Massless particle that have momentum not equal to 0

Question

I saw some problems in special relativity that use conservation of total energy and momentum and the conclusions are spectacular. My problem is the following: how can a massless particle like neutrino have mementum not equal to 0? If their rest mass is 0 then the momentum $$p = \frac{m_0v}{\sqrt{1-\frac{v^2}{c^2}}}$$ should be also 0.

Answer 1

That formula for momentum is only true for massive particles.

Here's what is always true: A particle with a mass $m$ ($\geq 0$) can have an arbitrary momentum $p$ (in some direction, with magnitude $\geq 0$). The energy of such a particle is

$$ E = \sqrt{m^2c^4 + p^2c^2}$$

The velocity of a particle is equal to

$$ v = \frac{pc^2}{E} $$

When $m = 0$, $E = pc$ and so $v=c$ -- the particle must travel at the speed of light. For $m \neq 0$, you can solve this for $p$ and find that

$$ p = \gamma m v $$

where $\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$.