Is there any gravitational force between two photons?
Using classical physics, no.
However, with Einstein's general relativity, I believe the answer to this question
is yes. One of the key ideas we get from general relativity is that 'gravity' is a
curvature of space-time caused by an energy density. And since light does carry
energy, and is affected by spacetime curvature, it makes sense that the two photons
will be 'pulled' _**slightly**_ towards each other, although since photons have
zero mass, 'force' is a bad word to use because Newton's 'F=ma' does us no good.
When speaking casually of general relativity, usually one hears 'mass' rather than
'energy density' as being the cause of space-time curvature, but that is really
mostly because what we refer to as mass is (also according to general relativity) a
_very_ dense form of energy. How dense? By taking a look at the famous E=mc2, we
see that a kilogram of mass is associated with 8.99x1016 joules, or enough energy
to supply a common USA resident with energy for ~350,000 years. The energy
associated with mass is so much greater than other types of energy that we normally
just think of gravity as a phenomenon caused by mass.
Now photons, though technically massless, do carry energy. The energy of a photon
with a wavelength of 560 nanometers ('yellow light') is given by E=hc/ where E is
the energy, h is planck's constant (6.626x10-34 J*s), c is the speed of light, and
is the wavelength of the light. The energy of our photon is thus 3.55x10-19 J.
Note that this is 35 or so orders of magnitude below mass energy.
If I were to break all kinds of rules, and calculate an 'equivalent mass' for our
friendly yellow photon, I would find that 3.55x10-19 J would be the energy
inherent in 3.95x10-36 kg of mass, or .0000000024 times the mass of a proton.
Again I want to note that because photons are massless, Newtonian kinematics and
'force' do us absolutely no good here. That said, it seems to me that a photon in
the presence of another photon would be bent or 'attracted' as if it were passing
through the gravitational field caused by a mass on the order of a billionth of the
mass of a proton.
Gregory Ogin, Physics Undergraduate Student, UST, St. Paul, MN
'I have no special talents. I am only passionately curious.'