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Black holes are solutions to Einstein's Field Equations. Black holes contain a singularity hidden from the outer universe. As I understand, these singularities are infinitesimal and contain a very large amount of matter, that's what is generally stated.

Do I have a correct understanding of a singularity? If yes, can I stuff infinite mass into a small region called singularity? Isn't there a limit to the mass I can stuff inside the singularity or given region of the space? From this perspective, is singularity really infinitesimal?

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  • $\begingroup$ There is no limit to the amount of mass you can stuff into a black hole and we don't know what the singularity looks like. $\endgroup$
    – zeta-band
    Commented Jun 17, 2019 at 17:25
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    $\begingroup$ Related: physics.stackexchange.com/q/18981/2451 and links therein. $\endgroup$
    – Qmechanic
    Commented Jun 17, 2019 at 17:52
  • $\begingroup$ @safesphere: These are valid only outside the horizon, because the metric inside is not static. No, the solution is valid inside the horizon as well. $\endgroup$
    – user4552
    Commented Jun 18, 2019 at 13:40
  • $\begingroup$ @safesphere I agree. I am completely confused with the singularity and I guess, I am mixing up my day to day experiences with GR. Thank you for pointing the misconceptions out. Perhaps this is because I have understood these concepts from popular literature and haven't technically solved or understood it. It would be interesting to chat with you and have some clarity on some of my doubts if you can spare a few moments on chat room. $\endgroup$
    – orionphy
    Commented Jun 21, 2019 at 16:26

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There are two questions in one here:

  1. Is it possible to concentrate mass more than is achieved by the horizon of a black hole?

  2. Is it possible to concetrate mass more than is achieved by the singularity of a black hole?

The answer to (1) is "no" and the answer to (2) is "it is not known".

To expand:

(1). The Schwarzschild black hole has a spherical horizon whose area is equal to $4 \pi r^2$ where $r = 2 G M/c^2$ and $M$ is the mass of the black hole, as indicated by the amount of gravity it presents in the far-field or asymptotic limit. If further matter enters the hole in such a way that $M$ grows, then $r$ grows too, and it is not possible to concentrate a larger $M$ within a spherical surface of smaller surface area than this. (In this point of view we are associating the whole of $M$ with the region within the horizon, which is arguably correct for a Schwarzschild black hole, but it would not be correct for a black hole with angular momentum or electric charge or both.)

(2). The field equation of general relativity places no restriction on the amount of mass that can go into a black hole, and any mass that passes the horizon proceeds to the singularity in a finite amount of proper time. The singularity of a Schwarzschild black hole is spacelike, not timelike, so it is not quite right to think of it as having a volume. Rather, it is a sort of edge to spacetime.

Two further things are relevant.

First, as the mass gets larger, so does the horizon area, so such a black hole would have a horizon that swells to engulf a larger and larger proportion of the rest of the universe.

Secondly, general relativity employs a classical (i.e. not quantum mechanical) approach to physics, and this is known to fail when the scale of distance and momentum approaches the Heisenberg uncertainty limit. Therefore general relativity almost certainly no longer applies when the Schwarzschild radial coordinate is below some finite amount. In other words, the singularity will not be correctly described by general relativity.

We do not at present have a theory sufficiently well tested for us to have any confident knowledge of what happens at such a location in spacetime.

enter image description here

This diagram shows a spacetime in which a Schwarzschild black hole forms as a result of gravitational collapse of the matter shown in grey. The coordinates (Kruskal) have been chosen in such a way that horizontal lines are spacelike and vertical ones are timelike. I added the diagram in order to illustrate that in some sense the singularity has an infinite spatial length, yet it fits inside the horizon! One can also find a coordinate system in which the whole singularity is at one moment in time. The light cone above the horizon contains the singularity, but I would be unsure how to say what volume it has.

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  • $\begingroup$ I have a quick follow up question, if I may. If the horizon area gets larger with an increase in mass for the black hole, then wouldn't that mean there is a limit to the amount of mass we could add in a certain region of space? Also, I am confused with the statement that if the singularity is spacelike, we should think in terms of the edge of spacetime, does that mean there exist a horizon area without a volume? $\endgroup$
    – orionphy
    Commented Jun 17, 2019 at 17:45
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    $\begingroup$ The horizon has a well-defined surface area, and you can think of its 'contents' as a sort of volume, and then the black hole has the highest mass that could be within such a volume. The region of spacetime beyond the horizon is like the contents of a future light cone. The singularity closes off the 'top' of this light cone. I added a picture to my answer to illustrate this. $\endgroup$ Commented Jun 17, 2019 at 18:51
  • $\begingroup$ @safesphere The original question concerned the concentration of mass near or at the singularity, and asks whether the latter is infinitesimal. Please state explicitly the part of my answer which is wrong, so that I can correct it. $\endgroup$ Commented Jun 18, 2019 at 7:53
  • $\begingroup$ @safesphere ok with the benefit of the comments it is clear that the question was asking about both aspects so I edited my answer to clarify this. $\endgroup$ Commented Jun 18, 2019 at 15:31
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There is no such limit in general relativity, but a black hole is the wrong example. When a gravitational field crushes matter to infinitesimal size, this is what's known in GR as a strong curvature singularity. GR allows strong curvature singularities, but a black hole singularity isn't a strong curvature singularity. The volume of a black hole singularity is not zero, it's undefined.

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  • $\begingroup$ @safesphere: Not sure what you mean here. The volume on a spacelike surface spanning the horizon depends on the surface you choose, so it's not really well-defined. A "moment of proper time" is not a well-defined concept here. $\endgroup$
    – user4552
    Commented Jun 18, 2019 at 20:01
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As a more practical example, you can probably google and find the density of a neutron star. Neutron stars are composed of nuclear matter compressed to a very small size. This may be close to the highest density of a natural occuring object. This source lists the mass as 1.4 times the mass of the sun compressed into an estimate spherical ball of radius 10 km.

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There doesn't seem to be any limit to the amount of mass which can disappear into a singularity. According to the Big Crunch-Big Bounce theory of the end of the universe(now fallen from favour but not completely abandoned), the universe will be halted by gravity and become blue-shifted, eventually collapsing into the kind of singularity from which it emerged and appearing again as a white hole, repeating the cycle ad infinitum Anyone who claims to know what goes on within singularities are deluding themselves, but there must be mass in there for it to give away its presence by its gravitational field

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    $\begingroup$ You seem to be mixing up black hole/white hole solutions with cosmological solutions. They're different. $\endgroup$
    – user4552
    Commented Jun 18, 2019 at 13:42

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