Gravitational plane waves are described as "non-flat solutions of

Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...

’s empty spacetime field equation". They are a special class of a vacuum pp-wave spacetime. In

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

, the may be defined in terms of Brinkmann coordinates by

ds^2= (u)(x^2-y^2)+2b(u)xy u^2+2dudv+dx^2+dy^2

Here,

a(u), b(u)

can be any

smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...

s; they control the

waveform In electronics, acoustics, and related fields, the waveform of a signal is the shape of its Graph of a function, graph as a function of time, independent of its time and Magnitude (mathematics), magnitude Scale (ratio), scales and of any dis ...

of the two possible

polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...

modes of

gravitational radiation Gravitational waves are oscillations of the gravitational field that travel through space at the speed of light; they are generated by the relative motion of gravitating masses. They were proposed by Oliver Heaviside in 1893 and then later by ...

. In this context, these two modes are usually called the plus mode and cross mode, respectively.

References

{{relativity-stub Exact solutions in general relativity

See also

References