Does p=mc Apply To Photons? | A.P. French's Special Relativity

[Aaron121]

In A.P. French's Special Relativity, the author said the following,

For photons we have

$E=pc$​

and

$m=E/c^{2}.$

​

Combining these, we have

$m= p/c$​

​

As I understand, photons are massless, so I don't think the last equation above applies to photons, but then, when deriving it, he used an equation proper to photons ($E=pc$).

So in which context is $m=p/c$ valid?

 

[Dale]

The generally valid formula is $m^2 c^2 = E^2/c^2-p^2$. So the only context where that would be valid is for $E=0$, in which case I don't think you actually have a particle.

 

[Orodruin]

In no context apart from old textbooks where relativistic mass (now typically considered an outdated concept) is used. Do not use outdated textbooks.

 

[Aaron121]

@Orodruin So if I understood correctly, $m$ in $p=mc$ is the relativistic mass $\gamma m_{0}$?

But then this is in contradiction with the way the author describes $m$ in $E=mc^{2}$. He said,

If, further, we suppose that $m=E/c^{2}$ describes a universal equivalence of energy and inertial
mass,...

 

[Orodruin]

@Orodruin So if I understood correctly, $m$ in $p=mc$ is the relativistic mass $\gamma m_{0}$?

But then this is in contradiction with the way the author describes $m$ in $E=mc^{2}$. He said,

That statement is false. I suggest changing textbook.

 

[PeroK]

@Orodruin So if I understood correctly, $m$ in $p=mc$ is the relativistic mass $\gamma m_{0}$?

But then this is in contradiction with the way the author describes $m$ in $E=mc^{2}$. He said,

The problem is that Physics Forums generally doesn't use the concept of relativistic mass. There is a page about why not here:

https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

If you are going to learn from French's book this is a problem if you post questions on here. For me, it's a bit like posting arithmetic problems with the old British pounds, shillings and pence. Relativistic mass (like the old British monetary system) feels like something from a bygone era.

I learned SR without relativistic mass. Maybe I'm biased, but it does seem like a truly terrible idea!

 

[Dale]

That statement is false. I suggest changing textbook.

@Aaron121 I agree with @Orodruin here. I would be highly suspicious of this textbook. I would strongly recommend a better source. Any source that uses relativistic mass as anything other than a historical curiosity is automatically highly suspect.

I am not aware of French's "Special Relativity", but Taylor and Wheeler's "Spacetime Physics" is well regarded: https://www.amazon.com/dp/0716723271/?tag=pfamazon01-20

 

[Vanadium 50]

Taylor and Wheeler is excellent. (Disclaimner: I was Ed Taylor's student)
Relativistic mass was coined in 1906 and discarded in 1908. Yet it still marches on.

 

[vanhees71]

In A.P. French's Special Relativity, the author said the following,

​

As I understand, photons are massless, so I don't think the last equation above applies to photons, but then, when deriving it, he used an equation proper to photons ($E=pc$).

So in which context is $m=p/c$ valid?

Look for another book then. One should not use textbooks, where it is claimed that energy and mass are (besides a conversion factor $c^2$) the same. This is highly misleading. Energy is the temporal component of the energy-momentum four-vector, and mass is a scalar.

A massless classical particle is defined by the energy-momentum relation $E=|\vec{p}|c$ in any inertial reference frame. These are are fictitious notion though, because there are no massless classical particles known.

Photons are one-quantum Fock states of the electromagnetic field; for the generalized momentum-helicity eigenstates the dispersion relation looks as if they were massless classical particles though photons are the least particle-like elementary quantum one can think of. It has not even a position observable in the usual sense.

 

[PeterDonis]

So if I understood correctly, $m$ in $p=mc$ is the relativistic mass $\gamma m_{0}$?

No, $m$ in $p = m c$ is the "relativistic mass" of a photon, for which $\gamma$ is undefined and $m_0$ is zero, so you can't use the formula $m = \gamma m_0$ the way you can for the relativistic mass of an ordinary object. In other words, for a photon the concept of "relativistic mass", while it can technically be defined, is even less useful and more problematic than it is for ordinary objects with $m_0 > 0$. Which is yet another reason to not learn SR from this textbook.

 

[PeterDonis]

this is in contradiction with the way the author describes $m$ in $E=mc^{2}$.

Yes, it is, since the concept of "inertial mass" can't be applied to photons anyway, yet the author invokes this formula in defining $m$ for photons. So, another reason not to use this textbook.

 

[SiennaTheGr8]

French book is outdated but very good in some ways. Emphasizes experimental verification, and he even gets into Terrell rotation. But not a first-choice today, for sure.

 

[PeroK]

@Aaron121 David Morin has put the first chapter of his book for free on line:

http://www.people.fas.harvard.edu/~djmorin/Relativity Chap 1.pdf

I think it looks good. You could take a look.

Alternatively, I like:

https://www.goodreads.com/book/show/6453378-special-relativity

 

[Dale]

Relativistic mass was coined in 1906 and discarded in 1908. Yet it still marches on.

Discarded, but recycled for the next 112 years and counting.

 

[Dale]

So in which context is m=p/c valid?

The generally valid formula is $m^2 c^2 = E^2/c^2-p^2$. So the only context where that would be valid is for $E=0$, in which case I don't think you actually have a particle.

I just realized that it isn't even valid in the $E=0$ case because of the sign. For $E=0$ we get $m^2 c^2 = -p^2$. This only holds if $E=0$ and either $m$ is imaginary or $m=p=0$.

He must be using $m$ as relativistic mass instead of invariant mass. Using relativistic mass is bad enough, but using it and just calling it "mass" is even worse. The unqualified term "mass" should always refer to the invariant mass and never the relativistic mass.

 

[SiennaTheGr8]

He must be using $m$ as relativistic mass instead of invariant mass. Using relativistic mass is bad enough, but using it and just calling it "mass" is even worse. The unqualified term "mass" should always refer to the invariant mass and never the relativistic mass.

Using "mass" for "relativistic mass" should probably disqualify a book from consideration for a beginner's first choice today, but I won't let it stop me from picking the brains of Rindler, Feynman, French, Bondi, or Schwartz every now and then!