3.4.2 Black Hole Temperature#

Prompts

What is the Schwarzschild metric, and why is the singularity at the horizon \(r_s = 2GM/c^2\) a coordinate artifact rather than a true curvature singularity?
What does the Wick rotation \(t \to -\mathrm{i}\tau\) do to the Schwarzschild line element? Why is the resulting Euclidean geometry the natural object for asking thermal questions about a black hole?
Near the Schwarzschild horizon, why does the Euclidean \((\tau, r)\) geometry reduce to flat 2D space in polar coordinates? What plays the role of the radius and the angle?
Why does the requirement that the Euclidean horizon be smooth (no conical singularity) fix the period of imaginary time to \(\Delta\tau = 8\pi GM/c^3\)?
How does identifying the imaginary-time period with \(\hbar\beta\) deliver the Hawking temperature \(T_H = \hbar c^3/(8\pi G M k_B)\)? Why are larger black holes colder than smaller ones?

Lecture Notes#

Overview#

Black holes have a temperature. Hawking’s original discovery used quantum fields on curved spacetime, but the Gibbons-Hawking argument reveals the same temperature from a cleaner geometric idea. After Wick rotation, the black-hole horizon becomes the origin of a Euclidean polar coordinate system. Smoothness at that origin forbids a conical singularity, so the imaginary-time direction must have one specific period. Since §3.4.1 taught us that the period of imaginary time is inverse temperature, the geometry itself tells us the black hole’s temperature: larger black holes have longer imaginary-time periods and are therefore colder.

The Schwarzschild Metric#

Outside a static, spherically symmetric, uncharged black hole of mass \(M\), spacetime is described by the Schwarzschild metric

(129)#\[ \mathrm{d}s^2 \;=\; -f(r)\,c^2\,\mathrm{d}t^2 \;+\; \frac{\mathrm{d}r^2}{f(r)} \;+\; r^2\,\mathrm{d}\Omega^2,\qquad f(r) \;=\; 1 - \frac{r_s}{r}, \]

where \(r_s \equiv 2GM/c^2\) is the Schwarzschild radius, \(\mathrm{d}\Omega^2 = \mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\varphi^2\) is the round metric on the unit 2-sphere, and \(f(r)\) is the lapse function. The surface \(r = r_s\) is the event horizon: the lapse vanishes there, \(g_{tt} = -f c^2 \to 0\), and the radial component \(g_{rr} = 1/f\) blows up. The horizon is not a true curvature singularity — geometric invariants like \(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\) remain finite there. The blow-up at \(r = r_s\) is a coordinate artifact, no more physical than the apparent singularity at the north pole of a longitude/latitude chart.

We will not derive (129); we treat it as a fixed background and only need two facts:

\(f(r_s) = 0\) (horizon).
\(f(r)\) is smooth and positive for \(r > r_s\).

Wick Rotation of the Metric#

Apply the Wick rotation \(t \to -\mathrm{i}\tau\) from §3.4.1. Then \(\mathrm{d}t = -\mathrm{i}\,\mathrm{d}\tau\) and \(\mathrm{d}t^2 = -\mathrm{d}\tau^2\), so the temporal sign flips:

(130)#\[ \mathrm{d}s_E^2 \;=\; +f(r)\,c^2\,\mathrm{d}\tau^2 \;+\; \frac{\mathrm{d}r^2}{f(r)} \;+\; r^2\,\mathrm{d}\Omega^2. \]

The signature changes from Lorentzian \((-,+,+,+)\) to Euclidean \((+,+,+,+)\); this is the Euclidean Schwarzschild geometry. Outside the horizon (\(r > r_s\)) it is a perfectly ordinary positive-definite Riemannian metric on \(\mathbb{R}_\tau\times(r_s,\infty)\times S^2\). The interesting question is what happens at the horizon \(r = r_s\).

Near-Horizon Geometry#

Near the horizon, the Schwarzschild radial coordinate \(r\) is not the best variable: the metric component \(\mathrm{d}r^2/f(r)\) looks singular even though the horizon itself is smooth. Use instead the proper distance from the horizon,

(131)#\[ \rho \;\equiv\; \int_{r_s}^{r}\frac{\mathrm{d}r'}{\sqrt{f(r')}}. \]

In this coordinate, the near-horizon \((\tau,r)\) part of the Euclidean metric becomes

(132)#\[ \mathrm{d}s_E^2\big\vert_{\mathrm{near}} \;=\; \mathrm{d}\rho^2 \;+\; \rho^2\,\mathrm{d}\theta^2, \qquad \theta \;\equiv\; \frac{c\,\tau}{2\,r_s}. \]

This is just flat 2D Euclidean space in polar coordinates: \(\rho\) is the radius and \(\theta\) is the angle. The horizon is the origin \(\rho=0\).

