Quantum Field Theory inCurved Spacetime and Horizon
Thermodynamics
Thesis Submitted in partial fulfillment of the requirementsof BITS C421T/422T Thesis
By
Aditya BawaneID No. 2006B5B4368P
Under the supervision of
Prof. T PadmanabhanDistinguished Professor, IUCAA, Pune
Birla Institute of Technology andScience
Pilani, Rajasthan
December 3, 2010
Contents
1 Introduction 1
2 Aspects of gravity and thermodynamics of horizons 3
2.1 Rindler horizons in flat and curved spacetimes . . . . . . . . . 32.2 Horizons in static spacetime . . . . . . . . . . . . . . . . . . . 42.3 Generalized models of gravity . . . . . . . . . . . . . . . . . . 6
3 Quantum effects in external electric field 13
3.1 Schwinger effect . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Quantum effects in external gravitational field 18
4.1 Unruh effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.1 Rindler metric revisited . . . . . . . . . . . . . . . . . 184.1.2 Quantum fields in accelerated frame . . . . . . . . . . . 22
4.2 Hawking Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Horizon temperature from path integrals . . . . . . . . . . . . 294.4 Black Hole Thermodynamics . . . . . . . . . . . . . . . . . . . 30
4.4.1 Lifetime and heat capacity of a black hole . . . . . . . 304.4.2 Black hole entropy . . . . . . . . . . . . . . . . . . . . 31
5 Cosmological particle production 33
5.1 Classical scalar field in FRW spacetime . . . . . . . . . . . . . 335.2 Correspondence with Particle Production in Electric Field . . 375.3 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 higherdimensional spacetime
• Quantum Field Theory in the background of electric field; Schwingereffect
• Quantum Field Theory in the background of curved spacetime; Unruhand Hawking effect
• Quantum fields in FRW metric; cosmological particle production
Apart from these, we outline the methods pursued to study the problem ofparticle production in a model universe.
Chapter 1
Introduction
The thesis aims at studying Quantum Field Theory in Curved Spacetimesand understanding what it tells us about Horizon Thermodynamics. Besidesthe fact that the study of quantum fields in curved spacetimes (or in general,non-inertial coordinates) reveals many new phenomena that cannot possiblybe foreseen classically, study in this area reveals several pointers as to whata quantum theory of gravity could possibly be like, i.e., what phenomenamust a candidate theory be able to explain in the limiting case of a quan-tum field evolving in a classical gravitational background. Indeed, a perfectexplanation to all the intriguing peculiarities that emerge out of the studyof spacetime thermodynamics remain a holy grail for physicists of the dayseeking a consistent quantum theory of gravity.
The thesis is divided in 4 chapters, excluding the introduction. Chapter2 explores some general notions about spacetime horizons and thermody-namics associated with them. Particular emphasis is on horizons that canbe 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 thesehorizons arise in cases of physical interest. The latter part of this chapterpresents a fairly detailed study of Lanczos-Lovelock theories of gravity andhow the many thermodynamic aspect of Einstein gravity smoothly generalizeto this wide class of theories. This therefore also serves to demonstrate thatmany 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 classicalelectric field. In particular, we discuss what is known as Schwinger effect—theproduction of charged particle-antiparticle pair in a sufficiently strong, con-stant electric field, treated classically. The formal correspondence betweentime-dependent electric fields and FRW universes is also highlighted.
Chapter 4 deals with quantum fields in the background of classical grav-
1
itational fields. Two phenomena, which arise solely due to the considerationof quantum fields in non-inertial coordinates — Unruh effect and Hawkingeffect — are dealt with in some detail. Derivation for the simplified case ofa massless scalar field in 1+1 dimensional spacetime is provided. Motivatedby the results obtained from the study of Hawking effect, Chapter 4 alsoexplains the laws of black hole thermodynamics, particularly the generalizedsecond law of thermodynamics.
In Chapter 5 we study quantum fields in FRW universes. Calculationsare again simplified by considering a scalar field in a spatially flat FRWspacetime. It is demonstrated how this problem can be translated to thatof a scalar field in Minkowski spacetime with time-dependent mass. Wealso study particle production in the case of a universe which is radiation-dominated 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 5.2on [4]. Section 5.3 is is a summary of a problem, and the approaches used tosolve 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 horizonscan be approximated locally as a Rindler horizon. In section 3, we discussLanczos-Lovelock gravity and some results related to them. Much of thediscussion in sections 1 and 2 is based on [3] and section 3 is based on [1].
2.1 Rindler horizons in flat and curved space-
times
Consider a flat spacetime with the metric
ds2 = −dT 2 + dX2 + dL2⊥
(2.1)
The line X = ±T divide the X − T plane into four quadrants which wecall past P, future F , right R and left L wedges. Let us now introduce acoordinate transformation in all four quadrants, with l and t being the newvariables:
κT =√
2κl sinh(κt); κT = ±√
2κl cosh(κt) (2.2)
for |X| > |T | with positive sign in the right wedge and negative sign in theleft , 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 writtenas
ds2 = −2κldt2 +dl2
2κl+ dL2
⊥ (2.4)
3
We note that the coordinates t and l are timelike and spacelike in the rightand left wedges (where l > 0) respectively; the reverse is true in the futureand past wedges (where l < 0). Also, a given set of coordinates (t, l) corre-sponds 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 actslike a horizon for observers in the right wedge. Observers stationary in thenew coordinates (l = constant and x⊥ = constant) move on X2 − T 2 = 2l/κin the X −T plane. This is the trajectory of an observer moving with a con-stant 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 calledRindler 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 anylocal region. This is done by first transforming to the local inertial framearound that event and then introducing local Rindler frame as above.
Another way by which one may locally introduce Rindler observers inpossible in metrics of the form:
ds2 = −f(r)dt2 +dr2
f(r)+ dL2
⊥(2.5)
Here f is a function such that f(r) has a simple zero at some point r = a andwith a non-zero first derivative f ′(a) ≡ 2κ. Performing a Taylor expansionof f about r = a up to first order gives f ≈ 2κl were l = r− a. We thereforehave 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 Rindlermetric in Eq. (2.4) at l = 0. As in the Rindler case, one can eliminate thesingularity in this case as well with suitable coordinate transformations. Forexample
κX = eκξ cosh κt; κT = eκξ sinh κt; ξ ≡∫
dr
f(r)(2.6)
effects a transformation from (t, r) to (T, X). The resulting metric is
ds2 =f
κ2(X2 − T 2)(−dT 2 + dX2) + dL2
⊥ (2.7)
The factor f/(X2−T 2) remains finite at the horizon, even though the horizonr = a is now mapped to X2 − T 2 = 0.
2.2 Horizons in static spacetime
In this section we show how a large class of metrics corresponding to staticspacetimes with horizons can also be mapped to the Rindler form near the
4
horizon. Firstly, we observe that performing a coordinate transformationthat takes (l, t) to (x, t) according to the rule l = (1/2)κx2 reduces the formof the Rindler metric to:
ds2 = −κx2dt2 + dx2 + dL2⊥ (2.8)
and the Cartesian coordinates in terms of these new coordinates are
T = x sinh(κt); X = ±x cosh(κt) (2.9)
Consider now a static spacetime with the following properties (i) The met-ric is static in the given coordinate system, g0α = 0, gab(t,x) = gab(x) (ii)g00(x) = −N2(x) vanishes on some 2-surface H defined by the equationN2(x) = 0 (iii) ∂αN is finite and nonzero on H and (iv) all other metriccomponents and curvature remain finite and regular on H. With these as-sumptions the line element now is
ds2 = −N2(xα)dt2 + γαβ(xα)dxαdxβ (2.10)
The comoving observers in this frame have trajectories x = constant, four-velocity ua = −Nδ0
a and four acceleration ai = uj∇jui = (0, a) where aα =
(∂αN)/N . The unit normal nα to the N = constant surface is given bynα = ∂αN(gµν∂µN∂νN)−1/2 = aα(aβaβ)−1/2. Let us define a(x) as
N(nαaα) = (gαβ∂αN∂βN)1/2 ≡ Na(x) (2.11)
As we take the limit of x going to the horizon N = 0, the quantity Na hasa 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 yA transverse to these level surfaces, form a coordinatesystem. The line element in this new coordinate system is:
ds2 = −N2dt2 +dN2
(Na)2+ σAB(dyA − aAdN
Na2)(dyB − aBdN
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 metricreduces can be approximated near the horizon as
ds2 = −N2dt2 +dN2
(Na)2+ dL2
⊥ ≃ −N2dt2 +dN2
κ2+ dL2
⊥ (2.13)
This therefore shows that a large class of static spacetimes with horizonscan be mapped to the Rindler form in a region around the horizon. Thisform of the metric will be essential in calculating the temperature associatedwith a horizon. As a corollary, a temperature can be associated with almostall horizons. We derive the expression for this temperature in terms of thesurface 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):
AEH =
∫
V
dDx√−gR ≡
∫
V
dDx√−gLEH (2.14)
The Lagrangian LEH can be rewritten as
LEH = Q bcda Ra
bcd (2.15)
where
Q bcda =
1
2(δc
agbd − δd
agbc) (2.16)
is the only fourth rank tensor that can be constructed using only the metricand has all the symmetries of the curvature tensor. Moreover, this tensor isdivergence-free in all of its indices: ∇aQ
abcd = 0. These properties, suitablymodified, yield generalized theories of gravity which have many interestingproperties as will be seen later.
