Topological Black Holes with Some Applications in Ads/CFT Correspondence

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Topological Black Holes with Some Applications in AdS/CFT Correspondence

Ong Yen Chin HT080889Y

An academic exercise presented in partial fulﬁllment for the degree of Master of Science in Mathematics

Supervisor: Professor Brett T. McInnes

Department of Mathematics

Faculty of Science

National University of Singapore

2010 1

Dedicated to the 4th Aegean Summer School on Black Holes, Mytilene, Island of Lesvos, Greece. 17-22 September 2007, where I discovered the fun of black holes and immersed in the tranquility of the Aegean Sea. Contents

Abstract iv

Disclaimer v

Acknowledgement vi

1 Introduction: From Black Holes to String Theory and AdS/CFT 2

1.1 A Brief History of Black Hole Research ...... 2

1.2 String Theory: What is it really good for? ...... 7

2 From de Sitter Space to Anti-de Sitter Space and Conformal Field Theory 10

2.1 From Einstein’s Static Universe to the de Sitter Solution ...... 10

2.2 Geometry of de Sitter Space ...... 15

2.3 AdS: Anti-de Sitter Space ...... 22

2.4 Conformal Compactiﬁcation ...... 23

2.5 Conformal Field Theory ...... 25

3 Black Holes in de Sitter and Anti-de Sitter Space 30

3.1 Black Holes with Cosmological Constant ...... 30 CONTENTS ii

3.2 Curvature of the Event Horizon ...... 42

3.3 Physics of Topological Black Holes ...... 42

3.3.1 Positively Curved Uncharged AdS Black Holes ...... 44

3.3.2 Flat Uncharged AdS Black Holes ...... 46

3.3.3 Negatively Curved Uncharged AdS Black Holes ...... 49

3.3.4 Flat Electrically Charged Black Holes in AdS ...... 50

4 Stability of Anti-de Sitter Black Holes 55

4.1 Thermodynamics Instability ...... 55

4.1.1 Phase Transition for Flat Uncharged AdS Black Hole ...... 61

4.2 Non-perturbative Instability of AdS Black Holes ...... 64

5 Estimating the Triple Point of Quark Gluon Plasma 66

5.1 An Introduction to Quark Gluon Plasma ...... 66

5.2 Estimating The QGP Critical Point ...... 73

5.3 Charging Up Black Holes in 5 Dimension ...... 76

5.4 Transition to Conﬁnement at Low Chemical Potential ...... 79

5.5 Stringy Instability at High Chemical Potential ...... 80

5.6 From the Critical Point to the Tripple Point ...... 88

5.7 Caveat: QCD Dual in AdS/CFT ...... 93

6 Dilaton Black Holes in Anti-de Sitter Space 94

6.1 Asymptotically Flat Spherically Symmetric Dilaton Black Holes . . . . 94

6.2 Topological Dilaton Black Holes ...... 97 CONTENTS iii

6.2.1 Seiberg-Witten Action for Flat AdS Dilaton Black Holes . . . . 98

6.3 Holography of Dilaton Black Holes in AdS ...... 102

Conclusion 103

A Penrose Diagram 105

B Black Hole Temperature: A Primer 124

Bibliography 127 Abstract

In this thesis, we attempt to review and understand the properties of topological black holes in asymptotically Anti-de Sitter (AdS) space. These black holes have the property that the horizon is an Einstein manifold of positive, zero or negative curvature. We then study Maldacena’s conjecture in string theory, called the AdS/CFT correspondence, which says that gravity in AdS bulk corresponds to conformal ﬁeld theory (CFT) deﬁned on its boundary. That is, the string theory under discussion lives not in our (3+1)- spacetime, but in the corresponding 5-dimensional AdS bulk. We study the geometry of stringy black holes in the bulk and uses it to understand the physics of quark gluon plasma. Since black holes in string theory can have scalar hair in addition to electrical charges, it is only natural that we also study stringy black holes with dilaton charge and their possible applications in AdS/CFT correspondence. Disclaimer

To keep this thesis to reasonable length, knowledge in typical ﬁrst course of general relativity, thermodynamics and some particle physics are assumed, and many results from the literature are taken for used without proof, which might upset the pure mathematicians. As a disclaimer, despite the fact that this thesis is a work done under the Department of Mathematics, I happen to agree with the Russian mathematician V. I. Arnol’d (best known for solving Hilbert’s 13th Problem in 1957) who once remarked that [1]:

It is almost impossible for me to read contemporary mathematicians who, instead of saying ‘Petya washed his hands,’ write simply: ‘There is a t1 < 0 such that the image of t1 under the natural mapping t1 7→ Petya(t1) belongs to the set of dirty hands, and a t1 < t2 ≤ 0 such that the image of t2 under the above-mentioned mapping belongs to the complement of the set deﬁned in the preceding sentence.

I have thus avoided abstract formulations in the typical deﬁnition-lemma-theorem- proof-corollary style.

The convention for our metric is not ﬁxed - most of the time the temporal coeﬃcient will be negative but not always. Anyway we will always give the explicit metric before calculations are carried out so readers will know the convention used. We will use the natural unit c = 1 = G, although sometimes we do explicitly restore them in their rightful places. Acknowledgment

I would like to take this opportunity to express my heartfelt gratitude to my supervisor Professor Brett McInnes for his invaluable guidance over all these years ever since my Honours year at the Department of Mathematics, National University of Singapore. I also appreciate his useful advise regarding life in general.

Most part of this thesis work was done in a nice oﬃce room at the Center for Math- ematics and Theoretical Physics at National Central University of Taiwan during my short visit from 17 March to 24 March 2010. I would like to thank Associate Professor Chen Chiang-Mei, Dr. Sun Jia-Rui and their research group, including Professor James M. Nester for their hospitality and various help they have rendered to me during the visit. I have enjoyed the academic discussions with them. I would also like to thank Professor Chen Pisin for his treat of a very nice dinner during my brief visit to LeCosPA (Leung Center for Cosmology and Astrophysics) at National Taiwan University, and especially for his willingness to accept me as his PhD student. CONTENTS 1

I also did part of this thesis work high up on Mt. John at Lake Tekapo in New Zealand, surrounded by colourful Lupins which contributed to the natural ﬂoral fragrance that ﬁlled the air. I am grateful to Alan Gilmore, the superintendent of Mt. John observatory for meeting me at Lake Tekapo upon my arrival and driving me up to the observatory; as well as all the friendly people at the observatory who helped me in one way or another. Here I had the best view into the starry night sky, a source of inspiration that I have always cherished. The visit to Mt. John took place during my visit to University of Canterbury to attend the 5th Australasian Conference on General Relativity and Gravitation: The Sun, the Stars, the Universe and General Relativity in December, 2009. I must thank Associate Professor David Wiltshire for his hospitality during the visit.

Last and not least, I would like to thank my beloved parents and all people in my life who in one way or another, made this thesis possible. Chapter 1

Introduction: From Black Holes to String Theory and AdS/CFT

In this chapter, I shall give a (mostly) non-technical account of the historical develop- ment of the subject.

1.1 A Brief History of Black Hole Research

Black hole, a term coined by the late John Wheeler to describe the state of gravitational collapse, has always been an intriguing subject in the ﬁeld of astronomy. Perhaps the reason that the general public are fascinated by the idea of black holes is that it sounds like science ﬁction or even fantasy, yet black holes are supposedly real objects that exist in our universe. In certain ways, black holes blur our boundaries between what is real and what is fantasy. There are many aspects that one may choose to explore when it comes to black holes, for example, one can study astrophysical black holes and their related phenomena such as gamma-ray bursts and active galactic nuclei; or one can use numerical methods to study properties of black holes merger. This thesis will not touch on either of these aspects; its approach is purely mathematical, or as Albert Einstein put it, “physics by pure thoughts”. In his own words in 1933,

I am convinced that we can discover by means of purely mathematical con- structions the concepts and the laws... which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it... In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed... 1.1 A Brief History of Black Hole Research 3

Concept similar to black hole predates Einstein’s General Theory of Relativity: Laplace [1796] and Mitchell [1783] independently speculated on the existence of stars so massive as to appear dark from observers far away since light corpuscles would not be able to escape the stars; instead they get pulled back to the surface just like a stone would fall to the surface of the Earth after being thrown into the air. Such an idea however was dismissed after Young’s double slit experiment [1801] convinced Laplace that light is a wave instead of particle (although Nature is far more mysterious than anyone would have guessed, as evident by the discovery of particle-wave duality!). It is a remarkable coincidence that the radius of such dark star in relation to the mass of the star and the speed of light is exactly the same as the Schwarzschild radius obtained using General Relativity. The Schwarzschild solution [1915] was found by Karl Schwarzschild only a few months after Einstein published his General Relativity which models gravity as eﬀect due to curvature of spacetime. This solution describes the spacetime outside of a spherical, non-rotating black hole. This was soon followed by Reissner [1916] and Nordstr¨om [1918] solution that incorporated electrical charges.

While on board a ship from India to Cambridge, England for his graduate study, Sub- rahmanyan Chandrasekhar calculated using general relativity that a non-rotating body of electron-degenerate matter above 1.44 solar masses (the Chandrasekhar limit) could not be stabilized against gravity by electron degeneracy pressure and would collapse further [1930]. Indeed it was later found that any star that exceeds the Chan- drasekhar limit but less than approximately 4 solar masses (the Tolman-Oppenheimer- Volkoff limit, or TOV limit for short) would collapse into neutron star which is stabilized by neutron degeneracy pressure. Robert Oppenheimer and Snyder [1939] finally concluded that any star exceeding the TOV limit would undergo complete gravitational collapse and forms a black hole. Oppenheimer-Snyder collapse model describes how pressureless dustball collapses under its own gravity to form a black hole. While initially met with suspicion since it is overly simplified, computer models incorporating pressure and other conditions showed that the same conclusion holds: gravitational collapse is unavoidable.

Note that the masses of the stars that end up as a black hole should in fact be much higher in astrophysical context, perhaps 25 solar masses, as stars do shed oﬀ large portion of their mass during the red supergiant and supernova stage. So to have around 4 solar masses in the end means it had a much higher mass to start with!

The misunderstanding regarding the nature of Schwarzschild radius led many to believe that time comes to a stop at the “surface” of a black hole. As such, black hole was ﬁrst called a “frozen star”, as an outside observer would see the surface of the star frozen in time at the instant where its collapse takes it inside the Schwarzschild radius. It was not until David Finkelstein introduced a more suitable coordinate system [1958] that people came to realize that the singularity at the Schwarzschild radius is merely due to bad choice of coordinate system, and that for the infalling traveler, he would reach 1.1 A Brief History of Black Hole Research 4 the black hole singularity in ﬁnite time. Shrouded by the “surface” of the black hole, observer outside of the black hole cannot see what happens to the infalling traveler, who seemed to fade out of existence as light red-shifted and time slowed down with respect to the outside world. Wolfgang Rindler then coined the term event horizon to refer to this peculiar “surface”.

Mathematician Roy Kerr found the exact solution for a rotating black hole [1963], which marked an important advancement in black hole physics, especially when we consider the fact that astrophysical bodies rotate and thus Kerr solution would be a more realistic one to model astrophysical black holes. While Reissner-Nordstr¨omsolution is not physical since any electrically charged black hole will quickly be neutralized, the structure of Reissner-Nordstr¨omblack hole is similar to that of Kerr black hole, and so is important in theoretical study because Kerr solution is quite complicated for computations.

The term black hole is ﬁnally coined by John Wheeler [1967], who was initially reluctant to accept the possible existence of black holes, but gradually accepted it. In his famous lecture Our Universe: The Known and the Unknown in December 1967, Wheeler stated:

The light is shifted to the red. It becomes dimmer millisecond by millisecond, and in less than a second is too dark to see... [The star,] like the Cheshire cat, fades from view. One leaves behind only its grin, the other, only its gravitational attraction. Gravitational attraction, yes; light, no. No more than light do any particle emerge. Moreover, light and particles incident from outside... [and] going down the black hole only add to its mass and increase its gravitational attraction.

In the same year, Jocelyn Bell Burnell discovered the ﬁrst pulsar: highly magnetized, rotating neutron star that emits a beam of electromagnetic radiation, thus promoting interests on compact astrophysical objects.

The term black hole was then adopted throughout the community, although it was resisted for a few years in France, where trou noir (black hole) has obscene connotations.

On the observational side of the story, the X-ray source called Cygnus XR-1 (also known as Cygnus X-1) was proposed to be the ﬁrst black hole candidate [1971]. More recently astronomers at the Max Planck Institute for Extraterrestrial Physics present evidence for the hypothesis that Sagittarius A* is a supermassive black hole at the center of our Milky Way galaxy [2002], where stars seem to be orbiting a compact massive object with tremendous speed.

With the solutions of black holes found, research started to focus on the properties of various black holes. Stephen Hawking found that the surface area of classical black 1.1 A Brief History of Black Hole Research 5 hole is non-decreasing [1972], which is analogous to the second law of thermodynamics where entropy is non-decreasing. In the same year, together with James Bardeen and Brandon Carter, Stephen Hawking proposed the four laws of black hole mechanics in analogy with the laws of thermodynamics:

(1) The 0-th Law: The horizon has constant surface gravity for a stationary black hole. (2) The 1-st Law: We have κ dM = dA + ΩdJ + ΦdQ 8π where M is the mass, A is the horizon area, Ω is the angular velocity, J is the angular momentum, Φ is the electrostatic potential, κ is the surface gravity and Q is the electric charge. (3) The 2nd-Law: The horizon area is, assuming the weak energy condition, a non-decreasing function of time. (4) The 3rd-Law: It is not possible to form a black hole with vanishing surface gravity.

Also in 1972, Jacob Bekenstein suggested that black holes have an entropy SBH proportional to their surface area. The Bekenstein-Hawking entropy formula reads A S = . BH 4 Surprisingly, Stephen Hawking further showed that contrary to what people had believed until then, black holes can radiate [1974]. This is a consequence of quantum ﬁeld theory applied to black hole spacetime. In fact due to what is now called Hawk- ing radiation, black holes would eventually evaporate, though the rate is very slow for astrophysical black holes.