Derivation: Schwarzschild horizon to polar coordinates

Write \(r=r_s+\epsilon\) with \(\epsilon\ll r_s\). To first order near the horizon,

(133)#\[ f(r)=1-\frac{r_s}{r}\approx \frac{\epsilon}{r_s}. \]

The proper distance is then

\[ \rho =\int_0^\epsilon \sqrt{\frac{r_s}{\epsilon'}}\,\mathrm{d}\epsilon' =2\sqrt{r_s\epsilon}. \]

Thus

(134)#\[ \rho \;=\; 2\sqrt{r_s\,\epsilon},\qquad \epsilon \;=\; \frac{\rho^2}{4r_s},\qquad f(r) \;=\; \frac{\rho^2}{4r_s^2}. \]

Now keep only the \((\tau,r)\) block of (130):

\[ f(r)c^2\,\mathrm{d}\tau^2+\frac{\mathrm{d}r^2}{f(r)}. \]

By definition of \(\rho\), the radial part is \(\mathrm{d}\rho^2\). Using (134), the temporal part is

\[ f(r)c^2\,\mathrm{d}\tau^2 =\rho^2\left(\frac{c\,\mathrm{d}\tau}{2r_s}\right)^2. \]

Defining \(\theta=c\tau/(2r_s)\) gives the polar form (132). The angular sphere \(r^2\mathrm{d}\Omega^2\) stays finite at \(r=r_s\) and does not affect the smoothness condition.

No Conical Singularity → Periodicity#

A flat polar metric \(\mathrm{d}\rho^2 + \rho^2\,\mathrm{d}\theta^2\) describes a smooth 2D plane only if the angular coordinate \(\theta\) has period exactly \(2\pi\). If the period were any other value \(\Delta\theta\ne 2\pi\), the geometry at \(\rho = 0\) would be a cone with a sharp tip carrying a delta-function curvature: a conical singularity, of the kind you create by cutting a wedge out of a sheet of paper and gluing the cut edges together (a coffee filter).

The vacuum Schwarzschild geometry is smooth at the horizon (curvature invariants are finite there), and the Euclidean continuation must respect that. Therefore the apparent singular point \(\rho = 0\) in our polar chart must really be just an axis of polar coordinates, not a genuine conical defect. This forces \(\theta\) to have period \(2\pi\).

Smoothness at the horizon forces periodicity

The angle \(\theta = c\tau/(2 r_s)\) in the near-horizon polar form (132) has period \(2\pi\) if and only if imaginary time has period

(135)#\[ \Delta\tau \;=\; \frac{4\pi r_s}{c} \;=\; \frac{8\pi G M}{c^3}. \]

Any other value would produce a conical singularity at the horizon — physically inadmissible.

The Euclidean black-hole geometry is therefore not a 4D Euclidean space at all: it is periodic in imaginary time, with a definite period set by the mass.

The Hawking Temperature#

The Euclidean black-hole geometry now satisfies exactly the same trace condition as the partition function of §3.4.1: paths of any quantum field on this background are periodic in \(\tau\) with period \(\hbar\beta\). Identifying

\[ \hbar\beta \;=\; \Delta\tau \;=\; \frac{8\pi G M}{c^3} \]

solves for the temperature:

Hawking temperature

A Schwarzschild black hole of mass \(M\) has temperature

(136)#\[ T_H \;=\; \frac{\hbar\,c^3}{8\pi\,G\,M\,k_B}. \]

Larger black holes are colder; smaller ones are hotter.

For a solar-mass black hole, \(T_H \approx 6\times 10^{-8}\,\mathrm{K}\) — colder than the cosmic microwave background (\(\approx 2.7\,\mathrm{K}\)), so an astrophysical black hole absorbs far more than it emits. For a primordial black hole of mass \(\sim 10^{12}\,\mathrm{kg}\), \(T_H \sim 10^{11}\,\mathrm{K}\), hot enough to radiate gamma rays and approach explosive evaporation.

