Manuscript Em06-A@1 · Em06-A-v1.tex · proposition · ee9923c6

Proposition: Gauge equality in the free scalar quotient

An exact compact scalar test · lines 523–550 · proof included or linked

Em06-A@1#prop:testgauge, Em06-A@1#eq:q-gauge

For $ξ \in g_{U}$ ,

q_{f, h + K ξ} - q_{f, h} = P_{m} (X_{ξ} u_{f}), X_{ξ} u_{f} \in C_{c}^{\infty} (M) .

Thus the tests define the same class modulo $P_{m} C_{c}^{\infty} (M)$ .

Proof. Use $χK ξ = K ξ$ , $w [K ξ] = ξ$ , and $P_{m} u_{f} = f$ :

q_{f, h + K ξ} - q_{f, h} = - Q_{K ξ} u_{f} + \partial_{μ} (f ξ^{μ}) .

Writing $d_{ξ} = \partial_{μ} ξ^{μ}$ , (eq:QK) gives

- Q_{K ξ} u_{f} = - d_{ξ} f + P_{m} (X_{ξ} u_{f}) - X_{ξ} f .

The remaining divergence is $d_{ξ} f + X_{ξ} f$ , proving (eq:q-gauge). Compactness follows from the compact support of $ξ$ .

Displayed formulas in this passage

Em06-A@1#eq:q-gauge · line 526
$q_{f, h + K ξ} - q_{f, h} = P_{m} (X_{ξ} u_{f}), X_{ξ} u_{f} \in C_{c}^{\infty} (M) .$
unlabeled · line 538
$q_{f, h + K ξ} - q_{f, h} = - Q_{K ξ} u_{f} + \partial_{μ} (f ξ^{μ}) .$
unlabeled · line 543
$- Q_{K ξ} u_{f} = - d_{ξ} f + P_{m} (X_{ξ} u_{f}) - X_{ξ} f .$