Comments on Superluminal Travel

A popular online personality – let’s call her Betty – has frequently posted her reasons for believing that superluminal travel is “entirely compatible” with “Einstein’s theories” of special and/or general relativity. I think her reasons are not valid. Here’s a summary of those “reasons”, along with my comments.

Where the Idea Comes From

Betty’s defense of her thesis that faster-than-light travel is “entirely compatible” with special relativity begins with the following comment:

The idea that the speed of light is a limit comes from special relativity. If you want to accelerate an object until it reaches the speed of light, you need an infinite amount of energy. This is where the idea comes from that the speed of light is a limit that you can’t cross.  

This is not a complete description of why, according to special relativity, speeds in excess of the speed of light (i.e., the speed of massless energy in vacuum) have no possibility of existence. For example, Einstein first mentioned the impossibility of superluminal speeds in his 1905 EMB paper in his discussion of length contraction, noting that the ubiquitous factor √(1 – (v/c)²) is imaginary for any v > c. This factor also appears in the expression for the proper time along an interval, and it goes to zero at v = c. For any v > c the proper time along an interval would be imaginary, not commensurate with physical proper time. Hence Einstein remarked

For superluminal velocities our considerations become meaningless; we shall see in the considerations that follow that in our theory the velocity of light physically plays the part of infinitely great velocities.

This by itself suffices to disprove Betty’s thesis. The problem is not just related to the work required to accelerate an object, nor that some quantities are singular at v = c, the problem is that the physical quantities such as elapsed proper time become imaginary for all v > c.  Tellingly, when Betty was asked what elapsed proper time would show on the wrist watch of a purported superluminal traveler along a given space-like interval, she was unable to answer, and professed to not even understand the question.

Limits and Barriers

Later in Einstein’s 1905 paper, when the expression for kinetic energy of a particle of rest mass m is given, again involving the above factor, he notes that for any positive m the kinetic energy increases without limit as v approaches c, and he remarks that “As in our previous results, superluminary velocities have no possibility of existence”. Again, the problem is not just at v = c, the problem is with any v > c.  Obviously the energy comprising a given rest mass, if unbound, can (indeed, must) propagate at v = c, but this is entirely consistent with special relativity. One could even say it is the very foundation of special relativity. In contrast, the premise that energy (bound or unbound) can propagate at v > c is incompatible with special relativity, as shown by the fact that every single relativistic relation, not just the expression for kinetic energy, involves imaginary quantities in that condition. Nevertheless, Betty presses on with further specious reasoning:

But this doesn’t mean faster than light travel is forbidden in special relativity.  [It just means] you can’t accelerate from below the speed of light to above the speed of light. It’s more like a barrier than a limit.

Again, that’s clearly not correct. The equations of special relativity are unphysical for any v > c, as noted above. Yes, for an object with positive rest mass m, the work required to accelerate it increases without limit as v approaches c (which is daunting enough), and the expression is singular at v = c, but for any v > c the factor discussed above is imaginary, and hence the kinetic energy would be complex, and the elapsed proper time and contracted spatial length, etc., would be imaginary, all of which are unphysical, i.e., incommensurate with the respective physical quantities.

Comical Interlude

Using Betty’s cavalier disregard for the distinction between real and imaginary quantities, we could imagine a comically hapless character in a science fiction version of an O. Henry story.  He first thinks of traveling to a star 25 light years away at nearly the speed of light, so his elapsed round-trip time would be essentially zero, but then realizes that his loved ones on earth would all have aged 50 years.  So, he decides to use one of Betty’s faster-than-light ships that magically conflate imaginary time with real time in the ship. He then makes the journey in essentially zero time for his loved ones, but discovers to his dismay that his elapsed time is 50 years.

An Artifact of Infinity

Undeterred, Betty presses on, suggesting that the relativistic relation between velocity and kinetic energy for a positive mass doesn’t mean that faster than light travel is incompatible with special relativity, but her reasoning is self-evidently invalid and self-defeating.  She says:

On all other occasions when physicists see some quantity go to infinity, they’ll tell you that infinity is unphysical and a sign that the maths doesn’t properly work.  They’ll say it’s a mathematical artefact and not real. They don’t say that in this case, but I think they should.

This is specious, because it amounts to arguing that special relativity (local Lorentz invariance) breaks down at sufficiently high energies, whereas her thesis is that superluminal travel is entirely consistent with special relativity.  Her reasoning points to exactly the opposite of what she claims, i.e., she is asserting here that special relativity is not compatible with superluminal travel, because it would need to break down (be violated) at some energy level in order for superluminal travel to be possible. This contradicts rather than supports Betty’s thesis, and commits her to the denial of local Lorentz invariance, which of course is a corner stone of all modern physics, from special and general relativity to quantum field theory.  It’s basically saying “If special relativity is wrong, then maybe superluminal travel is possible”.  But this is not the thesis she has proposed, which is that “superluminal travel is totally compatible with special relativity”.

