This chapter describes the general covariant modulation (GCM) procedure in detail. It considers an axially symmetric polarized spacetime region R foliated by two functions (u, s) such that: on R, (u, s) defines an outgoing geodesic foliation as in section 2.2.4. The chapter then outlines the elliptic Hodge lemma. It also looks at the deformations of S surfaces, frame transformations, and the existence of GCM spheres. It recalls the transformation formulas recorded in Proposition 2.90, before rewriting a subset of these transformations in a more useful form. In the proof of existence and uniqueness of GCMS, one needs, in addition to the equations derived so far, an equation for the average of α. Finally, the chapter discusses the construction of GCM hypersurfaces.
Keywords: general covariant modulation procedure, spacetime, geodesic foliation, elliptic Hodge lemma, S surfaces, frame transformations, general covariant modulation spheres, general covariant modulation hypersurfaces
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