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DeWitt-Morette Analysis, Manifolds and Physics II North Holland 1989. 16. Y. R. Acad. Sci. Paris 308, Serie I 1989, 167-170. 17. D. Christodoulou and A. Talvilar-Zadeh On the regularity of spherically symmetric wave maps, Comm. Pure Ap. Math. 46 , 1993, 1041-1091. 18. S. Klainerman and M. Machedon Spacetime estimates for null forms and the local existence theorem, Com. Pure App. Math. 46, 1993, 1221-1268. 19. C. Sogge On local existence for wave equations satisfying variable coefficients null conditions, Comm.
The Pfaff or convective derivatives ∂α with respect to θα are ∂ − β i ∂i ∂t ∂ ∂i ≡ ∂xi ∂0 ≡ (2) In this coframe, the metric g reads ds2 = gαβ θα θβ ≡ −N 2 (θ0 )2 + gij θi θj . (3) The t-dependent scalar N and space vector β are called the lapse function and shift vector of the slicing. These quantities were explicitly identified in [7] and play prominent roles in all subsequent 3 + 1 formulations. Any spacetime tensor decomposes into sets of time dependent space tensors by projections on the tangent space or the normal to Mt .
General formulation and physical interpretation. Phys Rev D10:428-436. 36. Teitelboim C. (1973) How commutators of constraints reflect the spacetime structure. Ann Phys (NY) 79:542-557. 37. Teitelboim C. (1977) Supergravity and square root of constraints. Phys Rev Lett 38:1106-1110. 38. Friedrich H. (1991) On the global existence and the asymptotic-behaviour of solutions to the Einstein-Maxwell-Yang-Mills equations. J Diff Geom 34:275345. 39. , Noutchegueme N. (1986) Syst`eme hyperbolique pour les ´equations d’Einstein avec sources.