KM Institute, Nagoya — Febraury 7 2012
Sergio Zerbini, Trento University and
Gruppo Collegato INFN Padova, Italy

Hawking Radiation, Tunneling Method andQuantum Thermometers

Contents:

Introduction
The Hamilton-Jacobi tunneling method
Applications to the static and dynamical cases
QFT in action: the Unrhu-de Witt andBuchholz detectors
Conclusions

KM Institute, Nagoya — Febraury 7 20121

The Unknown Nature of Gravity
Four known elementary Interactions
Electromagnetic, Weak and Strong Interactions are unified
at quantum level:
Standard Model of elementary particles well understood

The fourth fundamental interactions, Gravity is the oldest one(Newton)
but also the only one not completely understood.
Question:Has the Gravity to be quantized ?
Probably yes (Covariant and Path Integral approaches,SuperGR, String Theory, Loop Gravity) unsolved till now
Other questions (Jacobson, Panamandhan and many others)
Is Gravity an emergent phenomena?
Is the Gravity the thermodynamics of the space-time ?
Not easy to answer

KM Institute, Nagoya — Febraury 7 2012
Aim of this talk: making use of
Semiclassical methods: classical gravity and quantum matter
Try to answer to specific the question:
How hot is our (expanding) universe?
main idea: Universe as Dynamical Black HoleFor the FRW de Sitter universe:
ds2 = −dt2 + e2H0td2~
x
the answer is well know:

Temperature
H0 =

q

T =

H0
2π

(Gibbon-Hawking 77)

Λ , Hubble parameter, cosmological constant Λ > 0.
3

2

KM Institute, Nagoya — Febraury 7 20123

We will re-derive this famous result. The key point isthe existence also of the static dS patch:
ds2 = −(1 − H02r 2)dt2 +

dr2
2 dΩ2
+
r
1 − H02r 2)

a Static Spherical Symmetric BH space-time.
Static Hawking effect well understood.
Recall dS space-time is particularly important in modern Cosmology:
Inflation and Dark Energy
What about FRW dS (Dynamical situation)?
Necessity to discuss on general ground:
Hawking effect, and Hawking temperature forSpherically Symmetric Dynamical BHs.

KM Institute, Nagoya — Febraury 7 2012

4

Hawking radiation

Hawking radiation is one of the most important predictionsof quantum field theory in curved space-time.
The Hawking effect is associated with the presence of Black HolesClassically BHs are compact gravitational objects from which matter(massless and massive) enter but not escape.

If quantum theory enters the game, something can escape:Thermal Hawking radiation.
It turns out that (stationary) BHs are thermodinamical objects, andat least three fundamental issues have to be addressed:

KM Institute, Nagoya — Febraury 7 2012

5

i. BH entropy issueii. BH energy or mass issueiii. BH temperature issue
We will mainly concentrate on the temperature issue.
Within covariant gravitational Lagrangian theories, the Entropy issue
is well understood by the Wald method:a robust recipe for evalutating BH entropy starting fromthe Lagrangian of the model.
For Ex: in f (R) modified gravity, Wald method leads to′
AH f H
Generalized Area Law SBH = 4G

The energy issue is more problematic, we only recall :ADM mass in asymptoticaly flat BH space-times.
For spherically symmetric BHs in Lovelock theories:(d = 4, R − 2Λ, d = 5, R − 2Λ + αG, G Gauss-Bonnet)
there is the Misner-Sharp quasilocal energy. Others quasilocal
energies can be defined (Brown-York, Bartnich,..).

KM Institute, Nagoya — Febraury 7 2012
The Hawking effect is kinematics: dynamics plays no role
Several derivations have been proposed,
among them the most popular:
i. Bogoliubov transformation method Hawking 75
ii. Damour-Ruffini method 1976
iii. Path-Integral in Kruskal gauge Hartle-Hawking 76
iv. Tunneling method, Parikh and Wilczek 2000
In the following, we will review a variant of tunneling method:v. Hamilton-Jacobi Method Padmanabahn,our group 03-05
covariant and extendable to the dynamical case
(Hayward, Di Criscienzo, Nadalini, Vanzo, S.Z.)

