# Gravitational plane wave

In general relativity, a gravitational plane wave is a special class of a vacuum pp-wave spacetime, and may be defined in terms of Brinkmann coordinates by

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

Here, ${\displaystyle a(u),b(u)}$ 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.

Gravitational waves exist and travel at the speed of light. Every form of matter and energy with which we are familiar has the same gravitational effect positive attraction. Gravitational waves are very weak. Consider two neutron stars of one solar mass (MNS =M) and radius rNS = 6 km orbiting each other essentially in contact. (This is for example, clearly tidal forces will have a very disruptive effect on the two neutron stars.)

The characteristic acceleration at the neutron stars is

${\displaystyle a={\frac {2GM\odot }{(2rNS)^{2}}}\simeq 9\times 10^{11}m/s^{2}\simeq 10^{11}g}$[1]

## References

1. ^ Problem Set 12: GRAVITATIONAL WAVES G. F. SMOOT Department of Physics, University of California, Berkeley, USA 94720