Download Quantum Field Theory in Curved Spacetime and Horizon

Quantum Field Theory in
Curved Spacetime and Horizon
Thermodynamics
Thesis Submitted in partial fulfillment of the requirements
of BITS C421T/422T Thesis
By
Aditya Bawane
ID No. 2006B5B4368P
Under the supervision of
Prof. T Padmanabhan
Distinguished Professor, IUCAA, Pune
Birla Institute of Technology and
Science
Pilani, Rajasthan
December 3, 2010
Contents
1 Introduction
1
2 Aspects of gravity and thermodynamics of horizons
2.1 Rindler horizons in flat and curved spacetimes . . . . . . . . .
2.2 Horizons in static spacetime . . . . . . . . . . . . . . . . . . .
2.3 Generalized models of gravity . . . . . . . . . . . . . . . . . .
3
3
4
6
3 Quantum effects in external electric field
13
3.1 Schwinger effect . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Quantum effects in external gravitational field
4.1 Unruh effect . . . . . . . . . . . . . . . . . . . . .
4.1.1 Rindler metric revisited . . . . . . . . . .
4.1.2 Quantum fields in accelerated frame . . . .
4.2 Hawking Effect . . . . . . . . . . . . . . . . . . .
4.3 Horizon temperature from path integrals . . . . .
4.4 Black Hole Thermodynamics . . . . . . . . . . . .
4.4.1 Lifetime and heat capacity of a black hole
4.4.2 Black hole entropy . . . . . . . . . . . . .
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5 Cosmological particle production
33
5.1 Classical scalar field in FRW spacetime . . . . . . . . . . . . . 33
5.2 Correspondence with Particle Production in Electric Field . . 37
5.3 An example: Particle production in a model universe . . . . . 38
6 Conclusion
41
1
Abstract
This thesis involves a review of existing results in
• Rindler Horizons and Horizon Temperature
• Lanczos-Lovelock models of gravity and some related results in higher
dimensional spacetime
• Quantum Field Theory in the background of electric field; Schwinger
effect
• Quantum Field Theory in the background of curved spacetime; Unruh
and Hawking effect
• Quantum fields in FRW metric; cosmological particle production
Apart from these, we outline the methods pursued to study the problem of
particle production in a model universe.
Chapter 1
Introduction
The thesis aims at studying Quantum Field Theory in Curved Spacetimes
and understanding what it tells us about Horizon Thermodynamics. Besides
the fact that the study of quantum fields in curved spacetimes (or in general,
non-inertial coordinates) reveals many new phenomena that cannot possibly
be foreseen classically, study in this area reveals several pointers as to what
a quantum theory of gravity could possibly be like, i.e., what phenomena
must a candidate theory be able to explain in the limiting case of a quantum field evolving in a classical gravitational background. Indeed, a perfect
explanation to all the intriguing peculiarities that emerge out of the study
of spacetime thermodynamics remain a holy grail for physicists of the day
seeking a consistent quantum theory of gravity.
The thesis is divided in 4 chapters, excluding the introduction. Chapter
2 explores some general notions about spacetime horizons and thermodynamics associated with them. Particular emphasis is on horizons that can
be approximated in a Rindler form in a neighbourhood around them. It,
however, turns out that the set of all such horizons is very large and these
horizons arise in cases of physical interest. The latter part of this chapter
presents a fairly detailed study of Lanczos-Lovelock theories of gravity and
how the many thermodynamic aspect of Einstein gravity smoothly generalize
to this wide class of theories. This therefore also serves to demonstrate that
many of the thermodynamical results obtained here are fairly robust, i.e.,
immune to the exact theory one is working with.
Chapter 3 briefly discusses quantum fields in the background of classical
electric field. In particular, we discuss what is known as Schwinger effect—the
production of charged particle-antiparticle pair in a sufficiently strong, constant electric field, treated classically. The formal correspondence between
time-dependent electric fields and FRW universes is also highlighted.
Chapter 4 deals with quantum fields in the background of classical grav1
itational fields. Two phenomena, which arise solely due to the consideration
of quantum fields in non-inertial coordinates — Unruh effect and Hawking
effect — are dealt with in some detail. Derivation for the simplified case of
a massless scalar field in 1+1 dimensional spacetime is provided. Motivated
by the results obtained from the study of Hawking effect, Chapter 4 also
explains the laws of black hole thermodynamics, particularly the generalized
second law of thermodynamics.
In Chapter 5 we study quantum fields in FRW universes. Calculations
are again simplified by considering a scalar field in a spatially flat FRW
spacetime. It is demonstrated how this problem can be translated to that
of a scalar field in Minkowski spacetime with time-dependent mass. We
also study particle production in the case of a universe which is radiationdominated at early times and de Sitter at late times.
Chapter 2 is based on material available in [3] and [4]. In particular,
Sections 2.1 and 2.2 are based on the former and Section 2.3 on the latter.
Chapter 3 is based on [4]. Sections 4.1 and 4.2 are based on [2], 4.3 on [3]
and 4.4 is derived from [2] and [6]. Section 5.1 is based on [2] and Section 5.2
on [4]. Section 5.3 is is a summary of a problem, and the approaches used to
solve the same, worked on by the author.
2
Chapter 2
Aspects of gravity and
thermodynamics of horizons
In sections 1 and 2 we discuss how a large class of metrics with horizons
can be approximated locally as a Rindler horizon. In section 3, we discuss
Lanczos-Lovelock gravity and some results related to them. Much of the
discussion in sections 1 and 2 is based on [3] and section 3 is based on [1].
2.1
Rindler horizons in flat and curved spacetimes
Consider a flat spacetime with the metric
ds2 = −dT 2 + dX 2 + dL2⊥
(2.1)
The line X = ±T divide the X − T plane into four quadrants which we
call past P, future F , right R and left L wedges. Let us now introduce a
coordinate transformation in all four quadrants, with l and t being the new
variables:
√
√
(2.2)
κT = 2κl sinh(κt); κT = ± 2κl cosh(κt)
for |X| > |T | with positive sign in the right wedge and negative sign in the
left , and
√
√
κT = ± −2κl cosh(κt); κT = −2κl sinh(κt)
(2.3)
for |X| < |T | with positive sign in the future and negative sign in the past,
with l < 0. In terms of these new coordinates, the metric can now be written
as
dl2
+ dL2⊥
(2.4)
ds2 = −2κldt2 +
2κl
3
We note that the coordinates t and l are timelike and spacelike in the right
and left wedges (where l > 0) respectively; the reverse is true in the future
and past wedges (where l < 0). Also, a given set of coordinates (t, l) corresponds to a pair of points in right and left wedge, or past and future wedge
(according as l > 0 or l < 0). Moreover, we see that the surface l = 0 acts
like a horizon for observers in the right wedge. Observers stationary in the
new coordinates (l = constant and x⊥ = constant) move on X 2 − T 2 = 2l/κ
in the X − T plane. This is the trajectory of an observer moving with a constant acceleration in the inertial frame (as will be shown in a later chapter),
and such an observer perceives a metric at l = 0. Such observers are called
Rindler observers and the metric in Eq. (2.4) is called the Rindler metric.
Rindler observers may be introduced in a curved spacetime as well, in any
local region. This is done by first transforming to the local inertial frame
around that event and then introducing local Rindler frame as above.
Another way by which one may locally introduce Rindler observers in
possible in metrics of the form:
ds2 = −f (r)dt2 +
dr 2
+ dL2⊥
f (r)
(2.5)
Here f is a function such that f (r) has a simple zero at some point r = a and
with a non-zero first derivative f ′ (a) ≡ 2κ. Performing a Taylor expansion
of f about r = a up to first order gives f ≈ 2κl were l = r − a. We therefore
have a Rindler approximation of metrics of the form in Eq. (2.5).
We note that the pathology in Eq. (2.5) is similar to that of the Rindler
metric in Eq. (2.4) at l = 0. As in the Rindler case, one can eliminate the
singularity in this case as well with suitable coordinate transformations. For
example
Z
dr
κξ
κξ
(2.6)
κX = e cosh κt; κT = e sinh κt;
ξ≡
f (r)
effects a transformation from (t, r) to (T, X). The resulting metric is
ds2 =
f
(−dT 2 + dX 2 ) + dL2⊥
− T 2)
κ2 (X 2
(2.7)
The factor f /(X 2 −T 2 ) remains finite at the horizon, even though the horizon
r = a is now mapped to X 2 − T 2 = 0.
2.2
Horizons in static spacetime
In this section we show how a large class of metrics corresponding to static
spacetimes with horizons can also be mapped to the Rindler form near the
4
horizon. Firstly, we observe that performing a coordinate transformation
that takes (l, t) to (x, t) according to the rule l = (1/2)κx2 reduces the form
of the Rindler metric to:
ds2 = −κx2 dt2 + dx2 + dL2⊥
(2.8)
and the Cartesian coordinates in terms of these new coordinates are
X = ±x cosh(κt)
T = x sinh(κt);
(2.9)
Consider now a static spacetime with the following properties (i) The metric is static in the given coordinate system, g0α = 0, gab (t, x) = gab (x) (ii)
g00 (x) = −N 2 (x) vanishes on some 2-surface H defined by the equation
N 2 (x) = 0 (iii) ∂α N is finite and nonzero on H and (iv) all other metric
components and curvature remain finite and regular on H. With these assumptions the line element now is
ds2 = −N 2 (xα )dt2 + γαβ (xα )dxα dxβ
(2.10)
The comoving observers in this frame have trajectories x = constant, fourvelocity ua = −Nδa0 and four acceleration ai = uj ∇j ui = (0, a) where aα =
(∂α N)/N. The unit normal nα to the N = constant surface is given by
nα = ∂α N(g µν ∂µ N∂ν N)−1/2 = aα (aβ aβ )−1/2 . Let us define a(x) as
N(nα aα ) = (g αβ ∂α N∂β N)1/2 ≡ Na(x)
(2.11)
As we take the limit of x going to the horizon N = 0, the quantity Na has
a finite limit κ, called the surface gravity of the horizon.
