Black Hole Evaporation
Huriye Gürsel
Submitted to the
Institute of Graduate Studies and Research
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Physics
Eastern Mediterranean University
September 2020
Gazimag˘usa, North Cyprus
Approval of the Institute of Graduate Studies and Research
Prof. Dr. Ali Hakan Ulusoy
Director
I certify that this thesis satisfies all the requirements as a thesis for the degree of Doctor
of Philosophy in Physics.
Prof. Dr. I˙zzet Sakallı
Chair, Department of Physics
We certify that we have read this thesis and that in our opinion it is fully adequate in
scope and quality as a thesis for the degree of Doctor of Philosophy in Physics.
Prof. Dr. I˙zzet Sakallı
Supervisor
Examining Committee
1. Prof. Dr. Muzaffer Adak
2. Prof. Dr. Mustafa Gazi
3. Prof. Dr. Mustafa Halilsoy
4. Prof. Dr. I˙zzet Sakallı
5. Prof. Dr. Kamuran Saygılı
ABSTRACT
The humankind perceives reality as the set of events of which are detectable by
human senses. Although it is not possible to possess a full knowledge of everything,
the human brain is capable of predicting the hidden constituents of the universe by
performing algorithms on observations. Furthermore, the falsifiability of theoretical
frameworks contributes to the flame of enthusiasm experienced by theorists. From
this standpoint, this thesis focuses on the interconnection of gravitational phenomena
and Planckian systems. The study examines the quantum nature of black holes, as
well as the hypothetical astronomical objects commonly referred to as black branes or
black strings within the context of Hawking radiation, which is the so-called black
hole radiation. Since the background geometry is curved, it gives rise to an effective
potential, which in turn results in a scattering process. Under this perspective, as an
example, the linear stability of a (2+ 1)-dimensional Mandal-Sengupta-Wadia black
hole is studied against small time-dependent perturbations. Subsequently, a
(4 + 1)-dimensional dilatonic black string is considered and it is shown that there
exists a resemblance between tachyonic particles and the fifth dimension, as the
greybody factor evaluations only allowed for imaginary masses to be present.
As the final step, a particular (3+1)-dimensional curved spacetime that might lead to
experimental studies is considered: z = 2 Lifshitz-like black brane (which is also
counted as a black hole) with hyperscaling violation. To analyze its radiation, we first
tackle the problem with the tools of general relativity and derive its complete
analytical blackbody radiation. Then, a particular holographic model is studied with
the purpose of deriving its analytical dissipative properties: η ∝ T 3/2, σDC ∝ T 3/2,
and ρDC ∝ T−3/2 which are the shear viscosity, the DC-conductivity, and the
iii
DC-resistivity, respectively. The aforementioned observables are achieved via the
fluid/gravity correspondence, built upon the two-point correlation function
GO+(ω,0) = −iω
(
r4++ω
2
)
/3r+. The metric dynamic critical exponent is originally
chosen as z = 2 in order for supporting superconducting fluctuations. However, this
choice has also determined the characteristics of the dual model living on the
three-dimensional boundary: a strongly-coupled, non-relativistic fluid exhibiting
Lifshitz-like symmetry. Any possible confirmation of the theoretically-obtained
dissipative parameters would act as a supplementary empirical evidence for the
quantum properties of spacetime.
Keywords: Hawking Radiation, Greybody Factor, Decay Rate, Absorption,
Evaporation, Fluid/Gravity Correspondence, Hyperscaling Violation,
Strongly-Coupled Fluid.
iv
ÖZ
I˙nsan beyni gerçeklig˘i, duyularla tanımlanabilen olaylar zinciri olarak algılamaya
eg˘imlidir. Evrendeki her s¸eyi tam anlamıyla bilmek mümkün olmasa da insan beyni
yapılan gözlemler ile algoritmalar olus¸turarak, evrendeki gizli biles¸enleri tahmin etme
yeteneg˘ine sahiptir. Bunun yanında, teorik temellerin yanlıs¸lanabilirlig˘i, tüm teorik
fizikçilerin deneyimlerinin ardındaki heyecana katkıda bulunmaktadır. Bu noktadan
hareketle; bu tezde, kütleçekimsel etkiles¸imler ile Planck sistemlerinin bag˘lantılarına
odaklanılmıs¸tır. Bu çalıs¸mada; kara delik, kara zar ve kara sicimlerin kuantum dog˘ası,
kara delik ıs¸ınımı olarak da bilinen Hawking radyasyonu bas¸lıg˘ı altında ele alınmıs¸tır.
Arka plan geometrisinin eg˘rilig˘inden dolayı olus¸an etkin potansiyel sonucunda,
saçılma olayları meydana gelmektedir. Bu bilgiler ıs¸ıg˘ında; örnek olarak,
(2 + 1)-boyutlu Mandal-Sengupta-Wadia kara delig˘inin stabilitesi, zamana bag˘lı
pertürbasyonlar aracılıg˘ıyla incelenmis¸tir. Daha sonra, (4+ 1)-boyutlu dilatonik kara
sicim ele alınarak, takyonik tanecikler ile olan ilis¸kisi bulunmus¸tur.
Son olarak, gerçekçi (3 + 1)-boyutlu bir model olan ve deneysel çalıs¸malara da
katkıda bulunabilecek yüksek ölçek ihlalli ve Lifshitz benzeri bir kara zar (aynı
zamanda kara delik olarak da nitelendirilebilir) incelenmis¸tir. Öncelikle bu modelin
radyasyonunu analiz edebilmek için genel görelilik prensipleri kullanılmıs¸ ve kara
cisim ıs¸ıması ile ilgili parametrelerin tümü analitik olarak bulunmus¸tur. Daha sonra
holografik modelin dag˘ıtıcı özellikleri olan akıs¸kanlık, DC-iletkenlik ve DC-direnç;
η ∝ T 3/2, σDC ∝ T 3/2 ve ρDC ∝ T−3/2 olarak bulunmus¸tur. Teorik yöntemlerle
hesaplanan bu gözlemsel parametreler, iki-nokta korelasyon fonksiyonu
GO+(ω,0) = −iω
(
r4++ω
2
)
/3r+ kullanılarak, akıs¸kan/kütleçekimi ilis¸kisi ile elde
edilmis¸tir. Metrik dinamik kritik katsayısı, süperiletken etkiles¸imleri ele almak için
v
spesifik olarak z = 2 olarak seçilmis¸tir. Ancak bu tercih aynı zamanda üç boyutlu
sınırda var olan dual modelin karakteristik özelliklerini ins¸a etmis¸tir; ki bu sınırlar da
kuvvetli bag˘lı ve relativistik olmayan Lifshitz simetrisine sahip akıs¸kanları
içermektedir. Bu tezde teorik hesaplara dayanarak elde edilen parametereler
gözlemlenildig˘i takdirde, uzay zamanın kuantum mekaniksel özelliklerinin de
deneysel kanıtları desteklenmis¸ olacaktır.
Anahtar Kelimeler: Hawking Radyasyonu, Gri Cisim Faktörü, Bozunma Katsayısı,
Emilim, Buharlas¸ma, Akıs¸kan/Genelçekim I˙lis¸kisi, Yüksek Ölçek I˙hlali, Kuvvetli
Bag˘lı Akıs¸kan.
vi
Dedicated to my grandmother Huriye Hüdaog˘lu and my grandfather Ahmet Gürsel
vii
ACKNOWLEDGMENTS
This thesis is a sequel of the effort and support of several people to whom I am
immensely grateful. First and foremost, I would like to express my deep gratitude to
my supervisor Prof. Dr. I˙zzet Sakallı for his valuable guidance and support alongside.
From reading early drafts to giving me exquisite advice, he has been as significant to
this thesis getting done as I have been. His constructive feedback kept my progress on
schedule and enabled me to construct the thesis to its current form. Furthermore, his
willingness to give his time generously for our regular meetings throughout my
graduate studies has been very much appreciated. It has been a privilege to work
under his supervision.
I would also like to acknowledge the members of Physics Department at Eastern
Mediterranean University whom have been giving their time and energy and for
showing me what it means to be a dedicated, each in their own unique way. I am
sincerely grateful to all the instructors from whom I had the privilege to take courses
during my graduate studies. My special thanks go to Prof. Dr. Mustafa Halilsoy for
all the inspiring discussions and also for sharing his valuable experience with us.
Moreover, I would like to thank the beloved administrative assistant of Physics
Department, Mrs. Çilem Aydıntan Aydınlık, for always being there with patience to
guide and support in the most effective way; and Physics laboratory technician Mr.
Res¸at Akog˘lu for his positive approach even in the most difficult times. My genuine
appreciation also goes to all the research assistants of the faculty, who has been there
for me in this challenging journey. The members of the Physics Department, as a
whole, have provided me with a friendly and inspiring environment to both work
efficiently and have joyful memories alongside. In addition, I would like to thank my
viii
friend Hidayet for always being there with her never-ending patience and also for
providing feedback via proof reading the thesis.
Last but certainly not least, I would like to thank my family: my husband, my brother,
my parents and my great uncle for their eternal support. Their presence, love and
guidance have played a vital role in the pursuit of this thesis. They have not only
provided me with counselling via proof reading the entire thesis and providing
feedback, but also inspired me mentally by reminding me that science is a philosophy
of discovery only made possible by open exchange of ideas initiated from pure
curiosity. I am indebted to them for encouraging me in all of my pursuits.
ix
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
ÖZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF TABLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
LIST OF SYMBOLS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Thesis Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Discovery of (2+1)-Dimensional Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Remarks on Scattering and Hawking Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Aspects of Lifshitz-like Black Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 STABILITY OF MANDAL-SENGUPTA-WADIA BLACK HOLES . . . . . . . . . . . . 15
2.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Geometrical Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Comments and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 EVAPORATION OF (4+1)-DIMENSIONAL BLACK STRINGS . . . . . . . . . . . . . . . . 25
3.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Background Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Wave Equation of a Massive Scalar Tachyonic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Evaporation of Dilatonic Black String . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4.1 Evaluation of Flux for r→ r+ and r→ ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
x
3.4.2 Greybody Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4.3 Absorption Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.4 Decay Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 (3+1)-DIMENSIONAL LIFSHITZ-LIKE BLACK BRANE HOLOGRAPHY . . 41
4.1 Applications of (3+1)-Dimensional Hyperscaling Violating Theories with z=2
and θ=−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Scalar Field Propagation in Lifshitz-like Black Branes . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1 Properties of the Bulk Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.2 Klein Gordon Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.2.1 Near Horizon Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.2.2 Asymptotic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.3 Quasinormal Modes and Stability Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.4 Radiation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Duality Between Bulk Observables and Strongly Coupled Systems . . . . . . . . . . 56
4.3.1 Some Key Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.2 Holographic Approach: Transport Coefficients of the Dual Model . . . . 60
4.3.2.1 Shear Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.2.2 DC-Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.3.2.3 DC-Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
xi
LIST OF TABLES
Table 1.1: The mapping between black hole and solid state physics cases. . . . . . . . 13
xii
LIST OF FIGURES
Figure 3.1: Plots of the greybody factor versus frequency for different l and k
values. The plots obey the relation (3.58) and the configuration of the dilatonic
five-dimensional black string goes as follows: µ = 3 and Q = 0.2. . . . . . . . . . . . . . . . . . . 37
Figure 3.2: Plots of the absorption cross-section versus frequency. The plots are
governed by Eq.(3.61). The configuration of the dilatonic five-dimensional black
string is as follows: µ = 3 and Q = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 3.3: Plots of the decay rate versus frequency. The plots are governed by
Eq.(3.63). The configuration of the dilatonic five-dimensional black string is chosen
as follows: µ = 3 and Q = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Figure 4.1: The behaviour of effective potential under the choice q = 1 and κ= 0. 53
Figure 4.2: The shear viscosity as a function of temperature . . . . . . . . . . . . . . . . . . . . . . . 63
Figure 4.3: The graph of analytical DC-conductivity versus temperature . . . . . . . . . 65
Figure 4.4: The graphical illustration of analytical DC-resistivity as a function of
temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
xiii
LIST OF SYMBOLS
~κ wave vector
˙ derivative with respect to time-coordinate
′ derivative with respect to radial-coordinate
 d’Alembertian
η shear viscosity
Γ decay rate
γ greybody factor
σabs absorption-cross section
σDC DC-conductivity
ρDC DC-resistivity
κs surface gravity
(z,θ) dynamic and hyperscaling violating exponents
d total number of spatial coordinates
D total number of spacetime coordinates
r∗ tortoise coordinate
r+ outer event horizon
r− inner event horizon
s entropy density
TC critical temperature
TH Hawking temperature
xiv
Chapter 1
INTRODUCTION
“The theory of gravitation deals with phenomena on a cosmic scale, whereas Yang
Mills theory is concerned with the opposite end - the smallest scale conceivable.