6

KM Institute, Nagoya — Febraury 7 2012

7

H-J tunneling method: introduction

The H-J Tunneling method is reasonably simple:
Gravity at classical level, QM of particle enters in WKB or
Eikonal approximation:i Ih̄Amplitude ∝ e(c = 1)

Then computation of the classical action I along a trajectory incurved space-time which includes the horizon: First key point: the presence of Horizons leads to Im I 6= 0 andsemiclassical emission rate,−2 Imh̄ I2
Γ ∝ |Amplitude| ∝ e

KM Institute, Nagoya — Febraury 7 20128

Second key point: Appearance of Boltzmann factor withlinear dependence on energy ω−2 Imh̄ IΓ∝e− βh̄ ω∝e

In the static case, the Boltzmann factor suggests
a radiation in thermal equilibrium, and
it is reasonable that T = βh̄ is the BH temperature.
From now on h̄ = 1
What about the dynamical case?
Remark: In the dynamical case it is crucial that the argument of
the exponent be a coordinate scalar (invariant quantity) otherwiseno physical meaning can be given to Γ: Life is simple withSpherical Symmetry and related Hayward Covar. Formalism

KM Institute, Nagoya — Febraury 7 20129

Kodama-Hayward formalism

Generic spherically symmetric (SS) space-time metricds2 = γij (xi)dxidxj + R2(xi)dS 2 ,i, j = 0, 1 ,First ingredient: 2-dim normal metric, xi related coordinatesdγ 2 = γij (xi)dxidxj
Second ingredient: the scalar quantity in γ R(xi) called
areal radius of sphere S 2.
Third ingredient: the further scalar in γ
χ(x) = γ ij (x)∂iR∂j R .

KM Institute, Nagoya — Febraury 7 201210

One has three cases: the two sphere with areal radius R is called
i. untrapped if χR < 0,ii. marginal if χR = 0,iii. trapped if χR > 0.
A hypersurface foliated by marginal spheres is called trapping horizon,and defined by
χ(xH ) = 0 ,∂iχH 6= 0 .

Trapping horizons:i time-like in evaporating BH and cosmology,
ii. null in Static (eternal) BH,
iii. space-like (accreting BH).

KM Institute, Nagoya — Febraury 7 201211

Fourth ingredient: scalar evaluated on the dynamical horizon√
11
κH = 2γ RH = √ ∂i( −γγ ij ∂j R)H .
2
2 −γ

called Hayward dynamical surface gravity.
One has three cases: a trapping horizon is said to bei. outer if κH > 0,ii. degenerate if κH = 0,iii. inner if κH < 0.
Fifth ingredient: conserved Kodama vector1
K i(x) = √ εij ∂j R ,−γandKθ = 0 = Kϕ

which gives a preferred flow of time, generalizing the flow of time
given by Killing vector ∂t in static cases.

KM Institute, Nagoya — Febraury 7 201212

Ex. 4-dim static Schwarzschild BH x = (t, r), R = r
dr2
22+ r 2dS 2
ds = −V (r)dt +V (r)with V (r) = 1 − 2M
r . Here2Mχ(r) = V (r) = 1 −, χ(rH ) = 0 → rH = 2Mrexample of singular gauge, not defined on the horizon.

VH′
1 . In GR, for spherical horizons,
The surface gravity kH = 2 = 2M

the Misner-Sharp quasi-local gravitational energy is defined byR(x)EM S (x) =(1 − χ(x)) .
2
This is an invariant quantity on the normal space. Note that on thehorizon EM S (RH ) = R2H = M , typically in GR the BH mass.

KM Institute, Nagoya — Febraury 7 2012

Vaidya black hole

As a simple example of dynamical BH:
the Vaidya black hole2m(v)2
)dv 2 + 2dv dr + r 2dΩ2ds = −(1 −rwhere m(v) is the mass function of advanced time v.
The invariant χ(v)2m(v)
rTrapping horizon is rH (v) = 2m(v) and Hayward surface gravity
χ(v) = 1 −kH (v) =

1
.
2m(v)

13

KM Institute, Nagoya — Febraury 7 2012

14

Intermezzo: The First Law in GR

The Kodama vector and κH are geometric kinematical quantities . If
the dynamics enters, one has
Lemma: within GR and on spherical dynamical horizon
1(2)+ 2πRH TH ,
κH =2RH
where the reduce stess-tensor trace scalar in the normal space
T (2) = γ ij Tij .
Proof: Use the Einstein Equations
1
Gµν = Rµν − gµν = 8πTµν
2and evaluated them on the spherical dynamical horizon.

KM Institute, Nagoya — Febraury 7 201215

Consequence: Introduce
2 ,AH = 4πRH

43 ,VH = πRH
3horizon area

horizon volume

Then,
κH
RH
dAH = d
8π
2




(2)

+ TH dVH .

