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There are two expressions regarding relativistic mass of photon; namely relativistic approach:

$$m = \frac {m_0} {({1 - \frac {v^2 } {c^2 }) } ^\frac {1 } {2} }. $$

And, energy-mass equivalence formula:

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

There common uses in physics books:

First formula is used to prove that rest mass of photon is zero.

Second formula is used to find out what is called kinetic mass of photon in motion.

MY QUESTION: first formula suggests that photon must have infinite mass in motion, and second gives finite mass (kinetic), why is this disperacy?

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None of the formulae you quote can be used to conclude that the mass of a photon is zero. The formula that should be used to do so is the generic special relativistic relation $E^2 - p^2c^2 = m^2c^4 $ in conjunction with the formula relating the energy and momentum of a photon $E=pc$. Putting this latter formula into the former immediately yields $m=0$, i.e. that mass of a photon is zero.

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  • $\begingroup$ To be clear, the symbol $m$ in your answer denotes the invariant mass while the symbol $m$ in the OP's question does not. This might lead to some confusion. $\endgroup$ – Alfred Centauri Nov 27 '17 at 17:53
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Well, by definition a photon (a uniparticle state of the quantized e-m field) is massless, i.e. has mass = 0. There is no photon at rest, so that "photon in motion" is a linguistic mistake called pleonasm.

The first formula you wrote is applicable to objects with non-zero mass (inertial mass, or rest mass - you can see that for photons these two terms do not apply, there's no rest state/IRF at rest for a photon and inertia as opposition to acceleration doesn't apply, as photons are not subject to acceleration/deceleration), so it is pointless to invoke it here.

Read more here: https://en.wikipedia.org/wiki/Mass#Definitions_of_mass or some wonderful articles by the late Lev Okun, such as this one: https://arxiv.org/abs/hep-ph/0602037

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