Someday the two will meet, when we come to grips with what is inside that perceived
singularity we call black hole.” - Kerson Huang
1.1 Thesis Framework
This thesis does not only examine the theoretical aspects of the scattering phenomena
of black holes/branes, but also inspects their dual models, which are regarded as
living on the boundary of the corresponding bulk model. Although there exist many
studies discussing the holography between gravitational and quantum field theory
models, few have actually focused on exact analytical methods for maintaining the
observables of each scenario. Drawing upon theories of general relativity and
holographic principle, this study provides information regarding perturbations of a
(2+ 1)-dimensional Mandal-Sengupta-Wadia black hole, tachyonic evaporation of a
(4+ 1)-dimensional dilatonic black string, wave dynamics of a (3+ 1)-dimensional
black brane with hyperscaling violation and its dual observables living on the
boundary. The dual system is analytically evaluated to hold a rather small shear
viscosity to entropy density ratio, which suggests that the system under consideration
is highly likely to be corresponding to a strongly-coupled, non-relativistic fluid in
(2 + 1) dimensions. The structure of the thesis goes as follows: Within this
introduction, firstly the motivation behind the research will be provided. In what
follows, some general information regarding certain gravitational phenomena will be
explained very briefly. Chapter 2 emphasises the effect of small perturbations on
1
Mandal-Sengupta-Wadia black holes, whereas Chapter 3 covers the evaporation
process of a (4 + 1)-dimensional black string. Generally speaking; any black hole,
black brane or black string (let me call them continuum objects for simplicity) display
dissipative properties, once they are disturbed from outside. This process is
commonly referred to as the membrane paradigm [1, 2]. The membrane paradigm is
suggested to be applied to black branes rather than black holes, as there exists no
translational invariance within the horizons of black holes [3]. This is the reason why
Lifshitz-like black branes are chosen in Chapter 4.
The membrane paradigm tends to reflect issues that are connected to the relationship
between a D-dimensional astronomical object and the (D− 1)-dimensional theory
living on the boundary. A detailed study on the evaporation of continuum objects has
directed us towards the following question: Can we possibly identify observational
evidence for the traces of one physical law valid at some specific scale in another,
which naively appears to be completely irrelevant? Consistent with previous research
carried out by scientists expertised in different areas, this question can be answered
with the aid of the fluid/gravity correspondence [4], which is a rather specific version
of the holographic principle. During the 1990s, the ideas mainly constructed by
Charles Thorn [5], Gerard ‘t Hooft [6], and Leonard Susskind [7] merged together
beautifully and gave birth to the insight known as the ‘holographic principle’, which
laid the foundations of finding an appropriate answer for the question of our concern.
With the purpose of searching for a fulfilling answer, we have written two
papers [8, 9] on hyperscaling violating Lifshitz-like black branes with z = 2 dynamic
exponent. The first article examines the bulk properties of the chosen model, whereas
in the second one, the holographic approach is adopted. Chapter 4 can be considered
as a combination of these two studies and it is noteworthy to stress that these papers
2
made it clear that there indeed exists a direct relation between general relativity and
quantum mechanics.
Note that in this work, the natural units are considered, i.e. c = G = kB = ~= 1.
1.2 Motivation
For centuries, humankind has been driven by the genuine curiosity about the logical
explanations and reasonings of the phenomena occurring in the universe that we live
in, and today, it still remains as an ongoing motivation behind an immense amount of
discoveries made possible. The endeavours of having a complete understanding of the
way the universe functions require establishing individual laws valid for different
energy (or distance) regimes, which would in turn characterise the degrees of freedom
of any arbitrary system from Planck scale to cosmic scale. Have you ever wondered
what the word “reality” actually corresponds to? Is reality limited by human
perception? If one imagines a hypothetical observer deprived of human perspicacity,
would reality change its self-representation, or would the observer’s conception
narrow down the entire picture? Treating the Big Bang as the initial cause behind
everything existing in today’s universe, one could argue that laws belonging to
different scales need to have a common root, albeit seeming to be completely
different at first glance. Thus, it is highly probable for a unique and consistent theory
of nature to be subsisting behind the scenes, which reduces to appropriate branches
for the cases when specific constraints are applied on the systems and observers of
concern. For a long time, physicists have been trying to construct this unified theory
and quantum gravity, which retains its popularity up to today, is regarded as the best
candidate proposed so far.
Very often, we find ourselves wondering how time can come to an end or how it was
3
created at the first place. Once the creation of the universe as we know it is tried to be
visualised, it is tempting to think of time as being the main focusing point.
Nevertheless, time in a sense is being created or destroyed within the dense
astronomical black structures. Our current knowledge addresses that once the black
hole interior is of concern, time and space lose their characteristics. This perspective
combined with the singularity point would lead one to conclude that rather than
thinking of time as the main concept, one could rather concentrate on the combination
of quantum and classical laws in the vicinity of black holes. Furthermore, string
theory suggests that space and time themselves may emerge in the theory at large
distances.
To be able to have a grisp on quantum gravity, one shall perhaps ask herself/himself
of the mapping between quantum systems and classical models. Would it be
misleading to treat our every day experiences as quantum mechanical phenomena
under specific constraints? John Wheeler had long been possessing curiosity
regarding the correspondence principle proposed by Niels Bohr in 1913. He had a
strong feeling that studying semi-classical (or alternatively semi-quantum) analysis of
scattering processes could exploit the question marks in his mind, which were mainly
about the transition from quantum laws to classical laws [10]. The great majority of
natural processes observable in our daily lives can be explained via laws of classical
mechanics and special theory of relativity. Considering that our minds are wired in
such a way to make sense of the phenomena we do experience in our daily lives, one
can suggest that our basic instincts make it tempting to think in a Newtonian or
Galilean perspective. However, it is beyond doubt that the complete picture needs to
be way more complicated than the ones that human mind can grasp.
4
In 1905, the proposal of Einstein’s postulates led to a revolutionary era where it would
seem that we live in a region of spacetime in which everything we see, experience and
think to be universal is only an approximation to all that has been happening through
the entire universe. Just like the Newtonian and Galilean perspectives were shown to
be approximations to general relativity, it would not be absurd to think that general
relativity itself can be thought as an approximation as well. What we do observe may
not always reflect what actually is taking place in the entire picture. Considering that
we, as the observers, can be treated as the low energy limit of the overall picture, it
would be of no surprise to view our perspectives as being equivalent to an
approximation of the actual phenomena. To be more precise, it is beyond doubt that
Pythagorean theorem correctly enables one to express the shortest distance between
two points as a function of the infinitesimal coordinate displacements of concern,
once the spacetime is flat. Nonetheless, when one wishes to conceptualise the entire
scenario, in other words when one does not impose any constraints on the curvature
of spacetime, the Pythagorean theorem can no longer be viewed as the correct
equation relating the points to each other. In this case, the metric tensor gµν is added
to the theory and any spacetime geometry can be described via
ds2 = gµνdx
µdxν. (1.1)
Equation (1.1) can be thought as the modified Pythagorean theorem, which guides us
through the right direction. This is only one example stressing that in order to be
equipped with a general interpretation for the laws of the universe, one should first
make sure that s/he eliminates the limitations on the theory. Approaching the problem
from this viewpoint enables one to see that the beauty and consistency of general
relativity is not adequate for referring it to as the complete theory of gravitation, as
5
there are constraints within the theory. Surely, this does not imply that Einstein’s
theory of general relativity is an inconsistent theory; in contrary, it is outstandingly
successful when phenomena such as deflection of starlight, the perihelion shift and
time dilation in a gravitational field are considered. However, it encounters problems
when small scales are of interest; whence it seems to be valid under some constraints
only. Thus, it would certainly be beneficial if one gets rid of the constraint on the
scale and attempts to modify general relativity in such a way that it would
automatically include laws of quantum mechanics as well.
A theory relating the astronomical scales to Planckian scales would also be
enlightening for understanding the mechanism behind the atomic nuclei. In 1931,
Dirac suggested that constructing a complete theory regarding the spooky behaviour
of atomic nuclei would be a burdensome task, since it would force us to revise our
fundamental understanding of nature. Dirac also stated that constructing a theory
directly from the observations would exceed the human intelligence. Hence, he
suggested the future theorists to search indirect ways of approaching the
problem [11]. Dirac’s perspective is supported by the studies conducted on
holographic principle: As the low energy behaviour of strongly coupled systems
seems to cause problems when attempted to be approached by quantum
chromodynamics, they are studied from a different perspective, which is commonly
referred to as the anti-de-Sitter/conformal field theory (AdS/CFT)
correspondence [12, 13]. Recently, many physicists are working on AdS/CFT
correspondence proposed by Maldacena [14], with the purpose of applying principles
of holography to strongly coupled systems. The dissipative properties of the horizons
of astronomical objects are investigated via membrane paradigm [1, 3] and the
resemblance between laws of hydrodynamics and general relativity are compared.
6
Furthermore, chiral symmetry breaking and quark confinement are unresolved
phenomena in low energy quantum chromodynamics, which await a consistent
explanation. It is highly probable for the holographic principle to hold the answers for
the ambiguous behaviour within these processes.
Currently, scientists are searching for ways of combining laws of high energy
phenomena occurring at small scales and concepts of general relativity under the title
quantum gravity and a construction, known as string theory, enable one to combine all
these perspectives together. The physical mechanism behind interactions at short
distances requires further examination, as there still exist open questions such as
quark confinement and chiral symmetry breaking. In this era, quantum gravity, which
still continues to challenge physicists, dominates the interactions.
The holographic principle which plays a vital role in gluing all the pieces together
and maintaining the most basic perspective. This would imply that there needs to be a
convincing scientific theory capable of describing phenomena occurring at the smallest
scales of large objects moving very close to speed of light.
1.3 Discovery of (2+1)-Dimensional Black Holes
(2+ 1)-dimensional theories of general relativity play a requisite role in constructing
a relatively simpler frame of mind for conceiving (3 + 1)-dimensional gravity.
Emerging mainly during the 1980s, theorists had been attempting to assemble a
(2 + 1)-dimensional theory of gravity with this purpose in mind; however, there
seemed to exist an inevitable challenge along the way: (2 + 1)-dimensional
spacetimes did not welcome black hole solutions [15]. In 1992; Banados, Teitelboim
and Zerilli (BTZ) managed to overcome this obstacle by proposing a rotating black
hole solution characterised by mass M, charge Q, and angular momentum J, as can be
7
seen from the following metric [16]
ds2 =−N2dt2 +N−2dr2 + r2(Nφdt +dφ)2, (1.2)
with N2 = −M + r
2
l2
+ J
2
4r2
and Nφ(r) = − J
2r2
. Note that l represents the radius, which
is directly linked to the negative cosmological constant via l−2 = −Λ. Solution (1.2)
is now widely known as BTZ black holes in (2+1)-dimensions. BTZ black holes can
be treated as the starting point for (2+1)-dimensional gravity theories. Furthermore,
it is also referred to as low energy string theory solution [17]. Low-dimensional
gravity theories have been extensively studied by physicists such as Witten [18, 19],
Rocek [20], Bardakçı et al [21], Chan and Mann [15], and many more. As desired,
these studies are now considered as valuable sources for studying conceptual aspects
of lower dimensional gravity.
1.4 Remarks on Scattering and Hawking Radiation
The scattering phenomena for atoms heavier than hydrogen atom dates back to 1933,
when John Wheeler published his first solo-authored article entitled with “Theory of
Dispersion and Absorption of Helium” [22]. This paper can be considered as the first
study that reveals the connection between absorption and scattering [10]. To be
equipped with a well-founded insight, it would be beneficial to first approach the
problem from an ordinary, classical scenario. Assume that a system with mass M
labelled as ‘the target’ is hit by another system with mass m and velocity ~v. Let say
that for simplicity, the target is treated as stationary. The incident system is said to be
scattered if it keeps on moving by experiencing a change in momentum, i.e.
∃~P′ : f or ∀ ~P ∃ ∆~P = ~P′−~P ∧ ∆~P 6= 0. (1.3)
In other words, there needs to exist a final momentum ~P′ such that for all initial
8
momentum ~P values, this difference is non-zero, under the condition that the incident
system keeps on moving. Now, one can relate the scattering process to the absorption
probability by continuing the aforementioned thought experiment. Imagine the target
is replaced with another system having a high absorption coefficient, such as an object
made up of sponge. Then, it would be trivial to see that the absorption and scattering
probabilities need to be related. If the incident system gets fully absorbed, then the
observer would not detect any scattering. The exact opposite would be true as well: If
no absorption takes place, there would be a 100% probability for the process of
scattering. Therefore, one can conclude that gathering information on scattering
amplitudes provides findings about the constituents and properties of the target
system.
