# Neutrinos - what is their mass when they travel so close c?

neutrinos -- what is their mass when they travel so close c?

I understand that neutrinos move at very near the speed of light and that they have a very small amount of mass. This being true, why do they not have a great deal of mass at that speed?

PeterDonis
Mentor
Neutrinos have a very small rest mass. The "mass" that gets very large when particles move close to the speed of light is "relativistic mass", better known as "energy".

jtbell
Mentor
When we say that neutrinos have a very small mass, we mean that they have a very small rest mass. When physicists talk about mass, they usually mean "rest mass", not "relativistic mass."

i'm missing something. Why does this rest mass not increase with the increase in velocity?

Nugatory
Mentor
i'm missing something. Why does this rest mass not increase with the increase in velocity?

Rest mass is the mass that is measured by an observer who is at rest with respect to the object. Thus, it's not affected by the fact that the object might be moving with respect to other observers - you are moving at .99c relative to some observer somewhere, but that doesn't mean that you think that your mass is any different.

1 person
BruceW
Homework Helper
yeah. what Nugatory said is the important thing to keep in mind. But I should also add that it is more complicated when you go into the quantum physics of it all.

PeterDonis
Mentor
Why does this rest mass not increase with the increase in velocity?

The formula for rest mass is ##m^2 = E^2 - p^2## (in units where the speed of light is 1). As the object increases in velocity, ##E## and ##p## (energy and momentum) both increase, but they do so in a way that keeps ##m## constant.

1 person
I always thought that the rest mass was the mass that a particle would have if it could be at rest. That would not change no matter what the speed of the particle.

Nugatory
Mentor
I always thought that the rest mass was the mass that a particle would have if it could be at rest. That would not change no matter what the speed of the particle.

That is correct and consistent with the answers in #5 and #7... But note that massless particle cannot be at rest ever.

That is correct and consistent with the answers in #5 and #7... But note that massless particle cannot be at rest ever.

I think I may of interpreted those answers differently, but thanks for reassuring me. I think the term at rest is merely an ideal which cannot truly be obtained by any particle, but that the term cannot be applied to massless particles. I assume that neutrinos can obtain sufficiently high energy levels since they are detected by the use of Cerenkov detectors.

jtbell
Mentor
the term [rest mass] cannot be applied to massless particles.

Indeed the term "rest mass" is semantically inconsistent with the behavior of massless particles. That's one reason why many physicists prefer the term "invariant mass". However, "rest mass" is so deeply ingrained in the popular and introductory literature about relativity that we'll never be able to eradicate it, so we simply have to live with it.

I like rest mass despite being a physicist.

I like rest mass despite being a physicist.

Where as I like invariant mass and I am not a physicist.
(Invariant). The bit that remains when at rest.
It's all a matter of preference.

BruceW
Homework Helper
this is a good trick: when you hear the phrase "rest mass", just think of the word "rest" as being completely unrelated to a lack of motion. That is what I do. (and therefore I do not mind the phrase "rest mass")

Dale
Mentor
2021 Award
Where as I like invariant mass and I am not a physicist.
+1

I like invariant because it points out the important part, which is that all reference frames agree. The fact that for massive objects it is also the mass in its rest frame is, to me, not nearly as important.

+1

I like invariant because it points out the important part, which is that all reference frames agree. The fact that for massive objects it is also the mass in its rest frame is, to me, not nearly as important.

I agree but there are deeper hidden factors behind that.

this is a good trick: when you hear the phrase "rest mass", just think of the word "rest" as being completely unrelated to a lack of motion. That is what I do. (and therefore I do not mind the phrase "rest mass")

I find that funny.I have too much rest.

strangerep
+1

I like invariant because it points out the important part, which is that all reference frames agree. The fact that for massive objects it is also the mass in its rest frame is, to me, not nearly as important.
+2

Bill_K
+1

I like invariant because it points out the important part, which is that all reference frames agree. The fact that for massive objects it is also the mass in its rest frame is, to me, not nearly as important.

+2
-2

I think both physicists and nonphysicists alike are well-advised to use standard terminology, even though they might consider it less than ideal. To do otherwise conveys, rightly or wrongly, a lack of familiarity with the subject, or else a preoccupation with its trivial aspects that others have long since moved past.

In my experience, real physicists reserve the term "invariant mass" for the combined energy-momentum of two or more particles.

Dale
Mentor
2021 Award
I think both physicists and nonphysicists alike are well-advised to use standard terminology, even though they might consider it less than ideal. ...

