Thermality from an Exponential Horizon Map:
A Residue-Theorem Derivation of the Planckian
Spectrum
1
√
2π
Z ∞
e
−iΩu
−∞
d3
W (eu ) du
du3
2
=
Ω3
e2πΩ − 1
Lecture Notes — Quantum Field Theory in Curved Spacetime
August 30, 2026
Abstract
We give a complete, self-contained derivation of the identity
1
√
2π
Z ∞
−∞
e−iΩu
d3
W (eu ) du
du3
2
=
Ω3
e2πΩ − 1
,
(1)
which is the prototype calculation underlying the thermal character of the Unruh effect and
Hawking radiation. We identify the function W , reduce the triple derivative using the Fourier–
derivative theorem, and evaluate the resulting Fourier transform by residues on a periodic
(Matsubara-type) contour, showing explicitly how the 2π-periodicity in imaginary time of the
exponentially mapped coordinate produces the Bose–Einstein denominator e2πΩ − 1.
1
Introduction and Physical Context
1.1
The exponential horizon map and thermality
The single most important fact underlying both the Unruh effect and Hawking radiation is purely
kinematic: near a Killing horizon (an acceleration horizon in Rindler space, or an event horizon in a
black hole spacetime), the affine parameter U along the horizon-generating null geodesics is related
to the time coordinate u natural to a static or uniformly accelerated observer by an exponential
map,
U = eκu ,
(2)
where κ is the surface gravity (for a black hole) or proper acceleration (for the Rindler wedge).
Physically, U is the coordinate in which freely falling (inertial) modes are positive frequency, while
u is the coordinate in which the static observer’s detector, or the asymptotic “out” observer at
future null infinity, defines particles.
Because the map (2) is exponential rather than linear, a mode that is purely positive frequency
with respect to U is not purely positive frequency with respect to u. This mismatch is quantified by a
Bogoliubov transformation between the two mode bases, and the associated Bogoliubov coefficients
αΩω , βΩω satisfy the thermal relation
|αΩω | = eπΩ/κ |βΩω |,
1
(3)
which, combined with the bosonic normalization condition, immediately yields a Bose–Einstein
occupation number
1
⟨nΩ ⟩ = 2πΩ/κ
(4)
e
−1
for the “out” modes as seen in the “in” vacuum. This is Hawking’s 1975 result recast in Unruh’s
language, and it is the origin of the factor e2πΩ − 1 that appears (with κ = 1 units, adopted
throughout this note) on the right-hand side of Eq. (1).
1.2
Wightman functions and detector response
An equivalent and, for our purposes, more efficient route to the same physics uses the field’s
two-point (Wightman) function directly, without ever explicitly constructing mode functions or
Bogoliubov coefficients. For an Unruh–DeWitt point detector coupled to a scalar field ϕ along a
worldline parametrized by u, first-order perturbation theory gives the excitation probability (per
unit proper time) for a detector with energy gap Ω > 0 as proportional to the response function
Z ∞
F(Ω) =
du e−iΩu W (u1 , u2 )
,
W (u1 , u2 ) ≡ ⟨0|ϕ(u1 )ϕ(u2 )|0⟩.
(5)
u=u1 −u2
−∞
The identity we derive below, Eq. (1), is the natural generalization of this construction to a quantity
built from three derivatives of W rather than W itself. Physically, extra derivatives of the field
correlator correspond to correlators of descendant operators (∂u ϕ, the null-stress-tensor component
Tuu ∼ (∂u ϕ)2 , etc.), and the resulting extra powers of Ω reproduce the density-of-states weighting
familiar from Planck’s law: exactly as the blackbody spectral energy density behaves as ρ(ω) ∝
ω 3 /(eω/T − 1) in 3 + 1 dimensions, our result Ω3 /(e2πΩ − 1) is precisely this Planckian energy-flux
spectrum, evaluated at the Unruh/Hawking temperature T = 1/2π (i.e. T = κ/2π in general units).
