Manuscript IX-E-2@1 · IX-E-2.tex · section · e70d9077

The off-shell adjoint and support bounds

Explicit inherited inputs and their limits › The off-shell adjoint and support bounds · lines 393–425

IX-E-2@1#eq:adjoint, IX-E-2@1#eq:sharp-adjoint, IX-E-2@1#eq:support-upper

For the adjoint variable $b$ associated with a test covector $a$ , the supplied VII-B identity is

q_{a} B^{α β} (v) = θ C^{*} b + δ (t) B (b ∣_{Σ}), = - n^{(α} v^{β)} + \frac{1}{2} η^{α β} n^{μ} v_{μ} .

Equivalently, in the inherited convention,

C^{*} (θ b) = θ C^{*} b + δ (t) B (b ∣_{Σ}) .

There is no $δ^{'} (t)$ term. The supplied on-shell smooth representative does not replace this off-shell derivative.

For $F = supp f$ and $G = supp g$ , the available support bounds are

supp_{fun} B_{f} \subset P_{F} \cup F, P_{F} = J^{-} (F) \cap {t \geq 0}, supp_{fun} W_{g} \subset G .

These are upper inclusions. They are not lower support theorems.

Displayed formulas in this passage

IX-E-2@1#eq:adjoint · line 397
$q_{a} B^{α β} (v) = θ C^{*} b + δ (t) B (b ∣_{Σ}), = - n^{(α} v^{β)} + \frac{1}{2} η^{α β} n^{μ} v_{μ} .$
IX-E-2@1#eq:support-upper · line 416
$supp_{fun} B_{f} \subset P_{F} \cup F, P_{F} = J^{-} (F) \cap {t \geq 0}, supp_{fun} W_{g} \subset G .$
IX-E-2@1#eq:sharp-adjoint · line 407
$C^{*} (θ b) = θ C^{*} b + δ (t) B (b ∣_{Σ}) .$