E. Chapitre, ESPACE-TEMPS Soit par l'absurde un point q du développement de Cauchy futur de ? + D(A) Alors tout point p dans I ? (q) ? I + (? + D(A)) appartient à l'intérieur du développement de Cauchy futur de ? + D(A)

D. Si-c-rentre-dans, A) par définition elle rencontre A Comme par hypothèse p n'appartient pas à D(A), il existe au moins une courbe causale inextensible passée ? : [0, ?[? M qui ne rencontre pas D(A)

. Soit, Soit T 1 On suppose T 1 = ? et qu'il est un maximum

?. Comme, A) est un sous-ensemble achronal edgeless d'après la Remarque 6

I. Soit-q-un-point-dans, Alors d'après la Proposition 2.7 l'application D est injective sur le passé I ? (q) et on a I ? (p) ? I ? (q) Il s'en

. Est-causalement-convexe, toutes les géodésiques lumières qui lient x à ? ?1 (x) sont contenues dans D(I ? (q)). D'

H. La-frontière, Il s'en suit que, d'après (6.2), S est un sousensemble achronal de R 1,n (D(p)). On obtient que S

. On-va-montrer-que-le-développement-de-cauchy-passé-de-s-est-vide, On suppose par l'absurde que D ? (S) est non vide. On définit la variété M = M D(S)/ ? où ? est la relation d'équivalence qui identifie les points de I ? (p) avec leurs image par D, Soit ? : M D(S) ? M la projection canonique au quotient

?. Soit and M. Une-hypersurface-de-cauchy-de, Comme toute courbe causale future inextensible dans R 1,n (D(p)) issue d'un point x dans D ? (S) rencontre S, toute courbe causale inextensible de M issue du point ?(x) rentre dans ?(M), donc rencontre ?(?) Cela montre que ?(?) est une hypersurface de Cauchy de M

D. Le-théorème, 2 on obtient que le développement de Cauchy de S est un ouvert régulier G de R 1,n (D(p)) Or un ouvert régulier de R 1,n est soit passé complet, soit futur complet, ) est vide on obtient que G [1] Galloway G.J. Andersson L. dS CFT and spacetime topology, pp.307-327, 2002.

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