13.09.10 · gr-cosmology / microlocal-qft-curved-spacetimes

Hartle-Hawking and Unruh states on Schwarzschild

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Anchor (Master): Hartle and Hawking, *Phys. Rev. D* 13 (1976) 2188; Israel, *Phys. Lett. A* 57 (1976) 107; Kay and Wald, *Phys. Rep.* 207 (1991) 49; Wald, *Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics* (Chicago, 1994), Ch. 5 and 8; Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 10

Intuition Beginner

A black hole is not quite black. Stephen Hawking discovered in 1974 that a black hole glows with a faint thermal radiation, at a temperature fixed by its mass: the lighter the hole, the hotter it shines. To make this precise, one has to say which quantum state the field around the hole is in, because the same Schwarzschild spacetime supports several inequivalent vacua. They disagree about what an empty region looks like, and only one of them describes a real evaporating black hole.

There are three states worth naming. The Boulware state is the honest static vacuum: it is what you get if you demand the field look empty to an observer standing still far from the hole. It works fine far away, but it goes badly wrong at the horizon — the energy density an infalling observer measures blows up there. So the Boulware state describes the outside of a static star, never a black hole with a horizon.

The Hartle-Hawking state is the equilibrium state. Picture the black hole sealed inside a perfectly reflecting box, sitting in a bath of its own radiation at exactly the Hawking temperature, with as much falling in as comes out. This state is smooth and well-behaved everywhere, including at the horizon, because nothing special is happening there — the hole is in balance with its surroundings. It is the black-hole version of a room that has reached a uniform temperature.

The Unruh state is the realistic one. It models a black hole that actually formed by collapse and is now radiating into empty space, with nothing falling back in. An observer far away sees a steady outgoing stream of thermal particles — the Hawking radiation — draining energy from the hole. This state is smooth where it needs to be, at the future horizon that infalling matter crosses, but not at the past horizon, which a collapse-formed hole does not really have.

The way to tell these three apart, made sharp by mathematicians, is a smoothness test at the horizon called the Hadamard condition. It asks whether the short-distance behaviour of the quantum field near any point looks like flat empty space. The Boulware state fails this test on both horizons, the Unruh state fails it on the past horizon, and the Hartle-Hawking state passes everywhere. Smoothness at the horizon is the dividing line between a sensible state and a singular one.

Visual Beginner

The picture to hold is the Kruskal diagram of the full Schwarzschild spacetime: a spacetime square divided by two diagonal horizon lines into four regions — our exterior universe on the right, a mirror exterior on the left, a black-hole interior on top, and a white-hole interior on the bottom.

Three features carry the meaning. The two crossing diagonals are the horizons, meeting at the central bifurcation point — the analogue of the origin in the flat-space wedge picture. The right wedge is the universe we live in, outside the hole; its hyperbolae are the worldlines of observers who hover at a fixed distance, much like the accelerated worldlines of the Rindler wedge. The Killing flow, drawn as arrows along these hyperbolae, is the static time translation that the Hartle-Hawking state turns into a thermal clock.

The remarkable content is that the three states fill in the quantum field differently across these horizons. The Hartle-Hawking state is the one smooth across the whole figure; the Unruh state matches it on the future horizon but not the past; the Boulware state matches neither. The crossing point and the two diagonals are where all the physics lives.

Worked example Beginner

Compute the Hawking temperature and the imaginary-time period of a Schwarzschild black hole, for a concrete mass, and check that the two ways of getting the temperature agree.

Step 1. State the surface gravity. A Schwarzschild black hole of mass $M$ has a horizon at radius $r = 2 M$ (in units where the gravitational constant and the speed of light are one). Its surface gravity, the quantity that measures how strongly the horizon pulls, is $$ \kappa = \frac{1}{4M}. $$ A smaller mass means a larger surface gravity: small black holes have fierce horizons.

Step 2. Get the temperature from the surface gravity. The Hawking temperature is the surface gravity divided by two times $π$ : $$ T_H = \frac{\kappa}{2\pi} = \frac{1}{8\pi M}. $$ This is the same $2 π$ that appears in the Unruh formula for an accelerated observer, because a black-hole horizon looks locally like the accelerated-observer wedge.

Step 3. Get the temperature a second way, from imaginary time. Replace ordinary time by imaginary time and ask how the geometry behaves near the horizon. The region just outside $r = 2 M$ becomes a flat plane in polar form, with imaginary time as an angle. A plane is smooth at its centre only if the angle turns a full $2 π$ . Matching the angle to imaginary time forces it to repeat with period $$ \beta = \frac{2\pi}{\kappa} = 8\pi M. $$ A field whose imaginary time repeats with period $β$ is thermal at temperature $1/ β$ , so $T_{H} = 1/ (8 π M)$ — the same answer.

Step 4. Plug in a number. For a black hole of one solar mass, $M$ is about $1.5$ kilometres in these geometric units, and converting to ordinary temperature gives $T_{H} \approx 6 \times 1 0^{- 8}$ kelvin — sixty billionths of a degree, far colder than empty space. A black hole the mass of a mountain, around $1 0^{12}$ kilograms, would instead glow at billions of degrees and evaporate quickly.

What this tells us: the equilibrium temperature of the Hartle-Hawking state and the period of imaginary time are two faces of one fact. The horizon forces imaginary time to be periodic, and a periodic imaginary time is a temperature. The same number, $κ / (2 π)$ , governs both the thermal bath the Hartle-Hawking state sits in and the radiation the Unruh state pours out to infinity.

Check your understanding Beginner

Exercise (easy, multiple choice).

Which of the three Schwarzschild states is smooth (Hadamard) everywhere, including on both horizons of the Kruskal extension?

A. The Boulware state B. The Unruh state C. The Hartle-Hawking state D. None of them is smooth on the horizons

Hint

The state that is smooth everywhere is the equilibrium state, the one describing a black hole in balance with a surrounding thermal bath.

Answer

C. The Hartle-Hawking state.

The Hartle-Hawking state is constructed precisely to be regular on the whole Kruskal manifold, including both horizons; it is the thermal-equilibrium state at the Hawking temperature. The Boulware state is singular on both horizons (option A is wrong), and the Unruh state is regular on the future horizon but singular on the past one (option B is wrong). Because the Hartle-Hawking state passes the smoothness test, option D is wrong.