Her argument is fallacious for other reasons as well. Notably, special relativity does not posit or entail the existence of any infinite energy, so this is not a case in which “physicists see some quantity go to infinity” (such as, say, at the singularity at the center of a black hole). To the contrary, the fact that the kinetic energy cannot go to infinity is understood to signify that the positive rest mass cannot be moving at speed c in terms of any standard system of inertial coordinates.  It is, instead, Betty herself who is positing that such a mass can be accelerated from below to above c (into the imaginary range), which would entail an actual infinite kinetic energy if done continuously, which she then argues is probably unphysical. Thus she is refuting her own hypothesis.

Incidentally, we refer to local Lorentz invariance to be strictly accurate, but this doesn’t appreciably limit the applicability, because it covers all phenomena for which the relevant metrical relations are sufficiently flat, which applies (for example) to motions between stars, and even between nearby galaxies. The fact that stars like our Sun have slightly curved metrical relations in their vicinity doesn’t significantly alter the essentially flat background for motions between them. None of Betty’s arguments for superluminal travel involve spacetime curvature, and she is not referring to “worm holes” or any similar science fiction ideas. (If she was, they could easily be dispensed with as well, but she isn’t.)

Unbound Energy

Betty then continues with a somewhat odd line of attack.  She says:

We have a counterexample to the claim that one needs an infinite amount of energy to reach the speed of light…   in the early universe none of the particles had masses [and] they were all moving with the speed of light. Later they were not. The energy that was released in this phase transition was finite… but the equation that we looked at earlier said that the difference in energy should have been infinite.

This is just an elementary misunderstanding (facilitated by a misrepresentation of what the kinetic energy formula “claims”), and we don’t need an elaborate digression back to the early universe and the Higgs mechanism, etc., to explain it.  Simply consider a box with mirrors lining the interior, and a photon bouncing around inside. That photon of energy E contributes m = E/c² to the rest mass of the box, and if we open a tiny window and let the photon fly out and propagate away at the speed of light, we have thereby “accelerated the mass m to the speed of light” with negligible work. Ha! Einstein’s equation is wrong! Well, not so fast. It is well known that aggregate rest mass generally consists of bound massless energy, and some or all of the energy can be unbound, in which case it propagates at speed c.  Rest mass is not conserved.  Energy is conserved, and massless energy propagates at the speed of light but does not (and cannot) propagate superluminally. Bound energy (i.e., energy with positive rest mass) cannot even propagate luminally, let alone superluminally.

If Betty’s point is that the energy comprising her body can travel at the speed of light by unbinding all that energy (including turning off the Higgs mechanism and letting the zig and zag parts of the Dirac fermions split apart), and then transmitting this energy at the speed of light to a distant star, well, sure. But this is just luminal, not superluminal, propagation. Nothing in modern science prohibits luminal propagation of energy. Also, rather than exploding every sub-atomic particle in her body so that the energy that used to be Betty could propagate at the speed of light, Betty might want to consider just scanning her body and transmitting the information luminally.  Again, none of this supports a belief in superluminal signaling, let alone travel, nor does it challenge the correctness of the relativistic expression for the kinetic energy of a given positive rest mass, nor the fact that the work required to accelerate a given rest mass to speed v increases without limit as v approaches c.

An Admission and Non Sequiturs

Oddly enough, Betty goes on to acknowledge that the discussion about rest mass consisting of bound energy really does nothing to support her thesis.  She says:

Even if we could, it’d basically evaporate the traveler, so this isn’t going to let us build a warp drive. But it shows that the argument that the speed of light is a barrier isn’t even technically correct.

Unfortunately, even in this acknowledgement, there are a couple of problems. First, this has nothing to do with a “warp drive”. As mentioned above, her discussion was purportedly showing that superluminal travel is compatible with special relativity, which is in the context of flat spacetime, so there are no “warp drives” here (not to mention that “warp drives” make no sense even in the context of curved spacetime).  Nothing she has said had anything to do with warping spacetime. Second, she claims that her debunked reasoning “shows that the argument that the speed of light is a barrier isn’t even technically correct”. As we’ve seen, that is simply not true. The fact that bound energy can become unbound does not show that anything in special relativity is technically incorrect. To the contrary, everything we’ve said is elementary special relativity, including the fact that, as Einstein said, superluminal speeds have no possibility of existence.

Time Travel Paradoxes

Betty then confronts the fact that, in the context of special relativity, any possibility of superluminal signaling (let alone travel) implies logical paradoxes and inconsistencies, e.g., we could arrange for a signal to be sent if and only if it is not sent. This was first pointed out by Einstein in his 1907 survey article, and further clarified in later writings. The reasoning is irrefutable. Unfortunately, Betty deploys some fundamental misunderstandings of various areas of physics, including thermodynamics, as well as elementary misunderstanding of special relativity, to present a completely fallacious pseudo-refutation of the plain facts… only to then confess that her reasoning is wrong.  After extensive prior discussions, Betty was convinced that special relativity does indeed imply the time loop paradoxes that result in logical contradictions, so she belatedly shifts her ground, and says

…the time-travel argument is correct in special relativity, [but] it is not correct in general relativity.

