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The (rather old) source I dispose of "Sexl, Urbantke : Gravitation and Cosmology" describes the radiation of gravitational waves only rather sketchy. So why gravitational waves are only generated by time-varying quadrupole moment tensor of a mass distribution ? Why other pole moments are not possible (like monopolar, dipolar and even higher poles like sextu ...)? And why does it require a non-zero third time derivative of the quadrupole moment tensor to produce a non-zero gravitational wave energy flux ? For completeness I give the definition of the quadrupole moment tensor of a mass distribution:

$$ Q_{\alpha\beta} = \int d^3x (x_\alpha x_\beta -\frac{1}{3}r^2 \delta_{\alpha \beta})$$

where the indices $\alpha$ and $\beta$ run only from $1\ldots 3$.

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    $\begingroup$ Well with gravitational dipole radiation, you would need a mass to oscillate up and down. To do this, its momentum needs to change. If only one mass is oscillating, this violates the conservation of momentum. If two masses are oscillating in equal and opposite ways, the conservation of momentum isn't violated and gravitational quadrupole radiation is created. $\endgroup$ – Laff70 Jul 19 at 17:01
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    $\begingroup$ Higher poles do contribute. Just not lower poles. $\endgroup$ – G. Smith Jul 19 at 17:29
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    $\begingroup$ You might want to split this into two questions: why there are no monopole or dipole waves, and why it's the third derivative that matters. $\endgroup$ – Ben Crowell Jul 19 at 17:43
  • $\begingroup$ This is what leads to the spin of the graviton being 2. to make consistent with the classical frame a quantized gravitational theory.. The photons are emitted by dipoles (and there are also higher poles with lower probability) , and spin two has to be a quadrupole or higher, (from the math afaik) $\endgroup$ – anna v Jul 20 at 16:24

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