Hawking–Page Transition

These notes combine a self-contained overview of the Hawking–Page (HP) transition with a step-by-step derivation of the renormalized on-shell Euclidean action in global AdS.

0. Historical roadmap

• Black holes (1916) → black-hole thermodynamics (1973–1976).
  Area–entropy $S=A/4G$, Hawking temperature $T=\kappa/2\pi$, and the four laws turn black holes into thermodynamic systems.
• Hawking–Page (1983).
  In global AdS, equilibrium between radiation and a black hole is possible; comparing Euclidean actions reveals a first-order transition between thermal AdS and a spherical AdS–Schwarzschild black hole.
• AdS/CFT (1997–1998).
  In the dual CFT on $S^{d-1}$, the HP transition is the confinement/deconfinement transition at large $N$ (Witten 1998).

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1. Hawking temperature from Euclidean regularity

For a static, spherically symmetric spacetime

$$ds^2=-f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2 d\Omega_{d-1}^2,$$

Wick rotate $t\to -i\tau$ to obtain

$$ds^2=f(r)\,d\tau^2+\frac{dr^2}{f(r)}+r^2 d\Omega_{d-1}^2.$$

Near the horizon $r=r_+$ with $f(r_+)=0$, expand $f(r)\approx f'(r_+)(r-r_+)$. The $(\tau,r)$ sector becomes

$$ds^2 \approx f'(r_+)(r-r_+)\,d\tau^2+\frac{dr^2}{f'(r_+)(r-r_+)}.$$

Define $\rho\equiv \tfrac{2\sqrt{r-r_+}}{\sqrt{f'(r_+)}}$, so

$$ds^2\approx d\rho^2+\rho^2\Big(\tfrac{f'(r_+)}{2}\,d\tau\Big)^2.$$

Smoothness at $\rho=0$ requires $\tau\sim\tau+\beta$ with period

$$\beta=\frac{4\pi}{f'(r_+)},\qquad T=\beta^{-1}=\frac{f'(r_+)}{4\pi}\equiv \frac{\kappa}{2\pi}.$$

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2. Warm-up: 4D Schwarzschild

With $f(r)=1-\tfrac{r_0}{r}$ ($r_0=2GM$):

$$T=\frac{1}{4\pi r_0}=\frac{1}{8\pi GM},\qquad S=\frac{A}{4G}=\frac{\pi r_0^2}{G}.$$

Using $M=\tfrac{r_0}{2G}$, the first law $dM=T\,dS$ holds. In the canonical ensemble, $F=M-TS$.

Path-integral viewpoint.

$$Z=\int\!\mathcal Dg\,e^{iS[g]}\xrightarrow{t\to -i\tau}\int\!\mathcal Dg\,e^{-S_E[g]} \simeq e^{-S_E[g_\ast]}=e^{-\beta F},$$

where the last step is the semiclassical saddle-point approximation (large parameter $\sim 1/G$).

Aside (Gaussian saddle). If $I_N=\int dx\,e^{Nf(x)}$ with $N\gg1$ and $f'(x_\ast)=0,f''(x_\ast)<0$, then

$$I_N\approx \sqrt{\frac{2\pi}{N|f''(x_\ast)|}}\,e^{Nf(x_\ast)}.$$

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3. Global AdS and boundary conditions

Work in global AdS$_5$ with radius $L$; the Euclidean boundary is $S^1_\beta\times S^3$ with fixed proper thermal circle length $\beta$. Asymptotically,

$$ds^2\to \frac{r^2}{L^2}d\tau^2+\frac{L^2}{r^2}dr^2+r^2 d\Omega_3^2,\qquad \tau\sim\tau+\beta.$$

Saddles with these boundary data:

1. Thermal AdS: $f(r)=1+\tfrac{r^2}{L^2}$, no horizon.
2. Small AdS black hole: unstable.
3. Large AdS black hole: stable.

Both black holes share

$$ds^2=f(r)\,d\tau^2+\frac{dr^2}{f(r)}+r^2 d\Omega_3^2,\qquad f(r)=1+\frac{r^2}{L^2}-\frac{\mu}{r^2},$$

with horizon at $r=r_+$ and

$$\mu=r_+^2\Big(1+\frac{r_+^2}{L^2}\Big).$$

Temperature and branches.

$$T(r_+)=\frac{1}{2\pi r_+}\Big(1+2\frac{r_+^2}{L^2}\Big),\qquad \beta(r_+)=\frac{2\pi L^2 r_+}{L^2+2r_+^2}.$$

$T$ has a minimum at $r_x=L/\sqrt{2}$:

$$T_{\min}=\frac{\sqrt{2}}{\pi L}.$$

For $T>T_{\min}$ both a small (unstable) and a large (stable) black hole exist at the same $T$. The horizon area $A=2\pi^2 r_+^3$ yields

$$S_{\rm BH}=\frac{A}{4G_5}=\frac{\pi^2 r_+^3}{2G_5}.$$

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4. Canonical ensemble and renormalized on-shell action (AdS$_5$)

