Download ABSTRACT ACCELERATION AND OBSERVER DEPENDENCE OF

ABSTRACT
ACCELERATION AND OBSERVER DEPENDENCE OF VACUUM
STATES IN QUANTUM FIELD THEORY
We derive a novel flat spacetime metric of a non-uniformly accelerating
observer (NUAO). Like the UAOs, these particular NUAOs see a
Bose-Einstein distribution. However, we show that these NUAOs, unlike
UAOs, experience no event horizons, no temperature, and no information loss.
This has implications for black hole evaporation, firewalls, and the
information paradox.
Alaric Doria
May 2016
ACCELERATION AND OBSERVER DEPENDENCE OF VACUUM
STATES IN QUANTUM FIELD THEORY
by
Alaric Doria
A thesis
submitted in partial
fulfillment of the requirements for the degree of
Master of Science in Physics
in the College of Science and Mathematics
California State University, Fresno
May 2016
APPROVED
For the Department of Physics:
We, the undersigned, certify that the thesis of the following
student meets the required standards of scholarship, format, and
style of the university and the student’s graduate degree program
for the awarding of the master’s degree.
Alaric Doria
Thesis Author
Gerardo Muñoz (Chair)
Physics
Douglas Singleton
Physics
Karl Runde
Physics
For the University Graduate Committee:
Dean, Division of Graduate Studies
AUTHORIZATION FOR REPRODUCTION
OF MASTER’S THESIS
I grant permission for the reproduction of this thesis in part or
in its entirety without further authorization from me, on the
condition that the person or agency requesting reproduction
absorbs the cost and provides proper acknowledgment of
authorship.
X
Permission to reproduce this thesis in part or in its entirety
must be obtained from me.
Signature of thesis author:
ACKNOWLEDGMENTS
I would like to thank my advisor and committee members, Professor
Gerardo Muñoz, Professor Douglas Singleton, and Professor Karl Runde, for
their helpful advice, guidance and knowledge.
TABLE OF CONTENTS
Page
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
INTRODUCTION AND BACKGROUND . . . . . . . . . . . . . . . . .
1
Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
ACCELERATION IN SPECIAL RELATIVITY . . . . . . . . . . . . . .
6
Derivation of Metric . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Minkowski Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . .
10
Light Cone Coordinates . . . . . . . . . . . . . . . . . . . . . . . . .
12
UNRUH RADIATION . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
Quantum Field Theory in the Uniformly Accelerated Frame . . . . .
14
Unruh Radiation Particle Eigenstates Expansion . . . . . . . . . . . .
21
QUANTUM FIELDS AND NON-UNIFORMLY ACCELERATING OBSERVERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
Quantum Fields and Non-uniformly Accelerating Observers . . . . . .
29
Expansion of the Minkowski Vacuum State . . . . . . . . . . . . . . .
34
CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
FIGURE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
Appendix A: Rindler Observers and Unruh Temperature . . . . . . .
42
Appendix B: Verification of Bose-Einstein Distribution of Particle
Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
LIST OF FIGURES
Page
Figure 1. The spacetime diagram for a non-uniformly accelerating
observer with coordinates (T, X) plotted with the corresponding
Minkowski coordinates (t, x) light cone. The hyperbolae cross the
light cone making explicit the absence of horizons. . . . . . . . . .
41
INTRODUCTION AND BACKGROUND
A complete description of accelerating observers in flat spacetime is
lacking in the literature. Observers moving with constant proper acceleration
(Rindler observers [1]) are the most widely used description of such behavior
so far, especially in discussions of the thermal properties of vacuum states in
quantum field theories [2]. In this work we derive the trajectory and the
metric of an observer moving with non-constant proper acceleration in
two-dimensional flat spacetime.
It is clear that Rindler observers are unphysical. They accelerate with
constant proper acceleration for all time, reaching speeds arbitrarily close to
the speed of light. For this reason, Rindler observers see the light cone as an
event horizon.
These horizons partition the spacetime into four regions [1]: the left
and right Rindler wedges, which we characterize by a metric
ds2 = X 2 dT 2 − dX 2 , and the Kasner-Milne [3, 4, 5] universe, which we
characterize by a metric ds2 = dT 2 − T 2 dX 2 . These regions are bounded by
the lines in flat spacetime at 45 degrees to the x- and t-axes (in units of
c = 1). Observers in the Rindler wedges are disconnected from some of the
other wedges and thus do not have complete access to the full spacetime. For
instance, an observer in the right Rindler wedge can receive signals from the
past of the Kasner-Milne universe (bottom wedge) and send signals to the
future Kasner-Milne universe (top wedge), but is completely disconnected
from the left Rindler wedge. This is because observers in the Rindler wedges
see event horizons corresponding to the light cone portions in their quadrants
2
of the spacetime. Minkowski (inertial) observers never experience this; an
event occuring in flat spacetime is always accessible to the Minkowski
observer provided the observer waits sufficiently long times.
The Kasner-Milne universe is a constantly expanding universe where
all points in spacetime are moving at different constant rates, with the
exception of the center of the universe. An observer in this spacetime would
expand outward, away from the central point of the universe, traveling with
constant velocity. The observer’s speed is determined by their initial starting
distance from the center of the universe. We are particularly interested in
investigating the relationship between vacuum states in quantum field theory
and non-inertial observers and therefore focus our attention towards metrics
with non-zero acceleration.
We begin with a brief introduction to special relativity. We write an
invariant measure called the spacetime interval ds consistent with the two
postulates of special relativity. As we show, the spacetime interval is
fundamental in defining all of the physical quantities in special relativity, such
as the proper time. The special relativistic gamma factor γ, which is
responsible for all deviations from Newtonian physics, follows naturally from
the definition of ds. We use the concept of the proper time to define 4-vectors
like the 4-velocity, and the 4-acceleration in analogy with Newtonian physics.
Invariant scalars are constructed by taking 4-vector products defined through
the metric tensor; an object whose sign signature alters the standard scalar
product of two vectors. Finally, we define the invariant proper acceleration.
This is the acceleration that is measured the same in all inertial reference
3
frames, that is, those whose proper acceleration is zero. Therefore, the proper
acceleration categorically describes the motion of a particle.
In the next section, we derive a metric which describes non-uniformly
accelerating observers with proper acceleration that decays to zero for long
times. Special cases of this metric include both the Kasner-Milne metric and
the Rindler metric. We derive the velocity, acceleration, proper acceleration,
and trajectory of the observers. We make explicit the fact that these
observers accelerate to terminal speeds less than the speed of light. We show
analytically, and graphically in Figure 1, that the hyperbolic trajectories these
observers experience cross the 45-degree lines representing the Minkowski
light cone; the same light cone which Rindler observers see as an event
horizon. We note the corresponding light cone coordinates which we use later.
Special Relativity
Everything in Special Relativity begins with two postulates: The first
postulate is that space and time are homogeneous and isotropic. The second
is that the Universe has a maximum speed limit; the speed of light. This
allows one to define an invariant spacetime interval ds.
ds2 = c2 dt2 − dx2 − dy 2 − dz 2
(1)
Invariant here means that the spacetime interval between any two events ds
takes the same value no matter which frame of reference measures it.
The dimensions of the spacetime interval are length; however, defining
4
an invariant time measure, the proper time, as ds = cdτ is natural. Isolating
the time dimension on the right hand side yields
"
2 #
~v
c2 dτ 2 = 1 −
c2 dt2
c
(2)
where the differentials have been replaced by the velocity ~v . We see that the
time measured by an inertial observer is related to the proper time by
γdτ = dt. Where γ is defined as
2 #− 12
~v
γ ≡ 1−
c
"
(3)
and takes values in the range [1, ∞). From this point on, we take the speed of
light equal to unity (c = 1).
The spacetime interval can be expressed equivalently in terms of the
4-vector displacement dxµ = (dt, d~x) when a metric ηµν = diag (1, −1, −1, −1)
is introduced.
ds2 = ηµν dxµ dxν = dxν dxν
(4)
Where we have defined dxν = ηµν dxµ , a 4-vector with a lowered index. Now
we define the 4-velocity as the derivative of the 4-displacement with respect to
the proper time; this guarantees that it is a 4-vector itself.
dxµ
≡ uµ = γ (1, ~v )
dτ
(5)
There is a corresponding condition on the 4-velocity uµ like in the case of dxµ .
5
The Lorentz invariant for uµ is then
uµ uµ = 1
(6)
which confirms the second postulate that the speed of light is the maximum
speed and is invariant. From the 4-velocity we define the 4-acceleration as the
derivative of the 4-velocity with respect to the proper time. This leads to
duµ
≡ aµ = γ γ 3 (~v · ~a) , γ 3 (~v · ~a) ~v + γ~a
dτ
(7)
where the γ factor multiplying the components of uµ has made the derivative
non-trivial. The vector ~a = d~v /dt represents the 3-acceleration. Again, we
construct an invariant from the new 4-vector
aµ aµ = a2pr = γ 6 a2
where we have labeled the invariant apr meaning proper acceleration.
Therefore, the proper acceleration, being an invariant, characterizes the
accelerating motion for all external observers.
(8)
ACCELERATION IN SPECIAL RELATIVITY
Without loss of generality it suffices to work with a two dimensional
metric; this is because the accelerating observers under consideration
accelerate in a fixed direction which can always be oriented to the x-direction
using a coordinate transformation.
Derivation of Metric
We begin with the flat spacetime metric
ds2 = dt2 − dx2
(9)
and change coordinates to the accelerating frame. The most general
coordinate transformation is t = h(ξ, ζ) and x = f (ξ, ζ). These coordinate
transformations lead to the following metric:
ds2 = (ḣ2 − f˙2 )dξ 2 + 2(ḣh0 − f˙f 0 )dξdζ + (h02 − f 02 )dζ 2
(10)
Here primes indicate derivatives with respect to ζ and dots indicate
derivatives with respect to ξ. We attempt to eliminate the cross term dξdζ
and consider a metric with a time-like and space-like coordinate; later, we
consider a case where the cross term survives which is likened to the light
cone coordinates. We choose ḣh0 = f˙f 0 and solve the partial differential
equation. We assume a solution that is separable, t = a(ξ)b(ζ) and
7
x = c(ξ)d(ζ). It follows that
ȧa
dd0
= 0
ċc
bb
(11)
Each side of the equation depends only on ξ or ζ and therefore must be equal
to a constant. Integrating both equations, we obtain relationships between
the functions a and c as well as d and b.



