system
Given the following post, respond as if you're commenting on Reddit:
human
Are all black holes the same density?
gpt
***This question actually has no answer,*** despite it often being asked and getting an answer that seems to make sense to most people. In short, we want to define density as ρ = M/V, where V is the volume where the mass is located. The mass *M* of the black hole is the same number for all observers. However, the volume *V* is observer-dependent, and can even be undefined in some coordinate charts. Hence there is no way to unambiguously define the density of a black hole. 

Nevertheless, if you google the question "what is the density of a black hole?" or search for it on this sub or simply read the other responses in this thread, you will see that you often get an answer anyway. In fact, you usually get one of two answers. This is unfortunate because the general public gets misled into thinking this question or similar questions are even well-defined. Some questions are just meaningless, no matter how hard you try to answer them. In this particular case, the question "what is the density of a black hole?" is often answered in an improper framework, one which typically attempts to apply Euclidean geometrical concepts to curved spacetimes.

Okay... so what answers do you usually get for this question? These are by far the two most common:

* **Infinite density.** The singularity has zero volume, so if the mass is non-zero, then the density is M/0 = infinity.

* **Density scales like 1/M^(2).** The radius of the black hole is R = 2M (with c = G = 1), and so the density scales like M/V ~ M/R^(3) ~ 1/M^(2).

Both of these explanations are very wrong. Why? For the first, the singularity is not part of the spacetime manifold, so it doesn't make sense to say that the mass of the black hole is located at the singularity. It's like saying "the mass of the black hole is not part of the universe". As for the second explanation, well that is usually taken to be the "correct" answer, but the reason it is wrong is a bit subtle and something that is not really well explained in popular science. So let's step back a bit and explain a few things.

A Schwarzschild black hole is an eternal, static black hole. It is usually described in so-called Schwarzschild coordinates (*t*, *r*, *φ*, *θ*). Although these coordinates strongly suggest that (*r*, *φ*, *θ*) is a Euclidean spherical coordinate system, that is absolutely wrong. The coordinate *r* is emphatically *not* the distance from the singularity. The Schwarzschild coordinates are characterized by the fact that they describe a spacetime which, for each constant time *t*, is spherically symmetric. If we imagine a spherical shell centered on the preferred central point, then that spherical shell has a surface area *A*. We then *define* the coordinate *r* to be that number such that *A* = 4π*r*^(2). In other words, we define the coordinate *r* via a Euclidean formula for surface area but this does *not* imply that the infinitesimal *dr* is the distance between nearby spheres. We sometimes call *r* an "areal coordinate" to emphasize this fact. The angular variables *φ* and *θ* are defined to be the usual azimuthal and polar angles on a sphere.

It then turns out that the metric for the Schwarzschild spacetime is

> ds^(2) = -Λdt^(2)+Λ^(-1)dr^(2)+r^(2)dΩ^(2)

where Λ = (1-r*_s_*/r) and r*_s_* is the so-called Scharzschild radius of the black hole. The term "r^(2)dΩ^(2)" is just the usual Euclidean metric on a sphere. Note the factor of Λ! If this were a perfectly flat universe, the metric would be

> ds^(2) = -dt^(2)+dr^(2)+r^(2)dΩ^(2)

The Schwarzschild radial coordinate is *distorted* in the Schwarzschild metric. This has two immediate implications:

* If a particle is at Schwarzschild radial coordinate *r*, then the particle is *not* a distance *r* from the singularity. In fact, we cannot actually define that distance in this coordinate chart because there is a coordinate singularity at the event horizon. The best we can say is that the distance to the event horizon is infinite. However, we *can* talk about the distance between two radially separated particles outside the event horizon, one at inner radius *r* = *a* and the other at outer radius *r* = *b*. The Euclidean distance between the two particles is *b*-*a*, but the true distance in the Schwarzschild metric is actually larger. ***This means that distances are coordinate-dependent!***

* Consider a sphere with radial coordinate *r* about the singularity. We *cannot* define the volume enclosed by this sphere in this coordinate chart. (Remember though that the surface area is just *A* = 4π*r*^(2) because that's how *r* is defined.) It turns out that if we write the spacetime with a different coordinate chart, then these volumes *can* be defined. For instance, if we use Gullstrand-Painleve coordinates which are just the coordinates of a free-falling observer, then the volume enclosed by a sphere of radius *r* is the usual Euclidean volume *V* = (4π/3)*r*^(3). (For reference, the Schwarzschild coordinates are the coordinates of a stationary observer at infinity.) ***This means that volumes are coordinate-dependent!***

Hence there is no way to answer the question "what is the density of a black hole?" There is no way to unambiguously define the enclosed volume, *a fortiori* there is no way to unambiguously define the density.

Surface area, on the other hand, is coordinate-invariant as long as the surface is a *null surface*. The event horizon of a black hole is a null surface, and so all observers will agree on its surface area. The Schwarzschild observer measures the event horizon to have radial coordinate *r**_s_* = 2M, and hence surface area *A* = 4π(2M)^(2) = 16πM^(2) (or 16πG^(2)M^(2)/c^(4) in dimensional units). So all observers measure that same surface area.