Numerical solution to the relative gravitational time dilation of induced dipolar gravitational fields

Question

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?

Answer 1

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.