Manuscript X-F-ST@1 · X-F-ST.tex · section · e71f03ef

Background de Donder gauge

F2: background gauge, kinetic correction, and retarded jet law › Background de Donder gauge · lines 821–875

X-F-ST@1#eq:backgroundgauge, X-F-ST@1#eq:C1, X-F-ST@1#eq:gf1

Let $q ˉ = η + κh$ . For a covariant fluctuation $r$ , put

r ˉ^{μν} C_{q ˉ}^{μ} [r] S_{gf} [r; q ˉ] F [r; q ˉ] = q ˉ^{μα} q ˉ^{ν β} r_{α β} - \frac{1}{2} q ˉ^{μν} q ˉ^{α β} r_{α β}, = \nabla ˉ_{ν} r ˉ^{ν μ}, = \frac{1}{4} \int J_{q ˉ} q ˉ_{μν} C_{q ˉ}^{μ} [r] C_{q ˉ}^{ν} [r] d^{4} x, = \frac{δ S _{gf}}{δ r} = F^{(0)} r + κ F^{(1)} [h] r + O (κ^{2}) .

Only the fluctuation is gauge fixed. Classical backgrounds remain off shell and unconstrained. No extra de Donder constraint ideal is imposed.

An explicit specification of $F^{(1)}$ is useful. Set $r ˉ_{0} = P_{tr} r$ , $H = tr_{η} h$ , and

β^{ρ ν} [h; r] C_{1}^{ν} [h; r] Γ_{1}^{ν}_{ρ λ} [h] = - h^{ρ α} r_{α}^{ν} - h^{ν α} r^{ρ}_{α} + \frac{1}{2} h^{ρ ν} tr_{η} r + \frac{1}{2} η^{ρ ν} h^{α β} r_{α β}, = \partial_{ρ} β^{ρ ν} + \frac{1}{2} (\partial_{λ} H) r_{0} ˉ^{λ ν} + Γ_{1}^{ν}_{ρ λ} [h] r_{0} ˉ^{ρ λ}, = \frac{1}{2} η^{ν α} (\partial_{ρ} h_{α λ} + \partial_{λ} h_{α ρ} - \partial_{α} h_{ρ λ}) .

Then $F^{(1)} [h] r$ is the Euler derivative in $r$ of the fully specified quadratic density

L_{gf}^{(1)} [h; r] = \frac{1}{4} b_{1} [h] C_{μ} C^{μ} + \frac{1}{4} h_{μν} C^{μ} C^{ν} + \frac{1}{2} η_{μν} C^{μ} C_{1}^{ν} [h; r] .

Thus no unspecified gauge vertex is hidden in the notation.

Displayed formulas in this passage

X-F-ST@1#eq:backgroundgauge · line 825
$r ˉ^{μν} C_{q ˉ}^{μ} [r] S_{gf} [r; q ˉ] F [r; q ˉ] = q ˉ^{μα} q ˉ^{ν β} r_{α β} - \frac{1}{2} q ˉ^{μν} q ˉ^{α β} r_{α β}, = \nabla ˉ_{ν} r ˉ^{ν μ}, = \frac{1}{4} \int J_{q ˉ} q ˉ_{μν} C_{q ˉ}^{μ} [r] C_{q ˉ}^{ν} [r] d^{4} x, = \frac{δ S _{gf}}{δ r} = F^{(0)} r + κ F^{(1)} [h] r + O (κ^{2}) .$
X-F-ST@1#eq:C1 · line 846
$β^{ρ ν} [h; r] C_{1}^{ν} [h; r] Γ_{1}^{ν}_{ρ λ} [h] = - h^{ρ α} r_{α}^{ν} - h^{ν α} r^{ρ}_{α} + \frac{1}{2} h^{ρ ν} tr_{η} r + \frac{1}{2} η^{ρ ν} h^{α β} r_{α β}, = \partial_{ρ} β^{ρ ν} + \frac{1}{2} (\partial_{λ} H) r_{0} ˉ^{λ ν} + Γ_{1}^{ν}_{ρ λ} [h] r_{0} ˉ^{ρ λ}, = \frac{1}{2} η^{ν α} (\partial_{ρ} h_{α λ} + \partial_{λ} h_{α ρ} - \partial_{α} h_{ρ λ}) .$
X-F-ST@1#eq:gf1 · line 867
$L_{gf}^{(1)} [h; r] = \frac{1}{4} b_{1} [h] C_{μ} C^{μ} + \frac{1}{4} h_{μν} C^{μ} C^{ν} + \frac{1}{2} η_{μν} C^{μ} C_{1}^{ν} [h; r] .$