What this derivation does and does not show

We have computed the temperature that the Euclidean path integral assigns to a black hole. We have not derived the Hawking radiation itself — the actual stream of particles an observer at infinity detects. That requires quantum field theory on the Schwarzschild background (Hawking, 1974), which shows the black hole emits a thermal flux at exactly the temperature (136). Two completely different calculations — quantizing fields on a curved spacetime, versus demanding smoothness of a Euclidean geometry — agree on \(T_H\). The agreement is strong evidence that the Euclidean approach captures real physics.

Black Hole Thermodynamics#

A black hole with a temperature is a thermodynamic object. Once \(T_H\) is accepted, the first law \(\mathrm{d}E = T\,\mathrm{d}S\) (with \(E = Mc^2\)) integrates to assign the black hole an entropy proportional to its horizon area,

(137)#\[ S_{\mathrm{BH}} \;=\; \frac{k_B\,c^3\,A}{4\,G\,\hbar},\qquad A = 4\pi r_s^2, \]

the famous Bekenstein–Hawking entropy. Three threads then unspool:

The information paradox. A thermodynamic black hole radiates; if the radiation is exactly thermal, what happens to the quantum information that fell in?
Holography. Entropy scaling with area (not volume) suggests the black hole’s degrees of freedom live on a lower-dimensional surface — the seed of the holographic principle and the AdS/CFT correspondence.
Quantum gravity. A correct theory of quantum gravity must reproduce \(S_{\mathrm{BH}}\) from a microscopic count of states. Certain stringy black holes do; the full picture remains open.

Discussion: is the Euclidean geometry physical?

The Wick-rotated Schwarzschild geometry is not a spacetime any observer ever experiences — there is no Lorentzian time, no causal structure, just a smooth 4D Riemannian manifold with one periodic direction. Yet it produces the right temperature for a real, observable phenomenon (Hawking radiation), and it does so through nothing more than a smoothness requirement. What does it mean for a geometry that no observer can be in to predict observable physics? Is this a deep statement about the structure of quantum gravity, or a calculational accident that happens to land on the right number?

Poll: why must \(\theta\) have period \(2\pi\)?

In the near-horizon Euclidean Schwarzschild metric \(\mathrm{d}s_E^2 = \mathrm{d}\rho^2 + \rho^2\,\mathrm{d}\theta^2\), smoothness at \(\rho = 0\) requires \(\theta\) to have period exactly \(2\pi\). What goes wrong if the period is anything else?

(A) The metric becomes Lorentzian instead of Euclidean.

(B) A conical singularity appears at \(\rho = 0\), with delta-function curvature concentrated at the tip — incompatible with a smooth vacuum geometry.

(D) Causality is violated.

Summary#

Schwarzschild metric: vacuum geometry outside a non-rotating black hole; horizon at \(r_s = 2GM/c^2\); lapse \(f = 1 - r_s/r\) vanishes there.
Euclidean Schwarzschild: Wick rotation \(t\to -\mathrm{i}\tau\) produces a positive-definite metric, smooth for \(r > r_s\).
Near-horizon polar form: in proper-distance coordinates \(\rho\), the geometry near the horizon is flat 2D Euclidean space with angular coordinate \(\theta = c\tau/(2 r_s)\).
Smoothness ⇒ periodicity: the no-conical-singularity condition forces \(\tau\) to have period \(\Delta\tau = 8\pi GM/c^3\).
Hawking temperature: identifying \(\Delta\tau = \hbar\beta\) gives \(T_H = \hbar c^3/(8\pi G M k_B)\) — the same result Hawking obtained from QFT on a curved background.
Bekenstein–Hawking entropy: integrating the first law yields \(S_{\mathrm{BH}} = k_B c^3 A/(4 G\hbar)\), an entropy proportional to horizon area.
Profound consequences: black-hole thermodynamics, the information paradox, and the holographic principle — all founded on the same periodic imaginary time as §3.4.1.

Homework#

1. Schwarzschild lapse at the horizon. For the Schwarzschild metric (129):

(a) Verify \(f(r_s) = 0\) at \(r_s = 2GM/c^2\) and that \(g_{tt} = -f c^2\) vanishes while \(g_{rr} = 1/f\) diverges there.

(b) Compute the curvature scalar \(R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\) (look up or quote the standard result for Schwarzschild) and confirm it stays finite at \(r = r_s\), so the horizon is a coordinate, not curvature, singularity.

