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Grant Hatch
2017-Feb-03, 11:00 PM
I .just read a confusing to me article by Corey Powell at Discover which claims that the radius of black holes increases in direct proportion to its mass. Double the mass and the diameter of the event horizon doubles as well. This leads to the density of the black hole dropping off rapidly as its diameter increases.


"The direct relationship between size and mass has a funny effect. The more massive a black hole is, the less dense it is–and the dropoff happens rapidly, as the square of the radius. (Again, I’m using the event horizon to define the surface of the black hole.) A solar-mass black hole crams the sun’s entire 865,000-mile-wide bulk into that 4-mile wide sphere, corresponding to a density 18 quadrillion times the density of water. It’s a staggering number. The black hole at the center of the Milky Way has a mass of 3.6 million suns, which means its density is (3.6 million x 3.6 million) times lower. That translates to about 1,400 times the density of water–still very high, and more than 100 times the density of lead, but no longer so incomprehensible.

Other black holes are much more massive than the central one in our galaxy, though, which means they are also much puffier. The galaxy M87 contains a monster black hole that astronomers have measured as having the mass of 6.6 billion suns. Its density is about 1/3,000th the density of water. That is similar to the density of the air you are breathing right now!

Now to the most mind-blowing part. If you keep going to higher masses, the radius of the black hole keeps growing and the density keeps shrinking. Let’s examine the most extreme case: What is the radius of a black hole with the mass of the entire visible universe? Turns out that its radius is…the same as the radius of the visible universe. Almost as if the entire universe is just one huge black hole." http://blogs.discovermagazine.com/outthere/2013/08/20/the-baffling-simplicity-of-black-holes/#.WJT_UfkrLIU

Really? How can something the density of air even have an event horizon?

Does this mean that mathematically (at least in this limited sense) there is no difference between our Universe and a black hole of our universe's mass?

WaxRubiks
2017-Feb-03, 11:09 PM
Does the mass of the universe, there, include dark matter?

grant hutchison
2017-Feb-03, 11:13 PM
Really? How can something the density of air even have an event horizon?Well, why not? Essentially all the space inside an event horizon is empty space, no matter what size of black hole you have.

Grant Hutchison

Grant Hatch
2017-Feb-03, 11:25 PM
Does the mass of the universe, there, include dark matter?

Good question. I was wondering the same thing.

I'm also wondering how they even came up with the diameter and mass thing.... direct observation? How can anyone get a good enough look at a black hole to conclude its diameter doubles with a doubling of its mass? So is it derived mathematically then?

Grant Hatch
2017-Feb-03, 11:34 PM
Well, why not? Essentially all the space inside an event horizon is empty space, no matter what size of black hole you have.

Grant Hutchison

How can an object with an average density within the event horizon = to or < than "air" create a gravity well/event horizon such that light cannot escape from it? Really? Something isn't tracking here....

grant hutchison
2017-Feb-03, 11:38 PM
So is it derived mathematically then?Yes, it's a simple mathematical relationship. Schwarzschild radius increases in proportion to mass. Density varies with mass/(radius cubed). So density inside the event horizon of a black hole varies with mass/(mass cubed), which is 1/(mass squared). Higher the mass, lower the density, and once you get supermassive the density is less than air.

The radius of a black hole event horizon is a problematic quantity, though, so it's not clear to me how useful the concept of a specific "density" for black holes really is.

The story about the mass and radius of the observable Universe matching those of a black hole has been around for a long time - I'd be surprised if it had been a continuously valid observation for the last five decades, given how our knowledge has changed in that time.

Grant Hutchison

Grant Hatch
2017-Feb-03, 11:57 PM
Yes, it's a simple mathematical relationship. Schwarzschild radius increases in proportion to mass. Density varies with mass/(radius cubed). So density inside the event horizon of a black hole varies with mass/(mass cubed), which is 1/(mass squared). Higher the mass, lower the density, and once you get supermassive the density is less than air.

The radius of a black hole event horizon is a problematic quantity, though, so it's not clear to me how useful the concept of a specific "density" for black holes really is.

The story about the mass and radius of the observable Universe matching those of a black hole has been around for a long time - I'd be surprised if it had been a continuously valid observation for the last five decades, given how our knowledge has changed in that time.

Grant Hutchison

I don't understand. How is the radius of the EH a problematical quantity if it is derived mathematically? Is the math problematical?

grant hutchison
2017-Feb-04, 12:51 AM
I don't understand. How is the radius of the EH a problematical quantity if it is derived mathematically? Is the math problematical?The maths is not problematic. You simply need to be very clear about what the maths does and doesn't mean. The Schwarzschild radius is derived from circumference/(2*pi), which is a true relationship in flat space, but not in curved spacetime. That doesn't matter when it is simply being used as a standard length in the Schwarzschild metric (which is a particular coordinate system used for non-rotating black holes). There's only a problem if you decide you want to interpret the Schwarzschild radius as a real distance between the event horizon and the singularity - in Schwarzschild coordinates the radius is timelike, so it doesn't resemble a conventional distance at all.

