Yu-Zhu Chen and colleagues discussed gravitational waves with the exact cylindrical gravitational wave solutions. They showed that, there are two kinds of singularities in gravitational waves: source singularity and resonance singularity. Their study recently appeared in the Journal Symmetry.
In the weak field or say, linear approximation, gravitational waves are regarded as linear waves, which ignores the spacetime singularities. Most results about gravitational waves are deduced in this approximation, such as the gravitational quadrupole radiation, the resonance between the gravitational wave and the detector, and the linear superposition of two gravitational waves. But, there’s one another interesting theory called nonlinear theory—exact wave solutions of the Einstein equation. When you consider this theory, some new properties of gravitational waves come into sight.
Now, Yu-Zhu Chen and colleagues discussed gravitational waves with the exact cylindrical gravitational wave solutions rather than gravitational wave solutions in the linear approximation.
“Our paper is motivated by problems such as, the behavior of singularities in gravitational wave solutions and the new physical effects of gravitational wave solutions in addition to, e.g., the reflexion and the transmission.”— they wrote.
Based on the exact solution, they analyzed singularities in gravitational waves. They showed that there are two kinds of singularities in gravitational waves.
The first kind of singularities lies at a fixed spatial position which corresponds to a source. They called it the “source singularity”.
While, by considering a cylindrical gravitational wave as a complete solution, they showed that singularities in cylindrical gravitational waves carry the information about the source. The second kind of singularities arise as time proceeds to infinity. They recognized this singularity as a resonance and called it the “resonance singularity”.
Unlike other researchers, who considered a resonance between gravitational radiation and the matter (especially the gravitational radiation detectors), Yu-Zhu Chen and colleagues suggested that a gravitational wave resonates with other gravitational waves. They mentioned that the resonance singularity only emerges when a gravitational wave with a source singularity and a gravitational wave without a source singularity possess the same frequency. Two gravitational waves with source singularities or two gravitational waves without source singularities do not resonate. The resonance also indicates that the gravitational wave with sources and the gravitational wave without sources are two of different kinds.
“We suppose that the resonance between gravitational waves is irrelevant to the symmetry of the system. In recent years, gravitational wave detection has produced rapid progress. We expect that the resonance between gravitational waves will be found in the future.”— they wrote.
Moreover, they investigated the interference of two gravitational waves. They showed, how the interference terms of two cylindrical gravitational waves behave. Interference appears in both the metric and the energy-momentum tensor. Specifically, they showed that the interference term in the source vanishes in the sense of time-averaging.
“A gravitational wave with a source should be regarded as a gravitational radiation. Gravitational radiations will result in the energy loss of the source. With the conservation law of the energy, we may define the energy of the cylindrical gravitational radiation in our framework. We can also consider the resonance between matter waves and gravitational waves based on our previous works on scattering.”— they concluded.
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Reference: Chen, Y.-Z.; Li, S.-L.; Chen, Y.-J.; Dai, W.-S. Cylindrical Gravitational Wave: Source and Resonance. Symmetry 2021, 13, 1425. https://doi.org/10.3390/sym13081425
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