You are thinking classical physics + General Relativity.
In the centre of a black hole is a gravitational singularity, a one-dimensional point which contains infinite mass in an infinitely small space, where gravity become infinite and space-time curves infinitely, and where the laws of physics as we know them cease to operate. As the eminent American physicist Kip Thorne describes it, it is "the point where all laws of physics break down".
In this sense, one can define the singularity as a trajectory in the space time reference frame of the earth, it is a moving point, but "crack" attributes characteristics to space time that it does not have. There are continuous distortions of space time due to the existence of the singularity and these have to be explored in the framework of General Relativity. It means that as t changes the functional forms in space of the distortions of the singularity change . There is no permanent space medium that would keep a record of the "passing" of the singularity. In the absence of mass/energy, space time is flat.
Current research is attempting to quantize gravity, which means that singularities become blurred, due to the uncertainties introduced by the probabilistic nature. Have a look at the current Big Bang model to get an idea of what happens to singularities when quantization of gravity is assumed.