Sort of a fun “what if?” question in physics – What if we considered gravity to be approximated by a weaker “secondary” force in addition to the usual attractive force … similar to how the magnetic field relates to the electric field, and let it obey similar equations?

The magnetic field inside a spherical rotating shell of charge is given by

\[ B = \frac{2}{3}\mu_0\omega\sigma R \]

where \(\omega\) is the angular velocity of rotation and \(\sigma\) is the surface charge density. (ref) We can integrate that for a sphere of uniform volume charge density to get -

\[ B = \frac{1}{3}\mu_0\omega\rho R^2 \]

… where \(\rho\) is the volume charge density. Taking the analogical form for the gravitational secondary field, treating the universe as a spherical mass, we have –

\[ B_g = \frac{1}{3}\mu_g\omega \frac{M_{\text{univ}}}{\frac{4}{3}\pi R_{\text{univ}}^3} R_{\text{univ}}^2 \]

(where \(\mu_g = 4\pi G / c^2\) by analogy with \(\mu_0\epsilon_0 = 1/c^2\), with \(\epsilon_g = 1/{4\pi G}\).)

which gives –

\[ B_g = \frac{\mu_g\omega M_{\text{univ}}}{4\pi R_{\text{univ}}} \]

If such a secondary force did exist and the universe were rotating, then we’d feel an equivalent force of this kind that is proportional to \(\vec{v} \times \vec{\omega}\). But wait, the Coriolis force is indeed of that form. So if a rotating frame of reference and a rotating universe were to produce the same effects, that would put a constraint on some properties of the universe.

So if we set the effect of the secondary gravitational field to be that of the Coriolis force \(2m\vec{v} \times \vec{\omega}\), we get –

\[ 2\omega = \frac{\mu_g \omega M_{\text{univ}}}{4\pi R_{\text{univ}}} \]

which gives (after substituting \(\mu_g = 4\pi G/c^2\)) –

\[ 2 = \frac{G M_{\text{univ}}}{R_{\text{univ}}c^2} \]

The Schwarzschild radius of a blackhole is given by \(R_s = 2GM/c^2\).

So this would tell us that our universe is a blackhole and we’re living inside of it? … since that equivalence gives us \(R_{\text{univ}} = R_s/4\).

Caveat

This is kind of a silly extrapolation along the lines of trying to calculate the mass-radius relationship of a blackhole by setting the escape velocity of a Newtonian-gravitational mass to the speed of light. But it seems to give another perspective on the equivalence between a rotational frame of reference and a non-rotating frame but where the universe is rotating in the opposite direction?

Anyway, just some musings :)