The Hawking–Page Phase Transition

In asymptotically anti–de Sitter (AdS) spacetimes, black holes exhibit thermodynamic behavior markedly different from their counterparts in flat space. The negative cosmological constant provides a confining potential that prevents hot radiation from escaping to infinity. In 1983, Stephen Hawking and Don Page discovered that there exists a critical temperature at which thermal AdS space undergoes a phase transition to a black hole dominated configuration. This Hawking–Page transition represents a gravitational analogue of the confinement–deconfinement transition in gauge theories and plays a central role in the AdS/CFT correspondence, where it maps to the onset of deconfinement in the dual conformal field theory.

For a four-dimensional Schwarzschild–AdS black hole, the transition occurs when the horizon radius equals the AdS curvature radius $L$. The black hole temperature as a function of horizon radius $r_{+}$ is given by $T=\frac{ħc}{4\pi kB{r}_{+}}\left(1+\frac{3{r_{+}}^2}{L^2}\right)$ Setting $r_{+}=L$ yields the critical temperature $T_{\mathrm{HP}}=\frac{ħc}{\pi kBL}$ demonstrating that the transition temperature is inversely proportional to the AdS curvature radius. Larger AdS spaces transition at lower temperatures, while smaller AdS boxes require hotter conditions to nucleate a black hole. The corresponding black-hole mass at the transition follows from the mass–radius relation $M=\frac{r_{+}\mathrm{c^2}}{2G}\left(1+\frac{{r_{+}}^2}{L^2}\right)$ which for $r_{+}=L$ simplifies to $M=\frac{L{c}^2}{G}$ (after restoring factors of $\mathrm{c^2}$).

The Hawking–Page transition underscores the thermodynamic consistency of black holes and the profound links between gravity and quantum field theory. In the AdS/CFT correspondence, the transition maps to a confinement–deconfinement phase change in the boundary theory: thermal AdS corresponds to the confined phase, while the AdS black hole represents the deconfined plasma. Thus, studying the gravitational transition sheds light on strongly coupled gauge theories and vice versa. The notion that spacetime itself can undergo a phase transition also raises deep conceptual questions about the nature of quantum gravity, horizon microstates, and the emergence of spacetime from more fundamental constituents.

Using this calculator, one can explore how the transition temperature and the associated black-hole mass vary with the AdS curvature radius. Enter the radius in kilometers, and the script computes the temperature in Kelvin and the mass in solar masses. The calculations assume a four-dimensional spacetime and neglect corrections from charge or rotation; generalizations to higher dimensions or charged/rotating black holes follow similar principles but involve modified formulas.

To illustrate the dependence on $L$, consider the example table below:

The transition temperature is exceedingly small for macroscopic AdS radii, highlighting the theoretical nature of the phenomenon. Nevertheless, its conceptual impact is vast: the Hawking–Page transition provided one of the earliest bridges between gravitational thermodynamics and quantum field theory, inspiring decades of research into holography and black-hole phase transitions.

The calculator's implementation is straightforward. After converting the input radius from kilometers to meters, the script evaluates the critical temperature and mass using fundamental constants. The AdS curvature sets the length scale, and the result is expressed in units familiar to astronomers. The design emphasizes clarity and accessibility, inviting users to manipulate parameters and observe outcomes instantly.

While the transition occurs in an idealized setting, its study continues to yield insights into quantum gravity, gauge/gravity duality, and thermalization in strongly coupled systems. By providing an interactive window into this classical calculation, the tool aims to demystify a cornerstone concept of modern theoretical physics and to spark curiosity about the deep connections between geometry, thermodynamics, and quantum fields.