How is it that energy of matter yields gravity if the amount of energy in a system is frame dependent while the force caused by gravity is not?

Question

I've been told that the gravitational field arises due to the energy density terms in the stress-energy tensor of matter and therefore that all energy of matter exerts a gravitational field effect, regardless of whether this is energy in the form of mass of something else.

However the total amount of energy of matter in the universe is frame dependent (because in different inertial frames of reference objects have different velocities and therefore kinetic energies) and equally the energy density of the universe is frame dependent too.

This would imply that the amount of gravitational force caused on a test mass by this energy differs in different frames of reference so its acceleration would be different depending on which inertial frame we do the calculation in which doesn't make sense. So where is my reasoning going wrong here?

Answer 1

The curvature of spacetime does not depend only on energy, but rather on a frame independent object called the stress energy tensor. This includes not only energy but also momentum and pressure. The components of this tensor are of course different in different frames, but the overall effect is the same. Loosely speaking the momentum terms cancel out the kinetic energy part (they have opposite signs).

Answer 2

I've been told that the gravitational field arises due to the energy density terms in the stress-energy tensor

This is the misconception. Gravity is not sourced exclusively by the energy density component (which is frame-dependent), but rather by the whole stress-energy-momentum tensor, which is frame-independent (although its components are frame-dependent). Since the Einstein equations, $$G_{ab} = 8 \pi T_{ab},$$ take into account the whole stress tensor, the changes in components due to changes of reference frames or coordinates are already taken into account.