It's well known that, in relativity, if you can go faster than light, you can go backwards in time and create a paradox.

Also, attempts to create "warp-drive" space-times in which something moves faster than light (the best known is the Alcubierre drive) usually require lots of "negative energy", something which in reality may only be available under rather special quantum-mechanical conditions (e.g. Casimir effect).

So one might suppose that the universe obeys an "energy condition" which provides a physical (and not just logical) barrier to paradox.

But lately there's a news story about an American physicist (Erik Lentz) who claims to have constructed a superluminal soliton using only positive-energy sources. The preprint was released last year and has now been published.

I have yet to find a technical analysis of the paper. The closest thing so far is a comment to a journalist by relativist Ken Olum, who proved one version of a relationship between faster-than-light travel and energy condition violation, and who thinks his theorem must apply to Lentz's soliton too.

At his blog, Lentz also mentions a recent review of warp-drive space-times that talks about slower-than-light positive-energy warp-drives, and faster-than-light warp-drives that violate energy conditions, but says they didn't analyze his own construction (which, to repeat, is meant to be a faster-than-light warp-drive that doesn't violate energy conditions).

So what's the story? What exactly is Lentz's new idea? Is there a reason why a sufficiently advanced civilization can't build a Lentz drive and change the past?

(Thanks to T.L. for bringing these works to my attention.)

  • 1
    $\begingroup$ Two big red flags on this are the unprestigious journal (a genuine solution of this form, even if totally impractical, would have huge theoretical importance and should be in Physical Review or something) and the fact that, at least in the preprint, he doesn't even mention the no-go theorems or explain how his solution evades them. A lot of claims to have created a free-energy device from magnets never get analyzed in detail for similar reasons, and that could happen to this claim. $\endgroup$
    – benrg
    Apr 27, 2021 at 18:00
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    $\begingroup$ @benrg Classical and Quantum Gravity is unprestigious? In what world? $\endgroup$
    – Kosm
    Apr 28, 2021 at 13:55
  • $\begingroup$ Another paper in support of positive energy density warp drives: Positive Energy Warp Drive from Hidden Geometric Structures. $\endgroup$
    – A.V.S.
    Apr 28, 2021 at 16:48
  • $\begingroup$ @Kosm I thought it attracted cranky papers, but maybe I'm thinking of some other journal. $\endgroup$
    – benrg
    Apr 28, 2021 at 20:57
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    $\begingroup$ @A.V.S. The authors of that paper think that any metric with $|\vec N|>c$ anywhere is superluminal, which is false (e.g. any metric where $\vec N$ is a function only of $t$ is just Minkowski space in weird coordinates). So their metric is probably just not superluminal in any physically meaningful sense. $\endgroup$
    – benrg
    Apr 29, 2021 at 1:56

2 Answers 2


This new question links a recent paper

  • J. Santiago, S. Schuster and M. Visser, 2021, “Generic warp drives violate the null energy condition”, arXiv:2105.03079.

which counts as “technical analysis” of Erik Lentz's solution as well as Bobrick & Martire's (already linked in OP) and Fell & Heisenberg's solutions (that I mentioned in a comment). According to the authors:

The key observation is that WEC requires all timelike observers to see positive energy density, whereas the analyses of references [1–3] only investigate the energy density as seen by one class of timelike observers (the co-moving Eulerian observers). Thus the claims made in references [1–3] are at best incomplete, and in many key specific details, wrong.

So, this “new idea” seems to be a non-starter.


Put shortly, there are physical subliminal warp drive solutions in the literature, while superluminal solutions are/have been known earlier to be problematic.

There were the following recent inputs on the topic:

-E. Lentz, in his 2021 paper, suggested a superluminal positive-energy-sourced solution satisfying WEC (https://arxiv.org/abs/2006.07125). However, in the refereed version of the paper (https://iopscience.iop.org/article/10.1088/1361-6382/abe692), he also mentions explicitly that such solutions violate DEC when superluminal, which means that they are problematic.

-Bobrick and Martire, 2021, proposed a general definition of warp drives and constructed subluminal positive-energy-sourced warp drive solutions (https://arxiv.org/abs/2102.06824, https://iopscience.iop.org/article/10.1088/1361-6382/abdf6e); these can satisfy all energy conditions. They also argued that superluminal solutions should violate DEC and that accelerating past the speed of light leads to pathologies such as infinitely-long warp bubbles from the comoving observer point of view.

-Fell and Heisenberg, 2021 (https://arxiv.org/abs/2104.06488) improved Lentz's work, made a deeper analysis and formulated two positive-energy superluminal solutions in a more clear analytic form compared to Lentz. I believe they mentioned that their solution violated SEC but not DEC.

-Santiago, Schuster and Visser, 2021 (https://arxiv.org/abs/2105.03079) showed that superluminal warp drive solutions violate the NEC and are not positive-energy sourced for all observers. This argument applies to superluminal solutions of Lentz, and Fell/Heisenberg, and more general Natario-like warp drives with a unit lapse function. The argument does not apply to the more general warp drives with the non-unit lapse function discussed in Bobrick/Martire (and some earlier works) and subluminal warp drive solutions.

There is a lot of relevant past literature on these things. I would recommend reading the recent review by Alcubierre & Lobo from 2017 on the topic (https://arxiv.org/abs/2103.05610) or the introduction sections from the above four 2021 papers. All four papers are valuable contributions, and there will surely be more, as the discussion is obviously ongoing.

P.S. DEC - dominant energy condition

NEC - null energy condition

SEC - strong energy condition

WEC - weak energy condition

A good summary of these conditions and relations between them may be found in Santiago et al. 2021 (https://arxiv.org/abs/2105.03079)


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