Japanese version
It is simply described on the classification by mass, angular momentum and charge, Ciama's formula, and 2+1 dimensional black holes. In 2+1 dimensional it contains the work of Brown and Henneaux, Cardy formula, and BTZ black holes. Why are 2+1 dimensional black holes important? Because they are very much resemble to 3+1 dimensional ones, in our universe.
General Relativity and Black Holes II
This article is translated into English from my Japanese document for explanation of the elementary part of the theory of Black Holes.
the original Japanese document:
General Relativity and Black Holes II (in Japanese)
1, Of course, I explained 2+1 D gravity in order to relate to Witten's talk and the two previous posts, which is the idea of the relationship between BTZ black holes and monstrous moonshine.
ReplyDelete2, I would like to get a chance to the work of Brown-Henneaux, Cardy's formula, BTZ B.H. and Strominger'w Kerr/CFT correspondence etc. They are very important, I think.
3, Let us subtitute c=1, the central charge of the CFT=1, then we get Hardy-Ramanujan's fomula. I will write down the relation between them. Because I did not write the detail of Cardy's formula in the near future I will write it with the detail of Cardy's formula.
(the knowledge of CFT is needed for this subject, I must consider something good way to describe it.)
This part is the cross road of Zeta function, Casimir operatorsm and Virasoro algebra. Then it is rather difficult to describe it.
4, Mr. Carlip seems to relate BTZ black holes to Liouville thory but Mr. Witten might have oposite opinion. (see Woit's blog "someone asked him this point and he answered so.)
But Witten said in 2007 about B.H.s in 3D with positive cosmological constant :
ReplyDelete"Currently, there is some suspicion that quantum gravity with Lambda>0 doesn't exist perturbatiovely (in any dimension) with positive cosmological constat. One reason for this is that it does not appear to be possible with Lambda>0, to define precise observables. This is natural if it is the case that a world with positive cosmological constant (like the one we may be living in) is always unstable.
If that is so, then a world with Lambda>0 doesn't really make sense as an exact theory in its own right but(like an unstable particle) must be studied as part of a larger system.
Whether that is the right interpretation or not, since I do not know how to define any precise observables, I don't know what it would mean to try to solve 2+1 dimensional gravity with Lambda>0, since it is not clear what we'd want to compute."