Hawking radiation [624274]

Hawking radiation
Padmanath M.
General Relativity 2009
Jan. 9, 2010.
Classical general relativity says arbitrary timelike paths stay inside their light cones inside the event horizon, and hence never
escape the horizon. But quantum field description allows the black holes to give out thermal radiation. The key factors that causes
this effect is the spacetime geometry near the event horizon of the black hole, and the fact that the notion of a “particle” is not an
invariant concept in such field theories. In this report, I give brief description of Unruh radiation and later argue under reasonable
assumptions that this implies the Hawking effect.
1 Introduction
Hawking demonstrated, that classical black holes radiate a thermal flux of quantum particles, and
hence evaporate, in his paper “Particle creation by black holes”in 1975 . This was against the
predictions made by classical general relativity. It is harder to solve wave equations in curved
spacetime than in flat spacetime, and hence we will carefully derive Unruh effect and extend the
idea to understand Hawking radiation. The first and the third sections of the report describes the
facts in flat spacetime, which are analogous to the key factors that causes Hawking radiation and
Unruh effect, and in the next section, Hawking radiation is explained.
2 Quantum field theory in flat spacetime
A quantum scalar field theory in flat spacetime
L=−1
2ηµν∂µφ∂νφ−1
2m2φ2
ηµν= (−1,1,1,1) EOM ∂µ∂µφ−m2φ= 0 (1)
with appropriate commutation relation between the fields φ,
[φ(t, /vector x), φ(t,/vectorx/prime)] = 0 ; [π(t, /vector x), π(t,/vectorx/prime)] = 0 ; [φ(t, /vector x), π(t,/vectorx/prime)] =iδ3(/vector x−/vectorx/prime) (2)
is solved by doing a Fourier expansion of the fields.
φ(t, /vector x) =Z
d3kh
ˆa/vectorkf/vectork+ ˆa†
/vectorkf∗
/vectorki
(3)
Defining the co-efficient operators( a/vectorkandˆa†
/vectork) of the basis modes as creation and annihilation oper-
ators and hence the vacuum state( |0f/angbracketright), we construct the Fock space and describe the spectrum in
terms of the Fock space. The f/vectorkmodes are said to be positive frequency, meaning they satisfy
∂tf/vectork=−iωf/vectork; ω >0 (4)
andf∗
/vectorkmodes are negative frequency modes satisfying
∂tf/vectork=iωf/vectork; ω >0. (5)
1

Let us now see how Fock space behaves under Lorentz transformations. We implicitly have
been taking advantage of the symmetries of Minkowski space, in using plane waves as a basis
for the solutions to the Klein Gordon equation. The crucial aspect of these modes is the ability
to distinguish between positive and negative frequencies, allowing for an interpretation of their
coefficients in the mode expansion of φas annihilation and creation operators. Under a boost
/vector v=d/vector x
dt, the time derivative of the modes in the boosted coordinates is
∂t/primef/vectork=dxµ
dt/prime∂µf/vectork
∂t/primef/vectork=γ(−iω)f/vectork+γ(i/vectork)·/vector vf/vectork
=−iω/primef/vectork(6)
where ω/primeis simply the frequency of the mode in the boosted coordinate. Clearly the state describ-
ing a collection of particles with certain momenta is boosted into a state describing same number
of particles, but with boosted momenta. Thus the total number operator coincide in both the coor-
dinates, and in particular the vacuum state coincide. Hence Fock space in all inertial coordinates
remains same, up to the boosting in momentum. The timelike Killing vector in the Minkowski
spacetime, aids in defining the sign of frequencies, while the invariance of Fock space is because
all the timelike Killing vectors are related by Lorentz transformation. Though the frequency of
a mode depends on the choice of the inertial coordinate, the decomposition as the positive and
negative frequencies is invariant.
3 Quantum field theory in curved spacetime
A similar approach of creation-annihilation operator formalism breaks down in a quantum theory of
fields in a curved spacetime. Because the choice of modes in different choice of coordinates doesn’t
keep the Fock space invariant. We can find a set of basis of modes, but the problem is that there
will be generally many such sets, with no way to prefer over the other. And the notion of vacuum
and particles sensitively depend on which set of modes one choose. Each observer measures the
proper time τalong his trajectory, and will define positive- and negative- frequency with respect to
that proper time.
