# Relative velocities of particles and photons

1. Jun 30, 2014

### DiracPool

According to my understanding of SR, a light photon traveling at c, of course, relative to me "experiences" no time. In other words, it is not traveling through, at least, the time dimension I am traveling through. A neutrino, say, moving close to c does, but it is traveling very slowly through time compared to me. Now we can say that the neutrino, in it's own frame, is experiencing it's own proper time as "normal," and experiences me as traveling very slowly through it's time dimension. So far so good.

But what about the "experience" of the photon. If I experience the photon as traveling at c relative to me, doesn't the photon necessarily have to experience me traveling at c relative to it? Doesn't this, then, necessarily mean that I have to be massless in order for the symmetry to hold? Last time I checked I weighed over 200 pounds. How do we reconcile this? Also, am I frozen in time from the perspective of the photon? What form does that take?

From the neutrinos perspective, traveling at say .99+ the speed of light, I must seem enormous, a huge, heavy structure of mass-energy. However, if I just travel a tiny bit faster and reach the speed of c, all of that mass vanishes, and I become massless. I'm a bit confused on what the nature of that bridge is, or means. Now I know that the objection is that I can't reach the speed of light compared to the neutrino so it's a null argument, and I'd agree, but that is why I brought up the relative speed compared to the photon in the beginning. What do I look like from the perspective of the photon?

2. Jun 30, 2014

### Staff: Mentor

Then your understanding is not quite correct. We have a FAQ on this:

Briefly, it would be correct to say that the Minkowski length of a photon's worldline is zero; but interpreting that length as "experienced time" for the photon is the problem, since the assumptions that ground that interpretation for timelike objects like you and me do not hold for photons.

This is incorrect too, even leaving aside the points made in the FAQ. If you insist on thinking of a photon as "traveling through the dimensions", then you would have to say it travels through space and time "at the same speed", whereas you and I, for example, travel through space much more "slowly" than we travel through time. But this is a very limited interpretation and I don't recommend it.

If you insist on this kind of interpretation, yes, the "rate of travel through time" is frame-dependent; it is not an invariant. (Which is why I don't recommend spending too much effort on this interpretation.)

No. The concept of "speed relative to a photon" makes no sense, because a photon does not have a "rest frame" the way you and I do. The FAQ entry goes into this.

The question has no answer because "the perspective of the photon" is not a well-defined notion.

Last edited: Jun 30, 2014
3. Jun 30, 2014

### ghwellsjr

I wouldn't say it that way because it implies that a photon "experiences" something, just not time. A better way is to say it is that "experience" does not apply to a photon just as time does not apply to a photon. Then the rest of your post doesn't make any sense.

I wouldn't say that either. You're not traveling through time. In your rest frame, you're not traveling at all but a photon is traveling at c, by definition.

A neutrino traveling at close to c in your rest frame is Time Dilated. It's a very simple concept. If you know its speed, you can calculate its Time Dilation according to your rest frame. You can transform to any other rest frame and it may have a new speed and a new Time Dilation. Why are you trying to complicate such as simple concept?

You can transform from your rest frame to the neutrino's rest frame and it will not be Time Dilated but you will be. What's with this "traveling through a time dimension"?

So far not so good.

A meaningless concept.

No, you can't transform from your rest frame to that of a photon and the rest of your questions are meaningless.

4. Jun 30, 2014

### DiracPool

5. Jun 30, 2014

### Vorpal

Your wordline is timelike. Light's wordline isn't. Thus, there no symmetry to be enforced.

Yet the FAQ is mistaken (bracketed addition mine):
"One of the key axioms of special relativity is that light moves at c in all [inertial?] reference frames."​
Without the additional qualifier of "inertial", the statement is false, and the thrust of the "no rest frame" unravels with it. In fact, neither lightcone coordinates nor null tetrads (frame fields) are problematic, so there's no problem with a frame in which a light ray is at rest. It's just not an inertial frame.

If the FAQ was up-front about interpreting "rest frame" as "rest inertial frame" and then followed-up with an explanation that light isn't at rest in any inertial frame, it would be right.

An electromagnetic wave behaves just like a stopped clock moving at c. One can of course define proper time as specifically applicable to timelike wordlines only, but... why? I'm not sure what utility there is in excluding light.

6. Jun 30, 2014

### Staff: Mentor

I'm not sure I understand what you mean by this. Can you elaborate?

I can think of a couple of reasons:

First, proper time for timelike worldlines is an affine parameter, and this gives an obvious mathematical grounding to the physical statement that different clocks can have different natural rates and that the "zero point" of time is arbitrary. But the Minkowski length of a null worldline obviously can't be used as an affine parameter. A somewhat more intuitive way of saying this is that, on a timelike worldline, different proper times label distinct events; but you obviously can't use the Minkowski length to label distinct events on a null worldline. Equating that zero Minkowski length with "zero time" invites the erroneous inference that light rays are somehow "everywhere at once", or something like that. (This has happened many times here on PF...)