Formal definition Intermediate+

Throughout, $(M, g)$ is the maximally extended Schwarzschild spacetime of mass $M > 0$ , a globally hyperbolic Lorentzian manifold in the sense of 13.09.01, with signature $(-, +, +, +)$ matching 13.09.02. In the exterior region $r > 2 M$ the metric in Schwarzschild coordinates is $$ g = -f(r),dt^2 + f(r)^{-1},dr^2 + r^2,d\Omega^2, \qquad f(r) = 1 - \frac{2M}{r}, $$ with $d Ω^{2}$ the round metric on $S^{2}$ . The static Killing field is $χ = \partial_{t}$ . The coordinate singularity at $r = 2 M$ is removed by passing to Kruskal-Szekeres coordinates $(U, V)$ , defined in the exterior by $$ U = -e^{-\kappa(t - r_*)}, \qquad V = e^{\kappa(t + r_*)}, \qquad r_* = r + 2M\ln!\left(\frac{r}{2M} - 1\right), $$ where $κ$ is the surface gravity introduced below and $r_{*}$ is the tortoise coordinate, in which the metric becomes $$ g = -\frac{32 M^3}{r},e^{-r/2M},dU,dV + r^2,d\Omega^2, $$ manifestly regular at $r = 2 M$ , i.e. at $U V = 0$ . The bifurcate Killing horizon is the locus $U V = 0$ : the future horizon $H^{+} = {U = 0}$ , the past horizon $H^{-} = {V = 0}$ , and the bifurcation surface $B = {U = V = 0}$ where $χ$ vanishes. The Killing field acts as the boost $U \mapsto e^{- κ s} U$ , $V \mapsto e^{κ s} V$ , with flow parameter $s = κ t$ in the exterior.

The surface gravity $κ$ is defined by the horizon-generator relation $$ \chi^a\nabla_a\chi^b = \kappa,\chi^b \quad\text{on } H^+, $$ equivalently $\nabla^{a} (χ^{b} χ_{b}) = - 2 κ χ^{a}$ on the horizon. For Schwarzschild a direct computation gives $$ \kappa = \frac{1}{2}f'(2M) = \frac{1}{4M}. $$

A quasi-free state for the Klein-Gordon field on $(M, g)$ is determined by its two-point function $Λ \in D^{'} (M \times M)$ , a bisolution of the field equation with antisymmetric part fixed by the causal propagator. The state is Hadamard ^{[Wald 1994]} when the wave-front set of $Λ$ satisfies the microlocal spectrum condition of 13.09.03: $$ \mathrm{WF}(\Lambda) = {(x, k; x', -k') : (x,k) \sim (x', k'),\ k \in \overline{V}^+_x}, $$ where $\sim$ is the relation of lying on a common null geodesic with cotangents parallel-transported into each other, and $k$ lies in the future cone. The Hadamard condition is local and is tested in any neighbourhood, in particular neighbourhoods of horizon points.

The three states are defined by their mode content.

Boulware state $ω_{B}$ ^{[Boulware 1975]}. The quasi-free state whose one-particle modes are positive-frequency with respect to the Killing time $t$ , i.e. annihilated by the modes $\propto e^{- iω t}$ with $ω > 0$ . It is static and regular for $r > 2 M$ but its two-point function fails the Hadamard condition on $H^{+}$ and $H^{-}$ .

Hartle-Hawking state $ω_{H H}$ ^{[Hartle-Hawking 1976]}^{[Israel 1976]}. The quasi-free state invariant under $χ$ whose two-point function is the boundary value of a function analytic in the Euclidean section, periodic in imaginary Killing time with period $β = 2 π / κ$ . Equivalently (Israel), it is the thermofield double: the cyclic vector in $H_{I} \otimes H_{II}$ over the two exterior wedges whose reduction to one wedge is the Gibbs state at $T_{H} = κ / (2 π)$ .

Unruh state $ω_{U}$ ^{[Unruh 1976]}. The quasi-free state whose modes are positive-frequency with respect to the affine parameter $U$ on $H^{-}$ and positive-frequency with respect to $t$ at past null infinity $I^{-}$ . It models collapse and carries an outgoing flux at $T_{H}$ .

Counterexamples to common slips

The Hawking temperature is not an independent thermodynamic input. It is forced by the surface gravity through $T_{H} = κ / (2 π)$ , and the $2 π$ is the period of the Euclidean angle, not a separately measured quantity. Changing the period of imaginary time away from $2 π / κ$ introduces a conical singularity at the bifurcation surface and breaks smoothness.
"Hadamard" is a statement about the short-distance singularity of the two-point function, not about the value of the field. All three states solve the same field equation with the same antisymmetric part; they differ only in the symmetric part, and the wave-front set of that symmetric part on the horizon is what separates them. A state can be perfectly finite in the exterior and still be non-Hadamard at the horizon if its mode functions oscillate infinitely as a point approaches $r = 2 M$ in Killing time.
The Boulware state is the literal static vacuum, yet it is the singular one. Demanding that the field look empty to a static observer right down to the horizon is incompatible with smoothness there, because a static observer near the horizon is enormously accelerated and the Unruh effect of 13.09.09 forces a thermal bath in any Hadamard state. Emptiness in Killing time and regularity at the horizon are mutually exclusive at $r = 2 M$ .
The Hartle-Hawking state exists for Schwarzschild but need not exist for a general bifurcate Killing horizon. Kay and Wald ^{[Kay-Wald 1991]} proved that a stationary Hadamard state invariant under the Killing flow is unique if it exists, but for Kerr the horizon-generating Killing field is spacelike in the ergoregion and no globally Hadamard equilibrium state exists. Schwarzschild is the favourable case where the construction goes through.

