quadrupole formula

{{Short description|Formula in general relativity}}

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

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

where $\backslash bar\{h\}\_\{ij\}$ is the spatial part of the trace reversed perturbation of the metric, i.e. the gravitational wave. $G$ is the gravitational constant, $c$ the speed of light in vacuum, and $I\_\{ij\}$ is the mass quadrupole moment.{{cite book

|title=Spacetime and Geometry

|first=Sean M.

|last=Carroll

|publisher=Pearson/Addison Wesley

|isbn=978-0805387322

|pages=300–307

|year=2004

}}

It is useful to express the gravitational wave strain in the transverse traceless gauge, by replacing the mass quadrupole moment $I\_\{ij\}$ with the transverse traceless projection $I\_\{ij\}^\{TT\}$, which is defined as:

:$\{I\}\_\{ij\}^\{TT\}\; =\; \backslash int\; \backslash rho(\backslash mathbf\{x\})\; \backslash left[r\_i\; r\_j\; -\; r\_n(r\_i\; n\_j\; +\; r\_j\; n\_i)\; +\; \backslash frac\{1\}\{2\}\; r\_n^2\; (n\_i\; n\_j\; +\; \backslash delta\_\{ij\}\; )\; +\; \backslash frac\{1\}\{2\}\; r^2\; (n\_i\; n\_j\; -\; \backslash delta\_\{ij\}\; )\; \backslash right]\; d^3\; r$

where $\backslash mathbf\{n\}$ is a unit vector in the direction of the observer, $r\_n\; \backslash equiv\; \backslash mathbf\{r\}\backslash cdot\backslash mathbf\{n\}$, and $r^2\; \backslash equiv\; \backslash mathbf\{r\}\backslash cdot\backslash mathbf\{r\}$.{{cite web |last1=Creighton |first1=Teviet |title=Formulae and Details |url=http://www.tapir.caltech.edu/~teviet/Waves/gwave_details.html}}

The total energy carried away by gravitational waves can be expressed as:

:$\backslash frac\{d\; E\}\{dt\}\; =\; \backslash sum\_\{ij\}\; \backslash frac\{G\}\{5\; c^5\}\; \backslash left(\; \backslash frac\{d^3\; I\_\{ij\}^\{T\}\}\{dt^3\}\; \backslash right)^2$

where $I\_\{ij\}^\{T\}$ is the traceless mass quadrupole moment, which is given by:

:$\{I\}\_\{ij\}^T\; =\; \backslash int\; \backslash rho(\backslash mathbf\{x\})\; \backslash left[r\_i\; r\_j\; -\; \backslash frac\{1\}\{3\}\; r^2\; \backslash delta\_\{ij\}\backslash right]\; d^3\; r.$

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).{{cite book

|title=Gravity:Newtonian, Post-Newtonian, Relativistic

|first1=Eric

|last1=Poisson

|first2=Clifford M.

|last2=Will

|publisher=Cambridge University Press

|isbn=9781107032866

|pages=550–563

|date=2014-05-29

}}