Also, we note that the Einstein-Hilbert Lagrangian may be written as
LEH = δcdabR
abcd; δcd
ab =1
2(δc
aδdb − δd
aδcb) (2.17)
where δcdab 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 andas terms which are total derivatives (’surface terms’). Let us refer to theseterms as Lbulk and Lsur respectively. That is,
√−gLEH = Lbulk + Lsur (2.18)
where
Lsur = 2∂c
[√−gQ bcda Γa
bd
]
; Lbulk = 2√−gQ bcd
a ΓadkΓ
kbc (2.19)
Note that the explicit form of Q bcda has not been used: only the property
∇aQabcd = 0 has been exploited to effect this separation. An interesting
result, which is a simple relation that allows Lsur to be completely determinedfrom Lbulk, follows:
Lsur = − 1
[(D/2) − 1]∂i
(
gab∂Lbulk
∂(∂igab)
)
(2.20)
Actions which satisfy the above relation have been referred to in recentliterature as being ”holographic.” We now study a more general class of
6
Lagrangians, called the Lanczos-Lovelock Lagrangians, which satisfy a gen-eralized version of the above holographic relation, besides sharing severalother properties of the Einstein-Hilbert Lagrangian.
We also note that the above holographic relation is linear, implying thata linear combination of holographic Lagrangians will also be holographic andhence simplifying their analysis.
It was noted earlier that separation of the Lagrangians of types () onlyused the condition that Q bcd
a is divergence-free in any one (and hence all,since Q bcd
a obeys all the symmetry relations of the curvature tensor) index.Let us further assume that, besides these conditions, Q bcd
a is constructedfrom gab and R bcd
a (as mentioned earlier, if it is to consist of only the former,the resulting action will be the Einstein-Hilbert action), and also impose thecondition that Q bcd
a is proportional to the curvature, so that the resultingLagrangian is quadratic in curvature. The resulting tensor is:
Qabcd = Rabcd − Gacgbd + Gbcgad + Radgbc − Rbdgac (2.21)
The resulting Lagrangian now is:
LGB =1
2
[
RabcdRabcd − 4RabRab + R2]
(2.22)
which is the so-called Gauss-Bonnet Lagrangian, whose variation is a puredivergence in 3+1 dimensions (but not in higher dimensions) and thereforedoes not contribute to the equations of motion.
The Einstein-Hilbert and the Gauss-Bonnet Lagrangian can be writtenas
LEH = δ1234R
2413; LGB = δ1357
2468R2413R
6857 (2.23)
where the numeral n stands for an index an. These can be generalized towhat would be the mth order term of the Lanczos-Lovelock Lagrangian:
L(m) = δ135...2k−1246...2k R24
13R6857...R
2k−22k2k−32k−1; k = 2m (2.24)
where k is an even number. We see that m = 1 gives the Einstein-HilbertLagrangian whereas m = 2 gives the Gauss-Bonnet Lagrangian. Since L(m)
is a homogeneous function of degree m in the curvature tensor Rabcd, it can be
expressed in the form
L(m) =1
m
(
∂L(m)
∂Rbcda
)
Rbcda ≡ 1
mP bcd
a Rabcd (2.25)
where P bcda ≡ (∂L(m)/∂R bcd
a ) so that P abcd = mQabcd. For these La-grangians:
∇aPabcd = 0 = ∇aQ
abcd (2.26)
7
These, therefore, are the generalizations that were referred to earlier, with adivergence free Qabcd, and therefore permit separation into a bulk term anda surface term in a similar fashion as Eq. (2.18).
One can linearly combine several such holographic Lagrangians L =∑m
i=1 ciL(i) where the ci are arbitrary coefficients. As noted earlier, anysuch combination is a holographic Lagrangian. Because of the antisymmetryof δ135...2k−1
246...2k , only terms with 4m ≤ D will have a nontrivial effect (as thevariation terms which do not satisfy this condition will reduce to a total diver-gence), and any terms not satisfying this condition may be ignored as beingphysically irrelevant. This therefore uniquely singles out Einstein-Hilbert La-grangian in 3+1 dimensions. This also explains the earlier remark regardingGauss-Bonnet Lagrangian being physically irrelevant in 3+1 dimensions.
It was mentioned that the Lanczos-Lovelock class of Lagrangians are in-teresting for the reason that they share several interesting properties withEinstein gravity (to which the Lanczos-Lovelock model uniquely conforms in3+1 dimensions, in the sense that the Einstein-Hilbert term is the only non-trivial term in the Lanczos-Lovelock Lagrangian). We now corroborate thisstatement, first by showing that the equations of motion for this generalizedclass is takes the form:
1√−g
∂√−gL
∂gab=
1
2Tab (2.27)
where the right-hand side is obtained by the variation of the matter part ofthe action. Indeed, the above equation reduces to the familiar Einstein’s fieldequation for L = LEH = R. The above equation can be expanded to makethis manifest:
(
∂L
∂Rklij
∂Rklij
∂gab− 1
2gabL
)
=
(
P ijkbR
kaij −
1
2gabL
)
=1
2Tab, (2.28)
where (∂√−g/∂gab) = −(1/2)
√−ggab has been used. The generalization ofEq. (2.3) can also be derived:
[(D/2) − m]L(m)sur = −∂i
[
gabδL
(m)bulk
δ(∂igab)+ ∂jgab
∂L(m)bulk
∂(∂i∂jgab)
]
(2.29)
The surface term in the action is assumes great significance in light of the factthat it is closely related to the entropy of the horizon, if a solution possessesone. In particular, one can obtain the result that if a solution to Einstein’sequation 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
Ssur = 2
∫
dDx∂c
[√−gQabcdΓabd
]
= 2
∫
dDx∂c
[√−gQabcd∂bgad
]
(2.30)
for the static metric in Rindler approximation
ds2 = −κ2x2dt2 + 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 ast −→ itE (where tE is real). It is immediately seen that tE becomes anangular coordinate with periodicity 2π/κ, with the metric now of the form
ds2E = x2k(κtE)2 + dx2 + dx2
⊥ (2.32)
The integral is now equivalent to an integral done over polar coordinates. Inorder to avoid the singularity at x = 0, the integral over tE is evaluated overthe range (0, 2π/κ). On thus integrating over t and x we get
Ssur = 2
(
2π
κ
)∫
dD−2x⊥
√−gncQabcd∂bgad (2.33)
where ni = (0, 1, 0, 0..) is the unit normal vector in the x-direction. Due tothe term ∂bgad, we see that only the b = x, a = d = 0 term contributes (asg00 is the only non-constant component, with dependence only on x. Withthis taken into account, the above expression we on simplification yields
Ssur = −8π
∫
H
dD−2x⊥
√σ(
Q0x0x
)
(2.34)
where σ is the determinant of the metric in the transverse space. For thecase of Einstein-Hilbert action where Qabcd = (1/32π)[gacgbd−gadgbc], we getQ0x
0x = 1/32π and
Ssur = −1
4A⊥ (2.35)
where the negative sign can be traced to the choice of the normal vector tothe horizon surface.
Relations like above, that relate the boundary term in the action to thehorizon entropy, continue to hold for Lanczos-Lovelock Lagrangians, as willnow be shown. We show the existence of a conserved current (i.e., a Noethercurrent) Ja that relates to the area of the horizon and therefore to the blackhole entropy. Given any Lagrangian L constructed from the Riemann tensor
9
and the metric tensor, the variation of the gravitational Lagrangian densitycan be written as
δ(L√−g) =
√−g[
Eabδgab + ∇a(δv
a)]
(2.36)
where ∇aEab = 0 holds off-shell. In the case of Einstein gravity, Eab is just
the Einstein tensor up to a constant factor.Let us now consider now an infinitesimal coordinate transformation xi −→
xi + ξi. It can be shown that the quantity
Ja ≡ (Lξa + δξva + 2Eabξb) (2.37)
is conserved, i.e., ∇aJa = 0. Here δξva represents boundary term which
arises for the specific variation of the metric in the form δgab = ∇aξb +∇bξa.We note that this conservation law is off-shell. Let us now consider a specialcase where ξa is an approximate Killing vector near an event P, i.e., ξa
satisfies ∇(aξb) = 0 and ∇b∇cξd = Rkbcdξk at P. In this case, δξv
a = 0 andthe current becomes
Ja = (Lξa + 2Eabξb) (2.38)
Note that the equations of motion have not been used in the preceding deriva-tion of the expression for Ja. With Tab defined as δAm/δgab = −(1/2)Tab
√−g(where Am stands for the matter part of the total action), the field equationsare obtained as 2Eab = Tab The Noether current, on shell, is therefore givenby
Ja = (T aj + gajL)ξj (2.39)
For any vector ka which satisfies kaξa = 0 we have,
(kaJa) = T ajkaξj (2.40)
Thus we see that when T aj changes by a small amount δT aj, kaJa changes
by δ(kaJa) = kaξjδT
aj . On the same lines as before, the time integration isdone over the range (0, β) where β = 2π/κ and κ is thesurface gravity of thehorizon.
