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Additional resources for Black Hole Thermodynamics in Semiclassical, Superstring Theory
This works only fine in the near extremal region because then there are so few right movers for example that one can neglect the interactions between left moving and the right moving gas. This is called the “dilute gas approximation” and it allows us to just add the entropies of the gases. However, one should recall that the simple formula for the entropy was deduced for large charges only. In order for these equations to be valid again we have to add not few but many anti charges. Thus we require for the left and right moving momentum for example: NL ≫ NR ≫ 1 where QN = NL − NR .
Therefore the configuration is a thermodynamic equilibrium of the contributing charge carriers; but only in this sense because the right and left moving gasses have different temperatures and are thought to be noninteracting. The different temperatures come from the form of equation (176) which is written inversely as follows: 1 ∝ TH p cosh αp ∝ ( cosh αp )( Nq + Nq¯) (183) p=q for example 1 1 1 1 = ( + ) TH 2 TL TR (184) From equation (165) follows S = SL + SR ⇒ A+ = SL + SR 4GN (185) For an extremal black hole we have A+ = A− and SR = 0 and deviating from extremality the inner and outer horizons separate.
In perturbative string theory there are no fundamental objects (perturbative string states) that have charges under the described fields. Therefore, both, the electric and the dual magnetic charges must be solitons and indeed the p-branes that are BPS turn out to be D-p-branes. : via string dualities) and that are described by conformal field theories turn out to be extremal p-branes in the supergravities that we obtained 1 (equation (86)), where the from the superstring theories. ). To summarize: The S-duality (electric-magnetic duality) leads to the following pairing of D-p- and D-(6-p)-branes: In IIA • Aµ : electric 0-brane and magnetic 6-brane • Cµνρ : electric 2-brane and magnetic 4-brane and in IIB which has the zero form χ as well • χ : electric (-1)-brane (the instanton in the Euclidean theory) and magnetic 7-brane ′ • Bµν : electric 1-brane and magnetic 5-brane • Aµνρσ : self dual 3-brane Of course, the above is the RR sector.