Gravitational plane wave

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Gravitational plane waves are described as "non-flat solutions of Albert Einstein’s empty spacetime field equation".{{cite journal | title=Gravitational waves in general relativity III. Exact plane waves | journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences | volume=251 | issue=1267 | date=1959-06-23 | issn=0080-4630 | doi=10.1098/rspa.1959.0124 | pages=519–533}}{{cite journal | last=Russo | first=Jorge G. | title=Exact gravitational plane waves and two-dimensional gravity | journal=Physics Letters B | publisher=Elsevier BV | volume=784 | year=2018 | issn=0370-2693 | doi=10.1016/j.physletb.2018.07.039 | doi-access=free | pages=142–145| arxiv=1805.08663 }}{{cite journal | last=Hogan | first=P. A. | title=Plane Gravitational Waves | journal=Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences | publisher=Royal Irish Academy | volume=80A | issue=1 | year=1980 | issn=00358975 | jstor=20489085 | pages=85–92 | url=http://www.jstor.org/stable/20489085 | access-date=2024-12-23}}{{cite book | last=Wang | first=Anzhong | title=Interacting Gravitational, Electromagnetic, Neutrino and Other Waves: In the Context of Einstein's General Theory of Relativity | publisher=WORLD SCIENTIFIC | date=2020 | isbn=978-981-12-1148-5 | doi=10.1142/9789811211492_0002 | page=}} They are a special class of a vacuum pp-wave spacetime.

In general relativity,{{cite journal | last=BONDI | first=H. | title=Plane Gravitational Waves in General Relativity | journal=Nature | publisher=Springer Science and Business Media LLC | volume=179 | issue=4569 | year=1957 | issn=0028-0836 | doi=10.1038/1791072a0 | pages=1072–1073}} the may be defined in terms of Brinkmann coordinates by

$ds^2=[a(u)(x^2-y^2)+2b(u)xy]du^2+2dudv+dx^2+dy^2$

Here, $a(u),\; b(u)$ can be any smooth functions; they control the waveform of the two possible polarization modes of gravitational radiation. In this context, these two modes are usually called the plus mode and cross mode, respectively.