Schwarzschild metric

[Quartile]

Could the Schwarzschild metric describe black bodies instead of black holes? Perfect black bodies seem to have characteristics similar to black holes.

 

Also Kruskal-Szekeres coordinates are interesting but I don't feel like I understand them very well. From wiki:

 

The apparent singularity at r = rs is an illusion; it is an example of what is called a coordinate singularity. As the name implies, the singularity arises from a bad choice of coordinates. By choosing another set of suitable coordinates one can show that the metric is well-defined at the Schwarzschild radius. See, for example, Lemaitre coordinates, Eddington-Finkelstein coordinates, Kruskal-Szekeres coordinates or Novikov coordinates.

 

What exactly is a coordinate singularity?

Edited October 24, 2008 by Quartile

[Edtharan]

Perfect black bodies seem to have characteristics similar to black holes.

There are several major differences:

 

1) Black holes absorb all radiation.

2) Black holes have certain amount of mass concentrated into a certain area that theory states causes them to be the way they are. Black bodies don't have to ahve a certain mass for its size.

3) Black Bodies emit a certain and very specific set of light frequencies which depend on it temperature. Black holes, while they don't emit radiation themselves, never the less cause radiation to be emitted from the infilling matter as it collides with other matter. This causes a very specific set of frequencies of light to be emitted which is different form black body radiation.

 

Therefore, Black holes and Black bodies can not be the same things.

[Quartile]

I found this on the wiki bio for Karl Schwarzchild:

 

Schwarzschild's second paper, which gives what is now known as the "Inner Schwarzschild solution" (in German: "innere Schwarzschild-Lösung"), is valid within a sphere of homogeneous and isotropic distributed molecules within a shell of radius r=R. It is applicable to solids; incompressible fluids; the sun and stars viewed as a quasi-isotropic heated gas; and any homogeneous and isotropic distributed gas.

[AlphaNumeric]

What exactly is a coordinate singularity?

As the wiki article says, it's an artifact of making a bad choice of coordinates. There is no physical singularity or barrier or whatnot in the space-time, despite what your coordinates might say.

 

For instance, consider (1+1) dimensional flat space-time. You can write the metric as [math]ds^{2} = -dt^{2}+dx^{2}[/math]. I now define a new time coordinate to be T = 1/t, so [math]t = \frac{1}{T}[/math] and [math]dt = -\frac{1}{T^{2}}dT[/math] so [math]ds^{2} = -\frac{dT^{2}}{T^{4}} + dx^{2}[/math]. Now this metric, if I'd given it you without saying how I'd produced it, would lead you to think there's some kind of singularity in the space-time, because T=0 definitely doesn't work. But you can clearly see that the original description was flat space-time (ie no singularities) and it was my poor choice of T = 1/t which made that 'singularity' appear. The transformation is obviously meaningless at t=0 or T=0. This is a coordinate singularity.

 

One way to check if there exists a coordinate choice which will get rid of a coordinate singularity is to compute coordinate independent quantities in terms of the metric. For instance, the Ricci scalar is coordinate independent, since it doesn't transform under a change of basis (clearly, given it's a scalar). Other things might be [math]R^{abcd}R_{abcd}[/math]. In the case of black hole solutions you find these quantities are fine at [math]r = R_{Hor}[/math] but screw up at r=0 (well, in the case of spinning black holes it's a little more complicated but the concept is the same). Hence r=0 is the only actual singularity in the space-time. Everything else is just poor choice of coordinates.