Since M = R2H , the Lemma gives: The First Law (Hayward)κAH(2)
dM = H d− TH dVH
2π
4
The viceversa is very interesting:Assuming the First law and existence of horizons, Einstein equations
follow: gravity emergent phenomena? (Jacobson, Padmanabhan,..)KM Institute, Nagoya — Febraury 7 201216

H-J tunneling method in action

Several ingredients:
i. Invariant definition of energy of the particle via Kodama vector
ω = −K µpµ ,pµ is the 4-momentum of the particle: Working in SS spaces
independence of a specific choice of spacetime coordinates.
ii. Relativistic Hamilton-Jacobi equation ( radial trajectories)
Recall pµ = ∂µI where I classical action
γ ij ∂iI∂j I + m2 = 0 .
The effective mass m here defines two important and complementary energy scales:

KM Institute, Nagoya — Febraury 7 201217

a. In the Hawking effect as horizon tunnelling we may neglect themass,
γ ij ∂iI∂j I = 0 .
Physics comes from horizon singularities.
b. Effects in the bulk away from the horizon, ex decay rate of composite particles, the effective mass becomes relevant as the energyof the particle can be smaller than the energy scale settled by m, andbranch cut singularity is present.
Then one make two other key assumptions:a. The near horizon approximation.
b. The null expansion assumption.
This last requirement force us to make use of regular gauges on thehorizon. For example, the usual static Schwarzschild gauge issingular on the horizon.

KM Institute, Nagoya — Febraury 7 201218

iii. Reconstruction of the action
I=

Z

γ

dxi ∂iI

solving the HJ equation and with γ being a suitable path in thenormal space, which includes the horizon.
iv. Feynman prescription
The classical action is divergent on the horizon and the imaginary
part comes from Feynman prescription (or contour deformation):
f (r)
f (r)
dx = iπf (rH ) + real part
dr →
r
−
r
r
−
r
−
i0
γ
γ
H
H

Z

Z

KM Institute, Nagoya — Febraury 7 2012

19

A generic spherically symmetric dynamical BH

Starting point: any spherically symmetric metric can be rewritten in
the Eddington-Filkenstein-Bardeen regular gauge
ds2 = −e2ΨCdv 2 + 2eΨdvdr + r 2dΩ2 ,where xi = (v, r) as coordinates, and C = C(v, r), Ψ = Ψ(v, r)smooth functions. Here simply R = r and χ = C,thus the dynamical horizon C(rH , v) = 0.
The Kodama vector K = (e−Ψ, 0), and Kodama energy ω = e−Ψ∂v I
Hayward invariant surface gravity κH = ∂r2CH .
First step: expansion on a null direction in the neighbour of thehorizon gives
0 = eΨH dvdr .
No temporal contribution to the imaginary part of the action.

KM Institute, Nagoya — Febraury 7 201220

Second step: from the H-J equation and Kodama energyω∂r I = 2C
Thus, within a neighborhood of γ containing the horizon whereC(rH , v) = 0
πωH
ω
ω
1
,
=
Im I = Im dr∂r I = 2 dr ≃ 2 drC
∂r C (r − rH − i0)
κH
γ
γ
γ
Z

Z

Z

where C = k2H (r − rH ) around the horizon along the null direction
and the Feynman prescription have been used. Thus
− κ2π ωHΓ≃eH

with κH and ωH scalars on the normal space: Γ invariant. This
result is valid for generic SS space-time, in particular for a static BH
space-time.

KM Institute, Nagoya — Febraury 7 2012

21

Here we present the operational interpretation.
Static observers in static BH become in the dynamical case
Kodama observers whose velocity
KiivK = √ ,j

i v = −1
γij vKKχ

Lemma:
Kodama observers are such that R = R0, constant areal radius
The energy measured by this Kodama observer at fixed R0 isωK i∂iIi=√
E = −vK ∂iI = − √χ0χ0The tunneling rate can be rewritten as
− κ2π

Γ≃e

H

√

χ0E

− TE

≃e

0

and the local quantity T0 at radial radius R0 is also invariant:√
it contains an invariant factor χTT0 = √ H ,χ0k
TH = H2π

KM Institute, Nagoya — Febraury 7 201222

In the static case χ = g rr = −g00 and recalling Tolman’s theorem:√
For a gravitational system at thermal equilibrium, T −g00 = constant,
H is the intrisic temperature of the BH:
it follows TH = κ2π
the Hawking temperature.
In the dynamical case and for slow changes in the geometry, thequestion is :H the dynamical Hawking temperature ?
Is still TH = κ2π
It is an unsolved issue till now.
However, in a generic SS space-time and in GR, Hayward surface
gravity κH and ωH are invariant quantities and recalling the AreaH
Law SH = A
4G (Bekenstein-Hawking Entropy), the First Law can be
rewritten as(2)

dM = TH dSH − TH dVH .

