Manuscript VII-B@1 · VII-B.tex · section · be632835

Exact sharp-Cauchy adjoint

Exact sharp-Cauchy adjoint · lines 647–679

VII-B@1#eq:ell-a, VII-B@1#eq:C-adjoint

Let

a \in C_{c}^{\infty} ({t > 0}; T^{*} M)

be a test covector, and define

ℓ_{a} [h] = \int a_{μ} (x) (L h)^{μ} (x) d^{4} x, b_{μ} = E_{A} a_{μ} .

The field $b$ is smooth. It is supported in $J^{-} (supp a)$ , and it solves

□ b_{μ} = a_{μ} .

In a full neighbourhood of $Σ_{0}$ , it solves the homogeneous wave equation.

The formal adjoint of $C$ , using the pairing $\int b_{μ} C^{μ} [h]$ , is

(C^{*} b)^{α β} = - \partial^{(α} b^{β)} + \frac{1}{2} η^{α β} \partial^{μ} b_{μ} .

Displayed formulas in this passage

VII-B@1#eq:ell-a · line 654
$ℓ_{a} [h] = \int a_{μ} (x) (L h)^{μ} (x) d^{4} x, b_{μ} = E_{A} a_{μ} .$
VII-B@1#eq:C-adjoint · line 672
$(C^{*} b)^{α β} = - \partial^{(α} b^{β)} + \frac{1}{2} η^{α β} \partial^{μ} b_{μ} .$
unlabeled · line 650
$a \in C_{c}^{\infty} ({t > 0}; T^{*} M)$
unlabeled · line 664
$□ b_{μ} = a_{μ} .$