Why do planetary all rings seem to be on the equatorial plane?
When I first saw a picture of Uranus and its rings, I was surprised by their inclination, 90 degrees from the axis of rotation.
This led me to believe that the drag of the planet on spacetime created a gravity well that the rings 'fell' into. My recent studies into frame-dragging seem to validate this idea.
Are there any other studies in this area I can look into?
You have a bunch of confused concepts here. First, it is not true that all ringed gas giants have rings at such an angle to their axis of rotation. Take Saturn, for instance:
Image in the public domain.
See how its rings align with the horizontal cloud bands on its surface? The rings are clearly aligned with the axis of rotation, not at all like the situation on Neptune.
Another misconception is that Uranus's rings are tilted relative to its poles. This is not strictly correct; rather, Uranus is tilted relative to its rings. The reason for this is thought to be a collision between Uranus and another planet early in its life. The rings themselves formed much later.
Frame dragging is completely unrelated. It is more pronounced in the case of massive bodies rotating quickly; Uranus certainly does not satisfy the criteria for a major observable manifestation of this.
After your edit, I think I better understand what you're saying.
The reason that planetary rings are typically situated around the equator of a planet is because of their formation. One way that rings form is when a moon strays too close to its planet and is torn apart by tidal forces, forming one or more rings. Another way is when a planet pulls in extra matter from the protoplanetary disk, which can form a ring system.
In both cases, the source material is roughly perpendicular to the axis of rotation, and thus the rings form roughly perpendicular to the axis of rotation. Moons generally orbit in their planet's orbital plane, and planet axes are generally (but not always) closer to perpendicular than parallel to the protoplanetary disk.
Therefore, rings will form on the planet's equatorial plane.
The other answer misses out an important mechanism explaining this phenomenon.
Indeed, a ring system not initially aligned with the equator of the planet will actually align to the planet's equatorial plane over time due to gravitational and collisional effects:
For a rotating planet the centrifugal effect causes it to bulge at the equator. This causes extra mass to be distributed around the equator of the planet, giving rise to a gravitational field whose potential can be approximated to high precision by:
$$ V(r, θ, ϕ) = \frac{GM}{r} +\frac{GJ_2 (1−3cos^2 θ)}{2r^3 } $$
In this equation, ${θ}$ represents the angle between the planet's axis of rotation and a given point.${J_2}$ is a parameter describing the planet's equatorial bulge. A derivation for this is here - https://farside.ph.utexas.edu/teaching/336k/Newton/node108.html
The first term gives rise to the familiar inverse square law, while the second term is a "quadrupole term". This term falls off much more rapidly with distance but gives rise to angular features of the field. In this case it causes any objects in orbits inclined relative to the equatorial plane to experience a torque.
The torque causes the orbit of particles in the ring system precess around the rotational axis of the planet; only the z-component of angular moment is conserved.
In the below diagram, the angular momentum of a particle in the ring (green arrow) rotates around the planet angular momentum vector (red arrow).
The torque and rate of orbital precession depends heavily on the semi-major axis of the orbiting particle. This is described in detail in Chapter 2.1 - https://arxiv.org/pdf/1606.00759.pdf.
But importantly, particles which start off in orbits with similar inclinations but different semi-major axis will gradually precess into orbits with different inclinations.
These particles collide over time. These collision dissipate energy but conserve angular momentum. The effect of this is to average out the angular momenta of particles into a single plane. This is the exact same effect which causes the solar system, galaxies and ring systems to be flat in the first place. The difference here is that the quadrupole force causes the components of angular momentum which were not aligned with the planet to precess, so the collisions between particles will tend to align the ring with the planet.
