# Why does AdS/CFT with non-zero temperature correspond to a black hole in the bulk?

Is there a good intuitive explanation on why AdS/CFT with non-zero temperature corresponds to a black hole in the bulk? And what is the role of temperature and chemical potential in this black hole?

There are a few answers to this question. The one that makes the most intuitive sense probably depends on your background.

First of all, from the Euclidean quantum gravity program it is known that a stationary or static spacetime can be put at finite temperature by analytically continuing the spacetime to Euclidean signature. The temperature is then identified as being the inverse length of the rotated time direction, now called the thermal circle. This approach is familiar from studying field theories at finite temperature, and was developed for the case of gravity by Hawking and others in the late 1970's, early 1980's. Spacetimes without any black hole horizons, like Minkowski space, can be put at any temperature. Once the analytic continuation has been done, the thermal circle can be compactified to any radius.

Black holes, however, can only be analytically continued to have a specific length for the thermal circle. This length is chosen so as to avoid a conical deficit (a type of singularity) in the spacetime. For example, Euclidean Schwarzschild is (with $\tau = i t$)

$ds^2 = \left(1-\frac{r_0}{r}\right) d\tau^2 + \left(1-\frac{r_0}{r}\right)^{-1}dr^2 + r^2 d\Omega_2^2.$

Near the horizon, this looks like

$ds^2 \approx \left(\frac{r-r_0}{r_0}\right) d\tau^2 + \frac{r_0}{r-r_0} dr^2 + r_0^2 d\Omega_2^2 \approx \rho^2 \frac{d\tau^2}{4r_0^2} + d\rho^2 + r_0^2 d\Omega_2^2$, after setting $r=r_0 + \rho^2/(4r_0^2)$.

In order for the $(\tau,\rho)$-part of the space to be locally flat space written in polar coordinates, the periodicity $\tau \sim \tau + 4\pi r_0$ is forced on us. This is just the inverse Hawking temperature.

So, if a field theory at finite temperature is to be dual to a geometry, it can be dual to one without a horizon and one with. The question of which one is the true dual is determined by thermodynamics. Generally, one would need to take all the solutions and compare their free energy. Interesting phase transitions are possible, for example the Hawking-Page transition (http://projecteuclid.org/euclid.cmp/1103922135). For the case of black holes in the Poincare patch of AdS, the usual starting point for AdS/CFT, there is no such transition and the black brane is always thermodynamically preferred over thermal AdS.

A second way to see that finite temperature corresponds to black holes in the bulk is to recall the black hole-black string correspondence (http://arxiv.org/abs/hep-th/9612146). It was known even before AdS/CFT that there was a strong connection between the p-brane solutions of supergravity and the D-branes of string theory. Both solutions can be heated up. In the regime of parameters where each solution makes sense, heated up p-branes (for $p=3$) correspond to Poincare patch AdS black holes (for small enough temperature). Heated up D-branes just correspond to SYM at a finite temperature. So the finite temperature extension of the original Maldacena duality is then fairly obvious from this perspective.

Lastly, the role of temperature and chemical potential (and any other possible thermodynamic potentials) in the black hole simply correspond to the same potentials in the dual field theory. For example, rotation in the black hole corresponds to considering a rotating field theory (http://arxiv.org/abs/hep-th/9908109).

The explanation of Surgical Commander, of imposing periodic boundary conditions in the Euclidean time, is very good, and in some sense the most correct way of seeing what you should get. I'll add here another way you can think about it, more physically and dynamically, for another perspective.

You can think of finite temperature as putting your system weakly in contact with some enormous bath of energy, and allowing it to equilibrate (weakly so you don't affect the physics of your system too much, you just allow it to slowly exchange energy). You can also allow it to exchange some conserved charge if you want a grand canonical ensemble with chemical potentials. In AdS/CFT, this amounts to having some coupling between the boundary conditions of your fields (AKA CFT operators) and the heat bath. You can think of this as allowing energy to go out of and come in from the boundary.

Start with empty AdS, with a compact boundary. If the heat bath is hot, energy will start pouring into the bulk from the boundary coupling, and energy of whatever fields you have will start sloshing around. At some point, with enough energy it will get dense enough somewhere to form a black hole. The black hole will Hawking radiate. If the Hawking radiation is of the same temperature as the heat bath, there won't be any net exchange of energy with the bath, and the Hawking radiation will be in equilibrium with the black hole.

If the Hawking radiation is cooler than the heat bath, there will still be stuff coming into the black hole faster than it radiates, so the black hole will grow. Eventually, the black hole will reach the same temperature as the bath and equilibrate.

If the Hawking radiation is hotter than the heat bath, it will leak out, and the black hole will shrink as it radiates. If it's big enough, it cools down as it does this, so may at some point settle into equilibrium with the bath. If it's too small (or the bath is too cool to equilibrate with even the coldest possible black hole), it has negative heat capacity, so will get hotter as it loses energy (!), and ultimately will evaporate. The bath might then equilibrate with just radiation and no black hole, if it's cool enough. (The energy of the radiation is zero in the classical limit, being suppressed by some power of the Planck length over the AdS length).

There are some circumstances where a black hole can be semiclassically dynamically stable like this, but is not thermodynamically favoured so will evaporate by tunnelling events.

You can add more rich behaviour by adding a chemical potential as you suggest, and allowing charge to come in and out too. You can still intuit most things from this sort of dynamical picture, adding more interesting physics as required. For example, a black hole might need to be charged to support an electric field, so that charged stuff doesn't want to come in from the boundary. But if this field has to be too strong, at the horizon you might get Schwinger pair production, so the black hole starts to throw out charged particles, which hang around in the bulk as charged `hair'.