Second, important physical quantities of interest on timelike worldlines are standardly expressed as derivatives with respect to proper time: the two obvious ones are 4-velocity and 4-acceleration. This doesn't work the same for null worldlines, since, as above, the Minkowski length can't be used as an affine parameter. You can still define a tangent vector for a null worldline (though it can't be a unit vector, as it can with a timelike worldline--this, btw, is IMO the key physical difference between timelike and null worldlines, which underlies all the other things I'm saying), and take its derivative in order to find the path curvature of the null worldline; but to do so, you have to pick a valid affine parameter, which breaks the relationship with the Minkowski length.

7. Jun 30, 2014

### Vorpal

If you have a clock traveling in the x-direction with speed v with not acceleration, then its internal state changes and can be parametrized by λ = x-vt (incidentally also an affine parameter that is definitely not the proper time). Whatever it's really doing is not particularly relevant, just that its state is changing. On the other hand, if you have an electromagnetic wave traveling in the x-direction in vacuum, then its profile is going to be a function of the u = x-ct light-cone coordinate, but this profile does not evolve. Rather, it's simply the same pattern moving along instead. Thus, it is like a stopped clock.

A key property of being an affine parameter is that not just the "zero point", but also the scale, is completely irrelevant. This conceptual identification between affine parameters and proper time invites trouble--yes, in the situation given, one can serve as the other, but that doesn't mean we should collapse the distinction. I don't understand in what way this gives "grounding to the physical statement that different clocks can have different natural rates". The "natural rate" of the clock is completely independent of the scale of the affine parameter, because we are always free to rescale by it by arbitrary factor regardless of any physical properties of the clock.

While agree that the possibility of having proper time be the affine parameter is important, I don't see why that should exclude light from having zero proper time. Rather, this connection is due to the possibility of having an instantaneously comoving inertial frame, which lightlike worldlines lack. Meanwhile, insufficient emphasis of the key qualifier "inertial" leads to wrong explanations, such as the linked FAQ entry.

An electromagnetic wave profile behaves exactly like a stopped clock, so the "zero time" is actually completely sensible. Furthermore, in light-cone coordinates it is both stationary and at every point along its trajectory at once (wrt one light-cone coordinate). What you're describing sounds like a confusion regarding the conceptual role of inertial frames, and so can be addressed directly by emphasizing that light has no inertial (rest) frame.

A tangent vector makes sense in any parametrization. For timelike worldlines, using proper time makes it normalized, but if that's the key property you're after, then there was no need to bring up affine parameters at all, because they don't generally do that either. In any case, of course one shouldn't expect a null worldline to have a normalizable four-velocity--one should expect a null four-velocity instead. This is completely orthogonal to the issue of interpreting worldline length as proper time. As four four-acceleration, this is also sensible generally: $a = \nabla_uu$.

8. Jun 30, 2014

### Staff: Mentor

No, it's just a constant factor (namely, $\sqrt{1 - v^2}$) times the proper time. In other words, it's just the proper time with the "rate" adjusted.

By "profile" you must mean "phase"; yes, the phase of an electromagnetic wave doesn't change along a given null worldline that is parallel to the wave's motion. But if you're going to view the light as a wave at all, its motion can't be restricted to a single null worldline; it must occupy a region bounded by distinct parallel null worldlines, and the phase of the wave *does* change from worldline to worldline within that region.

True.

I agree the distinction can be useful in some circumstances, yes.

Yes, but that's just saying that we are always free to pick a mathematical model that is consistent in the abstract but bears no relation to the actual physical system we are supposed to be modeling. You can do that, but why would you want to? My point is that being able to rescale and reset the zero point of an affine parameter *can* be used, mathematically, to reflect the physical fact that clocks can have different rates and zero points.

I'm not sure I understand. Are you saying we should allow the concept of "proper time" to apply to light, by breaking the connection between "proper time" and inertial frames? I suppose this would be just as valid a use of terminology, in the abstract, as the standard usage, but I'm still not sure I see why it's an improvement.

Which is why the FAQ entry is entitled "rest frame of a photon". Proper time comes into it because of the association, in standard usage, between proper time and inertial rest frames (and I agree that the FAQ entry should be explicit about the fact that it uses the term "frame" exclusively to mean "inertial frame").

And you will find plenty of people who object to the term "four-velocity" being used in the null case, because to them, "four-velocity" implies a unit vector (as opposed to the term "tangent vector", which doesn't). (This came up in a PF thread quite a while back.) Again, this is a matter of terminology and standard usage, but standard usage does have a purpose, even if it's only standardization.

To you, perhaps it is. But I don't think it is to everybody. Basically, you are viewing the differences between null vectors/worldlines and timelike vectors/worldlines as minor compared to the similarities; to you, the two kinds of vectors/worldlines are basically the same thing, just with some different quirks, so to speak. But to me, and I think to people who favor the "standard" usages I've been arguing for, the differences between null and timelike vectors/worldlines are fundamental, pointing to a fundamental physical difference between the two types of objects; and because of this, we want the terminology to emphasize that difference. I'll agree that this is a matter of preference and terminology, not physics; we'll make the same physical predictions either way, and that's what really matters.