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Manuscript VI@1 · VI.tex · figure · b5b60afb

Finite-time harmonic coordinates

Finite-time harmonic coordinates and second-order descent › Coordinate coefficients · lines 989–1007

VI@1#fig:VI-harmonic-coordinates

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□γXγμ=0,Xγμ=xμ+κwμ+κ2vμ\Box_\gamma X_\gamma^\mu=0,\qquad X_\gamma^\mu=x^\mu+\kappa w^\mu+\kappa^2 v^\muXμ∣0=xμ∣0,∂tXμ∣0=δ0μX^\mu|_0=x^\mu|_0,\qquad \partial_t X^\mu|_0=\delta^\mu_0
□ηwμ=D(h)μ\Box_\eta w^\mu=D(h)^\muwμ=G□,0D(h)μw^\mu=G_{\Box,0}D(h)^\mu
□ηvμ=D(j)μ−Mhxμ−Lhwμ\Box_\eta v^\mu=D(j)^\mu-\mathsf M_h x^\mu-L_h w^\mu
G□,0b=−∫0tsin⁡((t−s)∣D∣)∣D∣b(s) dsG_{\Box,0}b=-\int_0^t\frac{\sin((t-s)|D|)}{|D|}b(s)\,\mathrm ds
target

schematic truncated past

Finite-time local dependence; no infinite-past inverse. μ\mu labels four scalar coordinate functions, not vector components.

Finite-time harmonic coordinate jet. Zero-data wave problems determine the first and second coordinate coefficients uniquely. Their values on a compact target depend on a truncated past region. The index labels four scalar coordinate functions, and the construction requires neither an infinite-past inverse nor a global numerical coordinate map.

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