# The definition of volume inside black hole?

• I
What is the definition of volume inside a black hole? we know the grr element of Schwarzschild metric is negative inside event horizon, so how to define a volume inside event horizon? if there is no definition of volume, is there the definition of density?

Ibix
2020 Award
There isn't a unique way to define "space" inside the event horizon. The obvious way to do it is surfaces of constant Schwarzschild ##r##, in which case the volume element is $$\frac 1{\sqrt{|g_{(3)}|}}du\wedge d\theta\wedge d\phi$$where ##g_{(3)}## is the spatial metric and ##u## is the coordinate such that ##g_{uu}## has the same functional form inside the event horizon as ##g_{tt}## does outside. These spaces are infinite and don't contain any mass.

ohwilleke and sysprog
Buzz Bloom
Gold Member
These spaces are infinite and don't contain any mass.
Hi @Ibix:

I hope that you can explain to me what happens to the matter that falls into a black hole, or the matter that creates the back hole. If the matter no longer exists, what is the stuff that comprises the mass of the black hole? Is it photons?

I also hope that the information I am seeking above is not incorrectly worded.

Regards,
Buzz

Ibix
2020 Award
It strikes the singularity and leaves our models: here be dragons. But the singularity isn't a place in space - it's a moment in time that's in your future once you enter the horizon.

DennisN and russ_watters
Ibix
2020 Award
I hope that you can explain to me what happens to the matter that falls into a black hole, or the matter that creates the back hole.
Mixing my models in my last post.

In an eternal black hole, which is what I was talking about, there is no matter and never was. The black hole just is. It seems likely that a better theory of gravity will explain what's going on in the singularity and insist there's some mass there, but for now the comment above applies: the singularity is not a place in space but a place in time. There is no matter anywhere in spacetime.

A semi-realistic model of a star collapsing into a black hole does include the collapsing matter in sensible definitions of "now". However, unless you drop in right behind it (as the horizon forms) you cannot reach it before you hit the singularity. The interior volume remains infinite so the density is not well defined.

Buzz Bloom
Nugatory
Mentor
What is the definition of volume inside a black hole?
When you hear the term "volume inside a black hole" usually the speaker means ##\frac{4}{3}\pi R_S^3##, what we get when we calculate the volume of a hypothetical sphere with radius ##R_S## (where ##R_S## is the Schwarzschild radius). This is the "volume" that people are using when they calculate the "density" of a black hole, for example.

It's not a particularly meaningful concept because the event horizon is nothing like the surface of a sphere and the region inside the event horizon is nothing like the interior of a sphere.

ohwilleke, DennisN and Ibix
mathman
The basic problem with black holes is that no current physical theory can describe the interior. Standard model and general relativity are in conflict here.

weirdoguy
DennisN
Gold Member
2020 Award
I hope that you can explain to me what happens to the matter that falls into a black hole, or the matter that creates the back hole.
We all want to know the answer to that. But nobody knows.

If the matter no longer exists, what is the stuff that comprises the mass of the black hole?
Recently I saw a lecture by Kip Thorne where he said that it can be useful to think of the energy/mass of the black hole to be (in) the spacetime curvature itself (if I remember correctly). However, I don't know if this is a common view or not.

The basic problem with black holes is that no current physical theory can describe the interior.

That is not true, GR has no problem with describing the interior of a black hole, and Standard Model is based on flat Minkowski space of SR so has nothing whatsoever to say about black holes.

Nugatory
Mentor
The basic problem with black holes is that no current physical theory can describe the interior.
It might be better to say that no current theory is expected to describe all of the interior. The event horizon is a coordinate singularity with no local physical significance, so current theories that work in locally flat spacetime remain applicable to the interior. It is only in the neighborhood of the real singularity at ##r=0## that we expect some new and as yet unknown physics to emerge.

When you hear the term "volume inside a black hole" usually the speaker means ##\frac{4}{3}\pi R_S^3##, what we get when we calculate the volume of a hypothetical sphere with radius ##R_S## (where ##R_S## is the Schwarzschild radius). This is the "volume" that people are using when they calculate the "density" of a black hole, for example.

