Imagine the rings of saturn. Try to find a stable rotating configuration in the following preparation idea: Each particle rotates on a circle at constant 3d-speed. Each particle tries to maintain constant distances to its two neighbours on the same circle, and, of course, its distance to the neighbouring inner and outer circle. This poses no difficulty.
Next, two neighbouring particles on the same radius try to circle in constant distance with respect to their board watch readings. This will establish a constant rigid rotation in their local system of reference. Both partners determine the distance at the ticks of their respective clocks, that run on different speeds with respect the global time. But independent of board time speed, the aim is to keep the local position independent wrt. to the angular coordinate $\phi$.
In Newtonian mechanics, with board time synchronous to global time at the axis, this preparation is a model of a rigidly rotating disk with constant angular frequency $\omega(r) =\frac{2\pi}{T(r)} =v(r)/r$.
Now switch representation to flat Minkowski space.
The proper time at radius r is
$$\tau(r)= \int \sqrt{dt^2 - r^2 d\phi ^2 }
= \int \sqrt{1- r^2 \frac{d\phi^2 }{dt^2}} \ dt=\int \sqrt{1- r^2 \Omega^2} \ dt=\sqrt{1-r ^2\Omega ^2} \ t$$
with $\Omega$ representing the frequency of rotation with respect to time t at the axis at rest.
Appying this formula to the inverses yields the local angular frequency in terms of the global one
$$ \omega(r)^2\ = \ \frac{\Omega(r)^2}{1-\Omega(r) ^2} $$
with the inverse
$$\Omega = \frac{\omega }{\sqrt{\omega ^2+1}}$$
Since the stability of such a system in flat Minkowski space depends on the constancy of rigid space coordinate differences at global flat time , $\Omega(r)$ has to be constant, keeping all radial distances between particles rotating on the same radius constant, determined by simultaneity of global time.
Such a model represents a rigidly rotating disk with local time rate slowing with radius, with a limiting radius where local time stands still because $r \Omega = 1$.
The local board time is central in determining the radial forces, and, insofar, determines the radial mass distribution. But the stable configuration is determined in the global system, because the potential fields are of electromagnetic nature, best understood in a global nonrotating system.
