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Open AccessArticle

The Wave-Front Equation of Gravitational Signals in Classical General Relativity

1
Research Center for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava, Bezručovo nám.13, CZ-74601 Opava, Czech Republic
2
Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127 Trieste, Italy
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(2), 216; https://doi.org/10.3390/sym12020216
Received: 17 December 2019 / Revised: 9 January 2020 / Accepted: 19 January 2020 / Published: 2 February 2020
In this paper the dynamical equation for propagating wave-fronts of gravitational signals in classical general relativity (GR) is determined. The work relies on the manifestly-covariant Hamilton and Hamilton–Jacobi theories underlying the Einstein field equations recently discovered (Cremaschini and Tessarotto, 2015–2019). The Hamilton–Jacobi equation obtained in this way yields a wave-front description of gravitational field dynamics. It is shown that on a suitable subset of configuration space the latter equation reduces to a Klein–Gordon type equation associated with a 4-scalar field which identifies the wave-front surface of a gravitational signal. Its physical role and mathematical interpretation are discussed. Radiation-field wave-front solutions are pointed out, proving that according to this description, gravitational wave-fronts propagate in a given background space-time as waves characterized by the invariant speed-of-light c. The outcome is independent of the actual shape of the same wave-fronts and includes the case of gravitational waves which are characterized by an eikonal representation and propagate in a generic curved space-time along a null geodetics. The same waves are shown: (a) to correspond to the geometric-optics limit of the same curved space-time solutions; (b) to propagate in a flat space-time as plane waves with constant amplitude; (c) to display also the corresponding form of the wave-front in curved space-time. The result is consistent with the theory of the linearized Einstein field equations and the existence of gravitational waves achieved in such an asymptotic regime. Consistency with the non-linear Trautman boundary-value theory is also displayed.
Keywords: General Relativity Hamilton equations; General Relativity Hamilton–Jacobi theory; wave-front theory; speed of propagation General Relativity Hamilton equations; General Relativity Hamilton–Jacobi theory; wave-front theory; speed of propagation
MDPI and ACS Style

Cremaschini, C.; Tessarotto, M. The Wave-Front Equation of Gravitational Signals in Classical General Relativity. Symmetry 2020, 12, 216.

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