We evaluate

$$Z_{\rm grav}(\beta)=\int\mathcal D g\,e^{-I_E[g]}\simeq \sum_i e^{-I_E^{(i)}(\beta)},$$

with the renormalized Euclidean action

$$I_E[g]=-\frac{1}{16\pi G_5}\!\int_{M}\! d^5x \sqrt{g}\,(R+\tfrac{12}{L^2}) -\frac{1}{8\pi G_5}\!\int_{\partial M}\! d^4x \sqrt{\gamma}\,K +I_{\rm ct}[\gamma].$$

For a 4D boundary,

$$I_{\rm ct}[\gamma]=\frac{1}{8\pi G_5}\!\int_{\partial M}\! d^4x \sqrt{\gamma}\left(\frac{3}{L}+\frac{L}{4}R[\gamma]\right).$$

Below we compute $I_E$ for the spherical AdS–Schwarzschild saddle at a cutoff $r=R$ and then send $R\to\infty$. Denote $\Omega_3=\mathrm{Vol}(S^3)=2\pi^2$ so that $\Omega_3/(16\pi)=\pi/8$.

4.1 Bulk term

On shell $R=-20/L^2$, hence $R+12/L^2=-8/L^2$. With $\sqrt{g}=r^3$,

$$\begin{aligned} I_{\rm bulk}^{\rm (bh)}(R) &=-\frac{1}{16\pi G_5}\int d^5x\sqrt{g}\left(-\frac{8}{L^2}\right) =\frac{\beta\,\Omega_3}{8\pi G_5 L^2}\int_{r_+}^R r^3\,dr\\ &=\frac{\beta}{4G_5L^2}\,\pi\,(R^4-r_+^4). \end{aligned}$$

4.2 Gibbons–Hawking term

Writing $ds^2=N^2dr^2+\gamma_{ij}dx^i dx^j$ with $N^2=1/f$ and $\gamma_{ij}dx^i dx^j=f\,d\tau^2+r^2 d\Omega_3^2$, the extrinsic curvature is

$$K=\sqrt{f}\Big(\frac{f'}{2f}+\frac{3}{r}\Big),\qquad \sqrt{\gamma}=\sqrt{f}\,r^3.$$

Therefore

$$\begin{aligned} I_{\rm GH}^{\rm (bh)}(R) &=-\frac{\beta\,\Omega_3}{8\pi G_5}\,R^3\sqrt{f}\,\sqrt{f}\Big(\frac{f'}{2f}+\frac{3}{R}\Big)\\ &=-\frac{\beta\,\pi}{4G_5}\Big(\frac{R^3 f'}{2}+3R^2 f\Big)\\ &=-\frac{\beta\,\pi}{4G_5}\Big(\frac{4R^4}{L^2}+3R^2-2\mu\Big), \end{aligned}$$

using $f(R)=1+\frac{R^2}{L^2}-\frac{\mu}{R^2}$ and $f'(R)=\frac{2R}{L^2}+\frac{2\mu}{R^3}$.

4.3 Counterterms

For $\gamma_{ij}=\mathrm{diag}(f(R),R^2 g_{S^3})$, the intrinsic scalar curvature is $R[\gamma]=6/R^2$. Hence

$$I_{\rm ct}^{\rm (bh)}(R)=\frac{\beta\,\Omega_3}{8\pi G_5}\,R^3\sqrt{f}\left(\frac{3}{L}+\frac{3L}{2R^2}\right) =\frac{\beta\,\pi}{4G_5}\!\left(\frac{3R^3\sqrt{f}}{L}+\frac{3L}{2}R\sqrt{f}\right).$$

At large $R$,

$$\sqrt{f(R)}=\frac{R}{L}+\frac{L}{2R}-\frac{L^3}{8R^3}-\frac{\mu L}{2R^3}+\cdots,$$

so

$$I_{\rm ct}^{\rm (bh)}(R)=\frac{\beta\,\pi}{4G_5}\Big(\frac{3R^4}{L^2}+3R^2+\frac{3}{8}L^2-\frac{3}{2}\mu+\cdots\Big).$$

4.4 Finite on-shell action

Summing $I_{\rm bulk}+I_{\rm GH}+I_{\rm ct}$, the $R^4$ and $R^2$ divergences cancel. The finite remainder is

$$I_E^{\rm (bh)}=\frac{\beta\,\pi}{8G_5L^2}\left(r_+^2L^2-r_+^4+\frac{3}{4}L^4\right),$$

using $\mu=r_+^2(1+r_+^2/L^2)$.

4.5 Thermal AdS

Repeat with $f_0(r)=1+\frac{r^2}{L^2}$ (integrate from $r=0$). One obtains

$$I_E^{\rm (th)}=\frac{\beta\,\pi}{8G_5L^2}\left(\frac{3}{4}L^4\right).$$

A background-subtraction derivation yields the same difference and is equivalent after matching the proper length of the thermal circle at the cutoff.

