Quadrupole formula

In general relativity, the quadrupole formula describes the rate at which gravitational waves are emitted from a system of masses based on the change of the (mass) quadrupole moment. The formula reads

${\displaystyle {\bar {h}}_{ij}(t,r)={\frac {2G}{c^{4}r}}{\ddot {I}}_{ij}(t-r/c),}$

where ${\displaystyle {\bar {h}}_{ij}}$ is the spatial part of the trace reversed perturbation of the metric, i.e. the gravitational wave. ${\displaystyle G}$ is the gravitational constant, ${\displaystyle c}$ the speed of light in vacuum, and ${\displaystyle I_{ij}}$ is the mass quadrupole moment.^[1]

It is useful to express the gravitational wave strain in the transverse traceless gauge, which is given by a similar formula where ${\displaystyle I_{ij}^{T}}$ is the traceless part of the mass quadrupole moment.

${\displaystyle {I}_{ij}^{T}=\int \rho (\mathbf {x} )\left[r_{i}r_{j}-{\frac {1}{3}}r^{2}\delta _{ij}\right]d^{3}r,}$

The total energy (luminosity) carried away by gravitational waves is

${\displaystyle {\frac {dE}{dt}}=\sum _{ij}{\frac {G}{5c^{5}}}\left({\frac {d^{3}I_{ij}^{T}}{dt^{3}}}\right)^{2}}$

The formula was first obtained by Albert Einstein in 1918. After a long history of debate on its physical correctness, observations of energy loss due to gravitational radiation in the Hulse–Taylor binary discovered in 1974 confirmed the result, with agreement up to 0.2 percent (by 2005).^[2]

See also

• Multipole radiation
• Birkhoff's theorem (relativity)
• PSR J0737−3039

References

1. Carroll, Sean M. (2004). Spacetime and Geometry. Pearson/Addison Wesley. pp. 300–307. ISBN 978-0805387322.
2. Poisson, Eric; Will, Clifford M. (2014-05-29). Gravity:Newtonian, Post-Newtonian, Relativistic. Cambridge University Press. pp. 550–563. ISBN 9781107032866.