Adapting this thought experiment to the theory of astronomical black objects, one
needs to get rid of the classical limit and introduce quantum fluctuations. The scenario
is adapted as follows: Suppose that there exists a black hole with mass M with a
strong gravitational influence due to its intense density. According to Hawking [23], a
black hole radiates some energy off its surface. Hawking radiation can be thought as
an outcome of quantum fluctuations in the vicinity of ultra-dense black objects such
as black holes. It occurs as a consequence of the strong gravitational effects around
the event horizon. A collection of particle-antiparticle pairs are produced
continuously within the quantum foam by the boundary of the ultra-dense black
object; and subsequently, one of the particles gets sucked by the black object, whereas
the other escapes to infinity. Mathematically, this finite temperature can be defined as
TH =
κs
2pi
. (1.4)
9
The radiation can technically be measured, if the black object is small enough. In
fact, experimental attempts are being made for testing the theory of Hawking radiation
in laboratory analogues of black holes [24]. But how about the full spectrum that
a hypothetical observer would see? The classically black but quantum mechanically
radiating objects are able to absorb only a portion of the waves approaching them
from distant regions of spacetime. Likewise, they seem to be unable to emit the entire
thermal radiation produced off their surface all the way to a hypothetical observer
situated infinitely far away. Nevertheless, when the event horizon is of concern, it
appears that the spectrum they emit is Planckian. The prime cause behind this paradox
seems to be the fields in the background, including the spacetime itself. Since the
background geometry possesses curvature due to the presence of the ultra-dense object
and fields of concern, the curvature can be treated as a gravitational potential, which
in turn enables a scattering process to take place. Hence, if one desires to comprehend
the quantum properties of spacetime and its puzzling genesis, scattering phenomena
within these regions would be a good point to focus on, as empirical evidence of any
kind would also provide information regarding the quantum fluctuations giving rise to
Hawking’s elegant framework.
Thus, in brief, a hypothetical observer at spatial infinity would only be able to detect
the scattered portion of the original Hawking radiation produced by the black hole.
Consequently, the radiation quantities (greybody factor, absorption cross-section, and
decay rate) give information about the target object; whence black hole evaporation is
a key subject for understanding the linkage between quantum and classical mechanical
laws. In other words, black hole evaporation opens the way for approaching a classical
object with the tools of quantum mechanics. There exists a distinction in terms of
their relevant scales, and yet quantum mechanical laws can be used to describe the
10
phenomena considering them both. Moreover, it enables one to think delicately about
the commonly heard phrase “nothing can escape from black holes”.
Admittedly, it is highly probable for the black hole evaporation to contain information
regarding the unsolved remedies in many different areas of physics. For instance, in
Ref. [25], it is shown that there exists a mapping between the superconducting phase
in superconductors and a black hole close to its final state.
1.5 Aspects of Lifshitz-like Black Branes
In 1974, ’t Hooft made a proposition that for any gauge theory, there exists a dual
string model in the large N limit [26]. Thus, the Lifshitz-like black brane of our
concern may be linked not only to the (2+ 1)-dimensional field theory model that it
corresponds to, but also to the associated one-spatial dimensional dual string theory.
At this point, we shall emphasise that both the observational data and the theoretical
framework of the Veneziano model [27] indicate the likely possibility of having an
underlying string structure to hadronic matter [28]. Once these relations are
investigated, one cannot go without noticing the relevance of the theory of magnetic
monopoles and the hadronic matter. In 1974, Mandelstam [29] proposed a model in
which the experimentally required confinement condition was satisfied. During his
work, he combined the Nielsen-Olesen interpretation of ST [30] and Nambu’s
idea [31] of treating quarks as the carriers of magnetic charges.
Dirac proposed that if one can figure out why electrons and protons exhibit different
properties, s/he would automatically recognise the reason behind the differences
between electricity and magnetism [11]. The non-Abelian massless monopoles in low
energy effective action of supersymmetric theories seem to possess an active role in
low energy behavior of quantum chromodynamics. Furthermore, the ground state of
11
quantum chromodynamics can be treated as a dual superconductor [28–30, 32].
Therefore, in this thesis, it is suggested that the transport coefficients of the
Lifshitz-like black brane of our concern is highly likely carrying information about
the magnetic monopoles, superconductors and low energy behavior of quantum
chromodynamics.
The dynamic scaling exponent of the model is chosen to be z = 2 so as to support
superconducting fluctuations [33]. Furthermore, theories with z = 2 scaling describe
multicritical points in certain liquid crystals and have been shown to arise at quantum
critical points in toy models of the cuprate superconductors [34]. On the other hand,
the spatial dimension of our bulk spacetime is adopted as d = 3. The reason behind
this specific choice is to shift the perception of a three-spatial dimensional reality
created by our minds (as a direct consequence of observing the macroscopic world) to
a (2+ 1)-dimensional holographic scenario in which the two cases exhibit common
properties. In Ref. [6], ’t Hooft claims that to be able to construct a consistent
quantum gravity model, the observable degrees of freedom should be described as if
they were Boolean variables defined on a two-dimensional lattice, which also
coincides with our specific choice of dimensionality. In Refs. [35–38] these
phenomena are explained via numerical simulations; however, there seems to be a gap
in literature for exact analytical approaches, as the usual perturbative methods are not
applicable in this regime.
In fluid/gravity correspondence, the dynamic critical exponent z and the hyperscaling
violating factor θ play a vital role in both characterizing the properties of the bulk
model and determining the scaling behavior of the observables in the dual scenario.
On the gravitational side, besides being subject to an overall hyperscaling violation
12
factor, the metric also encounters a temporal anisotropy due to quantum critical
phenomena. Such Lifshitz-like spacetimes correspond to the dual models, which
experience continuous phase transitions [39].
Additionally, it would be beneficial to stress that in Ref. [40], an exact solution for the
Mott problem has been maintained, and moreover, it is shown that black holes are
good candidates for revealing information regarding the superconductivity. In
Ref. [41], an interesting solid state approach to black hole thermodynamics is
maintained. In particular, the framework of the proposals made in Ref. [41] suggests a
mapping between quantum physics of black holes and thermodynamic properties of
superconductors. The duality between the two perspectives can be tabulated as
follows:
Table 1.1: The mapping between black hole and solid state physics cases.
Black hole case Superconducting case
Speed of light Fermi velocity
Black hole temperature TC of superconducting condensate
Event horizon Metal-superconductor interface
Schwarzschild radius Coherence length
Quantum state of a black hole Bardeen, Cooper and Schrieffer (BCS)
Black hole evaporation Andreev reflection processes
Hayden and Preskill’s information mirror Entanglement swapping
Traversable Einstein-Rosen bridge Crossed Andreev reflections
As can be seen from Table 1.1, the black hole temperature of a concerned
bulk-gravitational model has a corresponding dual analogy: the critical temperature of
13
the superconducting condensate in a particular condensed matter system. The desire
of interpreting the relationship between some particular gravitational structures and
condensed matter systems such as superconductors has resulted in the greybody factor
evaluation for (3 + 1)-dimensional non-Abelian charged Lifshitz-like black branes
with z = 2 hyperscaling violation. Moreover, z = 2 models correlate gauge/string
theory correspondence and quantum mechanical systems in condensed matter
physics. The similarities between two phenomena are presented elaborately in [42].
14
Chapter 2
STABILITY OF MANDAL-SENGUPTA-WADIA BLACK
HOLES
2.1 Preliminary Remarks
A black hole can retain its name as long as it preserves its stability, just like any other
physical object nominated for a specific name by the humankind. The concept of
‘black hole stability’ was first addressed by Regge and Wheeler [43], followed by
Zerilli, [44]; and since then, a plentiful amount of studies have been conducted on this
issue, some of which can be found in Refs. [45–55]. Regge and Wheeler pursued the
problem from a pedagogical point of view, which is of no surprise, as Wheeler has a
reputation for his elegant and yet ‘easy to grasp’ explanations regarding advanced
topics. The intense curiosity driven by Wheeler and Regge enabled them to visualise
a Schwarzschild black hole like a sphere of water held together by gravitational
forces. They viewed the black hole from this perspective so as to be able to possess a
solid understanding by having a comparison with a familiar concept from our daily
lives. Equipped with this notion, they aimed to achieve a basic intuition on the
possible scenarios that could take place provided that the black hole were subject to a
small perturbation. Their interpretation goes as follows: Assume that a system in
equilibrium is disturbed by an external effect via being subject to a small perturbation.
If the initially small disturbance happens to grow exponentially in time, the system is
said to be ‘unstable’, whereas for the stability to be maintained, it needs to be only
15
oscillating around the equilibrium [43]. Keeping this intuition in mind, this chapter 1
will be concerned with the effect of small spacetime-dependent perturbations acting
on a (2 + 1)-dimensional electrically charged Mandal-Sengupta-Wadia black hole.
There exist three subsections within this chapter: In Sec. 2.2, the black hole structure
will be mentioned briefly, whereas Sec. 2.3 is reserved for the stability analysis of the
concerned model. And finally, the results are summarised throughout the last section;
Sec 2.4.
2.2 Geometrical Structure
Before one starts discussing the black hole structure of interest, it would be beneficial
to introduce the action that includes information regarding the dynamics of the system.
As already stated throughout the introduction, the Einstein-Maxwell-Dilaton action
has many implications in string theory. However, in this chapter, the subject will be
examined from a relativist’s perspective only. For applications of this action in string
theory, one may refer to [57–60] and the references therein.
The Einstein-Maxwell-Dilaton action in (2+1)-dimensions can be expressed as [15]
SEMD =
ˆ
d3x
√
−g
[
R−
B
2
(∇φ)2− exp(−4aφ)FµνF
µν+2exp(bφ)Λ
]
. (2.1)
In this action, Λ is the cosmological constant, φ and Fµν stand for the dilaton and
Maxwell fields, respectively. Furthermore, a, b and B are dimensionless constants
where a and b represent the coupling of dilaton with the Maxwell field and the
cosmological constant, respectively [15]. These constants play a key role in
determining the black hole structure. Applying variations in the metric, gauge and
1 This chapter is based upon the article entitled “Linear Stability of Mandal-Sengupta-Wadia Black
Holes” [56].
16
dilaton leads to
Rµν =
B
2
∇µφ∇νφ+ exp(−4aφ)
(
−gµνF
2 +2Fαµ Fνα
)
−2gµν exp(bφ)Λ, (2.2)
∇µ
(
exp(−4aφ)Fµν
)
= 0, (2.3)
and
B
2
(
∇µ∇µφ
)
+2aexp(−4aφ)F2 +bexp(bφ)Λ= 0. (2.4)
The Einstein-Maxwell-Dilaton action provides a solution which corresponds to
charged static dilaton black holes. These black holes are allowed to hold magnetic or
electric charges, whilst in this chapter, the electric case will be inspected only.
Equations (2.2), (2.3), and (2.4) represent the equations of motion of the theory. Since
the background geometry of interest is going to be the one under the influence of a
charged Mandal-Sengupta-Wadia black hole, one needs to apply the conditions
b = 4a = B2 = 4 and φ0(r) = −
1
4 ln
(
r
β
)
{β :constant} on the constants found in Eqs.
(2.2), (2.3) and (2.4). Note that φ0(r) represents the static dilaton field. As desired,
these specific choices give rise to [15]
ds2 =− f (r)dt2 +
dr2
f (r)
+βrdθ2, (2.5)
with f (r) = 8Λβr− 2m
√
r + 8Q2 and Q represents the electric charge belonging to
the electromagnetic vector potential. The metric function f (r) can alternatively be
presented in the form
f (r) = 8Λβ
(√
r− r+
)(√
r− r−
)
, (2.6)
where r+ stands for the outer and r− for the inner horizon. The horizons can compactly
17
be defined as
r± =
m±
√
m2−64ΛβQ2
8Λβ
. (2.7)
The non-zero Maxwell tensor components are Ftr =−Frt = e4φ
Q√
βr
=
Q
√
β
r
3
2
with Aµ =
2Q
√
β
r δ
t
µ. The condition m> 8Q
√
Λβ needs to be satified in order for having a black
hole solution. Clearly, if one wishes to examine the case in the absence of electric
charge, Eq.(2.7) needs to be subdivided as r+ = m4Λβ and r− = 0.
The general definition [61]
TH(general) =
1
4pi
d
dr
(−gtt)
√
−gttgrr
∣
∣
∣
∣√
r=r+
, (2.8)
gives birth to the unique Hawking temperature expression for the charged black hole
of our interest which reads
TH(charged) =
8Λβr+−m
4pir+
=
√
m2−64ΛβQ2
4pir+
. (2.9)
In order to maintain a real-valued observable temperature, one needs to fulfill the
requirement m> 8Q
√
Λβ . For the case with no Maxwell field within the model, Eq.
(2.9) becomes
TH(neutral) =
m
4pir+
=
Λβ
pi
. (2.10)
Note that the Hawking temperature of the uncharged model comes out to be positive
iff Λβ > 0 is satisfied. For systems with negative temperature, the reader is referred
to [62].
18
2.3 Linear Stability Analysis
Having discussed the model’s geometrical properties and the static solution, one can
now move on to the effect of small perturbations on the system. Therefore, this
section is reserved for inspecting the linear stability of an electrically charged static
Mandal-Sengupta-Wadia black hole in (2 + 1)-dimensions. The perturbed field
equations derived from action (2.1) lead to the solution
ds2 =−exp(2γ)dt2 + exp(2α)dr2 + exp(2η)dθ2, (2.11)
Ftr = qexp(α+ γ−η+4φ) , (2.12)
in which γ, α, and η are (t,r)-dependent metric functions and q = Q. Going back to
Eq. (2.2), one can evaluate the non-vanishing components of the Ricci tensor, which
results in
Rtr = Rrt = 4
.
