The Center of a Black Hole - What's Going On?

In summary: I really don't like talking about the singularity of a black hole, since I think most people thing that it is an ugly object whose singular nature will be removed by an eventual theory of quantum gravity. So just know that these discussions are limited to the classical theory of general relativity...
  • #1
cragar
2,552
3
At the very center of a black hole would the gravitational field strength be zero. I'm just thinking of Gauss's law. But I Know that the laws of physics break down at the singularity, but it seems like it would or am I crazy?
 
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  • #2
If you want to try and make some sort of argument like this, you must first acknowledge that all the mass is concentrated within a point at the center of the hole. Therefore, no matter how small you make a sphere, the enclosed mass is the same.

But Gauss' law is a Newtonian physics phenomenon, or, only an approximation in the weak-field limit of GR, so it is clearly not appropriate to invoke in the interior of an object such as a black hole!
 
  • #3
Nabeshin said:
If you want to try and make some sort of argument like this, you must first acknowledge that all the mass is concentrated within a point at the center of the hole. Therefore, no matter how small you make a sphere, the enclosed mass is the same.

I didn't realize this. Rather interesting.

I was under the impression black holes had a radius. Sort of like shrinking the Earth down to only a few km or so in size (only with a lot more matter).
 
  • #4
jarednjames said:
I didn't realize this. Rather interesting.

I was under the impression black holes had a radius. Sort of like shrinking the Earth down to only a few km or so in size (only with a lot more matter).

The event horizon has a radius i believe, which grows larger the more massive the black hole is, but the matter itself does not.
 
  • #5
So is a black hole a ball of compressed matter with a radius?
 
  • #6
jarednjames said:
So is a black hole a ball of compressed matter with a radius?

I THINK that as the mass of the black hole increases, the event horizon, which is the point where the gravitational influence is too large for light to escape, also increases in radius. Unless I'm wrong, I don't believe that the event horizon is a real thing (IE Physical thing with substance), but merely the point that the gravity becomes too strong for light. Even though the matter would all be in one spot in the center with no real size, the effects of gravity and the radius of the black hole would increase as the mass increased.

How's that sound?
 
  • #7
I understand the event horizon, I'm referring to the matter itself.

So it's a spot in the middle without real size. Makes sense I suppose... :uhh:
 
  • #8
Drakkith said:
I THINK that as the mass of the black hole increases, the event horizon, which is the point where the gravitational influence is too large for light to escape, also increases in radius. Unless I'm wrong, I don't believe that the event horizon is a real thing (IE Physical thing with substance), but merely the point that the gravity becomes too strong for light. Even though the matter would all be in one spot in the center with no real size, the effects of gravity and the radius of the black hole would increase as the mass increased.

You are correct.
 
  • #9
jarednjames said:
I understand the event horizon, I'm referring to the matter itself.

So it's a spot in the middle without real size. Makes sense I suppose... :uhh:

Well, to stretch your brain a bit further, does any particle actually have a "physical size"? If the gravity overcomes the forces holding particles apart, would their wave packets or whatever just start to overlap?
 
  • #10
jarednjames said:
I understand the event horizon, I'm referring to the matter itself.

So it's a spot in the middle without real size. Makes sense I suppose... :uhh:

I really don't like talking about the singularity of a black hole, since I think most people thing that it is an ugly object whose singular nature will be removed by an eventual theory of quantum gravity. So just know that these discussions are limited to the classical theory of general relativity...

In that case, a black hole is precisely a dirac-delta mass distribution. All the mass is concentrated precisely at a point. And, analogously to how one can solve Laplace's equation for a dirac-delta distribution around a point charge, you can do something similar with Einstein's equations. Out pops what we call a black hole. The event horizon is just, if you will, the surface around the hole where the gravitational force has a certain value (a value so large not even light can escape). You could set up something analogous for an electrical charge (although, I'm not sure you could come up with something with the same significance!). So, when you increase the mass of the delta-function particle, you increase the radius this surface is located at.

When people say "size of a black hole" usually they refer to the event horizon, since that's all we can ever observe anyways. But strictly speaking, a black hole is the mathematical solution to Einstein's Equations corresponding to a delta-function mass distribution.
 
  • #11
Ok, guys, enough philosophy. Schwarzschild metric can be used bellow event horizon, so long as we are careful with resulting imaginary coordinates. (Complex coordinates reflect the fact that relative to external frame the object can only travel faster than light. It's possible to write a metric that does not diverge at event horizon, allowing us to avoid imaginary coordinates, but for question of gravitational acceleration at center it's not necessary.)

So all you have to do is take a look at Christoffel Symbol at r=0 in Schwarzschild metric. The metric itself is singular, so our best bet is to look for limit r->0.

For an object at rest, some distance r away from singularity, the general equation for proper 4-velocity (c=1 units):

[tex]u = \left( \frac{1}{\sqrt{1-r_s/r}}, 0, 0, 0\right)[/tex]

And acceleration.

[tex]a^{\alpha} = \Gamma^{\alpha}_{\beta \gamma}u^{\beta}u^{\gamma}[/tex]

Or using the form of u from above.

[tex]a^{\alpha} = \Gamma^{\alpha}_{tt}\frac{1}{1 - r_s/r}[/tex]

The only non zero element is along r.

[tex]\Gamma^r_{tt} = -\frac{1}{2}g^{rr}\frac{\partial g_{tt}}{\partial x^r} = -\frac{r_s}{2r^2}\left(1-\frac{r_s}{r}\right)[/tex]

The 4-acceleration comes out to be very simple.

[tex]a = \left(0, -\frac{r_s}{2r^2}, 0, 0\right)[/tex]

Of course, the quantity we are interested in is |a|.

[tex]|a|^2 = a^{\alpha}a^{\beta}g_{\alpha \beta} = \left(\frac{r_s}{2r^2}\right)^2\frac{1}{1 - r_s/r}[/tex]

This diverges as rs/4r³ as r->0. So no, gravity at the center of a black hole is not zero. It diverges.
 
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  • #12
thanks for all of the replies .
 
  • #13
jarednjames said:
I didn't realize this. Rather interesting.

I was under the impression black holes had a radius. Sort of like shrinking the Earth down to only a few km or so in size (only with a lot more matter).


You might be thinking of the Schwarzschild Radius. If you squeeze the Earth (or anything) down to its schwarzschild radius, it will become a black hole. but, once you sqeeze it that far, it will continue to collapse into a singularity (or so they say).
 

1. What is the center of a black hole?

The center of a black hole is known as the singularity, which is a point of infinite density and zero volume. It is where the laws of physics, as we know them, break down.

2. What happens at the center of a black hole?

At the center of a black hole, the gravitational pull is so strong that it traps everything, including light. This creates a one-way path in which nothing can escape, not even light, making it impossible for us to see or know what is happening inside.

3. Can anything survive at the center of a black hole?

No, nothing can survive at the center of a black hole. The extreme conditions, such as infinite density and intense gravitational pull, make it impossible for any form of matter to exist.

4. How does the center of a black hole affect time?

The intense gravitational pull at the center of a black hole causes time dilation, meaning time moves slower in the presence of a strong gravitational field. This results in time passing at a different rate for an observer outside the black hole compared to someone falling into the black hole.

5. Will we ever be able to see inside a black hole?

Currently, it is not possible for us to see inside a black hole as nothing can escape from its event horizon. However, with advancements in technology and theoretical research, it is possible that we may one day find a way to observe or gather information from the center of a black hole.

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