Key theorem with proof Intermediate+

Theorem (the Hawking temperature from conical smoothness; the Hartle-Hawking state is KMS at $β = 8 π M$ ). Let $(M, g)$ be the Kruskal extension of Schwarzschild with surface gravity $κ = 1/ (4 M)$ . The unique quasi-free Hadamard state invariant under the horizon Killing flow $χ = \partial_{t}$ , the Hartle-Hawking state $ω_{H H}$ , satisfies the KMS condition at inverse temperature $β = 2 π / κ = 8 π M$ with respect to the Killing flow $\beta_s(A) = U(s),A,U(s)^ $,$ s = \kappa t $. T h e p er i o d o f ima g ina r y K i l l in g t im e i s f i x e d t o$ \beta$ by requiring the Euclidean section to be free of conical singularities at the bifurcation surface* ^{[Hartle-Hawking 1976]}^{[Gibbons-Hawking 1977]}.

Proof. The argument has two parts: the geometric part fixing the imaginary-time period from horizon smoothness, and the algebraic part identifying periodicity with the KMS condition.

Part 1: near-horizon geometry and the conical-deficit constraint. Expand the metric near $r = 2 M$ . Write $r = 2 M + ξ$ with $ξ$ small. Then $f (r) = 1 - 2 M / r = ξ / (2 M) + O (ξ^{2})$ , so $f \approx κ ξ \cdot 2 = ξ / (2 M)$ , using $κ = 1/ (4 M)$ . Introduce the proper radial distance from the horizon, $$ \rho = \int_0^\xi \frac{d\xi'}{\sqrt{f}} = \int_0^\xi \sqrt{\frac{2M}{\xi'}},d\xi' = 2\sqrt{2M,\xi}, \qquad \text{so } \xi = \frac{\rho^2}{8M},\quad f = \frac{\rho^2}{16 M^2} = \kappa^2\rho^2. $$ The exterior metric in the $(t, r)$ plane then reads $$ -f,dt^2 + f^{-1},dr^2 = -\kappa^2\rho^2,dt^2 + d\rho^2 + O(\rho^4), $$ the angular part decoupling. Continue to Euclidean Killing time $t = - i t_{E}$ : $$ \kappa^2\rho^2,dt_E^2 + d\rho^2, $$ which is the flat metric of a plane in polar coordinates $(ρ, θ)$ with angle $θ = κ t_{E}$ . The plane is smooth at the origin $ρ = 0$ — no conical deficit at the bifurcation surface — exactly when $θ$ has period $2 π$ , i.e. when $t_{E}$ has period $$ \beta = \frac{2\pi}{\kappa} = 8\pi M. $$ Any other period leaves a conical singularity at $ρ = 0$ , on which the curvature has a delta function and the field equation is ill-posed. The Hartle-Hawking two-point function, being the boundary value of the Euclidean Green's function on this smooth section, inherits the period $β$ in imaginary Killing time.

Part 2: periodicity in imaginary time is the KMS condition. Let $ω_{H H}$ be the state and $β_{s}$ the Killing flow with real parameter the Killing time $t$ (so $s = t$ here, absorbing $κ$ into the definition of $β = 2 π / κ$ for the $t$ -flow). For field operators $A, B$ localised in the exterior, the two-point function $$ G_{A,B}(t) := \omega_{HH}\big(A,\beta_t(B)\big) $$ extends to a function holomorphic in the strip $- β < Im t < 0$ , because the Euclidean Green's function is analytic in $t_{E} = Im t$ on $(0, β)$ and the Schwinger functions are its restriction. Periodicity of the Euclidean section, $t_{E} \mapsto t_{E} + β$ , together with the reflection that exchanges the two exterior wedges, gives the boundary relation $$ G_{A,B}(t - i\beta) = \omega_{HH}\big(\beta_t(B),A\big), $$ which is exactly the KMS condition at inverse temperature $β$ for the flow $β_{t}$ . Hence $ω_{H H}$ is a $β = 2 π / κ$ KMS state for the Killing flow, thermal at $T_{H} = κ / (2 π) = 1/ (8 π M)$ . $□$

Bridge. This theorem builds toward the modular-theoretic reading of black-hole equilibrium and appears again in 13.09.09, where the identical conical-smoothness period $2 π$ fixes the Unruh temperature of the Rindler wedge. The foundational reason the Hartle-Hawking temperature is $κ / (2 π)$ is the geometric identity that the near-horizon Euclidean section is a smooth plane only at one period: this is exactly the same analytic continuation that, in flat space, turns the boost into the modular operator $Δ = U (Λ_{1} (2 π i))$ . The construction here generalises the flat-space wedge to a curved bifurcate Killing horizon, and the bridge is the Kay-Wald theorem: the horizon-generating Killing field is dual to the Rindler boost, the exterior wedge is dual to the right Rindler wedge, and the Hartle-Hawking state is dual to the Minkowski vacuum. Putting these together, the central insight is that black-hole temperature is not a thermodynamic postulate but a smoothness condition: the surface gravity $κ$ sets the only period of imaginary time compatible with a horizon free of conical defects, and that period is the inverse temperature.

Exercises Intermediate+

Exercise 5 (medium, short-answer).

Explain why the Boulware state, defined as the vacuum of positive-frequency Killing modes $e^{- iω t}$ , fails the Hadamard condition at the horizon, while being perfectly regular far from the black hole.

Hint

Consider what happens to the Killing modes $e^{- iω t}$ expressed in the regular Kruskal coordinate $U \sim - e^{- κ t}$ as a point approaches the horizon $U = 0$ .

Answer

In the exterior, the Killing modes behave as $e^{- iω t}$ , and the regular affine coordinate on the future horizon is $U = - e^{- κ (t - r_{*})}$ , so $t = - κ^{- 1} ln (- U) + \dots$ Near $U = 0$ the modes become $e^{i (ω / κ) l n (- U)}$ , which oscillates infinitely fast as $U \to 0^{-}$ : the mode functions are not smooth across the horizon in the regular Kruskal coordinate. The resulting two-point function has a wave-front set on the horizon that violates the microlocal spectrum condition of 13.09.03 — it carries the wrong-sign frequencies relative to the affine parameter. Far from the hole, $U$ and $t$ differ only by a regular conformal factor and the Killing modes are smooth, so $ω_{B}$ is Hadamard there. Physically, an observer freely falling across the horizon measures a divergent energy density in $ω_{B}$ , the signature of the non-Hadamard singularity. Rubric: full credit for the $e^{i (ω / κ) l n (- U)}$ infinite-oscillation argument plus the horizon-singularity conclusion.

Exercise 6 (hard, short-answer).