H could be a sort of temperature.
A further hint that TH = κ2π

KM Institute, Nagoya — Febraury 7 201223

General static black hole space-time

The starting point may be a BH metric in the Schwarzschild static
singular gauge2drds2 = −V (r)dt2 ++ r 2dS 2
V (r)with V (rH ) = 0, VH′ 6= 0.
Horizon located at r = rH , and the Kodama vector coincides with theusual Killing vector (1, 0, 0, 0), Hayward surface gravity is the Killing

VH′
surface gravity, namely κH = κ = 2 .

One can use of regular Painleve’ gauge (Parikh-Wilczek)
ds2 = −V (r)dt2 −q

1 − V (r)dt dr + dr2 + r 2dS 2

KM Institute, Nagoya — Febraury 7 2012

24

But, the final result has to be gauge-invariant, and we use the complete and regular time dependent Kruskal gauge.
Introduce the tortoise coordinate
dr
dr∗ =
V (r)
−∞ < r ∗ < ∞. Thus
ds2 = V (r ∗)(−dt2 + (dr∗)2) + r 2(r ∗)dS 2
Then Kruskal-like coordinates
1 κr∗cosh κt ,
X= e
κ
with

1 κr∗
T = e
sinh κtκ

1 2κkr∗2
2− T + X = 2eκ
and a conformally flat normal metric appears∗

ds2 = V (r ∗)e−2κr (−dT 2 + dX 2) + r 2(T, X)dS 2
new coordinates being T and X while r ∗ = r ∗(T, X).

KM Institute, Nagoya — Febraury 7 201225

In this gauge, the metric is still spherically symmetric, regular on∗Ψ−2κr
the horizon: the conformal factor e = V (r∗)ewell defined forr = rH , but time dependent. The trapping horizon(∂T r)H = (∂X r)H ,equivalent to T = ±X and r ∗ → −∞. The Killing-Kodama vector∗

K = e−Ψ(r ) (∂X r, −∂T r )Hayward surface gravity is the Killing (recall κH is invariant):

Thus


VH′
e−ΨH  22=
κH =−∂T r + ∂X r
H
2
2
E
− ′4π ωH−Γ ≃ e V (rH )≃ e T0 ,

VH′
T0 =
√ ,
4π V0

T
T0 = √ H
V0

Important check of the H-J dynamical tunneling method.

KM Institute, Nagoya — Febraury 7 201226

The FRW Spacetime

FRW space-times are important in cosmology.
The flat FRW space as Spherical Symm. Dynamical BH
ds2 = −dt2 + a2(t)dr2 + [a(t)r]2dS 2
The normal reduced metric is diagonal, areal radius R = a(t)r andthe invariant χχ(t, r) = 1 − a(t)r 2H 2(t) ,ȧ(t)H(t) =a(t)The dynamical horizon is implicitly given by χH = 0,
RH := a(t)rH =1
H(t)KM Institute, Nagoya — Febraury 7 201227

Hayward surface gravity is1
κH = − H 2(t) + Ḣ(t) RH (t) ,2and the minus sign refers to the fact the Hubble horizon is, in
Hayward’s terminology, of the inner type.


1 , Hubble radius, and we rewrite
In the flat case RH = H(t)κH = − H(t) +

Ḣ(t)
2H(t)

!

KM Institute, Nagoya — Febraury 7 2012

28

Thus the rate has the exponential form
− (κ2π ) ωHΓ≃eH

− T E(t)≃e0the local time dependent “factor” at R0
T0(t) = q

TH
2H 2
1 − R0.

For slowly metric changing, it seems suggestive to interpret
H as dynamical temperature associated with FRW spaceTH = − κ2πtimes, explicitely
TH =For dS space T0 = q TH

Ḣ(t)1
H(t) +
2π
2H(t)

2H 2
1−R0
0

!

0
, TH = H
2π , real temperatures.

It should be important to have a QFT confirmation of
the H-J tunnelling result:
Quantum Field Theory in action.

KM Institute, Nagoya — Febraury 7 201229

Quantum field theory in conformally flat spacetimes
A quantum field theoretical approach, with a quantum probe is used
to unveil the intrinsic temperature associated with static space-times
with horizons.
The probe: a conformally coupled massless scalar field Φ
We work on conformally FRW flat space-time, conformal time ηdη = dta,ds2 = a2(η)(−dη 2 + d~x2 ) ,

x = (η, ~x)

The knowledge of Wightman function W (x, x′) is crucial
W (x, x′) =< Φ(x)Φ(x′) >=X~kf~k (x)f~∗(x′) ,k

Φ(x) =

X~kf~k a~k + h.c.

