Can someone tell me if I got this line from the Feynman lectures right?

Rishabh Narula
'Mass is found to increase with velocity, but appreciable increases
require velocities near that of light. A true law is: if an object moves with a
speed of less than one hundred miles a second the mass is constant to within one
part in a million. '
What does 'constant to within one part in a million' mean?
does it mean something like mass doesn't change by more than 1/10^6 of its
original value?or Del m (in this case) <= 1/10^6 times m not.
del m meaning change in mass and m not meaning original mass,sorry don't know
how to write it in symbols here without a pen.:3

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2022 Award
Sigh, not again :-(. One should not use this oldfashioned idea of relativistic mass. What's talked about here is the energy of a massive particle, including its rest energy,
$$E=m \gamma c^2, \quad \gamma=\frac{1}{\sqrt{1-\vec{v}^2/c^2}},$$
and ##m## is the only mass one uses in contemporary physics, the socalled invariant mass, which is a scalar under Poincare transformations. The energy is the temporal component of the four-momentum four-vector,
$$(p^{\mu})=\begin{pmatrix}E/c \\ \vec{p} \end{pmatrix}.$$
I've been always surprised that Feynman, the mastermind of relativistic QFT, used this confusing oldfashioned notion of "relativistic mass" (which is not even the full story since if you start using relativistic masses you have to introduce not only one kind of such very mysterious quantities but an entire function of quantities that depend not only on the speed but also the direction of velocity; that's why Einstein abandoned this idea pretty early again).

Now, if you have a non-relativistic situation, where ##|\vec{v}|\ll c## and thus ##\gamma=m c^2 + m \vec{v}^2 + \mathcal{O}(\vec{v}^4/c^4)## the relativistic energy doesn't deviate from the rest energy ##E_0=m c^2## by more than ##10^{-6}## if ##v^2/c^2 \lesssim 10^{-6}## or ##v/c \lesssim 10^{-3}##.

sysprog and weirdoguy
mitochan
Hi. Get the value of
$$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}\approx 1 + \frac{v^2}{2c^2}$$
for v/c = 100*1.6*1,000 / 300,000,000. Then you shall find $$\gamma-1 < 1/10^6$$.

sysprog
2022 Award
What does 'constant to within one part in a million' mean?
does it mean something like mass doesn't change by more than 1/10^6 of its
original value?
This.

Note that relativistic mass fell out of use in professional science decades ago. Modern usage is that mass is invariant - it does not change. Feynman, unfortunately, uses the older convention. You should be aware that anywhere you see a "mass" changing with speed you need to mentally insert the word "relativistic" before it. And be very careful any other time he mentions mass, to think about which mass he means.

sysprog and vanhees71
codelieb
I agree with mitochan's answer to the question being asked. However, I am going to put my two-cents in about the use of relativistic mass in The Feynman Lectures on Physics [FLP]. I hope I will be forgiven for being somewhat off-topic; my comments are a response to some comments in the other answers.

I was a friend of Lev Okun, since around 2000. (Lev unfortunately passed away a few years ago.) My name appears in the acknowledgments of several of his later papers on energy and mass in relativity theory because I proof-read and edited them. If you are familiar with the pedagogical debates revolving around the use of relativistic mass you may have heard of Lev, who was a very outspoken opponent and probably the most published. [Lev's ~30 papers on relativistic mass and related subjects are collected in his book Energy and Mass in Relativity Theory.] For much of the time I associated with Lev I was (and still am) Editor of The Feynman Lectures on Physics, in which I have been actively correcting errata since 2000. So, as you might imagine, Lev put some pressure on me to excise relativistic mass from FLP! I could not do so, however, because it was impractical and not aligned with our editorial policy: the concept of relativistic mass appears in several places in FLP, it was clearly Feynman's intention to teach relativity using it, not some mistake made by Feynman nor one of the other FLP authors! [In 2007 I suggested adding a footnote in FLP about relativistic mass, where it is first mentioned (Vol. I, Chapter 1, page 2!), but that idea was not popular with FLP coauthor Matt Sands, nor with Kip Thorne, then Caltech's Feynman Professor (now emeritus) who was responsible for overseeing the FLP projects at Caltech.]

