B Photon Energy Without Rest Mass – How?

[Uncle Thi]

TL;DR Summary

This post raises a question about the apparent contradiction between the equation E = mc^2 and the fact that photons, which have zero rest mass, still carry energy described by E = hf. It seeks clarification on whether E = mc^2 only applies to particles with rest mass, or if there is a broader framework that includes massless particles like photons.

According to E = mc², energy seems to depend on rest mass.

But photons have zero rest mass (m₀ = 0), and still carry energy, expressed as E = hf.

How should this be understood?
Does E = mc² only apply to particles with rest mass?
Is there a more general form that includes massless particles like photons?

Looking for clarity on how this works.

 

[PeterDonis]

Does E = mc² only apply to particles with rest mass?

It only applies to particles with rest mass that are at rest. See below.

Is there a more general form that includes massless particles like photons?

Yes. The general equation is

$$
E^2 = p^2 c^2 + m^2 c^4
$$

where $E$ is the energy, $p$ is the momentum, and $m$ is the rest mass. For light, $m = 0$, so we just have $E = pc$; light has momentum as well as energy. For the case of a particle with nonzero rest mass at rest, we have $p = 0$, so we get $E = m c^2$.

Note that I used the term "light", not "photon", in the above; the concept of "photon" is a quantum concept and should not be used in the context of classical relativity. (Even in a quantum context, a photon is not best viewed as a "massless particle" unless you are willing to discard virtually all of the intuitions that the term "particle" naturally gives rise to. But that's a topic for a separate thread.)

 

[PeterDonis]

energy, expressed as E = hf.

This is a quantum formula. It is not relevant in a classical discussion, as here.

 

[Hornbein]

This common confusion arises because the definition of mass used by Einstein is not the same as current usage. He was using "relativistic mass" for which this is true by definition. In his 1905 paper he wrote E/c² = m so he was defining mass in this way. Current usage is that "mass" means rest mass so a different equation is used.

I seem to recall in George Gamow's One, Two, Three, Infinity he wrote that airplanes approaching the speed of light would become so heavy they would crash, a completely bogus idea. No wonder he didn't get the Nobel Prize for predicting the Cosmic Microwave Background. Though that memory is from was sixty years ago so I don't trust the recollection.

 

[Orodruin]

Does E = mc² only apply to particles with rest mass?

It is an expression for the rest energy. A photon cannot be at rest.

 

[Herman Trivilino]

According to E = mc², energy seems to depend on rest mass.

That equation is written incorrectly for the general case. The correct equation is $E_o=mc^2$ where $E_o$ is the rest energy and $m$ is the ordinary mass, what some people call the rest mass.

Einstein's great discovery is the equivalence of rest energy and mass.

So a massless particle has zero rest energy. And, as @Orodruin points out, a massless particle can never be at rest.

 

[Herman Trivilino]

This common confusion arises because the definition of mass used by Einstein is not the same as current usage. He was using "relativistic mass" for which this is true by definition. In his 1905 paper he wrote E/c² = m so he was defining mass in this way. Current usage is that "mass" means rest mass so a different equation is used.

True, but note that soon after 1905, by 1907 if I remember correctly, Einstein eschewed the notion of relativistic mass and instead preferred to refer to the rest mass.

 

[Hornbein]

Einstein's great discovery is the equivalence of rest energy and mass.

It was discovered by maybe Leo Szilard. Einstein didn't believe it at first and had to be convinced. Albert thought that it would be a subtle, difficult to detect phenomenon.

 

[PeterDonis]

It was discovered by maybe Leo Szilard. Einstein didn't believe it at first and had to be convinced.

Do you have a reference for this? Einstein introduced the concept in one of his classic 1905 papers.

 

[Hornbein]

Do you have a reference for this? Einstein introduced the concept in one of his classic 1905 papers.

I have read that paper. Einstein suggests that the effect might be detectable with "radium salts." As for him not believing it that comes from something I read long ago about the development of the atomic bomb.

On July 12, 1939, Szilard and Wigner drove in Wigner's car to Cutchogue on New York's Long Island, where Einstein was staying.[10] When they explained the possibility of atomic bombs, Einstein replied: "Daran habe ich gar nicht gedacht" ("I completely did not even think about that").[11] -- https://en.wikipedia.org/wiki/Einstein–Szilard_letter

 

[PeterDonis]

I have read that paper. Einstein suggests that the effect might be detectable with "radium salts."

In other words, you agree that Einstein, in 1905, discovered the concept of mass being converted to energy--not "maybe Leo Szilard" when fission was discovered three decades later.

As for him not believing it that comes from something I read long ago about the development of the atomic bomb.

The quote you give says that Einstein had not thought of an atomic bomb as a possibility. It says nothing whatever about Szilard discovering that rest mass could be converted to energy and Einstein not believing it and having to be convinced, which is the claim you made that I asked for a reference for.