Partly due to advancement in String Theory, black hole solutions are sought for in higher dimensional spacetime. However the ﬁrst generalization to higher dimensional analog of Schwarzschild solution was already available much earlier, due to the work of Tangherlini [1963]. Tangherlini showed that the general form

ds2 = −V 2dt2 + (V 2)−1dr2 + r2dΩ2

2m with [V (n)]2 = 1 − where m is related to the mass of the black hole and dΩ2 rn−2 being the metric for (n − 1)-dimensional sphere describes the exterior spacetime of a 1.1 A Brief History of Black Hole Research 6 non-rotating spherical black hole in (n + 1)-dimensional spacetime. Myers and Perry then generalized the Kerr metric to describe rotating black holes in (4 + 1)-dimensional spacetime [1986] [2].

Black holes in classical general relativity in (3 + 1)-dimensional spacetime enjoys a uniqueness property called No-Hair Theorem, which says that black holes are only characterized by 3 parameters: mass, angular momentum and electrical charge. The ﬁrst hint to No-Hair Theorem came in from Vitaly Lazarevich Ginzburg [1964] during his research involving quasars. The proof of the No-Hair Theorem was presented by Werner Israel [1967]. The name “No-Hair Theorem” came from the phrase “A black hole has no hair” by again, Wheeler. By “hair” he meant any property other than charge, angular momentum and mass which a black hole possesses which can reveal the details of the object before it collapsed to form black hole. Kip Throne recalled in his book “Black Holes & Time Warps: Einsteins Outrageous Legacy” that

Wheelers phrase quickly took hold, despite resistance from Simon Pasternak, the editor-in-chief of the Physical Review, the journal in which most Western black hole research is published. When Werner Israel tried to use the phrase in a technical paper in late 1969, Pasternak fired off a peremptory note that under no circumstances would he allow such obscenities in his journal. But Pasternak could not hold back for long the flood of “no-hair” papers.

As a consequence of the No-Hair Theorem, a uniqueness theorem in (3+1)-dimensional spacetime says that the only stationary and neutral black hole is the Kerr black hole (with Schwarzschild black hole as a special case). One naturally wants to know whether similar result holds in higher dimensions. The answer is no. The Myers-Perry black hole is not the unique black hole solution to Einstein Fields Equations in higher dimensions. There exists a rotating ring-shaped solution in ﬁve dimensions with the horizon topology of S2 × S1 which may have the same mass and angular momentum as the Myers-Perry solution. This is known as a black ring [Emparan-Reall, 2006] [3].

In fact, one has a large number of black objects, e.g. black string, black brane, and in the case of multi-black hole system, even black saturn [Elvang-Figueras, 2007] [4]. The topology of black holes in higher dimensions is thus much richer than the one in (3+1)-dimension, which, by one theorem of Hawking (assuming appropriate energy condition), can only be of spherical topology [1975] [12].

The interesting thing is that we can actually obtain non-spherical black holes in (3+1)- dimension, and even (2+1)-dimension, provided that the universe has negative cosmological constant (being asymptotically Anti-de Sitter). To be more speciﬁc Banados, Teitelboim and Zanelli found that with a negative cosmological constant, there can be a black hole solution in (2+1)-dimension, now called BTZ black hole [1992] [5]. If 1.2 String Theory: What is it really good for? 7 cosmological constant is nonnegative, (2+1)-dimension does not admit any black hole solution. Simply put, a topological black hole is just a black hole with non-trivial topology, such as a torus (doughnut) or even of higher genus (has more than 1 hole). This does not contradict Hawking’s theorem of spherical black holes because Anti-de Sitter space does not satisfy the dominant energy condition. We will study the detailed properties of Anti-de Sitter space in the next chapter, followed by the study of black holes in asymptotically Anti-de Sitter space in Chapter 3.

1.2 String Theory: What is it really good for?

String theory is a 21st-century physics that had fallen by chance into the 20th century. - Daniele Amati.

Our story started from the study of strong interactions. In 1968, Veneziano proposed a formula to ﬁt some of the high-energy characteristics of the strong force, resulted in the so-called dual resonance model, the details of which do not concern us here.

In 1970, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskind realize that the dual theories developed in 1968 to describe the particle spectrum also describe the quantum mechanics of oscillating strings. They proposed the early version of string theory that aimed to describe the strong interaction. Unfortunately the model doesn’t work very well. After the conception of Quantum Chromodynamics (QCD), this early form of string theory was discarded. However, it was resurrected in the 1980s as a quantized theory of gravity by John H. Schwarz and Joel Scherk, and independently, Tamiaki Yoneya. They noticed that the theory has spin 2 excitation which is massless - a possible candidate of the hypothetical graviton - carrier of gravitational force.

The early string theory only describes bosons in their excitation spectrum, and is now known explicitly as “bosonic string theory”. Curiously bosonic string theory requires 26 spacetime dimension to be consistent. While the idea of extra dimension is not new (Kaluza-Klein theory in 1921 already proposed an extra spatial dimension in the attempt to formulate electromagnetic force as spacetime curvature). It is later found that fermions can be included into the theory by introducing supersymmetry, and string theory, as of 1980, becomes superstring theory. In most contexts nowadays, string theory means superstring theory unless otherwise stated. Since supersymmetry requires equal number of fermions and bosons, it is not an exact symmetry of Nature, but instead a broken one. Strings can be open with two ends or closed as a loop. In string theory, particles are now identiﬁed as a particular vibrational mode of an elementary microscopic string - an extreme reductionist approach! 1.2 String Theory: What is it really good for? 8

In 1984, the First Superstring Revolution was started by a discovery of anomaly can- cellation by Michael Green and John H. Schwarz in 1984. In a crude manner of speaking, they showed that certain seemingly threatening features that could render the theory inconsistent are in fact treatable. The anomalies cancel out in the three known types of superstring theory. In 1985, two more cases where the anomalies cancel out have been found and studied, giving rise to the heterotic strings which hold great promise for describing the standard model. There are now 5 seemingly distinct string theories, called Type I, Type IIA, Type IIB, Heterotic SO(32), and Heterotic E8 × E8. Again, the details should not concern us.

Anyway, this was a great concern because if string theory is to be a theory of everything that uniﬁes gravity with the other three fundamental forces, you don’t want to have 5 diﬀerent theories of everything while we only have one universe!

Starting in 1995, Edward Witten led the Second Superstring Revolution. It was discovered that the seemingly diﬀerent superstring theories are related by certain dualities and are just diﬀerent limits of a 11-dimensional theory called M-theory, although no one seems to know what the “M” really stands for - it ranges from Mother, Matrix, Mystery, Magic, Membrane to, jokingly, an inverted “W” that stands for none other than Witten himself.

Within this new unifying framework, new objects called branes were discovered as inevitable ingredients of string theory. An n-brane, short for n-dimensional membrane is extended object in string theory: strings themselves are one-dimensional object and is thus a 1-brane. A special type of brane called the D-brane allows open strings to attach their two ends on it with Dirichlet boundary conditions. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Horava in 1989.

In 1997, approaching the physics of black holes with the powerful mathematical tools of superstring theory, Juan Maldacena proposed the idea that is now known as AdS/CFT correspondence or the holographic principle in which he claimed that there is a deep relationship between pure non-gravitational theories and superstring theories [6]. The AdS/CFT correspondence says that string theory defined in Anti-de Sitter space (AdS) is equivalent to a certain conformal field theory (CFT, to be explained in more details later) defined on its boundary. The term correspondence was first used by Edward Witten when he elaborated on the idea in his classic 1998 paper.[7]

This is what String Theory is good for as of now - not as a theory of everything, but to probe high energy strongly coupled systems that otherwise remain outside our reach. It does not matter whether what we call particles in our own universe are made of tiny wiggling strings or not - the strings in AdS/CFT live in 5-dimensional AdS bulk, if you prefer you can think of this as a mathematical trick in the following sense: What we are interested in is to solve problems involving certain ﬁeld theory in our universe, AdS/CFT allows us to translate this problem to a gravitational theory in the 1.2 String Theory: What is it really good for? 9

5-dimensional AdS bulk which is easier to solve. We then translate our result back to the ﬁeld theory. This is very much like going over to the Fourier transfrom space to solve problems.

In Chapter 5 and Chapter 6, we will look at some applications of AdS/CFT, in particular, how the gravitational theory of topological black hole in the 5-dimensional AdS bulk can tell us something about the physics of quark gluon plasma living on the boundary (i.e. in our own universe!). To do so, we need to understand stability issues of black holes, which we will explore in Chapter 4. Chapter 2

From de Sitter Space to Anti-de Sitter Space and Conformal Field Theory

In a nutshell, we can think of Anti-de Sitter space as an emtpy universe (with neither matter nor radiation) with negative cosmological constant. It is pedagogical to ﬁrst review brieﬂy the concept of cosmological constant, and some properties of de Sitter space that arised out of cosmology.

2.1 From Einstein’s Static Universe to the de Sitter Solution

Consider a general spherical symmetric metric in (3+1) dimension,

ds2 = e2A(r)dt2 − e2B(r)dr2 − r2dΩ2.

Working through the standard but tedious steps, we obtain two Einstein’s Field Equa- tions:

2B0 1 1 − e−2B + = −8πρ (2.1) r r2 r2

2A0 1 1 e−2B + − = −8πp (2.2) r r2 r2 2.1 From Einstein’s Static Universe to the de Sitter Solution 11 where p and ρ denote the pressure and density of the ﬂuid described by the stress energy tensor Tµν = (ρ + p) uµuν − pgµν and prime denotes derivative with respect to r.

µν The general relativistic conservation of energy ∇νT = 0 yields, in particular,

1ν −2B 0 0 −2B 0 −2B ∇νT = e p + A e ρ + A e p = 0.

That is, p0 = −A0(ρ + p).

The requirement that the universe be isotropic and homogeneous means that the energy density and pressure are uniform throughout space, i.e. p0 = 0. Furthermore Einstein, like most people during his days, believed that the universe should be static, and so p and ρ is a constant both in space and in time.

Thus the previous equation reads

− A0(ρ + p) = 0 (2.3)

Because the density of matter and the radiation pressure are presumably positive, Einstein chose A0 = 0, i.e. A is constant. Homogeneity implies that the coeﬃcient of dt2 must be a constant, which we normalized to c2, which is 1 in our choice of unit. Thus e2A = 1. Also, the second ﬁeld equation then becomes:

2A0 1 1 e−2B + − = −8πp r r2 r2 1 1 ⇒ e−2B − = −8πp r2 r2 1 ⇒ e−2B − 1 = −8πp r2 2 −2B 2 r ⇒ e = 1 − 8πpr = 1 − 2 . aE

2 −1 where aE = (8πGp) .

The metric now takes the form dr2 ds2 = dt2 − − r2dΩ2. r2 1 − 2 aE 2.1 From Einstein’s Static Universe to the de Sitter Solution 12

Re-scale by

r → r. aE

We have the metric that describes the Einstein’s Static Universe

dr2 ds2 = dt2 − a2 + r2(dθ2 + sin2 θdφ2) E 1 − r2

2 with spatial part describing a 3-sphere with radius aE.

Einstein was looking for a more or less realistic model of cosmology. However this solution is not suitable because if we were to work out the Einstein’s Field Equatiosn from the metric, we will get  1 3 R0 − R = −  0 2 2 aE 1 1 2 1 3 1 1 R1 − R = R2 − R = R3 − R = − 2 2 2 2 aE

Now using dust approximation for the stress-energy tensor

( 0 T0 = ρ 1 2 3 T1 = T2 = T3 = 0 to describe matter dominated epoch, we will find that  3 − = −8ρ  a2  E  1 − = 0  2 aE which implies that matter density is zero - the universe is empty! The way to fix this is to introduce what is dubbed cosmological constant Λ into the Einstein’s Field Equations, which is also known as Einstein’s Greatest Blunder. The modified Einstein’s Fields Equations with full glory of G and c restored take the form:

1 8πG R − g R + g Λ = − T . µν 2 µν µν c4 µν 2.1 From Einstein’s Static Universe to the de Sitter Solution 13

Note that our current choice of sign for metric (−, +, +, +) is the reason for the negative sign on the right hand side of Einstein’s Field Equations.

In doing so the calculation above gets modiﬁed to  3 − + Λ = −8ρ  a2  E  1 − = −Λ  2 aE which is equivalent to saying 1 a2 = . E 4πGρ

Thus introduction of cosmological constant into the theory allows for non-empty, matter- dominated static universe. Alas, after Edwin Hubble’s observation that remote galaxies seem to move away from us and thus the universe is expanding instead of static, Ein- stein abandoned the cosmological constant, and claimed that it is the greatest blunder of his life. Einstein’s mistake, however, was not the introduction of cosmological constant. Instead, the most general form of Einstein’s Field Equations should contain the cosmological constant - whether or not the value is nonzero is to be determined by observational data. Hence Einstein was too quick to claim that cosmological constant is a blunder, for modern cosmology does require small value of cosmological constant to account for the acceleration of the universe.

Mathematically, equation 2.3 has another solution corresponding to ρ + p = 0. For realistic matter, this means ρ = p = 0. Without cosmological constant this would mean the universe is utterly devoid of matter and radiation and so has zero curvature - it must be a Minkowski space. However, the cosmological constant allows for interesting feature - curvature without matter and energy content. Putting the Λ term on the left hand side of the equation, cosmological constant becomes a property of spacetime itself independent of matter and energy.

Adding Equation 2.1 and Equation 2.2 we have 2 −8π(p + ρ) = e−2B (A0 + B0) r With ρ = p = 0 this implies A = −B + const., and by requiring that locally the metric reduces to that of special relativity, the constant term must vanish. Equation 2.1 and Equation 2.2 are now equivalent, and with cosmological constant, they become:

−2B0 1 1 e−2B + − + Λ = 0. r r2 r2 2.1 From Einstein’s Static Universe to the de Sitter Solution 14

We can check that this is satisﬁed by B = B(r) such that Λ e−2B = 1 − r2. 3

This yields the de Sitter Space

Λ dr2 ds2 = 1 − r2 dt2 − + r2dΩ2 (2.4) 3 Λ 2 1 − 3 r

3H2 or equivalently, with Λ ≡ c2 , and restoring c, H2R2 dr2 ds2 = c2 1 − dt2 − − r2(dθ2 + sin2 θdφ2). c2 H2R2 1 − c2 The reason for introducing H seemingly out of nowhere will become clear shortly.

