Manuscript VII-A@1 · VII-A.tex · section · 11da3fe6

Formal inversion

Unswitched harmonic coordinates › Formal inversion · lines 814–830

VII-A@1#eq:Xinverse

A map with leading term $id$ has a unique formal inverse. In the present notation,

(X_{g}^{- 1})^{μ} (y) = y^{μ} - κ w^{μ} (y) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (y) + O (κ^{3}) .

This is a formal statement. It does not assert that a nonperturbative harmonic-coordinate map is a global diffeomorphism.

Displayed formulas in this passage

VII-A@1#eq:Xinverse · line 818
$(X_{g}^{- 1})^{μ} (y) = y^{μ} - κ w^{μ} (y) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (y) + O (κ^{3}) .$