 

[Herman Trivilino]

I have read that paper. Einstein suggests that the effect might be detectable with "radium salts."

So, in 1905 Einstein is suggesting that the mass-energy equivalence might be detectable, yet you claim that he hadn't discovered the mass-energy equivalence!

As for him not believing it that comes from something I read long ago about the development of the atomic bomb.

Things like radium salts and the atomic bomb were, at that time, possible evidence of the mass-energy equivalence. Einstein had thought of the former in 1905, but not the latter until it was pointed out to him by Szilard in 1939.

And to you that means the mass-energy equivalence was discovered, not by Einstein in 1905, but "maybe" by Szilard in 1939?!

 

[SiennaTheGr8]

I don't interpret Einstein's original paper as using relativistic mass. Or I'm at least unsure. The thought experiment involves a body that gives off equally energetic radiation in opposite directions, such that its velocity is unaffected by the emission. That's the context that precedes the key concluding paragraphs, which I now quote (source: https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf):

If a body gives off the energy $L$ in the form of radiation, its mass diminishes by $L/c^2$. The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that

The mass of a body is a measure of its energy-content; if the energy changes by $L$, the mass changes in the same sense by $L/9 × 10^{20}$, the energy being measured in ergs, and the mass in grammes.

It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test.

If the theory corresponds to the facts, radiation conveys inertia between the emitting and absorbing bodies.

I interpret at least the first sentence as pertaining to a body's rest energy, simply because of what $L$ represents in the thought experiment (he was previously using $H$ and $E$ for the body's total energy in the two frames). After that, it's less clear to me. He's certainly generalizing the result, but is he intending to generalize it to situations where a body loses energy in a way that causes it to accelerate? Or is there instead an implied "restriction" to the kind of scenario described in the thought experiment? I suspect it's the latter, and that if he were "redefining" mass as a velocity-dependent quantity he'd be more explicit about it.

 

[Herman Trivilino]

I don't interpret Einstein's original paper as using relativistic mass.

In Lev Okun's article in the June 1989 issue of Physics Today, on page 34, he makes the point that in his original 1905 paper Einstein was indeed not using relativistic mass. But then in his 1906 paper he made the argument that total energy is equivalent to mass, where obviously mass would be relativistic mass, thus being inconsistent. But sometime between then and 1921 he had made it quite clear that the mass-energy equivalence refers to the equivalence of mass to rest energy, thus by then he had abandoned the use of relativistic mass.

The concepts of relativistic mass, transverse mass, and longitudinal mass were in use by physicists at that time. But of course, as you well know, they are all antiquated and have fallen out of use. The concept of relativistic mass hung on stubbornly until as recently as the 1990's when, having seemingly been of primary importance up until just before then, was removed from virtually every college-level introductory physics textbook.

It still lingers in pop-sci publications, even those written by physicists who didn't and don't use it in their own professional research.

 

[SiennaTheGr8]

The concepts of relativistic mass, transverse mass, and longitudinal mass were in use by physicists at that time. But of course, as you well know, they are all antiquated and have fallen out of use. The concept of relativistic mass hung on stubbornly until as recently as the 1990's when, having seemingly been of primary importance up until just before then, was removed from virtually every college-level introductory physics textbook.

It still lingers in pop-sci publications, even those written by physicists who didn't and don't use it in their own professional research.

There are still exceptions, though. Hobson/Efstathiou/Lasenby used it in their GR book (2006), and more recently d'Inverno/Vickers continued using it in the 2nd edition of theirs (2022). John Kogut (2018) uses it in deriving the relativistic momentum, but then tells the reader that he'll always instead use total energy thereafter to avoid confusion (doesn't make sense to me—wouldn't it be less confusing not to introduce it in the first place?). I've read large chunks of all three of these books, and IMO they're all very good!

 

[Herman Trivilino]

There are still exceptions, though. Hobson/Efstathiou/Lasenby used it in their GR book (2006), and more recently d'Inverno/Vickers continued using it in the 2nd edition of theirs (2022). John Kogut (2018) uses it in deriving the relativistic momentum, but then tells the reader that he'll always instead use total energy thereafter to avoid confusion (doesn't make sense to me—wouldn't it be less confusing not to introduce it in the first place?).

Well, you have to show that those authors are professional physicists who use it in their research for them to count as exceptions.

 

[SiennaTheGr8]

I just meant that they're serious physics textbooks, not pop-sci. Whether they use it in their research I don't know.

 

[Herman Trivilino]

I just meant that they're serious physics textbooks, not pop-sci. Whether they use it in their research I don't know.

In the cases I've seen, they do. But I've not done anything even close to exhaustive.

Take for example A Brief History of Time. Relativistic mass is used there throughout but I don't think Hawking used it in his research and professional publications.