Quantum Gravity

QUANTUM-GRAVITATIONAL ASPECTS 299that one can give a microscopic foundation for extremal black holes and blackholes that are close to extremality. ‘Extremal’ is here meant with respect to thegeneralized electric and magnetic charges that can be present in the spectrum ofstring theory. It is analogous to the situation for an extremal Reissner–Nordströmblack hole (Section 7.1). We shall be brief in the following and refer the readerto, for example, Horowitz (1998) and Peet (1998) for reviews.The key idea to the string calculation of (7.17) is the notion of S-dualitydiscussed at the end of the last subsection. A central role is played by so-called‘BPS states’ (named after Bogomolnyi, Prasad, and Sommerfield), which havethe important property that they are invariant under a non-trivial subalgebraof the full SUSY algebra. As a consequence, their mass is fixed in terms of theircharges and their spectrum is preserved while going from a weak-coupling limitof string theory to a strong-coupling limit. In the weak-coupling limit, a BPSstate can describe a bound state of D-branes whose entropy S s can be easilycalculated. In the strong-coupling limit, the state can describe an extremal blackhole whose entropy can be calculated by (7.17). Interestingly, both calculationslead to the same result. This was first shown by Strominger and Vafa (1996)for an extremal hole in five dimensions. It has to be emphasized that all thesecalculations are being done in the semiclassical regime in which the black holeis not too small. Its final evaporation has therefore not yet been addressed.We give here only some heuristic arguments why this result can hold andrefer to the above references for details. The level density d N of a highly excitedstring state with level of excitation N is (for open strings) in the limit N →∞given byd N ∼ e 4π√N ≈ e M/M0 , (9.70)where (9.21) has been used, and1M 0 ≡4π √ . (9.71)α ′The temperature T 0 ≡ M 0 /k B connected with M 0 is called ‘Hagedorn temperature’or ‘temperature of the hell’ because the free energy diverges when T 0 isapproached (signalling a phase transition). The expression for d N can be understoodas follows. Dividing a string with energy M into two parts with energiesM 1 and M 2 , respectively, one would expect that M = M 1 + M 2 .Thenumberofstates would then obeyd N (M) =d N (M 1 )d N (M 2 )=d N (M 1 + M 2 ) ,from which a relation of the form (9.70) follows. Using from statistical physicsthe formula d N =exp(S s ), one finds for the ‘string entropy’S s ∝ M ∝ √ N. (9.72)Since the gravitational constant depends on the string coupling (see (9.43)), theeffective Schwarzschild radius R S =2GM increases if g is increased and a black