Consider an observer Alice finds a set of solutions, f/vectork(xµ), that are orthonormal. Then the mode
expansion of φwith these modes is given by
φ=Z
d3k³
ˆa/vectorkf/vectork+ ˆa†
/vectorkf∗
/vectork´
(7)
The coefficients ˆa/vectorkandˆa†
/vectorkobey commutation relations
[ˆa/vectork,ˆa/vectork/prime] = 0 [ˆ a†
/vectork,ˆa†
/vectork/prime] = 0 [ˆ a/vectork,ˆa†
/vectork/prime] =δ3(/vectork−/vectork/prime) (8)
The vacuum state ( |0f/angbracketright) is annihilated by the annihilation operator.
ˆa/vectork|0f/angbracketright= 0 (9)
The number operator is given by
ˆn/vectorkf= ˆa†
/vectorkˆa/vectork(10)
Consider another observer Bob, with a different set of solutions, g/vectork(xµ), that are orthonormal.
Then the mode expansion of φwith these modes is given by
φ=Z
d3k³
ˆb/vectorkg/vectork+ˆb†
/vectorkg∗
/vectork´
(11)
2

The corresponding commutation relations are given by,
[ˆb/vectork,ˆb/vectork/prime] = 0 [ ˆb†
/vectork,ˆb†
/vectork/prime] = 0 [ ˆb/vectork,ˆb†
/vectork/prime] =δ3(/vectork−/vectork/prime) (12)
The vacuum state ( |0g/angbracketright) is defined as
ˆb/vectork|0g/angbracketright= 0 (13)
And the number operator is given by
ˆn/vectorkg=ˆb†
/vectorkˆb/vectork(14)
Here Alice and Bob needn’t agree in the number of particles they observe. Infact the vacuum
state with respect to Alice will be a state with non-zero number of particles for Bob. Let us make
this explicit. Expanding one mode in terms of the other.
g/vectork=Z
d3/vectork/prime(α(/vectork,/vectork/prime)f/vectork/prime+β(/vectork,/vectork/prime)f∗
/vectork), f /vectork=Z
d3/vectork/prime(α∗(/vectork/prime,/vectork)g/vectork/prime−β∗(/vectork/prime,/vectork)g∗
/vectork/prime) (15)
This set of transformation is called Bogoliubov transformation and the coefficients α(/vectork,/vectork/prime)and
β(/vectork,/vectork/prime)are called Bogoliubov coefficients. These coefficients can be determined using the or-
thonormality of the mode functions.
α(/vectork,/vectork/prime) =/angbracketleftf/vectork/prime, g/vectork/angbracketright β(/vectork,/vectork/prime) =/angbracketleftf∗
/vectork/prime, g/vectork/angbracketright (16)
Bogoliubov coefficients can be used to transform between the operators
ˆa/vectork=Z
d3/vectork/prime(α(/vectork/prime,/vectork)ˆb/vectork/prime+β(/vectork/prime,/vectork)ˆb†
/vectork/prime), ˆb/vectork=X
/vectork/prime(α∗(/vectork,/vectork/prime)ˆa/vectork/prime−β∗(/vectork,/vectork/prime)ˆa†
/vectork/prime) (17)
Now the vacuum state observed by Alice |0f/angbracketright, would be a state with /angbracketleft0f|R
d3/vectorkˆng/vectork|0f/angbracketrightnumber
of particles for Bob. That is given by
/angbracketleft0f|Z
d3/vectorkˆng/vectork|0f/angbracketright=/angbracketleft0f|Z
d3/vectorkˆb†
/vectorkˆb/vectork|0f/angbracketright
=ZZZ
d3/vectorkd3/vectork/primed3/vectork/prime/prime(−β(/vectork,/vectork/prime))(−β∗(/vectork,/vectork/prime/prime))/angbracketleft0f|ˆa/vectork/primeˆa†
/vectork/prime/prime|0f/angbracketright
=ZZ
d3/vectorkd3/vectork/primeβ(/vectork,/vectork/prime)β∗(/vectork,/vectork/prime)
=ZZ
d3/vectorkd3/vectork/prime|β(/vectork,/vectork/prime)|2(18)
In general, this quantity doesn’t vanish. This is explicit from (17). And hence vacuum state ob-
served by an observer will contain non-zero number of particles when observed by another ob-
server.
