# Hawking-Page transition The Hawking-Page transition is a phase transition between two equilibrium states: a black hole bathed in radiation v.s. pure radiation. More specifically, consider a black hole and a bunch of radiation in AdS (remember that AdS is like a box) and let the system evolve to equilibrium. What is the equilibrium state? In a canonical ensemble, where we fix the temperature, the most likely state is either a black hole surrounded by radiation, or pure radiation (the black hole completely evaporates). Which state is more likely depends on the environment. In the canonical ensemble where the system has fixed temperature, there is a transition between these two configurations as the temperature is varied. Here is a summary: ![[0012_sum.jpg]] The temperature-versus-size curve can be worked out by solving vacuum Einstein's equation (with a negative cosmological constant) for the spherically symmetric black hole. Here, $r_*$ is defined to be at the minimum temperature below which there is no black hole solution; $r_0$ is defined to be the size of the black hole below which the system (black hole + radiation) is unstable against small perturbations. It can be shown that $r_0\ll r_*$. In a **canonical ensemble**, the dominant saddle is the one that minimises the free energy, $F=-T \log Z$, where the partition function $Z$ is [[0127 Black hole thermodynamics|related]] to the Euclidean action $I$ of the saddle by $Z=e^{-I}$. It turns out that $F_\text{small BH} > F_\text{big BH}$ always, so small black holes never dominate the canonical ensemble. As for thermal radiation, $F_\text{therm AdS} > F_\text{big BH}$ iff $r_{+}>r_{2}=c_{2} r_{*}$, where $c_2$ is an order-one number greater than one ($c_2>1$, $c_2=O(1)$). In other words, as temperature is lowered to the one corresponding to $r_2$, thermal AdS become the dominant saddle whereas the large black hole always dominates above this temperature. In a **microcanonical ensemble**, the probability is related to the entropy: $P\sim e^S$. We therefore look for the state that maximises the entropy at any given energy. In this case, $S_\text{BH+rad}>S_\text{rad}$ iff $r_{+}>r_{1}=c_{1} r_{0}>r_{0}$, for some order-one constant $c_1$ greater than one (i.e. $c_1>1$, $c_1=O(1)$). In other words, as energy is lowered to the one corresponding to $r_1$, pure radiation becomes dominant; above this energy, the black hole dominates. Here is no distinguishing between large and small because there is only one size at each energy. Tip: imagine moving along the horizontal axis when considering the microcanonical ensemble and the vertical axis when canonical. ## Refs - originals - [[1983#Hawking, Page]] - in holography [[1998#Witten (Mar)]] - including charge - [[1999#Chamblin, Emparan, Johnson, Myers]] - planar [[AnabalonConchaOlivaQuijadaRodriguez2022]][](https://arxiv.org/pdf/2205.01609.pdf) - including domain wall - [[BachasPapadopoulos2021]][](https://arxiv.org/pdf/2101.12529.pdf) ## Its CFT dual - a phase transition between [[0441 Confinement-deconfinement transition|confined]] and deconfined phases - see [[2003#Aharony, Marsano, Minwalla, Papadodimas, van Raamsdonk]] ## Topologies - the usual one: $S^2\times S^1$ where $S^1$ is the compactified Euclidean time - a weird one, where the boundary topology is $S^{d_1}\times S^{d_2}$ - then there is not even a thermal circle - [[2019#Aharony, Urbach, Weiss]] - similarly [[2020#Kiritsis, Nitti, Preau]] - more generally, see [[0231 Bulk solutions for CFTs on non-trivial geometries]] ## Loop correction - [[2008#Giombi, Maloney, Yin]] ## Warning The microcanonical ensemble is more involved and non-universal; and things depend on details on the internal space. See e.g.: - [[Horowitz1999]] - [[2003#Aharony, Marsano, Minwalla, Papadodimas, van Raamsdonk]]. ## Charged \[*Acknowledgement: Big thanks to Xi Dong's String Theory class where I learned most of this, and some of the notes on this page are essentially his.*\]