Why massless particles must travel at c

[wtronic]

Homework Statement

give a logical argument for why a particle with m0 = 0 with nonzero energy has to move with a speed equal to c.

Homework Equations

The Attempt at a Solution

I attempted using the momentum equation but I always get a multiplication by 0 which doesn't make sense to me... please some help

thanks

 

[2Tesla]

You can relate p, E, and v without explicitly involving m. Do you know how?

 

[Mr.Brown]

A particle with zero mass has to travel with c because his mass is independant of the reference frame (it´s zero in every reference frame) hence it has no rest energy in any reference frame ( you can´t transfer to one frame where it has energy and another where there is no energy because there is no rest energy).
But for the notion of particle to make any sense at all, having any manifestation in the real world, it has to contain some sort of energy and while this energy has to be the same in any reference frame it has to move with c.
how about that ?

 

[wtronic]

2tesla, maybe since p = mv and E = mc^2 then I can write E = (c^2*p)/v...? but this equation doesn't justify why it has to go like speed c... because any v along with any p will give a value for energy..

am I wrong?

 

[wtronic]

you seem to make sense, but I am guessing my professor is looking for something more mathematical... I don't know, if anything I will try with that

 

[Mr.Brown]

i guess the mathematical idea behind this is that $$p_{\mu}p^{\mu}=-M^2=0$$ hence if the mass is 0 the momentum and velocity are on the light cone which is one definition for speed c.

 

[wtronic]

hi Brown, I am wondering where did you get that equation... and what mu means... thanks]

 

[2Tesla]

Just in case Mr.Brown's indices are unfamiliar to you, the equation he's referencing is:

$$E^2 = (pc)^2 + (mc^2)^2$$

So, if you combine this with m=0 and the v=c^2*p/E you discovered just now, you should find your answer.

P.S. It doesn't make a difference here, but for future reference, in relativity:

$$p = \gamma mv$$

and

$$E = \gamma mc^2$$

where

$$\gamma = (1 - (v/c)^2)^{-1/2}$$