# Black hole ?

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?

Nabeshin
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!

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 realise 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).

Drakkith
Staff Emeritus
I didn't realise 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.

So is a black hole a ball of compressed matter with a radius?

Drakkith
Staff Emeritus
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 dont 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?

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 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 dont 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.

Drakkith
Staff Emeritus
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?

Nabeshin
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.

K^2
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):

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

And acceleration.

$$a^{\alpha} = \Gamma^{\alpha}_{\beta \gamma}u^{\beta}u^{\gamma}$$

Or using the form of u from above.

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

The only non zero element is along r.

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

The 4-acceleration comes out to be very simple.

$$a = \left(0, -\frac{r_s}{2r^2}, 0, 0\right)$$

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

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

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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thanks for all of the replies .

I didn't realise 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).