The Unruh state is positive-frequency in the affine parameter $U$ on the past horizon, whereas the Hartle-Hawking state is positive-frequency in $U$ on $H^{-}$ and in $V$ on $H^{+}$ . Use this to explain why the Unruh state is Hadamard on the future horizon but not the past, and why this is the correct state for a collapsing star.

Hint

A collapse spacetime has a genuine future horizon but no past horizon; the past horizon of the eternal Kruskal extension is replaced by the collapsing matter and past null infinity. The Hadamard condition is a positive-frequency condition with respect to the affine (Kruskal) parameter at the horizon.

Answer

The Hadamard condition at a horizon requires the two-point function to be positive-frequency with respect to the affine parameter along the horizon generators, because that is the parameter in which the null geodesics are complete and the short-distance structure matches the local vacuum. The Hartle-Hawking state imposes this on both $H^{-}$ (affine $U$ ) and $H^{+}$ (affine $V$ ), so it is Hadamard across the entire bifurcate horizon. The Unruh state imposes affine-positive-frequency only on $H^{-}$ ; on $H^{+}$ it inherits whatever frequency content propagates up from $I^{-}$ in Killing time, which is not affine-positive-frequency there, so the wave-front set on $H^{+}$ is fine but the state is mismatched on the past horizon $H^{-}$ of the eternal extension. For a star that collapses, there is no past horizon at all — that region is occupied by the infalling matter and the asymptotically flat past — so the Unruh prescription (affine on the only physical horizon, Killing-positive-frequency at past infinity where spacetime is nearly flat) is exactly the data a collapse produces. The mismatch on the eternal $H^{-}$ is an artefact of comparing to the full Kruskal manifold; in the physical collapse spacetime the Unruh state is Hadamard everywhere it needs to be, which is the rigorous content proved by Dappiaggi-Moretti-Pinamonti ^{[Dappiaggi-Moretti-Pinamonti 2011]}. Rubric: full credit for the affine-vs-Killing positive-frequency contrast plus the collapse-has-no-past-horizon point.

Exercise 7 (hard, short-answer).

Show that the Hartle-Hawking state, restricted to the exterior wedge, is the thermofield double of Israel: a Gibbs state at $T_{H}$ purified by the second exterior. Sketch why the modular flow of the exterior algebra in this state is the Killing flow, linking to the Bisognano-Wichmann mechanism of 13.09.09.

Hint

The Kruskal extension has two exterior wedges I and II, related by the Killing reflection. The Hartle-Hawking vector lives in $H_{I} \otimes H_{II}$ ; trace out wedge II. Recall that for a wedge with a boost Killing field, Bisognano-Wichmann identifies the boost as the modular flow.

Answer

The Kruskal manifold has two causally disconnected exteriors, I (our universe, $U < 0 < V$ ) and II (the mirror, $V < 0 < U$ ), exchanged by the Killing reflection $(U, V) \mapsto (- U, - V)$ . The Hilbert space factorises as $H_{I} \otimes H_{II}$ , and the Hartle-Hawking vector $∣ HH ⟩$ is the cyclic, separating vector built from the Euclidean path integral over the lower half ( $t_{E} \in (0, β /2)$ ), which prepares an entangled state of the two wedges. Tracing out wedge II leaves the reduced density matrix $ρ_{I} \propto e^{- β H_{I}}$ , where $H_{I}$ generates the Killing flow restricted to wedge I — the Gibbs state at $T_{H} = κ / (2 π)$ . This is precisely Israel's thermofield-double construction ^{[Israel 1976]}. Because $∣ HH ⟩$ is cyclic and separating for the wedge-I algebra, Tomita-Takesaki theory supplies a modular operator $Δ$ ; the analyticity of the two-point function in imaginary Killing time, with period $β$ , forces $Δ^{i t} = U (Killing flow by - β t)$ , exactly as Bisognano-Wichmann gives $Δ^{i t} = U (Λ_{1} (- 2 π t))$ for the Rindler boost 13.09.09. The Killing flow is the modular flow, and the modular flow is universally KMS at $β = 1$ in its own parameter, hence KMS at $β = 2 π / κ$ in Killing time. Rubric: full credit for the thermofield-double reduction plus the modular-flow-equals-Killing-flow identification.

Advanced results Master

The three states sit inside a rigorous structure that fixes their existence, uniqueness, and Hadamard status on the Kruskal manifold. The following sharpen the basic theorem.

Theorem (Kay-Wald uniqueness and non-existence). Let $(M, g)$ be a globally hyperbolic spacetime with a bifurcate Killing horizon generated by a Killing field $χ$ with surface gravity $κ \neq = 0$ . If a quasi-free Hadamard state invariant under the flow of $χ$ exists, it is unique, and on each exterior wedge it is KMS at $T = κ / (2 π)$ with respect to the Killing flow. For Schwarzschild this state exists and is the Hartle-Hawking state. There is no $χ$ -invariant Hadamard state that is also invariant under the wedge reflection unless the two-point function has the precise thermal periodicity $β = 2 π / κ$ ^{[Kay-Wald 1991]}.

The proof combines a deformation argument — pushing any candidate state to the horizon, where the Killing flow acts as a pure boost — with the observation that Hadamard regularity on the horizon forces the restricted state to coincide with the universal "boost-invariant Hadamard" state on a null surface. Uniqueness then follows because a quasi-free state is determined by its restriction to the horizon data. The non-existence half is sharp: for Kerr the horizon generator $χ = \partial_{t} + Ω_{H} \partial_{ϕ}$ becomes spacelike in the ergoregion, the would-be thermal state has no positive-energy ground structure, and no globally Hadamard $χ$ -invariant state exists, which is the Kay-Wald obstruction to a Hartle-Hawking state for rotating black holes.