It's not a particularly meaningful concept because the event horizon is nothing like the surface of a sphere and the region inside the event horizon is nothing like the interior of a sphere.
Thank you for respond to my question, I believe the volume formula you mentioned here is only correct for flat spacetime, my question is how to define the volume inside event horizon.

Nugatory
Mentor
my question is how to define the volume inside event horizon.
You can't. The notion is meaningless, which is why people fall back on the Euclidean sphere definition instead.

The problem is that the volume of a region is derived by integrating ##\mathrm{d}V## across the entire region. For a Euclidean sphere that comes out to be the ##\frac{4}{3}\pi r^3## formula; chances are you did this as an an exercise in your first year or so of calculus. Do the same integral across the region inside of a black hole event horizon and you'll find that the integral diverges (if you're being sloppy) or is completely undefined (if you're being rgorous).

You can't. The notion is meaningless, which is why people fall back on the Euclidean sphere definition instead.

The problem is that the volume of a region is derived by integrating ##\mathrm{d}V## across the entire region. For a Euclidean sphere that comes out to be the ##\frac{4}{3}\pi r^3## formula; chances are you did this as an an exercise in your first year or so of calculus. Do the same integral across the region inside of a black hole event horizon and you'll find that the integral diverges (if you're being sloppy) or is completely undefined (if you're being rgorous).
So we can't define the density ether? the density is infinite at center of black hole is incorrect?

Ibix
2020 Award
So we can't define the density ether? the density is infinite at center of black hole is incorrect?
It doesn't have a center. The singularity is not a place in space, but more like a moment in time, which is one way of explaining why you can't avoid it once you are inside the horizon - it is literally the future.

So no, the density is not defined.

It doesn't have a center. The singularity is not a place in space, but more like a moment in time, which is one way of explaining why you can't avoid it once you are inside the horizon - it is literally the future.

So no, the density is not defined.
Sorry, I meant the center is r=0 region of black hole. Some articles often states infinite density at r=0 due to zero volume the mass occupied, so infinite density is incorrect worded?

Ibix
2020 Award
infinite density is incorrect worded
Yes. ##r=0## is not a place and doesn't have a volume, exactly. It's not even technically part of spacetime.

Future theories of gravity may revise that answer but it's the best we can do for now.

jbriggs444
jbriggs444
Homework Helper
Sorry, I meant the center is r=0 region of black hole. Some articles often states infinite density at r=0 due to zero volume the mass occupied, so infinite density is incorrect worded?
An eternal black hole has no mass located anywhere. Just vacuum. In technical language, it is called a "vacuum solution". That is, it is a solution to the Einstein field equation where there is nothing but vacuum everywhere.

When we think of a black hole as having mass, we are (roughly speaking) measuring the curvature of space-time in the neighborhood of the hole (i.e. the gravity from the hole) and comparing it to the mass that would ordinarily be required to have that same gravitational effect.

A one solar mass black hole is one that gravitates like a one solar mass star.

To understand a real singularity, one needs to dig into the mathematical tool that we use to describe space-time. This is the concept of a manifold.

At its simplest level, a manifold is little more than a cartesian coordinate system. Technically this coordinate system is called a "coordinate chart". A coordinate chart is a systematic assignment of coordinates like (x,y) to events within a region of space-time. You can think of it like a paper map.

For a two dimensional manifold, the coordinates will have two numbers. For a four dimensional manifold, they will have four, etc.

To get a manifold, you put together a [finite] set of coordinate charts until you have covered the entire area you are interested in. At this level, manifold is nothing more than a bunch of coordinate charts. Think of it like an old Rand McNally road atlas with a separate map for each of the fifty states (apologies for being U.S. centric here).

There are some additional technical requirements. The coordinate charts have to overlap at the edges. So, for instance, the map of Iowa has to extend a little bit into Illinois. And vice versa. There are also requirements that the mapping between coordinates and points (within any particular chart) is one to one and that the mapping is smooth.