In my experience, real physicists reserve the term "invariant mass" for the combined energy-momentum of two or more particles.
I agree, but that is not my experience. In my experience invariant mass is considered standard terminology. I have never come across such a restriction.

I don't think the terminology is so well defined but my experience is also like Bill_k's expedience. Invariant mass is usually applied to two or more particles and rest mass is applied to single particles.

Dale
Mentor
2021 Award
Interesting. I wonder why you two have that experience and mine is different.

For non-rigid systems there is no reference frame where the entire system is at rest, so certainly for systems it is much more common to say "invariant mass" than to say "rest mass", but I have never (until this thread) had even the slightest impression that the term "invariant mass" was in any way non-standard when applied to single particles.

I don't think there is anything wrong with using invariant mass for a single particle but many people (including me) prefer to say rest mass. To be truthful I really just say mass most of the time and only say rest mass if there is any possibility of confusion

strangerep
I think both physicists and nonphysicists alike are well-advised to use standard terminology, even though they might consider it less than ideal. To do otherwise conveys, rightly or wrongly, a lack of familiarity with the subject, or else a preoccupation with its trivial aspects that others have long since moved past.
Well, at least one physicist I know once said that a Hilbert space is necessarily infinite-dimensional.
But that's not standard terminology, and conveys a "lack of familiarity with the subject".

I don't think there's anything wrong with trying to tighten up the use and meanings of commonly used phrases. Though I consider myself to be more physicist than mathematician, the sloppiness of (some) hard core physicists is not something to be proud of. We can all improve.

[...] real physicists reserve the term "invariant mass" for the combined energy-momentum of two or more particles.
The term "invariant mass" has a well-defined meaning in both cases, hence no problem. OTOH, a rest frame for the photon field does not exist, so the phrase "photon rest mass" is an oxymoron. :uhh:

BruceW
Homework Helper
the trace of the EM tensor is zero. is this kindof analogous to the idea of rest mass? This seems to me to be the most similar quantity once we go from single photons to the classical view of an EM field.

jtbell
Mentor
In my experience, real physicists reserve the term "invariant mass" for the combined energy-momentum of two or more particles.

My own experience as a grad student in experimental HEP 30+ years ago agrees with this. In fact, one of my first tasks for my dissertation advisor was to make a histogram of invariant masses of pairs of pion and proton tracks from our bubble-chamber data, for selecting a sample of ##\Lambda^0## decays.

For single particles, everybody I worked with back then used the simple term "mass", and it always meant the quantity given by the equation in post #7. Nobody ever talked about "rest mass" or "relativistic mass," even though we dealt with highly relativistic particles.

After I started teaching and hanging out on forums like this one, I got in the habit of using the more specific terms to help avoid confusion.

vanhees71
Gold Member
2021 Award
Of course, also nowadays, in the high-energy particle and nuclear physics the mass of a particle is always the invariant mass in the sense explained in this thread. There is no need to relabel energy diveded by $c^2$ "mass", which was a misconception in the very early days of relativistic theory (as was speed-dependent temperature). This confusing ideas were obsolete as soon as Minkowski discovered the mathematical structure behind space-time (of special relativity).

As a rule of thumb you can remember that for massive quanta all intrinsic quantities as mass and spin are defined in their rest frame. Formally this is derived from the theory of the proper orthochronous Poincare group, i.e., the symmetry group of Minkowski space and its quantum equivalent, where the Lorentz subgroup is substituted by its covering group SL(2,C).

For masseless quanta it's a bit more complicated, because there is no rest frame. As it turns out, there are only two spin-like degrees of freedom for quanta of spin $s \geq 1/2$ (contrary to the case of massive quanta, where one has $2s+1$ spin-degrees of freedom).

BruceW is right in his idea on the tracelessness of the energy-momentum tensor in classical (!) field theory. The photon (quantum of the electromagnetic quantum field) is massless and thus the corresponding field equations of the classical field (which are nothing else than the good old Maxwell equations) have an additional symmetry, the scaling symmetry. From the Noether theorem of this theory it follows that the energy-momentum tensor is (covariantly) traceless.

However, in the full quantized theory scale invariance almost always is broken by an anomaly. In perturbation theory this can be understood by the fact that in a theory with massless particles, the divergent loop diagrams cannot be renormalized at the renormalization point, where all external four-momenta are taken to vanish, because there is a singularity of the corresponding vertex functions. One has to choose a point where the external momenta are space-like instead, and this introduces an energy-momentum scale into the the theory. This expclitly breaks scale invariance.