a2 − k 2 c2 = A2
(12)


d 2 − k 2 b 2 = B 2
The general solutions to the equations in (12) come in four cases. We
consider the trivial case first, A = B = 0. This leads to functions a = ±kc and
d = ±kb and forces ds2 = 0.
We take cases two and three to be A2 = 0 and B 2 6= 0, and vice versa.
First we take A2 = 0; this reduces the equations in (12) to a = ±kc and
(d/B)2 − (kb/B)2 = 1. The general solution to this equation is
d = B cosh φ(ζ) and b = Bk −1 sinh φ(ζ). Thus, our coordinate transformations
look like t = Bc(ξ) sinh φ(ζ) and x = Bc(ξ) cosh φ(ζ), and we see that case
two corresponds to the left and right Rindler wedges. With coordinate choices
X = Bc(ξ) and T = φ(ζ) the metric becomes
ds2 = X 2 dT 2 − dX 2
(13)
For case three, the condition A2 6= 0 and B 2 = 0 implies
t = Ab(ζ) cosh θ(ξ) and x = Ab(ζ) sinh θ(ξ), which describe the Kasner-Milne
8
universe. The coordinate choices X = θ(ξ) and T = Ab(ζ) now yield the
metric
ds2 = dT 2 − T 2 dX 2
(14)
Observers in the Kasner-Milne wedges move with constant velocity; the
distance between two observers decreases in the lower wedge and increases in
the upper wedge, leading some to refer to the upper and lower wedges as the
expanding and contracting universe, respectively [2].
The fourth – and most interesting – case is A2 6= 0 and B 2 6= 0. Then,
the solutions to equations (12) with the condition that A and B are both
non-zero become a = A cosh φ(ξ), c = Ak −1 sinh φ(ξ), and also
d = B cosh θ(ζ), b = Bk −1 sinh θ(ζ). With the appropriate choice of X and T
it follows that
t = α cosh
X
α
T
sinh
α
,
x = α sinh
X
α
T
cosh
α
(15)
where α = ABk −1 . The metric of the accelerating observer is therefore
2
ds = cosh
X +T
α
cosh
X −T
α
(dT 2 − dX 2 )
(16)
In Figure 1 we show a spacetime diagram with lines of constant T and
constant X. It is clear from this figure that the accelerating observer sees no
event horizons. The trajectories are hyperbolic but they are not bounded by
the light cones.
9
Kinematics
We select a fixed position X0 in the instantaneous rest frame of the
accelerating observer and calculate the velocity.
dx
= tanh
dt
X0
α
T
T
tanh
= v∞ tanh
α
α
(17)
The speed of the accelerating observer ranges from 0 to 1, the speed of
light, and we may adjust this speed by varying either the position of the
accelerating observer in his coordinate system or the parameter α. We define
the terminal velocity v∞ ≡ tanh(X0 /α) for convenience and physical insight,
and denote by γ∞ the relativistic gamma factor associated with this terminal
velocity, so that γ∞ = cosh(X0 /α). We then calculate the acceleration
a = d2 x/dt2 .
a=
v∞
α cosh
1
X0
cosh3
α
=
T
α
v∞
1
αγ∞ cosh3
T
α
(18)
The transformation law to obtain the proper acceleration is apr = γ 3 a.
apr
−3/2
v∞
T
2
−2
=
1 + γ∞ sinh
αγ∞
α
(19)
For completeness, we note from (16) or from the equivalent form
T
X
2
2
ds = cosh
+ sinh
(dT 2 − dX 2 )
α
α
2
that the relationship between the time T and the proper time τ of the
(20)
10
observer at X0 is
−1/2
T
2
2
dT = γ∞ + sinh
dτ
α
(21)
Unlike in the Rindler case, dT and dτ are related here by a time-dependent
factor.
We have yet to identify the meaning of the parameter α. To determine
what α is physically, we need to analyze the trajectory x(T ) which we obtain
from the original coordinate transformations (15).
x(T ) = α sinh
X0
α
1/2
T
2
1 + sinh
α
(22)
We also see from (15) that t = 0 ⇒ T = 0, and this is the exact condition we
need to determine α.
x0 = α sinh
X0
α
= αγ∞ v∞
(23)
Therefore, α = x0 /v∞ γ∞ is determined by the distance of closest approach x0
and the initial velocity v∞ . Alternatively, we could also fix α by specifying the
acceleration at T = t = 0 which leads to α = v∞ /γ∞ a0 , where
a0 = a(0) = apr (0). In contrast to Rindler, where observers require infinite
accelerations [1], X0 = 0 corresponds to stationary observers.
Minkowski Coordinates
We now return to our original kinematic equations and replace α with
physical parameters and restore Minkowski time. This allows us to make
comparisons with what we know about the equations of motion for a
11
uniformly accelerating observer. The trajectory becomes
"
x(t) = x0 1 +
v∞ t
x0
2 #1/2
(24)
For large t, we see that (24) reduces to the equation of a line x(t) = v∞ t with
slope v∞ . This explicitly shows that the hyperbolas cross over the 45-degree
lines that define event horizons for uniformly accelerating observers. The
same conclusion follows from the velocity
"
2 #−1/2
2
v∞ t
t
v∞
1+
v(t) =
x0
x0
(25)
by considering again large t behavior, where v becomes v∞ .
The acceleration is deceptively similar to that of a uniformly
accelerating observer.
"
2 #−3/2
2
v∞ t
v∞
1+
a(t) =
x0
x0
(26)
We recover the exact answer for the acceleration of a uniformly accelerating
observer by taking the limit as v∞ → 1 with x0 (equivalently, αγ∞ ) fixed. The
proper acceleration is
"
"
2 #−3/2
2 #−3/2
2
v∞
v∞ t
v∞
t
apr (t) =
1+
=
1+
2
x0
x0 γ∞
αγ∞
αγ∞
(27)
which clearly describes a non-uniformly accelerating observer. We see exactly
what happens in the case of uniform acceleration in equation (27) by taking
12
the same limit v∞ → 1 with x0 fixed. The proper acceleration reduces to the
constant apr = 1/x0 one would expect for a uniformly accelerating observer
because the γ∞ dividing the time coordinate becomes infinite as we take the
limit. The transformation apr = γ 3 a from acceleration to proper acceleration
is equivalent to dividing the time coordinate t by a single power of γ∞ due to
the fact that the gamma factor resulting from (25) is given by
"
1 + (v∞ t/x0 )2
γ(t) =
1 + (v∞ t/x0 γ∞ )2
#1/2
(28)
where we recover the exact γ factor for the uniformly accelerating observer in
the limit v∞ → 1 as
1/2
γ(t) = 1 + (t/x0 )2
(29)
where the entire denominator of (28) becomes one due to γ∞ → ∞.
Light Cone Coordinates
We denote by u = t − x and v = t + x the standard light cone
coordinates in Minkowski spacetime. We define U = T − X and V = T + X
and use hyperbolic identities to obtain the following.
U
u = α sinh
α
,
V
v = α sinh
α
(30)
These coordinates are completely invertible since the hyperbolic sine inverse is
a well defined function on the entire domain. This isolates the coordinates,
leaving u as a function of U only, and v as a function of V only. Note that
13
u = 0 ⇐⇒ U = 0 and v = 0 ⇐⇒ V = 0, and that both coordinate systems
cover the entire spacetime. In light cone coordinates the metric becomes
U
V
ds = cosh
cosh
dU dV
α
α
2
(31)
These results will be crucial in determining whether non-uniformly
accelerating observers can associate thermal properties with the Minkowski
vacuum [12].
UNRUH RADIATION
Introduction
Soon after the discovery, by Bekenstein and Hawking [6, 7], that black
holes possess entropy and temperature, Davies [8] and Unruh [9], following
the work of Fulling [10], showed that observers moving with constant proper
acceleration in flat spacetime see the Minkowski vacuum as a thermal bath
with temperature T = ~a/2πk [8, 9]. The constants ~ and k are Planck’s
constant and Boltzmann’s constant respectively. The Unruh effect has
deservedly become the prime flat-spacetime example illustrating a deep
connection between relativity, quantum mechanics, and thermodynamics.
