# Why do massless objects travel at c?

1. Apr 17, 2010

### particlemania

Why do photons, gluons, and all massless particles move at c?

and will this fail if the Equivalence principle is disproved?

2. Apr 17, 2010

### karkas

First of all it is critical to understand that it is a postulate that light travels at maximum universal speed c. Photons also have mass because they (obviously) have Energy E=hf, but they do not have a rest mass m0, because in the formula

$m=\frac{m_0}{\sqrt{1-(v/c)^2}}$ it would be m->inf unless m0=0.

In that formula it is shown that anything traveling with speed c will have no mass, only "mass" due to energy-mass equivalence. If you are talking about disproving mass-energy equivalence, then I guess we would have particles traveling at speed c and having no energy or mass, which seems absurd since light can be observed both as a particle and wave.

Last edited: Apr 17, 2010
3. Apr 17, 2010

### bcrowell

Staff Emeritus
There are some mistakes in karkas's #2. The square root is missing from the denominator of the equation's right-hand side. Also, karkas's argument shows that if a particle travels at c, it must have zero rest mass. But what particlemania asked for was a proof that if a particle has zero rest mass, it must travel at c. Finally, the statement that "it is a postulate that light travels at maximum universal speed c" is not necessarily incorrect, but it is somewhat misleading in the present context, because: (1) there are various axiomatic foundations for relativity, and some do not require constancy of c as an axiom (Rindler 1979); (2) regardless of the axiomatization, one can always prove that if there is some frame-independent velocity, then massless particles must travel at that velocity.

As an intuition-building warmup, imagine that in purely Newtonian physics, we had a particle with zero mass. Even the tiniest perturbation, with the most miniscule energy, would be sufficient to accelerate it to an infinite velocity. This is more or less how it works in relativity, except that the relevant limit is not v->infinity but v->c. For instance, neutrinos have almost no mass, and therefore essentially all neutrinos that we observe are moving at very close to c.

A more rigorous argument is that E^2-p^2=m^2 (in units where c=1). The case of zero mass gives |p|=E, and this is only possible if, in the limit m->0, we have $m\gamma v=m\gamma$, so v=1.

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

4. Apr 17, 2010

### dx

In classical relativistic mechanics, the action for a free particle is proportional to the proper time, the proportionality constant being -m. If m = 0, then the action would be the same for all paths, and the action principle would make no sense. So the idea of a massless particle is not meaningful in classical relativistic mechanics.

In quantum mechanics, the idea of velocity is replaced by the idea of 'group velocity' of wave-packets. For massless quantum mechanical particles with the Lorentz invariant dispersion relation ω² = k² (in c = 1 units), the group velocity of wavepackets is equal to the speed of light.

5. Apr 17, 2010

Thanx a lot!