Some depictions of black holes show space being warped into a singularity, with no end, e.g. as pictured below. Moreover, in Cosmos, Neil Tyson speculates with the possibility that Black Holes contain other "universes". I am not sure about what to make out of this, but it made me wonder:
- Can space be stretched with no limits by a blackhole?
- If there is a limit, is there a way to quantify how much space the black hole "stretches"?
- Is the amount of "space stretching" (e.g. "extra space") determined in any way by the mass or volume of the blackhole?
Best Answer
It's a difficult question to answer because in relativity, distance and volume are coordinate-dependent quantities. Even without a black hole, special relativity allows us to distort distances (Lorentz contraction) to an arbitrary degree by moving at speeds close to lightspeed.
That being said, if we limit ourselves to Schwartzschild coordinates (which basically represent how an observer hovering at a distance from the black hole sees things), there are some interesting facts to examine.
First, the coordinate distance to the event horizon is indeed greater than would be expected in flat space, meaning if you hovered over the black hole and let down a tape measure, you'd need a longer-than-expected tape measure, by a factor of over 2, to reach the event horizon. This means the volume of the region around the hole (outside the event horizon) would be greater than that contained in a region of flat space with the same circumference.
Diagrams like the ones you posted are basically slices of the spacetime at a constant Schwartzschild time coordinate, so the relationship of distances and volumes they depict is qualitatively accurate. For objects like neutron stars that are close to being a black hole but not quite, there is also significant additional distance and volume, but less than for black holes.
On the other hand, if you freely fall into a black hole then you hit the event horizon and then the singularity in a finite time from your point of view. There are other coordinate systems, namely Lemaître and Gullstrand–Painlevé, that describe the spacetime "from the perspective of falling observers" in some sense. If you measure the volume of a constant-coordinate-time slice in those coordinates, you'll get something time-dependent (for Lemaître) or equivalent to flat space (for G-P). This just goes to show that the "volume of space" in a curved spacetime region really isn't a well-defined quantity.