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## 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 

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<img src='content_image/314475.jpg'>

This yields in the near-horizon, or decoupling, limit of r ≪ R 

<img src='content_image/314476.jpg'>

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 

<img src='content_image/314477.jpg'>

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 