Now we re-write t → T and r → R to avoid confusion, and apply change of variables

(R2 = e2Htr2 h H2r2e2Ht−c2 i T = t − ln − 2H . A straightforward albeit tedious computation enables us to re-write the de Sitter metric as the inﬂationary ﬂat de Sitter universe

ds2 = c2dt2 − e2Ht dr2 + r2(dθ2 + sin2 θdφ2) (2.5)

This describes a spatially ﬂat universe with time-dependent scale factor a(t) = eHt, i.e. a universe that expands at exponential rate. Indeed, let us write a = eγt where γ are constants, instead of eHt. Then diﬀerentiating with respect to t, which we will consistently denoted by an overdot, we get

a˙ = γeγt = γa.

That is, a˙ = γ. a Buta/a ˙ is the deﬁnition for Hubble parameter, hence the choice of our symbol H. In general, Hubble parameter is time-dependent, but for de Sitter cosmology, it truly is a constant.

The event horizon of the de Sitter universe is ﬁxed. It can be obtained by setting recession velocity as the speed of light c in the Hubble law v = HD, which yield 2.2 Geometry of de Sitter Space 15

D = c/H. As the galaxies are carried by the Hubble flow and move out of the fixed horizon, we will see less and less things in our observable universe. We sometimes define 3 Λ = l2 so that with our previous definition 3H2 Λ = c2 we get c l = H to be the length scale of the de Sitter universe.

2.2 Geometry of de Sitter Space

It is very important to realize that while we can deﬁne coordinate transformation to get Equation 2.5 from Equation 2.4, they are not to be thought as equivalent, which is why I have chosen to call them by diﬀerent names. This will be important fact to take note of when we study physics on either de Sitter or Anti-de Sitter space, and we will again point this out to the reader when the time comes. For now, let us take a detailed look at why the two metrics are not exactly the same thing.

Mathematically, the de Sitter manifold can be deﬁned as follow: Consider a 5-dimensional Minkowski space M4+1. The de Sitter manifold is the hypersurface deﬁned by 0 1 2 3 4 4+1 2 2 2 3 4 2 0 dS4 = x = (x , x , x , x , x ) ∈ M : −x0 + x1 + x2 + x3 + x4 = l , x = t. which is a hyperboloid in the 5-dimensional Minkowski space.

In general, a d-dimensional de Sitter space can be embedded in a ﬂat (d+1)-dimensional spacetime. The global coordinates are given by the following [8]:  0 t x = l sinh  l  t xi = lωi cosh , i = 1, ..., d.  l where −∞ < t < +∞ and ωi’s for the spatial sections of constant t satisfy

d X (ωi)2 = 1. i=1 2.2 Geometry of de Sitter Space 16

i Indeed, ω ’s are related to the angle parameters θi:  ω1 = cos θ ,  1  2 ω = sin θ1 cos θ2, . . d−2 ω = sin θ1 cos θ2 ··· sin θd−3 cos θd−2  ωd−1 = sin θ cos θ ··· sin θ cos θ  1 2 d−2 d−1  d ω = sin θ1 cos θ2 ··· sin θd−2 sin θd−1

where θ1, θ2, ..., θd−2 ∈ [0, π) and θd−1 ∈ [0, 2π).

2 2 2 2 2 Inserting this into the Minkowski metric ds = −dt + dx1 + dx2 + ...dxd we get

t ds2 = −dt2 + l2 cosh2 dΩ2 l d−1 where

2 2 2 2 2 2 2 dΩd−1 = dθ1 + sin θ1dθ2 + ··· + sin θ1 ··· sin θd−2dθd−1 d−1 j−1 ! X Y 2 2 = sin θi dθj . j=1 i=1

This coordinate system covers the entire de Sitter manifold (except for trivial coordinate singularities at θi = 0 due to the use of polar coordinates). At any fixed time t, the spatial section of de Sitter manifold is that of a (d − 1)-dimensional sphere of radius l cosh(t/l) - which is hence compact. The size of the sphere started out infinite in the infinitely old past and gradually contracted to a minimum size before expanding again until it becomes infinite size as t → ∞. de Sitter original solution Λ dr2 ds2 = 1 − r2 dt2 − + r2dΩ2 3 Λ 2 1 − 3 r employs the static coordinate system (t, r, θa), a = 1, 2, ..., d − 2. The parameter r, satisfying 0 ≤ r < ∞ and r ≤ l, allows us to decompose the hyperboloid equation into q r 2 two constraints: a 2-dimensional hyperbola of radius 1 − l described by 2 x0 2 xd r2 − + = 1 − l l l 2.2 Geometry of de Sitter Space 17

Figure 2.1: d-dimensional hyperboloid illustrating de Sitter spacetime embedded in (d + 1)-dimensions. Diagram modified from [8].

r and a (d − 2)-dimensional sphere of radius l described by 2 x1 2 xd−1 r2 + ... + = . l l l Summing up these two equations give us the hyperboloid equation that we started with.

These equations are satisﬁed by r x0 r2 t = − 1 − sinh , l l l xi r = ωi, l l r xd r2 t = − 1 − cosh , l l l 2.2 Geometry of de Sitter Space 18

i where the ω are related to the d − 1 angle variables θi as before.

One can check that these coordinate transformation converts the (d + 1)-dimensional Minkowski metric into r2 dr2 ds2 = − 1 − dt2 + + r2dΩ2 , l r 2 d−2 1 − l where

2 2 2 2 2 2 2 dΩd−2 = dθ1 + sin dθ2 + ... + sin θ1 ··· sin θd−3dθd−2 d−2 b−1 ! X Y 2 2 = sin θa dθb b=1 a=1 is the usual spherical metric on Sd−2.

Note that there exist a cosmological horizon at r = l which corresponds to the vanishing r 2 of the term 1 − l . √ √ One also notes that −x0 + xd = − l2 − r2e−t/l ≤ 0 and x0 + xd = − l2 − r2et/l ≤ 0 1 and so the region r ≤ l only covers 4 of the whole de Sitter space. To draw the Penrose + − diagram, we ﬁrst switch to Eddington-Finkelstein coordinates (x , x , θα) deﬁned by r ± l 1 + l x = t ± ln r , 2 1 − l where the range of x± is (−∞, +∞). This transforms the metric into the following form: x+ − x− x+ − x− ds2 = − sech2 dx+dx− + l2 tanh2 dΩ2 2l 2l d−2 which covers the whole de Sitter space.

We now transform to Kruskal coordinates (U, V ) by introducing

x− U := −e l

−x+ V := e l , which further converts the metric into l2 ds2 = −4dUdV + (1 + UV )2dΩ2 . (1 − UV )2 d−2

Now UV = −e(x−−x+)/l. So we have 1 + UV 1 − e(x−−x+)/l r = = , 1 − UV 1 + e(x−−x+)/l l 2.2 Geometry of de Sitter Space 19

x− − x+ l + r by using the fact that = − ln . l l − r

From this, we can see that at the origin r = 0, we have UV = −1. On the other hand, at the cosmological horizon r = l, we have UV = 0, corresponding to the axis of either constant U or constant V . Finally, at spatial inﬁnity, UV = 1. This gives us the corresponding Kruskal diagram, of which after suitable conformal transformation, gives the following Penrose diagram (See Appendix for introduction to Penrose diagram).

Figure 2.2: Penrose diagram for Kruskal extension of de Sitter space covered by the original static coordinate.

Note that U = 0 corresponds to past inﬁnity t = −∞ while V = 0 corresponds to the future inﬁnity t = ∞. The topology of d-dimensional de Sitter space is Sd−1 × R. In dS4, as usual, generic points in the Penrose diagram represent 2-spheres, except for the points on the left and right edge of the square, which represent poles of the 3-spheres and hence are points. Now, an observer O at r = 0 is surrounded by cosmological horizon at r = l. This should not come as surprise since the original static coordinates covers only a quarter of the de Sitter space, which corresponds to the right triangle in the Penrose diagram. Regions III and IV are events which O will never be able to observe. Thus de Sitter space, unlike Minkowski space, has the property that even if you wait for eternity, there are events that you will not be able to observe.

The inflationary flat de Sitter solution employs the planar (inflationary) coordi- 2.2 Geometry of de Sitter Space 20 nates related to the Kruskal coordinates by

 r − t − e l  l U =  2  2 V =  t r  − l e + l

r 1 r r Note that V > 0. Also, l = U + V . We see that UV = −1 when either l = 0 or l = ∞ t 1 and V = 0. Furthermore, l = − ln V − U , so past inﬁnity t = −∞ has V = 0 i.e. this corresponds to the diagonal line in the Penrose diagram. Future inﬁntity t = ∞ 1 imples V −U = 0 or equivalently, UV = 1 so it corresponds to horizontal line at the top of the Penrose diagram. The planar coordinates cover half of the de Sitter manifold:

Figure 2.3: Penrose diagram for de Sitter space covered by planar coordinates. Diagram taken from [8].

Indeed, due to the maximal symmetry (for introduction to maximally symmetric spacetime, see p.139 of [9].) and the topology of the de Sitter manifold, all three possible FLRW cosmologies can be realized on the space by suitable choices of the coordinate 2.2 Geometry of de Sitter Space 21 systems (Fig. 2.4). However, we will not go into further details. The point is this: while different coordinate systems are good for different applications, we need to be careful that a given coordinate system may also be misleading. For example, the global coordinate and static coordinate show the true curvature of de Sitter space, which is positively curved. The inflationary coordinate on the other hand suggests that the spatial section is flat, and worse, there is coordinate system that suggests the spatial section to be negatively curved corresponding to the FLRW hyperbolic open universe. In our subsequent application of AdS/CFT, the global feature of AdS space becomes important, and one should not be misled by coordinates.

Figure 2.4: Different coordinates on the de Sitter manifold. Black curves represent hypersurfaces of constant cosmological time, while blue curves are timelike geodesics. Diagram taken from [10]. 2.3 AdS: Anti-de Sitter Space 22

2.3 AdS: Anti-de Sitter Space

A d-dimensional anti-de Sitter space, denoted by AdSd, is a maximally symmetric spacetime with constant negative curvature. Unlike the de Sitter case, AdS space corresponds to solution of Einstein Field Equations with negative cosmological constant, i.e. Λ < 0. The metric for a d-dimensional anti-de Sitter space can be obtained by embedding a (d + 1)-dimensional hyperboloid in a ﬂat (d + 1)-dimensional space with two time directions. I.e We can take AdS space as the hypersurface −(X0)2 + (X1)2 + ... + (Xd−1)2 − (Xd)2 = −l2. in semi-Riemannian manifold with coordinates (X0,X1, ..., Xd) where X0 and Xd are the time coordinates.

In similar way as we did for dS space, we can describe AdS using static coordinates and ended up with the following metric r2 dr2 ds2 = − 1 + dt2 + + r2dΩ2 , l r 2 d−2 1 + l where r X0 r2 t = 1 + cos , l l l Xi r = ωi, l l r Xd r2 t = 1 + sin , l l l with ωi, 1 ≤ i ≤ d − 1 deﬁned as before in dS case. Unlike dS case though, the static coordinate covers the entire spacetime.

Note that unlike dS space, the coeﬃcient of dr2 is regular for all r, so there is no ± r cosmological horizon. Using new coordinates deﬁned by x = t ± l arctan l , we can re-write the metric as

x+ − x− x+ − x− ds2 = − sec2 dx+dx− + l2 tan2 dΩ2 . 2l 2l d−2

We then proceed to do coordinate transformations in ways similarly to de Sitter case, which will ﬁnally give us the Penrose diagram for anti-de Sitter space.

So we see a peculiar behavior for light rays in anti-de Sitter space: a light ray can travel from the center (r = 0) to infinity, and bounced back in finite proper time of an observer in the center. 2.4 Conformal Compactification 23

Figure 2.5: Penrose diagram for AdS. Dotted curves denoted timelike geodesics, while red lines are null geodesics.

d−1 1 We note that the spacetime topology of AdSd is R × S . It admits closed timelike curves (CTCs) due to time having topology S1. Some physicists are uncomfortable with this potential acausality and speak of passing to the universal covering spacetime AdS]d instead. See for example, p.131 of [12]. That is to say, one unwraps the S1 representing time coordinate into its covering space R. And by “Anti-de Sitter” space, one actually means its universal cover AdS]d. See, however, [13] for more detailed discussion why nothing new is gained by doing so, and why the “demon of acausality” remains not exorcised.

2.4 Conformal Compactiﬁcation

In the appendix, we show that Minkowski metric can be transformed into

ds2 = Ω−2(T,R) −dT 2 + dR2 + sin2 R(dθ2 + sin2 θdφ2) ,

Thus we see that Minkowski metric ds2 is conformally related to ds˜2 by ds˜ 2 = Ω2(T,R)ds2 = −dT 2 +dR2 +sin2 R(dθ2 +sin2 θdφ2), with ranges given by 0 ≤ R < π and −π < T < π. The spatial part of this metric is a three-sphere with constant curvature. The universal cover of the conformal compactiﬁcation of Minkowski spacetime is thus the Einstein ∼ 3 Static Universe ESU4 = S × R, with 0 ≤ R ≤ π and −∞ < T < +∞. That is to say, Minkowski spacetime is conformally mapped into a subspace of ESU4. If we represent ESU4 as a cylinder where time runs vertically and each circle of constant time 2.4 Conformal Compactiﬁcation 24 represents a 3-sphere, then we can map Minkowski space to a portion of the cylinder.

Figure 2.6: The embedding of Minkowski spacetime into Eistein static universe protrayed as a portion of an infinite cylinder.

