Quadrupole formula explained

In general relativity, the quadrupole formula describes the gravitational waves that are emitted from a system of masses in terms of the (mass) quadrupole moment. The formula reads$$\backslash bar\_(t,r)\; =\; \backslash frac\; \backslash ddot\_\; (t-r/c),$$where

is the spatial part of the trace reversed perturbation of the metric, i.e. the gravitational wave. is the gravitational constant, the speed of light in vacuum, and is the mass quadrupole moment.^[1]

It is useful to express the gravitational wave strain in the transverse traceless gauge, by replacing the mass quadrupole moment

with the transverse traceless projection , which is defined as:$$\_^\; =\; \backslash int\; \backslash rho(\backslash mathbf)\; \backslash left[r\_i\; r\_j\; -\; r\_n\; \backslash left(r\_i\; n\_j\; +\; r\_j\; n\_i\backslash right)\; +\; \backslash tfrac\{1\}\{2\}\; r\_n^2\; \backslash left(n\_i\; n\_j\; +\; \backslash delta\_\{ij\}\; \backslash right)\; +\; \backslash tfrac\{1\}\{2\}\; r^2\; \backslash left(n\_i\; n\_j\; -\; \backslash delta\_\{ij\}\; \backslash right)\; \backslash right]\; d^3\; r$$where is a unit vector in the direction of the observer, , and .^[2]

The total energy carried away by gravitational waves can be expressed as:$$\backslash frac\; =\; \backslash sum\_\; \backslash frac\; \backslash left(\backslash frac\; \backslash right)^2$$where

is the traceless mass quadrupole moment, which is given by:$$\_^T\; =\; \backslash int\; \backslash rho(\backslash mathbf)\; \backslash left[r\_i\; r\_j\; -\; \backslash tfrac\{1\}\{3\}\; r^2\; \backslash delta\_\{ij\}\backslash right]\; d^3\; r.$$

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).^[3]

See also

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

Notes and References

1. Book: Carroll, Sean M. . Sean M. Carroll . Spacetime and Geometry: An Introduction to General Relativity . Spacetime and Geometry . Addison-Wesley . 2003 . 978-0-805-38732-2 . 300–307.
2. Web site: Creighton . Teviet . Formulae and Details .
3. Book: Gravity: Newtonian, Post-Newtonian, Relativistic . Eric . Poisson . Clifford M. . Will . . 9781107032866 . 550–563 . 2014-05-29 .