4 Unruh radiation
Unruh radiation states that an accelerated observer in the traditional Minkowski vacuum state will
observe a thermal spectrum of particles. The basic idea is that observers with different notions of
positive and negative frequency modes will disagree on the particle content of a given state. For
a uniformly accelerated observer in Minkowski space, the trajectory will move along the orbits of
a timelike Killing vector. Now if one expand the fields in modes appropriate to the accelerated
observer, and calculate the number operator in the ordinary Minkowski vacuum, he will find a
thermal spectrum of particles defined by his set of modes. Let us see into the details below.
3

Figure 1: Minkowski space in Rindler coordinates.
Consider a (1+1)D Minkowski space as seen by a uniformly accelerating observer. The metric in
inertial co-ordinates is given by ds2=−dt2+dx2. An observer is moving at uniform acceleration
of magnitude αin the x-direction. Then his trajectory is given by
xµ(τ); t(τ) =1
αsinh(ατ);
x(τ) =1
αcosh(ατ) (19)
Let us verify that this is a constant acceleration path. Four acceleration is given by
aµ=D2xµ
Dτ2=d2xµ
dτ2; at=αsinh (ατ);
ax=αcosh (ατ) (20)
The covariant and normal derivative are the same because all Christoffel symbols are zero in
Minkowski space. And the magnitude of acceleration is given by√aµaµ=α, which is a con-
stant. The path therefore corresponds to a constant acceleration of magnitude αand defines a
hyperboloid x2(τ) =t2(τ)+α2asymptoting to null paths, x=±t. That is an accelerated observer
travels from past null infinity to future null infinity rather than timelike infinity as would be reached
by geodesic observers. Choosing the coordinates ( η, ξ) on 2D Minkowski space
t(τ) =1
aeaξsinh(aη) x(τ) =1
aeaξcosh(aη) ( x >|t|)
−∞< η, ξ < +∞ (21)
⇒ η(τ) =α
aτ; ξ(τ) =1
aln(a
α) (22)
so that the proper time is proportional to ηand the spatial coordinate ξis constant. In particular an
observer with α=amoves along the path η=τ,ξ= 0.
The metric in these coordinates takes the form ds2=e2aξ(−dη2+dξ2).Region I in figure (1)
with this metric is called Rindler space. A Rindler observer is the one moving along a constant
acceleration path.
The metric components are independent of η, and hence ∂ηis a Killing vector.
∂η=∂t
∂η∂t+∂x
∂η∂x=a(x∂t+t∂x) (23)
4

This is the Killing field associated with a boost in the x-direction. It is clear from this expression
that this Killing field naturally extends throughout the spacetime. In regions II and III it is spacelike,
while in I and IV, it is timelike, with past-directed in region IV.The horizons x=±tare the Killing
horizons for ∂η. The redshift factor, defined as the magnitude of the norm of the Killing vector is
V=p
−KµKµ=eaξ. And the surface gravity is given by κ=p
∇µV∇µV=a. There is no
gravitational force, as we are in flat spacetime, but this surface gravity characterizes the acceleration
of Rindler observer.
One can also define coordinates ( η, ξ) in region IV by flipping the signs in the earlier definitions,
t(τ) =−1
aeaξsinh(aη) x(τ) =−1
aeaξcosh(aη) ( x >|t|)
−∞< η, ξ < +∞ (24)
The sign guarantees ∂ηand∂tpoint in opposite directions in region IV. Strictly speaking, one
cannot use ( η, ξ) simultaneously in regions I and IV, since the ranges of these coordinates are same
in each region. But it’s okay as long as one explicitly mention which region he is referring to.
Along the surface t= 0,∂ηis a timelike Killing vector orthogonal to the hypersurface t= 0,
except for the point x= 0, where it vanishes. This vector can therefore be used to define a set of
positive and negative frequency modes, on which one can build a Fock basis for a scalar field. The
massless KG equation in Rindler coordinates
¤φ=e−2aξ(−∂2
η+∂2
ξ)φ= 0 (25)
A normalized plane wave solution is gk= (4πω)−1
2e−iωη+ikξ, with ω=|k|. This solution ap-
parently have positive frequency in the sense that ∂ηgk=−iωgk. But we need our modes to be
positive frequency with respect to a future directed Killing vector and in region IV, the future di-
rected Killing vector is ∂−η=−∂η, rather than ∂η. Defining a new set of modes, to solve this
issue,
g(1)
k=1√
4πωe−iωη+ikξI
= 0 IV
g(2)
k= 0 I
=1√
4πωeiωη+ikξIV (26)
Each set of modes is positive frequency with respect to the appropriate future directed timelike
Killing vector.
