Manuscript XXII-E@1 · XXII-E.tex · proposition · 51c99d2f

Proposition: Linearized Bianchi necessity

A well-defined linearized comparison › Gauge and conservation tests · lines 799–850 · proof included or linked

XXII-E@1#prop:bianchi, XXII-E@1#eq:bianchi, XXII-E@1#eq:source-conservation

For every $C^{3}$ symmetric field $h$ ,

\partial^{μ} E_{η} [h]_{μν} = 0.

The identity also holds distributionally.

Consequently, if a field equation

E_{η} [h] = κ_{cmp} τ

holds with fixed nonzero $κ_{cmp}$ , then

\partial^{μ} τ_{μν} = 0.

A source violating (eq:source-conservation) cannot be matched by any $h$ on that field comparison domain.

Proof. Put

h = η^{α β} h_{α β}, A_{ν} = \partial^{μ} h_{μν} .

The Ricci formula gives

R^{(1)}_{μν} = \frac{1}{2} (\partial_{μ} A_{ν} + \partial_{ν} A_{μ} - □ h_{μν} - \partial_{μ} \partial_{ν} h) .

Taking a divergence,

\partial^{μ} R^{(1)}_{μν} = \frac{1}{2} \partial_{ν} (\partial^{μ} A_{μ} - □ h) = \frac{1}{2} \partial_{ν} R^{(1)} .

Subtracting the divergence of the trace term proves (eq:bianchi). Constant-coefficient derivatives commute on distributions as well. Taking the divergence of the matching equation proves (eq:source-conservation).

Displayed formulas in this passage

XXII-E@1#eq:bianchi · line 802
$\partial^{μ} E_{η} [h]_{μν} = 0.$
unlabeled · line 809
$E_{η} [h] = κ_{cmp} τ$
XXII-E@1#eq:source-conservation · line 813
$\partial^{μ} τ_{μν} = 0.$
unlabeled · line 823
$h = η^{α β} h_{α β}, A_{ν} = \partial^{μ} h_{μν} .$
unlabeled · line 829
$R^{(1)}_{μν} = \frac{1}{2} (\partial_{μ} A_{ν} + \partial_{ν} A_{μ} - □ h_{μν} - \partial_{μ} \partial_{ν} h) .$
unlabeled · line 837
$\partial^{μ} R^{(1)}_{μν} = \frac{1}{2} \partial_{ν} (\partial^{μ} A_{μ} - □ h) = \frac{1}{2} \partial_{ν} R^{(1)} .$

Proposition: Linearized Bianchi necessity

Displayed formulas in this passage

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