It is possible to employ the level surfaces of N as a coordinate. These,
along with coordinates y A transverse to these level surfaces, form a coordinate
system. The line element in this new coordinate system is:
ds2 = −N 2 dt2 +
dN 2
aA dN
aB dN
A
B
+
σ
(dy
−
)(dy
−
)
AB
(Na)2
Na2
Na2
(2.12)
where aA etc. are the components of the acceleration in the new coordinates.
Since, as N → 0 (i.e, as we approach the horizon) Na → κ, the metric
reduces can be approximated near the horizon as
ds2 = −N 2 dt2 +
dN 2
dN 2
2
2 2
+
dL
≃
−N
dt
+
+ dL2⊥
⊥
(Na)2
κ2
(2.13)
This therefore shows that a large class of static spacetimes with horizons
can be mapped to the Rindler form in a region around the horizon. This
form of the metric will be essential in calculating the temperature associated
with a horizon. As a corollary, a temperature can be associated with almost
all horizons. We derive the expression for this temperature in terms of the
surface gravity of the horizon in section 4.3.
5
2.3
Generalized models of gravity
We first observe that the Einstein-Hilbert action in D dimensions is given by
(ignoring the normalization factor):
Z
Z
√
D √
AEH =
d x −gR ≡
dD x −gLEH
(2.14)
V
V
The Lagrangian LEH can be rewritten as
LEH = Qabcd Rabcd
(2.15)
where
1
Qabcd = (δac g bd − δad g bc )
(2.16)
2
is the only fourth rank tensor that can be constructed using only the metric
and has all the symmetries of the curvature tensor. Moreover, this tensor is
divergence-free in all of its indices: ∇a Qabcd = 0. These properties, suitably
modified, yield generalized theories of gravity which have many interesting
properties as will be seen later.
Also, we note that the Einstein-Hilbert Lagrangian may be written as
1
cd
δab
= (δac δbd − δad δbc )
2
cd ab
LEH = δab
Rcd ;
(2.17)
cd
where δab
is the alternating tensor. Since LEH is linear in second derivatives,
it can be written as terms quadratic in the first derivative of the metric and
as terms which are total derivatives (’surface terms’). Let us refer to these
terms as Lbulk and Lsur respectively. That is,
√
−gLEH = Lbulk + Lsur
(2.18)
where
Lsur = 2∂c
√
−gQabcd Γabd ;
√
Lbulk = 2 −gQabcd Γadk Γkbc
(2.19)
Note that the explicit form of Qabcd has not been used: only the property
∇a Qabcd = 0 has been exploited to effect this separation. An interesting
result, which is a simple relation that allows Lsur to be completely determined
from Lbulk , follows:
1
∂Lbulk
Lsur = −
(2.20)
∂i gab
[(D/2) − 1]
∂(∂i gab )
Actions which satisfy the above relation have been referred to in recent
literature as being ”holographic.” We now study a more general class of
6
Lagrangians, called the Lanczos-Lovelock Lagrangians, which satisfy a generalized version of the above holographic relation, besides sharing several
other properties of the Einstein-Hilbert Lagrangian.
We also note that the above holographic relation is linear, implying that
a linear combination of holographic Lagrangians will also be holographic and
hence simplifying their analysis.
It was noted earlier that separation of the Lagrangians of types () only
used the condition that Qabcd is divergence-free in any one (and hence all,
since Qabcd obeys all the symmetry relations of the curvature tensor) index.
Let us further assume that, besides these conditions, Qabcd is constructed
from g ab and Rabcd (as mentioned earlier, if it is to consist of only the former,
the resulting action will be the Einstein-Hilbert action), and also impose the
condition that Qabcd is proportional to the curvature, so that the resulting
Lagrangian is quadratic in curvature. The resulting tensor is:
Qabcd = Rabcd − Gac g bd + Gbc g ad + Rad g bc − Rbd g ac
(2.21)
The resulting Lagrangian now is:
LGB =
1 abcd
R Rabcd − 4Rab Rab + R2
2
(2.22)
which is the so-called Gauss-Bonnet Lagrangian, whose variation is a pure
divergence in 3+1 dimensions (but not in higher dimensions) and therefore
does not contribute to the equations of motion.
The Einstein-Hilbert and the Gauss-Bonnet Lagrangian can be written
as
12 24
1357 24 68
LEH = δ34
R13 ; LGB = δ2468
R13 R57
(2.23)
where the numeral n stands for an index an . These can be generalized to
what would be the mth order term of the Lanczos-Lovelock Lagrangian:
135...2k−1 24 68
2k−22k
L(m) = δ246...2k
R13 R57 ...R2k−32k−1
; k = 2m
(2.24)
where k is an even number. We see that m = 1 gives the Einstein-Hilbert
Lagrangian whereas m = 2 gives the Gauss-Bonnet Lagrangian. Since L(m)
ab
is a homogeneous function of degree m in the curvature tensor Rcd
, it can be
expressed in the form
1
1 ∂L(m)
a
Rabcd ≡ Pabcd Rbcd
(2.25)
L(m) =
bcd
m ∂Ra
m
where Pa bcd ≡ (∂L(m) /∂Rabcd ) so that P abcd = mQabcd .
grangians:
∇a P abcd = 0 = ∇a Qabcd
7
For these La(2.26)
These, therefore, are the generalizations that were referred to earlier, with a
divergence free Qabcd , and therefore permit separation into a bulk term and
a surface term in a similar fashion as Eq. (2.18).
PmOne can linearly combine several such holographic Lagrangians L =
i=1 ci L(i) where the ci are arbitrary coefficients. As noted earlier, any
such combination is a holographic Lagrangian. Because of the antisymmetry
135...2k−1
of δ246...2k
, only terms with 4m ≤ D will have a nontrivial effect (as the
variation terms which do not satisfy this condition will reduce to a total divergence), and any terms not satisfying this condition may be ignored as being
physically irrelevant. This therefore uniquely singles out Einstein-Hilbert Lagrangian in 3+1 dimensions. This also explains the earlier remark regarding
Gauss-Bonnet Lagrangian being physically irrelevant in 3+1 dimensions.
It was mentioned that the Lanczos-Lovelock class of Lagrangians are interesting for the reason that they share several interesting properties with
Einstein gravity (to which the Lanczos-Lovelock model uniquely conforms in
3+1 dimensions, in the sense that the Einstein-Hilbert term is the only nontrivial term in the Lanczos-Lovelock Lagrangian). We now corroborate this
statement, first by showing that the equations of motion for this generalized
class is takes the form:
√
1 ∂ −gL
1
√
= Tab
(2.27)
ab
−g ∂g
2
where the right-hand side is obtained by the variation of the matter part of
the action. Indeed, the above equation reduces to the familiar Einstein’s field
equation for L = LEH = R. The above equation can be expanded to make
this manifest:
! kl
∂L ∂Rij
1
1
1
ij k
− gab L = Pkb R aij − gab L = Tab ,
(2.28)
kl ∂g ab
2
2
2
∂Rij
√
√
where (∂ −g/∂g ab ) = −(1/2) −ggab has been used. The generalization of
Eq. (2.3) can also be derived:
"
#
(m)
(m)
δL
∂L
(m)
bulk
bulk
[(D/2) − m]Lsur
= −∂i gab
(2.29)
+ ∂j gab
δ(∂i gab )
∂(∂i ∂j gab )
The surface term in the action is assumes great significance in light of the fact
that it is closely related to the entropy of the horizon, if a solution possesses
one. In particular, one can obtain the result that if a solution to Einstein’s
equation possesses horizons which can be approximated by a Rindler metric,
8
then one can obtain the result that the entropy per unit transverse area is
(1/4). Consider the surface contribution
Z
Z
√
√
D
abcd
Ssur = 2 d x∂c −gQ Γabd = 2 dD x∂c −gQabcd ∂b gad
(2.30)
for the static metric in Rindler approximation
ds2 = −κ2 x2 dt2 + dx2 + dx2⊥
(2.31)
We now integrate over t and x on the x = ǫ surface, where ǫ is infinitesimal.
To do this we analytically continue the coordinate t to imaginary values as
t −→ itE (where tE is real). It is immediately seen that tE becomes an
angular coordinate with periodicity 2π/κ, with the metric now of the form
ds2E = x2 k(κtE )2 + dx2 + dx2⊥
(2.32)
The integral is now equivalent to an integral done over polar coordinates. In
order to avoid the singularity at x = 0, the integral over tE is evaluated over
the range (0, 2π/κ). On thus integrating over t and x we get
Z
√
2π
Ssur = 2
dD−2 x⊥ −gnc Qabcd ∂b gad
(2.33)
κ
where ni = (0, 1, 0, 0..) is the unit normal vector in the x-direction. Due to
the term ∂b gad , we see that only the b = x, a = d = 0 term contributes (as
g00 is the only non-constant component, with dependence only on x. With
this taken into account, the above expression we on simplification yields
Z
√
(2.34)
Ssur = −8π
dD−2 x⊥ σ Q0x
0x
H
where σ is the determinant of the metric in the transverse space. For the
case of Einstein-Hilbert action where Qabcd = (1/32π)[g ac g bd − g ad g bc ], we get
Q0x
0x = 1/32π and
1
Ssur = − A⊥
(2.35)
4
where the negative sign can be traced to the choice of the normal vector to
the horizon surface.
Relations like above, that relate the boundary term in the action to the
horizon entropy, continue to hold for Lanczos-Lovelock Lagrangians, as will
now be shown. We show the existence of a conserved current (i.e., a Noether
current) Ja that relates to the area of the horizon and therefore to the black
hole entropy. Given any Lagrangian L constructed from the Riemann tensor
9
and the metric tensor, the variation of the gravitational Lagrangian density
can be written as
√
√ δ(L −g) = −g Eab δg ab + ∇a (δv a )
(2.36)
where ∇a E ab = 0 holds off-shell. In the case of Einstein gravity, E ab is just
the Einstein tensor up to a constant factor.