φφ
′, (2.13)
and
Rθθ =−2
[
Λ−q2 exp(−2η)
]
exp(4φ). (2.14)
Once γ, α, η, and φ are treated as perturbed versions of the static background fields
γ0(r), α0(r), η0(r), and φ0(r), one can express
γ≡ γ(r, t) = γ0 (r)+δγ, (2.15)
α≡ α(r, t) = α0 (r)+δα, (2.16)
η≡ η(r, t) = η0 (r)+δη, (2.17)
19
and
φ= φ(r, t) = φ0 (r)+δφ. (2.18)
The perturbations δγ , δα, δη, and δφ can be regarded as ‘very small’, provided that
δγ =∈ γ1 (r, t), δα =∈ α1 (r, t), δη =∈ η1 (r, t) and δφ =∈ φ1 (r, t) with ∈<< 1.
Comparing Eq. (2.5) with Eq. (2.11) and setting η1 (r, t) = 0 results in exp(2η) = βr.
Then, the non-zero Ricci components (2.13) and (2.14) turn out to be
Rtr =
.
αη
′, (2.19)
Rθθ =
[
η
′
(α
′
− γ
′
)− (η
′
)2−η
′′
]
exp(−2α) , (2.20)
and the Klein-Gordon equation (2.4) evolves into
exp(−2α)
[
φ
′′
−φ
′
(α
′
− γ
′
−η
′
)
]
− exp(−2γ)
[..
φ+
.
φ(
.
α−
.
γ)
]
+
exp(4φ)
[
−Λ−q2 exp(−2η)
]
= 0. (2.21)
From this point onwards, the main task is to linearize both the field and the Klein-
Gordon equations. When small perturbations are of concern, the linearized equations
will enable one to investigate the effect of the associated disturbance on the geometry,
as the first-order terms are dominant. To be more precise, taking Eqs. (2.15-2.18) as
well as the gauge η1 (r, t) = 0 into account and equating Eqs. (2.13) and (2.14) with
Eqs. (2.19) and (2.20), we obtain the following set of equations:
.
α1 +2
.
φ1 = 0, (2.22)
4
(
Λβr−Q2
)
(α1 +2φ1)+ r
(
4Λβr−m
√
r+4Q2
)
(α
′
1− γ
′
1) = 0, (2.23)
20
4
(
Λβr−Q2
)
(α1 +2φ1)+r
(
4Λβr−m
√
r+4Q2
)
(α
′
1−γ
′
1)+4r
2φ
′′
1
(
4Λβr−m
√
r+4Q2
)
−
r2
(4Λβr−m
√
r+4Q2)
..
φ1 +4r(6Λβr−m
√
r+2Q2)φ
′
1 = 0, (2.24)
One may notice that Eq. (2.23) is included within Eq. (2.24). Consequently,
φ
′′
1 +
[
6Λβr−m
√
r+2Q2
r(4Λβr−m
√
r+4Q2)
]
φ
′
1−
..
φ1
(8Λβr−2m
√
r+8Q2)2
= 0. (2.25)
If one applies Fourier transformation with respect to time, s/he obtains
φ1 (r, t) = φ1 (r)exp(−ikt) , (2.26)
with k representing the frequency. Thus, one can rewrite Eq. (2.25) in the form of an
effective Klein-Gordon equation
φ
′′
1 (r)+hφ
′
1 (r)− jφ1 (r) = 0, (2.27)
in which h and j read
h =
12Λβr−2m
√
r+4Q2
r f
, (2.28)
and
j =
−k2
f 2
. (2.29)
As aforementioned, the stability was first introduced into the theory of black holes by
Regge and Wheeler and throughout their analysis, they had checked the behaviour of
the associated one-dimensional Schrödinger-like equation, with the aid of introducing
a new variable which is now known as the tortoise coordinate. By definition, the
21
tortoise coordinate can be found from the metric function via the relation [63]
r∗ =
ˆ
dr
f
. (2.30)
For the case of our concern, it reduces to
r∗ =
1
4Λβ(r+− r−)
ln
(
(
√
r− r+)
r+
(
√
r− r−)
r−
)
(2.31)
The range r+ <
√
r < ∞ is analogous to −∞< u < ∞, as √r→ r+ leads to r∗→−∞.
As one gets closer to the black hole, the radial coordinate alters more slowly with the
tortoise coordinate due to drdr∗ → 0. Therefore, the main interest will be the region
that satisfies r > r+. In other words, the tortoise coordinate parametrizes the entire
region outside the black hole [64]. From Eq. (2.30) it can be seen that for the extremal
case where r+ = r−, the tortoise coordinate can no longer be defined. Subsequently,
the effective Klein-Gordon equation (2.27) can be expressed in terms of the tortoise
coordinate as
d2φ1(r∗)
dr2∗
+X
dφ1(r∗)
dr∗
+ k2φ1(r∗) = 0, (2.32)
in which
X = 4Λβ−
m
√
r
+
4Q2
r
. (2.33)
Substituting
φ1 (r∗) = R (r∗)r−
1
4 , (2.34)
22
in Eq. (2.32) brings about the desired Schrödinger-like one-dimensional wave equation
−
d2R
du2
+
[
Ve f f (r)− k
2]R (u) = 0, (2.35)
in which Ve f f (r) represents the effective potential
Ve f f (r) =
f
(
f +4(
√
rm−8Q2)
)
16r2
. (2.36)
Thus, having evaluated the effective potential, one can now check whether the static
Mandal-Sengupta-Wadia black hole is stable or not. The physical insight behind
checking the stability goes as follows: A bound state may be present iff Ve f f is
negative. If there exists a bound state, an unstable mode should be presenting as well.
Thus, it would be enough to check whether the effective potential admits any negative
solutions [64]. Recall that for the solution to be in the form of a black hole, there
existed a condition: m > 8Q
√
Λβ. Hence, it can be recognised that the effective
potential is positive definite, implying that static electrically charged
Mandal-Sengupta-Wadia black holes are linearly stable for s-mode perturbations.
2.4 Comments and Discussions
Throughout this chapter, non-rotating and time-independent electrically charged
Mandal-Sengupta-Wadia black holes are checked for their stability. In order for
achieving so, infinitesimally small (t,r)-perturbations are applied to the black hole,
thereby influencing the dilaton and the metric fields. The perturbed equations are then
linearized and reduced to one-dimensional Schrodinger-like equation via applying the
necessary constraints and introducing the tortoise coordinate. The analysis used is a
semi-analytical method which can also be accessed in Refs. [65, 66] and is built upon
the Fubini-Sturm theorem [67]. The results obtained in this work came out to be
consistent with Ref. [68] where it was stated that Mandal-Sengupta-Wadia black
23
holes are stable under small time-dependent perturbations. In conclusion, by checking
the sign of the effective potential under the black hole condition, it was observed that
the black hole of concern is linearly stable.
24
Chapter 3
EVAPORATION OF (4+1)-DIMENSIONAL BLACK
STRINGS
3.1 Prologue
This chapter will be covering how particles evaporating off a five-dimensional black
string derived from Einstein-Yang-Mills-Born-Infeld-Dilaton action behave, as they
propagate through the spacetime of concern 2 . In addition, it will be shown that
particles ejected as a consequence of Hawking radiation are compelled to be tachyons,
i.e. particles holding imaginary mass values. The reasons behind this limitation will
be clarified in the forthcoming sections.
It is of no doubt that our current literature is filled with studies conducted on the wave
dynamics of black holes. Yet, the analytical approaches to greybody factors of higher
dimensional black strings do not appear to be that many. For some references on this
concern, one may check [70–72]. One of the possible reasons behind this scantiness
could be the severity arising in the theory due to the stringy structure in D > 4
dimensions. Briefly speaking, further research is needed to be carried out on
five-dimensional black strings and their corresponding evaporation process. In this
chapter, the black string of concern includes a dilatonic field and the propagation of
massive tachyons will be examined under the concerned geometry.
2 This chapter is based upon the article named “Absorption Cross Section and Decay Rate of Dilatonic
Black Strings” [69].
25
3.2 Background Geometry
In this section, the geometrical properties of five-dimensional dilatonic black strings
will be investigated briefly. As the starting point, let us introduce the action of the
theory which goes as [73]
IEY MBID =−
1
16piG(D)
ˆ
M
ddx
√
−g

R −
4(5ψ)2
D−2
+4χ2e−bψ

1−
√
1+
Fe2b
2χ2



 .
(3.1)
Here, ψ represents the dilaton field, χ is the Born-Infeld parameter and b = − 4d−2α
where α = 1√
d−1
is the dilaton parameter. Furthermore, G(D) stands for the
D-dimensional Newtonian constant and it can be related to
(
G(4)
)
as follows:
G(D) = G(4)L
D−4. (3.2)
However, as stated in introduction, we will consider G(D) = 1. One shall record that L is
regarded as the upper limit of the compact coordinate, i.e.
(´ L
0 dz = L
)
. Furthermore,
R stands for the Ricci scalar and F = F(a)λρ F
(a)λρ where the two-form Yang-Mills field
is given by
F(a) = dA(a)+
1
2σ
C(a)
(b)(c)
(
A(b)∧A(c)
)
, (3.3)
with C(a)
(b)(c)
and σ being structure and coupling constants, respectively. The Yang-Mills
potential A(a)is defined by following the Wu-Yang ansatz [74]
A(a) =
Q
r2
(
xidx j− x jdxi
)
, (3.4)
r2 =
d−1
∑
i=1
x2i , 2≤ j+1≤ i≤ d−1,1≤ a≤ (d−1)(d−2)/2, (3.5)
in which Q denotes the Yang-Mills charge. The dilatonic field is in the form expressed
26
below:
ψ=−
(d−2)
2
α lnr
α2 +1
. (3.6)
Having mentioned the dynamics of the model, let us now introduce the background
geometry whose infinitesimal interval can be achieved from [73]
ds2 =− f (r)dt2 +
dr2
r f (r)
+βrdz2 +dθ2 + sin2θdφ2. (3.7)
Note that f (r) = r− r+ and β=
4Q2
3 . As in the previous chapter, r+ indicates the outer
horizon which obeys the relation
r
d(d−2)+2
d
+ =
32
Ld−4
(
Q2d
d−1
) d−2
2
, (3.8)
The theory of our choice possesses D = 5, or similarly d = 4,thereby reducing Eq.(3.8)
to
r+ = 4
(
4Q2
3
) 2
5
= 4.488Q4/5. (3.9)
In order to find Hawking temperature, one first needs to evaluate the surface gravity.
Recall Hawking radiation definition Eq.(1.4):
TH =
κs
2pi
. (3.10)
The surface gravity can then be found via [75]
κs = OµϒµO
µϒµ, (3.11)
27
where
ϒµ = [1,0,0,0,0] (3.12)
is the timelike Killing vector. In this case,the surface gravity can be derived by using
κs =
√
rd f/dr
2
∣
∣
∣
∣
r=r+
=
√
r+
2
. (3.13)
Finally, the Hawking temperature reads
TH =
√
r+
4pi
. (3.14)
If one compares Eq. (3.14) with what has been obtained in Ref. [73], s/he would notice
the two results do not match. This is because in Ref. [73], the spacetime geometry is
assumed to be symmetric. Nevertheless, from Eq. (3.7) it can be seen gtt 6= 1grr .
3.3 Wave Equation of a Massive Scalar Tachyonic Field
The purpose of this section is to find an exact solution for the Klein-Gordon equation
considered for massive tachyons and investigate its behaviour for the cases when r→
r+ and r→ ∞. Klein-Gordon equation can be presented as
[
− (iµ)2
]
Ψ(t,r) =
[
+µ2
]
Ψ(t,r) = 0. (3.15)
Considering the ansatz
Ψ(t,r) = R(r)Y ml (θ,φ)eikze−iωt , (3.16)
with Y ml (θ,φ) representing the spherical harmonics and k being constant, the radial part
28
of Eq.(3.15) reduces to
r f
R′′
R′
+
R′
R
(
f + r f ′
)
+
ω2
f
−
k2
βr
+µ2−λ= 0, (3.17)
where λ= l(l +1).
Eq.(3.17) can be regarded as a generator leading to a familiar equation derived by
Euler [76]: the hypergeometric differential equation, under the requirement that the
essential conditions are satisfied. The hypergeometric series, which is the generalised
version of the geometric series
1+ x+ x2 + ..., (3.18)
can be defined as [77]
1+
ab
c
x+
a(a+1)b(b+1)
2c(c+1)
x2 + ..., (3.19)
with the constant parameters a, b and c. The hypergeometric differential equation
emerges in a wide range of physical models. To be more specific, from the flow of
compressible fluids to the Schodinger equation for a symmetrical top molecule [77],
hypergeometric functions appear withing the theory. Thus, one needs to hold a firm
understanding on these concepts. In the most general case, the hypergeometric
differential equation is expressed in the form
x(1− x)
d2X
dx2
+[c− (1−a+b)x]
dX
dx
+abX = 0. (3.20)
29
which possesses the general solution
X(x) = A 2F1 (a,b;c;x)+Bx1−c 2F1 (a− c+1,b− c+1;2− c;x) , (3.21)
as long as 1− c ≤ 0. In order for making use of the hypergeometric function, the
variable of our case is changed via r = r+ − zr+, after multiplying Eq.(3.17) by
rβ f (r)R(r). Then, one can write
z(1− z)R′′+(1−2z)R′+
[
ω2
zr+
+
k2
β(1− z)r+
−µ2 +λ
]
R = 0. (3.22)
Eq.(4.20) can be demonstrated as a hypergeometric differential equation under the
condition
[
ω2
zr+
+
k2
β(1− z)r+
−µ2 +λ
]
=
A2
z
−
B2
1− z
+C, (3.23)
in which
A = −
ω
2κ
,
B =
ik
2κ
√
β
, (3.24)
C = λ−µ2.
