Manuscript IX-Free-2@1 · IX-Free-2.tex · section · abb2397a

Geometry and smooth operators

Release alternatives and inherited contract › Geometry and smooth operators · lines 282–329

IX-Free-2@1#eq:C, IX-Free-2@1#eq:generators

Let

M = R^{1 + 3}, M_{+} = {t > 0}, Σ = {t = 0}, X = M_{+}^{3},

with metric

η = diag (- 1, 1, 1, 1), □ = - \partial_{t}^{2} + Δ_{x} .

Metric configurations are smooth symmetric covariant tensors. Set

(C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}), L = E_{0} C, w = L h .

The operator $E_{0}$ is the inverse of $□$ with zero Cauchy value and zero first time derivative at $t = 0$ . The operator $G_{A}^{0}$ is the componentwise advanced inverse of the same massless wave operator.

All configurations and directions are smooth and off shell. Compact directions supported in $M_{+}$ are allowed. We use the usual smooth mapping properties, Cauchy uniqueness, and finite propagation of these fixed linear operators.

With

T = C_{c}^{\infty} (M_{+}; C),

the generators are

B_{f} [h, ϕ] = - \int f (x) (L h)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, W_{g} [h] = \int g (y) h_{00} (y)^{2} d^{4} y .

Complex tests define complex-valued functionals on real smooth configurations. Complexified directions can equivalently be used by multilinearity.

Displayed formulas in this passage

IX-Free-2@1#eq:C · line 296
$(C h)^{μ} = \partial_{ν} h^{μν} - \frac{1}{2} \partial^{μ} (η^{α β} h_{α β}), L = E_{0} C, w = L h .$
IX-Free-2@1#eq:generators · line 319
$B_{f} [h, ϕ] = - \int f (x) (L h)^{μ} (x) \partial_{μ} ϕ (x) d^{4} x, W_{g} [h] = \int g (y) h_{00} (y)^{2} d^{4} y .$
unlabeled · line 285
$M = R^{1 + 3}, M_{+} = {t > 0}, Σ = {t = 0}, X = M_{+}^{3},$
unlabeled · line 290
$η = diag (- 1, 1, 1, 1), □ = - \partial_{t}^{2} + Δ_{x} .$
unlabeled · line 315
$T = C_{c}^{\infty} (M_{+}; C),$