Let us now consider now an infinitesimal coordinate transformation xi −→
xi + ξ i . It can be shown that the quantity
J a ≡ (Lξ a + δξ v a + 2E ab ξb )
(2.37)
is conserved, i.e., ∇a J a = 0. Here δξv a represents boundary term which
arises for the specific variation of the metric in the form δg ab = ∇a ξ b + ∇b ξ a .
We note that this conservation law is off-shell. Let us now consider a special
case where ξ a is an approximate Killing vector near an event P, i.e., ξ a
satisfies ∇(a ξb) = 0 and ∇b ∇c ξd = Rkbcd ξk at P. In this case, δξ v a = 0 and
the current becomes
J a = (Lξ a + 2E ab ξb )
(2.38)
Note that the equations of motion have not been used in the preceding deriva√
tion of the expression for J a . With Tab defined as δAm /δg ab = −(1/2)Tab −g
(where Am stands for the matter part of the total action), the field equations
are obtained as 2Eab = Tab The Noether current, on shell, is therefore given
by
J a = (T aj + g aj L)ξj
(2.39)
For any vector ka which satisfies ka ξ a = 0 we have,
(ka J a ) = T aj ka ξj
(2.40)
Thus we see that when T aj changes by a small amount δT aj , ka J a changes
by δ(ka J a ) = ka ξj δT aj . On the same lines as before, the time integration is
done over the range (0, β) where β = 2π/κ and κ is thesurface gravity of the
horizon.
Z
Z
Z
√
√
√
D−1
a
D−1
aj
δ
d
x h(ka J ) =
d
x hka ξj δT = β
dD−2 x hka ξj δT aj
H
H
H
(2.41)
Since the integral over δT aj is the flux of energy δE through the horizon,
βδE is interpreted as the rate of change of entropy. This suggests that the
entropy must be given by
SNoether = βN
(2.42)
10
where the Noether charge N corresponds to a current J a for a Killing vector
field ξ a = (1, 0), ξa = ga0 . The entropy thus obtained is known as Wald
entropy. The fact that ∇a J a = 0 implies that there exists an antisymmetric
tensor J ab such that J a = ∇b J ab . The Noether charge N is given by
Z
Z
√
√
D−1
0
N =
d
x −gJ = dD−1 x∂b ( −gJ 0b )
t
Zt
√
(2.43)
=
dD−2 x −gJ 0r
t,rH
where contributions over transverse directions have been ignored, and in the
radial direction only the contribution at r = rH has been accounted for. It
can be shown that
J 0r = 2P 0rcd ∇c ξd = 2P cdr0∂d gc0
Therefore, the Noether charge N is given by
Z
Z
√
√
cdr0
D−2
∂d gc0 = 2m
dD−2 x −gQcdr0 ∂d gc0
N =2
d
x −gP
(2.44)
(2.45)
t,rH
t,rH
Wald entropy therefore is
SNoether = βN = 2βm
Z
t,rH
√
dD−2 x −gQcdr0 ∂d gc0
(2.46)
In contrast, on evaluation of the surface term we get
Z
Z
√
√
abcd
D
dD−2x −gQabr0 ∂b ga0 (2.47)
Ssur = 2 d x∂c −gQ ∂b gad = 2β
t,rH
where t has again been integrated over the range (0, β) and transverse directions have been ignored. Comparing the expressions for SNoether and Ssur , we
get
SNoether = mSsur
(2.48)
We now obtain an expression for the horizon entropy in Lanczos-Lovelock
0x
models. Consider the quantity Qx0
x0 = Q0x for the mth order LanczosLovelock Lagrangian, given by
Qx0
x0 =
1 1 x0a3 ...a2m b3 b4
b2m−1 b2m
δ
R
...R
a3 a4
a2m−1 a2m |x=ǫ
16π 2m x0b3 ...b2m
(2.49)
where the normalization factor has been included so as to recover EinsteinHilbert term for m = 1. Also, we define Qx0
x0 = 1/16π for the m = 0 case. We
11
x0a3 ...a2m
A4 ...A2m
note that δx0b
= δBA33B
with Ai , Bi = 2, 3, ...D − 1 since the terms
4 ...B2m
3 ...b2m
0
x
containing the factors δA and δA all vanish. Therefore,
1
1
1
B2m−1 B2m
A3 A4 ...A2m
B3 B4
x0
Qx0 =
δ
R
...R
(2.50)
B3 B4
A2m−1 A2m |x=ǫ
2 16π 2m−1 B3 B4 ...B2m
AB
AB
Since in the ǫ → 0 limit, RCD
|H =(D−2) RCD
|H we see that
1
Qx0
x0 = L(m−1)
2
(2.51)
That is, Qx0
x0 turns to be the same as the Lanczos-Lovelock Lagrangian of
the immediate lower order. Now, using Eq. (2.34), the quantity Ssur can be
readily evaluated.
The Wald entropy SNoether = mSsur becomes
Z
√
SNoether = −4πm dD−2 x⊥ σL(m−1)
(2.52)
H
We now see the rationale for defining Qx0
x0 = 1/16π for the m = 0 case—
the entropy for the mth order Lanczos-Lovelock Lagrangian is given by an
integral over the Lagrangian of the (m − 1)th order. Since L(0) is a constant, we recover the familiar result that the entropy is proportional to the
transverse area, along with the required constant of proportionality. For a
general Lanczos-Lovelock Lagrangian the entropy is easily found as the linear
combination
S=
K
X
m=1
cm S
(m)
=−
K
X
4πmcm
m=1
12
Z
√
dD−2 x⊥ σL(m−1)
H
(2.53)
Chapter 3
Quantum effects in external
electric field
We now consider quantum fields in the background of classical electric fields.
In particular, we study complex scalar fields in static electric field and show
that particle-antiparticle pairs can spontaneously form in such a situation.
Similar results apply for spinor-valued quantum fields as well, and therefore
electron-positron pairs can be created in a static electric field. This phenomenon, known as Schwinger effect, is on the verge of being experimentally
verified [5]. The discussion in the following section is based on [4]
3.1
Schwinger effect
We now derive the probability of production of particle-antiparticle pairs in
the context of scalar fields interacting with a background electric field. We do
this using the method of Effective Lagrangian. We shall derive the effective
lagrangian from the appropriate kernel of the path intergral formalism and
show that the effective lagrangian thus obtained has a nonzero imaginary
part, which yields the pair production rate.
The relevant kernel is
s
K(x, y; s) = hx| exp i (i∂ − qA)2 − m2 + iǫ |yi
(3.1)
2
Consider an electric field along the z-axis, which we assume has an arbitrary
time dependence for now, i.e., E(t) = E(t)ẑ, B = 0. We choose the potential
Ai such that
Ai = (0; 0, 0, A(t)); E(t) = −A′ (t)
(3.2)
13
The translational invariance along the spatial coordinates allows us to write
Z
d3 p 0
is 0
0
K(x , y ; x, x; s) =
(i∂t )2 − p2⊥ − (pz − qA(t))2 − m2 + iǫ |y 0 i
hx | exp
3
(2π)
2
Z
3
dp
is 2
is 2
2
=
p⊥ + m − iǫ hx0 | exp
−∂t − (pz − qA(t))2 |y 0 i
exp −
3
(2π)
2
2
Z
3
is 2
dp
2
G(x0 , y 0; s)
(3.3)
p
+
m
−
iǫ
exp
−
=
⊥
(2π)3
2
where G(t, t′ ; s) is the propogator for the one-dimensional quantum mechanical Hamiltonian
1 ∂2
1
H=−
− (pz − qA(t))2
(3.4)
2
2 ∂t
2
Consider now the special case of a uniform electric field. The potential for
such an electric field is A = −Et. Then
1 ∂2
1
1 ∂2
1
2
H=−
−
(p
+
qEt)
=
−
− q 2 E 2 ρ2
(3.5)
z
2
2
2 ∂t
2
2 ∂ρ
2
where ρ = t+(pz /qE). In this form, we see that the above Hamiltonian is that
of a quantum harmonic oscillator with unit mass and imaginary frequency
(iqE). The coincidence limit of the propagator for this Hamiltonian is given
by
"
1/2
2 #
pz
qE
qE
(cosh qEs − 1) t +
G(t, t; s) =
exp i
2πi sinh qEs
sinh qEs
qE
(3.6)
On substituting this back into Eq. (3.3) and evaluating the integral (over
px ,py and ω) we get
− is (m2 −iǫ)
e 2
1
qE
(3.7)
K=
2
(2π) 2is sinh(qEs/2)
The effective lagragian Leff can be obtained from the kernel using the relation
Z ∞
ds
Leff = −i
K(x, x; s)
(3.8)
s
0
which for the kernel at hand yields
Z ∞
is
2
ds
qE
e− 2 (m −iǫ)
Leff = −i
s (2π)2 (2is) sinh(qEs/2)
Z0 ∞
ds 1
qE
1
2
e−is(m −iǫ)
= −
2
2
4 0 (2π) s sinh qEs
Z ∞
qE
ds 1
−is(m2 −iǫ)
e
= −
4π 2 s2 sinh qEs
0
14
(3.9)
In order to evaluate the above integral, we analytical continue the variable of
integration s to imaginary values (s −→ −is) and rewrite the above integral
as
Z ∞
2
ds e−m s qEs
ImLeff = −
(3.10)
(4π)2 s2 sin qEs
0
where we have discarded the iǫ term in the exponent as being unnecessary
for our present method pf evaluation of the integral. We note that the above
integral is not well-defined as it divergence as s = 0. This is related to the
problem of renormalization, which is not dealt with in this thesis (and as we
would only be interested in evaluating the imaginary part of the above quantity). Another problem is the existence of poles along the path of integration
at s = (nπ)/(qE), where n = 1, 2, 3..., due to the sine function.
The poles at {(nπ/qE)}n=1,2... are avoided by choosing the path of integration that consists of small semicircles of radius ε in the upper half plane.