Lev asked me to research why Feynman taught freshmen this way, despite the fact that he never used "relativistic mass" nor "rest mass" in his work in quantum field theory. I found a good clue in Jagdish Mehra's biography, "The Beat of a Different Drum," which includes many interviews of Feynman Mehra made only a few weeks before Feynman's death. In one interview Feynman says,

"As for the lectures on physics, I have put a lot of thought into these things over the years. I've always been trying to improve the method of understanding everything. I had already tried to explain the results of relativity theory in my own way to my girlfriend, Arline, and then I used the same explanations in my lectures. These things are very personal, my own way of looking at things and I recognize them. I did everything—all of it—in my own way."

The fact that Feynman refers to Arline as his "girlfriend" (as opposed to "wife") suggests that he used relativistic mass to explain relativity theory to her when they were young (they started dating when they were 13). The fact that he talks about Arline (whose death while he was working on the Manhattan Project was the biggest tragedy of his life) and then says, "These things are very personal", suggests that Feynman used relativistic mass for sentimental, not pedagogical, reasons.

I speculate as follows: It is probable that Feynman, by age 13 or 14 (circa 1932), studied relativity theory (it was about the time he was studying calculus), he would probably have studied from a book borrowed from a public library, and in the popular books on relativity published up to that time velocity-dependent mass m(v) appeared frequently. Furthermore, even if Feynman at that tender age realized that mass is a 4-scalar, it is very unlikely that he would try to explain relativity theory to his non-scientist girlfriend in terms so abstract as Minkowski spacetime. It's more likely he would choose the less correct but more appealing to "common sense" way of teaching relativity theory, involving m(v). Thirty years later, when Feynman composed his freshmen lectures on relativity, it seems he reconstructed his lessons on relativity for Arline. This would explain the inconsistency between his work in QFT, in which the mass of a particle is always constant, and his freshmen relativity lectures, in which m(v) appears.

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vanhees71, kith, SiennaTheGr8 and 7 others
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However, I am going to put my two-cents ...

I'd say that's significantly more than two-cents worth!

vanhees71 and sysprog
codelieb
I'd say that's significantly more than two-cents worth!

- special deal for PF ;-)

sysprog and PeroK
Gold Member
Feynman is not the only physicist who wrote about relativistic mass but instead used ordinary mass when doing relativistic physics. Feynman even goes so far as to state in FLP that all one has to do is replace ##m## with relativistic mass to understand the basics of the role played by mass in special relativity.

That was, of course, in the early 1960's. In 1989 Lev Okun's famous paper appeared in Physics Today. In the 1990's we saw the concept of relativistic mass disappear from the introductory textbooks. There are now very very few college-level introductory physics textbooks that even mention relativistic mass.

vanhees71
codelieb
Feynman is not the only physicist who wrote about relativistic mass but instead used ordinary mass when doing relativistic physics. Feynman even goes so far as to state in FLP that all one has to do is replace ##m## with relativistic mass to understand the basics of the role played by mass in special relativity.

That was, of course, in the early 1960's. In 1989 Lev Okun's famous paper appeared in Physics Today. In the 1990's we saw the concept of relativistic mass disappear from the introductory textbooks. There are now very very few college-level introductory physics textbooks that even mention relativistic mass.

In the interest of fairness (and playing devil's advocate, perhaps ;-) let me mention some things about another friend, FLP coauthor Matt Sands, whom I met around the same time I met Lev Okun. Matt is well-known for his publications and work on particle accelerators. He was hired by Caltech in 1950 to work on their 1.5 GeV electron synchrotron, and in 1963 he left Caltech to become SLAC's first Deputy Director, supervising its construction. Matt is someone who dealt at a practical level with relativistic mechanics, and he felt the concept of relativistic mass was useful. I suppose he considered it a useful concept because he actually used it, to make calculations related to particle accelerators he helped build. He was surprised by the suggestion of purging it from FLP, and (speaking to me personally) characterized Lev's outspoken opposition to relativistic mass as "a witch hunt."

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vanhees71
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Matt is someone who dealt at a practical level with relativistic mechanics, and he felt the concept of relativistic mass was useful.

He's not alone. Look at the letters to the editor that appeared in Physics Today following Okun's June 1989 article. Especially those written by Wolfgang Rindler.

