Certainly, but both are encoded in the energy-momentum tensor and for better or worse that's usually considered to be what generates the metric (or perturbation

Certainly, but both are encoded in the energy-momentum tensor and for better or worse that's usually considered to be what generates the metric (or perturbations thereof, if you want to do things that way). In Newtonian mechanics, kinetic energy is rotationally but not Galilean invariant, sure. But in GR in a local inertial frame the pressure T^{ij}, i=j, i!=0 sure looks like kinetic energy \gamma m(v^i)^2. [1] Should we really strongly distinguish between the pressure and T^00 just because in a local inertial frame the latter looks like \gamma mc^2 ? Sharpen the question by considering vastly different frames of reference. - -- [1] https://en.wikipedia.org/wiki/Kinetic_theory_of_gases#Pressure_and_kinetic_energy https://en.wikipedia.org/wiki/Kinetic_theory_of_gases#Pressu...