∂ηg(1)
k=−iωg(1)
k ∂−ηg(2)
k=−iωg(2)
k (27)
These two sets with their conjugates form a complete set of basis modes for any solutions to the
wave equation throughout the spacetime. Denoting the associated annihilation operators as ˆb(1,2)
k,
the mode expansion in these modes can be written as
φ=Z
dk³
ˆb(1)
kg(1)
k+ˆb(1)†
kg(1)∗
k+ˆb(2)
kg(2)
k+ˆb(2)†
kg(2)∗
k´
(28)
This expansion is an alternative to our expansion in terms of the original Minkowski modes,
which in two dimensions takes the form, φ=R
dk³
ˆakfk+ ˆa†
kf∗
k´
. The difference between the
Minkowski modes and the Rindler modes occurs in the fact that an individual Rindler mode can
never be written as a sum of positive frequency Minkowski modes at t= 0. The Rindler mode have
only support on the half line, and such a function cannot be expanded in purely positive frequency
5

Minkowski plane waves. Thus the Rindler annihilation operators used to define |0R/angbracketrightare necessarily
superpositions of Minkowski creation and annihilation operators, so the two vacua cannot coincide.
Calculating the Bogoliubov coefficients relating the Minkowski and Rindler coordinates and
using them to determine the expectation value of the Rindler number operator in the Minkowski
vacuum, will give us the result. The calculation is straight-forward, but tedious. So we will find a
set of modes that share the same vacuum state as the Minkowski modes(although the description of
excited states may be different), but for which the overlap with the Rindler modes is more direct.
For this, we analytically extend the Rindler modes to the entire spacetime, and express this extended
modes in terms of the Rindler modes. We have,
e−a(η−ξ)=a(−t+x) I
=a(t−x) IV
ea(η+ξ)=a(t+x) I
=−a(t+x) IV (29)
⇒√
4πωg(1)
k=e−iωη+ikξ=e−iω(η−ξ)=aiω/a(−t+x)iω/a
√
4πωg(2)∗
−k=e−iωη+ikξ=e−iω(η−ξ)=aiω/aeπω/a(−t+x)iω/a(30)
Then√
4πωh√
4πω+e−πω/ag(2)∗
−ki
=aiω/a(−t+x)iω/a(31)
Now defining new normalized functions out of the Rindler modes as
h(1)
k=1p
2sinh(πω
a)³
eπω/2ag(1)
k+e−πω/2ag(2)∗
−k´
h(2)
k=1p
2sinh(πω
a)³
eπω/2ag(2)
k+e−πω/2ag(1)∗
−k´
(32)
It is explicit that these are well defined along the whole surface t= 0for both. Expanding the fields
in these new modes, we have
φ=Z
dk³
ˆc(1)
kh(1)
k+ ˆc(1)†
kh(1)∗
k+ ˆc(2)
kh(2)
k+ ˆc(2)†
kh(2)∗
k´
(33)
⇒ ˆb(1)
k=1p
2sinh(πω
a)³
eπω/2aˆc(1)
k+e−πω/2aˆc(2)†
−k´
ˆb(2)
k=1p
2sinh(πω
a)³
eπω/2aˆc(2)
k+e−πω/2aˆc(1)†
−k´
(34)
The original positive frequency Minkowski plane wave modes with k > 0,fk∝e−iω(t−x), are
analytic and bounded for complex (t, x)so long as Im(t−x)≤0. The same arguments hold for
h(1)
kso long as we take the branch cut for the imaginary power to lie in the upper half complex
(t−x)plane. Similar arguments apply to the h(2)
kmodes, which are analytic and bounded in the
lower half complex (t+x)plane, as are the positive frequency Minkowski plane wave modes with
k <0.
Consequently, unlike the original Rindler modes g(1,2)
k, the modes h(1,2)
kcan be expressed purely
in terms of positive frequency Minkowski modes fk. They therefore share the same vacuum state.
