# False predictions of relativistic mass

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## Main Question or Discussion Point

Does the concept of relativistic mass make any wrong predictions? One of the most senior professor of the best university here wrote a note on false predictions of relativistic mass trying to disprove it. However, I have seen criticism of it. But I didn't get most part of those counter arguments.
The professor used gravity and 4 vector analysis two disprove it in two different ways. I am kinda confused. Does relativistic mass produce false prediction?

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mfb
Mentor
If handled properly, it does not make wrong predictions, but it also doesn't help anywhere. It is simply the energy divided by the squared speed of light. There is no need to duplicate that concept with just a conversion factor. It is like using kilometers and miles together in calculations: why would you?

dextercioby
Dale
Mentor
Does the concept of relativistic mass make any wrong predictions?
Relativistic mass isn't a theory so it doesn't make predictions. However, you cannot simply substitute relativistic mass for mass in equations like Newton's laws.

If handled properly, it does not make wrong predictions, but it also doesn't help anywhere. It is simply the energy divided by the squared speed of light. There is no need to duplicate that concept with just a conversion factor. It is like using kilometers and miles together in calculations: why would you?
Of course, what you (rightly) say of ##E## and ##m_r## could also be said of ##E_0## and ##m##. (Did anyone ever express kinetic energy ##E_k## in units of mass, as ##m_k##?)

Funny that educators have settled on ##E## (and ##E_k##) on the one hand but mostly ##m## on the other (though ##E_0## still often appears in the literature). Perhaps they've done this so that these quantities correlate closely with what students know from Newtonian physics, but I'm of the opinion that students would benefit from (at least sometimes) using ##E_0## instead of ##m## once their equivalence has been established. Most textbooks are full of "official" (boxed) equations like:

##E = \gamma mc^2##

##E_k = E - mc^2 = (\gamma - 1)mc^2##

##\vec p = \gamma m \vec v##.

##(mc^2)^2 = E^2 - (\vec{p}c)^2##

But once you've established that ##E_0 = mc^2##, why not give primacy to equations where all energy (and momentum) terms are expressed in the same unit? Simplifies the concepts, if you ask me, and certainly makes the equations easier to remember:

##E = \gamma E_0##

##E_k = E - E_0 = E_0 (\gamma - 1)##

##\vec p c = E \vec \beta##

##E_0^2 = E^2 - (\vec{p}c)^2##.

It's nice to see that last one like this, too:

##E_0^2 = E^2 - (E \vec{\beta})^2##

Orodruin
Staff Emeritus
Homework Helper
Gold Member
why not give primacy to equations where all energy (and momentum) terms are expressed in the same unit?
For most advanced relativity courses, mass and energy have the same units. While perhaps better in hindsight, going mainly with "rest energy" instead of "mass" would be a bad idea as it is not prevalent in the current research literature and would therefore just be confusing.

vanhees71
Gold Member
2019 Award
A photon has invariant mass 0 and thus no frame where it is at rest. I'd not use the expression "rest energy" for at least this reason. If, however, you have a massive object, then you can always define a (momentary) rest frame of it's center of momentum, i.e., the frame where it's total momentum vanishes. In this reference frame the invariant mass is ##m=E_{\text{cm}}/c^2##.

Then you can express everything in a manifestly covariant way, i.e., for any frame the invariant mass is given by
$$m^2 c^2=p_{\mu} p^{\mu}=E^2/c^2-\vec{p}^2,$$
where ##E## is the total energy and ##\vec{p}## the total momentum of the object under consideration. The good thing is that this manifestly covariant expression also holds for massless objects.

Dale
For most advanced relativity courses, mass and energy have the same units. While perhaps better in hindsight, going mainly with "rest energy" instead of "mass" would be a bad idea as it is not prevalent in the current research literature and would therefore just be confusing.
I see your point, of course.

Certainly it's important for learners to understand the fundamental equivalence of mass and (rest) energy—i.e., that the property they've always known to be responsible for resisting acceleration (and for gravitation) in Newtonian physics turns out to be nothing but a measure of how much energy a system has in its rest frame.

Now, I'm definitely not saying that understanding that requires always using ##E_0## instead of ##m##. But I can tell you that what really drove that point home for me was actually seeing all the relevant equations with ##E_0## instead of ##m##. I even rewrote some Newtonian relations like that, for instance:

##f_g = \dfrac{G}{c^4} \, \dfrac{E_{0a} E_{0b}}{r^2}##

So for me, the "aha" moment was realizing that all of physics could be done with ##E_0## instead of ##m## (in the same way that one could use ##\beta## instead of ##v##). That makes me suspect that there are others out there who would likewise benefit from at least seeing the equations expressed with ##E_0## instead of ##m##.

