Manuscript IX-A-ST@1 · IX-A-ST.tex · section · 6e99d1b2

The sharp adjoint and its intrinsic boundary trace

The sharp adjoint and its intrinsic boundary trace · lines 530–555

IX-A-ST@1#eq:Cstar, IX-A-ST@1#eq:boundarytensor

Use the auxiliary advanced and retarded inverses of $□_{η}$ , denoted by $G_{A}$ and $G_{R}$ . They are employed only in the supported prescription.

For a covector source $a \in D (V; T^{*} U)$ , set

b = G_{A} a .

The formal adjoint of $C$ , under the stated tensor pairing, is

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

Let $n_{μ} = \partial_{μ} t$ , and define

B^{α β} (v) = - n^{(α} v^{β)} + \frac{1}{2} η^{α β} n_{μ} v^{μ} .

Displayed formulas in this passage

IX-A-ST@1#eq:Cstar · line 542
$(C^{*} b)^{α β} = - \partial^{(α} b^{β)} + \frac{1}{2} η^{α β} \partial_{μ} b^{μ} .$
IX-A-ST@1#eq:boundarytensor · line 549
$B^{α β} (v) = - n^{(α} v^{β)} + \frac{1}{2} η^{α β} n_{μ} v^{μ} .$
unlabeled · line 538
$b = G_{A} a .$