λEnter Calculus CompanionRegistered manuscripts, kept distinct

Manuscript III@2 · III.tex · figure · 82dad3e5

Unique retarded harmonic representative

The coupled first-order jet and retarded harmonic section › The coupled jet · lines 1281–1302

III@2#fig:retarded-jet-section

Back to manuscript
metricscalarquotient / invariantobstruction / errorsupport / scope
gauge orbitharmonic sliceone intersection
generic point x=(h,ϕ0,ϕ1)x=(h,\phi_0,\phi_1)
Pϕ1=−Lhϕ0P\phi_1=-L_h\phi_0
SRx=T−wR(h)x\mathcal S_{\mathrm R}x=T_{-w_{\mathrm R}(h)}x
(k,ϕ0,ψ1)(k,\phi_0,\psi_1)
k=h−KwR(h),ψ1=ϕ1−XwR(h)ϕ0k=h-Kw_{\mathrm R}(h),\quad \psi_1=\phi_1-X_{w_{\mathrm R}(h)}\phi_0
Pψ1=−Lkϕ0,Dk=0P\psi_1=-L_k\phi_0,\quad \mathsf D k=0
J\mathscr J
→
J/GR\mathscr J/\mathscr G_{\mathrm R}
bijection →
Y={Dh=0}\mathscr Y=\{\mathsf D h=0\}
Ff=∫f(ϕ0+κψ1) d4x\mathcal F_f=\int f(\phi_0+\kappa\psi_1)\,d^4x
Orbit-space statement, not a quotient-manifold theorem.
ψ1\psi_1 need not be free.

The complete scalar coefficient and the metric representative transform together. The commuting diagram uses the past-trivial retarded group, not the residual-wave group.

Nearby in this section