In the cylinder, Minkowski spacetime is the interior of the region bounded by the red curves. The boundaries themselves are not part of the original Minkowski spacetime, We call such boundary conformal infinity or conformal boundary. The union of the original spacetime with its conformal infinity is a manifold with boundary which we call the conformal compactification. We now perform analogous construction on AdSd:

Starting from the metric

r2 dr2 ds2 = − 1 + dt2 + + r2dΩ2 , l r 2 d−2 1 + l We have 2.5 Conformal Field Theory 25

  r2 dr2 r2 2  2 2  ds = 1 + −dt + 2 + 2 dΩd−2 . l  h r 2i h r 2i  1 + l 1 + l

Introduce new coordinate variable Z dr r ω = = tan−1 . r 2 l 1 + l

This transforms the metric into the form ω h ω i ds2 = sec2 −dt2 + dω2 + l2 sin2 dΩ2 l l d−2 which is manifestly also mapped into the Einstein cylinder. Note that spatial inﬁnity lπ r = ∞ corresponds to ω = 2 .

l Thus AdS also doesn’t map into the entire cylinder, but rather 2 of the cylinder. For π example, if l = 1, then it covers half of the cylinder, with ω runs from 0 to 2 instead of full rotation π. Well, really, to get to AdSd instead of its universal cover one still needs to identify the time coordinate modulo 2π.

2.5 Conformal Field Theory

d(d+1) The Anti-de Sitter space is a space of maximal symmetry, that is, AdSd has 2 symmetry transformations. Thus AdS5 has 15 symmetry transformations. In Malda- cena’s conjecture, gravity in AdS5 is dual to a Yang-Mills theory in the usual (3 + 1)- dimensional space, which has 10 symmetries (6 Lorentz transformations and 4 spacetime translations). Therefore not all Yang-Mills theory in (3+1)-Minkowski space is dual to AdS5, only certain Yang-Mills theories with additional symmetry contraints are possible. In turns out that such additional symmetries are the conformal symmetries which are symmetries under the conformal transformations dilatation and inversion, i.e. xµ xµ → λxµ and xµ → x2 respectively.

For introduction to Yang-Mills theory, see for example, [14].

Any quantum field theory that is invariant under the conformal transformations is called a conformal field theory. Any theory that is invariant under dilatation is said to 2.5 Conformal Field Theory 26 be scale invariant. Note that under inversion, the origin is mapped to infinity and vice versa.

The Yang-Mills theory, in addition, is invariant with respect to supersymmetry which pairs integer-spin particles with half-integer-spin particles. The gauge symmetry of the theory is SU(N). For the correspondence to be useful, N must be suﬃciently large. This can be seen as follows [15] [73]:

2 The ’t Hooft Coupling λ = gYMN where gYM denotes the Yang-Mills coupling, deter- mines the interaction strength of the ﬁeld theory. The local strength of gravity in the AdS bulk is determined by the curvature with characteristic length scale lc: smaller value of lc corresponds to greater curvature. Since the AdS bulk is actually string theoretical, there is another length scale ls related to the string: it is inversely related to the string tension. The ’t Hooft coupling of the boundary Yang-Mills theory satisﬁes 4 λ ∝ (lc/ls) . If ls lc, the strings are weakly coupled in the bulk. Correspondingly, the interactions in the Yang-Mills theory is strongly coupled. Conversely, weak ’t Hooft coupling in the Yang-Mills theory corresponds to strong string coupling in the bulk, and the dual gravitational theory will require full non-perturbative stringy calculations. It turns out that the Yang-Mills coupling is proportional to the string coupling, so if string coupling were to be small (for perturbative method to be useful), N must be large.

We quote the following result without rigorous proof:

A d-spatial dimensional Euclidean quantum ﬁeld theory is dual to (d+1)-spatial dimensional hyperbolic space; while a d-dimensional quantum ﬁeld theory is dual to (d + 1)-dimensional Anti-de Sitter space.

For a reason why this is true, note that we have shown above that the conformal ∼ d−2 boundary of AdSd is essentially (part of) ESUd−1 = S × R, so for example, a 4- dimensional quantum field theory defined on S3 × R is dual to gravity in 5-dimensional Anti-de Sitter space. To get the Euclidean version of the correspondece, we simply Wick-rotate the time coordinate into imaginary time t → τ = it which is a standard procedure of analytic continuation in quantum field theory.

But how do we see that the Wick-rotated version of Anti-de Sitter space is just the hyperbolic space? Armed with basic knowledge of 2-dimensional hyperbolic geometry (See the following box on “Some Basic Facts in Hyperbolic Geometry”), we shall consider the 3-dimensional case (higher dimension is similarly constructed). 2.5 Conformal Field Theory 27

Some Basic Facts in Hyperbolic Geometry

There are a few ways that we can study the 2-dimensional hyperbolic space, one is by using the upper half-plane model H2 = {(x, y)|y > 0}. The Upper Half Plane Model is the hyperbolic plane, as much as anything can be, but we call it a model of the hyperbolic plane because any surface isometric to H2 is equally entitled to the name. [19].

The H2 model has metric

2 2 2 2 dx + dy |dz| ds 2 = = . H y2 (Im(z))2

where we write dz = dx + idy, and so dx2 + dy2 = (dx + idy)(dx − idy) = dzdz = |dz|2. Note that the angles in H2 always look like its Euclidean counterparts: the infinitesimal distance ds = pdx2 + dy2/y is simply the Euclidean infinitesimal distance pdx2 + dy2 scaled by 1/y. Angles are ratios of side lengths of infinitesimal triangles, which are therefore the same since the scaling factor cancelled out. Geodesics on H2 turned out to be semicircles orthogonal to the real axis and the upper half lines Re(z) = const. (Recall from Complex Analysis that lines are degenerate circles).

In other words the geodesics are of the form (x − b)2 + y2 = r2 where for r = ∞ the equation corresponds to Euclidean line.

The other widely used model of hyperbolic plane is the Poincar´edisk model D2, also called the conformal disk model. It is the unit open disk {z ∈ C||z| < 1} in which geodesics are arcs of circle whose ends are perpendicular to the disk’s boundary and the diameters of the disk (corresponding to the degenerate circles). One can obtain D2 from H2 by well known M¨obiustransformation from Complex Analysis.

The metric on D2 is given by

2 2 2 2 dx + dy |dz| ds 2 = = D (1 − x2 − y2)2 (1 − |z|2)2.

As with H2, the D2-angles are also the same as Euclidean angles. This follows from the fact that the mapping from H2 to D2 is angle-preserving by properties of M¨obius transformation. This is why we call this model conformal disk model. In fact, there are other disk models for hyperbolic space which are not conformal.

The hyperbolic space H3 can be obtained from the usual stereographic projection of the hyperboloid (of two sheets) X2 + Y 2 + Z2 − U 2 = −1 from the point (0, 0, 0, −1) 2.5 Conformal Field Theory 28 to the hyperplane at U = 0 ([20]). The coordinates on the hyperplane is given by X Y Z x = , y = , z = . U + 1 U + 1 U + 1 Also, X2 + Y 2 + Z2 −1 + U 2 −1 + U r2 ≡ x2 + y2 + z2 = = = . (1 + U)2 (1 + U)2 1 + U

If we solve X,Y,Z and U in terms of the hyperplane coordinates instead, we get 2x 2y 2z 1 + r2 X = ,Y = ,Z = ,U = . 1 − r2 1 − r2 1 − r2 1 − r2

The intrinsic metric on the hyperbolic space is then 4 ds2 = (dx2 + dy2 + dz2), r < 1. (1 − ρ2)2

To be more specific this is the metric of hyperbolic space in the Poincaréball model, a direct generalization of the Poincarédisk in 2 dimensions: dx2 + dy2 ds2 = . D (1 − x2 − y2)2

Similarly, consider say, 4-dimensional Anti-de Sitter space as a quadric X2 + Y 2 + Z2 − U 2 − V 2 = −1 where U and V are the “temporal” directions.

We can also deﬁne stereographic projection on AdS space from say, the point (0, 0, 0, 0, −1) onto the hyperplane at V = 0: 2x 2y 2z 2u 1 + r2 X = ,Y = ,Z = ,U = ,V = 1 − r2 1 − r2 1 − r2 1 − r2 1 − r2 where r2 = x2 + y2 + z2 − u2 < 1.

We then obtain the intrinsic metric of AdS4 as 4 ds2 = (−du2 + dx2 + dy2 + dz2). AdS4 (1 − r2)2 Thus we see that Wick-rotating the time coordinate to the Euclidean version yield

2 4 2 2 2 2 ds 4 = (du + dx + dy + dz ) H (1 − r2)2 the Poincar´eball form of hyperbolic 4-space. 2.5 Conformal Field Theory 29

Readers may enjoy The Hyperbolic Chamber at

http://www.josleys.com/article show.php?id=83, a website that gives a good attempt to visualize how does it feel like to live in the hyperbolic space. Chapter 3

Black Holes in de Sitter and Anti-de Sitter Space

The Kottler metric in (3+1)-dimension is given by

2m Λr2 2m Λr2 −1 ds2 = − 1 − − dt2 + 1 − − dr2 + r2 dθ2 + sin2 θdφ2 r 3 r 3 which is a generalization of the Schwarzschild metric (Λ = 0). It is the unique spherically symmetric solution of Einstein’s vacuum ﬁeld equation with a cosmological constant Λ. It is also known as the Schwarzschild-de Sitter metric for the case Λ > 0 and the Schwarzschild-Anti-de Sitter metric for Λ < 0. The Kottler metric was found independently by F. Kottler in 1918 and by H. Weyl in 1919.

3.1 Black Holes with Cosmological Constant

The generalized Kottler metric in d-dimensional spacetime with coordinate labelled by xµ = (t, r, xi) where 1 ≤ i ≤ d − 2 is given by

2 2 −1 2 2 i j ds = −f(r)dt + [f(r)] dr + r hij(x)dx dx (3.1)

ω m r2 16πG where f(r) = k − d ± , for which k = −1, 0, +1, and ω = , rd−3 l2 d (d − 2)Vol(M d−2) 3.1 Black Holes with Cosmological Constant 31

√ Vol(M d−2) = R dd−2x h; while l is a length scale (radius of curvature) related to the (d − 1)(d − 2) cosmological constant Λ := ∓ . The notation follows [21]. 2l2

r2 The ± sign in front of the term l2 and the ∓ sign of the cosmological constant depends on whether the black hole is asymptotically Anti-de Sitter (AdS) or asymptotically de 2 i j Sitter (dS), respectively. The horizon metric dσ = hij(x)dx dx describes a constant curvature Einstein submanifold with scalar curvature k = −1, 0, +1 which correspond to negatively curved, ﬂat, and positively curved compact submanifolds, respectively. In fact, as we shall see, the Ricci curvature of the horizon satisﬁes Rij(h) = k(d − 3)hij.

The coeﬃcient of dr2 being [f(r)]−1 is not a wild assumption; instead, we can derive it.

The generalized Kottler metric describes a static, spherically symmetric spacetime exterior of the black hole. Thus the general form of the metric should be

2 2 2 2 i j ds = −f(r)dt + g(r)dr + r hijdx dx for some functions f(r) and g(r).

1 We will show that g(r) = f(r) . The derivation is similar to the derivation of the well known Schwarzschild metric.

We start with the Lagrangian

1 1 n o L(x˙σ, xσ) := g (xσ)x˙µx˙ν = −f(r)t˙2 + [f −1(r)]r ˙2 + r2h (x)x˙ix˙j . 2 µν 2 ij

Here we use dot to denote diﬀerentiation with respect to t and prime to denote diﬀer- entiation with respect to r.

From the Lagrangian, we obtain the Euler-Lagrange equations: ∂L  = 0  ∂t

∂L  = −tf˙ (r). ∂t˙ and 3.1 Black Holes with Cosmological Constant 32

∂L 1 g0 0 ˙2 2 ˙i ˙j  = − f (r)t + r˙ + rhijx x  ∂r 2 2

∂L  = g(r)r. ˙ ∂r˙ From the ﬁrst set of Euler-Lagrange equations, we then obtain the geodesic equation

d ∂L ∂L − = 0 ⇒ r˙tf˙ 0(r) + f(r)t¨= 0. du ∂t˙ ∂t

I.e.

f 0 t¨+ r˙t˙ = 0. (3.2) f

This gives us the Christoﬀel symbols:

1 f 0 Γt = Γt = . rt tr 2 f

From the second set of Euler-Lagrange equations, we get the second geodesic equation given by

1 g0(r ˙)2 g(r)¨r + g0(r ˙)2 + f 0(r)t˙2 − − rh x˙ ix˙ j = 0. 2 2 ij

I.e.

g0 1 f 0 g0(r ˙)2 rh r¨ + r˙2 + t˙2 − − ij x˙ ix˙ j = 0. (3.3) g 2 g 2g g

This gives the following Christoﬀel symbols:

1 f 0 Γr = tt 2 f 2g0 g0 g0 Γr = − = rr 2g 2g 2g rh Γk = − ij . ij g 3.1 Black Holes with Cosmological Constant 33

We proceed to compute the Riemann curvature tensor:

t t t t f t f Rrtr = ∂tΓrr − ∂rΓtr + Γtf Γrr − Γrf Γtr t t r t t = −∂rΓtr + ΓtrΓrr − ΓrtΓtr 1 f 0 1 f 0 1 g0 1 f 0 2 = −∂ + − r 2 f 2 f 2 g 2 f 1 ff 00 − f 0f 0 1 f 0g0 1 (f 0)2 = − + − 2 f 2 4 fg 4 f 2 1 f 00 (f 0)2 1 (f 0)g0 1 (f 0)2 = − + + − 2 f 2f 2 4 fg 4 f 2 1 f 00 1 (f 0)2 1 f 0g0 = − + + . 2 f 4 f 2 4 fg

t t So Rtrtr = gttRrtt = −fRrtr which gives the pair

 1 f 00 1 (f 0)2 1 f 0g0 Rt = − + +  rtr 2  2 f 4 f 4 fg (3.4)  1 1 (f 0)2 1 f 0g0 R = f 00 − − .  trtr 2 4 f 4 g

