# E = pc derivation

1. Apr 29, 2010

I was reading the derivation of this equation on hyperphysics.com and after searching all over the internet it seems that eveywhere seems to just accept this equation is valid for light, even though a rest mass of 0 would make p equal to 0.

Sorry if I'm being stupid or if this has been asked a million times before but I can't find an explanation anywhere as to why light having zero rest mass gives the equation E = pc, despite p being proportional to the (zero) rest mass of light.

2. Apr 29, 2010

### George Jones

Staff Emeritus
Re: E=pc

For any particle, massive or massless, the relativistic relation between mass, energy, and spatial momentum is given by

$$E^2 - \left( cp \right)^2 = \left( mc^2 \right)^2 .$$

Using $m = 0$ in this relation results in $E = cp$.

3. Apr 29, 2010

Re: E=pc

But surely if p = m0v/sqrt(1-v2/c2), that would make p equal to 0?

4. Apr 29, 2010

### Staff: Mentor

Re: E=pc

When v = c, that equation leads you to divide by zero, which has a mathematically undefined result. So it simply doesn't apply when v = c. The equation George showed you is more general.

5. Apr 29, 2010

Re: E=pc

Yeah sorry my mistake, but that still would mean that p is an undefined value, despite it being in the equation George posted.

6. Apr 29, 2010

### Ich

Re: E=pc

It means that your equation has nothing to say on a photon's momentum, while George Jones' has.

7. Apr 29, 2010

Re: E=pc

But you can't just ignore the equation for momentum and state that light has a momentum, p, purely for convenience.

Unless momentum can be defined as something else, other than the equation I posted.

8. Apr 29, 2010

### ansgar

Re: E=pc

E^2 = p^2 + m_0^2

is the general formula

E = gamma m_0

applies only to massive particles

where is your lack of understanding?

9. Apr 29, 2010

Re: E=pc

It was in the definition of momentum itself, I've found the answer the answer anyway in the de Broglie wavelength equation, which is something I knew already was just having trouble remembering it. Cheers for the help people.

10. Apr 29, 2010

### George Jones

Staff Emeritus
Re: E=pc

Even classically, the energy and momentum densities of electromagnetic waves are related by E = pc.

11. Apr 29, 2010

### closet mathemetician

Re: E=pc

I think he's saying that in the equation E=pc, p=mv, so that if m=0, then p=zero, then E=0.

Unless maybe saying m=0 means that relativistic mass is zero, not rest mass?