The contribution by the nth pole is
Z θ=0
(ǫeiθ idθ) −m2 sn
qE
In =
e
2
2
cos(nπ)ǫeiθ
θ=π (4π) sn
2
1
m2 π
n+1 (qE)
exp −
n
(3.11)
= i(−1)
16π 3 n2
qE
The total contribution to ImLeff is
ImLeff =
∞
X
n=1
1 (qE)2
(−1)n+1
2 (2π)3
m2 π
1
exp −
n
n2
qE
(3.12)
The quantity ImLeff is related to the probablity for the system to make
transitions from ground state to excited state. Since the excited state in
this case is a state with particles of the quantum field present, we interpret
2Im(Leff ) as the probability per unit volume per unit time for the production
of scalar particles, which is
2Im(Leff ) =
∞
X
(qE)2 (−1)n+1
n=1
(2π)3
n2
πm2
exp −
n
qE
(3.13)
which is the required result.
It may also be shown that a constant magnetic field does not give rise
to particle production. In particular, the effective Lagrangian in this case
turns out to be purely real. Consider for example a vector potential given by
15
Aj = (0; A(z), 0, 0), so that the magnetic field is given by By = (−∂A/∂z).
The kernel analogous to Eq. (3.3) is
Z 2
d p⊥ dω 0
is 2
K =
ω − p2y − (pz − qA)2 + ∂z2 − m2 + iǫ |y 0i
hx | exp
3
(2π)
2
Z 2
is 2
d p⊥ dω
2
2
ω − py − m + iǫ G(z, z; s)
(3.14)
exp
=
(2π)3
2
where G is now a propagator for the Hamiltonian
H=−
1 ∂
1
+
(px − qA(z))2
2 ∂z 2 2
(3.15)
As can be seen, unlike in the case of pure electric field, the frequency is purely
real. For the special case of uniform magnetic field, A = −Bz,
H=−
1
1 ∂2
1
1 ∂2
2
+
(p
+
qBz)
=
−
+ q 2 B 2 ρ2
x
2
2
2 ∂z
2
2 ∂ρ
2
(3.16)
where ρ = z + (px /qB). This is the Hamiltonian for a Harmonic oscillator
with unit mass and frequency (qB). The propagator therefore is
"
1/2
2 #
qB
px
qB
G(z, z; s) =
(cos qBs − 1) z +
exp i
2πi sin qBs
sin qBs
qB
(3.17)
Performing the integrations over px , py and ω we have
qB
K=
(2π)2
1
2is
s
2
e−i 2 (m −iǫ)
1
(qB/2) −i s 2
=
e 2 (m − iǫ)
2
sin(qBs/2)
(2π) is sin(qBs/2)
(3.18)
This yields the effective Lagrangian
Leff
s
∞
2
qB
e−i 2 (m −iǫ)
ds
= −i
s (2π)2 (2is) sin(qBs/2)
Z0 ∞
1
ds 1
qB
2
= −
e−i(m −iǫ)s
2
2
4 0 (2π) s sin(qBs)
Z ∞
qB
ds 1
2
= −
e−i(m −iǫ)s
2
2
4π s sin(qBs)
0
Z
(3.19)
Once again, the presence of sine function in the denominator introduces poles.
However, the definition the harmonic oscillator path integral requires us to
regard the frequency ω as a limiting case of (ω −iǫ). Therefore sinqBs should
16
be interpreted as the limit of the expression sinqBs(1 − iǫ). The poles are
therefore located at
2nπ
sn = ±
(1 + iǫ)
(3.20)
qB
The effective Lagrangian may be expressed along the imaginary axis, i.e., we
can analytically continue the the integration variable s to imaginary values
as s −→ is. The integral must be closed in the lower half plane to ensure
convergence due to the (−is(m2 /2) in the exponent. We get
Leff = −i
Z
∞
0
ds
K(s) =
s2
Z
=
Z
∞
0
∞
0
m2
ds ie− 2 s
s2 sinh(qBs/2)
qB/2
(2π 2 )i
m2
ds ie− 2 s
qB
1
2
2
s sinh(qBs/2) 2π
2
(3.21)
We see that the above expression is purely real, and consequently conclude
that a uniform magnetic field does not lead to particle production, unlike a
uniform electric field.
17
Chapter 4
Quantum effects in external
gravitational field
4.1
Unruh effect
The notion of particles depends on the definition of positive frequency modes,
which an inertial observer defines with respect to the time t of some inertial
reference frame. An accelerating observer, however, will define the positivefrequency modes with respect to the clock the observer carries, i.e., the proper
time. It is therefore to be expected that two observers, one inertial and one
accelerated, will not agree on the number and nature of particles they detect
when they are observing the same region of spacetime.
It was theoretically demonstrated by Fulling (1973) [7], Davies (1975) [8]
and Unruh (1976) [9] that an accelerating observer will observe a thermal
bath of particles where an inertial observer only sees a vacuum. This result
is called Unruh effect, and the temperature of the observed thermal bath,
known as Unruh temperature, is proportional to the acceleration:
T ≡
a
2π
(4.1)
We will now demonstrate this through a derivation for the simplified case of
a massless scalar field in 1+1 dimensional spacetime. The derivation in this,
and in the following section on Hawking effect, is based on that given in [2].
4.1.1
Rindler metric revisited
Consider the two-dimensional Minkowski spacetime:
ds2 = −dt2 + dx2 = ηab dxa dxb
18
(4.2)
The 2-velocity, defined as ui = dxi /dτ satisfies the relation ui ui = −1. Also
the condition for constant acceleration can be covariantly stated as ηij ai aj =
−a2
Let us now rewrite the Minkwoski metric in terms of the lightcone coordinates defined as
u ≡ −t + x,
v ≡t+x
(4.3)
so that the metric becomes
ds2 = dudv
(4.4)
We see that the coordinate transformation
u → ũ = cu, v → ṽ =
v
c
(4.5)
where c is a nonzero constant, leaves the line interval invariant and corresponds to a Lorentz transformation. This can be made more explicit by
expressing the c in terms of the relative velocity vr of two inertial frames:
1 − vr
c=
1 + vr
1/2
;
vr =
1 − c2
1 + c2
(4.6)
We wish to determine the trajectory of a uniformly accelerating observer
in an inertial frame. This is most easily done in terms of the lightcone
coordinates (u(τ ), v(τ )). Using the relations satisfied by proper velocity and
constant acceleration that were stated earlier
ü(τ )v̈(τ ) = a2
u̇(τ )v̇(τ ) = −1,
(4.7)
which after separation and integration give
v(τ ) =
A aτ
e + B;
a
u(τ ) =
1 −aτ
e
+C
Aa
(4.8)
where A, B and C are constants of integration. As noted earlier, the multiplication of u by a nonzero constant and v by the inverse of that constant
corresponds to a Lorentz transformation. We use this freedom of the choice
of Lorentz frame to set A = 1. Similarly, the freedom of choosing the origin
of this Lorentz frame allows us to set B = C = 0. Therefore the trajectory
of the accelerated observer in terms of lightcone coordinates, in a suitable
coordinate system, is given by
1
u(τ ) = e−aτ ,
a
1
v(τ ) = eaτ
a
19
(4.9)
and in terms of the original Minkowski coordinates is
t(τ ) =
1
sinh aτ,
a
x(τ ) =
1
cosh aτ
a
(4.10)
Eliminating the parameter τ in the above set of equations gives us the
worldline of the accelerated observer as the right branch of the hyperbola
x2 − t2 = a−2 . It is therefore easily inferred that in the reference frame that
was chosen (with the specific choices of A, B and C), the observer arrives
from infinity, momentarily comes to rest a x = a−1 and accelerates back to
infinity.
We now find a frame (ξ 0 , ξ 1) comoving with the accelerating observer.
We choose the frame to be such that the observer is at rest at ξ 1 = 0 and ξ 0
coincides with the the proper time τ along the observer’s worldline. It turns
out that such a comoving frame, along with the added bonus of conformal
flatness, can be found:
h
i
2
2 0 1
0 2
1 2
(4.11)
ds = Ω (ξ , ξ ) − ξ
+ ξ
where Ω(ξ 0 , ξ 1 ) is to be determined. To do so, we transform to the lightcone
coordinates of the comoving frame:
ũ ≡ −ξ 0 + ξ 1,
ṽ ≡ ξ 0 + ξ 1
(4.12)
in which the the metric takes the form:
ds2 = Ω2 (ũ, ṽ)dũdṽ
(4.13)
and the observer’s worldline
ξ 0 (τ ) = τ,
ξ 1 (τ ) = 0
(4.14)
takes the form
ṽ(τ ) = −ũ(τ ) = τ
(4.15)
Ω2 (ũ = −τ, ṽ = τ ) = 1
(4.16)
Since ξ 0 is the proper time with respect to the accelerating observer, at the
observer’s own location, we have:
Also since Eq. (4.13) and Eq. (4.4) describe the same Minkowski spacetime
in different coordinate systems we have:
ds2 = dudv = Ω2 (ũ, ṽ)dũṽ
20
(4.17)
The functions u(ũ, ṽ) and v(ũ, ṽ) can each depend only on one of two arguments ũ or ṽ since if this was not the case there would be dũ2 and dṽ 2 terms
in the right-most expression. To be definite, we choose
u = u(ũ),
v = v(ṽ)
(4.18)
If the above functions are determined, Ω2 (ũ, ṽ) can be read off from Eq.