Similarly,

1 f 0 rh Rt = Γt Γr = − ij . itj tr ji 2 f g

So we obtain the pair

 1 f 0 rh Rt = − ij  itj  2 f g (3.5) 0  1 rhijf R = .  titj 2 g 3.1 Black Holes with Cosmological Constant 34

Also, with a ranges over the {xi}, i.e a = 1, ..., d − 2,

r r r r a r a Rirj = ∂rΓji − ∂jΓri +ΓraΓji − ΓjaΓri | {z } =0 h rh g0 = − ij + ij + Γr Γr − Γr Γi g g2 rr ji ji ri h rh g0 g0 rh rh 1 = − ij + ij + − ij − − ij g g2 2g g g r rg0h = ij . 2g2

which gives the pair

 rg0h Rr = ij  irj 2  2g (3.6) 0  rg hij R = .  rirj 2g

Contracting the Riemann curvature tensors gives us the Ricci tensors:

rr aa Rtt = g Rrtrt + g Ratat 1 1 (f 0)2 1 f 0g0 1 rh f 0 = f 00 − − + r−2haa aa g 2 4f 4 g 2 g 1 (f 0)2 f 0g0 f 0 = f 00 − − + (d − 2) 2g 4fg 4g2 2rg 1 (f 0)2 f 0g0 f 0 = f 00 − − + (d − 2) . 2g 4fg 4g2 2rg and similarly,

λ Rrr = Rrλr tt ij = g Rtrtr + g Rirjr 1 1 (f 0)2 1 f 0g0 rg0h = − f 00 − − + r−2hij ij f 2 4f 4 g 2g 1 (f 0)2 f 0g0 g0 = − f 00 + + + (d − 2) . 2f 4f 2 4fg 2rg 3.1 Black Holes with Cosmological Constant 35

Now we would like to ﬁnd the expression of Rij. Using Gauss equations (also known as Gauss-Codazzi equations) that relates the Riemann curvature of a manifold and its embedded submanifold, we have (see, e.g. page 100 of [22]), with tilde denotes operations and quantities with respect to the manifold and those without tilde denotes 2 2 i j those associated with the submanifold with metric ds = r hij(x)dx dx ,

D E ReVW X,Y = hRVW X,Y i − hII(V,X),II(W, Y )i + hII(V,Y ),II(W, X)i where II denotes the second fundamental form.

Taking V = ∂i,W = ∂j,X = ∂k,Y = ∂l, we have:

T T T T 2 Rijkl = r Rijkl(h) − ∇gV X , ∇gW Y + ∇gV Y , ∇gW X

2 D E D E = r Rijkl(h) − ∇g∂i ∂k , ∇g∂j ∂l + ∇g∂i ∂l , ∇g∂j ∂k 2 m m m m = r Rijkl − Γik∂m, Γjl ∂m + Γil ∂m, Γjk∂m .

Here

m m Γik∂m, Γjl ∂m t r a t r a = Γik∂t + Γik∂r + Γik∂a, Γjl∂t + Γjl∂r + Γjl∂a r a r a = Γik∂r + Γik∂a, Γjl∂r + Γjl∂a

t t since Γik = 0 = Γjl.

Similarly,

m m r a r a Γil ∂m, Γjk∂m = Γil∂r + Γil∂a, Γjk∂r + Γjk∂a .

Indeed,

r r r r Γik∂r, Γjl∂r = ΓikΓjl h∂r, ∂ri

= (−rfhik)(−rfhjl)grr 2 2 = (−rfhik)(−rfhjl)g = r f ghikhjl.

r r 2 2 Similarly Γil∂r, Γjk∂r = r f ghilhjk. 3.1 Black Holes with Cosmological Constant 36

We recall fondly that the Christoﬀel symbol in coordinate form is given by

1 Γi = gim (∂ g + ∂ g − ∂ g ) . kl 2 l mk k ml m kl This gives

1 Γa = gam (∂ g + ∂ g − ∂ g ) ik 2 k mi i mk m ik 1 = gam (∂ g + ∂ g ) 2 k mi i mk 1 = gaa (∂ g + ∂ g ) 2 k ai i ak 1 1 = gii (∂ g ) + gkk (∂ g ) . 2 k ii 2 i kk Note that we get from the ﬁrst step to the second step by noting that since the submanifold is of constant curvature, the famous result by Riemann guarantees that we can always choose a coordinate system which metric tensor is diagonal, i.e. gij = 0 if i 6= j.

We then obtain 1 Γa = gam (∂ g + ∂ g − ∂ g ) jl 2 l mj j ml m jl 1 = gam (∂ g + ∂ g ) 2 l mj j ml 1 = gaa (∂ g + ∂ g ) 2 l aj j al 1 1 = gjj (∂ g ) + gll (∂ g ) . 2 l jj 2 j ll

Similarly,

1 1 Γa = gii (∂ g ) + gll (∂ g ) il 2 l ii 2 i ll and

1 1 Γa = gjj (∂ g ) + gkk (∂ g ) . jk 2 k jj 2 j kk 3.1 Black Holes with Cosmological Constant 37

a a a a This implies that ΓikΓjl − ΓilΓjk 1 ii jj ll 1 kk jj ll = 4 g (∂kgii) g (∂lgjj) + g ∂jgll + 4 g (∂igkk) g (∂lgjj) + g ∂jgll 1 ii jj kk 1 ll jj kk − 4 g (∂lgii) g (∂kgjj) + g ∂jgkk − 4 g (∂igll) g (∂kgjj) + g ∂jgkk 1 ii jj ll 1 ii jj kk = 4 g (∂kgii) g (∂lgjj) + g ∂jgll − 4 g (∂lgii) g (∂kgjj) + g ∂jgkk 1 ii jj ii jj = 4 [g g ∂kgii∂lgjj − g g ∂lgii∂kgjj] 1 jj ii ii jj = 4 [g g ∂kgjj∂lgii − g g ∂lgii∂kgjj] = 0.

Thus

2 m m m m Rijkl = r Rijkl − Γik∂m, Γjl ∂m + Γil ∂m, Γjk∂m .

That is, 2 2 2 Rijkl = r Rijkl(h) − r f g[hikhjl − hilhjk]. (3.7)

We now derive the Ricci tensor:

λ Rij = R iλj tt rr aa = g Rtitj + g Rrirj + g Raiaj 1 1 rh f 0 1 rg0h = − ij + ij + gaa r2R (h) − r2f 2g(h h − h h ) f 2 g g 2g aiaj aa ij aj ia r h f 0 rg0h = − ij + ij + R (h) − f 2g(d − 2)h + f 2g haah h 2f g 2g2 ij ij aj ia | {za } =δj r h f 0 rg0h = − ij + ij + R (h) − (d − 3)f 2gh . 2f g 2g2 ij ij

So summarizing what we know about the Ricci tensors Rµν, we have

 1 (f 0)2 f 0g0 f 0 R = f 00 − − + (d − 2)  tt 2  2g 4fg 4g 2rg    1 (f 0)2 f 0g0 g0 R = − f 00 + + + (d − 2) (3.8) rr 2f 4f 2 4fg 2rg     r h f 0 rg0h R = − ij + ij + R (h) − (d − 3)f 2gh .  ij 2f g 2g2 ij ij

We will now use the Einstein Field Equations of the following form: 3.1 Black Holes with Cosmological Constant 38

1 R − g R + Λg = 8πT . (3.9) µν 2 µν µν µν

We remind the readers that we have set G = 1 = c.

2 2 2 2 i j Proposition.The generalized Kottler metric ds = −f(r)dt +g(r)dr +r hij(x)dx dx necessarily satisﬁes g(r) = [f(r)]−1.

2 2 i j Proof. If the submanifold with metric ds = r hijdx dx is Ricci-ﬂat, corresponding to vanishing cosmological constant, then in particular Rtt = Rrr = 0. From 3.8, we have: f 0 g0 (d − 2) + = 0 2rf 2rg f 0 g0 ⇒ + = 0 f g ⇒ f 0g + g0f = 0 ⇒ (gf)0 = 0 ⇒ gf = const.

In this case the metric is asymptotically ﬂat and we need f(r) → c2 and g(r) → 1 as c2 1 r → ∞. Thus g = f , i.e. g = f in our geometrical unit where c = 1.

2Λ If there exist nonzero cosmological constant Λ, then Rµν = d−2 gµν. We have:  2Λ 2Λ Rtt = gtt = − f  d − 2 d − 2 (3.10)  2Λ 2Λ Rrr = grr = g d − 2 d − 2 So that again we obtain f 0 g0 (d − 2) + = 0 ⇒ gf = const. 2f 2rg

In the presence of negative cosmological constant, the spacetime is asymptotically anti- de Sitter, and consequently, ω m r2 r2 f(r) = k − d + → k + rd−3 l2 l2 and 3.1 Black Holes with Cosmological Constant 39

r2 −1 g(r) → k + . l2

r2 r2 Note that the asymptotical behavior k + l2 is the same as that of 1 + l2 , as required for Anti-de Sitter metric, since the constant value k is negligible compared to the dominate r2 term l2 for large r. We need to be careful with the asymptotically de Sitter case. This is because there is no spacelike asymptotic region in spatially spherical de Sitter, and consequently there is no “large r”. In fact by asymptotically de Sitter, we means that the spacetime should tend to de Sitter space in the far future, and so we should consider timelike inﬁnity instead of just a spacelike one.

For simplicity, set l = 1 in the defining equation of de Sitter space as hyperboloid embedded in 5-dimensional Minkowski space. Recall that de Sitter space first shrinks and upon reaching minimal radius, expands as time passes. The spatial section is S3 and thus finite, with radius of the sphere grows with r = cosh(T ), where T is the proper time measured by observer at rest on the 3-sphere, and related to coordinate t (of the 5-dimensional ambient Minkowski space it embedded in) by t = sinh(T ). As t approaches infinity, the radius of the 3-sphere also tends to infinite, and consequently,

ω m r2 r2 f(r) = k − d − → k − rd−3 l2 l2 and

r2 −1 g(r) → k − . l2

1 Since we have fg constant, as in the asymptotically ﬂat case, we must have g = f .

2 2 −1 2 2 i j Thus, we have ds = −f(r)dt + [f(r)] dr + r hij(x)dx dx , with 3.1 Black Holes with Cosmological Constant 40

 ω m r2 f(r) = k − d ±  d−3 2  r l    0 (d − 3)ωdm 2r (3.11) f (r) = d−2 ± 2  r l    −(d − 3)(d − 2)ω m 2 f 00(r) = d ± .  rd−1 l2

So we can further simplify the Ricci tensors to:

 1 00 1 0 Rtt = ff + ff (d − 2)  2 2r    00 00 1 f 1 f (3.12) Rrr = − − (d − 2)  2 f 2r f     0 Rij = Rij(h) − hij [(d − 3)f + rf ]

Therefore, 1 1 R = ff 00 + ff 0(d − 2) tt 2 2r 1 f 0 = f f 00 + (d − 2) 2 r 1 −(d − 3)(d − 2)ω m 2 (d − 3)(d − 2)ω m 2(d − 2) = f d ± + d ± 2 rd−1 l2 rd−1 l2 1 2(d − 1) (d − 1) = f ± = ± f 2 l2 l2 d − 1 2Λ = ∓ g = g . l2 tt d − 2 tt 3.1 Black Holes with Cosmological Constant 41

1 f 00 1 f 00 R = − − (d − 2) rr 2 f 2r f 1 (d − 2)f 0 = − f 00 + 2f r 1 −(d − 3)(d − 2)ω m 2 (d − 3)(d − 2)ω m 2(d − 2) = f d ± + d ± 2 rd−1 l2 rd−1 l2 1 2(d − 1) = − ± 2f l2 d − 1 d − 1 2Λ = ∓ = ∓ g = g . l2f l2 rr d − 2 rr

and ﬁnally,

0 Rij = Rij(h) − hij [(d − 3)f + rf ] ω m r2 (d − 3)ω m 2r = R (h) − h (d − 3) k − d ± + r d ± ij ij rd−3 l2 rd−2 l2 (d − 3)ω m (d − 3)r2 (d − 3)ω m 2r2 = R (h) − h k(d − 3) − d ± + d ± ij ij rd−3 l2 rd−1 l2 (d − 3)r2 2r2 = R (h) − h k(d − 3) ± ± ij ij l2 l2 (d − 1)r2 = R (h) − h k(d − 3) ± ij ij l2 2Λ = g + (R (h) − h k(d − 3)) . d − 2 ij ij ij

Thus we get an the Kottler metric describing an Einstein manifold, i.e. the Ricci tensor satisﬁes 2Λ R = g µν d − 2 µν provided that the horizon itself is an Einstein space with Ricci tensor satisfying

Rij(h) = k(d − 3)hij. (3.13) 3.2 Curvature of the Event Horizon 42

3.2 Curvature of the Event Horizon

2 2 −1 2 2 i j Consider again the metric ds = −f(r)dt +[f(r)] dr +r hij(x)dx dx . In the asymptotically de Sitter case, we have

ω m r2 f(r) = k − d − , rd−3 l2

(d − 1)(d − 2) corresponding to the cosmological constant Λ = . We claim that the 2l2 event horizon of this black hole solution must be positively curved. In fact, one notes that if k = 0 or k = −1, then assuming that mass parameter m is positive, we must have f(r) < 0 and the time coordinate t becomes spacelike while the coordinate r becomes timelike, which contradicts our purpose of using this metric to describe spacetime exterior to a black hole. So indeed the horizon of the black hole must be positively curved.

For the case asymptotically Anti-de Sitter case however, as shown in [21], the curvature of the event horizon can be ﬂat or even negative.

3.3 Physics of Topological Black Holes

This section is largely based on [17] and [18], where I have ﬁlled in some details of the calculations.

