If an electron was vibrated back and forth via oscillating electromagnetic fields, it would presumably produce a small gravitational wave. Can the gravity wave be theoretically calculated to determine its gravitational field strength?
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$\begingroup$ en.wikipedia.org/wiki/Quadrupole_formula $\endgroup$– G. SmithCommented Jan 21, 2021 at 2:19
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$\begingroup$ @G.Smith does an electron (accelerating or otherwise) even have a non-zero quadrupole moment? $\endgroup$– Nihar KarveCommented Jan 21, 2021 at 3:30
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$\begingroup$ @NiharKarve Yes, I think so. I don’t see any reason that it would be zero. But when I google I find only discussions of two linearly oscillating masses, so maybe I am missing something. $\endgroup$– G. SmithCommented Jan 21, 2021 at 3:37
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$\begingroup$ @G.Smith there is a discussion here $\endgroup$– Nihar KarveCommented Jan 21, 2021 at 5:34
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The radiated electromagnetic power from an non-relativistic accelerated charge is given by the Larmor formula P = $(2/3)(1/4\pi \epsilon 0)q^2a^2/c^3$. Given the gravitoelectromagnetic approximation and the correspondence between G and $(1/4\pi \epsilon 0)$, the gravitational radiation power from a non-relativistic accelerating mass will be P = $(2/3)Gm^2a^2/c^3$.
The gravitational power will be less than the electromagnetic power by a factor of $Gm^2/(q^2/4\pi \epsilon 0)$ = $G4\pi \epsilon 0/(q/m)^2$ = 2.4 x 10-43 for an electron.
[Edit responding to the comments below] In reality, the gravitational radiation must be very much less than this value. Because of action and reaction, gravitational waves from other parts of a closed system will tend to cancel the waves from the electron. The center of mass of a closed system cannot be accelerated and produce any gravitational waves. The gravitational waves that are produced arise only from the changing distribution of mass within the system. In the case of an electron oscillating due to electromagnetic radiation, the momentum carried in the radiation scattered by the electron must balance the momentum in the electron. So the gravitational radiation will be largely cancelled. The small amount that is radiated will be due to the fact that the electron and the scattered photons are not exactly colocated.
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2$\begingroup$ This is wrong. “correspondence between $G$ and $1/(4πϵ_0)$” only applies to static field. For radiation, where spin of the field is important situation is different. EM radiation has dipole character (as seen from Larmor formula). Gravitational radiation requires quadrupole moment variation. $\endgroup$– A.V.S.Commented Jan 21, 2021 at 5:50
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$\begingroup$ @A.V.S. Various sources seem to be suggesting otherwise: ligo.org/science/GW-Sources.php and en.wikipedia.org/w/…. Surely an electron orbiting a black hole will emit graviitational waves? $\endgroup$ Commented Jan 21, 2021 at 7:09
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$\begingroup$ Wikipedia: “The mass quadrupole moment is also important in general relativity because, if it changes in time, it can produce gravitational radiation, similar to the electromagnetic radiation produced by oscillating electric or magnetic dipoles and higher multipoles. However, only quadrupole and higher moments can radiate gravitationally.” You can also consult any textbook on GR. $\endgroup$– G. SmithCommented Jan 21, 2021 at 22:57
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1$\begingroup$ @OperationE Note that Roger has asked a question about whether it makes sense to consider gravitational waves from an oscillating mass dipole moment of a system in which momentum is not conserved, such as your oscillating electron. My current point of view is that the fact that there seems to be little literature on this is an indication that people consider it to be not physically meaningful. $\endgroup$– G. SmithCommented Jan 24, 2021 at 20:09