My thought process here is the following: The inside of an (eternal) black hole carries four (Schwarzschild) coordinates t, r, theta, phi – r now being timelike and confined to the interval (0, 2M) and t now being spacelike and being any real number. That is, depending on when (at what time t) you cross the event horizon, you end up at a different point in space. The singularity at r=0 is then a point in your future which, like your own death, you cannot actually see but which you will nevertheless hit in finite proper time.
So in this sense I'd say the volume is very finite (if we disregard the (trivially unbounded) spacelike coordinate t which, as mentioned before, simply corresponds to the time of entering the BH).
1. In the interior of the BH, the determinant of the metric is bounded since the Schwarzschild factors in the metric cancel out. So the volume measure doesn't do anything crazy as one gets closer to the singularity and boundedness of coordinates implies boundedness of the volume. Again, I'm disregarding the spacelike t coordinate because to me the relevant fact is that all matter reaches the singularity in finite proper time, so while we could theoretically stack lots of (actually, an infinite amount of) (massless) cubes inside a black hole, they would soon all get crushed.
2. Of course the situation is slightly different if we're talking about a growing black hole whose mass (and, therefore, radius) increases as we throw matter into it.
A thought experiment: imagine we have a clock, falling into Schwarzschild black hole. Obviously, any real clock would have some non-zero size in all space dimensions. Here we will be concerned with just two: r and any orthogonal one. So for simplicity let the clock be a simple rubber-like oscillating ring with a fixed k and infinite resistance to tearing. (You could also take infinite k, but I'd argue that would not be physically meaningful in this setup)
As the ring is closing to the r = 0, its oscillations will slow down and come to a halt due to physical stretching along the r dimension. What I am trying to say is that maybe these oscillations make more sense as the measure of time for the ring people, than what a numerical value of proper time tells us. In a similar way the time singularity at the horizon is nothing special for a freely falling observer.
I am unsure how to interpret the fact, that the number of oscillations per proper time unit is going down though. Seems to be quite the opposite of my original note about the volume, yet something is ringing.
×) Counterexample: Consider the manifold M := R³\B, where B is the closed unit ball, equipped with the standard Euclidean metric. This manifold is certainly not Cauchy-complete and we can reach the singularity at r=1 in finite time. Now, if we had to define the dimension of the singularity, what dimension n should it have? n=2 (a sphere)? Maybe. At least we could extend M by the unit sphere to make it complete. But could the singularity also be a point (i.e. n=1)? Yes, certainly. By diffeomorphism invariance, we could simply find new coordinates and map R³\B to R³\{0}, so the singularity would suddenly become a point. So, as you can see, interpreting the singularity as a point or set of points that have a topological dimension doesn't work.
https://physics.stackexchange.com/a/194947
(I'm not a physicist.)
>A black hole itself (the singularity) is 1 dimensional - a single infinitesimal point.
In Euclidean geometry,
A cube is 3 dimensions.
A plane is 2 dimensions.
A line is 1 dimension.
A point is ...
> In this diagram the singularity is a line in spacetime i.e. a one dimensional object in spacetime.
is wrong or at least very misleading – the answer does (correctly) say that asking for the "dimensionality of a singularity […] is a meaningless question because the spacetime geometry is undefined at a singularity".
A point is 0-dimensional. A line is 1-dimensional.
The singularity is 1-dimensional (more precisely, a ring) if the black hole is rotating. [1]
2. The event horizon is spherical. Mathematically we treat it as a point. Nobody knows.
3. It's a gravity well, so everything falls towards it like it does towards the earth. Since it's the center of a galaxy, and galaxies form a 2D spin plane, most matter will be circling it, like our solar system circling the sun, so most matter will come from that planar distribution.
We will never see anything fall into a black hole, as in our space-time, this would happen at infinite time. As things get closer, they red shift from our frame of reference due to time dilation, until the light frequencies are so low that we can no longer detect them
I am not a physicist, but from my lay understanding, a black hole (and all other forms of matter) exists in a four-dimensional space-time form that we can experience directly, plus a number of higher dimensions that we can only extrapolate, and observe indirectly via experiment (kind of how a picture of a sphere is not a sphere due to a missing dimension, but you can still tell it's a sphere based on other characteristics). The exact number of higher dimensions is the realm of string theory, and could be 10, 11, 26, or something else.
The singularity of the black hole is one-dimensional, I think (I could be wrong, I'm often wrong). The event horizon is... funny. The outside of the event horizon is in four-dimensional space-time and is more or less "spherical". The inside is a land of theory and debate, because by definition we can't directly observe it from our space-time. Do space and time even exist inside the event horizon? I dunno, ask someone who knows what they're talking about. The best I can do is an analogy. Imagine it's the surface of the ocean. Above the surface, you're in air. Below the surface, you're in water.
I ask because even in a brief history of time, the diagrams are very "single plane of space-time, pulled infinitely deep by the black hole"