Bekenstein postulated that the entropy of matter falling into a black
hole should contribute entropy to the black hole itself [7]. This eventually led
to the proportionality of the entropy S of a black hole to the area of it’s event
horizon A [6, 7]. Hawking showed that the proportionality went like S = A/4
and that the temperature associated with his entropy was T = ~κ/2πk where
κ is the surface gravity of the black hole.
Unruh showed that the same effect applied to Rindler observers in a
flat spacetime background.
Quantum Field Theory in the Uniformly Accelerated Frame
A uniformly accelerating observer in special relativity is characterized
by the Rindler observer. This observer accelerates arbitrarily close to the
speed of light and sees the light cone as an event horizon. Due to the presence
of the horizons in the accelerating frame the spacetime for the Rindler
15
observer is partitioned into four regions; the two regions, left and right,
characterize the uniformly accelerating observer, while the top and bottom
wedges characterize an observer with zero acceleration. We will focus on the
left and right regions where the observer is accelerating. Due to the
partitioning of the spacetime, the accelerating observer sees radiation in the
Minkowski vacuum state. The radiation has temperature T = ~α/2πk.
As discussed previously, we describe the Rindler observer by the
following metric.
ds2 = X 02 dT 02 − dX 02
(32)
We label the spatial coordinate X 0 for now, and reserve the label X for the
final spatial transformation; the same is done for T 0 . The coordinates X 0 and
T 0 are described in terms of the inertial x and t as follows:
t = X 0 sinh T 0
,
x = X 0 cosh T 0
(33)
We calculate the velocity v = tanh(T 0 ) which tends arbitrarily close to one.
Therefore, this observer sees the light cone as an event horizon. The proper
acceleration is then apr = x−1
0 = α where x0 is the point on the x-axis which
intersects t = 0. Uniformly accelerating observers are then solely described by
the parameter x0 which characterizes the magnitude of the proper
acceleration.
It will be most convenient to solve for the quantum field operator in
light cone coordinates because we will be considering a massless spin zero field
which corresponds to an operator differential equation like the wave equation
16
for functions. The wave operator, which generates the wave equation when
acting on functions (or operators) is invariant under conformal
transformations; this allows for an easily expressible general solution for the
quantum field operator. In order to put the metric into a conformal form, we
introduce two additional coordinate transformations: T 0 = αT and
X 0 = α−1 exp(αX).
ds2 = e2αX dT 2 − dX 2
(34)
However, the transformation X 0 = α−1 exp(αX) forces the X 0 coordinate to
be positive. Therefore, an additional case for X 0 = −α−1 exp(αX) is required
and addressed later when necessary.
Now we consider a massless Klein-Gordon scalar quantum field theory.
As previously described, the differential equation for the quantum field
operator is reduced to the wave equation for an operator. In the (X, T )
coordinates, the wave equation is
∂ 2 Φ̂
∂ 2 Φ̂
−
=0
∂T 2 ∂X 2
(35)
where we have divided out the conformal exponential factor because it is
always greater than zero. Notice that the field Φ̂ is a scalar field operator, not
a scalar function. Therefore, the physical quantum fields that are observed by
experiments are expectation values of the operator Φ̂. We write the solutions
for Φ̂ in terms of the light cone coordinates U = T − X and V = T + X.
Φ̂(U, V ) = Φ̂1 (U ) + Φ̂2 (V )
(36)
17
From here, we consider only the left moving modes Φ̂2 (V ) as the analysis for
the right moving modes is essentially identical. We relabel Φ̂2 (V ) = Φ̂(V ) as
we will only be working with Φ̂2 (V ). The solution can be written in terms of
the frequency modes given by the Fourier expansion as
∞
Z
Φ̂(V ) =
0
i
dΩ h R −iΩ(T +X)
R† iΩ(T +X)
√
b̂Ω e
+ b̂Ω e
4πΩ
(37)
R†
The symbols b̂R
Ω and b̂Ω are called the creation and annihilation operators.
i
h
R†
,
b̂
They satisfy the discretized commutator relationship b̂R
Ω Ω0 = δΩΩ0 where
δΩΩ0 is the Kronecker delta and is one when the frequencies Ω and Ω0 are the
same, and zero otherwise. These operators are labeled by the frequency Ω
which describes the frequency of the particles that they “create” or
“annihilate” when acting on particle eigenstates. The superscript R indicates
the region of the spacetime in which the operator acts; in this case, the
operator acts on particle eigenstates only in the right Rindler wedge. The
R†
action of b̂R
Ω and b̂Ω on particle eigenstates is of utmost importance when
discussing observer dependence of vacuum states and will be discussed in
more detail later in the thesis.
The case when X 0 = −α−1 exp(αX) will lead to a similar result;
however, the sign of X in equation (37) changes to a minus sign. This yields
the solution for the left wedge of the spacetime.
Z
Φ̂(V ) =
0
∞
i
dΩ h L −iΩ(T −X)
iΩ(T −X)
√
b̂Ω e
+ b̂L†
e
Ω
4πΩ
Notice that the label on the creation and annihilation operators is now L
(38)
18
meaning that these operators act on particle eigenstates only in the left
Rindler wedge. They satisfy a similar commutation relationship
h
i
L L†
b̂Ω , b̂Ω0 = δΩΩ0 . All of the commutators are equal to zero if the superscripts
h
i
R L†
R and L do not match, b̂Ω , b̂Ω0 = 0, because the operators act on particle
eigenstates in different parts of the Rindler spacetime.
Since the quantum field operator Φ̂ is a scalar operator, a similar
solution can be written in terms of the inertial coordinates and these solutions
must yield equality.
Z
0
∞
Z ∞
i
dω dΩ h R −iΩV
iΩV
−iωv
† iωv
√
√
b̂Ω e
+ b̂R†
e
âω e
+ âω e
= θ(V )
Ω
4πω
4πΩ
Z 0∞
i (39)
dΩ h L −iΩV 0
iΩV 0
√
+θ(−V )
b̂Ω e
+ b̂L†
e
Ω
4πΩ
0
Here, θ(V ) is a Heaviside function which is equal to unity when the argument
is positive and zero when the argument is negative. They essentially turn on
and off the left and right wedge solutions for Φ̂. We superimpose the two
solutions, which are valid in different regions, and obtain a solution which is
valid in both regions. The ω on the left hand side of the equation is the
frequency of a particle created by the creation and annihilation operators
which act on particle eigenstates in the inertial Minkowski spacetime.
We will solve for the creation and annihilation operators using
orthogonal functions. The following integral formulas are necessary:
Z
∞
−∞
0
e±iωv eiω v dv = 2πδ(ω ± ω 0 )
(40)
19
which is the delta function representation via Fourier transforms, and
Z
∞
θ(v)e
±iΩV iωv
e
−∞
Z
dv =
∞
(av)±iΩ/a eiωv dv
(41)
0
where we have replaced the coordinate V = a−1 ln(av). The integral in
equation (41) has a closed form in terms of Γ functions. To arrive at this
closed form we introduce a cutoff in the allowed frequencies ω and analytically
continue to the complex plane. The closed form is given by
Z
∞
±iΩ/a iωv
(av)
e
0
1 ±πΩ/2a ω −iΩ/a
Ω
dv = e
Γ i
a
a
a
(42)
Combining the results of equations (40) and (42) we isolate the creation and
annihilation operators for the Minkowski observer. There are two sets of
equations:

R∞


−πΩ/a L†
b̂R
b̂Ω = 1 − e−2πΩ/a 0 dωχâω
Ω −e
(43)
R∞


−πΩ/a L
b̂R†
b̂Ω = 1 − e−2πΩ/a 0 dωχ∗ â†ω
Ω −e
and

R∞


−2πΩ/a
b̂LΩ − e−πΩ/a b̂R†
=
1
−
e
dωχ∗ âω
Ω
0
(44)
R∞


L†
−2πΩ/a
b̂Ω
− e−πΩ/a b̂R
=
1
−
e
dωχâ†ω
Ω
0
where
1
χ=
2πa
r
Ω πΩ/2a ω −iΩ/a
Ω
e
Γ i
ω
a
a
(45)
and χ∗ is the complex conjugate of χ. Since âω is the annihilation operator for
the Minkowski particle eigenstates âω |0M i = 0 defines the vacuum state for
the Minkowski observer. The 0 in the particle eigenstate |0M i means that
20
there are zero particles contained in the state and the M subscript indicates
that this particle eigenstate is for the Minkowski observer. The integrals on
the right hand side of equations (43) and (44) diverge; however, the
Minkowski annihilation operator, when acting on the vacuum state, gives zero
and leads to

h
i


−πΩ/a L†
 b̂R
−
e
b̂
Ω
Ω |0M i = 0
i
h


 b̂LΩ − e−πΩ/a b̂R†
|0M i = 0
Ω
(46)
from which it follow that
N̂ΩR |0M i = N̂ΩL |0M i
(47)
R
L† L
L
where N̂ΩR = b̂R†
Ω b̂Ω and N̂Ω = b̂Ω b̂Ω are called number operators. We see
that there are equal numbers of particles in the left and right particle
eigenstates. This means that there is a one-to-one correlation between the
particles in the left and right Rindler wedges.
The number operator for the Minkowski observer N̂ω returns the
particle number as an eigenvalue of the particle eigenstates in the following
manner.
N̂ω |nM i = n|nM i , n = 0, 1, 2 ...
(48)
The particle eigenstate |nM i represents a quantum state with n Minkowski
particles. An obvious question to ask is ”How many particles does a Rindler
observer see in the Minkowski vacuum state?”. We calculate the expectation
value h0M |N̂ΩR |0M i using the relationships in equations (46) and recall that
21
the answer is the same for N̂ΩL .
−1
h0M |N̂ΩR |0M i = e2πΩ/a − 1
(49)
Therefore, a Rindler observer sees a Planckian distribution in the Minkowski
vacuum. This is the Unruh radiation. The particles have energy E = ~Ω.
Then it is natural to define a temperature by setting the factor
2πΩ/a = E/kT . This leads to the Unruh temperature T = ~a/2πk. In order
to see that this definition of temperature is valid, we show that the density
matrix is indeed thermal.
Unruh Radiation Particle Eigenstates Expansion
The Minkowski vacuum state is clearly not a vacuum state for the
accelerating observer. The observer dependence of the vacuum state is
manifest through the Unruh effect.
R
We use the notation |{nL , nR }i = |nLΩ1 , nLΩ2 ...i ⊗ |nR
Ω1 , nΩ2 ...i for the
particle eigenstates of the accelerating observer. We expand the Minkowski
vacuum state in terms of the accelerating particle eigenstates.
|0M i =
X
|{nL , nR }ih{nL , nR }|0M i
(50)
{nL ,nR }
Where the sum above runs over all possible combinations of nL and nR
corresponding to each particle with frequency Ω. However, recall that the
particle numbers in the left and right wedges are the same and therefore
nL = nR ≡ n. Consequently, every particle in the right Rindler wedge with
22
frequency Ω has a correlated particle with frequency Ω in the left Rindler
wedge. This leads to a simplification in notation that proves to be useful.
|0M i =
X
|{n, n}ih{n, n}|0M i
(51)
{n}
The Rindler particle eigenstates are built from the action of the Rindler
L†
creation operators b̂Ω
and b̂R†
Ω on the Rindler vacuum |0R i. Since the number
of particles with frequency Ωi in the left and right Rindler wedges are the
same, we use b̂†Ωi ≡ b̂†i where i indexes the frequency of the particles being
created and nΩi ≡ ni . We express the particle eigenstates for the Rindler
observer as
|{n, n}i =
Y 1 L† ni R† ni
b̂i
b̂i
|0R i
(n
)!
i
i
(52)
where |0R i is the Rindler vacuum state. Therefore, we calculate the inner
product h{n, n}|0M i in equation (51) by taking the adjoint of equation (52)
and multiplying on the right by the Minkowski vacuum state |0M i.
h{n, n}|0M i =
ni ni
Y 1
b̂R
h0R | b̂Li
|0M i
i
(n
)!
i
i
(53)
From equations (46) we write
ni ni
ni ni
−ni πΩi /a
|0
i
=
e
b̂Li
b̂L†
|0M i
b̂Li
b̂R
M
i
i
(54)
where we use the commutator for b̂Li and b̂L†
i to write the expression above in
23
terms of Number operators N̂iL .
ni ni
L†
L
b̂i
b̂i
|0M i = ni + N̂i (ni − 1) + N̂i ... 1 + N̂i |0M i
(55)
Now we rewrite equation (53) using equation (55) where the number
operators act to the left on the Rindler vacuum state. This leads to the
following expression.
h{n, n}|0M i =
∞
YX
i
e−ni πΩi /a h0R |0M i
(56)
ni =0
This is substituted into the expression for the Minkowski vacuum given in
equation (50).
|0M i =
∞
YX
i
e−ni πΩi /a h0R |0M i |nLi i ⊗ |nR
i i
(57)
ni =0
The normalization of h0M |0M i = 1 and the following summation
∞
X
e−2ni πΩi /a = 1 − e−2πΩi /a
(58)
(59)
ni =0
leads to the expression
|h0R |0M i|2 =
Y
i
1 − e−2πΩi /a
24
Finally, the Minkowski vacuum state is
∞
Yp
X
−2πΩ
/a
i
|0M i =
1−e
e−ni πΩi /a |nLi i ⊗ |nR
i i
(60)
ni =0
i
By factoring out part of the exponential underneath the square root, we
obtain
s
|0M i =
∞
Ωi X −(ni + 1 )πΩi /a L
2
e
2 sinh π
|ni i ⊗ |nR
i i
a n =0
Y
i
(61)
i
This form for the Minkowski vacuum state is useful in making a statistical
mechanics interpretation of Unruh radiation. Let
Z=
Y
1 − e−2πΩi /a
−1
(62)
i
and recall equation (52). This leads to an expression of the Minkowski
vacuum state in terms of the Rindler vacuum state.
|0M i = Z
−1/2
∞
YX