ˆc(1)
k|0M/angbracketright= ˆc(2)
k|0M/angbracketright= 0 (35)
6

The excited states will not coincide, but that is not required, as we are interested only in the vacuum
state. Now, an observer in region I will observe particles defined by operators ˆb(1)
k, the expected
number of such particles of frequency ωwill be
/angbracketleft0M|ˆn(1)
R(k)|0M/angbracketright=/angbracketleft0M|ˆb(1)†
kˆb(1)
k|0M/angbracketright
=1
2sinh(πω)
a/angbracketleft0M|e−πω/aˆc(1)
kˆc(1)†
k|0M/angbracketright
=1
e2πω/a−1δ(0) (36)
This is a Planck spectrum with temperature T=a
2π. The temperature T=a
2πis what would be
measured by an observer moving along the path ξ= 0, which feels an acceleration α=a. Then any
other path with ξ=a constant, feels an acceleration α(ξ) =ae−aξ, because of (22) and thus should
measure a temperature of α(ξ)/2π. The redshift between the observers is given by ω2=V1ω1/V2,
where V1andV2are the redshift factors. For our case under consideration, it is V1=eaξ1and
V2=eaξ2. Then ω2=eaξ1ω1/eaξ2. Thus if an observer at ξ= 0detects a temperature T=a/2π,
the observer at ξ=ξ2will see it to be redshifted to a temperature T=ae−aξ2/2π. In particular the
temperature redshifts all the way to zero as ξ→0. This explains that a Rindler observer at infinity
will nearly inertial and will define the same notion of vacuum and particles as ordinary Minkowski
observer.
5 Hawking Radiation
The same arguments as above holds in our new case, except that our new metric is that of a
Schwarzschild spacetime, which has a redshift factor(outside the event horizon) which goes to
a non-zero finite value at infinity. Consider a static observer at radius r1>2GM outside the
Schwarzschild black hole. Such an observer moves along orbits of the timelike Killing vector
K=∂t. The corresponding redshift factor V=p
−KµKµ=q
1−2GM
r1and the corresponding
magnitude of acceleration is given by a1=GM
r1√r1−2GM. The static observer in a Schwarzschild
spacetime is a constantly accelerating observer with respect to a freely falling observer. Hence the
vacuum observed by a freely falling observer near the event horizon, will be viewed by the static
observer as a state with thermal radiation coming out at a temperature, T1=a1/2π. Now an ob-
server at a very large distance, say r2> r1, will observe a radiation at temperature T2=a2/2π. The
radiation observed near r1, will propogate with a redshift of V2/V1, such that T2=V1T1/V2. And
hence the radiation from the event horizon will propagate to infinity with an appropriate redshift.
In the limits as follows
lim
r1→2GMV1a1→κ lim
r2→∞V2→1 (37)
where κis the surface gravity; for Schwarzschild black hole κ= 1/4GM. And hence the temper-
ature observed by an observer in Schwarzschild infinity is given by
T=κ
2π(38)
Unlike an accelerating observer in the flat spacetime, in Schwarzschild the static Killing vector has
a finite norm at infinity, and the radiation near the horizon redshifts to a finite value rather than
all the way to zero. And hence observers far away from the black hole, sees a thermal flux of
radiation emitted from the black hole at a temperature proportional to its surface gravity. This is
called Hawking effect, and the radiation is called Hawking radiation.
7

Figure 2: Vacuum fluctuations causing Hawking radiation
The presence of the Killing horizon is crucial for this effect. Infact a neutron star in the
Schwarzschild spacetime doesn’t emit Hawking radiation. The presence of the Killing Horizon(In
the Schwarzschild case it is the event horizon) causes the Killing vector to change it nature through
the spacetime.
One picturesque way to understand what is going on is to think of vacuum fluctuations being
represented by virtual particle-antiparticle pairs popping in and out of existence. Normally, the
pairs will always annihilate, and their effect is only indirect, through a renormalization of processes
coupled to virtual particles. In the presence of an event horizon, occasionally one member of a
virtual pair will fall into the black hole while its partner escapes to infinity. It is these escaping
particles, we observe as Hawking radiation. The total energy of the virtual pair must add to zero,
but then the infalling particle can have a negative energy as viewed from infinity. But this can be
expected as the timelike Killing vector is spacelike inside the event horizon.
The original calculation did by Hawking considers a black hole formed by gravitational collapse.