In general, a hypergeometric differential equation has its solution in the following
form:
R = ξ1 (−z)
iA (1− z)−B F (a,b;c;z)+ξ2 (−z)−iA (1− z)−B F (α,ς;η;z) . (3.25)
The relevant constants a,b and c obey the relations
a =
1
2
(
1+
√
1+4C
)
+ iA−B,
30
b =
1
2
(
1−
√
1+4C
)
+ iA−B, (3.26)
c = 1+2iA.
whereas α, ς and η read
α= a− c+1, (3.27)
ς= b− c+1, (3.28)
and
η= 2− c.
Let us visualise what might possibly be occurring during the evaporation process of
the black string. The strong gravitational effects due to the presence of the
astronomical object of our concern, a pair of virtual particles can pop in and out of
existence, spontaneously. Our interpretation goes as follows: Shortly after the
creation of the virtual particles, one of the pairs is pulled into the black string, while
in the mean time, its pair follows the exact opposite path. This would imply that when
one pair moves towards the center of the black string, the other travels toward spatial
infinity. In that case, for r → r+, one should only reckon with the radial solution
directed into the horizon, or in other words, the purely ingoing solution. Therefore,
the constant ς1 in Eq.(4.21) needs to be set to zero for achieving logical consistency.
Mathematically speaking, the radial solution needs to be in the form
R = ξ2 (−z)
−iA (1− z)−B F (α,ς;γ;z) . (3.29)
31
It should not go without saying that there exists another physical constraint that is
required to be applied on the radial solution. Based on the analytical evaluations carried
out,it has been seen that
√
1+4C present in constants (3.22) is obliged to be imaginary,
otherwise the greybody factor turns out to diverge.Hence, one needs to set
4µ2 > 4λ−1, (3.30)
so as to achieve
√
1+4C = iτ, (3.31)
with
τ=
√
4µ2−4λ−1,τ ∈ R. (3.32)
Having discussed the necessary physical conditions on the radial solution, let us now
further examine its behaviour at spatial infinity and in the near horizon regime. For
r→ r+ (or alternatively for z→ 0),we are only left with
RNH = ξ2 (−z)
−iA . (3.33)
If one wishes to deal with the full solution rather than the radial one only, s/he can
express
ψNH = ξ2e
−iω(r̂∗+t). (3.34)
32
Recall that the tortoise coordinate is defined as
r∗ =
ˆ
dr
√
r f
(3.35)
which enables one to write
r̂∗ = lim
r→r+
r∗ '
ln(−z)
√
r+
=
1
2κ
ln(−z) =⇒ z =−e2κr̂∗. (3.36)
Now, the behaviour of the radial function at spatial infinity can be checked. In the
asymptotic region where r → ∞ or z→ ∞, one could take advantage of the inverse
transformation property of hypergeometric functions which goes as [78]
F (α,ς;η;z) = (−z)−α Γ(η)Γ(ς−α)
Γ(ς)Γ(η−α)
F (α,α+1−η;α+1− ς;1/z)+
(−z)−ς
Γ(η)Γ(α− ς)
Γ(α)Γ(η− ς)
F (ς,ς+1−η;ς+1−α;1/z) . (3.37)
This property plays a crucial role in obtaining an analytical expression for the
asymptotic wave function. With the aid of Eq.(3.37), the radial function becomes
ΦSI w ξ2 (−z)
−iA−B (−z)−α
Γ(η)Γ(ς−α)
Γ(ς)Γ(η−α)
+ξ2 (−z)
−iA−B (−z)−ς
Γ(η)Γ(α− ς)
Γ(α)Γ(η− ς)
.
(3.38)
Although Eq.(3.38) represents the behaviour of the radial function at an infinite
distance away from the black string correctly, it can be presented in a more elaborate
form. Applying the simplifications
−iA−B−α=−
1
2
(1+ iτ) , (3.39)
33
and
−iA−B− ς=−
1
2
(1− iτ) , (3.40)
together with substituting x =−z, the asymptotic solution yields
ΦSI =
1
√
x
[
ξ2x
− iτ2
Γ(η)Γ(ς−α)
Γ(ς)Γ(η−α)
+ξ2x
iτ
2
Γ(η)Γ(α− ς)
Γ(α)Γ(η− ς)
]
. (3.41)
One can now apply the definition of the tortoise coordinate so as to be able to express
ΦSI as a function of r̂∗. Then,
r̂∗ = lim
r→∞
r∗ '−
2
√
r
, (3.42)
where
x = r− r+ x|r→∞ ' r = 4e
−2r̂∗. (3.43)
Furthermore, r̂∗ = lnr∗. In this case, the radial function becomes
ΦSI =
1
√
r
[
Λ1e
ir̂∗τ+Λ2e
−ir̂∗τ
]
, (3.44)
in which
Λ1 = 2
−iτξ2
Γ(η)Γ(ς−α)
Γ(ς)Γ(η−α)
, (3.45)
and
Λ2 = 2
−iτξ2
Γ(η)Γ(α− ς)
Γ(α)Γ(η− ς)
. (3.46)
34
Finally, the wave-function for r→ ∞ takes the form
ψSI =
1
√
r
[
Λ1e
i(r̂∗τ−ωt)+Λ2e
−i(r̂∗τ+ωt)
]
. (3.47)
3.4 Evaporation of Dilatonic Black String
3.4.1 Evaluation of Flux for r→ r+ and r→ ∞
This subsection carries significance in the sense that the outcome for greybody factor
is directly linked to the wavefunction forms in two regions, namely at spatial infinity
and within the near horizon region. As the wavefunctions will then be used to calculate
the associated flux values, one can recognize why the wavefunction forms are the key
concepts to focus on here. In the light of this information, let us first evaluate the flux
in the vicinity of horizon. By definition [61]
zNH =
ABH
2i
(ψNH∂r∗ψNH−ψNH∂r∗ψNH) . (3.48)
For our case, Eq.(3.48) takes the form
zNH =−4piβ |ξ2|
2 r+. (3.49)
Nonetheless,in the asymptotic region, the flux can be calculated by using
zSI =
ABH
2i
(ψSI∂r∗ψSI−ψSI∂r∗ψSI) , (3.50)
in which
ψSI =
r∗
2
[
Λ2e
−i(r̂∗τ+ωt)
]
. (3.51)
35
and ψ is the complex conjugate of Eq.(3.51). The flux in this regime reduces to
zSI =−4piβ |Λ2|
2 τ. (3.52)
it would be useful to recall that if the mass of the particles were chosen to be real, it
would be necessary to replace Eq.(3.51) with ψSI →
r∗
2
[
Λ2er̂∗τ−iωt
]
. Nonetheless, this
specific choice would be problematic, as the greybody factor evaluation would not be
possible in that case.
3.4.2 Greybody Factor
In the most general case, the greybody factor belonging to any black astronomical
object can be calculated via [70]
γl,k =
zNH
zSI
. (3.53)
For the specific five-dimensional dilatonic black string of our concern, Eq.(4.50)
becomes
γl,k =
|ξ2|
2 r+
|Λ2|
2 τ
. (3.54)
Substituting Eq.(3.46) into Eq.(3.54) and using [79]
|Γ(iy)|2 =
pi
ysinh(piy)
, (3.55)
|Γ(1+ iy)|2 =
piy
sinh(piy)
, (3.56)
and
∣
∣
∣
∣Γ(
1
2
+ iy)
∣
∣
∣
∣
2
=
pi
cosh(piy)
, (3.57)
36
the greybody factor attains
γl,k =
κr+
ω
(
e
2piω
κ −1
)
Ξ, (3.58)
with
Ξ=
e2piτ−1
[
e
pi
(
τ+ωκ−
k
κ
√
β
)
+1
][
e
pi
(
τ+ωκ+
k
κ
√
β
)
+1
] . (3.59)
The graphical analysis of Eq.(3.58) can be viewed in Figure 3.1.
Figure 3.1: Plots of the greybody factor versus frequency for different l and k values.
The plots obey the relation (3.58) and the configuration of the dilatonic
five-dimensional black string goes as follows: µ = 3 and Q = 0.2.
37
Having a closer look at the behaviour of the greybody factor for different l and k values,
one can comment that the l = k = 0 case is the one exhibiting an odd behaviour in the
long distance (low frequency) era. The greybody factor in this regime seems to possess
a divergent behaviour unlike the others. On the other hand, once the short distance
(high energy) era is of concern, the values that l and k take seem to be irrelevant.
Lastly, the peak values experience a fall, as l values are raised. Although Figure 3.1 is
plotted by using µ = 3 and Q = 0.2, the overall trends came out to be the same, once
the parameters were changed.
3.4.3 Absorption Cross-Section
In arbitrary D-dimensions, the optical theorem originated from the partial wave
expansion reduces to [80]
σl,k =
4pi(l +1)2
ω3
γl,k, (3.60)
which represents the absorption cross-section of a five-dimensional black string. For
the model of our choice, it takes the form
σl,k =
4pi(l +1)2κr+
ω4
(
e
2piω
κ −1
)
Ξ. (3.61)
which can in turn be used for evaluating the total cross section via [81]
σTotalabs =
∞
∑
l=0
σl,k. (3.62)
The dependence of absorption cross-section on l, k and ω is illustrated graphically
in Figure 3.2. The cross-section of the model possesses a similar behaviour as the
greybody factor, in the sense that within short distances (for ω→∞), the trends are the
same regardless of the values that k and l take.
38
Figure 3.2: Plots of the absorption cross-section versus frequency. The plots are
governed by Eq.(3.61). The configuration of the dilatonic five-dimensional black
string is as follows: µ = 3 and Q = 0.2.
For ω→ 0, however, all curves tend to infinity. Lastly, as can be seen from metric
(3.7), the black string of our interest is non-rotating implying there does not exist
superradiance [63]. This behaviour can be confirmed from the graph by noticing the
plots do not attain any negative cross section values.
3.4.4 Decay Rate
As the final step, the decay rate of the black string of concern will be obtained
analytically by using [82]
Γl,kDR =
σl,k
e
2piω
κ −1
=
4pi(l +1)2κr+
ω4
Ξ. (3.63)
In Figure 3.3, the behaviour of decay rate is presented for various energy values. From
the figure, it can be noticed that the decay rate vanish in the high energy era, whereas
it exhibits a divergent behaviour for ω→ 0.
39
Figure 3.3: Plots of the decay rate versus frequency. The plots are governed by
Eq.(3.63). The configuration of the dilatonic five-dimensional black string is chosen
as follows: µ = 3 and Q = 0.2.
There does not seem to exist unique and distinctive behaviour for different l and k
values. However, it can be noted that for ascending l values, the divergence of decay
rate occurs at smaller ω values.
40
Chapter 4
(3+1)-DIMENSIONAL LIFSHITZ-LIKE BLACK BRANE
HOLOGRAPHY
4.1 Applications of (3+1)-Dimensional Hyperscaling Violating
Theories with z=2 and θ=−1
As had already been stated in the introduction, the exponents z and θ occupy a central
role both in the structure of the astronomical object and in the dissipative properties
of the holographic model. When there exists hyperscaling violation within the theory,
the scale invariance is broken into covariance, which in turn gives birth to a power
law scaling of the thermodynamic parameters, compared to the ones belonging to a
conformal field theory [83]. Since the main focusing structure of this chapter 3 consists
of a model desired to match with the ordinary intuition of spacetime as experienced
by the humankind, the total dimensionality will be chosen as D = 4. The dynamical
exponent, on the other hand, is allowed to possess a wide range of values and one shall
stress that each z value has its own implications in not only general relativity, but also
for condensed matter systems. In this work, z = 2 is assigned to this exponent and there
exist several reasons behind this specific choice, some of which will now be explained
briefly.
It is worthy to point out that systems holding z = 2 are proved to exhibit the properties
3 This chapter is based upon the articles entitled “Greybody Factors of Holographic Superconductors
with z = 2 Lifshitz Scaling” [8] and “Holographic Dissipative Properties of Non-relativistic Black
Branes with Hyperscaling Violation" [9].
41
of a superconductor [33] which indicates it is highly likely for these structures to have
a great many of applications not only in physics, but also in other fields of science.
