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Let's start with the most major feature of General Relativity:

Freely falling objects (around which one will build inertial frames of reference, with the object always at the spatial origin) are in geodesic motion, with the choice of all possible geodesics determined by the whole spacetime, and the particular geodesic determined by "initial" position and velocity.

The Schwarzschild metric for a black hole generates a set of available geodesics such that there is a large set of innermost stable circular orbits (ISCOs) permitted by the spherical symmetry and the absence of any outside matter to break the time-symmetry ("eternal, never growing or shrinking"). There are many initial <position, velocity> pairs that leave a test particle on an ISCO forever, and the ISCOs will generally be found at 3R_s, where R_s is the Schwarzschild radius.

If we break the spherical symmetry and make the system cylindrically symmetrical or axisymmetric, there are fewer ISCOs available: just as there are many great circles around a sphere, there are fewer around a cylinder. Rotating black holes generally do not have spherical symmetry, even if they are fully time-symmetric ("never grows nor shrinks, never changes its angular momentum").

Very roughly, frame-dragging captures the difference in 'equatorial' orbits between a spherically symmetric black hole ("one cannot distinguish an equatorial orbit from a polar orbit") and an equatorial orbit around a black hole with reduced symmetry (and between equatorial and nonequatorial orbits).

A rotating black hole with an axis of rotation lets one distinguish between an orbit that goes above both poles and an equatorial orbit (and orbital planes in between those extremes). It also lets one distinguish between a prograde and retrograde orbit. This affects available ISCOs: for an initial <position, velocity> where the velocity is prograde, the ISCO will be closer to the rotating black hole than if the velocity were retrograde. Notably, as the black hole rotation is increased to a maximum, the prograde ISCO "just touches" the event horizon.

On every ISCO, the local physics of a test free-faller is that of Special Relativity. This is essentially the content of the weak and Einstein equivalence principles, which you can get a feel for in the two sections starting at <https://en.wikipedia.org/wiki/Equivalence_principle#The_weak...>. Free-fallers behave like astronauts in the ISS or other orbiting or Earth-Moon spacecraft. Things just float about weightlessly, but the speed of light, Planck's constant, and so forth are unchanged compared to a lab on the surface of the Earth.

Now, ignoring the further breaking of symmetry by introducing "fixed stars at great distances" to the spacetime, let's have a pair of counter-rotating equatorial ISCO-freefallers use a system of local East-North-Up coordinates (see <https://en.wikipedia.org/wiki/Local_tangent_plane_coordinate...>) where we choose an equatorial plane at random where zero black hole rotation does not pick one out. "Up" is normal to the black hole's surface. North follows the right hand rule convention. The observers can counter-rotate on the same orbit because they are magically transparent to each other ("test objects", which are mathematical tools used explore the geodesics of a spacetime).

Let's have the initial <position, velocity> of our counter-rotating observers differ only by whether they are travelling East or West in these coordinates, and with "Up" pointing directly to a bright distant star. On a high-radius circular equatorial orbit above minimally-rotating black hole, each "year" our two observers will meet and look Up to see the bright star immediately overhead. As we harden the orbit towards ISCO, there is a growing identical mismatch (the geodetic effect) between the locally maintained "Up" and the bright star that was initially "Up". The geodetic effect is not the subject of the paper press-released by northwestern, so I won't dwell on that here.

If the black hole has significant rotation, as we harden the initially-soft orbit (with a long "year") there is a further but opposite "annual" mismatch between "Up" and the direction of the bright star. This is the Lense-Thirring effect. One can think of this in many ways, but it is related to prograde/retrograde geodesic motion. The <position, velocity> and <position, -velocity> starting points can only take one to a stable circular equatorial orbit. The other, with the opposite sign, must accelerate to be on the same circle but in the opposite direction. Or conversely, it will evolve into an orbit that is a combination of less-circular, less-equatorial, and different-height. We stop having the "annual" meetings below the distant star; each time the opposite-direction observers on approximately the same orbit meet, they will notice that the distant star is offset by an increasing angle. Or conversely, if they measure the year by being directly under the distant star, the prograde/eastbound observer will decide that the retrograde observer will arrive later and later every year, and the retrograde/westbound observer will decide that the prograde observer will arrive earlier and earlier every year.

We might consider a simple accretion disc as a "dust" of many observers confined to the equatorial plane and circular orbits in the same direction. We have to make a perturbation like a dust-dust collision to accelerate a mote off its particular orbit and onto another one. This tends to cause an avalanche of interactions, so one then is inclined to treat the disc as a fluid. Additionally, one would draw some analogies between the orbital differences in the previous paragraphs and gyroscopic precessions or processions of electromagnetically-charged objects in a magnetic field. This is the root of "GRMHD", where the last three letters stand for magnetohydrodynamics, which is a technique used in the paper press-released by northwestern.

In this approach one would not want to consider the local coordinates attached to each microscopic fluid element, but would work in some other coordinates suitable for studying the bulk fluid, such as spherical polar coordinates on the black hole itself. In those coordinates, the local "East-North-Up" of individual microscopic fluid elements along a <-ct,r,\theta=towards_our_bright_star,\phi=equator> will have to unalign, e.g., t_BH!=t_particle or \theta is at an angle to local Up/Down. The unaligning is stronger for fluid elements closer to the spinning black hole, and stronger as the black hole's spin increases.

This unalignment is the 'general-relativistic "frame-dragging of the Kerr BH"' that the paper contemplates.

We have to add something else: the fluid elements couple to each other non-gravitationally.

The usual way of explaining this is to consider a figure skater who can extend and retract his arms, changing the velocity of his spin (while keeping angular momentum constant, since the latter is the former times the radius squared). If the figure skater were non-interacting dust, he would simply fly apart under rotation; it is the stickiness of the molecular forces in his body that keeps him intact, with his core preventing his fingertips from flying away by accelerating them back onto the core's preferred trajectory.

In a far orbit around a spinning black hole, a freely falling spinning object that can extend and retract arms will notice nothing special compared to our figure skater: the arms' extremeties will be pulled towards the core by molecular forces and the like. The whole extended object will follow a free-falling trajectory.

However, in a close orbit, the effects above manifest: the bits of the object closer to the black hole have to be accelerated more strongly back onto the freely-falling orbit of the centre of mass-momentum of the whole object compared to the bits further from the black hole. This relatively-higher drag on the closer-to-the-black-hole parts of the spinning generates a torque. If the orbiting object extends arms like a figure skater, but with a hand dipping closer to the black hole each rotation, the torque is increased. This is "paid for" by the whole spacetime, but for simplicity one usually points to the angular momentum of the black hole: the black hole's spin alters the spin of the (much much less massive) spinning orbiting object.

In the accretion disc fluid, there are lots of small cells which are like the spinning object in the previous paragraph, and they bump into each other transferring angular momentum. The result is that inner regions of the disc experience a greater torque. The torques create differential misalignments of the black hole spin vector (in the spherical polar coordinates) and the spin vectors transported by each fluid element (in their local East-North-Up coordinates) that one an treat as an orbital precession. But an element initially travelling Eastbound in the equatorial plane will keep doing so no matter what sort of coordinates one puts on the bulk of all elements + the black hole, unless the element is acted on by some non-gravitational force. And fluid-element-to-fluid-element coupling does exactly that. The result is that the unalignment is forcibly realigned by bulk interactions, with the result that the disc warps in some combination of all four coordinates. The warp in \phi can be dramatic if the bulk is initially moving in a non-equatorial plane: the inner regions will be forced into the equatorial plane, leading to what the authors call a "torn disk".

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