Can a particle have a space-like four-velocity? If not why not?

[andrewkirk]

I have been mulling over the idea of the oft-quoted 'prohibition' on particles traveling 'faster than light'. While this is often mentioned in the media and even in physics books, it rarely explains exactly what is meant by that, or precisely what laws of nature form the basis of the 'prohibition'. The media usually say that it is prohibited by 'relativity', but that is vague and not at all helpful.

An example of a type of 'prohibition' is SR's rule for velocity composition: s = (u+c)/(1+uv/c²), where u is the velocity of observer A relative to observer B, v is the velocity of an object X relative to observer A and s is X's velocity relative to observer B. In this case, provided u,v<c we will also have s<c.

After a fair bit of reflection I have reached a tentative idea that perhaps what 'relativity' says is that no particle can have a spacelike four velocity (ie g(v,v)>0). I like that characterisation of the 'prohibition' because it is coordinate-independent and seems to be consistent with what I understand about physics. It also avoids conflict with observations such as the superluminal recession of distant galaxies. But I have no idea whether that 'law', which I just made up and have never seen written down, is really what GR (or SR) says.

So my questions are:

1. When we say nothing can travel faster than light, are we really saying that no particle can have a space-like four-velocity? If not that, then what is the best characterisation of this 'law'.

2. How does this rule follow from the postulates of GR?

3. What about the additional rule that no massive particle can have a light-like four-velocity? (g(v,v)=0). Does that also follow from the postulates of GR? How?

 

[PeterDonis]

You might want to check out this entry in the Usenet Physics FAQ:

http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/tachyons.html

It's really talking about SR, not GR, but on this issue I think it's sufficient to consider it in the framework of SR; the only thing GR adds is curvature of spacetime, but that just means that "spacelike 4-velocity" translates to "4-velocity lying outside the local light-cones".

My personal take on it is this: the real prohibition, if you want to call it that, is on spacelike separated events being causally connected. The reason for that is that our concept of "causal connection" requires causally connected events to be ordered: the "cause" has to come before the "effect". But if a pair of events is spacelike separated, their ordering is frame-dependent; so spacelike separated events can't be causally connected. Since we assume that any pair of events on an object's worldline are causally connected, the above implies that an object's worldline cannot have a spacelike tangent vector ("4-velocity") at any event.

It's perhaps worth noting that the quantum field theory version of "spacelike separated events can't be causally connected" is "field operators must commute at spacelike separations", which is subtly different. The prohibition then arises from the assumption that causally connected events can't commute, because, as above, the "cause" has to come before the "effect"; if the ordering of the events is reversed, what happens at the events changes, meaning they don't commute. But it could be possible, in principle, that our concept of "causal connection" is too limited, that there can be causally connected events which do commute; we haven't been able to conceive of such a thing yet, but that may be a limitation of our minds, not reality. However, that doesn't really change anything in a practical sense.

 

[PeterDonis]

3. What about the additional rule that no massive particle can have a light-like four-velocity? (g(v,v)=0). Does that also follow from the postulates of GR? How?

This is really a separate question from the above, and has a much easier answer. The "rest mass" of an object is just the length of its 4-momentum; a lightlike object has a 4-momentum of length zero, by definition, therefore it has zero rest mass.

In other words, what you're thinking is something that must follow from the postulates of SR/GR is really a definition, a way of partitioning "objects" into two categories that have obvious physical differences in how they work: "timelike" objects (with 4-momentum of nonzero length) and "lightlike" objects (with 4-momentum of zero length).

 

[PAllen]

This is really a separate question from the above, and has a much easier answer. The "rest mass" of an object is just the length of its 4-momentum; a lightlike object has a 4-momentum of length zero, by definition, therefore it has zero rest mass.

In other words, what you're thinking is something that must follow from the postulates of SR/GR is really a definition, a way of partitioning "objects" into two categories that have obvious physical differences in how they work: "timelike" objects (with 4-momentum of nonzero length) and "lightlike" objects (with 4-momentum of zero length).

Maybe there is a little more to it. The underlying question is why a world line of a massive particle can't change from being timelike to lightlike, over its history. The conclusion is that to do so it must give up all of its rest mass. Without any quantum restrictions, this implies a 'rocket' must give up all its mass to reach light speed.

 

[Naty1]

...precisely what laws of nature form the basis of the 'prohibition'...

That's a good question...one I wondered about as I went thru 'relativity'...My short answer: Observational evidence shows everybody measures light [locally] at speed 'c'. One good argument [I think] is causality, which PeterDonis described. But I think the above short answer is even more fundamental.

