# Do all particles with zero rest mass and finite energy have to travel with speed of light in vacuum?

Suppose there exist particles with zero rest mass ($$m_0=0$$) which are not photons. Let them have a non-zero finite energy $$E$$. Do they also, just like photons, travel through space with the speed of light?

• Apr 10, 2020 at 7:43
• It's unfortunate that that Wikipedia article uses relativistic mass. (It also has some ungrammatical English and poor typography). Apr 10, 2020 at 7:59

Yes, it can be derived from the energy-momentum-relation of SRT. $$E^2 - (pc)^2 = (m_0c^2)^2$$ which for $$m_0 = 0$$ becomes $$E^2 - (pc)^2 = 0$$ $$\implies \frac E p = c \quad(*)$$ Since $$E = \gamma m_0c^2$$ $$p = \gamma m_0v$$ $$\implies \frac E p =\frac{c^2}{v} \quad(**)$$ $$(*) = (**)$$ yields $$v = c$$
• That derivation is a little shaky because it involves the indeterminate form $0/0$ in the 2nd last step. Apr 10, 2020 at 7:56
• No, that step is fine. $E$ and $p$ are nonzero. You can take the ratio for finite $m_0,\gamma$ and then the limit as $m_0 \to 0, \gamma \to \infty$ Apr 10, 2020 at 8:38