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In gravitoelectromagnetism, an approximation to general relativity in the weak field limit, Einstein's equations simplify into a form very similar to Maxwell's equations. In this field, traditional gravitational fields are referred to as "gravitoelectric" fields, and by changing can induce their equivalent to a magnetic field, gravitomagnetic fields. Conversely, a changing gravitomagnetic field can induce a gravitoelectric field.

Importantly, the gravitational fields induced by gravitomagnetic fields can be dipolar, with both attractive and repulsive poles. With all of that in mind, and with the proviso that since these fields are nonconservative (the field lines of the induced gravitational field form closed loops much like an induced electric field) and thus usual arguments concerning Newtonian potentials are inapplicable:

What is the relative gravitational time dilation of an observer situated vertically 1 meter (on the repulsive side) from the central point of a torus which is producing a dipolar gravitational field of 100g relative to a faraway observer? Specifically, since the field is repulsive, would it cause the clock of the observer situated close to the torus to tick faster relative to the faraway observer?

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Assuming that we're working under the weak field approximation, the gravitational potential should have the form: $$P=\frac{n\cos(\theta)}{r^2}$$ The field along the vertical axis is: $$g=\frac{2n}{r^3}$$ To find n's value, we use the fact that g=100 at r=1. $$n=\frac{gr^3}{2}=\frac{100\cdot1^3}{2}=50$$ Gravitational time dilation depends on gravitational potential. $$t_d=e^{\frac{P}{c^2}}=e^{\frac{n\cos(\theta)}{c^2r^2}}=e^{\frac{50\cos(\theta)}{c^2r^2}}$$ Now to find the rate at which time passes at said point $$t_d=e^{\frac{50\cos(0)}{c^2\cdot1^2}}=e^{\frac{50}{c^2}}=e^{\frac{50}{299792458^2}}=e^{5.5632503\cdot10^{-16}}=1.0000000000000005563250280268093708358133869390635833174567871473...$$ As you can see, time passes a little bit faster at this point than a point infinitely far away. Given that the potential is $50\frac{m^2}{s^2}$, I'd say the weak field approximation is valid here.

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  • $\begingroup$ Firstly, thank you for your response. Are you absolutely sure that using potential is valid here? I've mentioned above that the field is non-conservative, so would you mind elaborating a bit on why you know its useful to apply in this situation? $\endgroup$ – CuriousDroid Aug 20 '20 at 2:17
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    $\begingroup$ If you had a current loop with current $I=/frac{I_0}{A}$ and you decreased its area to 0, its magnetic moment would be unchanged. Its field would become equivalent to a simple dipole's. Since you didn't specify the loop of gravomagnetic current's radius, I set it to 0. As such, using a dipole's potential should be valid. $\endgroup$ – Laff70 Aug 20 '20 at 14:05
  • $\begingroup$ I see, thank you for your clarification; that's a very clever solution! Is it possible to make a similar argument when the radius of the loop is finite and nonzero? I would assume not, but I'd prefer to hear your analysis. $\endgroup$ – CuriousDroid Aug 20 '20 at 14:19
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    $\begingroup$ Probably not. Gravomagnetism is just the name of the effect on matter moving in differentially flowing space. The path of the mass gets bent depending on how it's moving similarly to a charge moving in a magnetic field. Due to how it works on a fundamental level, it may not be possible to have a gravomagnetic monopole. As such, gravomagnetic current may not be able to even theoretically exist. In that case, the construction of such a current loop would be impossible. I honesty don't know. I do know that relative time dilation is dependent on gravitational potential difference though. $\endgroup$ – Laff70 Aug 20 '20 at 14:47

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