Manuscript V@1 · V.tex · section · e7ddf37d

The classical solution spaces

Locked signs, solution spaces, and Green operators › The classical solution spaces · lines 511–543

V@1#eq:testspace

A smooth field is spacelike compact if its support is contained in $J (C)$ for some compact $C \subset M$ . Define

S = {h \in C_{sc}^{\infty} (S^{2} T^{*} M) : L_{E} h = 0}, S_{dD} = {h \in S : D (h) = 0} .

The retarded-prescription domain is

D_{ret} = {h \in C_{sc}^{\infty} (S^{2} T^{*} M) : D (h) = 0 for all sufficiently negative times}, S_{ret} = D_{ret} \cap S .

These spaces are not identified with the Schwartz TT plane used later.

The test-tensor space is

E = {J \in C_{c}^{\infty} (S^{2} T M; R) : \partial_{μ} J^{μν} = 0}, F_{J} (h) = \int h_{μν} J^{μν} d^{4} x .

Displayed formulas in this passage

V@1#eq:testspace · line 533
$E = {J \in C_{c}^{\infty} (S^{2} T M; R) : \partial_{μ} J^{μν} = 0}, F_{J} (h) = \int h_{μν} J^{μν} d^{4} x .$
unlabeled · line 515
$S = {h \in C_{sc}^{\infty} (S^{2} T^{*} M) : L_{E} h = 0}, S_{dD} = {h \in S : D (h) = 0} .$
unlabeled · line 521
$D_{ret} = {h \in C_{sc}^{\infty} (S^{2} T^{*} M) : D (h) = 0 for all sufficiently negative times}, S_{ret} = D_{ret} \cap S .$