The real pedagogical issue for me is understanding the Einstein mass-energy equivalence. Many authors claimed that the energy used to speed up a particle gets converted to the particle's relativistic mass, and that is what is meant by the equivalence. Instead it's the energy of the constituents of a composite body that contribute to the ordinary mass of the body. Authors weren't using the correct meaning, Feynman included.

vanhees71
codelieb
The real pedagogical issue for me is understanding the Einstein mass-energy equivalence. Many authors claimed that the energy used to speed up a particle gets converted to the particle's relativistic mass, and that is what is meant by the equivalence. Instead it's the energy of the constituents of a composite body that contribute to the ordinary mass of the body. Authors weren't using the correct meaning, Feynman included.

Feynman was the youngest member of the theoretical group at Los Alamos. His job was to calculate the energy released by the atomic bomb, which he did both theoretically and numerically, and to do that you need to understand something about mass-energy equivalence, including "the energy used to speed up" particles.

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Mentor
Many authors claimed that the energy used to speed up a particle gets converted to the particle's relativistic mass, and that is what is meant by the equivalence.

Did Feynman claim this? Not just that "energy used to speed of a particle gets converted to relativistic mass", but "that is what is meant by the equivalence"?

Instead it's the energy of the constituents of a composite body that contribute to the ordinary mass of the body.

By "ordinary mass" I assume you mean the invariant mass. What you say is true only if you are working in the body's rest frame, since that is the frame in which you have to evaluate the energy of the constituents.

Authors weren't using the correct meaning, Feynman included.

Can you give a specific Feynman reference to support this?

Gold Member
Feynman was the youngest member of the theoretical group at Los Alamos. His job was to calculate the energy released by the atomic bomb, which he did both theoretically and numerically, and to do that you need to understand something about mass-energy equivalence, including "the energy used to speed up" particles.

I have no doubt that Feynman understood it far better than I ever will. But I wasn't speaking of his understanding, I was speaking of the way he explained things in FLP.

Gold Member
Did Feynman claim this? Not just that "energy used to speed of a particle gets converted to relativistic mass", but "that is what is meant by the equivalence"?

Hmmm... Upon careful review I see that he never did make that claim. Quite the opposite. He devotes the last section of Chapter 15 to the equivalence of rest energy and mass.

Gold Member
By "ordinary mass" I assume you mean the invariant mass.

Yes, ala Lev Okun.

What you say is true only if you are working in the body's rest frame, since that is the frame in which you have to evaluate the energy of the constituents.

I agree. Sorry for not stating that explicitly because it is central to the argument.

Mentor
Matt [Sands] is well-known for his publications and work on particle accelerators. He was hired by Caltech in 1950 to work on their 1.5 GeV electron synchrotron, and in 1963 he left Caltech to become SLAC's first Deputy Director, supervising its construction. Matt is someone who dealt at a practical level with relativistic mechanics, and he felt the concept of relativistic mass was useful. I suppose he considered it a useful concept because he actually used it, to make calculations related to particle accelerators he helped build.
It appears that particle accelerator physicists in general held on to the concept of "relativistic mass" longer than most other particle physicists. When I was in graduate school in experimental high-energy physics in the late 1970s and early 1980s, the people I worked with never used "relativistic mass", only the invariant mass, without making a fuss over it; it was simply the way they did things. The one place where I did see "relativistic mass" was a course on particle accelerators, using the 1962 textbook by Livingston and Blewett. I think I remember the professor commenting on the difference in convention between the accelerator physicists and the other particle physicists.

vanhees71 and weirdoguy
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If you are familiar with the pedagogical debates revolving around the use of relativistic mass you may have heard of Lev, who was a very outspoken opponent and probably the most published
There is a thread on here somewhere where a poster by the name of levokun enquires about attitudes to relativistic mass and is advised to read the works of Lev Okun...

codelieb
There is a thread on here somewhere where a poster by the name of levokun enquires about attitudes to relativistic mass and is advised to read the works of Lev Okun...

Ha, ha! I found the post here. Thanks for reminding me of this (June 2013) thread. I participated in it, discussing the use of relativistic mass in FLP in some detail. Nearly everything I wrote in this thread was already stated in that one (so I could have saved my "two cents")!

