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Let a mass oscillate with relativistic acceleration (sinusoidal) by means which are irrelevant. What does the gravitational field look like a distance R away?
What if perhaps we had a mechanism that made it such that the mass oscillates with constant /omegaIf one considers a sinusoidal mass oscillating in isolation, one finds that ##\nabla_a T^{ab}## is not equal to zero, while ##\nabla_a G^{ab} = 0##. As a consequence one cannot satisfy Einstein's field equations ##T^{ab} = 8 \pi G^{ab}## as taking the covariant derivative of each side yields the result that ##\nabla_a T^{ab} = \nabla_a G^{ab}##, but this is not possible.
Thus one is lead to the conclusion that the means by which the mass is made to osscilate cannot be ignored.. Another way of saying this that may be simpler - one needs the source to conserve energy-momentum (the precise mathematical statement of this idea is that ##\nabla_a T^{ab} = 0## ) in order to be able to apply Einstein's field equations in the first place. And an oscillating mass doesn't do that by itself, it needs help.