I'm writing a Science Fiction tale, and I am trying to get something together on the FTL physics involved in the story.

By chance I came across this paper, "A General Local Causality Principle of Space-Time", a preprint, by Colombian astrophysicist B. Calvo-Mozo.

From what I understand ;), he refers to various velocity regimes c1, c2, c3, …, where c1 is subluminal and c2, etc., are superluminal/FTL. The first FTL, c2, is 5.2E26 c; which is -- I'm sure you will agree -- truckin'.

I'd like to use this in my story, but don't understand the transition from one regime to another, i.e., c1 to c2, etc. He mentions a discrete something, whatever that means. Could anyone explain in plain English, please?

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    $\begingroup$ The gist of the paper is that, in the opinion of the author, there may exist multiple possible intervals of allowed velocities, with the generally accepted $0 \le v \lt c$ being only one of them. Transitioning from one such interval to the next up is not explained, but it must obviously involve some sort of qualitative (= "discrete") jump, because reaching the upper bound gradually from below is not possible. $\endgroup$
    – AlexP
    Nov 3, 2019 at 23:33
  • $\begingroup$ So he doesn't explain the transition/jump in any explicit manner. Thanks. $\endgroup$ Nov 3, 2019 at 23:56
  • $\begingroup$ So he's saying that constant ε^2 will aid in the jump to the next velocity regime? Getting your starship up to the needed velocity is a whole different problem. $\endgroup$ Nov 4, 2019 at 0:19
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    $\begingroup$ @catsteevens there was a paper or just an article, that described the idea of the speed of light being some sort of potential wall. As you might know of quantum tunneling, the idea was that an object could maybe "tunnel" through this potential barrier and travel with FTL speeds. I can`t find the paper, but it looks like this paper describes something like that. $\endgroup$
    – PSquall
    Nov 4, 2019 at 0:37
  • $\begingroup$ @PSquall Quantum tunneling, I've heard of that. With the assist of the constant ε^2 mentioned in the paper. Interesting $\endgroup$ Nov 4, 2019 at 1:10

2 Answers 2


As objects with mass are acted upon by force they increase their velocity, as their velocity becomes relativistic, ie begins to approach the speed of light, for every incremental gain in velocity an increasingly large force must be imparted - more and more energy.

This process goes on indefinitely, the object's mass increasing slower and slower according to the Lorentz factor:

$$\gamma = \frac {1}{\sqrt{1 - v^2 / c^2}}$$

This means that no matter what increments of energy are added, the speed of light is never achieved, just an increased fraction of the speed. Somewhat like Zeno's paradox of Achilles and the Tortoise :

Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. Supposing that each racer starts running at some constant speed, one faster than the other. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say 2 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise.

The paper seems to go about saying that to bypass the need for an infinite amount of energy to be added, resulting in a continuum of change but never reaching $c$, a discrete (ie. finite) change would need to occur pushing the object from sub-luminal speeds to super-luminal ones. The author then goes on to fudge lots of different mathematical concepts and techniques to endeavour to demonstrate a self consistent logic to model what happens. (At no point is it suggested that a way has been found to make it happen, but he does reference Alcubierre at one point to add to the credibility of the argument).

Edit to add:

I neglected to mention that one upshot of the theory is that in the process of becoming super-luminal the object takes a vector (unspecified) in a second time-like dimension. (The practical upshot of this is not discussed, but could lead to some very strange effects both in Super-L travel and when returning to sub-lightspeed).

The three spatial dimensions seem to have been preserved, in fact the inversion of the Lorentz factor at super-luminal speed would indicate that the faster you go, the more the spatial compression (maybe temporal compression too) and increase in mass that occurred as light-speed was approached from the other side would be negated.

Another possible inference is that during acceleration after passing the super-luminal barrier, the object would shed photons - starting at the highest frequency gamma end and decreasing with increased velocity. A civilization like ours would perceive a streaking flash from high to low energy photons (ie. from gamma through x-ray then blue through the visible spectrum, infra-red, then radio to background noise).

Slowing down in super-luminal velocity mode could produce the opposite effect. Exiting from super-l to ordinary space-time constraints would require the same input of infinite energy or a "discrete" shift. It's a sort of mirror effect to special relativity. Weirdness again.


