Download Calculation of Hawking Radiation as Quantum Mechanical Tunneling

UPTEC F07 085
Examensarbete 20 p
Augusti 2007
Calculation of Hawking Radiation
as Quantum Mechanical Tunneling
Petra Lange
Abstract
Calculation of Hawking Radiation as Quantum
Mechanical Tunneling
Petra Lange
Teknisk- naturvetenskaplig fakultet
UTH-enheten
Besöksadress:
Ångströmlaboratoriet
Lägerhyddsvägen 1
Hus 4, Plan 0
Postadress:
Box 536
751 21 Uppsala
The Hawking radiation derived from quantum field theory shows a spectrum that is
precisely thermal. In this thesis, the Hawking radiation is derived as a tunneling
process through the event horizon of a black hole. The tunneling rate is related to the
imaginary part of the action of the tunneling particle. Since energy conservation is
respected in the calculation, the obtained tunneling rate correponds to a spectrum
which is not precisely thermal but has an additional correction term.
Telefon:
018 – 471 30 03
Telefax:
018 – 471 30 00
Hemsida:
http://www.teknat.uu.se/student
Handledare: Joseph Minahan
Ämnesgranskare: Ulf Lindström
Examinator: Tomas Nyberg
ISSN: 1401-5757, UPTEC F07 085
Contents
1 Introduction
2
2 Quantum Field Theory
2
2.1 Quantum Field Theory in Flat Spacetime . . . . . . . . . . . . 2
2.2 Quantum Field Theory in Curved Spacetime . . . . . . . . . . 6
2.3 Surface Gravity and Hawking Radiation . . . . . . . . . . . . 10
3 The WKB-approximation
12
4 Hawking Radiation as Tunneling
15
4.1 The Painlevé Coordinates . . . . . . . . . . . . . . . . . . . . 16
4.2 Particle Tunneling . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Anti-particle Tunneling . . . . . . . . . . . . . . . . . . . . . . 20
5 Discussion
21
1
1
Introduction
Classically, anything that enters a black hole is trapped, without any possibility to escape. In 1974, Stephen Hawking showed that, quantum mechanically, black holes aren’t really black but radiate energy continuously.
Hawking used the technique of quantum field theory in his calculation, but
the radiation can also be described as pair production near the horizon followed by quantum mechanical tunneling of one of the particles. The purpose
of this thesis is to carry out a detailed calculation of the Hawking radiation
from a Schwarzschild black hole, by tunneling of a massless shell through the
event horizon of the black hole.
The thesis consists of three main parts. The first part gives a description
of quantum field theory in curved spacetime and how Hawking radiation
arises because of the curvature of spacetime. This part is mainly based on
Carroll’s book Spacetime and Geometry [1] but also Peskin and Schroeder
[2] has been studied. The second part describes the WKB-approximation
method. This part is based on Bransden and Joachain’s book Quantum
Mechanics [3], where the method is described in detail. In the third part of
the thesis the calculation of Hawking radiation as tunneling is carried out.
This part is based on one paper of Parikh and Wilczek [4] and two papers of
Parikh [5] and [6].
2
Quantum Field Theory
In quantum mechanics, the quantization of single particles is described. This
description is suitable when the kinetic energy is much less than the rest
mass. At high energies, the relativistic theory makes creation and annihilation of particles possible and the description in terms of single particles fails.
Quantum field theory describes the quantization of fields and is a suitable
description at high energies, since creation and annihilation of particles are
allowed. A quantum field is a way to describe a system of infinitely many
particles. In this section quantum field theory in flat spacetime and curved
spacetime is treated. In curved spacetime a general metric and a static metric
are considered.
2.1
Quantum Field Theory in Flat Spacetime
In classical field theory, the simplest field is the real scalar field φ(xµ ). The
equation of motion for the real scalar field of mass m in Minkowski space is
2
the Klein-Gordon equation
η µν ∂µ ∂ν φ − m2 φ = 0.
(2.1)
This equation is solved by the plane wave
φ = φ0 e−ωt+ik·x .
(2.2)
The frequency ω is related to the wave vector k by the dispersion relation
ω 2 = k2 + m2 .
(2.3)
The dispersion relation only determines the frequency of the solution up to
a sign. Therefore, there is not just one single solution to the Klein-Gordon
equation, but an entire set of mode solutions. The set is parametrized by
the wave vector k and the sign of the frequency ω. If the set is complete
and orthonormal, it can be used to express the general solution to the KleinGordon equation. To make sense of “orthonormal”, an inner product must
be defined on the space of solutions. An appropriate inner product is
!
