The gravitational wave effective stress tensor derived from the GW action is the only one that is conserved with positive energy density, and the other candidates reduce to it after removing nonconserved parts.
Weinberg, Gravitation and Cosmology (John Wiley, New York, 1972)
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The Conserved Effective Stress Tensor of Gravitational Wave
The gravitational wave effective stress tensor derived from the GW action is the only one that is conserved with positive energy density, and the other candidates reduce to it after removing nonconserved parts.