From Einstein:

I happen to have his little book [for the public, I think] RELATIVITY, The special and the general theory, a 1952 edition. [I found it at a garage sale for $3 while spending a boating summer in Maine around 2007...! It was literally in somebody's garage. ]

In the book, early on, Einstein points out that 'we will find the classical addition of velocities, V = u +v, cannot be maintained'. On the next page he says the Dutch astronomer De Sitter 'was able to show the velocity of the propagation of light cannot depend on the velocity of motion of the body emitting the light' [based on observational evidence]. And states on the subsequent page 'we know with great exactness this velocity is the same for all colors" again based on astronomical observations [but he does not say who].

[From here he goes on to derive how space and time are not constant because the speed of light is constant.]

That's all you really need to conclude nothing can go faster than light...because no matter what you do, no matter how fast you go, no matter how fast a light source goes, light zips past you at velocity 'c'.

It's a very persistent constant: You can't get it to 'behave' as might have been expected before the relativity of space and time was exposed.

A way to avoid this 'prohibition' is the 'ether'... and that has been found to be an unnecessary complication.

The other 'law', if you want to use that term that is typically given, is that to get any mass accelerated to 'c' requires 'infinite' energy...due to the behavior of the gamma factor. I find this explanation less satisfactory because it derives from the constancy of the speed of light.

 

[PeterDonis]

The underlying question is why a world line of a massive particle can't change from being timelike to lightlike, over its history. The conclusion is that to do so it must give up all of its rest mass. Without any quantum restrictions, this implies a 'rocket' must give up all its mass to reach light speed.

Good point, we do know of processes like matter-antimatter annihilation which take something with a timelike worldline and convert it to something with a lightlike worldline. But as you say, that requires changing something with nonzero rest mass into something else with zero rest mass.

Another point: if you look at the system as a whole, as opposed to individual objects, the total rest mass (or "invariant mass" may be a better term in this connection) does not change. Say, for example, we have an electron and a positron that annihilate to create two photons. The combined momentum of the photons in the center of mass frame is zero (as the combined momentum of the electron and positron was), but the combined energy is not; so in the CoM frame, the system as a whole has invariant mass equal to the total energy. Since invariant mass is, well, invariant, that means the system as a whole has nonzero invariant mass. The individual photons have zero invariant mass each, so this shows that invariant mass is not additive, which is another important thing to keep in mind.

 

[PeterDonis]

That's all you really need to conclude nothing can go faster than light...because no matter what you do, no matter how fast you go, no matter how fast a light source goes, light zips past you at velocity 'c'.

This is true for a "continuous" process, but what about processes like matter-antimatter annihilation? Einstein's logic doesn't really include such cases; he assumes that to reach c, an object must be accelerated to it, but we know there are ways to create photons, moving at c, from matter moving slower than c without accelerating the matter. In principle, one could also imagine such processes that would "instantaneously" convert ordinary matter to tachyons, traveling faster than light, without any intervening "acceleration". We need something additional, like causality, to rule out such possibilities.

 

[Saw]

My personal take on it is this: the real prohibition, if you want to call it that, is on spacelike separated events being causally connected. The reason for that is that our concept of "causal connection" requires causally connected events to be ordered: the "cause" has to come before the "effect". But if a pair of events is spacelike separated, their ordering is frame-dependent; so spacelike separated events can't be causally connected. Since we assume that any pair of events on an object's worldline are causally connected, the above implies that an object's worldline cannot have a spacelike tangent vector ("4-velocity") at any event.

We discussed this in another thread and reached no mutual understanding. I do not think we would be able to make any progress in that discussion. But just for the benefit of the original poster, with all respect for your higher knowledge and warning I am much less knowledgeable, I would like to note there is another possible conceptual approach. Yes, the order of two spacelike events is frame-dependent. Yes, if two events of that kind turned out, after all, to be causally connected, we would have a problem. Then you infer that the problem should not arise. I think there is a logical leap therein. I would rather say that, in principle, the problem may perfectly arise, unless we find out some further reason that fills the said logical gap. In the absence of such reason, there is no problem for admitting the existence of a faster-than-light agent. Said this, I do believe there is a reason filling the gap: in order to accelerate massive things, you need a force whose mediating particle is a photon or (due to the basic analogy between all forces that may permit one day their unification) another particle traveling at the same velocity; how can the accelerated object exceed in speed the accelerating agent? The only way out of this conundrum would be the discovery of an accelerating agent, a force-mediating particle whose nature/mechanism is radically different from all those observed so far, a non-unifiable force that could travel faster-than-light, something that is quite unlikely. (Like by the way the recent neutrinos story has taught.)

