EAP - Proceedings of the Estonian Academy of Sciences

PUBLISHED
SINCE 1952

proceedings
of the estonian academy of sciences

ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)

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Deformations of embedded Einstein spaces; pp. 313–325

PDF | doi: 10.3176/proc.2010.4.10

Authors

Richard Kerner, Salvatore Vitale

Abstract

Many important Einsteinian space-times can be globally embedded into a pseudo-Euclidean flat space of higher dimension N. In this paper we analyse in detail the geometrical properties of infinitesimal deformations of embedded Einstein spaces. Embeddings are defined by N functions z^A(x^μ), A = 1, 2, …, N, μ = 0, 1, 2, 3. Their infinitesimal deformations can be developed in a power series of small parameter ε as follows: z^A → z^A = z^A + ε v^A + ε² w^A + … . All geometrical quantities can be then expressed in terms of embedding functions z^A and their deformations v^A, w^A, etc. Then we require the deformations to keep Einstein equations satisfied up to a given order in ε. This method can be used to construct approximate solutions of Einstein’s equations, and was first introduced in 1978 by one of the authors (RK).

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