δ
∫
H
dD−1x√
h(kaJa) =
∫
H
dD−1x√
hkaξjδTaj = β
∫
H
dD−2x√
hkaξjδTaj
(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 theentropy must be given by
SNoether = βN (2.42)
10
where the Noether charge N corresponds to a current Ja for a Killing vectorfield ξa = (1, 0), ξa = ga0. The entropy thus obtained is known as Waldentropy. The fact that ∇aJ
a = 0 implies that there exists an antisymmetrictensor Jab such that Ja = ∇bJ
ab. The Noether charge N is given by
N =
∫
t
dD−1x√−gJ0 =
∫
t
dD−1x∂b(√−gJ0b)
=
∫
t,rH
dD−2x√−gJ0r (2.43)
where contributions over transverse directions have been ignored, and in theradial direction only the contribution at r = rH has been accounted for. Itcan be shown that
J0r = 2P 0rcd∇cξd = 2P cdr0∂dgc0 (2.44)
Therefore, the Noether charge N is given by
N = 2
∫
t,rH
dD−2x√−gP cdr0∂dgc0 = 2m
∫
t,rH
dD−2x√−gQcdr0∂dgc0 (2.45)
Wald entropy therefore is
SNoether = βN = 2βm
∫
t,rH
dD−2x√−gQcdr0∂dgc0 (2.46)
In contrast, on evaluation of the surface term we get
Ssur = 2
∫
dDx∂c
[√−gQabcd∂bgad
]
= 2β
∫
t,rH
dD−2x√−gQabr0∂bga0 (2.47)
where t has again been integrated over the range (0, β) and transverse direc-tions have been ignored. Comparing the expressions for SNoether and Ssur, weget
SNoether = mSsur (2.48)
We now obtain an expression for the horizon entropy in Lanczos-Lovelockmodels. Consider the quantity Qx0
x0 = Q0x0x for the mth order Lanczos-
Lovelock Lagrangian, given by
Qx0x0 =
1
16π
1
2mδx0a3...a2m
x0b3...b2m
(
Rb3b4a3a4
...Rb2m−1b2m
a2m−1a2m
)
|x=ǫ (2.49)
where the normalization factor has been included so as to recover Einstein-Hilbert term for m = 1. Also, we define Qx0
x0 = 1/16π for the m = 0 case. We
11
note that δx0a3...a2m
x0b3...b2m= δA3A4...A2m
B3B4...B2mwith Ai, Bi = 2, 3, ...D − 1 since the terms
containing the factors δ0A and δx
A all vanish. Therefore,
Qx0x0 =
1
2
(
1
16π
1
2m−1
)
δA3A4...A2m
B3B4...B2m
(
RB3B4
B3B4...R
B2m−1B2m
A2m−1A2m
)
|x=ǫ (2.50)
Since in the ǫ → 0 limit, RABCD|H =(D−2) RAB
CD|H we see that
Qx0x0 =
1
2L(m−1) (2.51)
That is, Qx0x0 turns to be the same as the Lanczos-Lovelock Lagrangian of
the immediate lower order. Now, using Eq. (2.34), the quantity Ssur can bereadily evaluated.
The Wald entropy SNoether = mSsur becomes
SNoether = −4πm
∫
H
dD−2x⊥
√σL(m−1) (2.52)
We now see the rationale for defining Qx0x0 = 1/16π for the m = 0 case—
the entropy for the mth order Lanczos-Lovelock Lagrangian is given by anintegral over the Lagrangian of the (m − 1)th order. Since L(0) is a con-stant, we recover the familiar result that the entropy is proportional to thetransverse area, along with the required constant of proportionality. For ageneral Lanczos-Lovelock Lagrangian the entropy is easily found as the linearcombination
S =K∑
m=1
cmS(m) = −K∑
m=1
4πmcm
∫
H
dD−2x⊥
√σL(m−1) (2.53)
12
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 showthat particle-antiparticle pairs can spontaneously form in such a situation.Similar results apply for spinor-valued quantum fields as well, and thereforeelectron-positron pairs can be created in a static electric field. This phe-nomenon, known as Schwinger effect, is on the verge of being experimentallyverified [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 inthe context of scalar fields interacting with a background electric field. We dothis using the method of Effective Lagrangian. We shall derive the effectivelagrangian from the appropriate kernel of the path intergral formalism andshow that the effective lagrangian thus obtained has a nonzero imaginarypart, which yields the pair production rate.
The relevant kernel is
K(x, y; s) = 〈x| exp(
is
2
[
(i∂ − qA)2 − m2 + iǫ]
)
|y〉 (3.1)
Consider an electric field along the z-axis, which we assume has an arbitrarytime dependence for now, i.e., E(t) = E(t)z, B = 0. We choose the potentialAi 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
K(x0, y0;x,x; s) =
∫
d3p
(2π)3〈x0| exp
is
2
[
(i∂t)2 − p2
⊥ − (pz − qA(t))2 − m2 + iǫ]
|y0〉
=
∫
d3p
(2π)3exp
[
−is
2
(
p2⊥ + m2 − iǫ
)
]
〈x0| expis
2
[
−∂2t − (pz − qA(t))2
]
|y0〉
=
∫
d3p
(2π)3exp
[
−is
2
(
p2⊥ + m2 − iǫ
)
]
G(x0, y0; s) (3.3)
where G(t, t′; s) is the propogator for the one-dimensional quantum mechan-ical Hamiltonian
H = −1
2
∂2
∂t2− 1
2(pz − qA(t))2 (3.4)
Consider now the special case of a uniform electric field. The potential forsuch an electric field is A = −Et. Then
H = −1
2
∂2
∂t2− 1
2(pz + qEt)2 = −1
2
∂2
∂ρ2− 1
2q2E2ρ2 (3.5)
where ρ = t+(pz/qE). In this form, we see that the above Hamiltonian is thatof a quantum harmonic oscillator with unit mass and imaginary frequency(iqE). The coincidence limit of the propagator for this Hamiltonian is givenby
G(t, t; s) =
[
qE
2πi sinh qEs
]1/2
exp
[
iqE
sinh qEs(cosh qEs − 1)
(
t +pz
qE
)2]
(3.6)On substituting this back into Eq. (3.3) and evaluating the integral (overpx,py and ω) we get
K =qE
(2π)2
(
1
2is
)
e−is2
(m2−iǫ)
sinh(qEs/2)(3.7)
The effective lagragian Leff can be obtained from the kernel using the relation
Leff = −i
∫
∞
0
ds
sK(x, x; s) (3.8)
which for the kernel at hand yields
Leff = −i
∫
∞
0
ds
s
qE
(2π)2(2is)
e−is2
(m2−iǫ)
sinh(qEs/2)
= −1
4
∫
∞
0
ds
(2π)2
1
s2
qE
sinh qEse−is(m2−iǫ)
= −∫
∞
0
ds
4π2
1
s2
qE
sinh qEse−is(m2−iǫ) (3.9)
14
In order to evaluate the above integral, we analytical continue the variable ofintegration s to imaginary values (s −→ −is) and rewrite the above integralas
ImLeff = −∫
∞
0
ds
(4π)2
e−m2s
s2
qEs
sin qEs(3.10)
where we have discarded the iǫ term in the exponent as being unnecessaryfor our present method pf evaluation of the integral. We note that the aboveintegral is not well-defined as it divergence as s = 0. This is related to theproblem of renormalization, which is not dealt with in this thesis (and as wewould only be interested in evaluating the imaginary part of the above quan-tity). Another problem is the existence of poles along the path of integrationat 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 inte-gration that consists of small semicircles of radius ε in the upper half plane.The contribution by the nth pole is
In =
∫ θ=0
θ=π
(ǫeiθidθ)
(4π)2s2n
e−m2snqE
cos(nπ)ǫeiθ
= i(−1)n+1 (qE)2
16π3
[
1
n2exp
(
−m2π
qEn
)]
(3.11)
The total contribution to ImLeff is
ImLeff =
∞∑
n=1
(−1)n+1 1
2
(qE)2
(2π)3
1
n2exp
(
−m2π
qEn
)
(3.12)
The quantity ImLeff is related to the probablity for the system to maketransitions from ground state to excited state. Since the excited state inthis case is a state with particles of the quantum field present, we interpret2Im(Leff) as the probability per unit volume per unit time for the productionof scalar particles, which is
2Im(Leff) =∞∑
n=1
(qE)2
(2π)3
(−1)n+1
n2exp
(
−πm2
qEn
)
(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 caseturns 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
K =
∫
d2p⊥dω
(2π)3〈x0| exp
is
2
[
ω2 − p2y − (pz − qA)2 + ∂2
z − m2 + iǫ]
|y0〉
=
∫
d2p⊥dω
(2π)3exp
[
is
2
(
ω2 − p2y − m2 + iǫ
)
]
G(z, z; s) (3.14)
where G is now a propagator for the Hamiltonian
H = −1
2
∂
∂z2+
1
2(px − qA(z))2 (3.15)