 

[PeterDonis]

Yes, if two events of that kind turned out, after all, to be causally connected, we would have a problem. Then you infer that the problem should not arise.

No, I say that nobody has a theory which solves the problem; therefore, we have no way of analyzing what would happen if the problem arises. If you have a theory that solves the problem, I'm all ears.

I think there is a logical leap therein. I would rather say that, in principle, the problem may perfectly arise, unless we find out some further reason that fills the said logical gap. In the absence of such reason, there is no problem for admitting the existence of a faster-than-light agent.

Then please show us your theory that consistently incorporates such an agent with causality. Please give details. You couldn't in the other thread, which is why it left things where it did.

Please note that I am not saying, and never said, that FTL travel is logically inconsistent. I only said that nobody knows how to consistently deal with both FTL travel and causality in a single theory. Either you do know how, or you don't. If you do, as I said, please give us your theory. If you don't, then we're basically in agreement; the only difference between us is choice of words.

Said this, I do believe there is a reason filling the gap: in order to accelerate massive things, you need a force whose mediating particle is a photon or (due to the basic analogy between all forces that may permit one day their unification) another particle traveling at the same velocity; how can the accelerated object exceed in speed the accelerating agent?

If this were true, rockets would not be able to exceed their exhaust velocity. They can. For example, see this Wiki page:

http://en.wikipedia.org/wiki/Specific_impulse

The exhaust velocity of current liquid fuel rocket engines is given as 4400 m/s. Such rockets propel the Space Shuttle, for example, into low Earth orbit at 8,000 m/s.

 

[Fredrik]

The underlying question is why a world line of a massive particle can't change from being timelike to lightlike, over its history. The conclusion is that to do so it must give up all of its rest mass. Without any quantum restrictions, this implies a 'rocket' must give up all its mass to reach light speed.

On the other hand, in the context of QM, the (rest) mass is one of the numbers that identifies a particle species. So if a "particle" goes from p²>0 to p²<0 or the other way round, it's (by definition) not the same particle; we're dealing with the annihilation of one particle, and the creation of another.

 

[Saw]

Please note that I am not saying, and never said, that FTL travel is logically inconsistent. I only said that nobody knows how to consistently deal with both FTL travel and causality in a single theory. Either you do know how, or you don't. If you do, as I said, please give us your theory. If you don't, then we're basically in agreement; the only difference between us is choice of words.

If so, I tend to agree. But to quote just your recent words, what do the following sentences mean, if not what I understood?

the real prohibition, if you want to call it that, is on spacelike separated events being causally connected. The reason for that is that our concept of "causal connection" requires causally connected events to be ordered: the "cause" has to come before the "effect". But if a pair of events is spacelike separated, their ordering is frame-dependent; so spacelike separated events can't be causally connected. Since we assume that any pair of events on an object's worldline are causally connected, the above implies that an object's worldline cannot have a spacelike tangent vector ("4-velocity") at any event.

I had parsed it as follows:

“FTL travel is impossible…” = “spacelike separated events can't be causally connected…”

“for a reason of logical consistency” = “(a) the ordering of spacelike events is frame-dependent; (b) causally connected means cause precedes effect; hence (a) + (b) means that there are different versions, all of them equally valid, about what is the cause and what is the effect; which is logically inconsistent”

Sorry if I misunderstood.

If this were true, rockets would not be able to exceed their exhaust velocity. They can. For example, see this Wiki page:
http://en.wikipedia.org/wiki/Specific_impulse

The exhaust velocity of current liquid fuel rocket engines is given as 4400 m/s. Such rockets propel the Space Shuttle, for example, into low Earth orbit at 8,000 m/s.

Good point. I should either phrase the reason in some other way or look for an alternative approach. But the detailed explanation (based on how interactions work at low level) should go in that line…

 

[PeterDonis]

“FTL travel is impossible…” = “spacelike separated events can't be causally connected…”

These two propositions aren't logically equivalent. The only logical implication I can see is that the second statement implies the first--but even that requires an additional premise, that any pair of events along an object's worldline must be causally connected. I've never seen anyone question that premise, but if we're talking about logic, it's as well to make all premises explicit. See further comments below.

