A waterfall doesn't ask if you're ready to jump. Its flow accelerates, and at some point even light can't escape back. A McVittie black hole is such a waterfall, but not in an ordinary river; it's in an expanding universe: the water here is space itself, which continuously stretches, carrying galaxies apart from each other.

For a long time, it remained unclear whether one could even fall into this hole. Particles of the cosmic fluid, that same "water," don't fall into it; they fly off to infinity, as if the current carries them away. But a freely falling observer, like a stone thrown into a waterfall, does reach the horizon in finite proper time. This is proven in a new work: the trajectories of timelike geodesics, the paths of freely falling bodies, lead to the event horizon, the boundary beyond which there's no return.

But what awaits the daredevil at the very edge? The answer depends on the cosmological constant, the energy of empty space. If it's positive, all scalar curvature invariants, including the Jacobi fields (a measure of the stretching of nearby trajectories) and the curvature components in a parallel-propagated frame, remain finite at the horizon. A body crosses it without catastrophe. However, if the cosmological constant is zero, a crack appears in this picture: a second-order scalar diverges as the gamma factor, and some curvature components, such as R0i0i, blow up. This means that an extended body, despite the finiteness of the Jacobi fields, could be torn apart by tidal forces, like a swimmer caught in a whirlpool that doesn't twist them immediately, but on the second turn.

Previously, it was thought that if the Jacobi fields are finite, the observer survives. The new work shows that one must look at higher derivatives of curvature. It's like checking not only the force of a wave's impact, but also its jerk. For McVittie black holes with Λ=0, the jerk turns out to be fatal. The study fills an important gap: for the first time, it proves that massive particles can fall into such a hole, and it reveals a possible weak singularity at the horizon. There are no practical applications for this knowledge yet; it's pure physics, but it clarifies how black holes behave in an expanding universe.

The waterfall continues to flow, and now we know: the jump is possible, but not always safe. The next step is to study how extended bodies behave with multipole moments, and how test fields interact with the horizon. Perhaps then we'll understand where exactly the line between survival and disintegration lies in this cosmic flow.