When I saw the questions why matter-anti matter annihilation produces photons not gravitons, it suddenly occured to me that if the latter really happens, it means the stress energy tensor vanishes completely because gravitational wave has zero stress energy tensor. So is it allowed when the ordinary matter(particle matter or radiation matter) convert to gravitational wave in general relativity just as particle matter convert to radiation matter in special relativity? Does such process violate the Einstein's equation?
Something like this happens in Hawking radiation. Matter (particle-antiparticle pairs) is produced with the energy coming from the gravitational field. This implies a change in the stress-energy tensor.
It isn't clear to me whether annihilation to two gravitons is allowed, but assuming it is then yes the stress-energy tensor will change as a result.
Your description is implicitly semi-classical because you are equating a classical description to the expectation value of a quantum system. This is a perfectly reasonable approximation, and indeed it's how Hawking radiation was first described. But I doubt we can get any real understanding of the processes involved from what is an effective theory. To understand exactly what's happening will require a theory of quantum gravity that doesn't currently exist.