Silly question about Schwarzschild Black holes and scientific communication Well, in scientific communication literature often we encounter the following phrase about black holes:

A Black Hole is a body that have infinite curvature and infinite density

So, with basic GR we can formalize the "infinite curvature" idea with the Kretschmann scalar of Schwarzschild spacetime:
$$R_{\mu \nu \gamma \delta}R^{\mu \nu \gamma \delta} = \frac{48G^2M^2}{r^{6}} $$
Because with a simple analysis we can see that the whole function tends to infinity when $r \to 0$.
But how can I explain formally the idea of " a Black Hole is a body with infinite density"?
 A: A black hole is a stranger object than you might think. In particular the Schwarzschild black hole is a vacuum solution and contains no mass at all. Not at the singularity nor anywhere else. The mass $M$ that we use in equations such as the one for the Kretschmann scalar is actually a geometrical property called the ADM mass. So in this respect the density of the black hole is zero everywhere.
However the Schwarzschild geometry is an idealised object that cannot exist in reality, not least because it requires an infinite time to form. In a real black hole we can have matter falling into the black hole and any matter falling in must reach the singularity in a finite (generally very short!) time. That means any matter falling into the black hole quickly ends up compressed into a point of zero volume, giving an infinite density at that point.
The problem with this claim is that we cannot say the matter reaches the singularity because we cannot calculate what happens at the singularity. It would be more precise to say the radial distance $r$ tends to zero in a finite time, but we cannot calculate what happens at $r=0$ because the geometry is singular there and our equations cannot be applied. All we can say is that the density tends to infinity as the matter approaches the singularity.
