Hawking–Page temperature and transition mass
This calculator estimates the Hawking–Page transition temperature for a four-dimensional Schwarzschild–AdS black hole and also reports the black-hole mass associated with the transition. In plain language, it identifies the temperature at which anti–de Sitter spacetime changes from a thermal-AdS description to a black-hole-dominated phase. The input is the AdS curvature radius , entered in kilometers. The outputs are the critical temperature in kelvin and the transition mass in solar masses, connecting the thermodynamic threshold to an astrophysical mass scale.
The Hawking–Page transition matters because it is one of the clearest examples of black-hole thermodynamics behaving like an ordinary phase transition. In asymptotically flat space, hot radiation can disperse to infinity, but in anti–de Sitter space the negative cosmological constant effectively acts like a confining box. That confinement changes the thermodynamics. Below a certain temperature, thermal AdS is favored. Above it, an AdS black hole can dominate the ensemble. In the language of the AdS/CFT correspondence, that gravitational switch is famously related to a confinement–deconfinement transition in the dual field theory.
The Hawking–Page phase transition
For the Schwarzschild–AdS Hawking–Page transition in four spacetime dimensions, the horizon radius equals the AdS curvature radius . The black hole temperature as a function of horizon radius is given by Setting yields the critical temperature so 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 before the black-hole phase becomes favorable.
The Schwarzschild–AdS mass associated with the Hawking–Page point follows from the mass–radius relation which for simplifies to (after restoring factors of ). That means the transition mass scales linearly with . In other words, increasing the AdS radius makes the critical temperature smaller but the corresponding black hole more massive.
That contrast is one of the easiest ways to interpret the Hawking–Page outputs. If you double , the temperature drops by a factor of two, while the mass doubles. The calculator therefore highlights two linked physical ideas at once: AdS geometry sets a thermal scale, and that same geometric length scale also fixes the size and mass of the black hole that becomes thermodynamically preferred at the transition.
Using the Hawking–Page transition outputs
To calculate a Hawking–Page threshold, enter the AdS curvature radius in kilometers. The script converts that number to meters, inserts it into the critical-temperature formula, and then computes the transition mass using the same length scale. The temperature output is given in kelvin because that makes the thermodynamic meaning obvious. The mass output is converted into solar masses because raw kilograms are not very intuitive at astronomical scales. If you are scanning several values of , look for the pattern rather than any one number: large AdS boxes correspond to very cold transitions, while small AdS boxes correspond to hotter transitions.
It is also worth noticing what the Hawking–Page result does not say. This calculator is not predicting a black hole that can form in our everyday universe from a laboratory heat bath. The Hawking–Page transition belongs to an idealized AdS setting with a fixed negative cosmological constant and reflecting asymptotic behavior. Its importance is conceptual and theoretical. Even so, the numerical outputs are valuable because they make the scaling concrete. They show exactly how the critical temperature depends on geometry and why the transition became so influential in holography and gravitational thermodynamics.
To illustrate the Hawking–Page dependence on , consider the example table below. The values shown here are consistent with the formulas implemented in the calculator, using SI constants and then converting the mass to solar masses.
Example Hawking–Page transition scales for several AdS radii
| L (km) |
THP (K) |
MHP (M☉) |
| 100 |
7.288×10-9 |
6.774×101 |
| 1000 |
7.288×10-10 |
6.774×102 |
| 10000 |
7.288×10-11 |
6.774×103 |
These Hawking–Page examples make the scaling easy to see. The temperature quickly becomes extremely small for macroscopic AdS radii, but the mass grows at the same time. That pairing is the key physical takeaway from the calculator: colder transitions are associated with larger AdS boxes and more massive transition-point black holes.
Worked example: a 500 km AdS radius
For a Hawking–Page calculation with an AdS curvature radius of km, converting to meters gives m. Plugging this into the temperature expression yields approximately K. The corresponding mass becomes solar masses. This Hawking–Page example shows why the outputs can feel numerically very different: the temperature is tiny while the associated mass is large. That is precisely what the inverse temperature scaling and linear mass scaling predict for the same AdS length scale.
Beyond the basic Schwarzschild–AdS model
The Hawking–Page formula implemented here assumes a neutral, non-rotating black hole in four spacetime dimensions. Introducing electric charge or angular momentum changes the temperature curve and can shift the critical point. In higher-dimensional AdS spacetimes, the metric and thermodynamic relations change as well, so the simple equality used here is no longer the whole story. Researchers study Reissner–Nordström–AdS and Kerr–AdS solutions precisely because they reveal richer phase diagrams, metastable branches, and multiple transitions. This page therefore focuses on the cleanest textbook version of the effect, which is usually the right place to start.
That Schwarzschild–AdS simplicity is a strength for teaching the Hawking–Page transition. Once you understand that the Hawking–Page temperature scales like and the transition mass scales like , it becomes much easier to read more advanced papers. The basic calculator gives you intuition for how geometry sets the thermodynamic scale before additional effects such as charge, spin, boundary conditions, or higher-curvature corrections complicate the picture.
Hawking–Page calculation limitations and assumptions
This Hawking–Page calculator uses semiclassical gravity and standard physical constants. It ignores quantum-gravity corrections, backreaction beyond the classical Schwarzschild–AdS solution, and any microscopic details of the dual field theory. For very small AdS radii or near-Planckian regimes, those neglected effects may matter. The page also assumes that the asymptotic AdS boundary behaves as the standard reflecting container used in equilibrium thermodynamics. In realistic cosmological environments, that idealization need not apply.
Another limitation concerns the meaning of the Hawking–Page mass and temperature. The mass reported here is the mass of the black hole at the transition point in this idealized model, not the mass of an astrophysical black hole observed in our universe. Likewise, the temperature is the critical thermodynamic temperature for the canonical ensemble in AdS, not the cosmic microwave background temperature or a laboratory heat setting. If you keep that distinction in mind, the outputs are straightforward and very informative.
Hawking–Page transition historical notes
Hawking and Page’s 1983 analysis was remarkable because it showed that black-hole spacetimes have a genuine phase structure, not just a collection of formal analogies. More than a decade later, the AdS/CFT correspondence gave the result a deeper interpretation by connecting the gravitational transition to deconfinement in a gauge theory. That historical path is one reason the Hawking–Page transition appears so often in modern theoretical physics. It sits at the crossroads of black-hole thermodynamics, holography, quantum field theory, and the study of emergent spacetime.
Even when the Hawking–Page calculator is used only numerically, the transition remains conceptually powerful. It helped establish that classical gravitational backgrounds can compete the way familiar thermodynamic phases do. That insight continues to shape research on holographic plasmas, entanglement, strongly coupled systems, and the thermal behavior of quantum gravity.