66 Chapter 2: Motivation 2.3.4 Applications of AdS/CFT to Heavy Ion Collisions To make contact with thermal QCD physics we extend the conjecture to nonex- tremal D 3 branes, in which case conformality is broken. Starting with the nonextremal 3-brane metric Eq. (2.30), take R = r$_{−}$ , r$_{0}$ = r 4 $_{+}$− r 4 $_{−}$, and change coordinates to r 4 = ρ 4 − R $^{4}$; it then becomes This yields in the near-horizon, or decoupling, limit of r ≪ R We find the temperature associated with the horizon by requiring that the τ = it coordinate after Wick rotation has the correct periodicity for a path integral partition function. We could integrate τ in a circle around r = r$_{0}$ [330] but will rather follow [331]: for a metric of the form ds 2 = V dτ $^{2}$+ V − $^{1}$dr$^{2}$+ . . . the temperature is T = V $^{′}$|$_{r}$$_{=}$$_{r}$$_{0}$ / 4 π , where r$_{0}$ is the radial position of the horizon. Both methods yield ds 2 = 1 $^{√}$H ( r ) $^{[}$− h ( r ) dt 2 + d⃗ x 2 $^{]}$+ $^{√}$H ( r )$^{[}$ dr 2 h ( r ) + r $^{2}$d Ω 2 5 ] , (2.61) H ( r ) = 1 + R 4 r 4 ; h ( r ) = 1 − r 4 0 r $_{4}$. (2.62) ds 2 = r 2 R 2 $^{[}$− h ( r ) dt 2 + d⃗ x 2 $^{]}$+ R 2 r $^{2}$h ( r ) dr 2 + R $^{2}$d Ω 2 $_{5}$. (2.63) T = r$_{0}$ πR $_{2}$. (2.64) Using this equation for T one may follow [330, 332] to compare the entropies of weakly coupled N = 4 U ( N ) and strongly coupled gravity at the same