Manuscript XI-B@1 · XI-B.tex · section · 063b989e

Retarded response and its nonzero divergence

Residual gauge and the switched-source diagnostic › Retarded response and its nonzero divergence · lines 1004–1053

XI-B@1#eq:response, XI-B@1#eq:divergence, XI-B@1#eq:witness

For XI-A's source-response coefficient put

J^{μν} = \frac{1}{2} g T_{0}^{μν} [ϕ_{0}] .

The gauge-fixed retarded solution is

γ_{1}^{R} = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ_{0}]) .

Taking its divergence on the free scalar shell gives

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

For the named input

ϕ_{m} (t) = {sin (m t) / m, t, m > 0, m = 0,

one has $T_{0}^{00} = 1/2$ and $T_{0}^{0 i} = 0$ . Thus

D^{0} γ_{1}^{R} = - \frac{1}{4} E_{0}^{R} (\partial_{t} g), □ D^{0} γ_{1}^{R} = - \frac{1}{4} \partial_{t} g .

A nonzero compact interior switch has $\partial_{t} g \neq \equiv 0$ , so this divergence is nonzero.

Both (eq:response) and its divergence exist as smooth retarded responses. The failed requirement is $D γ_{1} = 0$ , not existence of the divergence. The ungauge-fixed Einstein equation would require conservation of $g T_{0}$ by its Bianchi identity. XI-A's empty $J_{g} (ϕ_{m})$ is therefore unchanged. This diagnostic is not a failure of the Green construction and uses no contraction with the interaction.

Displayed formulas in this passage

XI-B@1#eq:response · line 1011
$γ_{1}^{R} = - \frac{1}{2} P E_{0}^{R} (g T_{0} [ϕ_{0}]) .$
XI-B@1#eq:divergence · line 1018
$D^{ν} γ_{1}^{R} = - \frac{1}{2} E_{0}^{R} ((\partial_{μ} g) T_{0}^{μν} [ϕ_{0}]), □ D^{ν} γ_{1}^{R} = - \frac{1}{2} (\partial_{μ} g) T_{0}^{μν} [ϕ_{0}] .$
XI-B@1#eq:witness · line 1034
$D^{0} γ_{1}^{R} = - \frac{1}{4} E_{0}^{R} (\partial_{t} g), □ D^{0} γ_{1}^{R} = - \frac{1}{4} \partial_{t} g .$
unlabeled · line 1007
$J^{μν} = \frac{1}{2} g T_{0}^{μν} [ϕ_{0}] .$
unlabeled · line 1029
$ϕ_{m} (t) = {sin (m t) / m, t, m > 0, m = 0,$