(φ1 , φ2 ) = −i
(φ1 ∂t φ∗2 − φ∗2 ∂t φ1 ),
(2.4)
Σt
where Σt is a constant-time hypersurface. For an orthonormal set {fk }
(fk1 , fk2 ) = δ n−1 (k1 − k2 ),
(2.5)
so that the inner product vanishes unless the wave vectors k are equal for
both modes.
Since the frequency is only determined up to a sign by the wave vector,
a mode fk is said to be positive-frequency if it satisfies
∂t fk = −iωfk ,
ω > 0.
(2.6)
Complex conjugation of a mode changes the sign of the frequency, so the
complex conjugate fk∗ of the mode satisfies
∂t fk∗ = iωfk∗ ,
ω > 0,
(2.7)
and is therefore said to be negative frequency. The complex conjugate modes
are orthonormal to each other but with negative norm
(fk∗1 , fk∗2 ) = −δ n−1 (k1 − k2 ),
3
(2.8)
and they are orthogonal to the original modes
(fk1 , fk∗2 ) = 0.
(2.9)
The set is complete by including the conjugate modes.
Expanding the solution to the Klein-Gordon equation in terms of the
modes gives
!
φ=
d3 k (ck fk + c∗k fk∗ ) .
(2.10)
On quantization, the classical field φ(xµ ) becomes an operator that satisfies the canonical commutation relations
[φ(t, x), φ(t, x# )] = 0
[π(t, x), π(t, x# )] = 0
[φ(t, x), φ(t, x# )] = δ (n−1) (x − x# ).
The field operator φ(t, x) is then
!
"
#
φ(t, x) = dn−1 k âk fk (t, x) + â†k fk∗ (t, x) .
(2.11)
(2.12)
The coefficients of the modes in equation (2.12) can now be interpreted as
creation operators â†k and annihilation operators and âk . By inserting the
expansion into (2.11), the creation and annihilation operators have the commutation relations
[â
$ k , âk’ ]%= 0
↠, ↠= 0
(2.13)
$ k k’ %
âk , â†k’ = δ (n−1) (x − x# ).
The creation and annihilation operators can be used to define a basis, where
the basis states are the eigenstates of the number operator. The vacuum
state |0" is a state that is annihilated by all annihilation operators, i.e.
âk |0" = 0 for all k.
(2.14)
Excited states are created by repeated action by â†k . The number operator
is defined by
n̂k = â†k âk
(2.15)
and obeys
n̂ki |n1 , n2 , ..., ni , ..., nj " = ni |n1 , n2 , ..., ni , ..., nj " .
4
(2.16)
The basis formed by the eigenstates of the number operator is known as a
Fock basis. By means of these eigenstates and the number operator, the
number of particles seen by an observer can be described.
A Lorentz transformation by velocity v = dx/dt will lead to the new
!
coordinates xµ
t# = γt − γv · x, x# = γx − γvt,
(2.17)
√
where γ = 1/ 1 − v 2 . Inverting the transformation gives
t = γt# − γv · x# ,
x = γx# − γvt# .
(2.18)
The transformation of basis vectors is
∂xµ
∂ =
∂µ ,
∂xµ!
µ!
(2.19)
so the time derivative of the mode function can be expressed as
∂t! fk =
∂xµ
∂µ fk .
∂t#
(2.20)
From the inverse tranformation
∂t
=γ
∂t#
and
∂x
= γv,
∂t#
(2.21)
and differentiating the mode function with respect to t and x gives
∂t fk = −iωfk
and ∂x fk = ikfk .
(2.22)
The time derivative of the mode functions in the boosted frame is
∂t! fk = γ(−iω)fk + γv · (ik)fk = −iω # fk ,
(2.23)
ω # = γω − γv · k
(2.24)
where
is just the frequency in the boosted frame. The mode functions that were
positive frequency in the original frame, are still positive frequency in the
new boosted frame, but with boosted frequency. In Minkowski space, modes
are classified as positive- and negative-frequency with respect to the time
direction, which is a Killing vector. Since the Killing vectors are related
by Lorentz transformations in Minkowski space, a mode that is positivefrequency in one frame will still be positive-frequency in a boosted frame,
and observers will agree on the number of particles present.
5
2.2
Quantum Field Theory in Curved Spacetime
When the theory is generalized to curved spacetime, some of the ideas that
were essential in flat spacetime have to be given up. In this section a general
metric is considered and the field φ(xµ ) is the real scalar field of mass m.
The equation of motion for this field is
g µν ∇µ ∇ν φ − m2 φ = 0,
(2.25)
[φ(t, x), φ(t, x# )] = 0
[π(t, x), π(t, x# )] = 0
i
[φ(t, x), π(t, x# )] = √−g
δ (n−1) (x − x# ),
(2.26)
where g µν is the inverse metric. The symbol ∇µ denotes the covariant derivative, which is the curved spacetime generalization of the partial derivative
∂µ .
