Manuscript XI-A@1 · XI-A.tex · section · b8211ecc

Why gauge-fixed wave solvability is a different question

Why gauge-fixed wave solvability is a different question · lines 1202–1242

XI-A@1#eq:quadraticoperator, XI-A@1#eq:constraintbreaking

At quadratic order the sum of (eq:FP) and (eq:gf) is

S_{FP} + S_{gf} = \int (\frac{1}{2} \partial_{λ} γ_{μν} \partial^{λ} γ^{μν} - \frac{1}{4} \partial_{λ} τ \partial^{λ} τ) d^{4} x,

with Euler operator

2 G_{lin}^{μν} + e_{gf}^{μν} = - □ γ ˉ^{μν} .

In the source-response expansion, the gauge-fixed equation is

- □ γ ˉ_{1}^{μν} = \frac{1}{2} g T_{0}^{μν} [ϕ_{0}] .

Componentwise smooth Cauchy theory permits such an equation with a nonconserved smooth compact source. Taking its divergence gives

□ D^{ν} (γ_{1}) = - \frac{1}{2} (\partial_{μ} g) T_{0}^{μν} [ϕ_{0}] .

For the witness input, $D$ therefore cannot vanish identically.

Thus a gauge-fixed solution is not an Einstein solution in de Donder gauge unless the source compatibility is restored. Gauge fixing does not remove the Bianchi obstruction; it changes the equation being solved. Equation (eq:quadraticoperator) specifies an operator only. No graviton Green function, two-point function, propagator normalization, or state is constructed.

Displayed formulas in this passage

XI-A@1#eq:quadraticoperator · line 1215
$2 G_{lin}^{μν} + e_{gf}^{μν} = - □ γ ˉ^{μν} .$
XI-A@1#eq:constraintbreaking · line 1228
$□ D^{ν} (γ_{1}) = - \frac{1}{2} (\partial_{μ} g) T_{0}^{μν} [ϕ_{0}] .$
unlabeled · line 1206
$S_{FP} + S_{gf} = \int (\frac{1}{2} \partial_{λ} γ_{μν} \partial^{λ} γ^{μν} - \frac{1}{4} \partial_{λ} τ \partial^{λ} τ) d^{4} x,$
unlabeled · line 1222
$- □ γ ˉ_{1}^{μν} = \frac{1}{2} g T_{0}^{μν} [ϕ_{0}] .$