Theorem (Hadamard classification of the three states). On the Kruskal extension of Schwarzschild, the Boulware state $ω_{B}$ is non-Hadamard on $H^{+} \cup H^{-}$ ; the Unruh state $ω_{U}$ is Hadamard on $H^{+}$ and on the exterior and interior up to $H^{+}$ , but non-Hadamard on $H^{-}$ ; the Hartle-Hawking state $ω_{H H}$ is Hadamard on the entire manifold ^{[Kay-Wald 1991]}^{[Dappiaggi-Moretti-Pinamonti 2011]}. The rigorous construction of $ω_{U}$ and the proof of its Hadamard property on the physically relevant region was completed by Dappiaggi, Moretti, and Pinamonti by a bulk-to-boundary technique: the state is fixed by specifying its restriction to $H^{-}$ as the affine-vacuum two-point function and its restriction to $I^{-}$ as the Killing vacuum, then propagated into the bulk by the causal propagator; a wave-front-set calculus argument verifies the microlocal spectrum condition off the past horizon.

Theorem (stress-energy and the Hawking flux). The renormalised expected stress-energy tensor in the Unruh state, evaluated at future null infinity, is a steady outgoing null flux $$ \langle T_{uu}\rangle_{\omega_U} \xrightarrow{r \to \infty} \frac{\pi}{12},T_H^2 = \frac{\kappa^2}{48\pi} \quad (\text{per polarisation, } 1{+}1\text{ reduction}), $$ corresponding to a luminosity $L \sim ℏ c^{6} / (G^{2} M^{2})$ and the Hawking lifetime $t_{evap} \sim M^{3}$ . The Boulware state has $⟨ T_{uu} ⟩ \to 0$ at infinity (no flux, as befits a static configuration) but a stress-energy diverging as one approaches the horizon in a freely falling frame; the Hartle-Hawking state has equal ingoing and outgoing thermal fluxes (zero net flux) and a finite, regular stress-energy on the horizon. The difference of stress-energies between any two of these states is a finite, state-independent-renormalisation-free quantity, which is how the flux is computed unambiguously.

Theorem (Euclidean Schwarzschild and the partition function). The Euclidean section of Schwarzschild, the "cigar" geometry obtained by Wick-rotating $t$ and removing the region inside $r = 2 M$ , is a smooth complete Riemannian manifold precisely when $t_{E}$ has period $β = 8 π M$ ; its on-shell Einstein-Hilbert action, regularised by subtracting flat space, is $I_{E} = β^{2} / (16 π) = 4 π M^{2}$ , and the saddle-point partition function $ln Z = - I_{E}$ gives entropy $S = (1 - β \partial_{β}) ln Z = 4 π M^{2} = A /4$ ^{[Gibbons-Hawking 1977]}. This Gibbons-Hawking computation derives the Bekenstein-Hawking area law from the same conical-smoothness period that fixes the temperature, tying the Hartle-Hawking state to black-hole thermodynamics.

Synthesis. The Hartle-Hawking state is the foundational reason black-hole thermodynamics is a literal thermodynamics rather than an analogy: it is the equilibrium quantum state whose KMS temperature $T_{H} = κ / (2 π)$ is the temperature in the first law $d M = (κ /8 π) d A$ , and whose Euclidean action delivers the entropy $A /4$ . The central insight is that the discriminator separating equilibrium, collapse, and static-star physics is a single microlocal condition — Hadamard regularity at the horizon — the curved-space image of the positive-energy spectrum condition behind the flat-space Unruh effect. This is exactly the Bisognano-Wichmann mechanism of 13.09.09 on a bifurcate Killing horizon: the horizon-generating Killing field is dual to the Rindler boost, the exterior wedge is dual to the right Rindler wedge, and the Hartle-Hawking state is dual to the Minkowski vacuum, with the conical-smoothness period $2 π / κ$ generalising the $2 π$ of the wedge modular flow.

Putting these together, the three states fill in the quantum field across the horizons of one geometry in three ways, and the same analytic continuation — the boost to imaginary rapidity, the imaginary time to a smooth Euclidean angle — appears again in the modular operator, the conical-deficit constraint, and the Euclidean partition function, so that temperature, entropy, and Hadamard regularity are three faces of the surface gravity $κ$ . The Unruh state identifies the abstract Hawking flux with the stress-energy of the affine-vacuum boundary condition on the past horizon, and the Hartle-Hawking state identifies the thermofield-double equilibrium with the modular flow of the exterior algebra.

Full proof set Master

Proposition (surface gravity of Schwarzschild). For the Schwarzschild metric with $f (r) = 1 - 2 M / r$ and Killing field $χ = \partial_{t}$ , the surface gravity defined by $χ^{a} \nabla_{a} χ^{b} = κ χ^{b}$ on the horizon $r = 2 M$ equals $κ = 1/ (4 M) = \frac{1}{2} f^{'} (2 M)$ .

Proof. The norm of the Killing field is $χ^{a} χ_{a} = g_{tt} = - f (r)$ . The acceleration of the orbits of $χ$ is governed by $\nabla^{b} (χ^{a} χ_{a}) = \nabla^{b} (- f)$ . Using the Killing equation $\nabla_{a} χ_{b} = \nabla_{[a} χ_{b]}$ and the standard identity for a static Killing field, on the horizon $$ \kappa^2 = -\tfrac{1}{2}(\nabla^a\chi^b)(\nabla_a\chi_b)\big|_{r = 2M}. $$ For the diagonal static metric this reduces to $κ = \frac{1}{2} ∣ f^{'} (r) ∣$ evaluated at the horizon, the well-known formula for a static Killing horizon. With $f (r) = 1 - 2 M / r$ , $f^{'} (r) = 2 M / r^{2}$ , so $f^{'} (2 M) = 2 M / (4 M^{2}) = 1/ (2 M)$ , and $κ = \frac{1}{2} \cdot 1/ (2 M) = 1/ (4 M)$ . An equivalent route: the redshifted proper acceleration of a static observer is $a f = \frac{1}{2} f^{'} / f \cdot f = \frac{1}{2} f^{'}$ , whose horizon limit is $κ$ ; both give $1/ (4 M)$ . $□$

Proposition (conical-smoothness period). A two-dimensional Riemannian metric $d ρ^{2} + κ^{2} ρ^{2} d θ^{2}$ , with $ρ \geq 0$ and $θ$ an angular coordinate, is smooth at $ρ = 0$ if and only if $θ$ is periodic with period $2 π / κ$ . Any other period produces a conical singularity with deficit angle $2 π (1 - κ Δ θ /2 π)$ supported at $ρ = 0$ .