The theory is quantized by imposing the commutation relations
where g is the determinant of the metric.
An inner product on the solutions in curved spacetime is defined by
!
√
(φ1 , φ2 ) = −i (φ1 ∇µ φ∗2 − φ∗2 ∇µ φ1 )nµ γdn−1 x.
(2.27)
Σ
Here Σ is a spacelike hypersurface, nµ is a unit normal vector and γij is the
induced metric from gµν on the hypersurface.
Since a general metric is considered, the time direction is not necessarily
a Killing vector. Therefore, the solutions to the equation of motion might
not be separated into space-dependent and time-dependent parts, and the
modes cannot be classified as positive- and negative-frequency. Still, a set of
solutions {fi } can be found that are orthonormal
(fi , fj ) = δij ,
(2.28)
(fi∗ , fj∗ ) = −δij ,
(2.29)
and
for the conjugate modes. These modes can be chosen to be a complete set
and the field operator can be expanded in terms of these modes
#
&"
φ=
âi fi + â†i fi† .
(2.30)
By (2.26), the annihilation and creation operators have the canonical commutation relations
[â
$ i , âj ]% = 0
↠, ↠= 0
(2.31)
$ i j%
†
âi , âj = δij
6
and there will be a vacuum state |0f " that is annihilated by all the annihilation operators, i.e.
âi |0f " = 0 for all i.
(2.32)
By acting with the creation operator â†i , a Fock basis can be defined. The
number operator is defined by
n̂f i = â†i âi .
(2.33)
However, the basis modes fi are not unique. An alternative set of modes
{gi } that has the same properties as the fi -modes can be used to expand the
field operator
#
&"
φ=
b̂i gi + b̂†i gi∗ .
(2.34)
The creation and annihilation operators have the commutation relations
$
%
b̂i , b̂j = 0
%
$
(2.35)
b̂†i , b̂†j = 0
%
$
b̂i , b̂†j = δij
and there is a vacuum state |0g " for which
b̂i |0g " = 0 for all i.
(2.36)
A number operator can also be defined by
n̂gi = b̂†i b̂i .
(2.37)
A Fock basis for the g-observer can be constructed using this vacuum state
and the creation operator just as well.
In a general curved spacetime there is no reason to prefer one set of modes
to any other. Every observer classifies modes to be positive- and negativefrequency with respect to his proper time and defines particles with respect
to that set. There is no reason why they should agree on how many particles
are detected. To see this, each set of modes can be expanded in terms of the
other by the transformation
)
' (
gi = j (αij fj + βij fj∗)
'
(2.38)
∗
fi = j αji
gj − βji gj∗ .
Such a transformation is known as a Bogolubov transformation, and the
coefficients are known as Bogolubov coefficients. Since the mode functions
are orthonormal, the coefficients can be found by
αij = (gi , fj )
βij = −(gi , fj∗ ).
7
(2.39)
The Bogolubov coefficients can also be used to express the operators in terms
of each other,
#
' "
∗ †
âi = j αji b̂j + βji b̂j
#
(2.40)
' "
b̂i = j αik âj + βij∗ â†j .
If the f -observer is in a vacuum state where no particles are observed, it is
now possible to find out how many particles are observed by the g-observer.
This can be done by calculating the expectation value of the number operator
of the g-mode in the f -vacuum
&
%0f | b̂†i b̂i |0f " =
|βij |2 .
(2.41)
j
If any of the coefficients βij are not vanishing, the g-observer will detect
particles. This means that what looks empty to one observer might be filled
with particles to another.
In a static spacetime, the coordinates can be chosen so that the metric is
independent of the time coordinate and no time-space cross terms are present
∂0 gµν = 0
g0i = 0.
(2.42)
Since the metric is independent of the time coordinate, the time direction is a
Killing vector. If restrictions are made to only consider spherically symmetric
metrics, the modes can be expressed in terms of the time coordinate t and the
radial coordinate r. Now the equation of motion can be written on a form
that separates time-dependent operators from space-dependent operators.
The solution to such an equation can be separated into a time-dependent
and a space-dependent factor
fk (t, r) = e−iωk t fk (r).
(2.43)
Relative to t, these solutions can be described as positive-frequency
∂t fk (t, r) = −iωk fk (r),
ω>0
(2.44)
ω > 0.
(2.45)
and negative-frequency
∂t fk∗ (t, r) = iωk fk∗ (r),
In a static spacetime two different observers should be considered. A static
observer at infinity is one whose proper time τ is proportional to the Killing
time t. The static observer can then classify the modes as positive- and
8
negative-frequency with respect to t and use the modes to express the field
operator
!