4.6 Free energy and Hawking–Page crossing

With $F=I_E/\beta$, the scheme-independent difference is

$$\Delta F \equiv F_{\rm bh}-F_{\rm th} =\frac{\pi}{8G_5L^2}\Big(r_+^2L^2-r_+^4\Big) =\frac{\pi r_+^2}{8G_5}\Big(1-\frac{r_+^2}{L^2}\Big).$$

Thus the dominant saddle changes at $r_+=L$. Using $T(r_+)$ from above,

$$T_{\rm HP}=\frac{3}{2\pi L},\qquad \beta_{\rm HP}=\frac{2\pi L}{3}.$$

Thermodynamic checks. From $Z=e^{-I_E}$,

$$E=-\partial_\beta\log Z=\frac{3\pi}{8G_5}\left(\mu+\frac{L^2}{4}\right),\qquad S=(1-\beta\partial_\beta)\log Z=\frac{\pi^2 r_+^3}{2G_5},$$

so $E$ splits into the black-hole mass plus the $S^3$ Casimir energy, and $dE=T\,dS$ holds.

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5. Physical picture

• $T<T_{\rm HP}$: thermal AdS dominates (no horizon entropy), system behaves as a gas of singlet excitations; a black hole, if formed, evaporates back to thermal AdS.
• $T>T_{\rm HP}$: the large AdS black hole dominates; the system has $S\sim \mathcal O(L^3/G_5)$ and is thermodynamically stable.
• Contrast with asymptotically flat space: there the Schwarzschild heat capacity is negative and no canonical equilibrium exists; AdS provides a natural “box,” enabling the phase structure.

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6. Generalization to AdS$_{d+1}$ (spherical horizon)

With

$$f(r)=1+\frac{r^2}{L^2}-\frac{\mu}{r^{d-2}},$$ $$T(r_+)=\frac{1}{4\pi r_+}\Big[(d-2)+d\,\frac{r_+^2}{L^2}\Big],\qquad S=\frac{\Omega_{d-1}\,r_+^{d-1}}{4G_{d+1}}.$$

The HP point is always at $r_+=L$ with

$$T_{\rm HP}=\frac{d-1}{2\pi L},\qquad r_x=L\sqrt{\frac{d-2}{d}}\ \text{ at which }T\text{ is minimal.}$$

The free-energy difference (vs thermal AdS) is

$$\Delta F=\frac{\Omega_{d-1}}{16\pi G_{d+1}L^2}\,r_+^{\,d-2}\,(L^2-r_+^2).$$

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7. AdS/CFT interpretation

Via $Z_{\rm grav}[S^1_\beta\times S^{d-1}]\simeq Z_{\rm CFT}[S^1_\beta\times S^{d-1}]$ at large $N$ and strong coupling:

• Thermal AdS $\leftrightarrow$ confined phase.
  $F\sim \mathcal O(1)$ on $S^{d-1}$; sparse spectrum of singlets. Polyakov loop vanishes in theories with a center.
• Large AdS BH $\leftrightarrow$ deconfined phase.
  $F\sim \mathcal O(N^2)$; many color degrees of freedom excited. Polyakov loop nonzero; center symmetry broken.

Planar limit. On $\mathbb R^{d-1}$ the dual bulk solution is a black brane; its free-energy density is negative for all $T>0$, so there is no HP transition (the CFT is “deconfined” at any nonzero $T$ in infinite volume).

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9. Exercises

1. Euclidean regularity: starting from the near-horizon expansion, re-derive $T=\tfrac{f'(r_+)}{4\pi}$ by requiring the absence of a conical defect.
2. Thermodynamic consistency: using $I_E$, verify that $E=-\partial_\beta\log Z$ and $S=(1-\beta\partial_\beta)\log Z$ reproduce the ADM mass (including the $S^3$ Casimir) and $S=A/4G_5$.
3. Cutoff matching: at $r=R$, match the proper thermal circles of the BH and thermal AdS and show explicitly that background subtraction yields the same $\Delta F$.
4. General $d$: from $f(r)=1+\frac{r^2}{L^2}-\frac{\mu}{r^{d-2}}$, derive $T_{\rm HP}=(d-1)/(2\pi L)$ and the location of $T_{\min}$.

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10. References

• S. W. Hawking and D. N. Page, Thermodynamics of Black Holes in Anti–de Sitter Space, Commun. Math. Phys. 87 (1983) 577–588.
• J. M. Maldacena, The Large $N$ Limit of Superconformal Field Theories and Supergravity, Adv. Theor. Math. Phys. 2 (1998) 231.
• E. Witten, Anti–de Sitter Space, Thermal Phase Transition, and Confinement in Gauge Theories, Adv. Theor. Math. Phys. 2 (1998) 505.
• For counterterms and holographic renormalization: V. Balasubramanian and P. Kraus, Commun. Math. Phys. 208 (1999) 413; K. Skenderis, Class. Quant. Grav. 19 (2002) 5849.