Hodge-de Rham Laplacian and geometric criteria for gravitational waves

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Abstract

The curvature tensor R^ of a manifold is called harmonic, if it obeys the condition Δ(HR)R^=0, where Δ(HR)=DD∗+D∗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.

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1. Introduction. Harmonic curvature tensor In non-Euclidean spaces, in the formalism of differential forms, an external covariant differential
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About the authors

Olga V. 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 N. Frolov

Moscow Pedagogical State University

Author for correspondence.
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

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Copyright (c) 2023 Babourova O.V., Frolov B.N.

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