"
#
φ(t, r) = dk âk e−iωk t fk (r) + â†k eiωt fk∗ (r) .
(2.46)
These modes are singular at the horizon.
A free-falling observer detects nothing abnormal when crossing the horizon. The modes of this observer are denoted by gk (t, r). Since the modes are
regular at the horizon, a vacuum state |0g " can be defined that looks empty
to the free-falling observer . The field operator expanded in terms of these
modes is
!
"
#
φ(t, r) = dk b̂k gk (t, r) + b̂†k gk∗ (t, r) .
(2.47)
To calculate how many particles the static observer will see when the freefalling observer is in the vacuum state, the annihilation and creation operators of the static observer can be expressed in terms of the creation and
annihilation operators of the free-falling observer
!
"
#
âk = dk # αk! k b̂k! + βk∗! k b̂†k!
(2.48)
where
αkk! = (gk , fk! )
(2.49)
βkk! = −(gk , fk∗! ).
The average number of particles that will be observed by the static observer
is
!
†
n̄k = %0g | âk âk |0g " = dk # |βkk! |2 .
(2.50)
From statistical mechanics, the average number of particles with energy ωk
can be expressed as
1
n̄k =
.
(2.51)
exp(ωk /T ) − 1
A derivation of this expression can be found in Mandl [7]. By using the
density of states dωk /2π, the flux of outgoing particles with energies between
ωk and ωk + dωk is
1
dωk
.
(2.52)
F (ωk ) =
2π exp(ωk /T ) − 1
If |αkk! /βkk! | is independent of k # , the completeness relation
!
(
)
dk # |αkk! |2 − |βkk! |2 = 1
(2.53)
can be used to express the flux as
1
dωk
F (ωk ) =
.
2π |αkk! /βkk! |2 − 1
9
(2.54)
2.3
Surface Gravity and Hawking Radiation
The event horizon of a black hole is a null hypersurface. This hypersurface
separates the spacetime points that are connected to infinity by timelike
paths from those points that are not.
If a Killing vector field is null along some null hypersurface, the hypersurface is said to be a Killing horizon. In a static spacetime, the event horizon is
a Killing horizon and the Killing vector field is K µ = (∂t )µ . To every Killing
horizon, a quantity called surface gravity κ is associated. If the Killing vector
χµ is null on the null hypersurface Σ, then χµ is a normal vector to Σ and
obeys the geodesic equation
χµ ∇µ χν = −κχν .
(2.55)
The right hand side arises because ξ µ may not be affinely parametrized. The
parameter κ is the surface gravity, and it is constant over the horizon. In a
static spacetime, the time-translation Killing vector ∂t can be normalized by
Kµ K µ (r → ∞) = −1.
(2.56)
This fixes the surface gravity at the Killing horizon.
For a static observer, the four-velocity is proportional to the time-translation
Killing field K µ , where
K µ = V (x)U µ .
(2.57)
The four-velocity is normalized to U µ Uµ = −1, so the function
*
V = −K µ Kµ ,
(2.58)
which is the magnitude of the Killing field. By the normalization of K µ , V
is zero at the horizon and unity at infinity. V is called the ”redshift factor”
since it relates the emitted frequency to the observed frequency of a photon,
measured by a static observer. The conserved energy of a photon with fourmomentum pµ is E = −pµ K µ while the frequency observed by an observer
with four-velocity U µ is ω = −pµ U µ , therefore
ω=
E
.
V
(2.59)
The four-acceleration aµ can be expressed as aµ = U σ ∇σ U µ . The acceleration can be expressed in terms of the redshift factor
aµ = ∇µ ln V.
10
(2.60)
The magnitude of the acceleration is
*
a = V −1 ∇µ V ∇µ V .
(2.61)
This will go to infinity at the Killing horizon. The surface gravity can be
shown to be
*
κ = V a = ∇µ V ∇µ V
(2.62)
and this is evaluated at the horizon.
A free-falling observer near a black hole finds a scalar field to be in the
vacuum state. A static observer right outside the black hole will not find a
vacuum state but will see radiation. Since the static observer is very close
to the horizon, he can be considered to have an acceleration a. The redshift
factor at his position is V . The temperature of the observed radiation is
T = a/2π. As the radiation propagates to infinity it will be redshifted. The
temperature measured by the observer at infinity is
T∞ =
V
V a
T =
.
V∞
V∞ 2π
(2.63)
At infinity, the redshift factor V∞ → 1, so the observed temperature is
T∞ = lim
Va
κ
=
.
2π
2π
(2.64)
The radiation that an observer far from the black hole will see is the Hawking
radiation. The flux of radiation is thermal and it is proportional to the surface
gravity of the black hole.