We do not require that the mapping be perfectly faithful. Something like a Mercator projection where distances are stretched and distorted are fine. There is something called a "metric" that can account for any stretching that is done and let us make sure that measurements on the manifold match measurements made in real life. But the stretching has to remain finite.

Importantly, the regions covered are open sets. An open set is like an open interval on the real number line. Like the interval (0,1) which includes all the numbers between zero and one but does not include the end points. More generally, an open set does not include its boundary. An open set cannot be a singleton point. This condition is imposed so that our other mathematical tools work well. We do not have any pesky boundary conditions showing up because our coordinate charts contain no boundaries.

If we were to construct a manifold covering the surface of the Earth, we could use a coordinate chart based on lines of latitude and longitude. That chart could cover the entire Earth... except for the poles and a problem at the international date line. The problem at the poles is a coordinate singularity. It is a place where we would have (for this chart), infinite stretching. We could fix that up in any number of ways. One way is to come up with three additional charts. One covering the north pole, one covering the south pole and one covering the seam at the international date line. Bingo -- we have a manifold covering the entire Earth.

It turns out that when we create a manifold for the space-time region containing a black hole, we can extend coordinate charts to cover almost everything. But we have this pesky singularity. It is not just a coordinate singularity that we can paper over with another coordinate chart. It involves a neighborhood where some important and invariant things go infinite (this violates the smoothness requirement).

When we say that the singularity in a black hole is not part of our space-time, we mean that the manifold that we use to model space-time does not include the singularity. It is not covered by any coordinate chart.

You can find a more complete definition of "manifold" in Wikipedia.

Last edited:
pinball1970, Jando and berkeman
ohwilleke
Gold Member
What is the definition of volume inside a black hole? we know the grr element of Schwarzschild metric is negative inside event horizon, so how to define a volume inside event horizon?
The definition to use depends to a great extent upon why you want to know, in much the way that say, the radius of a proton has multiple definitions used for different purposes, e.g. a charge radius v. a strong force confinement radius.

From the point of view of an external observer, determining the amount of space in a particular region including a black hole that is not in a black hole, the sphere containing a Swarzchild radius to the event horizon volume is the most useful definition, and since the mass of the black hole can be inferred from the perspective of the outside observer, with this definition, the density of the black hole is well defined.

But, if you want to use another definition, it is going to be very helpful to know why you want to know.

For example, is there a theory (e.g. a hypothesis that there is a maximum density that space-time can allow, or analyzing Hawking radiation considering the relative distance of mass from the event horizon of various mass densities) that you want to test?

For some purposes, the question defined in particular ways may have an ill-determined answer, which usually means that you are asking a question that isn't very meaningful.

We all want to know the answer to that. But nobody knows.

Recently I saw a lecture by Kip Thorne where he said that it can be useful to think of the energy/mass of the black hole to be (in) the spacetime curvature itself (if I remember correctly). However, I don't know if this is a common view or not.
I think that would also be the case outside a black hole. The way I see it, mass/energy and spacetime curvature are just two different ways of describing the same 'thing'. Helps with a black hole though, since you have the problem that if mass is inside the event horizon, how can it affect anything outside. Whereas if you consider the mass to actually be the curvature of spacetime (ie the field itself), which spreads throughout the entire universe, the problem goes away (almost).

Last edited:
weirdoguy and PeroK
Hi @Ibix:

I hope that you can explain to me what happens to the matter that falls into a black hole, or the matter that creates the back hole. If the matter no longer exists, what is the stuff that comprises the mass of the black hole? Is it photons?

I also hope that the information I am seeking above is not incorrectly worded.

Regards,
Buzz

The spaces defined by the MATH used there are infinite. But, as for what really goes on in a black hole, that math tries to accurately describe it yet breaks down and cannot explain what occurs at the center.

Which means you can say that math is like maybe an approximation of another bigger more complex reality.

I'd suggest watching PBS space time on black holes. It's amazing. The fact that this kind of speculation can happen is tied to the fact that we don't know much about the inside of black holes and must speculate.