Grant Hutchison

Grant Hatch
2017-Feb-04, 02:28 AM
Does that mean that the "density" within the Schwarszchild EH is a contrived/non real value trying to describe the actual spherical curved spacetime properties of a non spinning BH? Why is it a two dimensional area rather than a three dimensional volume description/solution of a spherical black holes EH?

grant hutchison
2017-Feb-04, 01:29 PM
Does that mean that the "density" within the Schwarszchild EH is a contrived/non real value trying to describe the actual spherical curved spacetime properties of a non spinning BH? Why is it a two dimensional area rather than a three dimensional volume description/solution of a spherical black holes EH?I don't understand the question.

The volume used in calculating the "density" of a black hole rests on the assumption that spacetime is flat in the vicinity of a black hole (it isn't). It also assumes that there's something interesting about the mean density of an object that consists of a superdense core and a non-physical boundary. And it is undermined by the fact that different observers, using different coordinate systems, will measure the size and shape of an event horizon differently.

It just doesn't seem like a useful quantity, apart from for the purposes of causing excitement and puzzlement in pop sci writing. Black hole physicists are much more interested in the surface area of the event horizon, rather than its volume.

Grant Hutchison

Grant Hatch
2017-Feb-04, 06:25 PM
Sorry, I'll try again....Then why is the mentioned "schwar..." solution two dimensional (flat) rather than three dimensional? Is the two dimensional solution still correct for surface area even though it's actually a three dimensional phenomenon? If so, would not a solution for surface area then still result in a given volume and density? Is it not possible to have a solution for volume and density that takes into account everything from no spin to very rapid spin?

grant hutchison
2017-Feb-04, 07:03 PM
I've maybe confused you by saying "flat space", which just means a condition in which there is no spacial curvature, not that the solution is two dimensional. The spacetime around a black hole is intensely curved, which means that the usual relationship between the radius and circumference of a circle does not pertain.
The Schwarzschild metric is a four-dimensional picture of curved spacetime. The Schwarzschild radius of the black hole is the radius it would have if it were in "flat" (that is, not curved) spacetime. A volume calculated from that radius is therefore the volume of a sphere in "flat" spacetime, so it's intrinsically misleading about the volume of the event horizon of a real black hole.

Grant Hutchison

grant hutchison
2017-Feb-04, 07:12 PM
There's a paper here (https://arxiv.org/pdf/0801.1734v1.pdf) (440KB pdf) about the difficulty of calculating the "real" volume enclosed by a black hole event horizon, and how it varies according to the choice of coordinates (including being zero in some coordinates).

Grant Hutchison

WaxRubiks
2017-Feb-04, 10:03 PM
(including being zero in some coordinates).<br>
<br>
Grant Hutchison<br><br><br>
So in this coordinate system there effectivly is no EH, and the matter just falls towards a single coordinate? A small grain of dense matter?

grant hutchison
2017-Feb-04, 10:41 PM
So in this coordinate system there effectivly is no EH, and the matter just falls towards a single coordinate? A small grain of dense matter?No, it's the Schwarzschild coordinate system, in which an object falls towards the event horizon asymptotically and doesn't cross it. According to the paper's authors, there is zero volume inside the event horizon in those coordinates - but it still has radius equal to the Schwarzschild radius, and no object ever enters the space inside.
See why I don't think deriving a value for the "flat space" density using the Schwarzschild radius is particularly helpful?

Grant Hutchison

Hornblower
2017-Feb-04, 10:58 PM
We can compare the OP's question with asking how a large, low-density planet can have the same escape velocity from its surface as a small, high-density one. For low-energy situations a Newtonian calculation is a good approximation. Start with the equations for potential and kinetic energy and invoke conservation of energy.

U = -GMm/r
T = 0.5 mv^2
where G is the gravitational constant, M is the mass of the planet, m is the mass of a small object trying to escape, r is the radius of the planet.

If v is the escape velocity, then U + T = 0. Solving for v yields

v = sqrt(2GM/r)

Any planets for which M is proportional to r will have the same escape velocity. If we make the big one big enough, we can lower the density to that of air, or perhaps even a pretty good laboratory vacuum, and it will have the same escape velocity as the small one. In our thought exercise we can do this by making it hollow with a thin but strong shell, so it doesn't collapse under its own gravity. If we set v = c, we have something on the order of magnitude of a black hole event horizon. Perhaps not exactly the same radius, having used Newtonian rather than GR, but still a nice analogy.