In general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, the Bonnor beam is an exact solution

Exact solutions in general relativity

In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field....

 which models an infinitely long, straight beam of light

Light

Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

. It is an explicit example of a pp-wave spacetime

Pp-wave spacetime

In general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutions of Einstein's field equation. These solutions model radiation moving at the speed of light...

. It is named after William B. Bonnor

William B. Bonnor

William Bowen Bonnor is a mathematician and gravitation physicist best known for his research into astrophysics, cosmology and general relativity. For most of his academic career he has been a professor of mathematics at the University of London....

 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 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

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

 and extrinsic curvature tensor must agree.

The interior part of the solution is defined by


This is a null dust solution

Null dust solution

In mathematical physics, a null dust solution is a Lorentzian manifold in which the Einstein tensor is null...

 and can be interpreted as incoherent electromagnetic radiation

Electromagnetic radiation

Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

.

The exterior part of the solution is defined by


The Bonnor beam can be generalized to several parallel beams traveling in the same direction. Perhaps surprisingly, the beams do not curve toward one another. On the other hand, "anti-parallel" beams (traveling along parallel trajectories, but in opposite directions) do attract each other. This reflects a general phenomenon: two pp-waves with parallel wave vector

Wave vector

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...

s superimpose linearly, but pp-waves with nonparallel wave vectors (including antiparallel Bonnor beams) do not superimpose linearly, as we would expect from the nonlinear nature of the Einstein field equation.