Henceforth in this chapter we shall consider topological black holes in (4+1)-dimensional asymptotically Anti-de Sitter spacetime. Let us consider electrically neutral black hole of the following form:

r2 16πM dr2 g(AdSSch ) = − + k − dt2 + + r2dΩ2 (3.14) k 2 2 r2 16πM k L 3Γkr 2 + k − 2 L 3Γkr

2 where L is the radius of curvature of AdS5; dΩk is a metric of constant curvature k = {−1, 0, +1} on a compact 3-space Ck with area Γk. The conformal boundary has the topological structure of Ck × R. Note that the notation g(AdSSchk) means it is uncharged black hole with horizon curvature k; despite “Sch” being short form for Schwarzschild, this black hole need not have spherical topology. For example, for 3 2 3 2 C1, it can be S with Γ1 = 2π or RP with Γ1 = π . In fact, Γ1 is ﬁxed by the topology of the underlying space. Mathematically the horizon can then be one of the 3.3 Physics of Topological Black Holes 43 inﬁnitely many 3-manifolds of constant positive curvature, for example, the Lens space 3 L(p, q) := S /Zn.

Note that the space C0 is not uniquely deﬁned unlike C1: There are 6 possible topologies in the orientable case. They are called Torocosm (3-dimensional torus T 3), Dicosm 3 3 3 3 (T /Z2), Tricosm (T /Z3), Tetracosm (T /Z4), Hexacosm (T /Z6), and Didicosm (also 3 called Hantzsche-Wendt space T /(Z2 × Z2)). For details, see Theorem 3.5.5 of [23] or [24].

Unlike asymptotically Minkowski black holes that most of us are familiar with, 5- dimensional AdS black holes are not uniquely specified by their entropy [17]. That is, for any fixed value of entropy, there will be AdS black holes of the same entropy with either positively curved or flat event horizons, and within each class, there are yet many different black holes with that entropy. Let us look at this claim in more details. For the metric

r2 16πM dr2 g(AdSSch ) = − + k − dt2 + + r2dΩ2. k 2 2 r2 16πM k L 3Γkr 2 + k − 2 L 3Γkr

We have

4 2 2 2 3Γkreh + 3L Γkrehk − 16πML 2 2 = 0. (3.15) 3L Γkreh

That is the event horizon satisﬁes

1 2 4 2 2 2 2 2 −3L Γkk + [9L Γkk + 12Γk(16πML )] reh = ≥ 0. 6Γk

The entropy of the black hole is proportional to the area of the event horizon given by

3 Γeh = Γkreh. 3.3 Physics of Topological Black Holes 44

3.3.1 Positively Curved Uncharged AdS Black Holes

We have

3 " 1 # 2 2 4 2 2 2 −3L Γ1 + [9L Γ1 + 192Γ1πML ] Γeh = Γ1 6Γk

3 " 1 # 2 2 3 64πM 3 − 2 = L Γ12 −1 + 1 + 2 . 3Γ1L

The event horizon can have the topology of any non-singular quotient of the form S3/Γ where Γ is a ﬁnite group, e.g. any of the ADE ﬁnite subgroups of SU(2), namely the cyclic, quarternionic and binary polyhedral groups. If the order of Γ is |Γ|, then the 2π2 value of the area is Γ1 = |Γ| . As seen from the example of lens space Γ = Zn, |Γ| = n can be as large as we want. Correspondingly Γ1 can be arbitrary small, as for the lens 2π2 + space, it is Γ1 = n , n ∈ Z . Looking at the equation

3 " 1 # 2 2 3 64πM 3 − 2 Γeh = L Γ12 −1 + 1 + 2 , 3Γ1L we see that if the mass is small compared to L2, (that is to say, it has a value typical for a black hole which evaporates completely), then taking the quotient of the event horizon by small group Γ actually increases the entropy of the black hole; and taking quotient by large enough group decreases the entropy. However if the mass is large enough compared to L2, then taking quotient always decreases the entropy.

M Fixing the entropy (and hence the horizon), the dimensionless quantity L2 can be expressed as follows: 2 2 3 " 3 ! # M 3 Γ Γ − 1 1 = eh eh Γ 3 + Γ 3 . L2 16π L2 L2 1 1

Proof. From 3.15, we see that

3Γ r4 + 3L2Γ r2 16πML2 1 eh 1 eh = . 16πL4 16πL4 I.e. 2 3 Γ r 4 Γ r M 1 eh + 1 eh = . 16π L4 L2 L2 3.3 Physics of Topological Black Holes 45

Now

" 4 − 1 # 3 Γ 3 r4 Γ 3 Γ r2 LHS = 1 eh 1 + 1 eh 16π L4 L2

" 4 4 − 4 − 1 2 − 2 # 3 Γ 3 Γ 3 Γ 3 Γ 3 Γ Γ 3 Γ 3 = 1 eh 1 1 + 1 eh 1 16π L4 L2

" 4 − 1 2 1 # 3 Γ 3 Γ 3 Γ 3 Γ 3 = eh 1 + eh 1 16π L4 L2 = RHS.

The fact that M can always be found for any value of Γ1 implies that if Γ1 changes by taking quotients, the eﬀects of this can always be compensated by choosing M appropriately.

By elementary Calculus, one notes that

2 2 3 " 3 ! # M 3 Γ Γ − 1 1 = eh eh Γ 3 + Γ 3 L 16π L2 L2 1 1 as a function of Γ1 has global minimum at Γ Γ = eh , 1 L3 at which point we have M 3 Γ = eh . L 8π L3

Γeh 2 2 If L3 2π , then for a ﬁnite number of steps downwards from 2π , we need to reduce Γeh 2 M if we were to keep entropy constant. Conversely if L3 2π , then M has to be increased correspondingly. Thus, one conclude that

For each specified value of the entropy, there is a countable infinity of AdS black holes with positively curved event horizons with the specified entropy.

Two diﬀerent black holes with positively curved event horizons can have the same entropy and the same mass. For example, ﬁxing M, black holes with the following event hozirons 3 3 3 ˜ S /Z120,S /Q120,S /I 3.3 Physics of Topological Black Holes 46

˜ where Q120 is the quarternionic (binary dihedral) group, and I is the binary icosahedral π2 group, all have Γ1 = 60 and hence have the same entropy.

Remark: In algebraic geometry, a homology sphere is an n-dimensional manifold with the homology groups of n-sphere. Among these homology spheres, The Poincaréhomology sphere (also known as Poincarédodecahedral space) is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group. The fundamental group is precisely the binary icosahedral group with order 120.

3.3.2 Flat Uncharged AdS Black Holes

As mentioned previously, in the orientable case, there exist 6 possible topologies for flat AdS black holes. Conway and Rossetti call compact flat 3-manifolds the rather cute name platycosms for flat universe. In each case, there are continuous parameters which distinguish manifolds of the same topology which have different global geome- tries. For example, consider the 3-torus T 3, one can tile R3 with various lattices before identifying the edges to form torus, and there are 6 continuous parameters that describe the possible fundamental domain (See [24] and [25]). For example the cubical 3-torus R3/Z3 is different from the Two-storey 3-torus R3/(Z × Z × 2Z). In general, a 3-torus is parametrized by 3 angles (just like the usual donut is parametrized by 2 angles). T 3 is said to be cubic if all 3 angles have the same periodicity 2πK, where K is a dimensionless parameter.

For the other topologies resulted from taking quotients of the 3-torus, we have smaller number of continuous parameters. This is to be expected since the parameters have to be ﬁxed in order to perform the projection to the quotient. For details see [24]. Also see [26] for detailed interesting discussion on tetracosm and didicosm.

Consider the cubical 3-torus. The corresponding area is then

3 3 3 Γ0(T ) = 8π K , −∞ < K < +∞ where K is a continuous parameter. This equation also deﬁnes K, so that for arbitrary compact ﬂat 3-manifolds, K becomes a measure of the overall relative size of the space.

The metric for ﬂat uncharged AdS black hole is

r2 2M dr2 g(AdSSch ) = − − dt2 + + r2dΩ2. 0 L2 3π2K3r2 r2 2M 0 L2 − 3π2K3r2 3.3 Physics of Topological Black Holes 47

Figure 3.1: Number of continuous parameters of platycosms.

The event horizon is given by

1 1 16πML2 4 2ML2 4 reh = = 2 3 3Γ0 3π K which decreases with K.

The area of the horizon however increases with K:

3 4 2 3 3 3 3 3 Γ = 8 π 2 (MK) 4 L 2 ≈ 32.866(MK) 4 L 2 . eh 3

It follows that 3 3 Γeh M 4 L 2 3 = 2 3 3 3 4 2 4 3 L 8 3 π K L Thus 4 ! M Γ 3 3 = eh K−1 . L2 L4 32π2

Thus similar to the positively curvature case, we can adjust M to keep Γeh fixed as we vary K over the 6 sets of flat compact orientable 3-manifolds. That is, if we specifiy the area Γeh, then we can choose any of the 6 topologies, choose the parameters corresponding to the topology chosen, compute its corresponding K, and use the above equation to deduce M. 3.3 Physics of Topological Black Holes 48

Thus,

For each specified value of the entropy, there is a uncountable infinity of AdS black holes with flat event horizons with the specified entropy.

We shall derive the entropy-area relation of ﬂat uncharged black hole:

Let r2 2M 2r 4M f(r) = − ⇒ f 0(r) = + . L2 3π2K3r2 L2 3π2K3r3

0 f (reh) The Hawking temperatre (see Appendix B) is T = 4π , so

2 3 4 2 reh M 3π K reh + 2L M T = 2 + 3 3 3 = 3 2 3 3 . 2πL 3π K reh 6π L K reh

2 3 4 3π K reh Also, M = 2L2 . Thus we have

3π2K3r4 3π2K3r4 + 2L2 eh eh 2L2 reh T = 3 2 3 3 = 2 . 6π L K reh πL

r Consider T as a function of r, i.e. T (r) = . Similarly we consider M as a function πL2 3π2K3r4 of r, i.e. M(r) = 2L2 . The entropy S is given by integrating the ﬁrst law of thermodynamics T dS = dM:

Z reh 1 dM(r) S = dr 0 T (r) dr Z reh πL2 6π2K3r3 = 2 dr 0 r L Z reh = 6π3K3r2dr 0 3 3 3 = 2π K reh.

That is, 1 1 S = (8π3K3)r3 = Γ . 4 eh 4 eh So ﬂat uncharged black hole in AdS satisﬁes the usual entropy-area relation of black holes, namely the entropy is a quarter of the horizon area. The same is true for electrically charged case although the derivation is much more complicated. 3.3 Physics of Topological Black Holes 49

In addition, the Hawking temperature is

1 1 r S 3 1 S 3 T = 2 = 3 3 2 = 1 . πL 2π K πL 2 3 π2KL2

3.3.3 Negatively Curved Uncharged AdS Black Holes

There is a vast set of distinct compact manifolds of negative curvature, of which we will not go into the details. The horizon in this case would be quotient of hyperbolic space Hn/Γ where Γ is a discrete subgroup made of discrete boosts of SO(n, 1). The resulting space is a compact space of genus g with 4g sides and angle sum equal to 2π. [29].

Figure 3.2: A regular octagon in the Poincar´edisk such that its sum of angles equals 2π. The edges are geodesics. When opposing edges are identified we obtain a surface of genus two, with no singularities. Diagram is from [29]. 3.3 Physics of Topological Black Holes 50

The volume of compact negatively curved space is ﬁxed by the magnitude of the curvature and the topology of the space. It is interesting to remark that in 3 dimensions the unique smallest volume (also, just to remind the readers, closed and oriented as we always assume in our discussion) hyperbolic 3-manifold is the Weeks manifold. See [27] and [28] for discussions.

3.3.4 Flat Electrically Charged Black Holes in AdS

The metric of (n + 1)-dimensional Reissner-Nordstr¨om-Kottlerblack holes with cosmological constant Λ ([30], [31]) is given by

2 2 −1 2 2 2 ds = −fdt + f dr + r dΩk where

2 2 ωnM ωnQ r 16π f = f(M, Q, r) = k − n−2 + 2n−4 ± 2 , ωn := . r 2(n − 2)Γkr l (n − 1)Γk

So a (4+1)-dimensional charged ﬂat AdS black hole has the metric

r2 2M Q2 dr2 g(AdSRN ) = − − + dt2 + + r2dΩ2. 0 L2 3π2K3r2 48π5K6r4 r2 2M Q2 0 L2 − 3π2K3r2 + 48π5K6r4 We will always assume Q > 0 for simplicity.

Let x = r2 > 0, the event horizon satisﬁes

2ML2x Q2L2 x3 − eh + = 0. eh 3π2K3 48π5K6 Note that the cubic polynomial

2ML2x Q2L2 G(x) := x3 − + 3π2K3 48π5K6 has local minimum at r 2ML2 x = . min 9π2K3

Existence of event horizon thus requires G(xmin) ≤ 0.

