Approaches to Quantum Gravity

Gauge/gravity duality 175Here l s is the fundamental length scale of string theory, related to the string tensionμ by μ −1 = 2πl 2 s . Notice that the spacetime radii are large in string units (and sothe curvature is small) precisely when the ’t Hooft coupling 4πg s N c = gYM 2 N c islarge, in keeping with the heuristic arguments that we made in the introduction.It is also instructive to express the AdS radius entirely in gravitational variables.The ten-dimensional gravitational coupling is G ∼ gs 2l8 s , up to a numericalconstant. Thusl ∼ N 1/4c G 1/8 , G ∼ l8N 2 c. (10.5)In other words, the AdS radius is Nc1/4coupling is Nc−2 in AdS units.in Planck units, and the gravitational10.3 Lessons, generalizations, and open questions10.3.1 Black holes and thermal physicsThe fact that black holes have thermodynamic properties is one of the most strikingfeatures of classical and Quantum Gravity. In the context of AdS/CFT duality, thishas a simple realization: in the dual gauge theory the black hole is just a hot gasof gauge bosons, scalars, and fermions, the gauge theory degrees of freedom inequilibrium at the Hawking temperature.A black hole in AdS 5 is described by the Schwarzschild AdS geometry( rds 2 2=−l + 1 − r 2 ) (0 rdt 2 2+2 r 2 l + 1 − r 2 ) −10dr 2 + r 2 d 2 r 2 3 . (10.6)Denoting the Schwarzschild radius by r + , the Hawking temperature of this blackhole is T H = (l 2 + 2r 2 + )/2πr +l 2 . When r + ≫ l, the Hawking temperature is large,T H ∼ r + /l 2 . This is quite different from a large black hole in asymptotically flatspacetime which has T H ∼ 1/r + . The gauge theory description is just a thermalstate at the same temperature T H .Let us compare the entropies in the two descriptions. It is difficult to calculatethe field theory entropy at strong coupling, but at weak coupling, we have of orderN 2 c degrees of freedom, on a three sphere of radius l at temperature T H and henceS YM ∼ N 2 c T 3 H l3 . (10.7)On the string theory side, the solution is the product of (10.6)andanS 5 of radius l.So recalling that G ∼ gs 2l8 s in ten dimensions and dropping factors of order unity,the Hawking–Bekenstein entropy of this black hole isS BH =A4G ∼ r 3 + l5g 2 s l8 s∼ T 3 H l11g 2 s l8 s∼ N 2 c T 3 H l3 (10.8)