For a Schwarzschild black hole the redshift factor is found to be
+
2M
(2.65)
V = 1−
r
and the acceleration
a=
The surface gravity κ is
M
(
r2 1 −
)
2M 1/2
r
.
M
.
r2
Evaluating this at the event horizon gives the surface gravity
κ=Va=
κ=
1
4M
11
(2.66)
(2.67)
(2.68)
for a Schwarzschild black hole. The temperature of the radiation measured
by an observer at infinity is then
T =
1
8πM
(2.69)
and this corresponds to a flux
F (ω) =
3
1
dω
.
2π exp(8πM ω) − 1
(2.70)
The WKB-approximation
The WKB-approximation is a method that can be used to find the wave
function when the potential energy is a slowly varying function of position.
It is a suitable method to use when penetration of a potential barrier is
considered.
The equation of motion for a particle of mass m that moves in a potential
V (x) in one dimension is the Schrödinger equation
−
h̄2 d2 ψ(x)
+ V (x)ψ(x) = Eψ(x).
2m dx2
(3.1)
When the potential is a function of position, the solution is on the form
,
i
ψ(x) = ψ0 exp
S(x) ,
(3.2)
h̄
where ψ0 is a constant. Inserting this solution into the Schrödinger equation
(3.1) gives
ih̄ d2 S(x)
1
+
−
2
2m dx
2m
,
dS(x)
dx
-2
+ V (x) − E = 0.
(3.3)
This equation is non-linear and difficult to solve and therefore an approximation is needed. In the classical limit h̄ → 0. This means that h̄ can be
used as a parameter of smallness and the function S(x) can be expanded in
the power series
h̄2
S(x) = S0 (x) + h̄S1 (x) + S2 (x) + ...
2
12
(3.4)
In the classical limit S(x) = S0 (x). Inserting the expansion into equation
(3.3) gives
,
-2
1
dS0 (x)
+ V (x) − E = 0
(3.5)
2m
dx
dS0 (x) dS1 (x) i d2 S0 (x)
−
=0
(3.6)
dx
dx
2 dx2
,
-2
d2 S1 (x)
dS1 (x)
dS0 (x) dS2 (x)
−i
+
=0
(3.7)
+
2
dx
dx
dx
dx
Solving the first equation gives
S0 = ±
!
x
p(x# )dx# ,
(3.8)
*
where p(x) = 2m(E − V (x)) is the momentum at the point x. Using this
result in the second equation gives
S1 (x) =
i
ln p(x).
2
(3.9)
By (3.8) and (3.9) the WKB-approximation of the wave function in a classically allowed region (E > V ) is
, ! x
i
−1/2
#
#
ψ(x) = A (p(x))
exp
p(x )dx
(3.10)
h̄
,
!
i x
#
#
−1/2
+B (p(x))
exp −
p(x )dx .
h̄
The plus-sign corresponds to a wave moving in the positive x-direction and
the minus-sign corrsponds to a wave moving in the negative x-direction. An
approximation to the wave function can be obtained in classically forbidden
regions (E < V ) as well, and in this region the momentum p(x) is imaginary.
The wave function in this region is
,
!
1 x
−1/2
#
#
(3.11)
|p(x )| dx
ψ(x) = C |p(x)|
exp −
h̄
, ! x
1
−1/2
#
#
+D |p(x)|
exp
|p(x )| dx .
h̄
The criterion of validity for the WKB-approximation is
.
.
. h̄ dp(x) .
.
.
. (p(x))2 dx . << 1.
13
(3.12)
V (x) "
E
1
2
3
!
a
b
x
Figure 3.1: A potential barrier of height V(x). Incident particles have energy
E. The points a and b are classical turning points.
This criterion can be derived from the solution of equation (3.7) and is satisfied if the potential energy V (x) varies slowly, provided that the kinetic
energy E − V (x) is large enough. The WKB-solution will therefore not be a
suitable approximation close the points where E = V (x).
A potential barrier of height V (x) is described in Figure 3.1. The barrier
can be split into three different regions, which are denoted 1, 2 and 3 in
the figure. There are two classical turning points a and b where E = V .
The WKB-solutions (3.10) and (3.11) are only accurate far away from the
turning points. To find a solution that is valid near the turning points,
the classical momentum p2 (x) is approximated by p2 (x) ( A(x − x1 ). By
this approximation a “new” Schrödinger equation is obtained, which can
be solved exactly at the turning points and is a good approximation to the
solution of the Schrödinger equation far from the turning points. In this way,
the solutions to the new approximate Schrödinger equation are connection
formulae for the WKB-solutions.
