Do two photons traveling in opposite directions emit gravitational waves? If so, does it mean that any volume filled with photon gas will eventually degrade into graviton gas?
In other words, if flat non-expanding universe is filled with photon gas, will the photons over time annihilate into gravitons? Or will photons and gravitons somehow reach equilibrium?
This is a very hypothetical question.
a) we do not have a quantized theory of gravity working in conjunction with elementary particles ( the photon is an elementary particle).
b) we have not detected a graviton in the way we detect photons
If we forget a) and assume for b) that if gravitons exist they are the 0 mass carriers of the gravitational interactions the way the photon is the carrier of the electromagnetic interactions and also assume that what we know of quantum mechanics and kinematics of elementary particles pertains to the graviton too then
no, the photons cannot radiate gravitons and vice versa because both are moving at the velocity of light and cannot be accelerated or decelerated in order to give up radiation in the classical way.
In this hypothesis, as elementary particles, they can interact by scatterings, with a very very low probability due to the smallness of the gravitational interaction. Through higher order diagrams in the scattering crossection, they may change form and produce other elementary particles depending on their energy, or just photons and gravitons of different energies. There will be an equilibrium of sorts.
Without a solid theory of everything all this is speculation.
In short, photons perfectly interact with gravity (or, if you like it more, with still hypothetical gravitons). It includes planar EM waves (i.e. photons traveling in parallel directions), because the stress–energy tensor is non-zero. Even a single photon must have, theoretically, its gravitational field.
But it is not clear, why an equilibrium gas should give most of its energy to gravitational field, depriving photons of it. IMHO it depends on exact conditions (how hot and dense is the gas and in which cosmology is it supposed to exist).
Photons can not emit gravitons, only mass (matter) can. If the photons are energetic enough, some of them will be changed into matter (which will emit gravitons), so you would end up with a mixture of "weak" photons, particles with mass, and gravitons. The ratio of each, depends on how much energy the initial photons have. If they are weak, no mass will be created and therefore no gravitons. If this is the case, you end up with nothing but the photons you started with.
