Manuscript XIX-B@1 · XIX-B.tex · section · b5670709

Signs, tensor positions, and initial data

The common slab and the native prescriptions › Signs, tensor positions, and initial data · lines 173–222

XIX-B@1#eq:signs, XIX-B@1#eq:trace-reverse-cov, XIX-B@1#eq:Pplus

Fix $T > 0$ and $0 < ε < T$ . Set

M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3} .

All coefficient fields are smooth on a neighborhood of the slab. No spatial decay is imposed unless it is explicitly stated.

Keep both background signatures visible:

η_{-} = diag (- 1, 1, 1, 1), η_{+} = - η_{-}, □_{-} = - \partial_{t}^{2} + Δ, □_{+} = \partial_{t}^{2} - Δ.

For a covariant symmetric tensor $a$ , define

a_{\pm}^{μν} = η_{\pm}^{μα} η_{\pm}^{ν β} a_{α β}, tr_{\pm} a = η_{\pm}^{α β} a_{α β},

and

a_{\pm}^{μν} = a_{\pm}^{μν} - \frac{1}{2} η_{\pm}^{μν} tr_{\pm} a, D_{\pm} (a)^{μ} = \partial_{α} a_{\pm}^{α μ} .

For a contravariant tensor $B$ , trace reversal means

(P_{\pm} B)^{μν} = B^{μν} - \frac{1}{2} η_{\pm}^{μν} η_{\pm, α β} B^{α β} .

In four dimensions, $P_{\pm}^{2} = id$ .

The superscript $μ$ on a coordinate function labels one of four scalar functions. It does not turn that function into a vector component under a change of metric-sign convention.

Displayed formulas in this passage

XIX-B@1#eq:signs · line 184
$η_{-} = diag (- 1, 1, 1, 1), η_{+} = - η_{-}, □_{-} = - \partial_{t}^{2} + Δ, □_{+} = \partial_{t}^{2} - Δ.$
XIX-B@1#eq:trace-reverse-cov · line 200
$a_{\pm}^{μν} = a_{\pm}^{μν} - \frac{1}{2} η_{\pm}^{μν} tr_{\pm} a, D_{\pm} (a)^{μ} = \partial_{α} a_{\pm}^{α μ} .$
XIX-B@1#eq:Pplus · line 210
$(P_{\pm} B)^{μν} = B^{μν} - \frac{1}{2} η_{\pm}^{μν} η_{\pm, α β} B^{α β} .$
unlabeled · line 176
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3} .$
unlabeled · line 193
$a_{\pm}^{μν} = η_{\pm}^{μα} η_{\pm}^{ν β} a_{α β}, tr_{\pm} a = η_{\pm}^{α β} a_{α β},$