×

Gravitational fields as generalized string models. (English) Zbl 1176.83132

Summary: We show that Einstein’s main equations for stationary axisymmetric fields in vacuum are equivalent to the equations of motion for bosonic strings moving in a special nonflat background. This new representation is based on the analysis of generalized harmonic maps in which the metric of the target space explicitly depends on the parametrization of the base space. It is shown that this representation is valid for any gravitational field which possesses two commuting Killing vector fields. We introduce the concept of dimensional extension which allows us to consider this type of gravitational fields as strings embedded in \(D\)-dimensional nonflat backgrounds, even in the limiting case where the Killing vector fields are hypersurface-orthogonal.

MSC:

83E30 String and superstring theories in gravitational theory
83C15 Exact solutions to problems in general relativity and gravitational theory
83C57 Black holes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] F. J. Ernst, New formulation of the axially symmetric gravitational field problem, Phys. Rev. 167, 1175 (1968); F. J. Ernst, New Formulation of the axially symmetric gravitational field problem II, Phys. Rev. 168, 1415 (1968). · doi:10.1103/PhysRev.167.1175
[2] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact solutions of Einstein’s field equations (Cambridge University Press, Cambridge UK, 2003). · Zbl 1057.83004
[3] H. Quevedo and B. Mashhoon, Exterior gravitational field of a rotating deformed mass, Phys. Lett. A 109, 13 (1985); H. Quevedo, Class of stationary axisymmetric solutions of Einstein’s equations in empty space, Phys. Rev. D 33, 324 (1986); H. Quevedo and B. Mashhoon, Exterior gravitational field of a charged rotating mass with arbitrary quadrupole moment, Phys. Lett. A 148 (1990) 149; H. Quevedo, Multipole Moments in General Relativity-Static and Stationary Solutions, Fort. Phys. 38, 733 (1990); H. Quevedo and B. Mashhoon, Generalization of Kerr spacetime, Phys. Rev. D 43, 3902 (1991). · Zbl 0972.83538 · doi:10.1016/0375-9601(85)90381-0
[4] D. Maison, Are the stationary, axially symmetric Einstein equations completely integrable?, Phys. Rev. Lett. 41, 521 (1978). · doi:10.1103/PhysRevLett.41.521
[5] C.W. Misner, Harmonic maps as models for physical theories, Phys. Rev. D 18, 4510 (1978). · doi:10.1103/PhysRevD.18.4510
[6] D. Korotkin and H. Nicolai, Separation of variables and Hamiltonian formulation for the Ernst equation, Phys. Rev. 74, 1272 (1995).
[7] D. Nuñez, H. Quevedo, and A. Sánchez, Einstein’s equations as functional geodesics, Rev. Mex. Phys. 44, 440 (1998); J. Cortez, D. Nuñez, and H. Quevedo, Gravitational fields and nonlinear sigma models, Int. J. Theor. Phys. 40, 251 (2001).
[8] H. Nishino, Stationary axisymmetric black holes, N = 2 superstring, and self-dual gauge or gravity fields, Phys. Lett. B 359, 77 (1995). · doi:10.1016/0370-2693(95)01033-M
[9] H. Nishino, Axisymmetric gravitational solutions as possible classical backgrounds around closed string mass distributions, Phys. Lett. B 540, 125 (2002). · Zbl 0996.83059 · doi:10.1016/S0370-2693(02)02128-7
[10] A. Ya. Burinskii, Some properties of Kerr solution to low-energy string theory, Phys. Rev. D 52, 5826 (1995). · doi:10.1103/PhysRevD.52.5826
[11] H. Weyl, Zur Gravitationstheorie, Ann. Physik (Leipzig) 54, 117 (1917). · JFM 46.1303.01 · doi:10.1002/andp.19173591804
[12] T. Lewis, Some special solutions of the equations of axially symmetric gravitational fields, Proc. Roy. Soc. London 136, 176 (1932). · Zbl 0005.26905 · doi:10.1098/rspa.1932.0073
[13] A. Papapetrou, Eine rotationssymmetrische Lösung in de Allgemeinen Relativitätstheorie, Ann. Physik (Leipzig) 12, 309 (1953). · Zbl 0052.44302 · doi:10.1002/andp.19534470412
[14] J. Polchinski, String Theory: An Introduction to the Bosonic String (Cambridge University Press, Cambridge, UK, 2001). · Zbl 1075.81054
[15] L. Patiño and H. Quevedo, Topological quantization of gravitational fields, J. Math. Phys. 46, 22502 (2005). · Zbl 1076.83013 · doi:10.1063/1.1828586
[16] V. A. Belinsky, I. M. Khalatnikov, and E. M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys. 19, 525 (1970). · doi:10.1080/00018737000101171
[17] A. Sánchez, A. Macías, and H. Quevedo, Generating Gowdy cosmological models, J. Math. Phys. 45, 1849 (2004). · Zbl 1071.83013 · doi:10.1063/1.1695448
[18] H. J. de Vega and N. G. Sanchez, A new approach to string quantization in curved space-times, Phys. Lett. B 197, 320 (1987); H. J. de Vega, I. Giannakis, and A. Nicolaidis, String quantization in curved space-times: Null string approach, Mod. Phys. Lett. A 10, 2479 (1995); M. Maeno and S. Sawada, String field theory in curved space: A nonlinear sigma model analysis, Nucl. Phys. B 306, 603 (1988); I. Bars, Heterotic string models in curved space-time, Phys. Lett. B 293, 315 (1992); N. G. Sanchez, Advances in string theory in curved backgrounds: A synthesis report, Int. J. Mod. Phys. A 18, 2011 (2003). · doi:10.1016/0370-2693(87)90392-3
[19] R. Gowdy, Gravitational waves in closed universes, Phys. Rev. Lett. 27, 826 (1971); Vacuum space-times with two-parameter spacelike isometry groups and compact invariant hypersurfaces: Topologies and boundary conditions, Ann. Phys. (N.Y.) 83, 203 (1974). · doi:10.1103/PhysRevLett.27.826
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.