# Bonnor beam

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In [general relativity](/source/General_relativity), the **Bonnor beam** is an [exact solution](/source/Exact_solutions_in_general_relativity) which models an infinitely long, straight beam of [light](/source/Light). It is an explicit example of a [pp-wave spacetime](/source/Pp-wave_spacetime). It is named after [William B. Bonnor](/source/William_B._Bonnor) who first described it.

The Bonnor beam is obtained by matching together two regions:

- a uniform plane wave interior region, which is shaped like the [world tube](/source/World_tube) of a solid cylinder, and models the electromagnetic and gravitational fields inside the beam,
- a vacuum exterior region, which models the gravitational field outside the beam.

On the "cylinder" where they meet, the two regions are required to obey matching conditions stating that the [metric tensor](/source/Metric_tensor) and [extrinsic curvature](/source/Extrinsic_curvature) tensor must agree.

The interior part of the solution is defined by

- \left\{ \begin{array}{lr} ds^2 = -8 \pi m r^2 \, du^2 - 2 \, du \, dv + dr^2 + r^2 \, d\theta^2,\\-\infty < u,\\ v < \infty,\\ 0 < r < r_0,\\ -\pi < \theta < \pi.\\ \end{array} \right.

This is a [null dust solution](/source/Null_dust_solution) and can be interpreted as incoherent [electromagnetic radiation](/source/Electromagnetic_radiation).

The exterior part of the solution is defined by

- \left\{ \begin{array}{lr} ds^2 = -8 \pi m r_0^2 \left( 1 + 2 \log(r/r_0) \right) \, du^2 - 2 \, du \, dv + dr^2 + r^2 \, d\theta^2 \\ -\infty < u,\\ v < \infty,\\ r_0 < r < \infty,\\ -\pi < \theta < \pi.\\ \end{array} \right.

The Bonnor beam can be generalized to several parallel beams travelling in the same direction. Perhaps surprisingly, the beams do not curve toward one another. On the other hand, "anti-parallel" beams (travelling along parallel trajectories, but in opposite directions) *do* attract each other. This reflects a general phenomenon: two pp-waves with parallel [wave vectors](/source/Wave_vector) superimpose linearly, but pp-waves with nonparallel wave vectors (including antiparallel Bonnor beams) do *not* superimpose linearly, as we would expect from the [nonlinear](/source/Nonlinear) nature of the [Einstein field equation](/source/Einstein_field_equation).

## References

- Faraoni, V. & Dumse, R. M. (1999). "The gravitational interaction of light: from weak to strong fields". *Gen. Rel. Grav.*. **31** (1): 91–105. [arXiv:gr-qc/9811052](https://arxiv.org/abs/gr-qc/9811052). [Bibcode:1999GReGr..31...91F](https://ui.adsabs.harvard.edu/abs/1999GReGr..31...91F). [doi:10.1023/A:1018867405133](https://doi.org/10.1023/A:1018867405133). [S2CID 18584608](https://api.semanticscholar.org/CorpusID:18584608). See also Faraoni & Dumse (1999). "The gravitational interaction of light: from weak to strong fields". *General Relativity and Gravitation*. **31** (1): 91–105. [arXiv:gr-qc/9811052](https://arxiv.org/abs/gr-qc/9811052). [Bibcode:1999GReGr..31...91F](https://ui.adsabs.harvard.edu/abs/1999GReGr..31...91F). [doi:10.1023/A:1018867405133](https://doi.org/10.1023/A:1018867405133). [S2CID 18584608](https://api.semanticscholar.org/CorpusID:18584608)
- Bonnor, W. B. (1969). ["The gravitational field of light"](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-13/issue-3/The-gravitational-field-of-light/cmp/1103841572.pdf). *Comm. Math. Phys.*. **13** (3): 163–174. [Bibcode:1969CMaPh..13..163B](https://ui.adsabs.harvard.edu/abs/1969CMaPh..13..163B). [doi:10.1007/BF01645484](https://doi.org/10.1007/BF01645484). [S2CID 123398946](https://api.semanticscholar.org/CorpusID:123398946)

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Adapted from the Wikipedia article [Bonnor beam](https://en.wikipedia.org/wiki/Bonnor_beam) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Bonnor_beam?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.