I guess I’m trying to understand the difference between a rock orbiting earth, that would radiate gravitational waves. And say a photon orbiting a black hole that is just following a straight line path. Why does one radiate and the other doesn’t? Best Regards, Andy
Questions of quantum gravity aside, viewed as moving blob of energy-momentum, a photon moving around a black hole should produce gravitational waves just like a massive particle would. In practice, this is a negligible amount. The fraction of the photon's (wavepacket's) energy that gets converted to gravitational waves is proportional to the ratio of the photon's energy to the mass of the black hole ($c=1$ obviously). For any realistic photon/black hole pair this is a mind boggling small number.
What exactly happens when view the photon as a quantum mechanical particle, requires understanding of how quantum mechanics interacts with gravity, i.e. a theory of quantum gravity. I kind of suspect that this should have a somewhat universal answer within perturbative quantum gravity.
It is very important to understand the difference between a static gravitational field, and GWs.
According to GR, every object that does have stress-energy (not mass contrary to popular belief), does have a static gravitational field, and does curve spacetime.
Now a GW is disturbance in the curvature of spacetime, generated by an accelerated mass, that propagate as waves outwards from their source.
In general terms, gravitational waves are radiated by objects whose motion involves acceleration and its change, provided that the motion is not perfectly spherically symmetric (like an expanding or contracting sphere) or rotationally symmetric (like a spinning disk or sphere). A simple example of this principle is a spinning dumbbell. If the dumbbell spins around its axis of symmetry, it will not radiate gravitational waves; if it tumbles end over end, as in the case of two planets orbiting each other, it will radiate gravitational waves. The heavier the dumbbell, and the faster it tumbles, the greater is the gravitational radiation it will give off. In an extreme case, such as when the two weights of the dumbbell are massive stars like neutron stars or black holes, orbiting each other quickly, then significant amounts of gravitational radiation would be given off.
Now the answer to your question is that photons do have stress-energy, they do have a gravitational field, but they do not create themselves GWs (except is they accelerate, and orbit another object).
For a photon to have acceleration, they would need to move around (orbit) a massive object.