Is Feynman's third-order Lagrangian density correct for the gravitational field?

The condition Feynman's cubic term must meet

Feynman's lectures build gravity as a field theory in flat spacetime: the gravitational field is a symmetric tensor hμν living on a Minkowski background, and the action is expanded order by order in hμν. The quadratic piece is the Fierz-Pauli Lagrangian density. The next term, third order in hμν, is not free: it has to satisfy a condition that Feynman derived from the equation of motion of a point mass and from the divergence of that system's energy-momentum tensor.

That condition is the perturbative form of the nonlinear Bianchi identity. In the paper's notation it is written as an equation for the divergence of the second-order field equation, ηλμ∂νχμν(2) = −Γλμνχμν(1), and it must hold at the level of the Lagrangian density, not merely for the resulting equations of motion. The paper works under five stated axioms: locality and Lorentz invariance, at most two derivatives in the field equations, the equivalence principle as universal coupling to the conserved stress tensor, the linear Bianchi identity at second order, and the Bianchi identity.

The reference point is Einstein's own third-order Lagrangian density L(3)E, obtained by expanding the Einstein Lagrangian density. The paper's claim is that any acceptable cubic term must equal L(3)E up to a total derivative, written L(3) ≃ L(3)E, where A ≃ B means A and B differ by ∂μCμ.

Sixteen terms and two free parameters

The most general cubic candidate built from hμν and one derivative per field has 16 independent structures, labelled T_1 through T_16. The paper computes the second-order field equation χμν(2) for each structure and then the divergence (∂χ[i])μ, assembling the condition as a linear combination Σ gi(∂χ[i])μ = Vμ, where Vμ is fixed by the quadratic Lagrangian density.

Solving that system gives L(3) = L(3)E + x(T_3 + T_9 − T_13 − T_15) + y(T_6 + T_8 − T_12 − T_16), with x and y arbitrary real constants. The two bracket combinations vanish up to a total derivative, so every choice of x and y is equivalent to L(3)E. This is the concrete sense in which the cubic term is fixed: the freedom is a surface-term freedom, not a physical one.

The paper also carries the expansion one order further. The fourth-order candidate contains 43 structures, and the solution is given with five arbitrary real constants C1, C12, C13, C18 and C21; the third-order field equation χμν(3) does not depend on them. Setting C1 = 1, C12 = −1, C13 = 1/2, C18 = 0 and C21 = −1 reproduces L(4)E, and the paper notes a single error in the {7}/2 term of the earlier Goroff-Sagnotti expression.

Where Feynman's expression departs from Einstein's

Feynman's printed cubic density, written with the trace-reversed field h̄μν = hμν − ½ημνh, can be re-expanded in the same 16-structure basis. The result matches L(3)E in most coefficients but differs in two: Feynman's version carries −4T_8 + 2T_12 where Einstein's carries −2T_8 + 2T_13. The difference is therefore L(3)Feynman/g − L(3)E/g ≃ −2T_8 + 2T_12, which the paper states is not zero.

That single line is the paper's central technical result. Because the difference is not a total derivative, Feynman's cubic Lagrangian density is not a solution of the condition it was meant to satisfy. The paper is explicit that this is a statement about the Lagrangian density itself, not about a derived equation of motion that might coincidentally agree.

A note added in the arXiv v3 version records that a different source, Huggins' 1962 thesis, prints a cubic Lagrangian density whose third term is −2hαβhγδ∂γ∂βh̄δα rather than Feynman's −2hαβhδβ∂γ∂δh̄γα. That variant is correct, satisfying L(3) ≃ L(3)E. The discrepancy is thus located in the widely read lecture expression, not in the general framework.

Why the perihelion shift still comes out right

The perihelion shift is computed for a spherically symmetric, static system: a particle orbiting a star of mass M, with hμν taken diagonal and depending only on the radius. The second-order Lagrangian density alone cannot account for the observed precession; the cubic term is needed. Working to the required order, the precession per orbit is δ ≈ π(2GNM)²/L² (α² + αβ − a), where α, β and a come from expanding the time and spatial components of hμν in powers of φ = −2GNMu.

Solving the field equations order by order in GN gives (α⟨1⟩, β⟨1⟩, a⟨1⟩, b⟨1⟩) = (1, 1, 0, 0) at first order and (α⟨2⟩, β⟨2⟩, a⟨2⟩, b⟨2⟩) = (1, 1, 1/2, −3/8) at second order, the latter agreeing with Ortín and Nikishov. These produce δ⟨1⟩ = π(2GNM)²/L² · 2 and δ⟨2⟩ = π(2GNM)²/L² · 3/2, so δ⟨1⟩ = (4/3)δ⟨2⟩. Only δ⟨2⟩ agrees with experiment.

The reason Feynman's incorrect density still passes this test is stated directly: in this background χμνT_8[h⟨1⟩] = χμνT_12[h⟨1⟩] holds, so the two offending structures contribute equally and their difference drops out. The agreement is a property of the restricted static, spherically symmetric sector, not evidence that the two Lagrangian densities are equivalent.

Where the two densities should part company

The paper lists the situations in which the difference is expected to matter: genuinely time-dependent settings such as radiation, processes sensitive to the full off-shell cubic vertex such as scattering amplitudes, and post-Newtonian orders beyond the restricted static sector tested by perihelion precession. In each case the cancellation that saves the perihelion calculation is not available, so the choice between Feynman's expression and L(3)E becomes physically visible.

These are stated as expectations about where the discrepancy should show up, not as computed discrepancies. The paper does not supply a numerical prediction for a radiation process or a scattering amplitude; it identifies the class of observables in which the −2T_8 + 2T_12 difference cannot be expected to cancel.

A further scope condition is stated alongside the main result: the analysis adopts the identification gμν = ημν + hμν and the resulting form of the Bianchi condition. A broader equivalence that includes local field redefinition would require separate discussion, so the verdict on Feynman's expression is tied to this specific framework.

What the correction changes for a reader

The practical upshot is a split between two kinds of use. If the cubic Lagrangian density is being used to reproduce the classical perihelion shift in a static, spherically symmetric system, Feynman's printed expression gives the same answer as the correct one, and the error is harmless there. If it is being used as an input to anything that probes the cubic vertex off shell, or to time-dependent and higher-order gravitational calculations, the paper's result says the printed expression should be replaced by L(3)E, or equivalently by the Huggins variant.

The paper's own framing of its value points the same way: it offers specific corrections to a widely read lecture text, uses Feynman's construction to display the modern spin-2 understanding of why general relativity is unique, and organizes explicit third- and fourth-order Lagrangian densities in a reference-friendly form. The fourth-order solution, with its five free constants and the corrected {7}/2 term, is presented as a technical reference rather than as a new physical claim.

The priority for a reader is therefore not that Feynman was wrong, but which test can and cannot detect the error. Perihelion precession, the one observable computed here, is blind to it because of a background-specific cancellation; the paper's contribution is to say exactly which two structures differ and to name the settings where that cancellation should fail.

Sources used in this answer