There's so much popsci nonsense floating around about "converting" mass to energy and vice versa. I think that somewhere in the back of my mind, that little nugget of misinformation was stopping me from grasping the true significance of the mass–(rest) energy equivalence. If mass and energy can be "converted into" each other, then they must be "different things," right? Well, no: mass really is (rest) energy. I had to unlearn the popsci stuff, which was buried rather deep. Seeing ##E_0## literally replace ##m## is what got me there.

A photon has invariant mass 0 and thus no frame where it is at rest. I'd not use the expression "rest energy" for at least this reason.
I disagree: I think that using the term "rest energy," it makes perfect sense that a photon has none of it, precisely because a photon can never be at rest. No rest frame? No rest energy.

vanhees71
Gold Member
2019 Award
But a photon has a definite invariant mass, namely 0 :-)!

SiennaTheGr8
Staff Emeritus
2019 Award
Fundamentally, this is a question of terminology. We can call gamma*m "giant purple kumquat" if we like, and predictions will be the same. The problem with the terminology "relativistic mass" is that it implies one can take a Newtonian equation and make it relativistic simply by swapping "mass" for "relativistic mass". Nothing could be further from the truth.

vanhees71
Mister T
Gold Member
In my opinion the real confusion surrounds a distinction between matter and mass. The mass-energy equivalence is exemplified as a change of matter to energy. All nonsense of course. We perpetuate this confusion when we refer to an object as a mass rather than an object that has as one of its many properties mass.

SiennaTheGr8 and martinbn
vanhees71
Gold Member
2019 Award
The irony is that, of course, Einstein got it right from the very beginning in his letter of 1905.

Invariant mass is defined as the total energy of a system in its (momentaneous) center-of-momentum frame (divided by ##c^2## if you use men-made units). If there is intrinsic energy like, e.g., heat or the electromagnetic field of a charged capacitor the corresponding inner energies contribute to the total center-of-momentum energy, and thus to the invariant mass of the body. It's as simple as that. You only have to abandon the very confusing concept of "relativistic mass" and work with the Minkowski covariant quantities like the energy-momentum vector (or the energy-momentum-stress tensor in fieldtheoretical descriptions) and invariant mass, which is a scalar by definition.

In my opinion the real confusion surrounds a distinction between matter and mass. The mass-energy equivalence is exemplified as a change of matter to energy. All nonsense of course. We perpetuate this confusion when we refer to an object as a mass rather than an object that has as one of its many properties mass.
Yes, many who should know better sometimes refer to matter as mass or to radiation as energy ("a photon is pure energy..."). So right off the bat, people get the wrong impression that mass and energy are "things" rather than properties, just as you say. Combine that with talk of "converting mass to energy" and you've reinforced the notion that mass and energy are entirely different "things," to boot.

(And honestly, I'm not fond of "converting matter to radiation" either, since both words have multiple definitions. It's a step in the right direction, though.)

Then in low-level physics, mass is sometimes taught as "amount of matter," which works heuristically at the beginning but can cause confusion later on. Even the more correct "measure of inertia" / "resistance to acceleration" concept must be modified in SR. And the "gravitational charge" concept gets modified in GR...

Come to think of it, pretty much everything we learn about mass in Newtonian physics must be partially unlearned. This is another reason why I rather like the "rest energy" concept. After many years of being confident that you know what "mass" is (amount of matter, measure of inertia / resistance to acceleration, gravitational charge), you're suddenly told that actually all of that is only sort of true. To top it off, you may have been exposed to the "relativistic mass" concept at some point ("mass increases with speed"). So you're faced with the challenge of suppressing all these old associations you have with the word, while you redefine it and work your way toward understanding why those old associations are valid in the Newtonian limit. On a psychological level, I think having an alternate term available during this process (rest energy) can be helpful. If you think "mass," you can easily inadvertently fall back on old associations. Brains are funny that way. If you think "rest energy," you can circumvent those old associations.

vanhees71
Gold Member
2019 Award
For me the epiphany was when I learnt about the symmetry principles, applied to QT, and particularly the representation theory of the Galileo and Poincare groups underlying Newtonian and special-relativistic spacetime, respectively.

From this point of view the notion of mass are surprisingly pretty different concepts in Newtonian and Minkowskian physics. In the former, the mass is a non-trivial central charge of the Galileo group's Lie algebra, giving rise to its conservation and the mass-superselection rule. In the latter it's simply a Casimir operator of the Poincare group's Lie algebra and as such doesn't need to be conserved. Sometimes (or maybe even always!) abstract math helps to get "breakthrough insights" into what's behind the natural laws!

Maybe you find the following video interesting

FactChecker
mfb
Mentor
Invariant mass is defined as the total energy of a system in its (momentaneous) center-of-momentum frame (divided by ##c^2## if you use men-made units).
As opposed to the divine MeV?

It is a matter of science education only. In particle physics, all masses are given in units of energy, and energy is given in units of energy as well. The idea of a relativistic mass doesn't even come up because it would be exactly the same as energy.

vanhees71
vanhees71