e−ni πΩi /a L† ni R† ni
b̂i
b̂i
|0R i
(ni )!
i n =0
(63)
i
The summation is evaluated and we obtain
|0M i = Z −1/2
Y
R†
exp e−πΩi /a b̂L†
b̂
|0R i
i
i
(64)
i
The exponential within the summation of equation (61) can be rewritten to
1
1
e−(ni + 2 )πΩi /a = e−(ni + 2 +ni + 2 )πΩ/2a
1
(65)
25
Thus making explicit the Hamiltonian structure for two independent
oscillators; however, the exponent is missing a factor of i. In any case, a
prescription for going from a quantum field theory to a statistical mechanics
interpretation is to analytically continue β = (kT )−1 = it/~. This analytic
continuation associates the statistical mechanical factor exp (−βEi ) to the
quantum mechanical evolution operator exp (−iHi t/~) for the harmonic
oscillators. Then, from the statistical mechanical point of view, we define a
temperature T by associating
βEi =
2πΩi
a
(66)
with the appropriate energies Ei = ~Ωi . This leads to the same temperature
as before T = ~a/2πk. This is also apparent when we note that equation (59)
can be identified as the reciprocal of the partition function Z −1 , where the
partition function
Z=
Y
1 − e−2πΩi /a
−1
(67)
i
is that of a gas of massless particles. This further motivates the association in
equation (66).
This becomes especially clear in the density matrix formulation. The
density of states matrix is
ρ = |0M ih0M |
(68)
which is representative of a pure quantum state; the density matrix is pure, as
opposed to thermal. This is also clear when we note ρ2 = ρ, which is
26
characteristic of a pure density matrix. The expectation value of an arbitrary
operator A is of course h0M |A|0M i. Consider the case when A is restricted to
the right wedge. From equation (64) we write
|0M i = Z −1/2
X
e−
P
ni πΩi /a
i
|{nL }i ⊗ |{nR }i
(69)
{n}
and the expectation value of A is
h0M |A|0M i = Z −1
X
e−
P
i
(ni +n0i )πΩi /a h{n0L }| h{n0R }|A|{nR }i |{nL }i (70)
{n,n0 }
where the first summation is over all possible combinations of sets of n and n0 .
Since the expectation value is just a constant, the product between the left
states gives a Kronecker delta in n and n0 . The summation then reduces to a
single summation over all possible combinations of sets of n. This leads to the
following.
h0M |A|0M i = Z −1
X
e−
P
i
ni 2πΩi /a
h{nR }|A|{nR }i
(71)
{n}
Here, we associate 2πΩi /a = βEi and bring the exponential inside the product
of the states. Since we are working with eigenstates of the Hamiltonian, we
write
h0M |A|0M i = Z −1
X
R
h{nR }|e−βH A|{nR }i
(72)
{n}
where H R is the Hamiltonian that acts on the eigenstates in the right wedge.
27
This is exactly the trace of the operators.
R
tr e−βH A
h0M |A|0M i =
tr e−βH R
(73)
The partition function Z has been replaced with an equivalent expression in
terms of the trace.
This makes explicit the thermal nature of the expectation value of an
operator which is restricted to one region of the spacetime. The expectation
value is no longer that of a pure quantum state but is instead a thermal
average.
QUANTUM FIELDS AND NON-UNIFORMLY ACCELERATING
OBSERVERS
Introduction
In this thesis we show that Rindler observers (i.e., observers moving in
flat spacetime with constant proper acceleration) are rather special in their
identification of the Minkowski vacuum as a thermal state. We make use of
the non-uniformly accelerating observer (NUAO) solution derived in equation
(16), where observers accelerate to terminal velocities less than the speed of
light, to study the properties of the Minkowski vacuum seen by this
accelerating observer.
We give a brief review of quantum field theory in non-inertial reference
frames, followed by a detailed calculation of the creation and annihilation
operators seen by our NUAOs. We show that these accelerating observers see
a Bose-Einstein distribution of particles, which strongly suggests the
definition of a pseudo-temperature in analogy with the Unruh temperature.
We justify that this is indeed a pseudo-temperature. We solve for the
complete expansion of the Minkowski vacuum in terms of the accelerating
observer’s particle number states. Furthermore, we prove that our NUAOs
experience the Minkowski vacuum as a single-mode squeezed state and not
the typical two-mode squeezed state seen by Rindler observers. This
single-mode pure quantum state does not allow for a natural reduction to a
mixed state, and the thermal state characteristic of the Unruh effect never
enters the physics detected by the NUAO. The absence of a thermal state
invalidates the seemingly natural definition of temperature suggested by the
29
Bose-Einstein distribution of particles measured by the NUAO.
Finally, a review concerning the differences between Rindler observers
and our NUAO is given in the conclusion of the thesis. Additionally, We
discuss the physics of our NUAOs in the context of squeezed states,
entanglement, black hole physics, and the physical implications thereof. A
final remark on general accelerations is given.
Quantum Fields and Non-uniformly Accelerating Observers
Like the case with constant proper acceleration discussed above, we
begin with a massless scalar field and show that non-uniformly accelerating
observers see a Bose-Einstein distribution for the expectation value of the
number operator hNΩ i in the Minkowski vacuum.
∂ 2 Φ̂ ∂ 2 Φ̂
−
=0
∂t2
∂x2
(74)