As can be seen, unlike in the case of pure electric field, the frequency is purelyreal. For the special case of uniform magnetic field, A = −Bz,
H = −1
2
∂2
∂z2+
1
2(px + qBz)2 = −1
2
∂2
∂ρ2+
1
2q2B2ρ2 (3.16)
where ρ = z + (px/qB). This is the Hamiltonian for a Harmonic oscillatorwith unit mass and frequency (qB). The propagator therefore is
G(z, z; s) =
[
qB
2πi sin qBs
]1/2
exp
[
iqB
sin qBs(cos qBs − 1)
(
z +px
qB
)2]
(3.17)Performing the integrations over px, py and ω we have
K =qB
(2π)2
(
1
2is
)
e−i s2(m2−iǫ)
sin(qBs/2)=
1
(2π)2is
(qB/2)
sin(qBs/2)e−i s
2 (m2 − iǫ) (3.18)
This yields the effective Lagrangian
Leff = −i
∫
∞
0
ds
s
qB
(2π)2(2is)
e−i s2(m2−iǫ)
sin(qBs/2)
= −1
4
∫
∞
0
ds
(2π)2
1
s2
qB
sin(qBs)e−i(m2−iǫ)s
= −∫
∞
0
ds
4π2
1
s2
qB
sin(qBs)e−i(m2−iǫ)s (3.19)
Once again, the presence of sine function in the denominator introduces poles.However, the definition the harmonic oscillator path integral requires us toregard 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 aretherefore located at
sn = ±2nπ
qB(1 + iǫ) (3.20)
The effective Lagrangian may be expressed along the imaginary axis, i.e., wecan analytically continue the the integration variable s to imaginary valuesas s −→ is. The integral must be closed in the lower half plane to ensureconvergence due to the (−is(m2/2) in the exponent. We get
Leff = −i
∫
∞
0
ds
s2K(s) =
∫
∞
0
ds
s2
ie−m2
2s
sinh(qBs/2)
(
qB/2
(2π2)i
)
=
∫
∞
0
ds
s2
ie−m2
2s
sinh(qBs/2)
1
2π2
(
qB
2
)
(3.21)
We see that the above expression is purely real, and consequently concludethat a uniform magnetic field does not lead to particle production, unlike auniform 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 inertialreference frame. An accelerating observer, however, will define the positive-frequency modes with respect to the clock the observer carries, i.e., the propertime. It is therefore to be expected that two observers, one inertial and oneaccelerated, will not agree on the number and nature of particles they detectwhen 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 thermalbath of particles where an inertial observer only sees a vacuum. This resultis 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 ofa 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 = ηabdxadxb (4.2)
18
The 2-velocity, defined as ui = dxi/dτ satisfies the relation uiui = −1. Alsothe condition for constant acceleration can be covariantly stated as ηija
iaj =−a2
Let us now rewrite the Minkwoski metric in terms of the lightcone coor-dinates defined as
u ≡ −t + x, v ≡ t + x (4.3)
so that the metric becomesds2 = dudv (4.4)
We see that the coordinate transformation
u → u = cu, v → v =v
c(4.5)
where c is a nonzero constant, leaves the line interval invariant and corre-sponds to a Lorentz transformation. This can be made more explicit byexpressing the c in terms of the relative velocity vr of two inertial frames:
c =
[
1 − vr
1 + vr
]1/2
; vr =1 − c2
1 + c2(4.6)
We wish to determine the trajectory of a uniformly accelerating observerin an inertial frame. This is most easily done in terms of the lightconecoordinates (u(τ), v(τ)). Using the relations satisfied by proper velocity andconstant acceleration that were stated earlier
u(τ)v(τ) = −1, u(τ)v(τ) = a2 (4.7)
which after separation and integration give
v(τ) =A
aeaτ + B; u(τ) =
1
Aae−aτ + C (4.8)
where A, B and C are constants of integration. As noted earlier, the mul-tiplication of u by a nonzero constant and v by the inverse of that constantcorresponds to a Lorentz transformation. We use this freedom of the choiceof Lorentz frame to set A = 1. Similarly, the freedom of choosing the originof this Lorentz frame allows us to set B = C = 0. Therefore the trajectoryof the accelerated observer in terms of lightcone coordinates, in a suitablecoordinate system, is given by
u(τ) =1
ae−aτ , v(τ) =
1
aeaτ (4.9)
19
and in terms of the original Minkowski coordinates is
t(τ) =1
asinh aτ, x(τ) =
1
acosh aτ (4.10)
Eliminating the parameter τ in the above set of equations gives us theworldline of the accelerated observer as the right branch of the hyperbolax2 − t2 = a−2. It is therefore easily inferred that in the reference frame thatwas chosen (with the specific choices of A, B and C), the observer arrivesfrom infinity, momentarily comes to rest a x = a−1 and accelerates back toinfinity.
We now find a frame (ξ0, ξ1) comoving with the accelerating observer.We choose the frame to be such that the observer is at rest at ξ1 = 0 and ξ0
coincides with the the proper time τ along the observer’s worldline. It turnsout that such a comoving frame, along with the added bonus of conformalflatness, can be found:
ds2 = Ω2(ξ0, ξ1)[
−(
ξ0)2
+(
ξ1)2]
(4.11)
where Ω(ξ0, ξ1) is to be determined. To do so, we transform to the lightconecoordinates of the comoving frame:
u ≡ −ξ0 + ξ1, v ≡ ξ0 + ξ1 (4.12)
in which the the metric takes the form:
ds2 = Ω2(u, v)dudv (4.13)
and the observer’s worldline
ξ0(τ) = τ, ξ1(τ) = 0 (4.14)
takes the formv(τ) = −u(τ) = τ (4.15)
Since ξ0 is the proper time with respect to the accelerating observer, at theobserver’s own location, we have:
Ω2(u = −τ, v = τ) = 1 (4.16)
Also since Eq. (4.13) and Eq. (4.4) describe the same Minkowski spacetimein different coordinate systems we have:
ds2 = dudv = Ω2(u, v)duv (4.17)
20
The functions u(u, v) and v(u, v) can each depend only on one of two argu-ments u or v since if this was not the case there would be du2 and dv2 termsin the right-most expression. To be definite, we choose
u = u(u), v = v(v) (4.18)
If the above functions are determined, Ω2(u, v) can be read off from Eq.(4.17). Since
−au(τ) =du(τ)
τ=
du(u
du
u(τ)
dτ=
du(u)
du(−1) (4.19)
we haveu = C1e
au (4.20)
and similarlyv = C2e
av (4.21)
Here C1 and C2 are constants of integration related by Eq. (4.7). ThereforeC1C2 = a2. The exact values of C1 and C2 cannot be determined, and are infact not needed. The line interval, in light cone coordinates, is therefore
ds2 = dudv = ea(v+u)dudv (4.22)
and in terms of the coordinates (ξ0, ξ1) is
ds2 = e2aξ1[
−(
dξ0)2
+(
dξ1)2]
(4.23)
One can also express the Cartesian coordinates (x, t) in the inertial frame interms of (ξ0, ξ1) as
t(ξ0, ξ1) =1
aeaξ1
sinh aξ0; x(ξ0, ξ1) =1
aeaξ1
cosh aξ0 (4.24)
We can see that for the ranges −∞ < ξ0 < ∞ and −∞ < ξ1 < ∞, thesecoordinates cover only the right wedge of the 1+1 dimensional Minkowskispacetime. This coordinate system is therefore incomplete. The acceleratingobserver cannot observe more than a−1 in the direction opposite to the accel-eration. Consider a hypersurface of constant time ξ0. We see that the infiniterange of spacelike coordinate −∞ < ξ1 spans a finite physical distance
d =
∫ 0
−∞
eaξ1
dξ1 =1
a(4.25)
Therefore no frame comoving with an accelerating observer can cover theentire Minkowski spacetime. Since events beyond the right wedge cannotbe observed by the acceleration observer, the boundary of the right wedgeis a horizon. We have therefore show the existence of a horizon within thecontext of special relativity.
