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Joint linearization instabilities in general relativity
1.For instance, see the following papers and the references cited therein: J. M. Arms, J. E. Marsden, and V. Moncrief, Commun. Math. Phys. 78, 455 (1981);
1.D. R. Brill and S. Deser, Commun. Math. Phys. 32, 291 (1973);
1.D. R. Brill, “Linearization stability,” in Spacetime and Geometry, edited by R. Matzner and L. Shepley (U. Texas Press, Austin, 1982);
1.A. E. Fischer, J. Marsden, and V. Moncrief, Ann. Inst. H. Poincare 33, 147 (1980).
2.D. R. Brill, O. Reula, and B. Schmidt, “Local linearization stability” (to be published);
2.GR11 abstracts, RESO, Stockholm, 1986.
3.R. Geroch and L. Lindblom, J. Math. Phys. 26, 2581 (1985).
4.In the present discussion metric shall mean metric tensors; that is a description of the geometry in a particular coordinate system. Metrics with a given symmetry form a smooth submanifold of all metrics. In contrast, in the space of geometries, where diffeomorphic metrics are identified, the symmetric geometries lie on strata.
5.A. E. Fischer and J. E. Marsden, Proc. Sympos. Pure Math. 27, 219 (1975).
6.See, for example, Chap. 21 in C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).
7.See, for example, N. J. Hicks, Notes on Differential Geometry (Van Nostrand, Princeton, NJ, 1965).
8.This name was originally given to finite‐dimensional spaces whose points are symmetric three‐geometries [C. W. Misner, in Magic Without Magic: John Archibald Wheeler, edited by J. Klauder (Freeman, San Francisco, 1972), pp. 441–473]. Here we consider a finite‐dimensional space of symmetric initial data that have not all been identified by diffeomorphisms. It can be regarded as a “minimomentum space” or as a minisuperspace of four‐geometries (generated by the Einstein time development equations for all initial data, whether solutions of the constraints of not).
9.The family of metrics used by Geroch and Lindblom, parametrized by λ, also corresponds to points in this minisuperspace if we identify those metrics that are equivalent under a z‐translation. Three independent parameters that label these metrics smoothly are (related to the diagonal components of ).
10.V. Moncrief, Phys. Rev. D 18, 983 (1978).
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