We change to light cone coordinates u = t − x and v = t + x. The wave
equation becomes
∂ 2 Φ̂
=0
∂u∂v
(75)
Φ̂ = Φ̂1 (u) + Φ̂2 (v)
(76)
The general solution to (75) is
As discussed previously, the left (Φ̂1 (v)) and right (Φ̂2 (u)) moving solutions to
the field equation are non-interacting; again, we analyze the left-moving
solution Φ̂2 (v) = Φ̂(v). We expand the field in terms of Fourier modes as
30
follows:
Z
Φ̂(v) =
0
∞
dω √
âω e−iωv + â†ω eiωv
4πω
(77)
where the creation (â†ω ) and annihilation (âω ) operators for the Minkowski
observer obey the standard (discretized) commutation relation [âω , â†ω0 ] = δωω0 .
The non-uniformly accelerating observer (NUAO) under consideration
sees the following metric:
2
ds = cosh
X +T
α
cosh
X −T
α
(dT 2 − dX 2 )
(78)
where the parameter α is related to the acceleration and the initial velocity of
the observer. For light cone coordinates U = T − X and V = T + X in the
accelerating frame, the metric becomes
U
V
cosh
dV dU
ds = cosh
α
α
2
(79)
The relationship between (U, V ) and the corresponding Minkowski
coordinates (u, v) is given by [11]
U
u = α sinh
α
,
V
v = α sinh
α
(80)
Note that both coordinate systems (u, v) and (U, V ) cover the entire spacetime
and there is no singularity located at the origin in the accelerating frame [11].
In other words, the acceleration of the NUAO does not become infinite as we
approach the origin. Instead, the acceleration tends to zero. The singularity
at the origin for a Rindler observer is responsible for the partitioning of the
31
spacetime and thus the necessity of the left and right creation and
annihilation operators. Therefore, in the absence of a singularity, only a single
pair of creation and annihilation operators for this NUAO is necessary.
The equation satisfied by the scalar field in the accelerating observer’s
coordinates U and V is formally identical to (75).
∂ 2 Φ̂
=0
∂U ∂V
(81)
The solution is then analogous to (76). Selecting again the left-moving
solution we write
Z
∞
Φ̂(V ) =
0
i
dΩ h
√
b̂Ω e−iΩV + b̂†Ω eiΩV
4πΩ
(82)
in terms of the Fourier modes. The creation (b̂†Ω ) and annihilation (b̂Ω )
operators in the accelerating frame satisfy the same commutation relation
[b̂Ω , b̂†Ω0 ] = δΩΩ0 and act on all regions of the spacetime.
We identify equations (77) and (82) and solve for the creation and
0
annihilation operators in the accelerating frame. We multiply by e−iΩ V and
integrate over V to obtain b̂†Ω .
b̂†Ω
√ Z ∞
Z ∞
i
dω h
Ω
√ âω e−iωv e−iΩV + â†ω eiωv e−iΩV
dV
=
2π −∞
ω
0
(83)
We refer to equations in (80) and replace v = α sinh(V /α) in the exponentials
inside the integral. The integral formula for the modified Bessel function of
32
the second kind [13]
1
Kν (x) = e−iνπ/2
2
Z
∞
dt e−ix sinh t−νt
(84)
−∞
is valid when x > 0 and −1 < <(ν) < 1. Therefore, using the fact that
K−ν = Kν , we have
b̂†Ω
α√
=
Ω
π
Z
∞
0
dω
√ KiΩα (ωα) e−πΩα/2 âω + eπΩα/2 â†ω
ω
(85)
This integral is divergent and a regularization procedure must be adopted.
However, the details do not matter for this calculation, since our results will
be independent of the specific regularization procedure and only depend on
action of the operators on vacuum states. Denoting by χ(ω) the combination
α
χ(ω) ≡
π
r
Ω
KiΩα (ωα)
ω
(86)
we write the expression for b̂†Ω more compactly as
b̂†Ω
=e
−πΩα/2
Z
∞
πΩα/2
Z
∞
dωχ(ω) â†ω
dωχ(ω) âω + e
0
(87)
0
Similarly, taking the adjoint of equation (87) gives the following expression for
b̂Ω .
πΩα/2
Z
b̂Ω = e
∞
dωχ(ω) âω + e
0
−πΩα/2
Z
∞
dωχ(ω) â†ω
(88)
0
We may now isolate the operator âω that annihilates the Minkowski vacuum
33
|0M i by solving the system of equations (87) and (88) for âω .
e
πΩα
b̂Ω −
b̂†Ω
= e
2πΩα
−1
Z
∞
dωχ(ω) âω
(89)
0
Therefore, the combination b̂Ω − e−πΩα b̂†Ω annihilates the Minkowski vacuum,
b̂Ω − e−πΩα b̂†Ω |0M i = 0
(90)
We construct the number operator NΩ = b̂†Ω b̂Ω for the accelerating observer
and determine the average particle number h0M |b̂†Ω b̂Ω |0M i. Using equation
(90) we find that the norm of the state b̂Ω |0M i is
h0M |b̂†Ω b̂Ω |0M i = h0M | e−2πΩα b̂Ω b̂†Ω |0M i
(91)
Finally, we use the commutator [b̂Ω , b̂†Ω ] = 1 on the right side of equation (91)
to obtain
−1
h0M |b̂†Ω b̂Ω |0M i = e2πΩα − 1
(92)
Equation (92) shows that the non-uniformly accelerating observer sees a
Bose-Einstein distribution in the Minkowski vacuum. One would therefore
expect to be able to define a temperature from 2πΩα = E/kT and E = ~Ω as
T =
~
2πkα
(93)
directly from the Bose-Einstein statistics. Although the association of the
state with a temperature seems justified, we show that this is not the case.
34
Expansion of the Minkowski Vacuum State
We expand the Minkowski vacuum state in terms of the accelerating
observer’s number eigenstates.
|0M i =
X
|{ni }ih{ni }|0M i
(94)
{ni }
The states |{ni }i are calculated by repeated action of the b̂†Ω creation
operators on the accelerating observer’s vacuum state |0i. We abbreviate
b̂Ωi = b̂i and write h{ni }|0M i in terms of products of annihilation operators
acting to the left.
h{ni }|0M i = √
1
h0|(b̂1 )n1 (b̂2 )n2 ...|0M i
n1 ! · n2 !...
(95)
Equation (90) in combination with repeated applications of the commutator
[b̂Ω , b̂†Ω ] = 1 yields
bn |0M i = e−πΩα (N̂ + n − 1)b̂n−2 |0M i
(96)
where N̂ = b̂† b̂ is the number operator. Iterating the process for b̂n−2 , b̂n−4 , . . .
we find
h{ni }|0M i =

