orbit - Orbiting around a black hole

For the purposes of comparison, here's flat, Minkowski spacetime in spherical coordinates:
$$mathrm{d}s^2 = -mathrm{d}t^2 + underbrace{mathrm{d}r^2 + r^2(mathrm{d}theta^2+sin^2theta,mathrm{d}phi^2)}_text{Euclidean 3-space}text{.}$$

The soure of misunderstanding was that I wasn't clear enough, how black holes work. I always imagined them as "suckholes" like whirlpools in the water.

That is not entirely incorrect. The Schwarzschild spacetime of an uncharged, nonrotating black hole in the Gullstrand-Painlevé coordinates is
$$mathrm{d}s^2 = -mathrm{d}t^2 + underbrace{left(mathrm{d}r + sqrt{frac{2M}{r}},mathrm{d}tright)^2}_text{suckhole} + r^2(mathrm{d}theta^2+sin^2theta,mathrm{d}phi^2)text{.}$$
Where it deviates from ordinary, flat Minkowski spacetime is entirely in the middle square term. Here, the time coordinate $t$ is not the Schwarzschild time, but rather the time measured by an observer free-falling from rest at infinity. The last bit, if adjoined with the $mathrm{d}r^2$ term one would get by multiplying out the middle part, is ordinary Euclidean $3$-space written in spherical coordinates.

If you recognize from Newtonian gravity the quantity $sqrt{2M/r}$, or $sqrt{2GM/r}$ in ordinary units, as the escape velocity, then the picture is very peculiar indeed: according to an observer free-falling from rest at infinity, Euclidean space is sucked into the singularity at the local escape velocity. The event horizon is the surface at which the speed at which space is "falling" at the speed of light.

This is an additional reason why sonic black holes are good analogues to their gravitational counterparts. In a sonic black hole, there can be an actual "suckhole" that drains a low-viscosity fluid at an increasing velocity, up to and faster than the speed of sound in that fluid. This forms an acoustic event horizon that is one-way to sound and is expected to have an analogue of Hawking radiation.

The corresponding structure for charged black holes is similar, and for a rotating one more complicated, although can still be described as "sucking" with a certain additional twist that rotates the free-falling observers.