A straightforward calculation yields

5 3 3 2 2 2 2 M 2 L Q L G(xmin) = − 9 + 5 6 . 27π3K 2 48π K 3.3 Physics of Topological Black Holes 51

Therefore 2 2 5 3 3 Q L 2 2 M 2 L G(xmin) ≤ 0 ⇒ 5 6 ≤ 9 48π K 27π3K 2 I.e. 5 2 5 2 √ Q 48π 2 64 2 3 ≤ 3 = 2π ≈ 99.255. (3.16) L(KM) 2 27π 9

A The entropy of the black hole satisﬁes the usual S = 4 relationship, namely 3 3 3 S = 2π K reh. In terms of M,L and K, the entropy of an extremal black hole is given by

√ 3  q M  2 2L K3 S = 2π3K3 E  3π 

3 " # 4 2L2 M = 2π3K3 K3 9π2

3 " 4 4 4 2 M # 4 2 3 π K 2L = K3 9π2

3 " 7 2 4 2 # 4 2 3 π K L MK = 9

2 In the ﬁrst step we have used the fact that for extremal hole, reh = xmin.

That is, 3 " 1 2 2 # 4 4(2) 3 π ML K S = . (3.17) 9

We can also compute the temperature of the charged ﬂat AdS black hole as we did for the uncharged case. Starting with r2 2M Q2 f(r) := − + L2 3π2K3r2 48π5K6r4 we get 2r 4M Q2 f 0(r) = + − . L2 3π2K3r3 12π5K6r5 3.3 Physics of Topological Black Holes 52

Therefore f 0(r ) T = eh 4π 2 1 2reh 4M Q = 2 + 2 3 3 − 5 6 5 4π L 3π K reh 12π K reh " 1 ! 3 3 2 3 3 5 # 1 2 S 3 4Mπ K 2 Q (2π K ) 3 = 2 1 + 2 3 − 5 4π L 2 3 πK 3π K S 12π5K6S 3

" 2 1 3 2 5 # 1 2 3 S 3 4Mπ 2 Q 2 3 = 2 + 2 − 5 4π πKL 3π S 12KS 3 " 2 1 2 2 2 3 2 2 5 # 1 2 3 S 3 8π reh Q 3π K reh Q 2 3 = 2 + 2 + 5 6 4 − 5 4π πKL 3S L 48π K reh 2 12KS 3 " 2 1 3 3 4 2 2 2 2 2 5 # 1 2 3 S 3 4π K S 3 8Q 2 3 π K Q 2 3 = 2 + 4 + 5 − 5 4π πKL SL22 3 π4K4 96π2K3S 3 12KS 3  1 2 2  S 3 2 3 + 2 3 2 2 5 ! 1 Q 8(2) 3 2 3 =  2 + 5 −  4π πKL KS 3 96 12

" 1 2 2 2 # 1 S 3 2 3 2 2(2) 3 Q = 1 5 2 − 5 . 2 3 π2 3 πKL 24KS 3

That is, " 1 2 # 1 S 3 Q T = 1 2 − 5 . (3.18) 2 3 π πKL 24KS 3

Let us now explore the relation between entropy S and the charge Q. We substitute the entropy-area relation 3 3 3 S = 2π K reh into the event horizon deﬁning equation

r6 2Mr2 Q2 eh − eh + = 0. L2 3π2K3 48π5K6 3.3 Physics of Topological Black Holes 53

This yields

2 S2 2M S 3 Q2 − + = 0. 4π6K6L2 3π2K3 2π3K3 48π5K6 2 1 2 2 S 2 3 MS 3 Q ⇒ − + = 0. 4π6K6L2 3π4K5 48π5K6 2 1 2 2 2 2 2 12S − 16(2) 3 MS 3 π KL + Q πL ⇒ = 0. 48π6K6L2 2 2 1 2 2 2 2 ⇒ πQ L = 16(2) 3 MS 3 π KL − 12S .

Note that for any Q2, there are two corresponding S, we shall choose the larger value of S corresponding to the entropy of the horizon, instead of the smaller one corresponding to the inner horizon.

The lower bound of entropy is obtained from 3.17:

3 " 1 # 4 4(2) 3 2 2 2 2 3 S ≥ S := π ML K ≈ 0.647322(π ML K) 4 . E 9

On the other hand the upper bound of the charge is given by

 1 3  1 ! 2 1 ! 2 2 2 1 1 2 2 4(2) 3 2 2 4(2) 3 2 2 Q ≤ Q = 2 3 16π ML K π ML K − 12 π ML K E πL2  9 9 

" 7 3 # 1 1 2 6 1 3 4 2 1 3 3 3 2 2 2 2 2 3 2 3 2 = 2 (2) 16π ML πM LK − 12 3 2 π M L K πL 3 9 2 9 √ ! 16(2) 6 32 2 2 3 3 = − π M 2 LK 2 3 9 √ √ ! 32 2(3) 32 2 2 3 3 = − π M 2 LK 2 9 9 √ 64 2 3 3 = 2π M 2 LK 2 . 9

This of course agrees with 3.16. 3.3 Physics of Topological Black Holes 54

The temperature of the black hole from 3.18 can be expressed in terms of M,S,K and L:

" 1 2 # 1 S 3 Q T = 1 2 − 5 2 3 π πKL 24KS 3

" 1 1 2 2 2 2 # 1 S 3 2 3 16π ML KS 3 − 12S = 1 2 − 5 2 3 π πKL πL2(24KS 3 ) " 1 1 1 # 1 S 3 2πM2 3 S 3 = 1 2 − + 2 . 2 3 π πKL 3S 2πL K

That is, 1 3S 3 2M T = 1 − . (3.19) 2 3 2π2KL2 3S

We see that for large value of S, i.e. small value of Q, the temperature of the black 1 hole varies as S 3 .

We note from the above expressions that the entropy of black holes in AdS is nonzero. As we throw in more and more electrical charges, the temperature decreases until it reaches zero temperature, and the entropy reaches SE. An extremal hole thus has zero temperature but nonzero entropy, a peculiar fact that also presents in the case of charged spherical black holes in asymptotically Minkowski spacetime. Chapter 4

Stability of Anti-de Sitter Black Holes

There are various (in)stability issues in the study of Anti-de Sitter black holes, for example: thermodynamical stability, various perturbative stabilities and non-perturbative stability. In this chapter we will only deal with thermodynamical stability and non- perturbative stability. For perturbative stability, one may see for example [32] where the author studies gravitational perturbations of scalar, vector as well as tensor type; and [33] which also deals with non-perturbative stability.

4.1 Thermodynamics Instability

Black holes in Anti-de Sitter space have thermodynamical properties such as possessing temperature and entropy equal to one quarter of the horizon area, just as their better known asymptotically ﬂat counterparts. An obvious stability issue to consider for a thermodynamical system is whether there can be phase transition where the system changes its physical behaviour drastically - e.g. water solidifying into ice. More precisely, phase transition is classiﬁed by the behaviour of the order parameter around the transition temperature. Recall that for example, in water-ice and water-vapour transition the density is the order parameter, while the order parameter in the case of ferromagnetic transition is the magnetization.

In this section we will focus on the uncharged case only. Thermodynamic stability of charged black holes will be discussed in the next chapter when we explore the application to quark gluon plasma.

Consider a spherically symmetric uncharged black hole in (n + 1)-dimensional Anti-de 4.1 Thermodynamics Instability 56

Sitter space, the temperature and mass as functions of r are given by (see e.g [34] and [35]). Ω rn−1 S(r) = n−1 4 and Ω (n − 1)[rnL−2 + rn−2] M(r) = n−1 . 16π where Ωn−1 is the area of unit (n − 1)-dimensional sphere.

For a ﬁxed temperature T , consider the function f(r, T ) deﬁned by

f(r, T ) ≡ M − TS.

We have

f(r, T ) = M − TS (n − 1)[rnL−2 + rn−2] T rn−1 = Ω − n−1 16π 4 Ω = n−1 (n − 1)[rnL−2 + rn−2] − 4πT rn−1 . 16π

Diﬀerentiating with respect to r yields ∂f Ω = n−1 (n − 1) nrn−1L−2 + (n − 2)rn−3 − 4πT rn−2(n − 1) ∂r 16π Ω (n − 1)rn−3 = n−1 nr2L−2 + (n − 2) − 4πT r . 16π

We see that setting this derivative to zero yields the temperature of the black hole with event horizon at r = reh:

2 −2 2 2 nrehL + (n − 2) nreh + (n − 2)L TE ≡ = 2 . 4πreh 4πL reh Consider T as a function of r: nr2 + (n − 2)L2 T (r) = . 4πL2r 4.1 Thermodynamics Instability 57

Then the Helmholtz free energy is given by H ≡ f(r, T ) Ω nr2 + (n − 2)L2 = n−1 (n − 1) rnL−2 + rn−2 − 4πrn−1 16π 4πL2r Ω = n−1 (n − 1)[rnL−2 + rn−2] − L−2rn−2(nr2 + (n − 2)L2) 16π Ω = n−1 (n − 1)rn−2 − (n − 2)rn−2 + (n − 1)rnL−2 − L−2rnn 16π Ω = n−1 rn−2 − rnL−2 16π Ω r2 = n−1 rn−2 1 − . 16π L2

Thus we can see that  r2 > 0 if < 1 ⇔ 0 < r < L,  L2 H =  r2 < 0 if > 1 ⇔ r > L. L2

Let us look at the explicit case for n = 4 case, and set L = 1 for simplicity. Then

π2r3 S(r) = 2 and 2π2(3)[r4 + r2] 3π(1 + r2)r2 M(r) = = . 16π 8

We have 3π(r2 + r4) π2r3T f(r, T ) = − . 8 2 and so ∂f 6πr + 12πr3 3r2π2T = − . ∂r 8 2 Setting this partial derivative to zero gives us

3 2 6πreh + 12πreh 4reh + 2 TE = 2 2 = . 12π reh 4πreh

Now 3π(r2 + r4) − 4π2r3T πr2[3 + 3r2 − 4πrT ] f(r, T ) = = . 8 8 4.1 Thermodynamics Instability 58

For f(r, T ) = 0, we have either r = 0 or √ √ 4πT ± 16π2T 2 − 36 2πT ± 4π2T 2 − 9 r = = . 6 3 The obvious constraint for having real nonzero roots is 4π2T 2 ≥ 9, that is 3 T ≥ . 2π 3 Evidently when T = Tc ≡ 2π , we have, 2πT r = = 1 = L. 3 The Helmholtz free energy is 3π(r2 + r4) π2r3 4r2 + 2 H = − . 8 2 4πr 3π(r2 + r4) π(4r2 + 2)r2 = − 8 8 π(3r2 + 3r4 − 4r4 − 2r2) = 8 π = r2(1 − r2). 8

Thus we see that  > 0, if 0 < r < 1 = L.  H = < 0, if r > 1 = L.

Note that H = 0 for r = 0 holds in any dimension. This is the no black hole phase or AdS phase since r is the horizon radius (which acts as the order parameter).

We can plot the curve of f(r, T ) against r. We note that for any ﬁxed curvature scale L, the range r > L corresponds to black hole phase while 0 < r < L corresponds to AdS phase. This is because AdS has H = 0; for r > L, we have H < 0 which is energetically favourable state than H = 0 and vice versa. We note that the local minimum shifts from H > 0 region into H < 0 region as we increase the temperature. At T > Tc the black hole phase minimizes the Helmholtz free energy and so AdS black holes are stable in those regime. At T < Tc black hole phase is unstable and we have the background Anti-de Sitter space which is energically favorable. At T = Tc both the black hole phase and the AdS phase coexist. This is a way of looking at the Hawking-Page phase transition [36]. 4.1 Thermodynamics Instability 59

Figure 4.1: A plot of f(r, T ) with horizontal r-axis. The curves from top to bottom correspond to temperatures 0.417, 3 0.467, 2π ≈ 0.47 and 0.497. The plot is lifted from [35].

Another interesting fact to note is that from the formula of the temperature nr (n − 2) T = + , 4πL2 4πr we see that setting the derivative with respect to r to zero yields

n (n − 2) rn − 2 = ⇒ nr2 = (n − 2)L2 ⇒ r = L. 4πL2 4πr2 n That is to say, a spherical uncharged black hole in Anti-de Sitter space has a minimum temperature which occurs when its size is of the order of the characteristic radius of the Anti-de Sitter space. Larger black holes have greater temperature as measured from infinity. Thermodynamically this means that AdS black holes are very different from asymptotically flat cousins - they have positive specific heat and can be in stable equilibrium with thermal radiation at a fixed temperature. Below the allowed minimum temperature there is no stable black hole solution. Indeed, as evidenced from a typical 1 plot of β ≡ T against r, for any fixed nonzero temperature (except the minimum temperature), there are two corresponding black hole solutions.

The large black holes, which by large we means larger than the AdS curvature scale, the temperature grows linearly with radius, while the small black holes have temperature 4.1 Thermodynamics Instability 60

Figure 4.2: A typical plot of the inverse temperature plotted against horizon radii for spherical uncharged AdS black holes. There is a minimum temperature below which there are no black hole solutions. The plot is lifted from [37]. inversely proportional to their size (i.e. they have negative speciﬁc heat), a feature shared by the asymptotically ﬂat case. This we can check:

Taking the limit L → ∞ gives us the corresponding result for asymptotically ﬂat case, in which the temperature is inversely proportional to radius: n − 2 T = . L→∞ 4πr In the familiar (3+1)-dimensional Schwarzschild case, since 4π(2)r r M = = , 16π 2 this reduces to the well-known Hawking-Bekenstein formula (See Appendix B): 1 T = . 8πM

Written in full glory this reads c3 T = ~ . 8πGMk

Therefore 4.1 Thermodynamics Instability 61

Asymptotically ﬂat spherical uncharged black holes have temperature inversely proportional to their mass (and hence size): large black holes are cold while small black holes are hot. They have negative speciﬁc heat and so evaporates. Furthermore, they heat up as they evaporate.

Asymptotically Anti-de Sitter spherical uncharged black holes behave diﬀer- ently: due to positive speciﬁc heat, large black holes do not evaporate away completely but come to equilibrium with their own Hawking radiation, in other words, they are eternal. On the other hand, small hot black holes evaporate while cold black holes are not stable and decay into the Anti-de Sitter background.

4.1.1 Phase Transition for Flat Uncharged AdS Black Hole

From our previous discussion on the physics of flat AdS black hole, we showed that in the case of uncharged black hole, the temperature grows linearly with the horizon radius r T = πL2 so these black holes also have positive specific heat much like the spherical case. How- ever there is a crucial difference: there is no Hawking-Page transition into the AdS background for cold black holes. For details see [21]. Nevertheless if we choose the background to be Horowitz-Myers soliton [38] instead of AdS, then there is a transition. The following discussion follows that of [41].

The (n + 1)-dimensional Horowitz-Myers AdS soliton takes the metric of the form

dr2 r2 A ds2 = −r2dt2 + + − dφ2 + r2h dθidθj. s s r2 A L2 rn−2 ij L2 − rn−2 where hij is a Ricci flat metric on the horizon equipped with coordinates {θi} and A is some parameter that need not coincide with that of the black hole. In view of comparison with the flat black hole, we consider the case whereby hij is a metric on the torus Rn−2/Γ where Γ is some discrete group action. Similarly to the Wick-rotated black hole metric, we need to avoid conical singularity and impose regularity condition: 4πL2 the angle φ needs to be identified with period βs = where rs+ is the zero of (n−1)rs+ r2 A 1 Vs(r) = L2 − rn−2 . Recall that here β = T is the reciprocal of the temperature. See Appendix B for the method of such regularization. Note that as pointed out in [41], the AdS soliton has a flat conformal boundary of topology R × S1 × Rn−2/Γ instead of positively curved topology of R × Sn−2 in the case of global Anti-de Sitter space. The Wick-rotated version is of course 4.1 Thermodynamics Instability 62

dr2 r2 A ds2 = r2dτ 2 + + − dφ2 + r2h dθidθj. s r2 A L2 rn−2 ij L2 − rn−2 where the imaginary time τs, in view of matching solutions for regularization later on, has the same period as that of the Euclidean black hole, say 2πP .