The modes functions f~k (x) satisfy the conformally invariant equation(R being the curvature scalar) sastisfied by Φ(x)R
2−f~k (x) = 0 .
6KM Institute, Nagoya — Febraury 7 201230

The solution of this equation is a vacuum choicee−iηk −i~k·~xef~k (x) = √2 ka(η)k = |~k|W (x, x′) can be computed in an exact way
Wε(x, x′) =

1
1
.
2
′
′
2
′
2
4π a(η)a(η ) |~x−~x | − |η − η − iǫ|As is usual in distribution theory we shall leave understood the limit
as ǫ → 0+. However, it has been shown by Takagi and Schlicht that
it is necessary the covariant form
Wε(x, x′) =

1
1
.
2
′
′
′
2
4π a(η)a(η ) [(x − x ) − iǫ(ẋ + ẋ )]where an over dot stands for derivative with respect to proper timeKM Institute, Nagoya — Febraury 7 201231

The Unruh-De Witt detector

The Unruh-De Witt detector approach is a well known and usedtechnique for exploring quantum field theoretical aspects in curved
space-time.
The transition probability per unit proper time of the detector depends on the response function per unit proper time which, for radial
trajectories, at finite proper time τ , and this depends on Wightman
functionZ τ −τ
01−iEsW (x(τ ), x(τ − s))dseReḞ (E, τ ) =ε
2π 20where τ0 is the initial detector proper time, E is two-level positive
detector transition energy. The Wightman function W (x(τ ), x(τ − s))is singular at s = 0 because x → x′ is singular and ǫ → 0+ cures the
singularity.

KM Institute, Nagoya — Febraury 7 201232

Subtracting the leading divergence at s = 0 and using the normalisation conditiongµν ẋµẋν ≡ [a(τ )ẋ(τ )]2 = −1 ,introducing the geodesic distanceσ 2(τ, s) ≡ a(τ )a(τ − s)[x(τ ) − x(τ − s)]2 ,which for small s
σ 2(τ, s) = −s2[1 + s2d(τ, s)] .
one may take the limit ǫ → 0 (Louko, Satz 08)

KM Institute, Nagoya — Febraury 7 201233

Final expression
Ḟ (E, τ ) =

Z ∞

1
1
1
+
ds
cos(Es)2π 2 0σ 2(τ, s)s2!

+ Jτ (E) ,where the “tail” or finite time fluctuating term is∞cos(E s)1
ds 2.
Jτ (E) := − 22π ∆τσ (τ, s)In the important stationary cases:Zσ(τ, s)2 = σ 2(s) = σ 2(−s)

Ḟ (E, τ ) =

Z ∞

1
1
1
−iEs
ds
e
+
4π 2 −∞σ 2(s)
s2!

+ Jτ = Ḟ (E) + Jτ (E) .

The first term is τ independent, and all the time dependence iscontained only in the fluctuating tail, and Jτ (E) → 0 for large τKM Institute, Nagoya — Febraury 7 201234

Quantum thermometers in static and stationary
spacesApplications:

Rindler space (Unruh effect)

Generic static black hole (Hawking effect)
I will talk only on Hawking effect. The important case of the de Sitter
space, independence of the coordinates choice (gauge independence)
will be explicitly verified.

KM Institute, Nagoya — Febraury 7 201235

The generic static black hole

Recall for a static black hole reads
dr2
22
ds = −V (r)dt ++ r 2dS 2 ,V (r)the horizon is located at r = rH
the surface gravity, is κH = κ = VH′ /2.
Recall the Kruskal gauge∗

ds2 = e−2κH r V (r ∗)[−dT 2 + dX 2] + r 2(T, X)dS 2 ,coordinates are T and X, r ∗ = r ∗(T, X), and the normal metric turns
out to be conformally flat. For Kodama observers the areal radiusr(T, X) and r ∗ are constant, say r = r0.

KM Institute, Nagoya — Febraury 7 201236

the proper time along Kodama trajectories reads∗dτ 2 = V0 dt2 = V0e−2κr0 (dT 2 − dX 2) = a2(r0)(dT 2 − dX 2) .
√Thus t = τ / V0 and, as a function of proper-time τ!

