# Can the gravitational wave of an electron be theoretically calculated to determine its gravitational field strength?

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?

• en.wikipedia.org/wiki/Quadrupole_formula Commented Jan 21, 2021 at 2:19
• @G.Smith does an electron (accelerating or otherwise) even have a non-zero quadrupole moment? Commented Jan 21, 2021 at 3:30
• @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. Commented Jan 21, 2021 at 3:37
• @G.Smith there is a discussion here Commented Jan 21, 2021 at 5:34

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.
• 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. Commented Jan 21, 2021 at 5:50