If an electron was vibrated back and forth via oscillating electromagnetic fields, it would presumably produce a small gravitational wave. Can the gravitational wave be theoretically calculated to determine its gravitational field strength?
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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.
Roger Wood
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