First, your formulation of kinetic energy is Newtonian, not relativistic. When relativistic velocities are involved, you have to use the relativistic version:
which can be expanded in an infinite binomial expansion:
(I think the "approximately equals" should be "equals", since this is the infinite series, not a truncation). Anyhow, when v = 0 you get E = m
0c
2, which you may have seen before. When v/c is small enough, we can ignore all the terms with c in the denominator
and we're always calculating
differences in kinetic energy at different speeds so the m
0c
2 terms cancel out, and there's your classical kinetic energy: (delta E) = m
0(v
A2 - v
B2)/2.
But that's all irrelevant; m
0 is the "rest mass", and the rest mass of photons is exactly zero, so their kinetic energy is exacly zero. They do carry energy (just not in kinetic form) and they do (in manner of speaking) have and respond to gravitational fields.
Yet you could pose the question with particles that do have non-sero rest mass traveling
close to the speed of light, so it's a valid question.
Consider all the ways energy can be stored in the system. Offhand I come up with electromagnetic energy (if the particles are photons), kinetic energy (if the particles have non-zero rest mass), and gravitational potential energy. As the particles separate, the gravitational potential energy
increases. The kinetic energy of non-zero-rest-mass particles
decreases (they slow down) or the electromagnetic energy of photons
decreases (they get longer wavelength and are red-shifted). Without doing the calculations, I bet the decrease exactly equals the increase in all cases.
As an aside, this sort of thing sort of explains the idea that the Universe may be a quantum fluctuation with zero overall mass. There's some reason for assigning potential energy (which can be measured relative to any position) the value 0 when everything is an infinite distance apart. If you do that, and you already know that the act of separating things
increases gravitational potential energy, it follows that the gravitational potential energy of the Universe that we see is negative*. In fact, it's not
terrifically different, as such things go, from the negative of the amount of other positive energy we see ... and maybe they're exactly equal in magnitude but opposite in sign, and exactly cancel each other out. Maybe.
Be very careful of drawing new conclusions from the above; it's really a hopeless oversimplification. To draw conclusions you need the math, and I've long forgotten the math and wouldn't try to teach it here if I remembered it.
Finally, modern physicists dislike the term "rest mass" and try to avoid it. I'm just an old fart. Sylas is probably going to discredit it all.
Say, anybody seen Eta_Carinae?
-------------------------
* Say the current gravitational potential energy of the Universe is X. Getting everything infinitely far apart would increase X and make it zero, therefore X is negative now.
{edited to fix classical K.E. formula)
This message has been edited by JonF, 01-16-2005 21:30 AM