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Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952

Proceedings of the Estonian Academy of Sciences

ISSN 1736-7530 (electronic)   ISSN 1736-6046 (print)
Formerly: Proceedings of the Estonian Academy of Sciences, series Physics & Mathematics and  Chemistry
Published since 1952
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Deformations of embedded Einstein spaces; pp. 313–325

(Full article in PDF format) 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 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).

Keywords

embeddings, Einstein spaces, gravitational waves.

References

  1. Kerner , R. Deformations of the embedded Einstein spaces. Gen. Relat. Grav. , 1978 , 9 , 257–270.
doi:10.1007/BF00759378

  2. Wesson , P. S. An embedding for general relativity with variable rest mass. Gen. Relat. Grav. , 1984 , 16 , 193–203.
doi:10.1007/BF00762447

  3. Giorgini , B. and Kerner , R. Cosmology in ten dimensions with the generalised gravitational Lagrangian. Class. Quant. Grav. , 1988 , 5 , 339–351.
doi:10.1088/0264-9381/5/2/013

  4. Kerner , R. and Martin , J. Change of signature and topology in a five-dimensional cosmological model. Class. Quant. Grav. , 1993 , 10 , 2111–2122.
doi:10.1088/0264-9381/10/10/019

  5. Kerner , R. , Martin , J. , Mignemi , S. , and van Holten , J.-W. Geodesic deviation in Kaluza–Klein theories. Phys. Rev. D , 2001 , 63 , 027502.
doi:10.1103/PhysRevD.63.027502

  6. Rosen , J. Embedding of various relativistic Riemannian spaces in pseudo-euclidean spaces. Rev. Mod. Phys. , 1965 , 37 , 204–214.
doi:10.1103/RevModPhys.37.204

  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.
doi:10.1103/PhysRevLett.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.
doi:10.1103/PhysRevD.57.R5948

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.
doi:10.1088/0264-9381/17/24/306

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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Current Issue: Vol. 68, Issue 4, 2019




Publishing schedule:
No. 1: 20 March
No. 2: 20 June
No. 3: 20 September
No. 4: 20 December