It still doesn't sound physically distinct any more than distinguishing any continuous quantity as being zero or nonzero. If we measure something that looks like 0, we can't be sure if it's just below the sensitivity of our instruments.
For neutrinos, even if we accelerated an rocket and somehow checked if a neutrino was at rest relative to it, we might find that it's not. That means we won't know if we need more speed or if it's impossible. I suppose it's a bit easier than that because we only have to accelerate the rocket fast enough that the neutrino's speed becomes measurably less than c, rather than 0. But still, what if we can't even get it to go fast enough for that? No way to prove that it's travelling at c, it seems.
I'd like to add that even photons have a nonzero upper bound to their possible rest mass. At least they used to. Is there any way, in principle, to show that it's exactly zero, and thus falls into this distinct category?
If you try what I described with a light ray, it will be moving away from you at c no matter how much you accelerate in its direction.
If you try it with a massive object, even a neutrino with a very, very tiny invariant mass, that will not be the case; its speed relative to you will decrease as you accelerate after it, eventually to zero.
There is no continuum between those two possibilities; they are distinct and discrete. The only continuum is in the latter case, where the final speed of the object relative to you will depend continuously on how long you accelerate.
> even if we accelerated an rocket and somehow checked if a neutrino was at rest relative to it, we might find that it's not. That means we won't know if we need more speed or if it's impossible
Yes, you will know, because you will know if the neutrino's speed relative to you has decreased or not. If it has, it's possible to bring it to rest relative to you. If it hasn't, it's not. See above.
> I suppose it's a bit easier than that because we only have to accelerate the rocket fast enough that the neutrino's speed becomes measurably less than c, rather than 0.
Exactly.
> But still, what if we can't even get it to go fast enough for that?
That's basically the position we are in now: we have no way of building a rocket or other device that can accelerate after a neutrino long enough to tell whether its speed relative to the rocket is measurably decreasing. So we have to resort to indirect measurements. But as I said before, that doesn't change the principle.
> even photons have a nonzero upper bound to their possible rest mass
Yes, because, as I said, practically speaking we can't run the obvious and straightforward experiment I described, to confirm that a photon moves away from you at c no matter how much you accelerate after it. So we have to resort to indirect measurements, like trying to measure its invariant mass by other means. But that doesn't change the principle.
> Yes, you will know, because you will know if the neutrino's speed relative to you has decreased or not. If it has, it's possible to bring it to rest relative to you. If it hasn't, it's not. See above.
I don't think this quite works because of relativistic addition of velocities. Naively, it seems to. For example, if the object was travelling at 0.999999c relative to you (appears to be 1.0c according to your limited instruments), then you accelerate to 0.50c in its direction, you'd see its speed reduce to 0.50c (same 2s.f. instrument), which would clearly prove it's not massless. But velocities don't add like that relativistically and I think you'd still see it as travelling at 1.0c because it only decreased a tiny amount, below what you instrument can detect. If you use a more precise instrument or a faster rocket, you might measure it as 1.00000c but then you still won't know if it's exactly c or a smidgen less.
Maybe I've got my relativistic velocity addition wrong? But it still looks like the same measurement problem as trying to prove a classical object has a speed of exactly 0, which can't be done no matter how accurate our instruments are.
So if your accuracy is, say, 1 part in 100,000, you wouldn't be able to see the difference. But with an accuracy of 1 part in 500,000, you would, even though you wouldn't have been able to see the difference before the acceleration.
Also, suppose you accelerated for a second increment equal to the first; you would get
As you can see, the differences in velocity grow fairly quickly for each equal increment of acceleration; the growth is not at all linear. And, as I said in my other post just now, for any given measurement accuracy, it would be simple to calculate how much acceleration you would need to be able to distinguish moving at exactly c from moving at 0.999999c (or any other speed less than c that you choose) to that accuracy.
> velocities don't add like that relativistically and I think you'd still see it as travelling at 1.0c
Not indefinitely. Sure, if you accelerated for a short enough time in the direction of the neutrino, you might still be within your measurement error and so not have learned anything. But that just means you need to accelerate for a longer time. For any given measurement accuracy, you will be able to calculate how long you need to accelerate, by your clock, to definitely distinguish the two cases. Relativistic velocity addition does not change that fact. All it changes is the details of that calculation; yes, for a given measurement accuracy, you need to accelerate for a longer time, by your clock, to definitely distinguish the cases than you would if velocity addition were linear. But that doesn't mean relativistic velocity addition makes it impossible to distinguish the cases at all, ever. It doesn't.
With neutrinos we might find that it's not but it'd be impossible to catch a photon as it would always have the same speed of c in our reference frame.
> I'd like to add that even photons have a nonzero upper bound to their possible rest mass. At least they used to.
Not sure what you're talking about, their momentum?
No object with mass can reach the speed of light and we know they're travelling at that exact speed.
How do we know they're travelling at exactly c? That's my concern. Last I heard, a couple of decades ago, physicists would occasionally measure a new maximum possible rest-mass for photons. It would be very tiny, of course, but they couldn't say it's exactly zero.
> How do we know they're travelling at exactly c? That's my concern.
We don't, strictly speaking. The measurements you refer to aren't even measuring the speed of photons. They're measuring their rest mass.
> physicists would occasionally measure a new maximum possible rest-mass for photons. It would be very tiny, of course, but they couldn't say it's exactly zero.
Based on just those measurements, no. The most they can say is that the photon rest mass is zero to within some error bar, and the size of the error bar keeps getting smaller. (The current error bar, IIRC, is 10^-52 grams, or about 24 orders of magnitude smaller than the electron mass.)
However, we have a ton of indirect evidence that photons are massless; the most extensive body of such evidence is all the evidence for the gauge invariance of electromagnetism. If photons had a nonzero rest mass, that would break electromagnetic gauge invariance. So photons having a nonzero rest mass would be a huge issue for our current theories, in the way that neutrinos having a nonzero rest mass would not; there is no important symmetry coresponding to electromagnetic gauge invariance that is broken by neutrinos having a nonzero rest mass.
For neutrinos, even if we accelerated an rocket and somehow checked if a neutrino was at rest relative to it, we might find that it's not. That means we won't know if we need more speed or if it's impossible. I suppose it's a bit easier than that because we only have to accelerate the rocket fast enough that the neutrino's speed becomes measurably less than c, rather than 0. But still, what if we can't even get it to go fast enough for that? No way to prove that it's travelling at c, it seems.
I'd like to add that even photons have a nonzero upper bound to their possible rest mass. At least they used to. Is there any way, in principle, to show that it's exactly zero, and thus falls into this distinct category?