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The local nonsingularity theorem
1.D. Christodoulou and N. O’Murchadha, Comm. Math. Phys. 80, 271 (1981);
1.D. Christodoulou, J. Math. Pures Appl. 60, 99 (1981).
2.See, for example, S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space‐Time (Cambridge U.P., Cambridge 1973), Chap. 8.
3.S. Sobelev, Applications of Functional Analysis to Physics, Translations of Mathematical Monographs (Am. Math. Soc., Providence, R.I., 1963).
4.See, for example, R. Geroch, J. Math. Phys. 11, 437 (1970).
5.See Ref. 3, pp. 137.
6.See, for example, Ref. 2, pp. 221;
6.R. Geroch, in Asymptotic Structure of Space‐Time, edited by F. P. Esposito and L. Witten (Plenum, New York, 1977).
7.See, for example, Ref. 2, Chap. 7.
8.E. Kasner, Am. J. Math. 43, 217 (1921).
9.E. Witten, Comm. Math. Phys. 80, 391 (1981).
10.That is, this norm bounds the surface integrals in (20). See Ref. 3.
11.This is seen by means of examples. Let vanish except near one sphere over which the average is taken in (20), there having a “bump” of amplitude κ and width λ. Then smallness of just the first two terms in the norm (4) bounds while the contribution of the second term in the average in (20) is of the order But, for as one can make as large as one wishes, while holding as small as one wishes.
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