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Gravitational waves as deformations of embedded Einstein spaces

Richard Kerner and Salvatore Vitale

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We show how approximate radiative solutions of Einstein's equations can be constructed using small deformations of Einstein spacetimes embedded into a pseudo-Euclidean flat space of higher dimension. Infinitesimal deformations are seen then as vector fields in EN. All geometrical quantities can be then expressed in terms of embedding functions zA and their deformations v^A, z^A \rightarrow {\tilde{z}}^A = z^A + \varepsilon \, v^A + \varepsilon ^2 \, w^A +\cdots . Then we require the deformations to keep Einstein equations satisfied up to a given order in ε. The system obtained is then analyzed in particular cases of the Minkowski and Schwarzschild manifolds taken as a starting point, and solutions of deformations of Einstein's equations displaying radiative behavior are found up to the third order of expansion in small parameter ε.


PACS

04.30.-w Gravitational waves

95.30.Sf Relativity and gravitation

04.20.Gz Spacetime topology, causal structure, spinor structure

MSC

83C35 Gravitational waves

83C05 Einstein's equations (general structure, canonical formalism, Cauchy problems)

Subjects

Gravitation and cosmology

Astrophysics and astroparticles

Dates

Issue 23 (7 December 2009)

Received 15 April 2009, in final form 28 September 2009

Published 6 November 2009



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