Q


j −1)!!
h0|0M i j e−nj πΩj α/2 (n√
(nj even)


 0
(nj odd)
nj !
(97)
35
We relabel ni by 2ki where ki is the number of pairs of particles; this allows us
to write a summation over all ki . The expansion of the Minkowski vacuum
becomes
X Y (2ki − 1)!!
−ki πΩi α
p
e
|{2ki }i
|0M i = h0|0M i
(2ki )!
i
{ki }
(98)
We calculate the transition amplitude h0|0M i by left-multiplying by
h0M |, replacing h0M |{2ki }i with our result from equation (97) and switching
the order of product and summation.
2
1 = h0M |0M i = |h0|0M i|
∞
YX
i
e−2ki πΩi α
ki =0
[(2ki − 1)!!]2
(2ki )!
(99)
Each sum in equation (99) is given by the closed form
∞
2
X
−1/2
−2πΩi α k [(2k − 1)!!]
(e
)
= 1 − e−2πΩi α
(2k)!
k=0
(100)
Substitution of the closed form (100) into equation (99) leads to
h0|0M i =
Y
1 − e−2πΩi α
1/4
≡ Z −1/2
(101)
i
where we have dropped an irrelevant phase in extracting the square root.
Replacing this result for the vacuum-to-vacuum transition amplitude into
equation (98) determines the expansion of the Minkowski vacuum in terms of
pairs of bosons to be
|0M i = Z
−1/2
X Y (2ki − 1)!!
−ki πΩi α
p
|{2ki }i
e
(2ki )!
i
{ki }
(102)
36
We may now elucidate the relationship between the two vacua. Writing
equation (102) as an operator relationship between the two vacuum states is
easily accomplished,
|0M i = Z
−1/2
Y
i
1 −πΩi α † 2
exp
e
bΩi
|0i
2
(103)
With some effort, one can show that equation (103) is equivalent to
|0M i =
Y
i
exp
n
1
− ln tanh
4
πΩi α
2
h 2 io
2
†
bΩi − bΩi
|0i
(104)
Note that the operator acting on |0i is unitary and is in fact the product of
squeezing operators for single-mode states [14]. This shows explicitly that the
Minkowski vacuum |0M i is a single-mode squeezed |0i vacuum, and equation
(92) may now be seen to be in agreement with a well-known result in
quantum optics.
CONCLUSIONS
We have a metric that describes observers moving with non-constant
proper acceleration. Our solution reduces to the Rindler case in the limit
v∞ → 1 with x0 fixed. For v∞ 6= 1, the proper acceleration vanishes as
|t| → ∞ and is at most the Rindler value apr = 1/x0 . These observers
asymptotically approach speeds less than the speed of light, characterized by
v∞ , and therefore do not experience event horizons, temperature, or
information loss.
For Rindler observers, the Minkowski vacuum is perceived as a pure
two-mode squeezed state. However, Rindler observers see an event horizon
which forces a tracing, in either the left or right wedge, over inaccessible
entangled degrees of freedom. The loss of entangled information reduces the
pure density matrix ρ = |0M ih0M | to a thermal mixture. This is due to the
unphysical nature of constant proper acceleration; such an observer requires
an infinite amount of energy to maintain its motion, in stark contrast to the
NUAO which requires finite energy. Restricting the constant acceleration to a
finite time interval solves the problem of infinite energies, but it does not
guarantee a valid definition of temperature. As a matter of fact, not even
constant acceleration for an infinite amount of time guarantees the existence
of a temperature [17]. A well known example illustrating this fact is uniform
circular motion [18].
We have shown that the Minkowski vacuum is perceived as a pure
single-mode squeezed state by these NUAOs. In this case there is no need for
an equivalent tracing over inaccessible degrees of freedom, or even a
38
motivation for doing such a thing given that this observer sees no event
horizon and has complete knowledge of all information over the entire
spacetime. Therefore, defining a temperature T = ~/2πkα is not justified
despite the fact that the Bose-Einstein nature of the particle number
distribution appears to suggest a thermal state.
Thermal states should not be associated with accelerations in general,
but only with accelerations capable of introducing horizons in the frame of the
accelerating observer.
This seems to suggest that the information paradox in black hole
physics is an artifact of interpreting measurements of radiation seen by an
unphysical stationary observer just outside the event horizon of a black hole.
The observers considered in this thesis are in a sense “intermediate” between
freely falling and stationary observers. These observers should shed light on
the issues of black hole complementarity, firewalls [15, 16], and the
information paradox.
REFERENCES
[1] W. Rindler, “Kruskal space and the Uniformly Accelerated Frame”, Am.
J. Phys. 34, 1174-1178 (1966).
[2] L. C. B. Crispino, A. Higuchi, and G. Matsas, “The Unruh effect and its
applications”, Rev. Mod. Phys, 80 787-838 (2008).
[3] E. Kasner, “Solutions of the Einstein Equations Involving Functions of
Only One Variable”, Trans. Amer. Math. Soc., 27 155-162 (1925).
[4] E. A. Milne, “World-Structure and the Expansion of the Universe”,
Zeitschrift für Astrophysik, 27 1-95 (1933).
[5] W. O. Kermack and W. H. McCrea, “On Milne’s Theory of World
Structure”, Monthly Notices of the Royal Astronomical Society, 93
519-529 (1933).
[6] S. W. Hawking, “Black-hole explosions”, Nature (London) 248, 3031
(1974); “Particle creation by black-holes”, Commun. Math. Phys. 43,
199220 (1975).
[7] J. D. Bekenstein, “Black Holes and Entropy”, Phys. Rev. D 7, 2333
(1973); “Generalized second law of thermodynamics in black-hole
physics”, Phys. Rev. D 9, 3292 (1974).
[8] P. C. W. Davies, Particle production in Schwarzschild and Rindler
metrics, J. Phys. A: Math Gen. 8, 609-616 (1975).
[9] W. G. Unruh, “Notes on black-hole evaporation”, Phys. Rev. D 14,
870-892 (1976).