21
4.1.2 Quantum fields in accelerated frame
We now consider a massless scalar field in 1+1 dimensional spacetime withminimal coupling to gravity. The action is given by
S[φ] =1
2
∫
gabφ,aφ,b
√−gd2x (4.26)
It is easily seen that this action in 1+1 dimensions is conformally invariant.Under a conformal transformation
gab → gab = Ω2(x)gab (4.27)
the quantities√−g and gab change as
√−g →√
−g = Ω2√−g, gab → gab = Ω−2gab (4.28)
The factors of Ω2 cancel and the action remains conformally invariant. Itmust be noted that it was necessary for the dimension of spacetime to be1+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 acceleratedframes:
S = −1
2
∫
[
− (∂tφ)2 + (∂xφ)2] dtdx (4.29)
= −1
2
∫
[
− (∂ξ0φ)2 + (∂ξ1φ)2] dtdx (4.30)
In terms of lightcone coordinates,
S = 2
∫
∂uφ∂vφdudv = 2
∫
∂uφ∂vφdudv (4.31)
The field equations in the lightcone coordinates are
∂u∂vφ = 0, ∂u∂vφ = 0 (4.32)
have solutions
φ(u, v) = A(u) + B(v), φ(u, v) = A(u) + B(v) (4.33)
where A,A,B and B are arbitrary smooth functions. In particular,
φ ∝ e−iωu = eiω(t−x) (4.34)
22
describes a right-moving, positive-frequency mode with respect to the Minkowskitime t, while
φ ∝ e−iΩu = eiω(ξ0−ξ1) (4.35)
describes a right-moving, positive-frequency mode with respect to the propertime τ = ξ0 with respect to the accelerating observer. Similarly the solutionsφ ∝ e−iωv and φ ∝ e−iΩv describe left-moving modes. In the right wedgeof the Minkowski spacetime where both coordinate systems overlap, one canwrite the mode expansion for the field operator φ as
φ =
∫
∞
0
dω
(2π)(1/2)
1√2ω
[
e−iωua−
ω + eiωua+ω
]
+ (left-moving) (4.36)
=
∫
∞
0
dΩ
(2π)(1/2)
1√2Ω
[
e−iΩub−Ω + eiΩub+Ω
]
+ (left-moving) (4.37)
where both sets of operators satisfy standard commutation relations:
[
a−
ω a+ω
]
= δ(ω − ω′),[
b−Ω b+Ω
]
= δ(Ω − Ω′) etc. (4.38)
The vacuum state associated with the annihilation operators a−ω is called the
Minkowski vacuum |0M〉 and the one associated with b−Ω operators is calledthe Rindler vacuum |0R〉. Therefore
a−
ω |0M〉 = 0 b−Ω |0R〉 = 0 (4.39)
The Minkowski vacuum is the physical vacuum that pertains to the iner-tial observer while the Rindler vacuum pertains to the accelerating observer.That is, an inertial observer will register no particles in the Minkowski vac-uum as will an accelerating observer in the Rindler vacuum. However, to theaccelerating observer the Minkowski vacuum will appear to be a state popu-lated with particles. We now calculate the occupation number of particles inMinkowski 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 + m2χ = 0 (4.40)
where the prime ′ stands for derivative with respect to η. One can Fourierexpand χ as
χ(x, η) =
∫
d3k
(2π)3/2χk(η)eikx (4.41)
The equation of motion satisfied by the Fourier modes χk(η) is obtained as
χ′′
k+ ω2
k(η)χk = 0 (4.42)
23
where ω2k(η) ≡ k2 + m2. As the above equation is a linear, homogeneous,
second-order differential equation, the general solution to these equations canbe expressed as the linear combination of two particular solutions, which wechoose to be complex conjugates of each other (vk(η) and v∗
k(η)), as
χk(η) =1√2
[
a−
kv∗
k(η) + a+−k
vk(η)]
(4.43)
We also normalize these functions vk(η) and v∗
k(η) such that their Wron-skian (v′
kv∗
k −vkv∗
k′) = 2i (the Wronskian is guaranteed to be nonzero if vk(η)
and v∗
k(η) are linearly independent, and therefore the normalization can beeffected). The functions are called ”mode functions”.
The field χ(x, η) can now be written as
χ(x, η) =1√2
∫
d3k
(2π)(3/2)
[
a−
kvk ∗ (η) + a+
−kvk(η)
]
eikx (4.44)
=1√2
∫
d3k
(2π)(3/2)
[
a−
kvk ∗ (η)eikx + a+
−kvk(η)e−ikx
]
(4.45)
It turns out that the equations obeyed by the mode function (i.e., theequation of motion and the normalization condition) do not serve to selectthe mode functions uniquely. It is easily verified that
uk(η) = αkvk(η) + βkv∗
k(η) (4.46)
where αk and βk are time independent complex coefficients satisfying |α2k| −
|β2k| = 1, also satisfy both the equation of motion for mode functions and
the normalization condition, and therefore can be used as mode functionsinstead of vk(η), i.e., either of the two set of functions can be used to expandthe field χ. The coefficients αk and βk are called Bogolyubov coefficients.
In a general case where positive and negative frequency modes with re-spect to the inertial observer contributes to the positive frequency modeswith respect to positive frequency modes in the accelerated frame we have
b−Ω =
∫
∞
0
dω[
αΩωa−
ω − βΩωa+ω
]
(4.47)
where αωΩ and βΩω are functions of ω and Ω that satisfy the normalizationcondition
∫
∞
0
dω (αΩωα∗
Ω′ω − βΩωβ∗
Ω′ω) = δ(Ω − Ω′) (4.48)
such that the commutation relations Eq. (4.38) hold. The above normal-ization 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
1√ω
e−iωu =
∫
∞
0
(
αΩ′ωe−iΩ′u − β∗
Ω′ωeiΩ′u)
(4.49)
We multiply both sides by exp(±iΩu), integrate over u to solve for αΩω andαΩω to obtain
αΩω =
∫ +∞
−∞
e−iωu+iΩudu =1
2πa
√
Ω
ωe
πΩ
2a exp
(
iΩ
aln
ω
a
)
Γ
(
−iΩ
a
)
(4.50)
and similarly
βΩω =
∫ +∞
−∞
e+iωu+iΩudu = − 1
2πa
√
Ω
ωe−
πΩ
2a exp
(
iΩ
aln
ω
a
)
Γ
(
−iΩ
a
)
(4.51)Therefore we have the relation
|αΩω|2 = e2πΩ
a |βΩω|2 (4.52)
We now compute the occupation number 〈NΩ〉, i.e., the mean number ofparticles with frequency Ω, of the Minkowski vacuum as measured by aRindler observer. This is just the expectation value of the number oper-ator NΩ ≡ b+
Ω b−Ω with respect to the Minkowski vacuum:
〈NΩ〉 ≡ 〈0M |b+Ω b−Ω |0M〉
= 〈0M |(∫
dω[
α∗
ωΩa+ω − β∗
ωΩa−
ω
]
)
×(∫
dω′[
α∗
ω′Ωa−
ω − β∗
ω′Ωa+ω′
]
)
|0M〉
=
∫
dω|βωΩ|2 (4.53)
The normalization condition Eq. (4.48) for Ω′ = Ω becomes
∫
∞
0
dω(
|αΩω|2 − |βΩω|2)
= δ(0) (4.54)
and therefore using Eq. (4.52) we have
〈NΩ〉 =
∫
∞
0
dω|βωΩ|2 =
[
exp
(
2πΩ
a− 1
)
− 1
]−1
δ(0) (4.55)
The δ(0) factor is a result of the fact that the field was quantized in infinitevolume 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 densityof particles with frequency Ω is
nΩ =〈NΩ〉V
=
[
exp
(
2πΩ
a− 1
)
− 1
]−1
(4.56)
The above calculation has only been done for right-moving modes. The resultfor left-moving modes can be similarly derived. We see that massless particlesin Bose-Einstein distribution are observed by the accelerating observer inMinkowski vacuum, with the (Unruh) temperature
T =a
2π(4.57)
4.2 Hawking Effect
A nonrotating black hole with zero electric charge is in (3+1) dimensions isdescribed in natural units (G = ~ = c = 1)by the Schwarzschild metric,
ds2 = −(
1 − 2M
r
)
dt2 +dr2
1 − 2Mr
+ r2(
dθ2 + dφ2 sin2 θ)
(4.58)
Let us first consider the simpler case of a 1+1-dimensional black hole, assum-ing that its metric is given by the time and radial part of the Schwarzschildmetric:
ds2 = gabdxadxb = −(
1 − rg
r
)
dt2 +dr2
1 − rg
r
(4.59)
where rg = 2M . We now introduce the so-called tortoise coordinates:
dr∗ =dr
1 − rg
r
(4.60)
Therefore
r∗(r) = r − rg + rg ln
(
r
rg− 1
)
(4.61)
The metric takes the form
ds2 =
(
1 − rg
r (r∗)
)
[
−dt2 + dr∗2]
(4.62)
As can be seen from Eq. (4.61), the tortoise coordinates are defined only forr > 0. Introducing the tortoise lightcone coordinates:
u ≡ −t + r∗, v ≡ t + r∗ (4.63)
26
and the metric now takes the form
ds2 =
(
1 − rg
r(u, v)
)
dudv (4.64)
There, however, is another coordinate system that covers the entire space-time. These are Kruskal-Szekeres coordinates. The Kruskal-Szekeres light-cone coordinates are defined as
u = 2rg exp
(
u
2rg
)
, v = 2rg exp
(
v
2rg
)
(4.65)
in which the metric takes the form
ds2 =rg
r(u, v)exp
(
1 − r(u, v)
rg
)
dudv (4.66)
We see that the metric now is singular at r = rg. Thus the singularity inthe 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 analyticallycontinued to u < 0 and v < 0 so that the Kruskal-Szekeres coordinates spanthe entire spacetime. One may express the original Schwarzschild coordinatest and r in terms of the Kruskal-Szekeres coordinates as follows
uv = 4r2g exp
(
r∗
rg
)
= 4r2g exp
(
r
rg
− 1
)
exp
(
r
rg
− 1
)
(v
u
)2
= exp
(
2t
rg
)
(4.67)These equations are valid even beyond the applicability of the coordinatesas in (), via analytic continuation. We see from the first of the above pair ofequations 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 (). Butbefore doing so, we pause to point out the mathematical similarity betweenthe relation the Minkowski and Rindler coordinates (in the case of an ac-celerating observer), and the tortoise and Kruskal-Szekeres coordinates (inthe case of a Schwarzschild black hole) bear with one another. In order tohighlight this similarity, we reproduce the transformation between the cor-responding lightcone coordinates below
u = a−1 exp(au), u = κ−1 exp(κu)
v = a−1 exp(av), v = κ−1 exp(κv) (4.68)
27
where κ = (2rg)−1 is called the surface gravity of the horizon. Note also, that
the Kruskal-Szekeres coordinates cover the entire spacetime, like Minkowskicoordinates and the tortoise coordinates cover only the region of spacetimeoutside the horizon, like the Rindler coordinates.
