Proposition: Basic admissions

The space ${\mathcal{P}}_{\Sigma }$ contains:

1. every bulk polynomial $F\in {\mathcal{P}}_{\mathrm{mc}}$;

2. every $\mathsf{J}(a)$ with $a\in {C_c}^{\infty }(M_{+};T^{\ast }M)$;

3. every $B_f$ with $f\in {C_c}^{\infty }(M_{+})$.

For the sharp insertion,

If $\eta =(a,\sigma )$, then

The first term is a regular scalar linear functional. For ${\eta }^{\prime }=(a^{\prime },{\sigma }^{\prime })$,

and derivatives of order three vanish.

Proof. Bulk admission follows from Lemma lem:bulkderivative. The metric derivative of $\mathsf{J}(a)$ is $q_a$, and further directional derivatives are constants.

For $B_f$, differentiation in the metric gives the first formula by the adjoint identity. Integration by parts in the scalar variation gives (eq:Bphi); it is smooth and compact for each smooth metric configuration.

Differentiation in both fields gives (eq:DB) and (eq:DDB). The coefficient ${\partial }_{\mu }(f(La{)}^{\mu })$ of the scalar linear term is smooth and compact. All metric derivatives involved are therefore either zero, bulk linear kernels, or kernels $q_a$ with a smooth compact covector source.