(4.17). Since
−au(τ ) =
du(τ )
du(ũ ũ(τ )
du(ũ)
=
=
(−1)
τ
dũ dτ
dũ
(4.19)
we have
u = C1 eaũ
(4.20)
v = C2 eaṽ
(4.21)
and similarly
Here C1 and C2 are constants of integration related by Eq. (4.7). Therefore
C1 C2 = a2 . The exact values of C1 and C2 cannot be determined, and are in
fact not needed. The line interval, in light cone coordinates, is therefore
ds2 = dudv = ea(ṽ+ũ) dũdṽ
and in terms of the coordinates (ξ 0, ξ 1 ) is
h
2
2 i
ds2 = e2aξ1 − dξ 0 + dξ 1
(4.22)
(4.23)
One can also express the Cartesian coordinates (x, t) in the inertial frame in
terms of (ξ 0 , ξ 1) as
1 1
1 1
t(ξ 0 , ξ 1) = eaξ sinh aξ 0 ;
x(ξ 0 , ξ 1 ) = eaξ cosh aξ 0
(4.24)
a
a
We can see that for the ranges −∞ < ξ 0 < ∞ and −∞ < ξ 1 < ∞, these
coordinates cover only the right wedge of the 1+1 dimensional Minkowski
spacetime. This coordinate system is therefore incomplete. The accelerating
observer cannot observe more than a−1 in the direction opposite to the acceleration. Consider a hypersurface of constant time ξ 0. We see that the infinite
range of spacelike coordinate −∞ < ξ 1 spans a finite physical distance
Z 0
1
1
(4.25)
d=
eaξ dξ 1 =
a
−∞
Therefore no frame comoving with an accelerating observer can cover the
entire Minkowski spacetime. Since events beyond the right wedge cannot
be observed by the acceleration observer, the boundary of the right wedge
is a horizon. We have therefore show the existence of a horizon within the
context of special relativity.
21
4.1.2
Quantum fields in accelerated frame
We now consider a massless scalar field in 1+1 dimensional spacetime with
minimal coupling to gravity. The action is given by
Z
√
1
S[φ] =
g ab φ,a φ,b −gd2 x
(4.26)
2
It is easily seen that this action in 1+1 dimensions is conformally invariant.
Under a conformal transformation
gab → g̃ab = Ω2 (x)gab
(4.27)
−g and g ab change as
p
√
√
−g → −g̃ = Ω2 −g, g ab → g̃ ab = Ω−2 gab
(4.28)
√
the quantities
The factors of Ω2 cancel and the action remains conformally invariant. It
must be noted that it was necessary for the dimension of spacetime to be
1+1 for this to happen. Since the metric in accelerated frame Eq. (4.23)
is conformally flat, the action looks similar in both inertial and accelerated
frames:
Z
1 − (∂t φ)2 + (∂x φ)2 dtdx
(4.29)
S = −
2
Z
1 = −
(4.30)
− (∂ξ0 φ)2 + (∂ξ1 φ)2 dtdx
2
In terms of lightcone coordinates,
Z
Z
S = 2 ∂u φ∂v φdudv = 2 ∂ũ φ∂ṽ φdũdṽ
(4.31)
The field equations in the lightcone coordinates are
∂u ∂v φ = 0,
∂ũ ∂ṽ φ = 0
(4.32)
have solutions
φ(u, v) = A(u) + B(v), φ(ũ,
ṽ) = Ã(ũ) + B̃(ṽ)
(4.33)
where A,Ã,B and B̃ are arbitrary smooth functions. In particular,
φ ∝ e−iωu = eiω(t−x)
22
(4.34)
describes a right-moving, positive-frequency mode with respect to the Minkowski
time t, while
0
1
φ ∝ e−iΩũ = eiω(ξ −ξ )
(4.35)
describes a right-moving, positive-frequency mode with respect to the proper
time τ = ξ 0 with respect to the accelerating observer. Similarly the solutions
φ ∝ e−iωv and φ ∝ e−iΩṽ describe left-moving modes. In the right wedge
of the Minkowski spacetime where both coordinate systems overlap, one can
write the mode expansion for the field operator φ̂ as
Z ∞
dω
1 −iωu −
√
φ̂ =
e
âω + eiωu â+
ω + (left-moving) (4.36)
(1/2)
(2π)
2ω
Z 0∞
i
1 h −iΩũ −
dΩ
iΩũ +
√
e
b̂
+
e
b̂
=
Ω
Ω + (left-moving) (4.37)
(2π)(1/2) 2Ω
0
where both sets of operators satisfy standard commutation relations:
i
h
− +
′
+
etc.
(4.38)
âω âω = δ(ω − ω ′ ),
b̂−
b̂
Ω Ω = δ(Ω − Ω )
The vacuum state associated with the annihilation operators â−
ω is called the
−
Minkowski vacuum |0M i and the one associated with b̂Ω operators is called
the Rindler vacuum |0R i. Therefore
â−
ω |0M i = 0
b̂−
Ω |0R i = 0
(4.39)
The Minkowski vacuum is the physical vacuum that pertains to the inertial observer while the Rindler vacuum pertains to the accelerating observer.
That is, an inertial observer will register no particles in the Minkowski vacuum as will an accelerating observer in the Rindler vacuum. However, to the
accelerating observer the Minkowski vacuum will appear to be a state populated with particles. We now calculate the occupation number of particles in
Minkowski vacuum as observed by an accelerating observer. Before doing so,
however, we introduce the necessary notion of Bogolyubov transformation.
Consider a classical scalar χ(x, η) that satisfies the equation of motion
χ′′ − ∆χ + m2 χ = 0
(4.40)
where the prime ′ stands for derivative with respect to η. One can Fourier
expand χ as
Z
d3 k
χk (η)eikx
(4.41)
χ(x, η) =
(2π)3/2
The equation of motion satisfied by the Fourier modes χk (η) is obtained as
χ′′k + ωk2(η)χk = 0
23
(4.42)
where ωk2 (η) ≡ k 2 + m2 . As the above equation is a linear, homogeneous,
second-order differential equation, the general solution to these equations can
be expressed as the linear combination of two particular solutions, which we
choose to be complex conjugates of each other (vk (η) and vk∗ (η)), as
1 ∗
+
χk (η) = √ a−
k vk (η) + a−k vk (η)
2
(4.43)
We also normalize these functions vk (η) and vk∗ (η) such that their Wronskian (vk′ vk∗ − vk vk∗ ′ ) = 2i (the Wronskian is guaranteed to be nonzero if vk (η)
and vk∗ (η) are linearly independent, and therefore the normalization can be
effected). The functions are called ”mode functions”.
The field χ(x, η) can now be written as
Z
ikẋ
1
d3 k −
χ(x, η) = √
ak vk ∗ (η) + a+
(4.44)
−k vk (η) e
(3/2)
(2π)
2
Z
d3 k −
1
−ikẋ
ak vk ∗ (η)eikẋ + a+
(4.45)
= √
−k vk (η)e
(3/2)
(2π)
2
It turns out that the equations obeyed by the mode function (i.e., the
equation of motion and the normalization condition) do not serve to select
the mode functions uniquely. It is easily verified that
uk (η) = αk vk (η) + βk vk∗ (η)
(4.46)
where αk and βk are time independent complex coefficients satisfying |αk2 | −
|βk2 | = 1, also satisfy both the equation of motion for mode functions and
the normalization condition, and therefore can be used as mode functions
instead of vk (η), i.e., either of the two set of functions can be used to expand
the field χ. The coefficients αk and βk are called Bogolyubov coefficients.
In a general case where positive and negative frequency modes with respect to the inertial observer contributes to the positive frequency modes
with respect to positive frequency modes in the accelerated frame we have
Z ∞
−
+
b̂Ω =
dω αΩω â−
(4.47)
ω − βΩω âω
0
where αωΩ and βΩω are functions of ω and Ω that satisfy the normalization
condition
Z ∞
∗
∗
′
dω (αΩω αΩ
(4.48)
′ ω − βΩω βΩ′ ω ) = δ(Ω − Ω )
0
such that the commutation relations Eq. (4.38) hold. The above normalization condition is analogous to the normalization condition stated earlier,
where the Wronskian of their mode functions was set to 2i.
24
Substituting Eq. (4.47) into Eq. (4.37) we get
Z ∞
1 −iωu
′
′
√ e
=
αΩ′ ω e−iΩ ũ − βΩ∗ ′ ω eiΩ ũ
ω
0
(4.49)
We multiply both sides by exp(±iΩũ), integrate over ũ to solve for αΩω and
αΩω to obtain
r
Z +∞
iΩ
Ω πΩ
iΩ ω
1
−iωu+iΩũ
2a
Γ −
(4.50)
e exp
ln
αΩω =
e
dũ =
2πa ω
a
a
a
−∞
and similarly
βΩω =
Z
+∞
−∞
1
e+iωu+iΩũ dũ = −
2πa
r
Ω − πΩ
e 2a exp
ω
iΩ ω
ln
a
a
iΩ
Γ −
a
(4.51)
Therefore we have the relation
|αΩω |2 = e
2πΩ
a
|βΩω |2
(4.52)
We now compute the occupation number hN̂Ω i, i.e., the mean number of
particles with frequency Ω, of the Minkowski vacuum as measured by a
Rindler observer. This is just the expectation value of the number oper−
ator N̂Ω ≡ b̂+
Ω b̂Ω with respect to the Minkowski vacuum:
−
hN̂Ω i ≡ h0M |b̂+
Ω b̂Ω |0M i
Z
Z
∗ −
∗ +
+
′
∗
∗
−
×
dω αω′ Ω âω − βω′ Ω âω′ |0M i
= h0M |
dω αωΩ âω − βωΩ âω
Z
=
dω|βωΩ |2
(4.53)
The normalization condition Eq. (4.48) for Ω′ = Ω becomes
Z ∞
dω |αΩω |2 − |βΩω |2 = δ(0)
(4.54)
0
and therefore using Eq. (4.52) we have
hN̂Ω i =
Z
0
∞
2
dω|βωΩ | = exp
−1
2πΩ
−1 −1
δ(0)
a
(4.55)
The δ(0) factor is a result of the fact that the field was quantized in infinite
volume rather than in a finite box. In the latter case the divergent factor
25
would be replaced by the volume of the box V . Therefore the mean density
of particles with frequency Ω is
−1
hN̂Ω i
2πΩ
nΩ =
= exp
−1 −1
(4.56)
V
a
The above calculation has only been done for right-moving modes. The result
for left-moving modes can be similarly derived. We see that massless particles
in Bose-Einstein distribution are observed by the accelerating observer in
Minkowski vacuum, with the (Unruh) temperature
T =
4.2
a
2π
(4.57)
Hawking Effect
A nonrotating black hole with zero electric charge is in (3+1) dimensions is
described in natural units (G = ~ = c = 1)by the Schwarzschild metric,
2M
dr 2
2
2
2
2
2
ds = − 1 −
+
r
dθ
+
dφ
sin
θ
(4.58)
dt2 +
r
1 − 2M
r
Let us first consider the simpler case of a 1+1-dimensional black hole, assuming that its metric is given by the time and radial part of the Schwarzschild
metric:
dr 2
rg 2
2
a
b
dt +
ds = gab dx dx = − 1 −
(4.59)
r
1 − rrg
where rg = 2M. We now introduce the so-called tortoise coordinates:
dr ∗ =
dr
1 − rrg
Therefore
∗
r (r) = r − rg + rg ln
The metric takes the form
ds = 1 −
2
(4.60)
r
−1
rg
rg
−dt2 + dr∗2
r (r∗)
(4.61)
(4.62)
As can be seen from Eq. (4.61), the tortoise coordinates are defined only for
r > 0. Introducing the tortoise lightcone coordinates:
ũ ≡ −t + r ∗ , ṽ ≡ t + r ∗
26
(4.63)
and the metric now takes the form
2
ds = 1 −
rg
r(ũ, ṽ)
dũdṽ
(4.64)
There, however, is another coordinate system that covers the entire spacetime. These are Kruskal-Szekeres coordinates. The Kruskal-Szekeres lightcone coordinates are defined as
ũ
ṽ
u = 2rg exp
, v = 2rg exp
(4.65)
2rg
2rg
in which the metric takes the form
rg
r(u, v)
ds =
dudv
exp 1 −
r(u, v)
rg
2
(4.66)
We see that the metric now is singular at r = rg . Thus the singularity in
the Scharwzschild metric is a coordinate singularity which can be removed,
as we have, by a suitable coordinate transformation. Also, as defined above,
the Kruskal-Szekeres coordinates vary in the intervals 0 < u < +∞ and 0 <
v < ∞, covering only the exterior of the black hole, they can be analytically
continued to u < 0 and v < 0 so that the Kruskal-Szekeres coordinates span
the entire spacetime. One may express the original Schwarzschild coordinates
t and r in terms of the Kruskal-Szekeres coordinates as follows
∗
r
r
v 2
2t
r
2
2
= 4rg exp
− 1 exp
−1
= exp
uv = 4rg exp
rg
rg
rg
u
rg
(4.67)
These equations are valid even beyond the applicability of the coordinates
as in (), via analytic continuation. We see from the first of the above pair of
equations that the black hole horizon r = rg corresponds to u = 0 and v = 0.