Ibix
Rishabh Narula
thanks for all the replies.will read some more and maybe come back to this thread when ACTUALLY get it.

Homework Helper
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It appears that particle accelerator physicists in general held on to the concept of "relativistic mass" longer than most other particle physicists. When I was in graduate school in experimental high-energy physics in the late 1970s and early 1980s, the people I worked with never used "relativistic mass", only the invariant mass, without making a fuss over it; it was simply the way they did things. The one place where I did see "relativistic mass" was a course on particle accelerators, using the 1962 textbook by Livingston and Blewett. I think I remember the professor commenting on the difference in convention between the accelerator physicists and the other particle physicists.

I could do with some help from the accelerator physicsts over here:

Them and their relativistic mass! Once the seed is sown that a particle physically changes as its speed increases, it's a hard idea to shift.

In some ways the OP over there has a valid point. If mass really is increasing, why doesn't the particle eventually explode? Like the epicure in Python's The Meaning of Life!

codelieb
If mass really is increasing, why doesn't the particle eventually explode? Like the epicure in Python's The Meaning of Life!

- hasn't eaten the mint yet.

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Well, I'd also not go so far to "edit relativistic mass" out of the Feynman lectures. One should leave textbooks, particularly those of such a historical caliber as these. Of course, it's good to eliminate trivial typos and such things, but one should not alter the content. The Feynman Lectures as they are, are among the best theoretical-physics textbooks ever written, despite I think "relativistic mass" is really confusing the issue more than it helps, as this debate also shows again.

The real pedagogical issue for me is understanding the Einstein mass-energy equivalence. Many authors claimed that the energy used to speed up a particle gets converted to the particle's relativistic mass, and that is what is meant by the equivalence. Instead it's the energy of the constituents of a composite body that contribute to the ordinary mass of the body. Authors weren't using the correct meaning, Feynman included.
This is correct though it may be misunderstood, because it's a bit vague, and indeed it was misunderstood:
By "ordinary mass" I assume you mean the invariant mass. What you say is true only if you are working in the body's rest frame, since that is the frame in which you have to evaluate the energy of the constituents.
It's all much simpler when you correctly use the word mass only for the invariant mass, which is a scalar quantity (which is why it's called "invariant mass"), and "invariant mass" is not necessarily the same as "rest mass", as this example shows!

Invariant mass and rest mass are the same as long as you consider a situation, where you can use the concept of "point mass" (which in relativity is a very delicate issue in itself; if you ask me it doesn't exist in a literal sense at all), i.e., a body, where intrinsic excitations are irrelevant for the physics problem at hand. Then using the covariant formulation of energy and momentum, i.e., using the energy-momentum four-vector ##p=(E/c,\vec{p})## and ##p^2=(E/c)^2-\vec{p}^2=m^2 c^2## shows that indeed if the particle is at rest (meaning in the reference frame, where ##\vec{p}=0##) ##E=E_0=m c^2##.

Now, if you deal with macrosocpic objects, and that you do if you use the classical-particle picture, these objects consist of very many "particles". If you deal with a physical situation, where you can consider such a macroscopic object as "closed system", it's energy and momentum is conserved, and again the "on-shell condition" ##p^2=m^2 c^2## holds, where now ##p## is the total four-momentum of the object, and indeed ##E_0=mc^2## is the total energy in the center-momentum frame. One should note that here the "invariant mass" ##m## indeed depends on the internal energy of the object, and this is indeed the amazing and significant content of Einsteins "mass-energy equivalence". Indeed if you consider a body which is in thermal equilibrium, you have internal energy in the thermal motion of its constituents.

An extreme example, where this is important, is black-body radiation. Though it consists of massless photons as constituents the electromagnetic energy ##\propto V T^4## of the thermal-radiation field inside a cavity contributes to the invariant mass of the cavity by the corresponding "mass equivalent" ##\propto V T^4/c^2##.

Another important example is the "mass defect" of atomic nuclei. Even in the ground state the invariant mass of a nucleus is not the sum of the invariant masses of the nucleons it consists of, but it's less by the binding energy divided by ##c^2## (an effect, from which e.g., the Sun gets its energy through nuclear fusion).