Superconductors can be noticed in our daily lives as mechanicsms that help to
minimize the energy consumed by human beings. They play a crucial role in the latest
technology for transportation and make it possible for one to travel a relatively large
distance in a rather short period via eliminating friction. Some examples for uses of
superconductors can be seen in Ref. [84]. Superconductors are also mandatory if one
wishes to reconstruct the conditions for high-energy physics with the purpose of
figuring out what precisely has happened during the Big Bang. The experimental
verifications of the open questions such as the unified theory of everything, cosmic
inflation, cosmological problem, existence of magnetic monopoles and so on are all
being tested and pursued within a particle accelerator. However, without the use of a
superconductor, gathering observational evidences does not seem to be possible, at
least with the current technology. Last but not least, all living things on Earth seem to
have the chance to benefit from superconductors, under the threat of a health
condition (up to the extent that is cared enough by the humanity). Superconductors
are found in magnetic resonance imaging scanners, whence they can be used for
diagnosis of certain diseases for both humans and other living beings. For instance, in
veterinary medicine, one can check for existence of a respiratory cyst of a dog [85] or
scan her/his brain to see whether there exists the condition known as the hereditary
polioencephalomyelopathy [86]. In brief, the choices of z = 2 and D = 4 are not
random, but rather compulsory in order for being able to explore a strongly coupled
system perceivable by human intuition, whose bulk geometry can dispel some
ambiguous characteristics of these systems. This can, in turn, contribute to
challenging phenomena such as figuring out how to build high-temperature
superconductors, relating aspects of magnetic monopoles to other
42
seemingly-irrelevant theories, getting a fulfilling insight on the strange behaviour of
second order phase transitions in liquid crystals or other systems and so on. This can
be carried out by using the main tools of the holographic principle. The exact
opposite scenario would also be of advantage: For the cases when some specific
astronomical objects are desired to be investigated further, their dual models (which
may, for instance, be as easily accessible as a liquid crystal) could in principle be
adequate to interprete the concerned astronomical property. Surely, this could be
possible iff the mapping between the two (or more) theories is precise enough,
supported by an extensive range of examples and experimental evidences. Here, we
mainly aim for contributing to the examples regarding the realisation of holographic
principle in nature.
This chapter consists of the following structure: Sect. 4.2 includes details on the
propagation of a massless scalar field in a non-Abelian, electrically charged
Lifshitz-like black brane with hyperscaling violation. Accordingly, the greybody
factor, absorption cross-section and decay rate of the concerned brane is maintained
by means of flux evaluations. In Sec. 4.3, the bulk results are linked to the boundary
model and dissipative properties of the event horizon is investigated.
4.2 Scalar Field Propagation in Lifshitz-like Black Branes
4.2.1 Properties of the Bulk Model
The evolution of the dynamics of a system is encoded in the affiliated Lagrangian [87],
thus providing the Lagrangian form of concern would be a good starting point.
L =
N
λ
tr(∂µΦ∂µΦ+ ...), (4.1)
where the fields of the theory, i.e. Φk, are large N ×N matrices and the associated
43
interactions can be expressed via [88]
O = tr(Φk1Φk2...Φkm), (4.2)
at which k ∈ N. The arbitrary constant λ in the denominator of Eq.(4.1) represents
the ’t Hooft coupling and it carries a vital importance when it comes to determining
whether a system is coupled weakly or strongly.If the system under consideration is
strongly coupled, λ is obliged to be large. For the model of interest within this chapter,
one shall imagine a situation where the dilatonic field φ is coupled to not only the
gravitational field and the cosmological constant Λ, but also to the Maxwell A and N
SU(2) Yang-Mills Aak fields (a = 1,2, ...,N) . The dynamics of such a system can be
mathematically expressed via the Lagrangian [89]
L =
√
−g
[
R−V (φ)−
1
2
(∂φ)2−
N
∑
k=1
1
4g2k
eλφF2k −
1
4
eλφFaµνF
aµν
]
(4.3)
where Λ=−[D(z−1)2+ z−1], A = (ϕ0+qr)dt (ϕ0 represents the gauge parameter),
Faµν = ∂µA
a
υ− ∂υA
a
µ + ε
abcAbµA
c
υ, V (φ) = Λe
−λφ. Moreover, R implies the usual Ricci
curvature and gk determines the coupling strength to the Maxwell field. For the spin−0
dilatonic field, the following ansatz is established:
φ=
θ
λ
logr. (4.4)
From Eq.(4.4), one can comment that the hyperscaling violating parameter is
introduced as a consequence of the presence of the dilaton. Thus, the exponent θ of
the condensed matter system on the boundary determines the strength of the dilatonic
field in the bulk, and vice versa. The holography suggests the action of the bulk
theory in (d + 1)-dimensions has a direct influence on the observables of the
44
boundary, which live in d-dimensions. The field equations of Lagrangian (4.3)
inholds a family of solutions which can compactly be presented as ,
ds2 = rθ
(
−r2z f (r)dt2 +
dr2
r2 f (r)
+ r2
D−2
∑
i=1
dx2i
)
, (4.5)
in which the metric function reads
f (r) = 1−
q2
2(z−1)r2(z−1)
, (4.6)
with q being the parameter determining the electric charge of the brane via Q = ω16piq
and θ = 2D−2 [z− (D− 1)]. These solutions are commonly referred to as hyperscaling
violating Lifshitz-like black branes carrying non-Abelian electric charges.
For D = 4 and z = 2, the violation exponent comes out to be θ=−1, and metric (4.5)
reduces to
ds2 =−N(r)dt2 +
dr2
N(r)
+ r
2
∑
i=1
dx2i , (4.7)
where N(r) = r3 f (r). As a result, the metric function becomes f (r) = 1− q
2
2r2
. From
Eq.(4.7), one can notice that the spacetime of concern is non-relativistic. In literature,
a vast number of studies on non-relativistic backgrounds can be found, amongst which
some can be accessed from Refs. [90–95]. The surface gravity of the model can be
written as
κs =
1
2
dN(r)
dr
∣
∣
∣
∣
r=r+
= r2+, (4.8)
in which the event horizon is r+ = q/
√
2. Substituting Eq.(4.8) into definition (1.4),
45
the Hawking temperature becomes
TH =
r2+
2pi
=
q2
4pi
. (4.9)
4.2.2 Klein Gordon Equation
Let us now inspect how scalar particles with no mass propagate through the model
of our concern. To achieve this, one needs to introduce the massless Klein-Gordon
equation, which can be demonstrated as
1
√
−g
∂µ
(√
−ggµυ∂νϕ
)
= 0. (4.10)
The separation of variables enables one to express the wavefunction in the form [96]
ϕ(t,−→x ) =Φ(r)ei~κ.~xe−iωt , (4.11)
where~κ and~x stand for the wave and spatial vectors, respectively. Moreover, one can
get an idea about the energy of the emitted radiation from the frequency, ω. For the
case when no specific values are assigned to the exponents, i.e. for metric (4.5), Eq.
(4.10) transforms into
d
dr
[
f (r)r2+η˜−θ
dΦ
dr
]
+
1
r2+θ−η˜
(
ω2
r2(z−1) f (r)
−κ2
)
Φ(r) = 0 , (4.12)
when only the radial part is of concern. Note that η˜= θD2 + z+D−3.
If one wishes to evaluate the effective potential of the theory, tortoise coordinate would
need to be introduced. Suppose that
Φ(r) = F (r)r−ξ, (4.13)
46
in which ξ= (D−2)(2+θ)4 . Then, the tortoise coordinate r∗ reads
r∗ =
ˆ
r−(1+z)
dr
f (r)
, (4.14)
which results in
d2F (r∗)
dr2∗
−Ve f fF (r∗) =−ω2F (r∗). (4.15)
Consequently, the effective potential is found to be
Ve f f (r) = r
2(z−1) f (r)
[
q2
2
ξr3−z +ξ(ξ+ z)r2 f (r)+κ2
]
. (4.16)
Due to the reasons specified in the previous section, the differential equation will be
solved for z = 2, D = 4 and θ = −1. At this point, it would be worthy to emphasise
that these black branes are stable. A detailed analysis for the stability will be provided
in the upcoming subsection.
Going back to the wave-function analysis, the radial differential equation (4.12) with
specific exponents turn into
N(r)
d2Φ
dr2
+(4r2−2r2+)
dΦ
dr
+
(
ω2
N(r)
−
κ2
r
)
Φ(r) = 0. (4.17)
Let us introduce a new variable z˜ such that z˜ = r−2(r2− r2+). Combining this with the
ansatz
Φ(z˜) = z˜α(1− z˜)βG(z˜), (4.18)
47
with β= 3/2 results in
z˜(1− z˜)
d2G
dz˜2
+
(
1−
7z˜
2
−
iω(1− z˜)
r2+
)
dG
dz˜
+
[
5iω−6r2+−κ
2
4r2+
]
G = 0, (4.19)
Checking the resemblance between Eq. (4.19) and the general hypergeometric
differential equation (3.20) [79]
z˜(1− z˜)
d2G
dz˜2
+[c− (1+a+b)z]
dG
dz˜
−abG = 0 (4.20)
yields
G(z˜) =C1 2F1 (a,b;c; z˜)+C2z˜1−c 2F1 (a− c+1,b− c+1;2− c; z˜) , (4.21)
where
a = α+
5
4
∓
√
κ2s −4κsκ2−4ω2
4κs
, (4.22)
b = α+
5
4
±
√
κ2s −4κsκ2−4ω2
4κs
, (4.23)
c = 1+2α. (4.24)
Note that α=± iω2κs . In this study, we pick
α=−(iω/2κs),
a =
5
4
−
i
2κs
(ω+ ω̂) , (4.25)
b =
5
4
−
i
2κs
(ω− ω̂) , (4.26)
48
by establishing
ω̂=
√
ω2 +κs
(
κ2−
κs
4
)
, (4.27)
for simplicity. Consequently, Eq. (4.24) takes the form
c = 1−
iω
κs
. (4.28)
Substituting Eq.(4.21) into the ansatz (4.18)
Φ(z˜) = z˜α(1− z˜)β
[
C1 2F1 (a,b;c; z˜)+C2 z˜1−c 2F1 (a− c+1,b− c+1;2− c; z˜)
]
(4.29)
is obtained. As previously mentioned, the examination of the radial solution for two
specific regions (for r→ r+ and r→ ∞ ) is rather important, in case one desires to
obtain analytical solutions for the radiation parameters of (z = 2,θ=−1) black branes.
Hence, let us explore the radial behaviour for different points of spacetime.
4.2.2.1 Near Horizon Solution
The first step will be checking the behaviour of the wavefunction in the near horizon
era. For z˜→ 0, 2F1 (a,b,c;0) = 1, leading to
ΦNH =C1e
α ln z˜ +C2e
−α ln z˜. (4.30)
One shall first introduce the tortoise coordinate in this region, which can be denoted as
r∗(NH) =
ln
√
1− z˜−1
2r2+
. (4.31)
Based on the Hawking radiation intuition, one needs to impose the constraint C2 = 0 in
49
order for the theory to admit waves approaching the horizon only. This is because one
of the virtual particles will fall into the black brane, whilst the other reaches to infinity.
Then, the general solution becomes
Φ(z˜) =C1z˜
α(1− z˜)β 2F1 (a,b;c; z˜) , (4.32)
Setting C2 to zero and using α=−(iω/2κs), one obtains
ΦNH =C1e
−iω ln z˜
2r2+ = C˜1e
−iωr∗(NH). (4.33)
for the near horizon region, with C˜1 = C1eωpi/2r
2
+ . Inevitably,the wave function (4.11)
reads
ϕNH =C1e
−iω
(
t+ ln z˜
2r2+
)
= C˜1e
−iωr∗NH e−iωt . (4.34)
4.2.2.2 Asymptotic Behaviour
Amongst a numerous ways for calculating the flux, the method followed in Ref. [97]
will be followed here. Thus, firstly, the behaviour of the wave function for r→∞ needs
to be examined delicately. For the evaluation of the outgoing flux, one may use
zSI =
√
−ggrr
2i
(Φ∗SI∂rΦSI−ΦSI∂rΦ
∗
SI). (4.35)
which implies that in order for obtaining ϕSI , the asymptotic behaviour of
hypergeometric solution should be studied. For z˜→ ∞, the general solution (4.32)
becomes
ΦSI(z˜) =C1z˜
α
[
A1(1− z˜)
β
2F1 (a,b;a+b− c+1;1− z˜) +
A2 (z˜) 2F1 (c−a,c−b;c−a−b+1;1− z˜)] . (4.36)
50
At this point, one can derive benefit from the inverse transformation property obeyed
by hypergeometric functions, which indicates that in general
2F1(a,b,c;u) = A1 2F1(a,b,a+b− c+1;1−u) (4.37)
+A2(1−u)
c−a−b
2F1(c−a,c−b,c−a−b+1;1−u).
(4.38)
Record that the constants are expressed in terms of gamma functions as follows.
A1 =
Γ(c)Γ(c−a−b)
Γ(c−a)Γ(c−b)
, (4.39)
A2 =
Γ(c)Γ(a+b− c)
Γ(a)Γ(b)
. (4.40)
In brief, for r→ ∞, the radial solution (4.36) exhibits the behaviour
ΦSI =C1
[
A1
(r+
r
)3
+A2
]
. (4.41)
Our main interest is the low energy greybody factor, since the Lagrangian involves
strong coupling, and furthermore, β ∈ R turns it into a severe challenge to apprehend
between the ingoing and outgoing fluxes [97]. Therefore, in the asymptotic region, Eq.
