Thursday, October 11, 2007

Black hole basics part 1

To fill in those of you who may not know a black hole is an object with such a great density of mass that the gravitational acceleration near enough to the center of mass is so great that light is unable to escape. So more or less a black hole is an object whose escape velocity is the speed of light c. Since a massive body would have to have infinite energy to achieve a speed of c there exists a region around a black hole such that once an object has entered that region it cannot again escape it. The surface of this region of no return is called the event horizon of the black hole. When one talks of the "radius" of a black hole one means the radius of the event horizon.

The event horizon is actually always perfectly spherical and so the radius of the event horizon is actually a perfect description of its geometry. The reason the surface of the region of no return is called the event horizon is that events that happen on the outside of it may have an effect on other events that happen farther away but events that happen inside of the event horizon cannot affect anything outside of it. This is because information can only travel at the speed of light and so information can't get transmitted from inside to outside a black hole.

The event horizon of course has no physical substance and is not the surface of a black hole in the sense that a star has a surface. In fact we do not know if there is a material surface of a black hole or not. It is generally considered that because the gravitational acceleration inside the event horizon exceeds the speed of light no other force could be sufficient to overcome it and therefore all matter inside a black hole must collapse to a single point. This point is known as the singularity. However it is possible that because of quantum effects or effects due to string theory there may be a point beyond which it is not possible to compress matter in which case there would be a tiny nugget of unimaginably dense material at the core of each black hole. As a point of reference it is interesting to note that the density of a neutron star is somewhere in the neighborhood of 1017 Kg/m3 and the primary agent against further collapse of neutron stars is in fact the pauli exclusion principle. So quantum effects are already needed to keep neutron stars from collapsing to a singularity.

The model of a black hole that I have been describing up to this point is what is called the non-rotating or Schwarzschild black hole. It is a solution to Einstein's equations for the case where there is no rotation and no charge. both rotating and charged black holes are fascinating critters but for the moment I am just going to keep on ignoring them maybe I will include a post about them later.

You may have heard the phrase that "black holes have no hair" this simply means that black holes have only three independent properties, mass, angular momentum, and charge. They have no hair in the sense that unlike any other macroscopic object they are totally indistinguishable but for those three properties. Since we are looking at non-rotating and non-charged black holes the only item of interest is the radius of the event horizon and its relation to mass. Interestingly enough the radius that classical mechanics suggests for such an object turns out to be the correct radius. This radius of the black hole for the non-rotating case is called the Schwarzschild radius.

If you compress a mass down to close to its Schwarzschild radius it will collapse into a singularity under its own gravity. The equation for the Schwarzschild radius is R = 2GM/c2 , where G is the gravitational constant and c is of course the speed of light. Now for the grand finale of the post we apply this equation to find the approximate Schwarzschild radius for Jupiter as being 2*6*10-11*1027/(9*1018). Which is about 1.3 meters. Now compressing the entirety of jupiter's mass into a volume of only 1.3 meters may seem just a tad on the side of implausible and I would tend to agree. However keep in mind that this is the radius that one would need to put matter in in order to make it collapse under its own gravitational field alone. At any rate though we have our first rough calculations of what it would take to make jupiter a black hole.