Proof. Set $ϕ := κ θ$ . The metric becomes $d ρ^{2} + ρ^{2} d ϕ^{2}$ , the flat Euclidean plane in polar coordinates. This is a smooth manifold at the origin precisely when $ϕ$ is a standard angle of period $2 π$ , i.e. $ϕ \in [0, 2 π)$ with $0$ and $2 π$ identified; in Cartesian coordinates $x = ρ cos ϕ$ , $y = ρ sin ϕ$ the metric is $d x^{2} + d y^{2}$ with no singularity. Period $2 π$ in $ϕ$ means period $2 π / κ$ in $θ$ . If instead $θ$ has period $Δ θ \neq = 2 π / κ$ , then $ϕ$ ranges over $[0, κ Δ θ)$ , and the total angle around $ρ = 0$ is $κ Δ θ \neq = 2 π$ ; the surface is a cone with deficit angle $2 π - κ Δ θ$ . The Gaussian curvature acquires a distributional term $2 π (1 - κ Δ θ /2 π) δ^{(2)} (ρ)$ by the Gauss-Bonnet theorem applied to a small disc around the apex. Smoothness, the absence of this delta function, holds iff $Δ θ = 2 π / κ$ . Applied to Euclidean Schwarzschild with $θ = t_{E}$ this gives $β = 2 π / κ = 8 π M$ . $□$

Proposition (KMS from imaginary-time periodicity). Let $ω$ be a state invariant under a one-parameter flow $β_{t} = Ad (e^{i H t})$ , and suppose every two-point function $G_{A, B} (t) = ω (A β_{t} (B))$ extends holomorphically to the strip $- β < Im t < 0$ , is bounded and continuous on its closure, and satisfies $G_{A, B} (t - i β) = ω (β_{t} (B) A)$ . Then $ω$ is KMS at inverse temperature $β$ for $β_{t}$ .

Proof. Define $F_{A, B} (z) := G_{A, B} (- z)$ . Then $F_{A, B}$ is holomorphic on $0 < Im z < β$ , continuous on the closure. On the lower edge $Im z = 0$ , $F_{A, B} (t) = G_{A, B} (- t) = ω (A β_{- t} (B))$ ; relabelling $t \mapsto - t$ inside the KMS strip (the flow $β_{t}$ is a group, so this is a reparametrisation) gives the standard form $ω (A β_{t} (B))$ on the real axis. On the upper edge, $F_{A, B} (t + i β) = G_{A, B} (- t - i β) = G_{A, B} ((- t) - i β) = ω (β_{- t} (B) A)$ , which after the same relabelling is $ω (β_{t} (B) A)$ . Thus $F_{A, B}$ is holomorphic on the strip with boundary values $ω (A β_{t} (B))$ and $ω (β_{t} (B) A)$ , the defining data of the KMS condition at $β$ . The hypothesis that the Schwinger (Euclidean) functions of $ω_{H H}$ are analytic and periodic with period $β = 2 π / κ$ in imaginary Killing time, established in Part 1 of the Key theorem by the smoothness of the Euclidean section, supplies exactly the strip analyticity and the boundary relation, so $ω_{H H}$ is KMS at $β = 8 π M$ . $□$

Theorem (Hadamard failure of the Boulware state on the horizon), with proof. The Boulware two-point function $Λ_{B}$ violates the microlocal spectrum condition of 13.09.03 at every point of $H^{+} \cup H^{-}$ .

Proof. On the future horizon the regular (affine) coordinate is $U$ , with the Killing time related by $t = - κ^{- 1} ln (- U)$ in the exterior. The Boulware modes $u_{ω} \propto e^{- iω t}$ , expressed in $U$ , become $u_{ω} \propto (- U)^{iω / κ}$ as $U \to 0^{-}$ . The Fourier content of $(- U)^{iω / κ}$ in the affine variable $U$ is a power-law distribution whose wave-front set at $U = 0$ contains both signs of the affine frequency: the function is not the boundary value of a function holomorphic in the lower affine-frequency half-plane only. Concretely, $(- U)^{iω / κ} = e^{(iω / κ) l n (- U)}$ has $WF$ at $U = 0$ equal to ${(0, ξ) : ξ \neq = 0}$ — both $ξ > 0$ and $ξ < 0$ — because the logarithmic phase oscillates without a definite affine-frequency sign. The microlocal spectrum condition demands that the singular support of the two-point function carry only the future-pointing (positive affine-frequency) covectors along the horizon generators; the Boulware state carries both, so $WF (Λ_{B})$ at horizon points includes the forbidden past-pointing covectors. Hence $Λ_{B}$ is non-Hadamard on $H^{+}$ ; the identical argument with $V$ on $H^{-}$ gives non-Hadamard there. The physical corollary is that $⟨ T_{ab} ⟩_{ω_{B}}$ , computed by the Hadamard point-splitting subtraction, diverges in a freely falling orthonormal frame as the horizon is approached, since the subtraction is calibrated to a Hadamard reference the state does not match. $□$

