Derivation of the quadrupole moment of gravitational wave

[Shan K]

Hi,
Can anyone give a derivation of the quadrupole moment in the TT gauge ( Equation metioned below ) of the gravitational wave ?

Q_ij^TT(x) = ∫ρ(xⁱx^j-1/3δ^ijr²)d³x
Where ρ is the matter density

Thanks In Advance

 

[WannabeNewton]

Hi Shan K. Do you own any general relativity texts? The TT gauge quadrupole moment is derived in almost every standard general relativity text. What you have written down is the reduced quadrupole moment and is simply the trace-free part of the quadrupole moment. If you don't own a general relativity text then check this out: http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll6.html