Personally, I thought the paper was absolute nonsense - it introduced irrelevant variables purely for the purpose of justifying the equations (and one which is completely fantastical quite early on). The math is not consistent with math as we know it, it frankly seems like a "mock-up" by physics graduates to use as a joke on undergrads. It didn't offer anything substantial regarding a new understanding.

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    $\begingroup$ @catsteevens I've edited in something that could lead to weirdness in the story-telling if the control systems in your ship preventing it go awry. $\endgroup$ Nov 4, 2019 at 0:45
  • $\begingroup$ A second time-like dimension? I don't know what to make of that. I do now remember reading that in the paper. And negation of mass increases and spatial plus temporal compressions? Wow. Thanks $\endgroup$ Nov 4, 2019 at 0:48
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    $\begingroup$ @catsteevens No, I've no clue either, there's nothing like it in the real world that I'm aware of, it's all pretty speculative. Fuel for the imagination though. $\endgroup$ Nov 4, 2019 at 1:22
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    $\begingroup$ Maybe, it's up to you as the writer....... I can't stop thinking about this darned silly paper now. I'll make an additional edit. $\endgroup$ Nov 4, 2019 at 6:32
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    $\begingroup$ @Zwuwdz That's at the root of my annoyance with the paper, there's no real-world analogue (it's imaginary in all senses). Then again, I'm happy to have read it as it made me think. $\endgroup$ Dec 1, 2019 at 2:09

One of the first rules of writing science-fiction is to never explain more than you know. Remarks in your question indicate you're are not a scientist, or if you are you're not a physicist.

The simplest explanation, the one I can derive from a cursory examination of the paper, is that a material object might pass from the subluminal velocity regime to the next superluminal velocity regime not by changing its velocity continuously, but could do so in a jump.

In science-fiction terms, assume a spaceship accelerates to a velocity very, very close to lightspeed it will reach a point where its velocity is high enough and its velocity will change from a subluminal velocity into a superluminal one. Crudely put, the spaceship "jumps" over the lightspeed barrier. One moment it is moving at a subluminal velocity just below lightspeed and the next it is traveling at a superluminal velocity just above lightspeed. The top speed is this superluminal velocity regime is 5 x 10^26 c.

The energy requirements to do so are unthinkably colossal. But you're writing science-fiction, so you can to ignore them or not.

PS: The more scientifically minded WBers will undoubtedly want to excoriate my simplified explanation. But this is it in simple language.

  • $\begingroup$ You’re right, I’m not a scientist, I am (claim) a writer of fiction, so all I can do is glance over the preprint article and try to decipher something out of it. The different velocity regimes is new to me, and a cool concept I think. Have some starship accelerate up to some doable speed (.94c? He mentioned that) and transition/jump to the next so-called regime. Some high tech rocket/ warp engine physically employs that tiny constant ε^2 and the ship jumps into the next regime. Using a mole of electrons that he mentioned. :) $\endgroup$ Nov 3, 2019 at 23:54
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    $\begingroup$ If you're writing fiction, as you are, show what happens & how it happens. It's not your job to explain the physics, not even to the satisfaction of any physicists who read your story. Show the spaceship reaching a critical velocity where it can transition from below to above lightspeed. Any technical details you can glean from Calvo-Mozo's paper the better the plausibility will be. Even physicist-readers will be happy with that, for the rest just get on with the story. Even if the physics isn't right, it's enough for a plausible version of FTL travel. $\endgroup$
    – a4android
    Nov 4, 2019 at 0:28
  • $\begingroup$ You are right, of course. The 32 billion light years from earth to the galaxy GN-z11 in a couple nanoseconds! Thanks. $\endgroup$ Nov 4, 2019 at 0:40
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    $\begingroup$ @catsteevens You must have taken the scenic route. You miss all the interesting bits along the way when you take the short cut. $\endgroup$
    – a4android
    Nov 4, 2019 at 1:11
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    $\begingroup$ @a4android most of the universe is the equivalent of "flyover country"... $\endgroup$ Nov 4, 2019 at 12:50

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