1 κH r ∗τ
0X(τ ) =
ecosh κH √κH
V0!
τ
1 κH r ∗0 sinh κH √e.
T (τ ) =κHV0

KM Institute, Nagoya — Febraury 7 201237

The geodesic distance reads
i
∗h
2
2
2
−2κr
0 − (T (τ ) − T (τ − s)) + (X(τ ) − X(τ − s))σ (τ, s) = V0e,

and one gets

4VκH sσ 2(τ, s) = − 20 sinh2
√
2 V0
κH

!

.

Stationarity: σ 2(τ, s) = σ 2(s) = σ 2(−s), Ḟ can be exactly computedḞ (E) =

1
E
 √

.
V
E
2π
2π exp0−1
κH

Planck distribution: Unruh-DeWitt detector in the generic SSS BH
space-time detects a quantum system in thermal equilibrium at the
local temperature
κH
T0 =
√ .
2π V0

KM Institute, Nagoya — Febraury 7 201238√
Again Tolman’s theorem: T −g00 = constant.
Generic static black hole has intrinsic constant temperature, theHawking temperature, i.e.
VH′
κ
TH =
=
.
2π
4π
Another form of Tolmann factor: the Killing-Kodama observers withr = r0 constant, have an invariant acceleration
V0′2
2
a0 =
,
4V0
where aµ = uν ∇ν uµ, uµ being the observer’s four-velocity. Thus, thelocal equilibrium temperature can be rewritten in the form
T0 = TH

2a0
.
′
V0

KM Institute, Nagoya — Febraury 7 201239

de Sitter space in the static patch
The static de Sitter metric is a particular case with
V (r) = 1 − H02r 2 ,Λ
2
H0 = .
3

The horizon is located at rH = H0−1 and the surface gravity is
kH = H0
H04r02
2, thus
The acceleration at fixed r0 reads a0 =1−H 2r 2
0 0

2
a2
0 + H0 =H02.
2
2
1 − H0 r0

and de Sitter local temperature felt by the Unruh detector is(Narnhofer-Thirring )1q 2
TdS (r0) =a0 + H02 .
2π

KM Institute, Nagoya — Febraury 7 201240

de Sitter space in FRW coordinates
In this case the de Sitter metric is a SS dynamical BH
ds2 = −dt2 + e2H0t(dr2 + r 2dS 2) ,a(t) = eH0t.

with H(t) = H0 constant. For Kodama observers r = R0e−H0t and2denoting V0 = 1 − H02R0
τ =
and

q

a(τ ) = e

V0 t ,

H0
√
τ
V0

H

1 − √V0 τ
0 ,η(τ ) = −
e
H0

,
H
V0

−√ 0 τ

r(τ ) = R0 e

so, the geodesic distance isσ 2(τ, s) = −4 V0
2 H0 s
sinh
√
2
2 V0
H0

!

.

,

KM Institute, Nagoya — Febraury 7 201241

Again stationarity: σ 2(τ, s) = σ 2(s) = σ 2(−s), andḞ (E) =

1
2π

2π

√

E
V0 E

,

e H0 − 1
In the FRW de Sitter space one detects a quantum system in thermal
equilibrium at a temperature T0 = 2πH√0V .
0
Recall that 4-acceleration of a Kodama observer in a FRW spacetime is2222 Ḣ(t) + (1 − H (t)R0 )H (t) 2µ2
a = a a µ = R0 
32) 2
(1 − H 2(t)R0
where aµ := uν ∇ν uµ, uµ being the 4-velocity of the detector.
KM Institute, Nagoya — Febraury 7 201242

As a result, for dS space in a time dependent patch we havea20=
showing that

2H 4
R0
0
,
2
2
1 − R0 H0

H0q2
1 − H02R0=

q

H02 + a20

in agreement with the dS static patch calculation.
This is an important check of the approach, since
it shows the coordinate independence of the result for the importantcase of de Sitter space.
When R0 = 0, one has V0 = 1 and the classical Gibbons-Hawking
0 is recovered.
result TdS = H
2π

KM Institute, Nagoya — Febraury 7 201243

Wath about a generic F RW space-time?
In the stationary case, in the limit τ → ∞, one hasḞ (E) =

Z ∞

1
1
1
−iEs
ds
e
+
4π 2 −∞σ 2(s)
s2

!

