Hodge-de Rham Laplacian and geometric criteria for gravitational waves

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The curvature tensor \(\hat{R}\) of a manifold is called harmonic, if it obeys the condition \(\Delta^{\text{(HR)}}\hat{R}=0\), where \(\Delta^{\text{(HR)}}=DD^{\ast} +
D^{\ast}D\) is the Hodge–de Rham Laplacian. It is proved that all solutions of the Einstein equations in vacuum, as well as all solutions of the Einstein–Cartan theory in vacuum have a harmonic curvature. The statement that only solutions of Einstein’s equations of type \(N\) (describing gravitational radiation) are harmonic is refuted.

作者简介

Olga Babourova

Moscow Automobile and Road Construction State Technical University

Email: ovbaburova@madi.ru
ORCID iD: 0000-0002-2527-5268

Professor, Doctor of Sciences in Physics and Mathematics, Professor at Department Physics

64, Leningradsky pr., Moscow, 125319, Russian Federation

Boris Frolov

Moscow Pedagogical State University

编辑信件的主要联系方式.
Email: bn.frolov@mpgu.su
ORCID iD: 0000-0002-8899-1894

Professor, Doctor of Sciences in Physics and Mathematics, Professor at Department of Theoretical Physics, Institute of Physics, Technology and Information Systems

29/7, M. Pirogovskaya str., Moscow, 119435, Russian Federation

参考

  1. G. de Rham, Differentiable manifolds: forms, currents, harmonic forms. Berlin, Heidelberg, New York, Tokyo: Springer-Verlag, 2011, 180 pp.
  2. M. O. Katanaev, “Geometric methods in mathematical physics,” in Russian. arXiv: 1311.0733v3[math-ph].
  3. A. L. Besse, Einstein manifolds. Berlin, Heidelberg: Springer-Verlag, 1987.
  4. J.-P. Bourguignon, “Global riemannian geometry,” in T. J. Willmore and N. J. Hitchin, Eds. New York: Ellis Horwood Lim., 1984, ch. Metric with harmonic curvature.
  5. D. A. Popov, “To the theory of the Yang-Mills fields,” Theoretical and mathematical physics, vol. 24, no. 3, pp. 347-356, 1975, in Rusian.
  6. V. D. Zakharov, Gravitational waves in Einstein’s theory of gravitation. Moscow: Nauka, 1972, 200 pp., in Rusian.
  7. O. V. Babourova and B. N. Frolov, “On a harmonic property of the Einstein manifold curvature,” 1995. arXiv: gr-qc/9503045v1.
  8. D. A. Popov and L. I. Dajhin, “Einstein spaces and Yang-Mills fields,” Reports of the USSR Academy of Sciences [Doklady Akademii nauk SSSR], vol. 225, no. 4, pp. 790-793, 1975.

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