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
'The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.'