Photons are energy. According to general relativity they should bend space.
Assuming two photons pass one another in a large void of empty space how would they gravitationaly affect each other exactly? Would there be a change in their path, a change in color, both, neither or something entirely different?
General Relativity – How Do Photons Affect Each Other Gravitationally?
general-relativitygravitylinearized-theoryphotonsscattering-cross-section
Best Answer
One can quantize linearized spacetime perturbations in General Relativity and compute the effect of photons scattering elastically by exchanging virtual gravitons. This theory isn’t consistent at Planck-scale photon energies but is believed to be fine at the energies of photons we observe... even very high-energy gamma rays.
All the energy coming in has to come out. In the center-of-momentum frame the two photons each enter with energy $E$ and exit with energy $E$. Thus in this frame there is no change in their frequency (“color”).
Their direction does change (but the effect is tiny). There is a probability of scattering through different angles, and this is described as usual by a differential cross-section $d\sigma/d\Omega$ which depends on the scattering angle $\theta$.
The details of the calculation are in this 1967 paper: Gravitational Scattering of Light by Light.
The differential cross section for unpolarized photons found in this paper — and then corrected in an erratum — is
$$\frac{d\sigma}{d\Omega}=\frac{32G^2E^2}{c^8\sin^4{\theta}}\left(1+\cos^{16}{\frac{\theta}{2}}+\sin^{16}{\frac{\theta}{2}}\right).$$
As you can guess, $G$ is Newton’s gravitational constant and $c$ is the speed of light.
Try computing the area $G^2E^2/c^8$ for a visible photon (or a gamma-ray photon) to see how tiny and unmeasurable this scattering effect is!