If particles are assumed to be incident from the left, some of the them
will be reflected back into region 1, others will be transmitted through region
2 and emerge into region 3. In the third region the wave function is
, ! x
i
#
#
−1/2
(3.13)
p(x )dx , x > b
ψ3 (x) = A (p(x))
exp
h̄ b
which describes a wave moving in the positive x-direction. By the connection
formulae, the solution in region 2 is
, ! b
1
−1/2
#
#
ψ2 (x) = −i A |p(x)|
exp
|p(x )| dx , a < x < b
(3.14)
h̄ x
14
This wave function can be rewritten as a decreasing exponential for x > a
,
!
−1 x
−1/2 Λ
#
#
|p(x )| dx , a < x < b (3.15)
ψ2 (x) = −i A |p(x)|
e exp
h̄ a
where
1
Λ=
h̄
!
b
a
|p(x)| dx.
The solution in region 1 turns out to be
,
!
i x
−1/2 Λ
#
#
ψ1 (x) = −i (p(x))
e exp ±
|p(x )| dx ,
h̄ a
(3.16)
x < a.
(3.17)
The plus-sign is a right moving wave and corresponds to the incoming wave
and the minus-sign is a left moving wave which corresponds to the reflected
wave.
The probability current density associated with this wave is
j = v(|A|2 − |B|2 ) = v # |C|2 ,
(3.18)
where A, B and C are amplitudes of an incident, reflected and transmitted
wave respectively. v and v # are magnitudes of the velocities of the incident
and reflected waves. From this relation
|B|2 v # |C|2
+
= 1.
v |A|2
|A|2
(3.19)
The transmission coefficient T is defined to be
T =
v # |C|2
.
v |A|2
(3.20)
By using the transmitted and incident wave functions (3.13) and (3.17), the
transmission coefficient can be evaluated. The ratio of the velocities is equal
to the ratio of the corresponding momenta. This gives the transmission
coefficient
,
!
2 b
−2Λ
T =e
= exp −
|p(x)| dx .
(3.21)
h̄ a
4
Hawking Radiation as Tunneling
Although Stephen Hawking used the technique of quantum field theory in
his calculation, the radiation is often pictured as tunneling. According to
15
this picture, the radiation arises by pair creation just outside or inside the
event horizon followed by tunneling of one of the particles through the horizon. In this section, this schematic picture will be used to actually calculate
the Hawing radiation. A description of the appropriate coordinates is given
before the calculation is carried out.
4.1
The Painlevé Coordinates
Since the tunneling process takes place at the horizon, it is important that
the coordinate system behaves well there. The Schwarzschild metric
,
-−1
,
2M
2M
2
2
ds = − 1 −
dts + 1 −
dr2 + r2 dΩ2
(4.1)
r
r
is singular at the horizon since some of its components diverge and is therefore
not suitable to use. By defining a new time-coordinate
√
√
√
r − 2M
√
t = ts − 2 2M r + 2M ln √
(4.2)
r + 2M
and inserting it into the Schwarzschild metric, the line element
+
,
2M
2M
2
2
ds = − 1 −
dt + 2
dtdr + dr2 + r2 dΩ2
r
r
(4.3)
is obtained. The coordinates are known as the Painlevé coordinates. In
these coordinates, the metric is well-behaved at the horizon. Since all metric
components are independent of the time coordinate t, the time direction is
a Killing vector. There are, however, time-space cross terms so the metric
is stationary but not static. The Painlevé time t is precisely the proper
time τ of a radially free-falling observer. Modes can therefore be classified
as positive- and negative-frequency relative to t, and a vacuum state can be
defined which annihilates modes that are negative frequency with respect to
t. Since the proper time τ is proportional to t, the free-falling observer will
see the vacuum state as empty.
The radial null geodesics can be found by
+
,
2M
2M
dt2 + 2
dtdr + dr2 ,
(4.4)
0 = ds2 = − 1 −
r
r
which leads to

2
 +1 − 2M
2 r
ṙ =
 −1 − 2M
r
outgoing geodesics
ingoing geodesics.
16
(4.5)
When the particle is inside the black hole, r decreases for both ingoing and
outgoing trajectories so the particle is trapped inside the horizon.
When self-gravitation of the particle is taken into account, these equations
are changed. Self-gravitating particles were studied by Kraus and Wilczek [8].
By treating the particle as a spherical shell and using Hamiltonian methods
to solve the constraints of the theory, the gravitational degrees of freedom
were eliminated and an expression for the effective action of the shell was
obtained,
!
S = dt(pr ṙ − M+ ).
