Deriving Feynman's Consistency Condition from His Abandoned Energy--Momentum Approach
Feynman briefly considered a nonlinear theory of gravity in flat spacetime by imposing energy--momentum conservation on the sum of the energy--momentum tensor of matter and that of the gravitational field. He used the symmetrized canonical energy-momentum tensor as the energy--momentum tensor of the gravitational field.He reported that the resulting construction gave an incorrect value for Mercury's perihelion precession and abandoned this route in favor of a variational argument based on dynamical consistency. We reconsider the abandoned route. The key step is to identify the gravitational energy--momentum tensor not with the conventional Belinfante tensor, which becomes symmetric upon use of the field equations, but with its off--shell Belinfante--Rosenfeld tensor, which contains explicit Euler--Lagrange terms. If we apply the law of conservation of energy and momentum to the sum of this tensor and the matter energy-momentum tensor, we obtain Feynman's consistency condition.Thus Feynman's dynamical-consistency argument and the reconstructed energy--momentum argument, although based on different physical assumptions, yield the same identity. The Euler--Lagrange terms are essential to this result.