Connections Master

Hadamard states via the wave-front-set criterion 13.09.03. The microlocal spectrum condition is the exact instrument that separates the three states. The Boulware state carries both signs of affine frequency on the horizon generators and fails the criterion on $H^{+} \cup H^{-}$ ; the Unruh state fails it on $H^{-}$ only; the Hartle-Hawking state satisfies it on the whole Kruskal manifold. The Dappiaggi-Moretti-Pinamonti construction of the Unruh state is a wave-front-set calculation in the sense of 13.09.03, propagating boundary data from $H^{-}$ and $I^{-}$ into the bulk and verifying the singularity structure. The Hadamard condition is therefore not a technical nicety but the physical discriminator between equilibrium, collapse, and static-star states.
Unruh effect via the Bisognano-Wichmann theorem 13.09.09. The Hartle-Hawking state is the curved-space image of the Minkowski vacuum, and the exterior Schwarzschild wedge is the image of the right Rindler wedge. The horizon-generating Killing field $χ = \partial_{t}$ plays the role of the Rindler boost, the surface gravity $κ$ plays the role of the proper acceleration $a$ , and the modular flow of the exterior-algebra in the Hartle-Hawking state is the Killing flow, giving a KMS state at $T_{H} = κ / (2 π)$ by the same Tomita-Takesaki argument. The conical-smoothness period $2 π / κ$ here is the $2 π$ of the wedge modular operator $Δ = U (Λ_{1} (2 π i))$ there.
Hawking radiation 13.06.04. The outgoing thermal flux at future null infinity in the Unruh state is the Hawking radiation. The Bogoliubov-coefficient Planck spectrum derived in 13.06.04 is the late-time mode content of the state that is positive-frequency in the affine parameter on the past horizon, and the luminosity $L \sim M^{- 2}$ and evaporation time $t_{evap} \sim M^{3}$ are properties of the Unruh-state stress-energy. This unit supplies the state-theoretic home of the Hawking effect and identifies its temperature with the surface gravity through the conical-smoothness period.
Black-hole thermodynamics and the area theorem 13.06.03. The Hartle-Hawking state is the equilibrium state whose KMS temperature $T_{H} = κ / (2 π)$ is the temperature in the first law $d M = (κ /8 π) d A$ , and whose Euclidean Gibbons-Hawking action yields the entropy $S = A /4$ . The quantum state constructed here is what upgrades the thermodynamic analogy of 13.06.03 — area as entropy, surface gravity as temperature — into a literal thermodynamics with a partition function and a heat bath.
Globally hyperbolic Lorentzian manifolds 13.09.01. The entire construction rests on the global causal structure of the Kruskal extension as a globally hyperbolic spacetime in the sense of 13.09.01: the existence of a Cauchy surface guarantees a well-posed Klein-Gordon Cauchy problem and a unique causal propagator, the bifurcate Killing horizon is a feature of this global structure, and the comparison of states across $H^{+}$ and $H^{-}$ is meaningful only because the maximal extension is causally complete. Without global hyperbolicity neither the quasi-free states nor their horizon restrictions would be well-defined.

Historical & philosophical context Master

The three Schwarzschild states crystallised in the two years following Hawking's 1974/1975 discovery of black-hole radiance. Hawking's original derivation ^{[Hawking 1975]} computed the Bogoliubov coefficients between the past and future modes of a star collapsing to a black hole and found a Planck spectrum at $T_{H} = κ / (2 π)$ . The result raised an immediate question: which quantum state is the field actually in? David Boulware ^{[Boulware 1975]} examined the natural static vacuum, defined by positive frequency with respect to the Killing time, and showed that its stress-energy diverges at the horizon in a freely falling frame, so it cannot describe a real black hole. James Hartle and Stephen Hawking ^{[Hartle-Hawking 1976]} then constructed, by a Euclidean path integral over the section periodic in imaginary time, a state regular on the entire Kruskal manifold; Werner Israel ^{[Israel 1976]} independently recognised it as the thermofield double, the entangled vacuum of the two Kruskal exteriors whose reduction is a Gibbs state. William Unruh ^{[Unruh 1976]}, in the same paper that introduced the Unruh effect, defined the state appropriate to genuine collapse, positive-frequency in affine time on the past horizon, and showed it carries the steady outgoing flux.

The rigorous, observer-independent foundation came from the algebraic and microlocal communities. Geoffrey Sewell ^{[Sewell 1982]} identified the horizon Killing flow as the Tomita-Takesaki modular flow of the exterior-wedge algebra in the Hartle-Hawking state, transporting the Bisognano-Wichmann mechanism to the curved setting. Bernard Kay and Robert Wald ^{[Kay-Wald 1991]}, in a long Physics Reports memoir, proved that a stationary Hadamard state invariant under the horizon Killing flow is unique if it exists, classified the Hadamard failures of the Boulware and Unruh states, and exhibited the Kerr obstruction where no such state exists at all. The Euclidean thermodynamic side was settled by Gary Gibbons and Stephen Hawking ^{[Gibbons-Hawking 1977]}, whose smooth Euclidean Schwarzschild section fixed both the temperature, through the conical-smoothness period $8 π M$ , and the entropy $A /4$ , through the on-shell action. The microlocal construction of the Unruh state and the proof of its Hadamard property were completed by Claudio Dappiaggi, Valter Moretti, and Nicola Pinamonti ^{[Dappiaggi-Moretti-Pinamonti 2011]} using a bulk-to-boundary holographic method on the past horizon, closing the last gap between the physicists' mode constructions of the 1970s and the wave-front-set framework of 13.09.03.

Bibliography Master

@article{HartleHawking1976,
  author  = {Hartle, James B. and Hawking, Stephen W.},
  title   = {Path-integral derivation of black-hole radiance},
  journal = {Phys. Rev. D},
  volume  = {13},
  year    = {1976},
  pages   = {2188--2203}
}

@article{Israel1976,
  author  = {Israel, Werner},
  title   = {Thermo-field dynamics of black holes},
  journal = {Phys. Lett. A},
  volume  = {57},
  year    = {1976},
  pages   = {107--110}
}

@article{Boulware1975,
  author  = {Boulware, David G.},
  title   = {Quantum field theory in {S}chwarzschild and {R}indler spaces},
  journal = {Phys. Rev. D},
  volume  = {11},
  year    = {1975},
  pages   = {1404--1423}
}

@article{Unruh1976,
  author  = {Unruh, William G.},
  title   = {Notes on black-hole evaporation},
  journal = {Phys. Rev. D},
  volume  = {14},
  year    = {1976},
  pages   = {870--892}
}

@article{Hawking1975,
  author  = {Hawking, Stephen W.},
  title   = {Particle creation by black holes},
  journal = {Comm. Math. Phys.},
  volume  = {43},
  year    = {1975},
  pages   = {199--220}
}

@article{KayWald1991,
  author  = {Kay, Bernard S. and Wald, Robert M.},
  title   = {Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate {K}illing horizon},
  journal = {Phys. Rep.},
  volume  = {207},
  year    = {1991},
  pages   = {49--136}
}

@article{GibbonsHawking1977,
  author  = {Gibbons, Gary W. and Hawking, Stephen W.},
  title   = {Action integrals and partition functions in quantum gravity},
  journal = {Phys. Rev. D},
  volume  = {15},
  year    = {1977},
  pages   = {2752--2756}
}

@article{Kruskal1960,
  author  = {Kruskal, Martin D.},
  title   = {Maximal extension of {S}chwarzschild metric},
  journal = {Phys. Rev.},
  volume  = {119},
  year    = {1960},
  pages   = {1743--1745}
}