(4.6)
Here pr is the canonical momentum conjugate of r and M+ is the total mass
of the gravity-shell system, as seen from infinity. The solution r(t) to the
equations of motion that are obtained by extremizing (4.6), are the radial
null geodesics of the metric
3
42
+
2M
+
ds2 = −dt2 + dr +
dt + r2 dΩ2 .
(4.7)
r
This is the same line element as in (4.3), but with M replaced by M+ . When
the black hole mass M is held fixed and the total mass M+ is allowed to vary,
the total mass as seen from infinity is M+ = M + ω, where ω is the energy of
the shell. When instead the total mass is held fixed and the black hole mass
is allowed to vary, the mass M in (4.3) is replaced by M − ω.
4.2
Particle Tunneling
When a virtual particle pair is created just inside the horizon, the positive
energy particle tunnels outwards through the horizon. Outside the horizon
it materializes as a real particle and can escape to infinity, where it appears
as radiation. By energy conservation, the mass of the black hole must go
down when the particle is radiated. For black holes, the mass is related to
the radius. When the mass goes down, the hole shrinks and the radius of the
horizon decreases.
Tunneling through a potential barrier was described in section 3. By
(3.21), the tunneling rate is related to the imaginary part of the action in the
classically forbidden region. The tunneling rate of a particle out of a black
hole is therefore
,
! rout
Γ ∼ exp −2 Im
pr dr .
(4.8)
rin
17
The barrier arises because the horizon shrinks and the particle starts just
inside the initial position rin = 2M of the horizon and materializes just
outside the final position rout = 2(M − ω) of the horizon.
When the outgoing wave is traced back towards the horizon, its characteristic wavelength is infinitely blueshifted. This means that the geometry is
slowly varying over the wavelength and therefore the WKB-approximation
can be applied.
To calculate the tunneling rate the imaginary part of the action
! rout
Im S = Im
pr dr
(4.9)
rin
is evaluated. The radial momentum pr can be written as an integral
! rout
! rout ! pr
Im S = Im
pr dr = Im
dp#r dr.
(4.10)
rin
rin
0
By Hamiltons equation,
dH
= ṙ
dpr
the variable is changed from momentum to energy
! rout ! M −ω
dH
Im S = Im
dr.
ṙ
rin
M
The variable is changed again from H to ω #
! rout ! +ω
−dω #
dr.
Im S = Im
ṙ
rin
0
(4.11)
(4.12)
(4.13)
The particle moves on the outgoing geodesic obtained in equation (4.5), but
with the mass M replaced by M − ω # . When the total energy is held fixed
and the black hole energy is allowed to vary, it is the metric inside the shell
that determines the motion. Inserting ṙ from (4.5) and changing the order
of integration gives
! +ω ! rout
dr
2
dω # .
(4.14)
Im S = −Im
!
0
rin
1 − 2(Mr−ω )
By the substitutions
√
r = u and 2u du = dr, the action can be written as
! ω ! uout
2u2 du
*
Im S = −Im
dω # .
(4.15)
#
u − 2(M − ω )
0
uin
18
*
This integral has a pole at u = 2(M − ω # ). The integral is evaluated by
first making u complex and then displacing the pole into the complex plane
and deforming the contour around the pole. Since the particle has positive
energy, the wave function is on the form
!
e−iω t f (r).
(4.16)
A displacement ω # → ω # − i/ of the energy makes sure that positive energy
solutions decay in forward time
!
!
!
e−iω t f (r) → e−i(ω −i$)t f (r) = e−iω t e−$t f (r).
*
Taylor expansion of 2(M − ω # + i/) in / gives
*
2(M − ω # + i/) ≈
*
1
2(M − ω # ) + *
i/ + ...
2 2(M − ω # )
and by rescaling the second term
*
*
2(M − ω + i/) ≈ 2(M − ω) + i/ + ...
The pole has now been pushed into the upper half complex u-plane
! ω ! uout
2u2 du
*
Im S = −Im
dω # .
#
u − ( 2(M − ω ) + i/)
0
uin
(4.17)
(4.18)
(4.19)
(4.20)
By deforming the contour around this pole and using residue calculus, the
integral is evaluated to
! ∞
*
2u2 du
*
= −2πi( 2(M ω # ) + i/)2 ≈ −4πi(M − ω # ).
#
−∞ u − ( 2(M − ω ) + i/)
(4.21)
A detailed description of residue calculus can be found in Mathews and Howell [9].
The result in (4.21) gives the action integral
! +ω
Im S = +Im
4πi(M − ω # )dω # .
(4.22)
0
Solving this last integral gives
"
ω#
.
Im S = 4πω M −
2
(4.23)
The tunneling rate for a particle outwards through the horizon thus turns
out to be
"
"
ω ##
Γ ∼ exp −8πω M −
.