2. Wick rotation of the metric. Apply \(t \to -\mathrm{i}\tau\) to the Schwarzschild line element (129) and verify that the resulting line element (130) has Euclidean signature \((+,+,+,+)\) for \(r > r_s\). Identify which sign change(s) come from \(\mathrm{d}t^2 \to -\mathrm{d}\tau^2\).

3. Proper distance integral. Near the horizon, \(f(r) \approx (r - r_s)/r_s\). Compute the proper distance

\[ \rho \;=\; \int_{r_s}^{r}\frac{\mathrm{d}r'}{\sqrt{f(r')}} \]

explicitly using the linearized \(f\), and verify \(\rho = 2\sqrt{r_s\,(r - r_s)}\). Invert to obtain \(r - r_s = \rho^2/(4 r_s)\) and \(f = \rho^2/(4 r_s^2)\).

4. Reduction to polar coordinates. Starting from (130), change variables from \(r\) to \(\rho\) using HW 3.4.2.3 and verify that the \((\tau, r)\) block reduces to \(\mathrm{d}\rho^2 + \rho^2\,\mathrm{d}\theta^2\) with \(\theta = c\tau/(2 r_s)\). State explicitly which approximation (linearization of \(f\)) is used and where it breaks down.

5. Conical defect angle. Suppose imaginary time is given some period \(\Delta\tau \ne 8\pi GM/c^3\). The angular variable \(\theta = c\tau/(2 r_s)\) then has period \(\Delta\theta = c\,\Delta\tau/(2 r_s)\).

(a) Compute the deficit angle \(\delta = 2\pi - \Delta\theta\) in terms of \(\Delta\tau\).

(b) The Gauss–Bonnet theorem assigns a delta-function curvature \(\delta\cdot\delta^{(2)}(\rho)\) to a cone of deficit angle \(\delta\). Argue that any \(\Delta\tau \ne 8\pi GM/c^3\) produces a non-vanishing curvature at \(\rho = 0\) — incompatible with the vacuum Einstein equations \(R_{\mu\nu} = 0\) at the (otherwise smooth) horizon.

6. Hawking temperature: numbers. Compute \(T_H\) for three black holes:

(a) A solar-mass black hole, \(M = 1\,M_\odot = 2\times 10^{30}\,\mathrm{kg}\).

(b) The supermassive black hole at the center of the Milky Way, Sagittarius A\(^*\), \(M \approx 4\times 10^6\,M_\odot\).

In each case compare \(T_H\) to (i) the cosmic microwave background temperature \(T_{\mathrm{CMB}} \approx 2.7\,\mathrm{K}\), and (ii) the surface temperature of the Sun. Comment on which black holes net-emit and which net-absorb radiation in our universe today.

7. Hawking lifetime estimate. Treat a black hole as a blackbody radiating at \(T_H\) from horizon area \(A = 4\pi r_s^2\).

(a) Use Stefan’s law \(L = \sigma A T_H^4\) with \(\sigma\) the Stefan–Boltzmann constant. Show \(L \propto 1/M^2\), so smaller black holes radiate more powerfully.

(b) From \(\mathrm{d}(Mc^2)/\mathrm{d}t = -L\), derive a differential equation for \(M(t)\) and integrate to obtain the lifetime \(\tau_{\mathrm{life}} \propto M_0^3\).

(c) Estimate \(\tau_{\mathrm{life}}\) for a solar-mass black hole and for the primordial black hole of HW 3.4.2.6(c). Comment on the cosmological implication: which primordial black holes (if any) would be evaporating today?

8. Bekenstein–Hawking entropy. Treat \(E = Mc^2\) as the black hole’s energy and use the first law \(\mathrm{d}E = T_H\,\mathrm{d}S\):

(a) Derive \(\mathrm{d}S/\mathrm{d}M = 8\pi G M k_B/(\hbar c)\).

(b) Integrate from \(M = 0\) to obtain the Bekenstein–Hawking entropy (137): \(S_{\mathrm{BH}} = k_B c^3 A/(4 G\hbar)\) with \(A = 4\pi r_s^2\).

(c) Estimate \(S_{\mathrm{BH}}\) for a solar-mass black hole and compare to the entropy of the Sun (\(\sim 10^{58}\,k_B\)). What does the comparison say about the information content of a stellar collapse?