(4.17) comes up to
d2Φ
dr2
+
4
r
dΦ
dr
= 0. (4.42)
The second-order Cauchy-Euler equation (4.42) admits the solution
ΦSI = D1 +
D2
r3
. (4.43)
As the final step, Eq. (4.41) can be compared to Eq. (4.43) in order to obtain the
arbitrary constants. They come out as D1 = A2C1 and D2 = A1C1r3+. Ultimately, the
51
asymptotic flux (4.35) reads
zSI = 3
(
|Dout |
2−|Din|
2
)
, (4.44)
where
Dout =
D1 + iD2
2
, (4.45)
and
Din =
D1− iD2
2
. (4.46)
4.2.3 Quasinormal Modes and Stability Check
Throughout this subsection, the stability and quasinormal mode analysis will be carried
out for the (2,−1) Lifshitz-like black branes by considering spin-0 perturbations. As
already stated in introduction, when a black astronomical object is externally disturbed,
let say due to a particle crossing the event horizon, the object radiates energy. In the
region r > r+, there exists an effective potential, which can analytically be derived by
inspecting the Zerilli & Regge-Wheeler equations. Note that the effective potential
will always be produced in the same form, as long as the astronomical black object of
concern is not changed. When this barrier is of interest, the composition of test fields
perturbing the black object becomes irrelevant. The effective potantial is in such a form
that it experiences an exponential decay for r→ r+ (r∗→−∞) and r→ ∞ (r∗→ ∞)
In this regard, quasinormal modes can be treated as electromagnetic or gravitational
perturbations of astronomical spacetimes [98]. During quasinormal inspection, one
needs to consider how effective potential behaves.
52
From Eq. (4.16), one can see that the effective potential of the four-dimensional non-
Abelian charged Lifshitz-like black brane with (2,−1) metric exponents yields
Ve f f (r) = N(r)
(
5r
4
−
r2+
4r
+
κ2
r
)
. (4.47)
One can notice that lim
r→∞
Ve f f (r)→∞. Thus, quasinormal modes of concern obey the
required boundary conditions which states that the spin 0 field ϕ is obliged to be
purely ingoing in the vicinity of event horizon; and furthermore, it needs to vanish
asymptotically far away (a similar case can be recognised in Ref. [99]).
Figure 4.1: The behaviour of effective potential under the choice q = 1 and κ= 0.
As the asymptotic behaviour of the radial function had already been determined in Eq.
(4.36), one can express
ΦSI(z˜) ≈ C1A1(1− z˜)
β+C1A2,
∼= C1A2 =C1
Γ(c)Γ(a+b− c)
Γ(a)Γ(b)
. (4.48)
This indicates that Eq.(4.48) should be forced to vanish for z˜→ 1. The asymptotic
53
wave function can vanish if and only if a =−n or b =−n for n = 0,1,2, ... The latter
provides us with the precise quasinormal spectrum which goes as
ω=−i
q2(n+1)(2n+3)+κ2
5+4n
. (4.49)
From Eq.(4.49) one can notice that the over damping of the system is assured, as
the quasinormal modes are found to be purely negative and imaginary. Hence, the
four-dimensional electrically charged Lifshitz-like black brane with z = 2 dynamic
exponent is shown to be stable under massless spin 0 perturbations.
4.2.4 Radiation Parameters
This subsection is reserved for maintaining exact results for greybody factor,
absorption cross-section and decay rate of the thermal radiation emitted by the black
brane (4.7)
Based on the discussions in section 1.4, the greybody factor can be found via [97]
γ= 1−ℜ=
2i(D−D∗)
DD∗+ i(D−D∗)+1
, (4.50)
in which ℜ= |Dout |
2 / |Din|2 and D= D1/D2. To be more specific, one can write
D=
3
8
Γ
(
−14 − iX
)
Γ
(
−14 − iY
)
Γ
(
5
4 − iY
)
Γ
(
5
4 − iX
)
r3+
, (4.51)
which yields
DD∗ =
2304
pi4r6+
[Γ(3/4)]8
∞
∏
n=0
εn
[
1+
(
Y
n−1/4
)2
][
1+
(
X
n−1/4
)2
] , (4.52)
D−D∗ =
24
pi2r3+
Ξ [Γ(3/4)]4
∞
∏
n=0
εn. (4.53)
54
The expressions above are maintained in these neat forms by using
X =
ω− ωˆ
2r2+
, (4.54)
Y =
ω+ ωˆ
2r2+
, (4.55)
εn =
[
1+
(
Y
n+5/4
)2
][
1+
(
X
n+5/4
)2
]
, (4.56)
and
Ξ=
(sinθ1 sinθ2− sinθ3 sinθ4)
sinθ1 sinθ2 sinθ3 sinθ4
. (4.57)
Moreover, the angles of interest are θ1 = pi
(
5
4 − iX
)
, θ2 = pi
(
5
4 − iY
)
, θ3 = pi
(
5
4 + iY
)
and θ4 = pi
(
5
4 + iX
)
. One of the key relations which helped us during our analytical
steps goes as
Γ(x+ iy)Γ(x− iy)
[Γ(x)]2
=
∞
∏
n=0
[
1+
(
y
x+n
)2
]−1
, (4.58)
together with the reflection formula
1
Γ(Z)
1
Γ(1−Z)
=
sinpiZ
pi
, (4.59)
where Z ∈ C [100]. Armoured by the properties above, the final form of the greybody
factor is stated as
γ=
2i Ξ
∞
∏
n=0
εn
96 [Γ(3/4)]4
pi2r3+
∞
∏
n=0
εn[
1+
(
Y
n−1/4
)2
][
1+
(
X
n−1/4
)2
] + i Ξ
∞
∏
n=0
εn +
pi2r3+
24[Γ(3/4)]4
. (4.60)
After achieving an exact solution for the greybody factor, one can now calculate the
55
absorption cross-section which results in [101]
σabs =
∞
∑
l=0
ipi
ω2
2(2l +1) Ξ
∞
∏
n=0
εn
96 [Γ(3/4)]4
pi2r3+
∞
∏
n=0
εn[
1+
(
Y
n−1/4
)2
][
1+
(
X
n−1/4
)2
] + i Ξ
∞
∏
n=0
εn +
pi2r3+
24[Γ(3/4)]4
. (4.61)
Finally, the decay rate of the associated black brane can be written as
Γ=
i Ξ
∞
∏
n=0
εn d3k
4pi3(eω/TH −1)

96 [Γ(3/4)]4
pi2r3+
∞
∏
n=0
εn[
1+
(
Y
n−1/4
)2
][
1+
(
X
n−1/4
)2
] + i Ξ
∞
∏
n=0
εn +
pi2r3+
24[Γ(3/4)]4


.
(4.62)
4.3 Duality Between Bulk Observables and Strongly Coupled
Systems
4.3.1 Some Key Aspects
Throughout the preceding sections, the bulk characteristics such as quasinormal
modes, greybody factor, absorption cross-section and decay rate were explored via
analytical methods. Now, in this section, the main focusing point will be the
associated holographic model. In the dual picture, the model is built on the boundary
of the bulk spacetime located at r → ∞. Therefore, the fields present in the
gravitational picture are mapped onto the dual operators of the holographic field
theory of two-spatial boundary dimensions. In the astronomical picture, the field was
picked to be in the massless spin 0 form, or in other words, as
ϕ(t,r,~x) =Φ(r)ei~κ.~x e−iωt . This implies that Φ of bulk theory will represent OΦ on the
boundary, where OΦ represents a well-defined boundary value for Φ.
Quasinormal modes (4.49) can be used to maintain the diffusion constant of the
fluctuating horizon via membrane paradigm. The membrane paradigm suggests that
56
small oscillations of a stretched horizon possess resemblance with the diffusive
properties of a conserved charge in simple fluids [1,93,102]. To rephrase, a dispersion
relation of the form ω=−iDq2 puts forward the existence of diffusion of a conserved
charge [102]. Comparing the dispersion relation with Eq. (4.49), one can write
D =
(n+1)(2n+3)
5+4n
, (4.63)
in which D represents the shear mode diffusion constant [103]. The diffusion
constant is rather crucial in fluid/gravity duality, since it can be used to derive the
ratio of shear viscosity to entropy density, denoted by η/s. Note that for n = 0, i.e. for
the fundamental quasinormal mode, the diffusion constant becomes D = 3/5. Recall
the geometrical structure of black brane (4.5)
ds2 = rθ
(
−r2z f (r)dt2 +
dr2
r2 f (r)
+ r2
2
∑
i=1
dx2i
)
. (4.64)
Applying r→ 1/r˜ and θ→−θ˜ yields
ds˜2 = r˜θ˜
(
−
f (r˜)
r˜2z
dt2 +
dr˜2
r˜2 f (r˜)
+
2
∑
i=1
dx2i
r˜2
)
. (4.65)
Metric (4.65) is in the same form as the family of non-relativistic branes mentioned
in [104]. Before we start the analysis for η/s based on the universal relation derived
by Kolekar, Mukherjee and Narayan, recall that for the model of our interest; d =
D−1 = 3, z = 2 and θ˜= 1. These specific choices obey the null energy conditions
(d−1− θ˜)((d−1)(z−1)− θ˜)> 0, (z−1)(d−1+ z− θ˜)> 0. (4.66)
In their work [104], the authors proposed that there exists a universal relation for all
57
hyperscaling violating theories which goes as
η
s
=
d− z+1
4pi
D r˜2−zH . (4.67)
Thus, based on their proposal, if one can evaluate the diffusion constant from the
dispersion relation of the astronomical black object, s/he can use tools of membrane
paradigm to get the viscosity-to-entropy ratio of the holographic fluid model.
Furthermore, it would be beneficial to stress that this ratio has played a vital role in
experimental verification of AdS/CFT correspondence.
Going back to Eq. (4.67), although the authors have suggested that this relation would
act as a universal law for theories based on background (4.65), in another study of
theirs [93], they stated that the situation may as well have slight differences in the
vicinity of a charge. Since η/s ratio evaluation via holographic principle is still a
pending research question, we wanted to evaluate this ratio both from the universal
relation (4.67) - as claimed by the authors- and from another method that will become
apparent in the upcoming sections.
For the brane-fluid system of our concern, substituting d = 3, z = 2 and D = 3/5 into
Eq. (4.67) leads to
η
s
=
3
10pi
. (4.68)
Note that the so-called universal Kovtun-Son-Starinets bound seems to be satisfied,
i.e. η/s > 14pi [102]. From this result, one can conclude the following: If this value
can experimentally be verified, it would be implied that the model under
consideration would be carrying information regarding both (3 + 1)-dimensional
58
Lifshitz-like black branes and strongly coupled, non relativistic three-dimensional
fluids. Moreover, any empirical evidence for Eq. (4.68) would confirm that the
formulation proposed by Kolekar, Mukherjee and Narayan is also valid for charged
hyperscaling violating Lifshitz-like backgrounds as well. It is also noteworthy
addressing that the Kovtun-Son-Starinets bound is suspected to be an inherent
property of semi-classical gravitational theory [102].
Let us now describe the framework developed by Gubser-Klebanov-Polyakov-Witten
(GKPW) [105]. GKPW method states that under the consideration of an infinitesimal
distance ε away from the boundary of the bulk spacetime, the equations of motion
remain the same, albeit the perturbations in the action. As a result, the ultraviolet
divergence is avoided and imposing ε→ 0 gives birth to a well-defined boundary value
for Φ. Accordingly, the flux factor is defined as [106]
z(~κ,ω) = lim
r→ε
√
g grrΦ(r) ∂rΦ(r), (4.69)
representing the momentum-space two-point correlation function. Alternatively,
z(~κ,ω) = 〈OΦ(~κ,ω) OΦ(−~κ,−ω)〉 . (4.70)
One may check Refs. [106–108] for further regards. The two-point correlation function
plays a key role in a wide range of experiments such as particle physics experiments
including scattering processes and the transitions between states; and furthermore, it
benefits from non-relativistic perturbation theory [109]. Consequently, the differential
cross-section is achieved by using
dσ=
1
z
∣
∣M
∣
∣2 dΦ˜, (4.71)
59
where M and dΦ˜ respectively stand for the matrix element and the phase factor.
4.3.2 Holographic Approach: Transport Coefficients of the Dual Model
4.3.2.1 Shear Viscosity
The shear viscosity of the holographic model can be evaluated via [105, 110, 111]
η=− lim
ω→0
1
ω
Im
[
GO+(ω,0)
]
, (4.72)
which is the so-called Kubo formula. Eq.(4.72) represents the linear response of an
object after being subject to infinitesimal perturbations. Note that GO+ stands for the
two-point correlation function which can be calculated by inspecting how the radial
function, namely Eq. (4.17), behaves for r→ ∞. Based on the relation GO+ = D2/D1
[112], the retarded Green’s function becomes
GO+(ω,0) =
8
3
Γ
(
5
4 − iY˜
)
Γ
(
−14 − iY˜
)
Γ
(
5
4 − iX˜
)
Γ
(
−14 − iX˜
)r3+, (4.73)
in which X˜ = X |κ→0 =
ω−
√
ω2−r4+/4
2r2+
and Y˜ = Y |κ→0 =
ω+
√
ω2−r4+/4
2r2+
.
