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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?

Qmechanic
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OperationE
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1 Answers1

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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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    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. – A.V.S. Jan 21 '21 at 05:50
  • @A.V.S. Various sources seem to be suggesting otherwise: https://www.ligo.org/science/GW-Sources.php and https://en.wikipedia.org/w/index.php?title=Gravitational_wave&action=edit§ion=4. Surely an electron orbiting a black hole will emit graviitational waves? – Roger Wood Jan 21 '21 at 07:09
  • 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. – G. Smith Jan 21 '21 at 22:57
  • A mass orbiting a black hole radiates gravitationally, but the power it radiates is not given by your incorrect formula. The correct formula is (2.29) here. – G. Smith Jan 21 '21 at 22:59
  • @G. Smith Agreed. However, the quadrupole arguments only apply to closed systems where momentum is conserved. This paper does discuss situations where the dipole effects are significant: https://arxiv.org/ftp/arxiv/papers/1002/1002.0351.pdf ,but doesn't directly give a formula a relatively isolated accelerating mass. (I did edit my answer earlier to try to indicate the limitations.) – Roger Wood Jan 21 '21 at 23:44
  • That 10-year-old preprint does not seem to have ever gotten published, and the author is an independent researcher currently unassociated with any academic institution or research institute. In my opinion some skepticism is warranted. In the OP’s example, whatever is causing the electron to oscillate is going to oscillate in the other direction, conserving momentum. – G. Smith Jan 22 '21 at 01:28
  • @G. Smith there's another subtlety when I re-read the question. The electron is oscillating in an electromagnetic field. So presumably the scattering caused by the electron will conserve momentum in some localized region. I suppose the same is true even for a DC field, the momentum of the Larmor radiation will balance the changing electron momentum. – Roger Wood Jan 22 '21 at 01:39
  • thanks @G.Smith. To confirm what is the most precise set of equations to use? As the system would theoretically not be a closed system. Is it more precise to use the gravitational potential for the mass quadrupole? – OperationE Jan 24 '21 at 19:20
  • @OperationE If the electron is oscillating, whatever is producing the electric field making it oscillate is going to also oscillate to conserve momentum, and you should not ignore its gravitational radiation. The only calculations I have seen are for closed systems where momentum is conserved (except for whatever momentum gets radiated away). – G. Smith Jan 24 '21 at 19:57
  • @OperationE Common cases I have seen calculated include a rotating rod, two masses connected by a spring, two masses orbiting each other, two masses falling toward each other, two masses on fly-by trajectories, etc. These systems all conserved momentum and the calculations all used the quadrupole formula. – G. Smith Jan 24 '21 at 20:05
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    @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. – G. Smith Jan 24 '21 at 20:09
  • thanks @G.Smith. Overall, the set up I envision is a mini particle accelerator. Instead of straight acceleration, the particle (or particles if required) are vibrating at high enough frequencies that in principle radiate gravity waves. Given my limited background, I know that gravity couples to mass, and since we have good control over manipulating and generating electromagnetic energy; I thought it would make the most sense to try and use electromagnetic fields to control the movement of small masses to intentionally produce, albeit small amplitude, high frequency gravitational waves. – OperationE Jan 26 '21 at 00:38
  • @OperationE If it were possible to generate measurable gravitational waves in a lab environment, physicists would be doing that. I am confident that the amplitude is far too small. With current detector technology, we have to rely on astrophysical sources. – G. Smith Jan 26 '21 at 01:38
  • @OperationE The quadrupole formula says that the metric perturbation would be of order $Gma^2\omega^2/(c^4r)$ where $m$ is the mass of the electron(s) you are shaking, $a$ and $\omega$ are the amplitude and angular frequency of its/their oscillation, and $r$ is the distance at which you detect the wave. Try putting in some values. LIGO can detect a perturbation of around $10^{-21}$, I think, but not a high-frequency one. – G. Smith Jan 26 '21 at 01:55
  • thanks again @G.Smith for entertaining my naive questions. This one is more abstract. If gravitational vacuum fluctuations exist, then there must be all sorts of gravity waves (albeit weak) in the quantum foam.

    Assuming in the far distant future we have the ability to generate gravity waves at whatever frequencies we desired, could it be possible to amplify the magnitude of the existing gravity waves in the quantum foam via resonance?

     – OperationE Jan 26 '21 at 02:00
  • @OperationE That doesn’t seem possible to me, but explaining why would not be brief. I don’t want to have an extended discussion about a distantly-related topic in the comments on Roger’s answer. You could post that as a question. I won’t necessarily answer it, but someone might. – G. Smith Jan 26 '21 at 04:02
  • @G.Smith the expression $Gma^2ω^2/(c^4r)$ is for 2 bodies with opposing motions in a circle (orbit) or simple harmonic motion. This following expression is pure guesswork: I replaced one amplitude with $a = E.q/(mω^2)$ for the response of charge, $q$, to electric field, $E$, and the other with $a = c/ω$ to try capture the scale in which momentum is conserved with the scattered photons. The result is $(G/c^3).(E.q/ω).(1/r)$. This says the gravitational waves are proportional to the electric field and electron charge and independent of mass - which seems reasonable. But not sure about the $1/ω$? – Roger Wood Jan 27 '21 at 02:18