=

E
1
 
.
E
2π exp−1
T0This is a sort of Fluctuation-Dissipation Theorem, since Ḟ (E) is a
Fourier transform of a renormalized correlation function (Wightman
function). Note that in this case Detailed Balance Principle holds
Ḟ (E)− TE=e 0,
Ḟ (−E)Ḟ (−E) − Ḟ (E) =E
2π

Viceversa, if the above relations hold then Ḟ (E) is the Plank distribution. Thus we can define the local equilibrium temperature:

AlternativelyET0 =ln Ḟ (−E) − ln Ḟ (E)
T0 =

E


ln 1 +E
2π(Ḟ (E))



KM Institute, Nagoya — Febraury 7 2012

44

Thus, in general, we may define the local effective temperature by− T E(τ )Ḟ (E, τ )=e 0 .
Ḟ (−E, τ )

Thus

or

E
T0(τ ) =ln Ḟ (−E, τ ) − ln Ḟ (E, τ )T0(τ ) =Eln 1 +E
2π(Ḟ (E,τ )−J(τ ))In the stationary case, Ḟ is time independent, and for large ∆τ
T0 =

E


ln 1 +E
2π(Ḟ )

KM Institute, Nagoya — Febraury 7 201245

dS asymptotically space-times
Important physical ex: flat FRW universe with Λ > 0 and matterp = 0. The solution of Friedman eq. ( asymptotically dS)3
2/3
a(t) = a0 sinhh t → eh t t → ∞
2


√



q

with h ≡ ΩΛH0, Ωm + ΩΛ = 1; H0 = 8πρcr /3.
Take for simplicity the comoving observer R0 = 0. Expansion in ∆τ
∞
X



3n−1
X



1
1
k hs  ,2
e−3hn∆τ
g(n,
k)
e
−
h
=
2 (s)
σ 2(τ, s)
σdS
n=1
k=1

g(n, k) computable numerical coefficients. Note pure de Sitter contribution,
2 (s) = −4h−2 sinh2 hs
σdS2√with the effective Hubble constant h = ΩΛH0.





KM Institute, Nagoya — Febraury 7 201246

Direct computation∞ 3n−1X
h2 X−3n h∆τ ×g(n,k)
eḞτ (E) = ḞdS (E) + JdS,τ (E) −2π 2 n=1 k=1(1)ehk∆τ (hk cos(E∆τ ) + E sin(E∆τ )) − hk
h2 k 2 + E 2

JdS,τ is the dS tail. The numerical coefficients g(n, k) appear inanother tail which decays exponentially in time withoscillating terms vanishing in the limit ∆τ → ∞.
Thus the detector clicks close to a de Sitter response and reaches
thermalization through decaying oscillations as ∆τ is sufficiently large
and the local effective temperature goes to dS temperature
TdS = h =qΩΛH0

KM Institute, Nagoya — Febraury 7 2012

47

Buchholz Quantum Thermometer

Another proposal to detect local temperature associated with stationary space-time admitting an event horizon has been put forward
by Buchholz and collaborators. The idea may be substantiated by
the following argument.
Let us start with a free massless quantum scalar field Φ(x) in thermal
equilibrium at temperature T in flat space-time. It is well known that
finite temperature effects may be investigated by considering thescalar field defined in the Euclidean manifold S1 × R3,
with imaginary time τ = −it, compactified in the circle S1, with period
β = T1 .

KM Institute, Nagoya — Febraury 7 201248

Let us consider the local quantity < Φ(x)2 >. Formally, this is a divergent quantity: product of valued operator distribution in the same
point x: regularisation and renormalisation are required.
Within zeta-function regularisation (Dowker, Hawking, and manyothers)< Φ(x)2 >= ζ(1|Lβ )(x) ,where ζ(z|Lβ )(x) is the analytic continuation of the local zetafunction associated with the operator elliptic self-adjoint Lβ
Lβ = −∂τ2 − ∇2 ,defined on S1 × R3. The local zeta-function is defined with Re z
sufficiently large by means∞
1dttz−1Kt(x, x) ,ζ(z|Lβ )(x) =Γ(z) 0where the heat-kernel is ( spectral theorem)Z

Kt(x, x) =< x|e−tLβ |x >=1

2

X − 4π2 n2β(4πt)3/2 neβKM Institute, Nagoya — Febraury 7 201249

Jacobi-Poisson formula leads to
X − n2β 21
e 4t .
Kt(x, x) =2(4πt) nR∞
The term n = 0 leads to divergent integral 0 dttz−3, but within

Gelfand analytic continuation it vanishes and one hasΓ(2 − z)ζ(z|Lβ )(x) =8π 2Γ(z)

!z−2
2
β

4

ζR (4 − 2z) ,where ζR (z) is the Riemann zeta-function. This analytic continuation2π
is regular at z = 1, recalling that ζR (2) = 6 , one has< Φ(x)2 >=

1
T2
.
=
2
12β
12

The regularised vacuum expectation value of the observable Φ2 gives
the temperature of the quantum field in thermal equilibrium:a quantum thermometer.

