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Deformations of embedded Einstein spaces; pp. 313–325

Full article in PDF format | doi: 10.3176/proc.2010.4.10

Richard Kerner, Salvatore Vitale

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 zA(xμ), 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: zA → zA = zA + ε vA + ε2 wA + … . All geometrical quantities can be then expressed in terms of embedding functions zA and their deformations vA, wA, 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).

  1. Kerner, R. Deformations of the embedded Einstein spaces. Gen. Relat. Grav., 1978, 9, 257–270.

  2. Wesson, P. S. An embedding for general relativity with variable rest mass. Gen. Relat. Grav., 1984, 16, 193–203.

  3. Giorgini, B. and Kerner, R. Cosmology in ten dimensions with the generalised gravitational Lagrangian. Class. Quant. Grav., 1988, 5, 339–351.

  4. Kerner, R. and Martin, J. Change of signature and topology in a five-dimensional cosmological model. Class. Quant. Grav., 1993, 10, 2111–2122.

  5. Kerner, R., Martin, J., Mignemi, S., and van Holten, J.-W. Geodesic deviation in Kaluza–Klein theories. Phys. Rev. D, 2001, 63, 027502.

  6. Rosen, J. Embedding of various relativistic Riemannian spaces in pseudo-euclidean spaces. Rev. Mod. Phys., 1965, 37, 204–214.

  7. Damour, T. and Deruelle, N. General relativistic celestial mechanics of binary systems I. The post-Newtonian motion. Ann. Inst. Henri Poincaré, 1985, 43, 107–132.

  8. Damour, T. and Deruelle, N. General relativistic celestial mechanics of binary systems II. The post-Newtonian timing formula. Ann. Inst. Henri Poincaré, 1986, 44, 263–292.

  9. Blanchet, L., Damour, T., Iyer, B. R., Will, C. M., and Wiseman, A. G. Gravitational-radiation damping of compact binary systems to second-post-Newtonian order. Phys. Rev. Lett., 1995, 74, 3515.

10. Jaranowski, P. and Schäfer, G. Nonuniqueness of the third post-Newtonian binary point-mass dynamics. Phys. Rev. D, 1998, 57, 5948–5950.

11. Balakin, A., van Holten, J. W., and Kerner, R. Motions and worldline deviations in Einstein–Maxwell theory. Class. Quant. Grav., 2000, 17, 5009–5024.

12. Kerner, R., van Holten, J. W., and Colistete, R. Jr. Relativistic epicycles: another approach to geodesic deviations. Class. Quant. Grav., 2001, 18, 4725–4742; arXiv:gr-qc/0102099.

13. Gal¢tsov, D. V., Melkumova, E. Yu., and Kerner, R. Axion bremsstrahlung from collisions of global strings. Phys. Rev. D, 2004, 70, 045009.
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