(4.24)
2
19
4.3
Anti-particle Tunneling
When the pair is created outside the horizon, the negative energy particle
tunnels inwards into the black hole. For negative energy particles, time goes
backwards so the time has to be reversed in the equations of motion. A time
reversal corresponds to t → −t and the line element with time reversed is
+
,
2M
2M
ds2 = − 1 −
dt2 − 2
dtdr + dr2 + r2 dΩ2 .
(4.25)
r
r
The radial null geodesics are obtained by
+
,
2M
2M
2
2
0 = ds = − 1 −
dt − 2
dtdr + dr2 ,
r
r
which leads to

2
 +1 + 2M
2 r
ṙ =
 −1 + 2M
r
outgoing geodesics
ingoing geodesics
(4.26)
(4.27)
The negative energy particle sees a geometry of a fixed black hole mass.
Taking self-gravitation into account will in this case replace M by M + ω
instead of M − ω. The action for an ingoing particle with negative energy is
! rin
! rin ! M
Im S =
pr dr = Im
dp#r dr.
(4.28)
rout
rout
M −ω
Using Hamiltons equation
dH
= ṙ
dpr
to change variables from momentum to energy gives
! M +ω ! rin
dH
Im S =
dr.
M
rout ṙ
(4.29)
(4.30)
Changing the variables from H to ω # and inserting ṙ from (4.27) gives for the
negative energy particle
! −ω ! rin
dr
2
Im S =
dω # .
(4.31)
!
2(M +ω )
0
rout −1 +
r
√
Substituting r = u and dr = 2u du gives
! −ω ! uin
2u2 du
*
dω # .
(4.32)
Im S = −Im
#
0
uout u − ( 2(M + ω ))
20
In this case the energy must be displaced into the upper complex plane so
#
#
that the negative energy
* wave function is decreasing, so ω → ω + i/. By
Taylor expansion of 2(M + ω # + i/)
*
*
1
2(M + ω # + i/) ≈ 2(M + ω # ) + *
i/ + ...
(4.33)
2 2(M + ω # )
and rescaling the second term gives
*
*
2(M + ω + i/) ≈ 2(M + ω) + i/ + ...
This pole is also pushed into the upper complex u-plane
! −ω ! uin
2u2 du
*
Im S = −Im
dω # .
#
0
uout u − ( 2(M + ω ) + i/)
(4.34)
(4.35)
By deforming the contour around the pole and using residue calculus
! −∞
2u2 du
*
= 4πi(M + ω).
(4.36)
#
+∞ u − ( 2(M + ω ) + i/)
This result gives the action integral
!
Im S = −Im
−ω
4πi(M + ω # )dω #
(4.37)
0
and the imaginary part of the action is
"
ω#
Im S = 4πω M −
.
(4.38)
2
The tunneling rate of a negative energy particle inwards through the horizon
is found to be
"
"
ω ##
Γ ∼ exp −8πω M −
.
(4.39)
2
Both particle and anti-particle tunneling contributes to the tunneling
rate, so that their amplitudes should be added. This however, only contributes with a pre-factor and does not change the exponential in (4.39).
5
Discussion
The tunneling rate (4.39) obtained in this calculation will reduce to Γ ∼
exp(−8πM ω) if the ω 2 term is neglected. This is a Boltzmann factor for a
particle with energy ω at the Hawking temperature T = 1/8πM . The correction term ω 2 arises since the energy conservation was taken into account.
This means that when energy conservation is enforced the spectrum of the
radiation is not precisely thermal.
21
Acknowledgements
I am very grateful to my supervisor, Joseph Minahan, for taking the time
and answering my questions. Without his patience and positive attitude I
would not have been able to finish this thesis. I also would like to thank Ulf
Lindström.
22
References
[1] Sean Carroll, Spacetime and Geometry, Addison Wesley, 2004.
[2] Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum
Field Theory, Westview, 1995.
[3] B.H. Bransden and C.J. Joachain, Quantum Mechanics, Prentice Hall,
2000.
[4] Maulik Parikh and Frank Wilczek, Hawking Radiation as Tunneling,
2001, arXiv:hep-th/9907001.
[5] Maulik Parikh, A Secret Tunnel Through The Horizon, 2004, arXiv:hepth/0405160.
[6] Maulik Parikh, Energy Conservation and Hawking Radiation, 2004,
arXiv:hep-th/0402166.
[7] F. Mandl, Statistical Physics, Wiley, 1988.
[8] Per Kraus and Frank Wilczek, Self-Interaction Correction to Black Hole
Radiance, 1994, arXiv:hep-th/9408003.
[9] John H. Mathews and Russell W. Howell, Complex Analysis for Mathematics and Engineering, Jones and Bartlett Publishers, 1997.
23