KM Institute, Nagoya — Febraury 7 201250

Another derivation (more traditional)
Quantum thermal Euclidean Wigthman function periodic of period β
in the imaginary time isWβ (x, x′) =< Φ(x)Φ(x′) >=1 X
1~′|2 + (τ − τ ′ + nβ)2
4π 2 n |~x−x

The term n = 0 is singular when x → x′.
Subtracting this term (renormalization), one has in the coincidence
limit
∞
X
1
1
2
Wβ (x, x) =< Φ(x) >=2π 2β 2 n=1 n2

Again< Φ(x)2 >=

1
T2
=
2
12β
12

KM Institute, Nagoya — Febraury 7 201251

Motivated by this argument, consider again the quantum probe in a
FRW conformally flat space-time.
Off-diagonal Wigthman function isW (x, x′) =< Φ(x)Φ(x′) >=

1
1
,
2
2
′
4π σ (x, x )where
σ 2(x, x′) = a(x)a(x′)(x − x′)2 ,recall a(x) being the conformal factor. In the limit x → x′, formally
one has< Φ(x)2 >= W (x, x) ,but W (x, x) is ill defined:again regularisation and renormalisation

KM Institute, Nagoya — Febraury 7 201252

Use point splitting regularization: take W (x, x′) and evaluate the limit
x′ → x. Then subtract the leading divercence and define the local
temperature as
T2=< Φ(x)2 >R = F.P. lim W (x, x′) ,12x′ →x
where F.P. stands for finite part prescription, the two points arejoined by a Kodama trajectory parametrized by the proper time.
In the stationary space-times we have considered, the relevant
Wightman functions of the probe all have the specific form
α210W (s, s + ε) = − 2
α0 .
2
4π 4 sinh (ε 2 )KM Institute, Nagoya — Febraury 7 2012
In fact,
for a generic static black holeκH
α0 = √ ,
V0

VH′
κH =
,
2

and for the de Sitter space-time in flat FRW form

In invariant form

α0 = qH0
2H 2
1 − R00κH
α0 = √
χ0.

53

KM Institute, Nagoya — Febraury 7 2012

54

Within point splitting regularization

2
α2
1
α
1
1
0
0
=− 2 2+W (s, s + ε) = − 2+ O(ε) .
α)2
4π 4 sinh (ε 0
4π ε
12 2π
2

Naive Renormalization (F.P. prescription) gives
α0
TH
T0 =
=√
2π
χ0
in agreement with H-J tunneling and Unruh-DeWitt detector technique.
Again, the local dependence comes from the localisation of the Kodama observer at R = R0, R being the areal radius of BH.

KM Institute, Nagoya — Febraury 7 2012

55

Summary and Concluding remarks
Aim of the talk: understand thetemperature-versus-surface gravity paradigm.
The tunnelling in its H-J covariant version leads to a very simple
result for a spherically symmetric dynamical black holeω

− TH

RateΓ ≃ e

− TE

H ≃ e

0 ,

T
T0 = √ H
χ0

where kH is the Hayward dynamical surface gravity. To what extend
one may talk of dynamical temperature?
Answer has been tested with QFT techniques as the Unruh-DeWitt
detector and the Buchholz proposal.
In the stationary cases ( static black holes, dS-FRW ) complete
agreement with QFT methods.
In the dynamical case, positive answer for asymptotically dS spacetimes. In general, still an unsolved issue. Work in progress.

M. Angheben, M. Nadalini, L. Vanzo and S. Zerbini, JHEP 0505, 014(2005); M. Nadalini, L. Vanzo and S. Zerbini, J. Physics A: Math.
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R. Di Criscienzo, M. Nadalini, L. Vanzo, S. Zerbini and G. Zoccatelli,Phys. Lett. B657, 107 (2007).
S. A. Hayward, R. Di Criscienzo, L. Vanzo, M. Nadalini and S. Zerbini,Class. Quant. Grav. 26 , 062001 (2009).
R. Di Criscienzo, S. A. Hayward, M. Nadalini, L. Vanzo and S. Zerbini,Class. Quant. Grav. 27, 015006 (2010).
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G. Acquaviva, R. Di Criscienzo, L. Vanzo, S. Zerbini, “Unruh-DeWitt
detector and the interpretation of the horizon temperature in spherically symmetric dynamical space-times, [arXiv:1101.5254 [hep-th]].