“for a reason of logical consistency” = “(a) the ordering of spacelike events is frame-dependent; (b) causally connected means cause precedes effect; hence (a) + (b) means that there are different versions, all of them equally valid, about what is the cause and what is the effect; which is logically inconsistent”

First of all, I would state the logic here a bit differently: (a) the ordering of spacelike separated events is frame-dependent; (b) the ordering of causally connected events must be invariant; therefore (c) causally connected events cannot be spacelike separated.

Second, as I just stated the argument, (a) and (b) do logically require (c). But premise (b) is an independent assumption; as far as I can see, it's not a logical implication of anything else in SR. So the logical inconsistency is only there if you accept premise (b), and acceptance of premise (b) isn't based on logic, it's based on our intuitions about causality and what's physically reasonable. I certainly don't see a proof here that FTL travel is logically inconsistent, period; it's perfectly possible, logically, that there is some consistent theory, which we just haven't figured out, that matches the rest of SR in the domain where we've tested it, but violates premise (b) and therefore allows some type of FTL travel.

In fact, the Usenet Physics FAQ entry on tachyons talks about attempts to construct just such a theory, or at least the skeleton of one, and work out its implications. Nothing in there is logically inconsistent with SR. However, some of the implications do not seem physically reasonable: for example, any theory of interacting tachyons leads to "runaway" reactions that release arbitrary amounts of energy, and in the quantum version, they make the vacuum unstable. So again, it's not that FTL travel is logically inconsistent; it's that, as far as we can tell in our present state of knowledge, assuming that FTL travel is possible leads to physically unreasonable predictions. But logically, it's perfectly possible that our present state of knowledge is incomplete, and future knowledge will show us either how to construct a theory of FTL travel that doesn't lead to the physically unreasonable predictions, or how the predictions aren't physically unreasonable after all. I think the latter, at least, is highly unlikely, but that's not based on logic, it's based on my judgment about what's physically reasonable.

 

[Demystifier]

As shown in Sec. 8.2 of
http://xxx.lanl.gov/abs/1205.1992
the (in)ability to accelerate particle to a superluminal velocity is a matter of force acting on the particle. It cannot be done with vector potential (electromagnetic force) or tensor potential (gravitational force), but it could be done with a force generated by a scalar potential.

 

[PeterDonis]

http://xxx.lanl.gov/abs/1205.1992

One question about this: you say the nonlocal "relativistic Newton equation for many particles" (equation 8.29) is covariant because "s" is a Lorentz scalar. But earlier you defined "s" as a parameter along a single worldline, not a general parameter that has a value at every event in the whole spacetime. You also say that "s" has a priori no physical meaning. But your nonlocal force equation says that the force on one particle depends on the positions of other particles "at the same s". How is "at the same s" defined physically, if "s" is a parameter on a single worldline and has no physical meaning?

 

[Demystifier]

you say the nonlocal "relativistic Newton equation for many particles" (equation 8.29) is covariant because "s" is a Lorentz scalar.

True.

But earlier you defined "s" as a parameter along a single worldline, not a general parameter that has a value at every event in the whole spacetime.

So? Scalar means invariant under the change of spacetime coordinates x -> x'. s is a scalar in the same sense in which proper time is a scalar.

You also say that "s" has a priori no physical meaning.

Yes, but I do not say that it does not have an a posteriori physical meaning. For example in the 1-particle case, one special choice of s corresponds to the proper time, which certainly does have a physical meaning.

But your nonlocal force equation says that the force on one particle depends on the positions of other particles "at the same s". How is "at the same s" defined physically, if "s" is a parameter on a single worldline and has no physical meaning?

The point is that "at the same s" does NOT need to be defined physically, as long as the only goal is to calculate the trajectories in spacetime. It is defined mathematically, and that's enough. You may think of s as an auxiliary mathematical tool. In fact, s can even be eliminated from the equations, as is well known from the mathematical theory of integral curves of vector fields. You can calculate the trajectories in spacetime even without using parameter s. See e.g. Eq. (30) of
http://xxx.lanl.gov/abs/quant-ph/0512065

 

[PeterDonis]

You may think of s as an auxiliary mathematical tool. In fact, s can even be eliminated from the equations, as is well known from the mathematical theory of integral curves of vector fields.

So basically, a given solution of the equations will be a set of integral curves, and it's the same set of curves regardless of how we define s along each one?

 

[Demystifier]

So basically, a given solution of the equations will be a set of integral curves, and it's the same set of curves regardless of how we define s along each one?

Yes.