[10] S. A. Fulling, “Nonuniqueness of canonical field quantization in
Riemannian space-time”, Phys. Rev. D 7, 2850 (1973).
[11] A. Doria and G. Muñoz, “Acceleration without Horizons”,
arXiv:1502.05093 [gr-qc] (2015).
[12] A. Doria and G. Muñoz, “Acceleration without Temperature”,
arXiv:1503.01152 [gr-qc] (2015).
[13] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed.
(Cambridge University Press, 1966).
40
[14] G. S. Agarwal, Quantum Optics (Cambridge University Press, 2013).
[15] A. Almheiri, D. Marolf, J. Polchinski, and J. Sully, “Black Holes:
Complementarity or Firewalls?”, J. High Energy Phys. 02, 062 (2013).
[16] Y. Nomura, J. Varela, and S. Weinberg, “Black holes or firewalls: A
theory of horizons”, Phys. Rev. D 88, 084052 (2013).
[17] S. Deser and O. Levin, “Mapping Hawking into Unruh thermal
properties”, Phys. Rev. D 59, 064004 (1999).
[18] J. R. Letaw and J. D. Pfautseh, “Quantum scalar field in rotating
coordinates”, Phys. Rev. D 22, 1345 (1980).
41
FIGURE
t
X =X1
X = X2
4
T =T 2
2
T =T 1
-4
2
-2
4
-2
-4
Figure 1. The spacetime diagram for a non-uniformly accelerating observer
with coordinates (T, X) plotted with the corresponding Minkowski
coordinates (t, x) light cone. The hyperbolae cross the light cone making
explicit the absence of horizons.
x
APPENDICES
Appendix A: Rindler Observers and Unruh Temperature
We have already noted that equation (93) cannot be interpreted as the
temperature of the particle distribution detected by our accelerating observer.
Of course, a similar definition is warranted in the case of a Rindler observer.
This naturally raises the question as to whether the Unruh temperature can
be obtained from our results.
At first sight it may appear that we have two incompatible expressions,
since the Rindler observer is the α → 0 limit of our NUAO. However, as
pointed out in [11], the limit should be taken as the NUAO’s incoming
velocity v∞ → 1 with the product αγ∞ fixed, where γ∞ is the gamma factor
associated with v∞ . Furthermore, in this limit the relationship between the
observer’s time dT and the proper time dτ becomes γ∞ dT = dτ .
This implies that energies and temperatures defined with respect to
the proper time must include a gamma factor; in particular,
Tτ =
~
2πkαγ∞
(105)
replaces equation (93). Since αγ∞ = 1/apr , as v∞ → 1 [11], the temperature
becomes
Tτ =
~ apr
2πk
and we have agreement with the Unruh temperature.
(106)
43
Appendix B: Verification of Bose-Einstein Distribution of Particle Numbers
We check our expansion of the vacuum state by an independent
calculation of the expectation value for the number operator. From equation
(102), h0M |NΩj |0M i is
h0M |Z
−1/2
X Y (2ki − 1)!!
−ki πΩi α
p
e
NΩj |{2ki }i
(2ki )!
i
{ki }
(107)
The number operator acting on the states gives NΩj |{2ki }i = 2kj |{2ki }i.
Substitution of h0M |{2ki }i from (97) then yields
h0M |NΩj |0M i = Z
−1
2
X Y (2ki − 1)!!
−ki πΩi α
p
e
2kj
(2ki )!
i
{ki }
(108)
Equations (100) and (101) imply that the infinite product will produce a
cancellation of every Zi factor in the product, with a corresponding term in
the normalization, except when i = j. Hence
h0M |NΩj |0M i = Zj−1
∞
X
[(2kj − 1)!!]2 −2kj πΩi α
e
2kj
(2k
j )!
k =0
(109)
j
The remaining sum is easily evaluated by taking a derivative of (100). The
result is
h0M |NΩj |0M i = (e2παΩj − 1)−1
This is the same Bose-Einstein distribution found in Section 2.
(110)
Fresno State
Non-Exclusive Distribution License
(to archive your thesis/dissertation electronically via the library’s eCollections database)
By submitting this license, you (the author or copyright holder) grant to Fresno State Digital
Scholar the non-exclusive right to reproduce, translate (as defined in the next paragraph), and/or
distribute your submission (including the abstract) worldwide in print and electronic format and
in any medium, including but not limited to audio or video.
You agree that Fresno State may, without changing the content, translate the submission to any
medium or format for the purpose of preservation.
You also agree that the submission is your original work, and that you have the right to grant the
rights contained in this license. You also represent that your submission does not, to the best of
your knowledge, infringe upon anyone’s copyright.
If the submission reproduces material for which you do not hold copyright and that would not be
considered fair use outside the copyright law, you represent that you have obtained the
unrestricted permission of the copyright owner to grant Fresno State the rights required by this
license, and that such third-party material is clearly identified and acknowledged within the text
or content of the submission.
If the submission is based upon work that has been sponsored or supported by an agency or
organization other than Fresno State, you represent that you have fulfilled any right of review or
other obligations required by such contract or agreement.
Fresno State will clearly identify your name as the author or owner of the submission and will not
make any alteration, other than as allowed by this license, to your submission. By typing your
name and date in the fields below, you indicate your agreement to the terms of this
distribution license.
Embargo options (fill box with an X).
X
Make my thesis or dissertation available to eCollections immediately upon
submission.
Embargo my thesis or dissertation for a period of 2 years from date of graduation.
Embargo my thesis or dissertation for a period of 5 years from date of graduation.
Alaric Doria
Type full name as it appears on submission
April 22, 2016
Date