A Tony Rothman Questions E=mc^2 in American Scientist

[mathman]

TL;DR Summary

Amer. Scientist Article by Tony Rothman questions $E=mc^2$. Questions?

In the latest issue of American Scientist there is an article by Tony Rothman questioning $E=mc^2$. Are the assertions made in the article valid? Are there comments by other physicists available?

 

[Dale]

The famous formula is valid, but it is also frequently misunderstood.

That said, I have no knowledge about the specific article in question. Maybe you can identify a particular question that concerns you from the article?

 

[Nugatory]

The article is behind a paywall at American scientist.org. The teaser is not reassuring: https://www.americanscientist.org/article/the-curse-of-emc2

 

[TSny]

The author has a homepage where there is a link to the article. I don't know if accessing the article through this link violates copyright law.

 

[Dale]

Are the assertions made in the article valid?

The article seemed mostly historical. I don’t know the history, so I have no reason to dispute it. Was there something you found particularly concerning?

 

[jartsa]

From the principle of relativity we know that the very weak radiation coming out of the rear of a very fast moving heater and the very strong radiation coming out of the front of the same heater exert opposite forces on the heater.

The physicists that tried to calculate the radiation reaction force on a moving heater did not make use of that simple fact. I mean they did not notice that if the calculated forces cause the heater to accelerate, then the forces have been miscalculated.

 

[mathman]

The article seemed mostly historical. I don’t know the history, so I have no reason to dispute it. Was there something you found particularly concerning?

The author cited (pre 1905) papers where people derived $E=\frac{3}{4}mc^2$ rather than the usual. He wondered whether $E=mc^2$ was valid at high energies.

 

[PeterDonis]

He wondered whether $E=mc^2$ was valid at high energies.

I strongly suspect he has never published such claims in a peer-reviewed journal--because he knows the reviewers would call bullshit.

In the second bullet under "Quick Take" on the page you linked to, it says: "Einstein's famous 1905 relativity paper is valid only for low velocities", which is false. Einstein's derivation of the Lorentz transformation in that paper is valid for any relative velocity (less than $c$, of course). His derivation of $E = mc^2$ in that paper, of course, is not valid for any velocity, but that's because that formula itself is only valid for zero velocity--see next paragraph.

It then says "in six further attempts he never succeeded in producing a universal derivation of $E = mc^2$". I'm not sure what "six further attempts" he's talking about, but since the equation $E = mc^2$ is only valid at rest (zero velocity) to begin with, of course Einstein, or anyone else, could never produce a "universal derivation" of the formula. So this statement strikes me as highly disingenous.

 

[Dale]

The author cited (pre 1905) papers where people derived $E=\frac{3}{4}mc^2$ rather than the usual. He wondered whether $E=mc^2$ was valid at high energies.

$E=mc^2$ is only valid for $p=0$. For non-zero momentum the relationship is $m^2 c^2=E^2/c^2-p^2$. The factor of 3/4 is not right.

 

[PeterDonis]

This article by Rothman in Scientific American in 2015 appears to have similar content to the one referenced in the OP:

https://www.scientificamerican.com/article/was-einstein-the-first-to-invent-e-mc2/

 

[PeterDonis]

This paper by Boughn also appears relevant (the paper references Rothman, and Rothman references his work with Boughn in the SciAm article I linked to):

https://arxiv.org/abs/1303.7162

 

[Histspec]

In the second bullet under "Quick Take" on the page you linked to, it says: "Einstein's famous 1905 relativity paper is valid only for low velocities", which is false. Einstein's derivation of the Lorentz transformation in that paper is valid for any relative velocity (less than $c$, of course). His derivation of $E = mc^2$ in that paper, of course, is not valid for any velocity, but that's because that formula itself is only valid for zero velocity--see next paragraph.
It then says "in six further attempts he never succeeded in producing a universal derivation of $E = mc^2$". I'm not sure what "six further attempts" he's talking about, but since the equation $E = mc^2$ is only valid at rest (zero velocity) to begin with, of course Einstein, or anyone else, could never produce a "universal derivation" of the formula. So this statement strikes me as highly disingenous.

Rothman wasn't referring to Einstein's "Electrodynamics of moving bodies", but to Einsteins follow-up paper in which he derived E=mc², namely the paper “Does the Inertia of a Body Depend on Its Energy Content?”. In general, Rothman's statements were based on Ohanian's analysis of Einstein's multiple attempts of deriving E=mc²:

1. Ohanian, H. 2009. Did Einstein prove E = mc2? Studies in History and Philosophy of Modern Physics 40(2):167–173. For a similar preprint version see: https://arxiv.org/abs/0805.1400

So it wasn't about the question whether the formula E=mc² is correct or not, but whether Einstein correctly derived it or not. Namely, do we have to consider stresses etc. in order to conclusively derive E=mc² ? Indeed, it was Planck as early as in 1907 who argued that Einstein's derivation, is only correct to first approximation, while a complete statement of mass-energy equivalence requires the consideration of internal processes too.

 

[vanhees71]

The author cited (pre 1905) papers where people derived $E=\frac{3}{4}mc^2$ rather than the usual. He wondered whether $E=mc^2$ was valid at high energies.

This has been done in a time, where SR hasn't really existed, and even today more than 115 years later people don't know von Laue's theorem. If $j^{\mu}$ is a current, then
$$Q^{\mu}=\int \mathrm{d}^3 x j^{\mu}(t,\vec{x})$$
is only a four-vector if $\partial_{\mu} j^{\mu}=0$.

Since for an electromagnetic field with non-vanishing $j^{\mu}$ the energy-momentum tensor of the em. field alone is not conserved,
$$P^{\mu}=\int \mathrm{d}^3 x T^{\mu 0}$$
is not a four-vector, and that's the origin of the famous "4/3 problem".

Nevertheless you can always choose some preferred inertial reference frame depending on the physical situation, specify the spatial 3D hypersurface there, and then define
$$P^{\mu}=\int \mathrm{d}^3 \sigma_{\nu} T^{\mu \nu}$$
in a manifestly covariant way, no matter whether $\partial_{\mu} T^{\mu \nu}=0$ or not. See the corresponding discussion in Jackson, Classical Electrodynamics.

 

[PeterDonis]

Rothman wasn't referring to Einstein's "Electrodynamics of moving bodies", but to Einsteins follow-up paper in which he derived E=mc², namely the paper “Does the Inertia of a Body Depend on Its Energy Content?”.

If that's the case, Rothman probably shouldn't have said "Einstein's famous 1905 relativity paper", since that is going to be taken by most readers to refer to the "Electrodynamics of moving bodies" paper.

So it wasn't about the question whether the formula E=mc² is correct or not, but whether Einstein correctly derived it or not. Namely, do we have to consider stresses etc. in order to conclusively derive E=mc² ? Indeed, it was Planck as early as in 1907 who argued that Einstein's derivation, is only correct to first approximation, while a complete statement of mass-energy equivalence requires the consideration of internal processes too.

Then I would say that, based on our modern view, the fundamental error here is in considering the formula $E = m c^2$ to be a general expression of "mass-energy equivalence", instead of just the $p = 0$ special case of the general energy-momentum relation $E^2 = p^2 c^2 + m^2 c^4$. In the modern view, "mass-energy equivalence" is not expressed by any single formula in relativity; depending on what you take that phrase to mean, it could be expressed by choosing units in which $c = 1$, or by the equations describing processes in which particles with nonzero rest mass are converted to photons, or vice versa. There simply isn't any general notion of "energy" and "mass" for which the formula $E = m c^2$ applies in all cases.