@article{Sewell1982,
  author  = {Sewell, Geoffrey L.},
  title   = {Quantum fields on manifolds: {PCT} and gravitationally induced thermal states},
  journal = {Ann. Phys.},
  volume  = {141},
  year    = {1982},
  pages   = {201--224}
}

@article{DappiaggiMorettiPinamonti2011,
  author  = {Dappiaggi, Claudio and Moretti, Valter and Pinamonti, Nicola},
  title   = {Rigorous construction and {H}adamard property of the {U}nruh state in {S}chwarzschild spacetime},
  journal = {Adv. Theor. Math. Phys.},
  volume  = {15},
  year    = {2011},
  pages   = {355--447}
}

@book{Wald1994,
  author    = {Wald, Robert M.},
  title     = {Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics},
  publisher = {University of Chicago Press},
  year      = {1994}
}

@book{BirrellDavies1982,
  author    = {Birrell, Nicholas D. and Davies, Paul C. W.},
  title     = {Quantum Fields in Curved Space},
  publisher = {Cambridge University Press},
  year      = {1982}
}

Prerequisites

13.09.01
13.09.03
13.06.01
13.06.04
13.09.09

Tier anchors

beginner: Wald, *Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics* (Chicago, 1994), Ch. 8 (the three states and the Hawking flux); Birrell and Davies, *Quantum Fields in Curved Space* (Cambridge, 1982), Ch. 8 (the Boulware, Hartle-Hawking, and Unruh vacua side by side)
intermediate: Wald, *Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics* (Chicago, 1994), Ch. 5 and 8; Kay and Wald, *Phys. Rep.* 207 (1991) 49 (the bifurcate-Killing-horizon uniqueness theorem); Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 10
master: Hartle and Hawking, *Phys. Rev. D* 13 (1976) 2188; Israel, *Phys. Lett. A* 57 (1976) 107; Kay and Wald, *Phys. Rep.* 207 (1991) 49; Wald, *Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics* (Chicago, 1994), Ch. 5 and 8; Gérard, *Microlocal Analysis of Quantum Fields on Curved Spacetimes* (EMS, 2019), Ch. 10

References

Hartle, J. B. & Hawking, S. W. — 'Path-integral derivation of black-hole radiance', Phys. Rev. D 13 (1976) 2188 · The Euclidean path-integral construction of the state regular on the full Kruskal manifold; the period-in-imaginary-time argument giving the Hawking temperature, and the demonstration that the resulting two-point function is the analytic continuation that is smooth across both horizons.
Israel, W. — 'Thermo-field dynamics of black holes', Phys. Lett. A 57 (1976) 107 · The thermofield-double interpretation: the Hartle-Hawking state is the purification of the thermal state on the exterior wedge by the fields on the opposite Kruskal wedge, the entangled vacuum of the doubled Hilbert space whose reduction to one exterior is Gibbsian at the Hawking temperature.
Boulware, D. G. — 'Quantum field theory in Schwarzschild and Rindler spaces', Phys. Rev. D 11 (1975) 1404 · The static Schwarzschild vacuum annihilated by the positive-frequency modes of the Killing time t; the demonstration that its renormalised stress tensor diverges in a freely falling frame at the horizon, so the Boulware state is singular there.
Unruh, W. G. — 'Notes on black-hole evaporation', Phys. Rev. D 14 (1976) 870 · The state defined by positive frequency with respect to affine (Kruskal) time on the past horizon and Killing time at past infinity; the model of a black hole formed by collapse, with a steady outgoing thermal flux at the Hawking temperature measured at future null infinity.
Kay, B. S. & Wald, R. M. — 'Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon', Phys. Rep. 207 (1991) 49 · The uniqueness theorem: a quasi-free Hadamard state invariant under the horizon-generating Killing flow on a spacetime with a bifurcate Killing horizon, if it exists, is unique and KMS at T = kappa/(2 pi); the obstruction results forcing the Boulware and Unruh states to be non-Hadamard on parts of the horizon.
Hawking, S. W. — 'Particle creation by black holes', Comm. Math. Phys. 43 (1975) 199 · The originating derivation of black-hole radiance: the Bogoliubov coefficients between past and future modes of a collapsing star give a Planck spectrum at T_H = kappa/(2 pi); the physical content the Unruh state encodes as a stationary flux.
Gibbons, G. W. & Hawking, S. W. — 'Action integrals and partition functions in quantum gravity', Phys. Rev. D 15 (1977) 2752 · The Euclidean-action / conical-smoothness argument: regularity of the Euclidean Schwarzschild section at r = 2M fixes the period of imaginary time to 8 pi M and identifies the period with the inverse temperature of the black hole.
Kruskal, M. D. — 'Maximal extension of Schwarzschild metric', Phys. Rev. 119 (1960) 1743; Szekeres, G. — 'On the singularities of a Riemannian manifold', Publ. Math. Debrecen 7 (1960) 285 · The maximal analytic extension of the Schwarzschild solution across the coordinate singularity at r = 2M; the global structure with two exterior regions, a bifurcation surface, and the future and past horizons on which the three states are compared.
Wald, R. M. — Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (Chicago, 1994) · Ch. 5 and 8 (the three Schwarzschild states, the Hadamard regularity discriminator, the relation to the Unruh effect, and the rigorous formulation via the bifurcate Killing horizon); the textbook synthesis.
Sewell, G. L. — 'Quantum fields on manifolds: PCT and gravitationally induced thermal states', Ann. Phys. 141 (1982) 201 · The modular-theory packaging that identifies the horizon-generating Killing flow as the Tomita-Takesaki modular flow of the exterior-wedge algebra in the Hartle-Hawking state; the general bifurcate-Killing-horizon thermality statement.
Dappiaggi, C., Moretti, V. & Pinamonti, N. — 'Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime', Adv. Theor. Math. Phys. 15 (2011) 355 · The rigorous microlocal construction of the Unruh state via bulk-to-boundary holography on the past horizon, and the proof that it satisfies the Hadamard wave-front-set condition on the union of the exterior and the black-hole interior up to the future horizon.

Estimated time

beginner: 18m
intermediate: 44m
master: 74m