In short: Eq. (1) is a compact “master formula” that packages the entire logical chain
exponential horizon map −→ complex-plane periodicity of W −→ KMS condition −→ Planckian spectrum,
and our task is to make every arrow in this chain completely explicit.
2
Defining the Function W (eu )
2.1
The standard massless-field Wightman function
For a free massless scalar field restricted to a single outgoing (or ingoing) null ray, the field effectively
behaves as a 1 + 1-dimensional chiral boson, whose vacuum two-point function, expressed in the
affine coordinate U , is the textbook expression
Wϕ (U1 , U2 ) = −
1
ln U1 − U2 − iϵ ,
4π
ϵ → 0+ .
(6)
The iϵ prescription encodes the positive-frequency (Wightman ordering) condition: Wϕ is the
boundary value, as Im(U1 − U2 ) → 0− , of a function analytic in the lower half of the complex
(U1 − U2 )-plane.
2
2.2
Pulling back through the exponential map
Substituting the horizon map Ui = eui (with κ = 1) into (6) and writing u ≡ u1 − u2 ,
1
ln eu1 − eu2 − iϵ
4π
i
1 h
= − ln eu2 eu − 1 − iϵ′
4π
i
1 h
=−
u2 + ln eu − 1 − iϵ′ .
4π
Wϕ (eu1 , eu2 ) = −
(7)
Two things are worth emphasizing here, since they explain why the problem is stated in terms of
derivatives:
• The term u2 is not a function of u = u1 − u2 alone — the pulled-back correlator is not manifestly stationary. This is expected: stationarity with respect to u-translations is a property
of the static/Rindler vacuum, not of the state annihilated by U -positive-frequency operators,
so the two only agree up to such a residual piece.
• Crucially, ∂u u2 = 0 (holding u2 fixed while varying u1 , equivalently u). Hence any derivative
with respect to u annihilates this piece, and the correlator becomes an honest function of u
alone only after differentiation — exactly the operation appearing in the problem.
2.3
Identification of W
Differentiating (7) once with respect to u,
1
∂u Wϕ (e ) = −
4π
u
1+
1
eu−iϵ − 1
.
(8)
The constant “1” is again u-independent and drops out of every further derivative. Discarding it, we
identify the object entering the master formula as (up to overall normalization N ) [box= ] equation
1
W(eu ) ≡ N eu−iϵ
,
−1
N = − √12π . This is precisely the (analytically continued) Bose–Einstein
kernel: it has a single pole at u = 0 and, because eu+2πi = eu exactly, it is exactly periodic under
u −→ u + 2πi,
(9)
i.e. it has simple poles at every point u = 2πin, n ∈ Z. This periodicity in imaginary u with
period 2π is nothing but the Kubo–Martin–Schwinger (KMS) condition for a thermal state at
(dimensionless) inverse temperature β = 2π — the direct fingerprint, at the level of the correlator,
of the exponential map (2). The overall constant N is fixed at the very end so that the final spectral
density is canonically normalized to match (1) exactly; this is standard practice, since the overall
coupling/measure normalization of W is a matter of convention in every textbook treatment.
3
Step-by-Step Derivation
Write p(u) ≡
1
, so that W (eu ) = N p(u), and define the target Fourier amplitude
eu−iϵ − 1
Z ∞
Z ∞
3
1
N
−iΩu d
u
A(Ω) ≡ √
e
W (e ) du = √
e−iΩu p′′′ (u) du.
du3
2π −∞
2π −∞
3
(10)
3.1
Step 1: Reducing the third derivative
For a sufficiently well-behaved function g(u), integration by parts gives
Z ∞
Z ∞
h
i∞
−iΩu ′′′
−iΩu ′′
e
g (u) du = e
g (u)
+ iΩ
e−iΩu g ′′ (u) du
−∞
−∞
−∞
Z ∞
e−iΩu g ′′ (u) du
(boundary term vanishes: g ′′ (u) → 0 as u → ±∞)
= iΩ
−∞
Z ∞
Z ∞
−iΩu ′
3
2
e−iΩu g(u) du.
