Warp Drives
Erik W. Lentz
2021 · arXiv
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From the abstract
Solitons in space-time capable of transporting time-like observers at superluminal speeds have long been tied to violations of the weak, strong, and dominant energy conditions of general relativity. This trend was recently broken by a new approach that identified soliton solutions capable of superluminal travel while being sourced by purely positive energy densities. This is the first example of hyper-fast solitons satisfying the weak energy condition, reopening the discussion of superluminal mechanisms rooted in conventional physics. This article summarizes the recent finding and its context in the literature. Remaining challenges to autonomous superluminal travel, such as the dominant energy condition, horizons, and the identification of a creation mechanism are also discussed.
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More recently we have the work of Erik Lentz to make these viable with positive energy (a bunch of it AFAIR) using Solitons[0]. The funny thing is there's actually a TNG episode, New Ground, where the crew test a Soliton wave for warp propulsion. [0]: https://arxiv.org/abs/2201.00652
This abstract says only that they've reduced the need for exotic matter by some orders of magnitude. Presumably it's still required. However, there were some recent papers describing warp drive geometries with only positive mass. Ex: https://arxiv.org/abs/2201.00652
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