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Basically the title. I came across this question while going through old exams. I thought that electromagnetism is stronger than gravity by 40 orders of magnitude more or less, at all scales, so this is confusing. Any help is appreciated.

As a hint this is provided: ignore the running of the coupling for electromagnetism (that is, the fine structure constant).

Qmechanic
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Razor
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  • How is electromagnetism stronger than gravity across intergalactic distances? Or even interstellar or interplanetary distances? My understanding is the complete opposite but I could be wrong. – DKNguyen Jun 18 '21 at 20:19
  • At a black hole, gravity can literally stop time. Is there any scale where electric force can achieve anything like that? – foolishmuse Jun 18 '21 at 20:26
  • What do you mean by scale? Charges come in both signs, so they tend to cancel over large distances, whereas gravity does not, but the tendency for large objects to be electrically neutral is not a fundamental law. Are you referring to the fact that gravity always dominates scattering interactions at sufficiently high center-of-mass energy, even if the particles are charged so that the electrostatic force dominates at lower energy? If so, then this post may be related: Does kinetic energy warp spacetime? – Chiral Anomaly Jun 18 '21 at 20:40
  • I'm not sure what the title is getting at, but perhaps the forces be equal near the surface of the earth if $F_{electrical}=QE=F_{gravity}=mg$ or $QE=mg$ – Bob D Jun 18 '21 at 21:08
  • @DKNguyen That's what I thought, I think the question is looking for a lengthscale much shorter than that, because clearly anywhere equal to a few Debye lengths in a plasma gravity would dominate over electromagnetic forces. – Razor Jun 18 '21 at 23:48
  • @ChiralAnomaly The question is looking for a lengthscale. Of course, theories of QG are supposed to be of the order of planck length thus very high energies, so yes, along with the hint, I do think that's the question is getting at. However, the ratio of the forces are independent of the scale, so I still don't know the answer – Razor Jun 18 '21 at 23:56
  • @foolishmuse That is not the point, and also is a frame-dependent fact. At long scales, gravity surely wins. So we are talking about small scales. – Razor Jun 18 '21 at 23:58
  • @BobD If you work that out, you'll find that the charge to mass ratios need to be the same for the particles, but still does not give a lengthscale. – Razor Jun 19 '21 at 00:00

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