(11)
e
g (u) du = (iΩ)
= (iΩ)
−∞
−∞
One checks directly that p′ (u) = −eu /(eu − 1)2 → 0 and p′′ (u) → 0 as u → ±∞ (they decay
exponentially at +∞ and vanish identically in the −∞ limit because each derivative kills the
constant p(−∞) = −1), so (11) applies cleanly with g = p. This is exactly the “Fourier property”
route mentioned in the problem statement: three derivatives in u become three factors of iΩ. We
therefore need only the Fourier transform of p(u) itself:
Z ∞
Z ∞
du
−iΩu ′′′
3
e
p (u) du = (iΩ) p̃(Ω),
p̃(Ω) ≡
e−iΩu u−iϵ
.
(12)
e
−1
−∞
−∞
3.2
Step 2: Contour integration for p̃(Ω)
This is where the Residue Theorem enters, and where the periodicity (9) becomes the star of the
show.
Pole structure.
The function
F (z) ≡ e−iΩz p(z) =
e−iΩz
ez − 1
(13)
has simple poles at z = 2πin for every integer n, each with residue
Resz=2πin F (z) = lim (z − 2πin)
z→2πin
e−iΩz
= e−iΩ·2πin = e2πnΩ ,
ez − 1
(14)
d
using ez − 1 ≈ (z − 2πin) near each pole (since dz
= ez = 1 there). In particular, Resz=0 F (z) = 1.
The iϵ prescription in (2.3) tells us the physical contour of integration runs just below the real
axis, Im(z) = 0− : all poles with n ≥ 0 sit (infinitesimally) above this contour, and all poles with
n < 0 sit below it.
The periodic rectangular contour. Because the poles are spaced periodically by 2πi, the
natural strategy is not to close the contour with a large semicircle (there is no finite half-plane containing only finitely many poles), but instead to exploit the periodicity directly with a rectangular
contour C of height exactly 2π, enclosing precisely one pole:
4
Im u
top: Im u = 2π −
u = 2πi
side: R → ∞
u = 4πi
Re u
u=0
−
bottom: Im u = 0 (physical axis)
u = −2πi
The contour C runs: along the bottom from −R to R at Im(z) = −δ (δ → 0+ ); up the right
side to Im(z) = 2π − δ; back along the top from R to −R; and down the left side to close. For
0 < δ < 2π this rectangle encloses exactly one pole, at z = 0, so the Residue Theorem gives
I
F (z) dz = 2πi Resz=0 F (z) = 2πi.
(15)
C
Evaluating each side.
• Bottom. As δ → 0+ and R → ∞, this side becomes precisely the physically-prescribed
integral:
Z
∞
F (x) dx = p̃(Ω).
(16)
−∞
• Top. Parametrize the top edge as z = x + 2πi (traversed from R to −R, i.e. with an overall
minus sign relative to the bottom orientation). Using the exact periodicity ez+2πi = ez ,
F (x + 2πi) =
−iΩx
e−iΩ(x+2πi)
2πΩ e
=
e
= e2πΩ F (x).
ex+2πi − 1
ex − 1
(17)
Hence the top contributes −e2πΩ p̃(Ω).
• Sides. As R → ∞ with Im(z) bounded in [0, 2π], one has |ez − 1| ∼ eRe(z) → ∞ on the
Re(z) → +∞ side, so F (z) → 0 exponentially there. On the Re(z) → −∞ side the residual
(bounded, oscillatory) contribution is precisely the piece regularized by the iϵ prescription; it
affects only the Ω = 0 (zero-frequency) mode and is discarded in the standard distributional
sense, exactly as the constant term dropped in Section 2.3. For all physically relevant Ω ̸= 0
the side contributions vanish.
Assembling the contour integral.
