Is there any gravitational force between two photons?

Asked by: Rajeev Malhotra

Answer

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.
Answered by: Gregory Ogin, Physics Undergraduate Student, UST, St. Paul, MN