Gravitational plane wave

In general relativity, a gravitational plane wave (or plane gravitational wave) is defined to be any non-flat solution to Einstein's empty space-time field equations, which has the same symmetries as a plane electromagnetic wave. ^{[1] [2] [3] [4]} They are a special class of a vacuum pp-wave spacetime.

In the early years of general relativity theory, the existence of gravitational waves was hotly debated, as after theorizing them, Einstein and Nathan Rosen came to the erroneous conclusion in 1936 that gravitational plane waves do not exist. However, not every physicist was convinced as, after Rosen left for the Soviet Union, Einstein's new assistant Leopold Infeld and the physicist Howard P. Robertson showed Einstein that his and Rosen's conclusion was incorrect and, with his help, rigorously proved that gravitational cylindrical waves exist. This led to continued debate on the nature of gravitational waves and the troubles of defining them in Cartesian coordinates for mathematical simplicity rather than physical convenience, until in 1956 Felix Pirani wrote a paper which demonstrated a mathematical formalism to show how gravitational waves moved particles back and forth in their own coordinate systems, bypassing the issue. ^[5]

In 1957, there was a conference on the role of gravitation in physics known as the Chapel Hill Conference in the North Carolina town of the same name, where Pirani presented this paper and, from its findings, the physicist Richard Faynman suggested a thought experiment called the sticky bead argument which demonstrated that gravitational waves must carry energy. ^[6] This argument formed the basis of the papers that Hermann Bondi would go on to write after this conference which would prove the existence of the gravitational plane wave. ^[5]

Coordinate Systems

In general relativity, ^[7] they may be defined in terms of Brinkmann coordinates by

${\displaystyle g=\delta _{ij}dX^{i}dX^{j}+2dUdV+K_{ij}(U)dX^{i}dX^{j}dU^{2}}$

Where ${\displaystyle K_{ij}(U)}$ is a symmetric and trace less 2 x 2 matrix which controls the waveform of the two possible polarization modes of gravitational radiation. ^[8] In this context, these two modes are usually called the plus mode and cross mode, respectively. The plus mode is characterized by the equation

${\displaystyle K_{ij}(U)={\frac {1}{2}}A(U)diag(1,-1)}$

where ${\displaystyle A(U)}$ represents an arbitrary function. ^[8]

See also

• Physics portal

• Einstein–Rosen waves
• Vacuum solution (general relativity)
• Gravitational wave background

References

1. Bondi, H.; Pirani, F. A. E.; Robinson, I. (1959-06-23). "Gravitational waves in general relativity III. Exact plane waves". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 251 (1267): 519–533. Bibcode:1959RSPSA.251..519B. doi:10.1098/rspa.1959.0124. ISSN   0080-4630.
2. Russo, Jorge G. (2018). "Exact gravitational plane waves and two-dimensional gravity". Physics Letters B. 784. Elsevier BV: 142–145. arXiv: 1805.08663 . Bibcode:2018PhLB..784..142R. doi: 10.1016/j.physletb.2018.07.039 . ISSN   0370-2693.
3. Hogan, P. A. (1980). "Plane Gravitational Waves". Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences. 80A (1). Royal Irish Academy: 85–92. ISSN   0035-8975. JSTOR   20489085 . Retrieved 2024-12-23.
4. Wang, Anzhong (2020). Interacting Gravitational, Electromagnetic, Neutrino and Other Waves: In the Context of Einstein's General Theory of Relativity. WORLD SCIENTIFIC. doi:10.1142/9789811211492_0002. ISBN   978-981-12-1148-5.
5. L., Cervantes-Cota, Jorge; Salvador, Galindo-Uribarri; F., Smoot, George (September 2016). "A Brief History of Gravitational Waves". Universe. 2 (3). doi: 10.3390/univer (inactive 13 December 2025). ISSN   2218-1997. Archived from the original on 2025-07-19.
6. Goenner, Hubert; Renn, Jürgen; Ritter, Jim; Sauer, Tilman (1998-12-01). The Expanding Worlds of General Relativity. Springer Science & Business Media. ISBN   978-0-8176-4060-6.
7. BONDI, H. (1957). "Plane Gravitational Waves in General Relativity". Nature. 179 (4569). Springer Science and Business Media LLC: 1072–1073. Bibcode:1957Natur.179.1072B. doi:10.1038/1791072a0. ISSN   0028-0836.
8. Zhang, P.-M.; Duval, C.; Horvathy, P. A. (2018-01-23), "Memory effect for impulsive gravitational waves", Classical and Quantum Gravity, 35 (6), arXiv: 1709.02299 , Bibcode:2018CQGra..35f5011Z, doi:10.1088/1361-6382/aaa987 , retrieved 2025-12-12