A Note on the Feynman Lectures on Gravitation

TL;DR

The note reframes gravity as a flat-spacetime theory of a massless spin-2 field, deriving explicit second-, third-, and fourth-order Lagrangian densities and examining the perturbative constraints (Bianchi identities and universal coupling). It shows that Feynman’s proposed cubic Lagrangian does not satisfy the cubic constraint in general, though it can reproduce the correct perihelion shift in certain static backgrounds; the authors construct the Einstein cubic and quartic Lagrangians (L_E^{(3)}, L_E^{(4)}) as weak-equivalent targets and provide explicit, reference-friendly forms. The perihelion-shift analysis demonstrates that higher-order terms are essential for matching observational data, while also clarifying where Feynman’s approach agrees with GR and where it may diverge in time-dependent or off-shell contexts. Overall, the work clarifies the limitations and domain of validity of Feynman’s flat-spacetime bootstrap picture and supplies detailed Lagrangian densities for rigorous comparison and further study.

Abstract

Following Feynman's lectures on gravitation, we consider the theory of the gravitational (massless spin-2) field in flat spacetime and present the third- and fourth-order Lagrangian densities for the gravitational field. In particular, we present detailed calculations for the third-order Lagrangian density. We point out that the expression for the third-order Lagrangian density which Feynman provided is not a solution of Feynman's condition that the third-order Lagrangian density must satisfy. However, Feynman's third-order Lagrangian density gives the correct perihelion shift.