Manuscript X-G@1 · X-G.tex · section · 01566e14

Geometry, switches, and tensor types

Frozen conventions and a support correction › Geometry, switches, and tensor types · lines 355–405

X-G@1#eq:separation, X-G@1#eq:twojet

Retain

M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.

Fix the X-E/X-F observation and scalar switch

f \in C_{c}^{\infty} ((0, T) \times R^{3}; R), g \in C_{c}^{\infty} ((0, T) \times R^{3}; [0, 1]) .

Let $U$ be relatively compact with

U \subset P_{g} := int {g = 1}, g_{U} = C_{c}^{\infty} (U; T M) .

Write $K_{f} = supp f$ . The separation condition is

K_{f} \cup supp g \cup U \subset {2 δ < t < T}, 0 < ε < δ .

The ramp $σ_{ε} (t)$ is smooth, zero for $t \leq 0$ , and one for $t \geq ε$ . Monotonicity is not needed for the obstruction.

The backgrounds $h_{cl}, k_{cl}$ remain arbitrary smooth off-shell tensors. In particular, (eq:separation) does not require them to vanish in $0 < t < ε$ .

Metrics and metric fluctuations are covariant symmetric tensors. Their Euler derivatives are contravariant tensor densities. The Lie derivative on either side of an operator acts in the appropriate type.

For a vector field $ξ$ define

(K ξ)_{μν} = (L_{ξ} η)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ} .

The plateau transformation is the inherited two-jet rule

h_{cl}^{'} = h_{cl} + K ξ, k_{cl}^{'} = k_{cl} + L_{ξ} h_{cl} + K ζ + \frac{1}{2} L_{ξ} K ξ, ξ, ζ \in g_{U} .

Displayed formulas in this passage

X-G@1#eq:separation · line 375
$K_{f} \cup supp g \cup U \subset {2 δ < t < T}, 0 < ε < δ .$
X-G@1#eq:twojet · line 399
$h_{cl}^{'} = h_{cl} + K ξ, k_{cl}^{'} = k_{cl} + L_{ξ} h_{cl} + K ζ + \frac{1}{2} L_{ξ} K ξ, ξ, ζ \in g_{U} .$
unlabeled · line 358
$M = R^{4}, η = diag (1, - 1, - 1, - 1), □ = \partial_{t}^{2} - Δ, P_{m} = □ + m^{2}, m \geq 0.$
unlabeled · line 365
$f \in C_{c}^{\infty} ((0, T) \times R^{3}; R), g \in C_{c}^{\infty} ((0, T) \times R^{3}; [0, 1]) .$
unlabeled · line 370
$U \subset P_{g} := int {g = 1}, g_{U} = C_{c}^{\infty} (U; T M) .$
unlabeled · line 393
$(K ξ)_{μν} = (L_{ξ} η)_{μν} = \partial_{μ} ξ_{ν} + \partial_{ν} ξ_{μ} .$