This formal similarity greatly simplifies the following analysis of the afore-mentioned massless scalar field. The conformal invariance of the action allowsus to write the solution of the scalar field equation in terms of the tortoiselightcone coordinates as
φ = A(u) + B(v) (4.69)
or in terms of the Kruskal-Szekeres lightcone coordinates as
φ = A(u) + B(v) (4.70)
where A, A etc. are arbitrary smooth functions. In a manner identical to theanalysis in Rindler spacetime, the quantized massless scalar field can now bemode-expanded in tortoise coordinates as
φ =
∫
∞
0
dΩ
(2π)1/2
1√2Ω
[
eiΩub−Ω + e−iΩub+Ω
]
+ (left − moving) (4.71)
which define the creation and annihilation operators b±Ω. The vacuum state
|0B〉 corresponding to these annihilation operators, that is b−Ω |0B〉 = 0, iscalled Boulware vacuum. Similarly, one may expand the field in Kruskal-Szekeres lightcone coordinates.
φ =
∫
∞
0
dω
(2π)1/2
1√2ω
[
eiωua−
ω + e−iωua+ω
]
+ (left − moving) (4.72)
which defines another set of creation and annihilation operators a±ω , which
in turn defines the Kruskal vacuum |0K〉 as the state satisfying a−ω |0K〉 = 0.
As is clear from the emphasized similarity between the present case andthat in the previous section, the Kruskal vacuum is on the same footing asthe Minkowski vacuum; the Boulware vacuum is on the same footing as theRindler vacuum. The derivation of the occupation number is identical tothat in the previous section. The remote (Boulware) observer sees particlesin a thermal spectrum
〈NΩ〉 ≡ 〈0K |b+Ω b−Ω |0K〉 =
[
exp
(
2πΩ
κ
)
− 1
]−1
δ(0) (4.73)
with the temperature
TH =κ
2π=
1
8πM(4.74)
28
4.3 Horizon temperature from path integrals
We now derive the temperature of a horizon using the path integral formal-ism. The discussion is based on [4].
Consider a spacetime with a horizon described with two different coordi-nate systems: one being a global coordinate system (T, X,x⊥) like Cartesiancoordinates for the case of flat spacetime or Kruskal coordinates for the caseof spherically symmetric metrics with horizons and the other (t, x,x⊥) cov-ers four different quadrants of the spacetime and is related to the first bytransformations like Eq. (2.6). Next, we analytically continue both T and tto imaginary values as T −→ iTE , t −→ itE (where TE and tE are real). Itis immediately seen that tE becomes an angular coordinate with periodicity2π/κ. Therefore, the field evolution from TE = 0 to TE → ∞, effected bythe inertial Hamiltonian HI , is mapped to the field evolution from tE = 0to tE → 2π/κ, which is effected by the Rindler Hamiltonian HR. For eitherHamiltonian, we assume that by adding an appropriate constant the energyof the ground state has been set to zero.
We now show that
〈vac|φL, φR〉 ∝ 〈φL|eπHR/κ|φR〉 (4.75)
where φL and φR are field configurations for X < 0 and X > 0 respectively.This follows since the ground state wave functional 〈vac|φL, φR〉 is expressibleas a path integral as
〈vac|φL, φR〉 ∝∫ 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 fieldconfiguration at tE = 0 is φ = φR and at tE = π/κ, φ = φL. Therefore theabove integral can also be written as
〈vac|φL, φR〉 ∝∫ κtE=π;φ=φL
κtE=0;φ=φR
Dφe−A (4.77)
This, however, is equivalent to a time evolution, effected by the RindlerHamiltonian HR from tE = 0 to tE = π/κ and therefore:
〈vac|φL, φR〉 ∝∫ κtE=π;φ=φL
κtE=0;φ=φR
Dφe−A = 〈φL|e−(π/κ)HR |φR〉 (4.78)
29
thus proving the claim in Eq. (4.75). This result, together with the imposi-tion of normalization condition help determine the proportionality constantin Eq. (4.75):
∫
DφL DφR 〈vac|φL, φR〉〈φL, φR|vac〉
= C2
∫
DφL DφR 〈φL|e−πHR/κ|φR〉 〈φR|e−πHR/κ|φL〉 = C2 Tr(
e−2πHR/κ)
= 1
Therefore we have
〈vac|φL, φR〉 =〈φL|e−πH/κ|φR〉[Tr(e−2πH/κ)]
1/2(4.79)
The density matrix for observations on the right Rindler wedge R can nowbe computed by taking the trace with respect to field configuration φL onthe left wedge L, which is the region behind the horizon:
ρ(φR, φ′
R) =
∫
DφL〈vac|φL, φR〉〈φL, φ′
R|vac〉
=
∫
DφL〈φR|e−(π/κ)HR |φL〉〈φL|e−(π/κ)HR|φ′
R〉Tr(e−2πHR/κ)
=〈φR|e−(2π/κ)HR |φ′
R〉Tr(e−2πHR/κ)
(4.80)
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 modelof a black hole that Hawking-radiates. We also note the behaviour of a blackhole 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 itsmass. 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 = 4πr2g = 16πM2 (4.81)
30
and temperature TH = (8πM)−1. Using the Stefan-Boltzmann law,
L = ΓγσT 4HA = Γγ
(
π2
60
)(
1
8πM
)4(
16M2)
=Γγ
15360πM2(4.82)
where Γ is the grey-body correction factor, γ counts the number of degreesof freedom and σ = π2/60 is the Stefan-Boltzmann constant in Planck units.The mass of the black hole therefore obeys the equation
dM
dt= −L = − Γγ
15360πM2(4.83)
the solution to which is
M(t) = M0
(
1 − t
tL
)1/3
tL ≡ 5120πM3
0
Γγ(4.84)
where M0 is the initial mass of the black hole. Therefore, an isolated blackhole has a finite lifetime tL ∝ M3
0 .It is easily shown that black holes have negative heat capacity. Since
E = M = (8πT )−1, we have
CBH =∂E
∂T= − 1
8πT 2< 0 (4.85)
It immediately follows that a black hole in an infinite heat bath can neverattain stable thermal equilibrium. Also, a black hole placed in a colder(infinite) heat bath will continue to emit radiation and get hotter; a black holekept in hotter surroundings, infinite in extent, will keep absorbing radiationand grow in size.
A stable equilibrium can, however, be attained if the black hole is placedin a finite heat bath.
4.4.2 Black hole entropy
In view of various conundrums raised by thought experiments involving blackholes (for instance, the experiment of pouring hot tea down a black holeand the possible decrease in entropy that results, as discussed by Wheeler[10]), Bekenstein conjectured that black holes must have nonzero entropySBH proportional to its surface area. This was, in part, motivated by atheorem 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 surfaceareas of the constituent black holes. Bekenstein could only however provide
31
a bound on the constant of proportionality in this relation, which was laterfixed following the discovery of Hawking radiation.