The first clause represents tremendous progress, since Betty now acknowledges that her original thesis, which was (and strangely still is, in most of her presentations) that superluminal travel is entirely compatible with special relativity, was wrong. After rehearsing her debunked reasoning one more time (for old time’s sake?), she admits that her critics were correct after all, and special relativity does indeed imply closed timelike loops if superluminal travel were possible. Excellent. However, the concluding clause is senseless, because local Lorentz invariance is one of the corner stones of general relativity. If, as she now admits, superluminal travel is incompatible with the local Lorentz invariance of special relativity, it is also incompatible with general relativity. Closed timelike curves are no more innocuous in general relativity than they are in special relativity.

A Preferred Frame, Denying Local Lorentz Invariance (Again)

At this point, Betty totally abandons any pretense of consistency with local Lorentz invariance, and argues for a return to the distinguished absolute rest frame of the 19^th century aetherists as follows:

…you can define absolute rest to be motion that has no relative velocity to the average of all that stuff… it’s called the “co-moving frame”.  It’s the reference frame that moves along with matter in the universe.

Here Betty is mixing cosmology and physics terminology, but in essence she is struggling to articulate the view that we can define the local “absolute rest” frame as the frame in terms of which the CMBR is maximally isotropic (which matches pretty well with the frame in terms of which the distant galaxies are isotropic), and we can assert that local physics takes a special form in terms of these coordinates. Specifically, in a laboratory at rest in the preferred frame, she asserts that we can experimentally transmit superluminal signals isotropically, but in a different inertial laboratory moving relative to the isotropic CMBR frame, we cannot. Therefore, by local experiments inside a sealed laboratory, we can (according to Betty) determine our absolute velocity, merely from the different form that the laws of physics take, depending on our absolute state of motion.

Needless to say, this is a flagrant denial of local Lorentz invariance, and hence incompatible with both special and general relativity. Betty has (again) refuted her own hypothesis. Of course, she could dispense with the charade of pretending that superluminal travel is compatible with relativity, and simply assert that relativity is wrong, i.e., that LLI fails in some way. But one can understand Betty’s reluctance to do this, because, as noted previously, LLI is the corner stone of modern physics, including not only special and general relativity, but also quantum field theory. Of course, none of this is to say that general relativity or QFT are correct – that’s an empirical question. But Betty’s thesis is not an empirical one, it is the claim that superluminal travel is compatible with special and general relativity, which it clearly is not.

The Know-Nothing Argument

After first jettisoning special relativity, Betty seems to begin to recognize that she must jettison general relativity as well.

…we know our current theory of space-time, General Relativity, can’t be correct because it doesn’t work together with quantum theory. We [also] know that causality and locality become really screwed up in quantum mechanics, and the same is probably the case in quantum gravity. This is why I think it’s extremely implausible that any argument about faster-than-light travel would survive in the to-be-found theory of quantum gravity.

This is the standard anti-scientific sophistry that “until we know everything we don’t know anything”, and of course it does nothing to support the thesis of compatibility. To the contrary, it is an acknowledgement that the thesis is false. Superluminal travel (or even signaling) is not compatible with local Lorentz invariance. If Betty wants to suggest that the unknown laws of some future theory of physics are not locally Lorentz invariant, then she is free to do so, but this refutes rather than supports her thesis. Also, her remark that “causality and locality become really screwed up in quantum mechanics” is just further refuting her compatibility thesis, and it also not particularly impressive, especially because even quantum entanglement is known to fully respect local Lorentz invariance. And, again, this is all a bait-and-switch argument, first claiming that superluminal travel is compatible with special and general relativity, only to admit that it is not, and then switching to the know-nothing claim that superluminal travel might be compatible with some presently unknown future theory, that violates special and general relativity – without providing even a hint of a reason for expecting such a fundamental principle of modern science to be violated. (Incidentally, the violation would have to be of a magnitude that is already abundantly falsified by observation, but that’s an empirical issue, separate from the theoretical compatibility thesis.)

The Real Reason?

Since all of Betty’s reasons are utterly specious, we may be puzzled by what motivates her steadfast belief in superluminal travel. The answer may possibly be found in her introduction, where she says:

I believe there’s intelligent life on other planets, and the most plausible reason why they haven’t contacted us is that we’re too boring. I mean, we haven’t even figured out how to send information faster than light. Pathetic…  If we want to properly evaluate how likely that is, we need to talk about the possibility of travelling faster than light, or at least sending information faster than light. Because if it’s possible at all, then that’s what the aliens are doing.

This might be intended as just a humorous introduction, not to be taken seriously, but it’s the closest thing to an actual reason in Betty’s presentations. Of course, it isn’t a technical reason at all, it’s more like Bayesian meta-reasoning. The argument is that the “most plausible” answer to Fermi’s question “Where is everybody?” is that our failure to figure out how to travel (or at least signal) superluminally somehow makes us repellant (too boring) to all the life forms zooming around the galaxy. (Is this really the most plausible reason?) In summary, the argument is:  We have not been contacted by aliens, ergo superluminal travel may be compatible with local Lorentz invariance – even though it self-evidently is not.

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