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$E$ = $mc^2$ And also $E$ = $hf$ (f - frequency)

And hence Einstein said $m$ = $hf\over c^2$ And so photons have mass

But later he also said

$M$ = $M_0\over \sqrt {1-v^2/c^2}$

Where if we put $v = c$ we get

$M = M_0/0 \leftrightarrow M=\infty$

And so photon travelling at the speed of light ($c$) have undefined mass (or $\infty$)

And so in one equation it is said photons have mass and in one it is said they have $\infty$ mass.

So do i have a problem in understanding or is there really any discrepancy ?

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    $\begingroup$ Don't apply formulae that pertain to massive objects (i.e. formulae in which $v \neq c$ appears) to massless objects. $\endgroup$
    – ACuriousMind
    Aug 10, 2014 at 18:46
  • $\begingroup$ $M_0=0$ for photons (they have no rest mass), so $M= M_0/\sqrt{1-v^2/c^2}=0/0$ is actually indeterminate. $\endgroup$
    – Andrew
    Aug 10, 2014 at 18:55
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    $\begingroup$ Photons have mass (=your formula) in the sense that when confined in a cavity for example, the total mass of the cavity will increase by that much $\endgroup$ Aug 10, 2014 at 19:01
  • $\begingroup$ Why do photons have 0 rest mass? $\endgroup$
    – NeilRoy
    Aug 10, 2014 at 19:24
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    $\begingroup$ There are endless version of this question in many guises already on the site. physics.stackexchange.com/q/3541 doesn't look the same but is, and in the sidebar of that you will find many more which also have the same answer. $\endgroup$ Aug 10, 2014 at 19:56

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And so photons have mass

No - photons don't have mass - they have momentum. And energy. But just because energy is equivalent to mass, doesn't mean they have mass. And they can only travel at the speed of light. A photon cannot travel at any other speed - so you cannot apply the Lorentz transformation to it. The Lorentz transformation applies to "rest mass" - but there is no such thing for a photon since it cannot be observed at rest in any frame of reference...

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$E$ = $mc^2$

A much better expression is $E^2 = (mc^2)^2 + (pc)^2$, where $m$ is the "mass" (also known as "intrinsic mass", also known "rest mass", but most physicists nowadays just use "mass") of the particle and $p$ is the particle's momentum. This reduces to $E=mc^2$ in the special case of a particle with zero momentum, but it also reduces to $E=pc$ in the case of a particle such as a photon with zero mass.

Using $E=mc^2$ as a general expression implies a rather different concept of mass, that of relativistic mass. Many physicists did indeed use the concept of relativistic mass early on in the development of relativity theory. At least initially, even Einstein was in that camp. Most of those physicists, Einstein included, abandoned that concept for the concept of "rest mass. " There are just too many problems with the concept of relativistic mass. The concept of "rest mass" (or "intrinsic mass" or just "mass") makes much more sense than does the concept of relativistic mass.

Note that the term "rest mass" is a bit contradictory for massless particles such as photons. There is no frame in which a photon is at rest. This apparent contradiction vanishes if you use the phrase "intrinsic mass" (or just "mass") in lieu of "rest mass."

There are a few hangers-on amongst professional physicists who still prefer the concept of relativistic mass over rest mass. These physicists are now few and far between. Eventually they'll die, and the concept of relativistic mass will eventually die with them.

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    $\begingroup$ While I agree with your last paragraph, it makes me sad... $\endgroup$
    – Floris
    Aug 13, 2014 at 1:27
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    $\begingroup$ @Floris - Max Planck once said that science advances one funeral at a time. $\endgroup$ Aug 13, 2014 at 1:37
  • $\begingroup$ I had not heard that - but it makes sense. $\endgroup$
    – Floris
    Aug 13, 2014 at 1:38

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