We start with the definition of entropy
dE = TdS (4.86)
We note that dE is equivalent to the quantity dM where M is the mass ofthe black hole. Also, as derived in the previous section, the temperature ofthe black hole is given by T = (1/8πM). Therefore we have
S =
∫
8πMdM (4.87)
which yields
SBH = 4πM2 =1
4A (4.88)
where A is the area of the black hole. The second law of thermodynamics isnow salvaged by asserting a generalized second law of black hole thermody-namics, which states that the total entropy—now defined as the sum of theentropy 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] andsection 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 aspatially flat FRW metric. The relevant metric, in coordinates that make itssymmetries manifest, is
ds2 = −dt2 + a2(t)δαβdxαdxβ (5.1)
We define a new coordinate, the conformal time:
η(t) ≡∫ t dt
a(t)(5.2)
in terms which the conformal equivalence of the metric to the Minkowskimetric ηab becomes manifest:
ds2 = a2(η)[
−dη2 + δαβdxαdxβ]
= a2(η)ηabdxadxb (5.3)
The action for a real massive scalar field φ(x) with minimal coupling to themetric is given by
S = −1
2
∫ √−gd4x[
gabφ,aφ,b + m2φ2]
(5.4)
33
which in terms of conformal time becomes
S = −1
2
∫
d3xdηa2[
−φ′2 + (∇φ)2 + m2φ2]
(5.5)
where prime denotes derivative with respect to conformal time. We nowdefine ξ ≡ a(η)φ and rewrite the action in terms of this new field, whileeliminating the total derivative terms, as
S =1
2
∫
d3xdηa2
[
ξ′2 − (∇ξ)2 −(
m2a2 − a′′
a
)
ξ2
]
(5.6)
The variation of the above action gives the equation of motion
ξ′′ − ∆ξ +
(
m2a2 − a′′
a
)
ξ = 0 (5.7)
Thus we see that the equation of motion of minimally coupled massive scalarfield is formally equivalent to that of a Klein-Gordon field in Minkowskispacetime, except that the mass is now a time-dependent effective mass:
m2eff(η) ≡ m2a2 − a′′
a(5.8)
We substitute the Fourier expansions of ξ
ξ(x, η) =
∫
d3k
(2π)3/2ξk(η)eikx (5.9)
into the equation of motion for ξ we find that the Fourier modes ξk(η) satisfya set decoupled ordinary differential equations
ξ′′k
+ ω2k(η)ξk = 0 (5.10)
where
ω2k(η) ≡ k2 + m2
eff(η) =2 +m2a2(η) − a′′
a(5.11)
In the above equation since ω2k(η) depends only on k ≡ |k| the general solu-
tion to the above equation may be written as
ξk(η) =1√2
[
a−
kv∗
k(η) + a+−k
vk(η)]
(5.12)
where vk(η) and its complex conjugate v∗
k(η) are two linearly independentsolutions of Eq. (5.10). The two complex constant of integration a±
kcan
depend on the direction of k as well. The reality of field ξ implies ξ∗k(η) =
34
ξ−k(η) which in turn implies a+k
=(
a−
k
)∗. It can be easily shown that vk and
v∗
k are linearly independent if and only if their Wronskian
W [vk, v∗
k] = v′
kv∗
k − vkv∗
k′ = 2iIm(v′v∗) (5.13)
is nonzero. Also, Eq. (5.10) implies that the Wronskian will be time-independent. Therefore, if W is nonzero, one can always normalize vk suchthat Im(v′v∗) = 1. In this case the complex solution vk(η) is called a modefunction. Substituting Eq. (5.12) in Eq. (5.9) we get
ξ(x, η) =1√2
∫
d3k
(2π)(3/2)
[
a−
kvk ∗ (η) + a+
−kvk(η)
]
eikx (5.14)
=1√2
∫
d3k
(2π)(3/2)
[
a−
kvk ∗ (η)eikx + a+
−kvk(η)e−ikx
]
(5.15)
One proceeds with quantization using the usual equal-time commutationrelations on the field operator ξ and its canonically conjugate momentumπ ≡ ξ′
[
ξ((x), η), π(y, η)]
= iδ(x − y) (5.16)[
ξ((x), t), ξ(y, t)]
= [π(x, t), π(y, t)] = 0 (5.17)
The Hamiltonian is given by
H(η) =1
2
∫
d3x
[
π2 +(
∇ξ)2
+ m2eff(η)ξ2
]
(5.18)
Alternatively one can impose the commutation relations on the constants ofintegration a±
k
[
a−
k, a+
k′
]
= δ (k − k′) ,[
a−
k, a−
k′
]
=[
a+k, a+
k′
]
= 0 (5.19)
together with the constraints that the mode functions obey Eq. (5.10) andthe normalization condition Im (v′
kv∗
k) = 1. The constants of integration a±
k
are now, therefore, interpreted as creation and annihilation operators. Thefield operator can now be expanded as:
ξ(x, η) =1√2
∫
d3k
(2π)(3/2)
[
a−
kv∗
k(η)eikx + a+−k
vk(η)e−ikx]
(5.20)
As was explained in the earlier chapter, the mode functions uk(η) (anduk(η)∗)can be expressed as linear combination of functions which serve equallywell as mode functions, as
uk(η) = αkvk(η) + βkv∗
k(η) (5.21)
35
where αk and βk are time independent complex coefficients satisfying |α2k| −
|β2k| = 1.Expanding the field operator in terms of the mode functions uk(η),
ξ(x, η) =1√2
∫
d3k
(2π)(3/2)
[
b−kv∗
k(η)eikx + b+−k
vk(η)e−ikx]
(5.22)
where b±k
are another set of creation and annihilation operators satisfying thecommutation relations. As we now have two expansions for the same field:
eikx[
u∗
k(η)b−k
+ uk(η)b+−k
]
= eikx[
v∗
k(η)a−
k+ vk(η)a+
−k
]
(5.23)
Using the above equation and Eq. (5.21) one has
a−
k= αk∗ b−
k+ βk b
+−k
, a+k
= αkb+k
+ βk∗ b−−k
(5.24)
The above relations are called the Bogolyubov transformations. Similarly,one can write b±
kas a linear combination of a±
k:
b−k
= αka−
k+ βka
+−k
, b+k
= αk∗ a+k
+ βk∗ a−
−k(5.25)
Each of the set of operators a±
kand b±
kdefine their respective vacuum
states |(a)0〉 and |(b)0〉 as
a−
k|(a)0〉 = 0 b−
k|(b)0〉 = 0 (5.26)
for all k. It is, however, not necessary that the vacuum state with respectto the a±
koperators, i.e., |(a)0〉 is a vacuum with respect to the b±
koperators.
This can be seen by explicit calculation:
〈(b)0|N (a)k
|(b)0〉 = 〈(b)0|a+ka−
k|(b)0〉
= 〈(b)0|(
αk b+k
+ βk∗ b−−k
)(
αk∗ b−k
+ βk b+−k
)
|(b)0〉= |βk|2δ(3)(0) (5.27)
The divergent factor of δ(3)(0) is a result of quantization in infinite volumerather than a closed box. The density of the a-particles in the mode k istherefore
nk = |βk|2 (5.28)
The total mean density of all particles is given by
n =
∫
d3k|βk|2 (5.29)
36
This quantity is finite only if |βk|2 decays faster than k−3 for large k.The state |(b)0〉 can be expressed as a superposition of the eigenvectors of
the N(a)k
operator as
|(b)0〉 =
[
∏
k
1
|αk|1/2exp
(
βk
2αka+ka+−k
)
]
|(a)0〉
=∏
k
1
|αk|1/2
(
∞∑
n=0
(
βk
2αk
)n
|(a)nk, n−k〉)
(5.30)
Quantum states which are defined as exponential of quadratic combinationof operators acting on vacuum state are called squeezed states. By virtueof this definition we see that the the a-vacuum is a squeezed state of theb-vacuum and vice versa.