After this point, one needs to take advantage of some mathematical tricks, in order
for achieving analytical results. With this purpose in mind, let us express the ratio of
complex Gamma functions as
Γ(b+ iy)
Γ(a+ iy)
=
Γ(b+ iy)
Γ(1−a+ iy)
Γ(1−a+ iy)
Γ(a+ iy)
Γ(a− iy)
Γ(a− iy)
, (4.74)
where a,b ∈ R. As can be noticed from Eq. (4.74), both the numerator and the
denominator are multiplied by Γ(1−a+ iy)Γ(a− iy). This simple trick have led to
intriguing conclusions: It enabled us to find exact expressions for the dissipative
properties of the dual model. Record that the complex Gamma functions obey the
60
relations [79, 113]
Γ(1−a+ iy)Γ(a− iy) =
pi
sin [pi(a− iy)]
, (4.75)
and
|Γ(a+ iy)|2 = Γ(a− iy)Γ(a+ iy) , (4.76)
which in turn result in
Γ(b+ iy)
Γ(a+ iy)
=
1
|Γ(a+ iy)|2
Γ(b+ iy)
Γ(1−a+ iy)
pi
sin [pi(a− iy)]
. (4.77)
To be more specific, Eq. (4.77) can now be used to evaluate the two-point correlation
function. The associated ratios take the form
Γ
(
5
4 − iX˜
)
Γ
(
−14 − iX˜
) =
1
∣
∣Γ
(
−14 − iX˜
)∣
∣2
pi
sin
[
pi
(
−14 + iX˜
)] , (4.78)
and
Γ
(
5
4 − iY˜
)
Γ
(
−14 − iY˜
) =
1
∣
∣Γ
(
−14 − iY˜
)∣
∣2
pi
sin
[
pi
(
−14 + iY˜
)] . (4.79)
Equipped with the tools of fluid/gravity correspondence, the region of interest will be
chosen as the one where low energy is of concern, i.e the condition w << r+ needs to
be imposed. The main reason behind this is because the dual system will be considered
to be living on the boundary, at which the bulk-gravitational model exhibits properties
61
of hydrodynamics equations. Then, X˜ and Y˜ become
X˜ ∼
ω
2r2+
−
i
4
,
Y˜ ∼
ω
2r2+
+
i
4
.
, (4.80)
respectively. Substituting the low energy behavior of X˜ and Y˜ into Eqs. (4.78) and
(4.79) together with the commonly referred relation zΓ(z) = Γ(1+ z), the modulus
squared terms can be replaced by
∣
∣
∣
∣Γ
(
−
1
4
− iX˜
)∣
∣
∣
∣
2
=
4r2+
r4++ω2
∣
∣
∣
∣Γ
(
1
2
−
iω
2r2+
)∣
∣
∣
∣
2
, (4.81)
and
∣
∣
∣
∣Γ
(
−
1
4
− iY˜
)∣
∣
∣
∣
2
=
∣
∣
∣
∣Γ
(
−
iω
2r2+
)∣
∣
∣
∣
2
. (4.82)
Taking the relations [113]
|Γ(ib)|2 =
pi
bsinh(pib)
, (4.83)
and
∣
∣
∣
∣Γ
(
1
2
+ ib
)∣
∣
∣
∣
2
=
pi
cosh(pib)
, (4.84)
into account, gamma functions (4.78) and (4.79) finally evolve into rather simple forms
that go as
Γ
(
5
4 − iX˜
)
Γ
(−1
4 − iX˜
) ∼
i
(
r4++ω
2
)
4r2+
coth
(
piω
2r2+
)
,
Γ
(
5
4 − iY˜
)
Γ
(−1
4 − iY˜
) ∼−
ω
2r2+
tanh
(
piω
2r2+
)
.
(4.85)
62
Recall that for our case, b = −ω/2r2H . Plugging the results obtained in Eq. (4.85) in
Eq. (4.73), the Green’s function becomes
GO+(ω,0) =−
iω
(
r4++ω
2
)
3r+
. (4.86)
Ultimately, applying the condition ω→ 0, one of the most significant observables of
the dual model comes out as
η=
r3+
3
. (4.87)
For the reasons that will become apparent, it is preferable to express the observables in
terms of temperature. Thus, recalling rH =
√
2piT , the shear viscosity can alternatively
be written as
η= ζT 3/2, (4.88)
in which ζ = 2
√
2pi3
3 . Eq. (4.88) can be investigated for observing how the specific
choice of metric exponents influenced the observables of the holographic image.
Below, one can find the graphical illustration of the shear viscosity as a function of
temperature.
η
T
Figure 4.2: The shear viscosity as a function of temperature
63
From Eq.(4.88) one can comment that the dual theory lives in an effective dimension,
which in general reads de f f = db− θ. Here, db = 2, as it represents the number of
spatial dimensions on the boundary. For the case concerned here, i.e. for (2,−1), the
effective theory seems to live in three dimensions. The results of the analytical
methods followed throughout this work suggest that for general (z,θ), the shear
viscosity is directly proportional to a scaled temperature which goes as η ∝ T (db−θ)/z.
It is noteworthy to stress that our findings match with the ones in
literature [104, 114–117].
4.3.2.2 DC-Conductivity
Another observable of the dual model is the so-called DC-conductivity. It can be
derived from optical conductivity
σi j (ω) =−
1
iω
〈Ji (ω)J j (ω)〉 (4.89)
which is the main bridge between the four-dimensional Lifshitz-like brane and the
holographic (2+ 1)-dimensional fluid. Eq. (4.89) is also known as Kubo’s formula
in the literature and it can be viewed as the key function to be evaluated. Note that Ji
denotes the current operator [118, 119]. The zero-frequency limit of Eq. (4.89) results
in DC-conductvity, or mathematically speaking
σDC = lim
ω→0
σi j(ω). (4.90)
For our case, Eq. (4.90) reduces to
σDC =
e2
3
(2pi)3/2 T 3/2, (4.91)
where e is the charge of an electron. The DC-conductivity of the non-relativistic,
64
strongly coupled fluid of our concern can be presented graphically as shown below.
σDC
T
Figure 4.3: The graph of analytical DC-conductivity versus temperature
4.3.2.3 DC-Resistivity
The DC-resistivity of a strongly-coupled, non-relativistic fluid supporting
superconducting fluctuations carries a vital importance, as its graphical representation
can provide some conceptual findings. The DC-resistivity can simply be maintained
via
ρDC =
1
σDC
=
3
e2
(2pi)−3/2 T−3/2. (4.92)
As can be seen from Eq. (4.92), the resistivity of the model exhibits an untrivial
behaviour which deserves further attention. To be able to comment further, let us plot
a graph presenting how resistivity behaves as a function of temperature.
From the graphical representation provided above, one can notice that there exists a
sharp decrease in resistivity, around a non-zero temperature point. It is highly probable
for this point to represent the critical temperature where a second-order phase transition
takes place. As the bulk metric was chosen in such a way to support superconducting
65
ρT
Figure 4.4: The graphical illustration of analytical DC-resistivity as a function of
temperature
fluctuations, this is one of the key consequences belonging to this section.
In Ref. [118], one can have an access to a similar diagram. The authors first used
analytical evaluations to figure out the holographic DC-conductivity of a system with
arbitrary (z,θ), and subsequently discussed the behavior of the DC-resistivity for a
non-relativistic system via numerical methods. Their analysis indicates that the
DC-resistivity possesses different scaling behavior for different temperature regimes;
namely, for T ≤ Tcritical and T > Tcritical . In our case, the holographic DC-resistivity
behaves as ρDC ∝ T−3/2 which seems to be only a small portion of a broader picture.
For studies on second order superfluid and superconducting phase transitions, one
may refer to Refs. [120, 121].
66
Chapter 5
CONCLUSION
The key outcomes obtained within this thesis can be summarised as follows: As the
starting point, a wide variety of gravitational interactions in the vicinity of
(2 + 1)-dimensional Mandal-Sengupta-Wadia black holes and (4 + 1)-dimensional
dilatonic black strings were investigated. Different dimensionalities and continuum
objects were picked strategically, with the purpose of constructing a firm
understanding on the semi-classical aspects of gravity for different scenarios. The
effective potential of the Mandal-Sengupta-Wadia black hole was evaluated to be
positive definite, whence assuring the linear stability. Besides, for the
(4 + 1)-dimensional case, the analysis has shown that there exists a resemblance
between tachyonic particles and the fifth dimension, as the greybody factor of the
scattering process only allowed for imaginary masses to be present.
For the (4+ 1)-dimensional black string of concern, one of the key results obtained
was that the greybody factor of the concerned object can be evaluated analytically, if
and only if the mass is chosen to be imaginary. This suggests that there exists a
resemblance between the fifth-dimension and the tachyons. As the last step, for the
(3 + 1)-dimensional brane model, the propagation of massless scalar particles are
analysed via Klein-Gordon equation; subsequently giving rise to analytical
expressions for the absorption cross-section, decay rate, and greybody factor of the
model. In what follows, the dual model is found to represent a strongly-coupled,
non-relativistic fluid displaying Lifshitz-type symmetry. Furthermore, the analytical
67
expressions obtained for the radiation parameters are linked to the theory living on the
boundary of the bulk theory, giving rise to a relationship between a one-less
dimensional theory and the gravitational theory of interest. Although the results
obtained are purely theoretical, there exist strong indication that the findings are
subject to experimentation, as the specific model of choice supports superconducting
fluctuations. Moreover, the brane of concern has many implications in both string
theory and condensed matter systems, as it is a solution of
Einstein-Yang-Mills-Maxwell theory.
The main motivation behind this thesis was inspecting a holographic system where
not only the bulk theory could be handled delicately, but also the dissipative
observables of the dual scenario would be explored via the tools of a very substantial
concept: the fluid/gravity correspondence. As aforementioned, one of the main tasks
was maintaining information regarding strongly-coupled systems of the
four-dimensional nature, as we perceive it. Having followed analytical methods, the
two-point correlation function was found as GO+(ω,0) = −iω
(
r4++ω
2
)
/3r+, which
resulted in η ∝ T 3/2, σDC ∝ T 3/2, and ρDC ∝ T−3/2. Note that these observables obey
the general relation; i.e. η ∝ T (db−θ)/z. These parameters correspond to the
observables of the strongly-coupled, non-relativistic fluid in three dimensions.
Although it is out of scope of this thesis, it would be inspiring to extend the study and
check whether the dual model corresponds to a high-temperature superconductor.
This can be achieved via experimental tools, and moreover, any possible confirmation
of the theoretically-obtained dissipative parameters would act as a supplementary
empirical evidence for the quantum properties of spacetime.
Furthermore, the analysis suggests that there exists a second order phase transition
68
around some critical temperature within the theory. This outcome appears to be rather
interesting,as the black brane of concern was originally chosen in such a way to
support superconducting fluctuations, or in other words, the dynamic metric exponent
was chosen as z = 2. The Lifshitz-like solution admitted by the
Einstein-Yang-Mills-Dilaton action constrained the hyperscaling violation factor to be
θ = −1. Therefore, the holographic system under these conditions seems to be
encrypted with a wealth of information not only on the theoretical aspects of the
underlying string models, but also the ambiguous behaviour during the second order
phase transitions of superconducting systems, which still remain as a peculiar
phenomenon in experimental physics. The results obtained in this work can lead to
interesting phenomena: the bulk spacetime includes Yang Mills and dilatonic fields,
and from the perspective of an experimentalist, it contains information about
superconducting phase transitions.
In this thesis, the effect of small perturbations on a (2 + 1)-dimensional
Mandal-Sengupta-Wadia black hole, the tachyonic evaporation of a
(4 + 1)-dimensional dilatonic black string, and the wave dynamics of a
(3 + 1)-dimensional black brane with hyperscaling violation are investigated.
Furthermore, in what follows, the dual observables living on the boundary of the
(3+1)-dimensional brane are evaluated via linear response theory. Strictly speaking,
different aspects of three and (4 + 1)-dimensional objects are first studied with the
purpose of getting equipped with bulk-gravitational insights concerning the most
dense objects of the universe. In order for not being limited to (3 + 1)-dimensions
only, the theories were chosen to have different dimensionality. The concepts covered
during the first three chapters are subsequently applied in Chapter 4, which is related
to hyperscaling violating Lifshitz-like black branes in four dimensions.
69
For future references, it would be intriguing to extend the studies inspected in this
thesis by checking the relevance of the findings with the theory of magnetic monopoles
and D-branes from string theory and check whether it can provide us with any useful
information regarding concepts like quark confinement and chiral symmetry breaking.
I would like to finalise the thesis with the words of Neil deGrasse Tyson: “We are
all connected; to each other, biologically; to the earth, chemically. To the rest of the
universe, atomically. I look up at the night sky, and I know that, yes, we are part of this
universe, but perhaps more important than these facts is that the universe is in us. When
I reflect on that fact, I look up—many people feel small, because they’re small and the
universe is big, but I feel big, because my atoms came from those stars. We are not
figuratively, but literally stardust. We are stardust brought to life, then empowered by
the universe to figure itself out and we have only just begun. I know that the molecules
in my body are traceable to phenomena in the cosmos. After all, what nobler thought
can one cherish than that the universe lives within us all? The more of us that feel the
universe and the connections within, the better off we will be in this world.”
70
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