What would source the gravitational radiation that would thermalize the two reservoirs? The metric sourced by a uniform cloud of hot gas doesn't really lend i

What would source the gravitational radiation that would thermalize the two reservoirs? The metric sourced by a uniform cloud of hot gas doesn't really lend itself to periodic change (under a slicing) unless it's collapsing far from spherically symmetrically, and even then the momentum-energy carried out by electromagnetic radiation would dwarf the gravitational radiation. Also, if you let the hot and cold clouds fall into each other, then electromagnetic scattering will do the bulk of the work of thermalization (or more likely the heating of both clouds of inelastically colliding matter; cf. collisonless dark matter clouds sliding through one another).