5.2 Correspondence with Particle Production
in Electric Field
We now point out an interesting formal correspondence between the problemof scalar fields in an FRW universe and those in a time-dependent electricfield. Consider the action for a scalar field Φ
A = −∫
d4x√−g
1
2Φ
[
∂i∂i + m2 +
R
6
]
Φ (5.31)
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). Thespacetime 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 actionas
A = −1
2
∫
d4xφ[
∂i∂iflat + m2a2(t)
]
φ (5.33)
where φ = aΦ. The relevant kernel for this action is
K(x, y, ; s) = 〈x|e−is 1
2[∂i∂i+m2a2(t)−iǫ]|y〉
=
∫
d3p
(2π)3〈t|e−is 1
2 [∂2t +p2+m2a2(t)−iǫ]|t′〉
=
∫
d3p
(2π)3e−is 1
2(p2−iǫ)G(t, t′; s) (5.34)
37
where is G is the propagator for the Hamiltonian
H = −1
2
∂2
∂t2− 1
2m2a2(t) (5.35)
This Hamiltonian, we observe, is identical to that of a particle in an electricfield if we are allowed the identification m2a2(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 fieldsand FRW universes. As an example, let us consider the case of constantelectric field. The above mentioned identification motivates the expansionfactor
a2(t) =1
m2(pz + qEt)2 ≡ α2(t + t0)
2, α ≡(
qE
m
)
(5.36)
In the ”comoving” coordinates, in which the metric takes the form
ds2 = dτ 2 − a2(τ)dr2 (5.37)
the expansion factor is
a(τ) = (2aτ)1/2 ∝ τ 1/2 (5.38)
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 ofparticle 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-dominatedat early times and de Sitter at late times. As in the previous section, theuniverse we deal with a spatially flat 3+1 dimensional FRW universe. Themetric therefore has the generic form in Eq. (5.1). We now evaluate theexpansion factor using the 00 component of Einstein’s equations along withthe appropriate source terms:
a2
a2=
8πG
3ρ (5.39)
In order to model a universe that is asymptotically de Sitter, we choose theρ as
ρ = A∗ +B∗
a4(5.40)
38
where A∗ and B∗ are constants. Substituting this in Eq. (5.39), we have
a2
a2= A +
B
a4(5.41)
where A and B are constants. The above equation is conveniently solvedby a substitution of the type a2 = y. The solution to the above equation isobtained as
a =
(
B
A
)1/4√
sinh (2A1/4t) (5.42)
It is readily verified that the above expansion factor behaves in the requiredfashion at both early and late times: When t ≈ 0, sinh(2A1/2t) ≈ 2A1/2t andtherefore a(t) ≈
√2B1/4t1/2; when t is large sinh(2A1/2t) ≈ (1/2) exp(2A1/2t)
and therefore a(t) ≈ (1/√
2)(B/A)1/4 exp(A1/2t). The conformal time η isgiven by
η =
∫ t dt
a(t)=
∫ t dt(
BA
)1/4√
sinh (2A1/2t)(5.43)
Inverting the above expression so as to obtain a(η) is difficult and the ap-proach of the previous section, namely that of using Eq. (5.7) is not readilyapplicable. We may, however, write the equation of motion of the scalar fieldin the original ’comoving’ coordinates. For a metric of the form in Eq. (5.1),the Laplacian operator
=1√−g
∂i
(√−ggik∂kφ)
(5.44)
is given by
= −∂2t − 3a2a
a3∂t +
∑
x,y,z
1
a2∂2
x (5.45)
The equation of motion for the scalar field φ = 0 then becomes
a2φ + 3aaφ −∇2φ = 0 (5.46)
The mode expansion, in comoving coordinates, for the field φ can now bewritten, like in Eq. (5.12). The mode functions will then satisfy the equation
a2v + 3aav + k2v = 0 (5.47)
On substituting for the expansion factor from Eq. (5.42) we have,
(
B
A
)1/2
sinh (2A1/2t)v + 3B1/2 cosh (2A1/2t)v + k2v = 0 (5.48)
39
We make the substitution 2A1/2t = τ to obtain
2 sinh τ v + 3 cosh τ v + Kv = 0 (5.49)
where the dot overhead now represents derivative with respect to τ andK = k2/2(AB)1/2.
Solving the above equation turns out to be extremely difficult. We there-fore use a technique that allows us to instead work with a second order lineardifferential equation with no first derivative terms. Consider a differentialequation of the form
y + P (x)y + Q(x)y = 0 (5.50)
Let y = A(x)B(x). Let us also assume A(x) is such that the coefficientof 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
A = exp
(
−∫
P (x)
2dx
)
(5.53)
The equation satisfied by B is therefore
AB′′(A′′ + P (x)A′ + Q(x)A)B = 0 (5.54)
which using A′ = (−1/2)AP and A′′ = (1/4)A(P 2 − 2P ′) becomes
B′′ + B
[
Q − P 2
4− P ′
2
]
= 0 (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
B′′ +
[
K∗cosech(τ) +3
16cosech2(τ) − 9
16
]
B = 0 (5.56)
We note that τ varies over the range (0,∞). But for this fact, the aboveproblem is equivalent to that of one-dimensional scattering of a quantumparticle, as dictated by the Schrodinger equation with an appropriate poten-tial. 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 usefamiliar techniques such as a WKB approximation in order to understandthe early and late times behaviour of the mode functions. This is currentlybeing pursued.
40
Chapter 6
Conclusion
Thermodynamical aspects of spacetime horizons, namely how a temperaturecan be ascribed to a large class of metric that can be approximated by aRindler form locally, was studied. However, it still to be seen how the notionof entropy can be attached to such horizons, as we saw was naturally thecase with the horizon of a Schwarzschild black hole. The Lanczos-Lovelockgeneralization of Einstein gravity to higher dimensions was studied. It wasunderstood, and strongly emphasized in the report, as to how this is an ex-tremely natural generalization of Einstein gravity, to which it uniquely boilsdown in 3+1 dimensions. It was also understood that several thermody-namical results regarding gravity were robust in the sense that this naturalgeneralization continued to uphold these thermodynamical results or somegeneralization thereof. The author has therefore understood that, from atheoretical standpoint, lot of clues about the thermodynamical aspects ofgravity are to be discovered from the study of this larger set of theories.
The author has also studied and understood Schwinger effect, i.e., theproduction of particle-antiparticle pair in a static electric field. Besides un-derstanding the effect itself, the general physical and mathematical methodsinvolved in obtaining the result were well-understood. Also, the correspon-dence between quantum effects in time-dependent electric field and the prob-lem of cosmological particle production in FRW universes was noted. Theauthor intends to study this formal similarity between the two situationsfurther.
Considerable time was spent in understanding quantum fields in classicalgravitational background. The two effects, Unruh effect and Hawking effectwere studied in considerable detail. It was understood how the underlyingidea behind the two effects is essentially the same: the most natural choice ofvacuum with respect to one observer appears populated with a thermal sea ofquantum particles with respect to another observer, with their corresponding
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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 surfacearea of the horizon. While this is a great relief with the sanctity of secondlaw of thermodynamics in mind and fits well the general relativistic resultthat net surface area of black holes never decreases during collisions andmergers, this raises several more questions regarding information loss fromblack holes, the resulting violation of the unitarity of quantum mechanicsand as to where does the black hole entropy ”live.” The author is currentlyreading further on these topics.
Lastly, quantum fields in spatially flat FRW metrics in 3+1 dimensionswere studied. The idea that the problem of a scalar field in curved spacetimecan be brought to a formal equivalence with that of a scalar field with time-dependent mass in flat spacetime was particularly appreciated. This methodwas employed in order to understand particle production in a universe thatis 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 severalpossible methods to achieve analytic or numeric, accurate or approximateresults to this problem.
To conclude, the author has learned a lot of new ideas regarding QuantumField Theory in Curved Spacetime and Horizon Thermodynamics, but hasalso fully understood that there is a lot more ground to be covered in thestudy of this subject, which he intends to do with time and hopes contributeto the subject itself.
Acknowledgments
I would like to extend my deepest gratitude to my guide Prof. T. Padman-abhan for all the motivation during the course of this work and for providingme the opportunity to work under his guidance at IUCAA. I thank IUCAAfor their kind hospitality, their library and computer facilities; latter twoof which were extremely crucial for the completion of this thesis. My deepgratitude also to BITS Pilani that provides its students the opportunity topursue their Masters thesis off-campus.
I also thank Dawood Kothawala, Sanved Kolekar and Suprit Singh formany fruitful discussions.
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Bibliography
[1] Padmanabhan T., 2010, Gravitation: Foundations and Frontiers
(Cambridge University Press)
[2] Mukhanov V. F., Winitzki S., 2007, Introduction to Quantum Effects
in Gravity (Cambridge University Press)
[3] Padmanabhan T., 2010, Thermodynamical Aspects of Gravity: New
Insights [arxiv:0911.5004v2]
[4] Padmanabhan T., 1997, Aspects of Quantum Field Theory in Geom-
etry, Fields and Cosmology ed. Iyer B. R. and Vishveshwara C. V.,(Kluwer)
[5] Blaschke D. B., et al, Dynamical Schwinger effect and high-intensity
lasers. Realising nonperturbative QED [arXiv:0811.3570]
[6] Jacobson T., Introductory Lectures on Black hole Thermodynamics
[arXiv:gr-qc/0308048]
[7] Fulling S.A., 1973 . Nonuniqueness of Canonical Field Quanti-
zation in Riemannian Space-Time. Physical Review D 7: 2850.doi:10.1103/PhysRevD.7.2850
[8] Davies P.C.W., 1975, Scalar production in Schwarzschild and Rindler
metrics, Journal of Physics A 8: 609. doi:10.1088/0305-4470/8/4/022
[9] Unruh W.G., 1976, Notes on black-hole evaporation, Physical ReviewD 14: 870. doi:10.1103/PhysRevD.14.870
[10] Wheeler J. A., 1990, A Journey into Gravity and Spacetime (ScientificAmerican Library, NY)
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