The Penrose diagram for the same is given in figure (3). In the far past, the spacetime is nearly
Minkowski, the largest gravitational effects being at the surface of the star, and we can assume that
the quantum effects is empty of in-particles near I−(|0/angbracketrightin). The star collapses to form a black hole.
Hawking found that near I+, the state |0/angbracketrightincontains a thermal flux of particles. The particles found
are known as Hawking radiation. The wave equation in the black hole spacetime is harder than in
Figure 3: Penrose diagram for a real Schwarzschild spacetime
8

the Minkowski case. In the black hole case, we don’t know global analytic solutions as we have
seen earlier.
Consider a wave packet peaked about frequency ωthat propagates inward from I−towards the
horizon of the black hole. Roughly, the wave scatters into two parts. A fraction 1−Γωof the packet
backscatters off the curved geometry, and propagates out to I+, without a change in frequency. The
remaining fraction Γωpropagates parallel to H−and is absorbed by the black hole horizon. It is the
second portion that leads to particle production. Hawking did the calculation by studying a wave
propagating backwards in time in the collapsing star spacetime.
A careful analysis gives us the resulting thermal spectrum as
/angbracketleftˆnω/angbracketright=Γω
e2πω/a−1(39)
where κis the surface gravity. Γωis called the grey body factor. This is equal to the fraction of
a wave which is absorbed by the black hole horizon H+for a wave which starts at I−. SoΓωis
just the classical absorption coefficient for scattering a classical scalar field off a black hole. In the
high frequency limit the wavelength will be small and backscattering can be neglected. At very
low frequencies the wavelength becomes greater than the Schwarzschild radius and backscattering
becomes significant. Though an analytic expression for the grey body factor is difficult to obtain,
limiting case behaviors can be obtained. For a scalar theory it obeys
Γ(ω)→1, ωGM /greatermuch1
Γ(ω)→A
4πω2, ωGM /lessmuch1 (40)
where A is the area of the black hole.
6 Comments on Hawking radiation
Hawking radiation consummates the marriage of black hole mechanics and thermodynamics. Sta-
tionary black holes act just like bodies of energy E=Min thermal equilibrium with temperature
T=κ/2πand entropy S=A/4G. This is a very large amount of entropy. From a statistical
mechanical point of view, this supports the enormous amount of microstates for a black hole. The
inclusion of quantum mechanics causes black hole to evaporate away. Hawking radiation escapes
to infinity, we may safely conclude that it will carry energy away from the black hole, which must
therefore shrink in mass. As the mass shrinks, the surface gravity increases, and with it the temper-
ature. This is evident from the negative specific heat of black hole due to the inverse proportionality
of the temperature with the mass and hence the energy. And hence there is a runaway process in
which the entire mass evaporates away in a finite time. But still this is a huge amount of time.
One other point worth noting is the information loss, which is known as information loss para-
dox. The fact that Hawking radiation is purely thermal, means that there is no way of conveying the
vast amount of information needed to specify the states implied by the huge value of entropy. Sup-
pose if we assemble two very different original states and collapse them into two black holes of the
same mass(in general same charge and same spin), they will radiate away into two indistinguish-
able clouds of Hawking particles. The information that went into the specification of the system
before it became a black hole seems to have erased. This is the famous information loss paradox.
Both quantum field theory and classical general relativity feature unitary evolution-the information
required to specify a state at early times is precisely equal to that needed to specify a state at later
times, since they are connected by equations of motion. But in the process of combining quantum
field theory and classical general relativity, this unitarity apparently seems to be violated.
9

7 Conclusions
The basic notions of particles and vacuum state in quantum field theory in flat spacetime is not
valid, for a quantum field theory in curved spacetime. The spacetime geometry properties of a
Schwarzschild spacetime near the event horizon, combined with the difficulty in the notion of
particles between the observers causes black holes to emit radiation, which the theory of classical
general relativity doesn’t allows. The peak temperature of the thermal radiation from the black hole
is found to be proportional to the surface gravity of the black hole near by the event horizon.
I acknowledge all my colleagues in C-338 for their help in understanding various materials
covered in the project. I also thank Prof. Sunil Mukhi for his exciting lectures, which helped me
understand the subject and be helpful in understanding the materials covered in the project.
References
[1] S. Carroll, Spacetime and geometry, An introduction to general relativity.
[2] J. Traschen, An introduction to black hole evaporation (2000), http://arxiv,org/gr-qc/0010055
10