We also see from the second of these equations that v = 0 (with nonzero u)
corresponds to t = −∞ and u = 0 (with nonzero v) corresponds to t = +∞.
These are referred to as the past and future horizons respectively.
We now analyze a massless scalar field with the action given by (). But
before doing so, we pause to point out the mathematical similarity between
the relation the Minkowski and Rindler coordinates (in the case of an accelerating observer), and the tortoise and Kruskal-Szekeres coordinates (in
the case of a Schwarzschild black hole) bear with one another. In order to
highlight this similarity, we reproduce the transformation between the corresponding lightcone coordinates below
u = a−1 exp(aũ),
v = a−1 exp(aṽ),
u = κ−1 exp(κũ)
v = κ−1 exp(κṽ)
27
(4.68)
where κ = (2rg )−1 is called the surface gravity of the horizon. Note also, that
the Kruskal-Szekeres coordinates cover the entire spacetime, like Minkowski
coordinates and the tortoise coordinates cover only the region of spacetime
outside the horizon, like the Rindler coordinates.
This formal similarity greatly simplifies the following analysis of the aforementioned massless scalar field. The conformal invariance of the action allows
us to write the solution of the scalar field equation in terms of the tortoise
lightcone coordinates as
φ = Ã(ũ) + B̃(ṽ)
(4.69)
or in terms of the Kruskal-Szekeres lightcone coordinates as
φ = A(u) + B(v)
(4.70)
where A, Ã etc. are arbitrary smooth functions. In a manner identical to the
analysis in Rindler spacetime, the quantized massless scalar field can now be
mode-expanded in tortoise coordinates as
Z ∞
i
dΩ
1 h iΩũ −
−iΩũ +
√
φ̂ =
(4.71)
e
b̂
+
e
b̂
Ω
Ω + (left − moving)
(2π)1/2 2Ω
0
which define the creation and annihilation operators b̂±
Ω . The vacuum state
|0B i corresponding to these annihilation operators, that is b̂−
Ω |0B i = 0, is
called Boulware vacuum. Similarly, one may expand the field in KruskalSzekeres lightcone coordinates.
Z ∞
dω
1 iωu −
−iωu +
√
e
â
+
e
â
+ (left − moving)
(4.72)
φ̂ =
ω
ω
(2π)1/2 2ω
0
which defines another set of creation and annihilation operators â±
ω , which
in turn defines the Kruskal vacuum |0K i as the state satisfying â−
|0
ω K i = 0.
As is clear from the emphasized similarity between the present case and
that in the previous section, the Kruskal vacuum is on the same footing as
the Minkowski vacuum; the Boulware vacuum is on the same footing as the
Rindler vacuum. The derivation of the occupation number is identical to
that in the previous section. The remote (Boulware) observer sees particles
in a thermal spectrum
−1
2πΩ
+ −
hN̂Ω i ≡ h0K |b̂Ω b̂Ω |0K i = exp
−1
δ(0)
(4.73)
κ
with the temperature
TH =
κ
1
=
2π
8πM
28
(4.74)
4.3
Horizon temperature from path integrals
We now derive the temperature of a horizon using the path integral formalism. The discussion is based on [4].
Consider a spacetime with a horizon described with two different coordinate systems: one being a global coordinate system (T, X, x⊥ ) like Cartesian
coordinates for the case of flat spacetime or Kruskal coordinates for the case
of spherically symmetric metrics with horizons and the other (t, x, x⊥ ) covers four different quadrants of the spacetime and is related to the first by
transformations like Eq. (2.6). Next, we analytically continue both T and t
to imaginary values as T −→ iTE , t −→ itE (where TE and tE are real). It
is immediately seen that tE becomes an angular coordinate with periodicity
2π/κ. Therefore, the field evolution from TE = 0 to TE → ∞, effected by
the inertial Hamiltonian HI , is mapped to the field evolution from tE = 0
to tE → 2π/κ, which is effected by the Rindler Hamiltonian HR . For either
Hamiltonian, we assume that by adding an appropriate constant the energy
of the ground state has been set to zero.
We now show that
hvac|φL , φR i ∝ hφL|eπHR /κ |φR i
(4.75)
where φL and φR are field configurations for X < 0 and X > 0 respectively.
This follows since the ground state wave functional hvac|φL, φR i is expressible
as a path integral as
hvac|φL , φR i ∝
Z
TE =∞;φ=(0,0)
TE =0;φ=(φL ,φR )
Dφe−A
(4.76)
We may now also evaluate the above integral using the coordinates (tE , x),
which are akin to polar coordinates, by varying tE from 0 to π/κ. The field
configuration at tE = 0 is φ = φR and at tE = π/κ, φ = φL . Therefore the
above integral can also be written as
Z κtE =π;φ=φL
hvac|φL , φR i ∝
Dφe−A
(4.77)
κtE =0;φ=φR
This, however, is equivalent to a time evolution, effected by the Rindler
Hamiltonian HR from tE = 0 to tE = π/κ and therefore:
hvac|φL, φR i ∝
Z
κtE =π;φ=φL
κtE =0;φ=φR
Dφe−A = hφL |e−(π/κ)HR |φR i
29
(4.78)
thus proving the claim in Eq. (4.75). This result, together with the imposition of normalization condition help determine the proportionality constant
in Eq. (4.75):
Z
DφL DφR hvac|φL, φR ihφL , φR |vaci
Z
2
DφL DφR hφL |e−πHR /κ |φR i hφR|e−πHR /κ |φL i = C 2 Tr e−2πHR /κ = 1
=C
Therefore we have
hvac|φL, φR i =
hφL |e−πH/κ |φR i
1/2
[Tr(e−2πH/κ )]
(4.79)
The density matrix for observations on the right Rindler wedge R can now
be computed by taking the trace with respect to field configuration φL on
the left wedge L, which is the region behind the horizon:
Z
′
ρ(φR , φR ) = DφL hvac|φL , φR ihφL , φ′R |vaci
Z
hφR |e−(π/κ)HR |φLihφL |e−(π/κ)HR |φ′R i
= DφL
T r(e−2πHR /κ )
hφR |e−(2π/κ)HR |φ′R i
(4.80)
=
T r(e−2πHR /κ )
We see that the above density matrix corresponds to a temperature β −1 =
(2π/κ).
4.4
Black Hole Thermodynamics
In this section, we obtain an estimate of the lifetime of our simplified model
of a black hole that Hawking-radiates. We also note the behaviour of a black
hole in an infinite heat bath.
4.4.1
Lifetime and heat capacity of a black hole
As seen in the previous section, a black hole emits radiation and loses its
mass. We now calculate the flux L of the radiated energy from a black hole,
which we treat as a spherical body with surface area
A = 4πrg2 = 16πM 2
30
(4.81)
and temperature TH = (8πM)−1 . Using the Stefan-Boltzmann law,
L=
ΓγσTH4 A
= Γγ
π2
60
1
8πM
4
16M 2 =
Γγ
15360πM 2
(4.82)
where Γ is the grey-body correction factor, γ counts the number of degrees
of freedom and σ = π 2 /60 is the Stefan-Boltzmann constant in Planck units.
The mass of the black hole therefore obeys the equation
dM
Γγ
= −L = −
dt
15360πM 2
(4.83)
the solution to which is
M(t) = M0
t
1−
tL
1/3
tL ≡ 5120π
M03
Γγ
(4.84)
where M0 is the initial mass of the black hole. Therefore, an isolated black
hole has a finite lifetime tL ∝ M03 .