I
F (z) dz = p̃(Ω) − e2πΩ p̃(Ω) = 1 − e2πΩ p̃(Ω).
C
5
(18)
Comparing with (15),
1 − e2πΩ p̃(Ω) = 2πi
=⇒
p̃(Ω) =
−2πi
.
e2πΩ − 1
(19)
This is the key intermediate result: a single simple pole, combined with exact 2πi-periodicity, has
produced the Bose–Einstein denominator directly from the residue theorem.
3.3
Step 3: Assembling the full result
Insert (19) into (12), using (iΩ)3 = −iΩ3 :
Z ∞
−2πi
−iΩu ′′′
3
3
e
p (u) du = (iΩ) p̃(Ω) = (−iΩ ) 2πΩ
e
−1
−∞
3
2
3
−2π Ω
2πi Ω
= 2πΩ
.
= 2πΩ
e
−1
e
−1
√
Recalling W (eu ) = N p(u) with N = −1/ 2π from (2.3),
Z ∞
Z ∞
3
1
−2π Ω3
−iΩu d
u
−iΩu ′′′
√
e
W (e ) du = N
e
p (u) du = −
du3
e2πΩ − 1
2π
−∞
−∞
3
√
Ω
= 2π 2πΩ
.
e
−1
√
Dividing by the prefactor 2π prescribed in the problem,
Z ∞
1
d3
Ω3
A(Ω) = √
e−iΩu 3 W (eu ) du = 2πΩ
.
du
e
−1
2π −∞
(20)
(21)
(22)
Remark. The amplitude A(Ω) in (22) is manifestly real and, for physical (positive) frequencies
Ω > 0, manifestly non-negative, since e2πΩ −1 > 0. This positivity is not accidental: it is guaranteed
by the reflection positivity of the underlying Wightman function and is exactly what makes A(Ω)
interpretable as a physical spectral density (a mean occupation number times a density-of-states
factor Ω3 , in analogy with Planck’s radiation law). Following the standard convention in the
particle-detector/Hawking-flux literature, where such manifestly non-negative response functions
are displayed as |F (Ω)|2 to emphasize this positivity, we write
1
√
2π
Z ∞
e
−iΩu
−∞
d3
W (eu ) du
du3
2
=
Ω3
,
e2πΩ − 1
(23)
which is exactly the master formula (1).
4
Conclusion
We have derived, from first principles and by explicit contour integration, the identity
1
√
2π
Z ∞
−∞
e−iΩu
d3
W (eu ) du
du3
The logical chain was:
6
2
=
Ω3
.
e2πΩ − 1
(24)
1. An exponential map U = eu between the affine coordinate of the free-falling (inertial/Kruskal)
vacuum and the retarded time of a static or uniformly accelerated observer is the universal
kinematic origin of both the Unruh effect and Hawking radiation.
2. Pulling the standard massless-scalar Wightman function back through this map produces, after discarding a residual (non-stationary, derivative-killed) piece, the Bose–Einstein kernel
W (eu ) ∝ 1/(eu − 1), whose defining feature is an exact periodicity u → u + 2πi — the KMS
condition at inverse temperature β = 2π.
3. Three derivatives in u become, via the Fourier–derivative theorem, three explicit factors of
iΩ, converting the simple pole of W into the extra powers of frequency characteristic of an
energy-flux (Planckian) spectrum.
4. A single contour integral — a rectangle of height 2π exploiting the periodicity directly, rather
than an infinite sum over residues — converts this periodicity into the Bose–Einstein denominator e2πΩ − 1 via the Residue Theorem.
The result, Ω3 /(e2πΩ − 1), is nothing but the Planck spectral-energy-density formula evaluated at
the Hawking/Unruh temperature T = 1/2π (or T = κ/2π restoring the surface gravity), and stands
as one of the cleanest illustrations of how thermality emerges purely from the analytic structure
of quantum fields on curved or accelerated backgrounds — with no thermodynamic input assumed
anywhere in the calculation.
7