How do we account for the 'one way' drag of moving space?

Question

As I understand it, the rotating space outside a Kerr black hole drags radially falling particles into circular motion. Similarly the river model posits that the inward flow of space ensures particles and light cannot escape from inside the event horizon of a black hole.

The question is, why is that when space moves past a stationary particle, there is drag, but when a particle moves relative to stationary flat space, there is no drag? An inertially moving object will continue forever at it current velocity without slowing down. How are the situations different, when the relative velocities of space versus particles are the same in both cases? Is there any formal explanation for this asymmetry?

Answer 1

In the body of the paper The river model of black holes, the authors do not use the term "drag" anywhere to describe the model. They describe it as a flow, but not a flow of a viscous material that would produce drag. The mathematics of drag do not correspond well to either the river model or to more standard GR in more general spacetimes.

One thing to keep in mind is that the river model is not generalizable. To my knowledge, it applies only for the Kerr spacetime and the Schwarzschild spacetime. The model used in general relativity is curvature, not flow. It only happens that in these specific spacetimes the curvature can be mapped to a flow concept. Even then, the flow concept for the Kerr spacetime is odd and is not really akin to a flowing liquid. So I think it is best to focus on curvature, rather than flow.

The formal explanation for the asymmetry that you mention is the curvature. The "flowing" spacetimes that you mention have curvature. As such, they have the geometric features that you mention, specifically that in the Kerr spacetime initially radial falling geodesics will curve into a tangential path, and any radial timelike path inside the horizon goes inward. The absence of curvature in flat spacetime explains the fact that an inertial object continues at a constant velocity.

The asymmetry is explained by the curvature and the curvature, in turn, is explained by the presence of a large amount of gravitating matter. Symmetries in the laws of physics, in general, need not be preserved in cases where the boundary conditions or sources violate that symmetry.

Answer 2

In FLRW spacetime, there is a drag-like contribution to the equation of motion of a test particle, $$\frac{\mathrm{d}\vec p}{\mathrm{d}t}=-H\vec p.$$ Here $\vec p$ is the momentum with respect to the local comoving observer and $H$ is the Hubble rate.

However, this is not a relativistic effect. It arises also in a Newtonian description and is most clearly understood as a consequence of self-sorting, as I described in another answer. It is due to the relative velocities of the comoving observers themselves. This effect can even arise in Minkowski space if you populate it with the appropriate comoving observers (giving rise to the Milne cosmological model, i.e. curvature-dominated FLRW).