It is easily shown that black holes have negative heat capacity. Since
E = M = (8πT )−1, we have
CBH =
1
∂E
=−
<0
∂T
8πT 2
(4.85)
It immediately follows that a black hole in an infinite heat bath can never
attain stable thermal equilibrium. Also, a black hole placed in a colder
(infinite) heat bath will continue to emit radiation and get hotter; a black hole
kept in hotter surroundings, infinite in extent, will keep absorbing radiation
and grow in size.
A stable equilibrium can, however, be attained if the black hole is placed
in a finite heat bath.
4.4.2
Black hole entropy
In view of various conundrums raised by thought experiments involving black
holes (for instance, the experiment of pouring hot tea down a black hole
and the possible decrease in entropy that results, as discussed by Wheeler
[10]), Bekenstein conjectured that black holes must have nonzero entropy
SBH proportional to its surface area. This was, in part, motivated by a
theorem in general relativity which asserts that when black holes combine,
the surface area of the new black hole is at least as large as the surface
areas of the constituent black holes. Bekenstein could only however provide
31
a bound on the constant of proportionality in this relation, which was later
fixed following the discovery of Hawking radiation.
We start with the definition of entropy
dE = T dS
(4.86)
We note that dE is equivalent to the quantity dM where M is the mass of
the black hole. Also, as derived in the previous section, the temperature of
the black hole is given by T = (1/8πM). Therefore we have
Z
S = 8πMdM
(4.87)
which yields
1
SBH = 4πM 2 = A
(4.88)
4
where A is the area of the black hole. The second law of thermodynamics is
now salvaged by asserting a generalized second law of black hole thermodynamics, which states that the total entropy—now defined as the sum of the
entropy of all black holes and that of ordinary matter—never decreases.
δStotal = δSmatter + δSBH ≥ 0.
(4.89)
Further details regarding black hole thermodynamics are available in [6]
32
Chapter 5
Cosmological particle
production
The discussion in section 1 is based on [2]. Section 2 is derived from [4] and
section 3 summarizes a study done independently.
5.1
Classical scalar field in FRW spacetime
We shall only consider the class of a scalar field minimally coupled to a
spatially flat FRW metric. The relevant metric, in coordinates that make its
symmetries manifest, is
ds2 = −dt2 + a2 (t)δαβ dxα dxβ
We define a new coordinate, the conformal time:
Z t
dt
η(t) ≡
a(t)
(5.1)
(5.2)
in terms which the conformal equivalence of the metric to the Minkowski
metric ηab becomes manifest:
ds2 = a2 (η) −dη 2 + δαβ dxα dxβ = a2 (η)ηab dxa dxb
(5.3)
The action for a real massive scalar field φ(x) with minimal coupling to the
metric is given by
Z
1 √
(5.4)
S=−
−gd4 x g ab φ,a φ,b + m2 φ2
2
33
which in terms of conformal time becomes
Z
1
S=−
d3 xdηa2 −φ′2 + (∇φ)2 + m2 φ2
2
(5.5)
where prime denotes derivative with respect to conformal time. We now
define ξ ≡ a(η)φ and rewrite the action in terms of this new field, while
eliminating the total derivative terms, as
Z
1
a′′
′2
2
2 2
3
2
S=
(5.6)
d xdηa ξ − (∇ξ) − m a −
ξ2
2
a
The variation of the above action gives the equation of motion
a′′
′′
2 2
ξ=0
ξ − ∆ξ + m a −
a
(5.7)
Thus we see that the equation of motion of minimally coupled massive scalar
field is formally equivalent to that of a Klein-Gordon field in Minkowski
spacetime, except that the mass is now a time-dependent effective mass:
m2eff (η) ≡ m2 a2 −
a′′
a
We substitute the Fourier expansions of ξ
Z
d3 k
ξ(x, η) =
ξk (η)eikx
3/2
(2π)
(5.8)
(5.9)
into the equation of motion for ξ we find that the Fourier modes ξk (η) satisfy
a set decoupled ordinary differential equations
ξk′′ + ωk2 (η)ξk = 0
(5.10)
where
a′′
(5.11)
a
In the above equation since ωk2 (η) depends only on k ≡ |k| the general solution to the above equation may be written as
ωk2(η) ≡ k 2 + m2eff (η) =2 +m2 a2 (η) −
1 ∗
+
ξk (η) = √ a−
k vk (η) + a−k vk (η)
2
(5.12)
where vk (η) and its complex conjugate vk∗ (η) are two linearly independent
solutions of Eq. (5.10). The two complex constant of integration a±
k can
∗
depend on the direction of k as well. The reality of field ξ implies ξk (η) =
34
− ∗
. It can be easily shown that vk and
ξ−k (η) which in turn implies a+
k = ak
∗
vk are linearly independent if and only if their Wronskian
W [vk , vk∗ ] = vk′ vk∗ − vk vk∗ ′ = 2iIm(v ′ v ∗ )
(5.13)
is nonzero. Also, Eq. (5.10) implies that the Wronskian will be timeindependent. Therefore, if W is nonzero, one can always normalize vk such
that Im(v ′ v ∗ ) = 1. In this case the complex solution vk (η) is called a mode
function. Substituting Eq. (5.12) in Eq. (5.9) we get
Z
ikẋ
1
d3 k −
+
ξ(x, η) = √
a
v
∗
(η)
+
a
v
(η)
e
(5.14)
k
k
−k
(2π)(3/2) k
2
Z
d3 k −
1
−ikẋ
ak vk ∗ (η)eikẋ + a+
(5.15)
= √
−k vk (η)e
(3/2)
(2π)
2
One proceeds with quantization using the usual equal-time commutation
relations on the field operator ξˆ and its canonically conjugate momentum
π̂ ≡ ξˆ′
h
i
ˆ
ξ((x),
η), π̂(y, η) = iδ(x − y)
(5.16)
h
i
ˆ
ˆ t) = [π̂(x, t), π̂(y, t)] = 0
ξ((x),
t), ξ(y,
(5.17)
The Hamiltonian is given by
Z
2
1
3
2
2
2
ˆ
ˆ
d x π̂ + ∇ξ + meff (η)ξ
Ĥ(η) =
2
(5.18)
Alternatively one can impose the commutation relations on the constants of
integration a±
k
− +
+ +
−
âk , ak′ = δ (k − k′ ) , â−
(5.19)
k , ak′ = âk , ak′ = 0
together with the constraints that the mode functions obey Eq. (5.10) and
the normalization condition Im (vk′ vk∗ ) = 1. The constants of integration a±
k
are now, therefore, interpreted as creation and annihilation operators. The
field operator can now be expanded as:
Z
1
d3 k − ∗
−ikẋ
ξ(x, η) = √
âk vk (η)eikẋ + â+
(5.20)
−k vk (η)e
(3/2)
(2π)
2
As was explained in the earlier chapter, the mode functions uk (η) (and
uk (η)∗ )can be expressed as linear combination of functions which serve equally
well as mode functions, as
uk (η) = αk vk (η) + βk vk∗ (η)
35
(5.21)
where αk and βk are time independent complex coefficients satisfying |αk2 | −
|βk2 | = 1.
Expanding the field operator in terms of the mode functions uk (η),
Z
h
i
d3 k
1
− ∗
ikẋ
+
−ikẋ
√
ξ(x, η) =
b̂ v (η)e + b̂−k vk (η)e
(5.22)
(2π)( 3/2) k k
2
where b±
k are another set of creation and annihilation operators satisfying the
commutation relations. As we now have two expansions for the same field:
i
h
+
ikẋ
∗
−
+
(5.23)
e
uk (η)b̂k + uk (η)b̂−k = eikẋ vk∗ (η)â−
k + vk (η)â−k
Using the above equation and Eq. (5.21) one has
−
−
+
+
+
â−
k = αk ∗ b̂k + βk b̂−k , âk = αk b̂k + βk ∗ b̂−k
(5.24)
The above relations are called the Bogolyubov transformations. Similarly,
±
one can write b±
k as a linear combination of ak :
−
+
+
+
−
b̂−
k = αk âk + βk â−k , b̂k = αk ∗ âk + βk ∗ â−k
(5.25)
±
Each of the set of operators â±
k and b̂k define their respective vacuum
states |(a) 0i and |(b) 0i as
â−
k |(a) 0i = 0
b̂−
k |(b) 0i = 0
(5.26)
for all k. It is, however, not necessary that the vacuum state with respect
±
to the â±
k operators, i.e., |(a) 0i is a vacuum with respect to the b̂k operators.
This can be seen by explicit calculation:
(a)
−
h(b) 0|N̂k |(b) 0i = h(b) 0|â+
k âk |(b) 0i
−
−
+
∗ b̂
∗ b̂
= h(b) 0| αk b̂+
+
β
α
+
β
b̂
k
k
k
k
−k
k
−k |(b) 0i
= |βk |2 δ (3) (0)
(5.27)
The divergent factor of δ (3) (0) is a result of quantization in infinite volume
rather than a closed box. The density of the a-particles in the mode k is
therefore
nk = |βk |2
(5.28)
The total mean density of all particles is given by
Z
n = d3 k|βk |2
36
(5.29)
This quantity is finite only if |βk |2 decays faster than k −3 for large k.
The state |(b) 0i can be expressed as a superposition of the eigenvectors of
(a)
the N̂k operator as
"
#
Y 1
βk + +
|(b) 0i =
|(a) 0i
exp
â â
|αk |1/2
2αk k −k
k
!
n
∞ Y 1
X
βk
=
|(a) nk , n−k i
(5.30)
|αk |1/2 n=0 2αk
k
Quantum states which are defined as exponential of quadratic combination
of operators acting on vacuum state are called squeezed states. By virtue
of this definition we see that the the a-vacuum is a squeezed state of the
b-vacuum and vice versa.
5.2
Correspondence with Particle Production
in Electric Field
We now point out an interesting formal correspondence between the problem
of scalar fields in an FRW universe and those in a time-dependent electric
field. Consider the action for a scalar field Φ
Z
R
1
i
2
4 √
Φ
(5.31)
A = − d x −g Φ ∂i ∂ + m +
2
6
in a spatially flat FRW spacetime. The above action, in 3+1 dimensions,
is conformally flat (thus justifying the presence of the (R/6) term). The
spacetime metric, in conformal coordinates, is given by
ds2 = a2 (t)(dt2 − dr2 )
(5.32)
The conformal flatness of the metric can be exploited to rewrite the action
as
Z
i
1
A=−
d4 xφ ∂i ∂flat
+ m2 a2 (t) φ
(5.33)
2
where φ = aΦ. The relevant kernel for this action is
K(x, y, ; s) = hx|e−is 2 [∂i ∂ +m a (t)−iǫ] |yi
Z
1
d3 p
2
2 2
2
=
ht|e−is 2 [∂t +p +m a (t)−iǫ] |t′ i
3
(2π)
Z
d3 p −is 1 (p2 −iǫ)
=
G(t, t′ ; s)
e 2
(2π)3
1
i
37
2 2
(5.34)
where is G is the propagator for the Hamiltonian
H=−
1 ∂2
1
− m2 a2 (t)
2
2 ∂t
2
(5.35)
This Hamiltonian, we observe, is identical to that of a particle in an electric
field if we are allowed the identification m2 a2 (t) ⇔ (pz − qA(t))2 . Therefore,
as far as quantization of scalar fields in classical background is concerned,
there is a one-to-one correspondence between time-dependent electric fields
and FRW universes. As an example, let us consider the case of constant
electric field. The above mentioned identification motivates the expansion
factor
1
qE
2
2
2
2
a (t) = 2 (pz + qEt) ≡ α (t + t0 ) , α ≡
(5.36)
m
m
In the ”comoving” coordinates, in which the metric takes the form
ds2 = dτ 2 − a2 (τ )dr2
(5.37)
a(τ ) = (2aτ )1/2 ∝ τ 1/2
(5.38)
the expansion factor is
which is the expansion factor for a radiation dominated universe.
We therefore infer that the problem of particle production in a given
model of an FRW universe can be equivalently studied as the problem of
particle production in an appropriately chosen time-dependent electric field.
5.3
An example: Particle production in a model
universe
We now consider particle production in a universe that is radiation-dominated
at early times and de Sitter at late times. As in the previous section, the
universe we deal with a spatially flat 3+1 dimensional FRW universe. The
metric therefore has the generic form in Eq. (5.1). We now evaluate the
expansion factor using the 00 component of Einstein’s equations along with
the appropriate source terms:
ȧ2
8πG
=
ρ
2
a
3
(5.39)
In order to model a universe that is asymptotically de Sitter, we choose the
ρ as
B∗
ρ = A∗ + 4
(5.40)
a
38
where A∗ and B ∗ are constants. Substituting this in Eq. (5.39), we have
ȧ2
B
=
A
+
a2
a4
(5.41)
where A and B are constants. The above equation is conveniently solved
by a substitution of the type a2 = y. The solution to the above equation is
obtained as
1/4 q
B
sinh (2A1/4 t)
a=
(5.42)
A
It is readily verified that the above expansion factor behaves in the required
fashion at both early
and late times: When t ≈ 0, sinh(2A1/2 t) ≈ 2A1/2 t and
√ 1/4
is large sinh(2A1/2 t) ≈ (1/2) exp(2A1/2 t)
therefore a(t) ≈ 2B t1/2
√ ; when t 1/4
and therefore a(t) ≈ (1/ 2)(B/A) exp(A1/2 t). The conformal time η is
given by
Z t
Z t
dt
dt
=
η=
(5.43)
p
B 1/4
a(t)
sinh (2A1/2 t)
A
Inverting the above expression so as to obtain a(η) is difficult and the approach of the previous section, namely that of using Eq. (5.7) is not readily
applicable. We may, however, write the equation of motion of the scalar field
in the original ’comoving’ coordinates. For a metric of the form in Eq. (5.1),
the Laplacian operator
√
1
−gg ik ∂k φ
= √ ∂i
−g
is given by
= −∂t2 −
X 1
3a2 ȧ
∂
+
∂x2
t
3
2
a
a
x,y,z
(5.44)
(5.45)
The equation of motion for the scalar field φ = 0 then becomes
a2 φ̈ + 3aȧφ̇ − ∇2 φ = 0
(5.46)
The mode expansion, in comoving coordinates, for the field φ can now be
written, like in Eq. (5.12). The mode functions will then satisfy the equation
a2 v̈ + 3aȧv̇ + k 2 v = 0
(5.47)
On substituting for the expansion factor from Eq. (5.42) we have,
B
A
1/2
sinh (2A1/2 t)v̈ + 3B 1/2 cosh (2A1/2 t)v̇ + k 2 v = 0
39
(5.48)
We make the substitution 2A1/2 t = τ to obtain
2 sinh τ v̈ + 3 cosh τ v̇ + Kv = 0
(5.49)
where the dot overhead now represents derivative with respect to τ and
K = k 2 /2(AB)1/2 .
Solving the above equation turns out to be extremely difficult. We therefore use a technique that allows us to instead work with a second order linear
differential equation with no first derivative terms. Consider a differential
equation of the form
ÿ + P (x)ẏ + Q(x)y = 0
(5.50)
Let y = A(x)B(x). Let us also assume A(x) is such that the coefficient
of B ′ (x) in the above equation (after the substitution of y = A(x)B(x))
vanishes. That is, with regard to the equation
A′′ B + 2A′ B ′ + AB ′′ + P (x)(AB ′ + A′ B) + Q(x)AB = 0
(5.51)
we require that A(x) is any function that satisfies
2A′ + P (x)A = 0
(5.52)
Therefore we have
Z
P (x)
A = exp −
dx
2
The equation satisfied by B is therefore
AB ′′ (A′′ + P (x)A′ + Q(x)A)B = 0
which using A′ = (−1/2)AP and A′′ = (1/4)A(P 2 − 2P ′ ) becomes
P2 P′
′′
B +B Q−
=0
−
4
2
(5.53)
(5.54)
(5.55)
For the equation at hand, P (τ ) = (3/2) coth(τ ) and Q(τ ) = K∗ cosech(τ ),
where K∗ = K/2. We therefore have the equation
9
3
2
′′
∗
B=0
(5.56)
B + K cosech(τ ) + cosech (τ ) −
16
16
We note that τ varies over the range (0, ∞). But for this fact, the above
problem is equivalent to that of one-dimensional scattering of a quantum
particle, as dictated by the Schrodinger equation with an appropriate potential. So as to have the independent variable vary over the entire real line,
one may use a new variable ρ defined by τ = exp(ρ). One may then use
familiar techniques such as a WKB approximation in order to understand
the early and late times behaviour of the mode functions. This is currently
being pursued.
40
Chapter 6
Conclusion
Thermodynamical aspects of spacetime horizons, namely how a temperature
can be ascribed to a large class of metric that can be approximated by a
Rindler form locally, was studied. However, it still to be seen how the notion
of entropy can be attached to such horizons, as we saw was naturally the
case with the horizon of a Schwarzschild black hole. The Lanczos-Lovelock
generalization of Einstein gravity to higher dimensions was studied. It was
understood, and strongly emphasized in the report, as to how this is an extremely natural generalization of Einstein gravity, to which it uniquely boils
down in 3+1 dimensions. It was also understood that several thermodynamical results regarding gravity were robust in the sense that this natural
generalization continued to uphold these thermodynamical results or some
generalization thereof. The author has therefore understood that, from a
theoretical standpoint, lot of clues about the thermodynamical aspects of
gravity are to be discovered from the study of this larger set of theories.
The author has also studied and understood Schwinger effect, i.e., the
production of particle-antiparticle pair in a static electric field. Besides understanding the effect itself, the general physical and mathematical methods
involved in obtaining the result were well-understood. Also, the correspondence between quantum effects in time-dependent electric field and the problem of cosmological particle production in FRW universes was noted. The
author intends to study this formal similarity between the two situations
further.
Considerable time was spent in understanding quantum fields in classical
gravitational background. The two effects, Unruh effect and Hawking effect
were studied in considerable detail. It was understood how the underlying
idea behind the two effects is essentially the same: the most natural choice of
vacuum with respect to one observer appears populated with a thermal sea of
quantum particles with respect to another observer, with their corresponding
41
creation and annihilation operators related by Bogolyubov transformations.
An inescapable consequence of Hawking effect is that black hole horizons
have an entropy associated with them, which is proportional to the surface
area of the horizon. While this is a great relief with the sanctity of second
law of thermodynamics in mind and fits well the general relativistic result
that net surface area of black holes never decreases during collisions and
mergers, this raises several more questions regarding information loss from
black holes, the resulting violation of the unitarity of quantum mechanics
and as to where does the black hole entropy ”live.” The author is currently
reading further on these topics.
Lastly, quantum fields in spatially flat FRW metrics in 3+1 dimensions
were studied. The idea that the problem of a scalar field in curved spacetime
can be brought to a formal equivalence with that of a scalar field with timedependent mass in flat spacetime was particularly appreciated. This method
was employed in order to understand particle production in a universe that
is radiation-dominated at early times and de Sitter at late times. However,
alternate methods were explored due to intractability in the earlier approach.
At the time of writing this thesis, the author is still working on several
possible methods to achieve analytic or numeric, accurate or approximate
results to this problem.
To conclude, the author has learned a lot of new ideas regarding Quantum
Field Theory in Curved Spacetime and Horizon Thermodynamics, but has
also fully understood that there is a lot more ground to be covered in the
study of this subject, which he intends to do with time and hopes contribute
to the subject itself.
Acknowledgments
I would like to extend my deepest gratitude to my guide Prof. T. Padmanabhan for all the motivation during the course of this work and for providing
me the opportunity to work under his guidance at IUCAA. I thank IUCAA
for their kind hospitality, their library and computer facilities; latter two
of which were extremely crucial for the completion of this thesis. My deep
gratitude also to BITS Pilani that provides its students the opportunity to
pursue their Masters thesis off-campus.
I also thank Dawood Kothawala, Sanved Kolekar and Suprit Singh for
many fruitful discussions.
42
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43