Horowitz and Myers proposed that in (4+1)-dimension any spacetime which asymptotically approaches the soliton metricg ¯µν, that is, the spacetime with metric gµν =g ¯µν +hµν −2 −4 −6 such that hαβ = O(r ), hαr = O(r ) and hrr = O(r ) for α, β 6= r, must have energy E ≥ 0 with respect to the soliton, where equality is attained only by the soliton itself. It has since been proven that with proper boundary conditions satisﬁed, the Horowitz-Myers soliton is indeed the conﬁguration of least energy [39],[40].

Let us compare the soliton metric with the ﬂat black hole metric r2 B dr2 ds2 = − − dt2 + + r2dψ2 + r2h dθidθj. L2 rn−2 b r2 B ij L2 − rn−2 where B is a parameter related to the mass of the black hole and the size parameter K. We have singled out arbitrarily an angle parameter ψ from the {θi}. Recall that the period of ψ is 2πK.

The conformal boundary of the flat black hole is the same as that of the soliton, and the r2 B event horizon reh of course satisfies Vb(reh) = 0 where Vb = L2 − rn−2 . Wick-rotating 4πL2 the metric and requiring regularity one obtain similarly that βb = . (n−1)reh We regularize thermodynamical quantities by matching the two solutions at some finite cutoff radius R and calculate the quantity as a function of R, and finally send R to infinity. This is similar to ultraviolet cutoff in quantum field theory. The matching conditions at finite R then requires that the metric on the two tori be the same, and furthermore p βs Vs(R) = R(2πK) and p βb Vb(R) = R(2πP ).

That is r R2 A β − = 2πRK s L2 Rn−3 and r R2 B β − = 2πRP. b L2 Rn−3 Sending R to inﬁnity yields in the limit,

βs = 2πLK, βb = 2πLP 4.1 Thermodynamics Instability 63 which is independent of the parameter A and B.

As shown in [41], there is indeed phase transition from ﬂat black hole to the soliton (provided the soliton parameter A is not the same as that of the black hole parameter B).

Indeed the regularized black action in the 5-dimensional case is given by [18]

I = αP K3 K−4 − P −4 where α is a positive number depending on L, of which the precise value can be computed [41] but should not concern us for later applications. It is clear that if A = B then the action vanishes and there is no phase transition. The phase transition is determined by K and P , i.e. by the precise shape of the 4-dimensional torus at the Euclidean conformal inﬁnity, by the extent to which it deviates from being cubic.

β 1 From the matching condition βb = 2πLP , we have P = 2πL = 2πLT . Thus I = αP K3 K−4 − (2πT L)4 . (4.1)

That is, 1 − (2πT LK)4 I = αP K3 . K4

From this, with free energy of the Horowitz-Myers soliton taken to be zero, we see that black hole is energetically favoured only if its temperature is not too low compared with 1 2πKL . Unlike spherical black holes however, area and temperature of AdS black holes are independent quantities. This distinction implies that the stability of black holes not only depend on temperature but also their size: black holes with suﬃciently big K can be stable even if T is small, i.e.

A very large but very cold AdS ﬂat uncharged black hole can be stable.

Conversely a suﬃciently small AdS ﬂat black hole can be unstable even if it is very hot.

While this property is richer than that of the spherical black holes, nevertheless there exists minimum temperature for these ﬂat black holes: for any ﬁxed K and L, stable black holes must statisfy the bound 1 T ≥ . (4.2) 2πKL which we will refer to hereinafter as the Horowitz-Myers phase transition temperature bound. 4.2 Non-perturbative Instability of AdS Black Holes 64

In other words lower temperature hole is unstable and is replaced by the AdS soliton instead. Thus just like the spherical AdS black holes, ﬂat AdS black holes have temperature that is bounded away from zero.

4.2 Non-perturbative Instability of AdS Black Holes

By non-perturbative instability or Seiberg-Witten instability, we mean nucleation of codimension-one branes in the AdS. This instability arises out of string theoretical considerations, hence to formulate this even semi-rigorously will require a great deal of eﬀort which will deviate too far oﬀ the track of this thesis. We therefore only state the results that are relevant to us.

In the notation of [42], the Seiberg-Witten action for a (D − 1)-brane carrying a charge q in a (D + 1)-dimensional spacetime is

 D T r0 √ 2φ 8 D − 2 2(D−4)  g (1 − q)φ D−2 + (∂φ)2 + φ2R + O φ D−2 for D > 2  D 2  2 (D − 2) 4(D − 1) S =  2 T r0 √ 2φ 2 −2φ  g (1 − q)e + 2 [(∂φ) + φR] − R + O(e ) for D = 2. 4 We will be focusing on D > 2 case, i.e. for spacetime dimension at least 4. The ﬁeld ϕ tends to inﬁnity as the conformal boundary is approached, and R is the scalar curvature of that boundary.

We state the following without proof: Non-perturbative instabilities arise if S becomes negative since brane nucleation leads to large branes having energy which is unbounded from below as they approach the boundary. Equivalently, stability requires that the scalar curvature at conformal infinity should remain non-negative. This is the case for AdS space itself, with posivitely curved conformal infinity. Obvious danger for the action to become negative is in the case of either q > 1 or R < 0. The first case does not happen if the background is supersymmetric, such as in AdS/CFT correspondence. For readable account of non-perturbative instability, see [33].

In particular, for any codimensional-one brane in any Euclidean asymptotically AdS5 space, the Seiberg-Witten action [42] is deﬁned by

S = Θ (Brane Area) − µ (Volume Enclosed by Brane) . (4.3) where Θ is related to the tension of the brane and µ relates to the charge enclosed by the brane. Note that the area contributes positively, but the volume negatively, to the 4.2 Non-perturbative Instability of AdS Black Holes 65 action; thus the volume must not grow too rapidly relative to the area if the action is to remain positive.

The most dangerous case is when the charge is saturated (much like extremal black hole), called the BPS case (BPS is short for Bogomol’nyi-Prasad-Summerﬁeld), which in 5-dimension, is given by 4Θ µ = . BPS L We will assume BPS condition for subsequent discussion.

It is also worth recalling that de Sitter and Anti-de Sitter space admit many diﬀer- ent coordinates that might or might not cover the whole manifold. Since the branes are sensitive to the global geometry of the spacetime, it is crucial that one does not use misleading coordinates which might suggest that the the conformal inﬁnity of the spacetime has the wrong scalar curvature than what it should be. Chapter 5

Estimating the Triple Point of Quark Gluon Plasma

In this chapter we will study the quark gluon plasma, specifically we will use our knowledge of black holes in 5 dimension to tell us something about quark gluon plasma in our 4 dimensional universe. We remind the readers that quark gluon plasma is a complicated thing to study from field theory perspective, what we are doing is to deal with its gravitational dual in the AdS bulk which is easier to study, and finally see what the solution entails about the original field theory problem. We also remind the readers that there need not be any physical basis in the mysterious 5 dimensional AdS bulk as one can simply treat it as a sort of transform space which need not be physical. The chapter is based on [18] and [44]. I fill in most of the calculation details.

5.1 An Introduction to Quark Gluon Plasma

A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. There are six types of quarks, known as ﬂavors: up, down, charm, strange, top, and bottom. Quarks have various intrinsic properties, including electric charge, colour charge, spin, and mass; they experience all 4 fundamental forces, namely electromagnetism, strong and weak nuclear force, as well as gravity. The quark model was proposed by physicists Murray Gell-Mann and George Zweig in 1964.

A proton for example, has 2 up quarks and 1 down quark while a neutron has 1 up quark and 2 down quarks. Since proton has electric charge +1 and neutron is electrically 5.1 An Introduction to Quark Gluon Plasma 67 neutral, this means that quarks have fractional electric charge: up quark has electric 2 1 charge + 3 while down quark − 3 . Quarks are conﬁned inside the hadron in colour- neutral state, i.e. the colour combination of the constituent quarks give “white”: for hadrons this typically mean the 3 quarks are red, green and blue; while for meson which is made up of 2 quarks, this means red and anti-red, blue and anti-blue or green and anti-green. Note that colour is merely a convenient label, they are not really optically colourful in the usual sense. Colour-conﬁnement means that we cannot observe free quark.

One way to think of colour-confinement is to consider say, a meson made up of one quark and one anti-quark. The early attempt to study quark models the force between them as flux tube, which is the original version of string theory. The colour force favors confinement because at a certain range it is more energetically favorable to create a quark-antiquark pair than to continue to elongate the colour flux tube. So as we try to pull the quark-anti-quark pair apart, we eventually end up with two pairs of them instead, i.e. 2 mesons instead of one. In standard model QCD, the colour force is mediated by particle called gluon, analogous to photon being the force-carrier for electromagentic force in QED. Unlike photons however, gluons carry colour charges and so interact among themselves. There are eight independent types of gluon in QCD.

In dealing with the nature of quark conﬁnement, one visualization is that of an elastic bag of hadron or meson which allows the quarks to move freely around, as long as you don’t try to pull them further apart. But if you try to pull a quark out, the bag stretches and resists, eventually the energy is so great as to allow pair production of new quark- antiquark pairs and we end up with bags of mesons in addition to the original bag. Thus the further we try to pull quarks apart, the greater the containment force they experience; inside the bag, they are free to move about, and we call that asymptotic freedom. These bags of hadrons and mesons are analogous to vapour bubbles in a liquid. The QCD vacuum exerts pressure B on the quarks and gluons. 5.1 An Introduction to Quark Gluon Plasma 68

Figure 5.1: The confinement of quarks. Cartoon is from [45].

Quark Gluon Plasma or QGP, or sometimes dubbed quark soup, is a phase of quantum chromodynamics (QCD) which exists at extremely high temperature or density. This phase consists of (almost) free quarks and gluons, i.e. the quarks are deconﬁned. Such a state is believed to exist in the early universe when the temperature is very high (T > 100GeV) and all the known particles were extremely relativistic. Back then even what we now call “strongly interacting” particles such as the quarks and gluons should interact weakly due to asymptotic freedom. They thus form hot weakly interacting colour-charged particles. As the universe cooled due to expansion of space, the quarks, antiquarks and gluons combined to form hadrons which eventually result in baryonic matter that we are familiar with today. Much of the physics about QGP and how this transition to hadronic matter happened are still not understood. 5.1 An Introduction to Quark Gluon Plasma 69

It is interesting to note that it is possible that the core of neutron star is dense enough for quarks to deconﬁne to exhibit QGP or other quark matter phase, although it is colder than the QGP in the very early universe and is made up of predominantly matter instead of almost equal mix of matter and antimatter [51] [52].

A neutron star is formed by gravitational collapse of a massive star which is nevertheless not massive enough to form black hole. The most famous neutron star is probably that at the center of the Crab Nebula (M1) in the constellation of Taurus, which is also a pulsar. The supernova explosion that produced the Crab Nebula was observed on Earth in 1054 A.D. and recorded down by the ancient Chinese and Arabs. Historical records revealed that a new star bright enough to be seen in the daytime was observed. Now, almost a thousand years later, a neutron star left behind by the explosion is seen spewing out a blizzard of high-energy particles into the expanding debris ﬁeld known as the Crab Nebula. X-ray data from Chandra telescope provides signiﬁcant clues to the workings of this mighty cosmic generator, which is producing energy at the rate of 100,000 suns.

Below the surface of the neutron star, the pressure due to gravity is so extreme that there are no longer any atoms: everything is compressed down to a liquid of neutrons. Even deeper into the core, if the density becomes high enough, the neutrons themselves will be crushed out of existence, thus liberating the quarks inside forming quark matter.

Since black holes in 5 dimensions may help us to understand the properties of QGP in 4 dimensions, one day we may be able to study the state of matter in the interior of neutron stars in this way. So neutron stars which are as closed as one can get before becoming black hole in 4 dimension is in this very roundabout way related to black holes in 5 dimension! 5.1 An Introduction to Quark Gluon Plasma 70

Figure 5.2: M1 Crab Nebula. The Chandra X-ray image is shown in blue, the Hubble Space Telescope optical image is in red and yellow, and the Spitzer Space Telescope’s infrared image is in purple. The X-ray image is smaller than the others because extremely energetic electrons emitting X-rays radiate away their energy more quickly than the lower-energy electrons emitting optical and infrared light.

More down to earth though, experimentally, a consensus has been reached that some form of a quark gluon plasma has been produced in the RHIC Au-Au collisions [48] [49] [50]. RHIC stands for Relativistic Heavy Ion Collider, a 2.4-mile-circumference particle accelerator at the U.S. Department of Energys (DOE) Brookhaven National Laboratory. At the temperature reached at RHIC, quark matter is still strongly interacting, and is sometimes called sQGP. In this work, we will simply refer to them as QGP: they are what QGP becomes once temperature and pressure is suﬃciently low, but still not low enough to undergo phase transition. The strong coupling behaviour makes perturbative ﬁeld theoretical approaches to explain the properties of the quark-gluon plasma in this temperature range a nearly impossible task. Three new experiments running on CERN’s Large Hadron Collider (LHC), ALICE, ATLAS and CMS, will continue studying properties of QGP.

A key objective of the study of QGP is to further understand the quark matter phase 5.1 An Introduction to Quark Gluon Plasma 71

Figure 5.3: Various models of neutron stars and other hypothetical compact stars. This diagram is from [47].

diagram. As of now we don’t know the details of the various phases, although we do have a very simpliﬁed qualitative picture of it [52] that we reproduce in the Figure 5.4. The authors in [52] referred to the study of matter at ultra-high density as the condensed matter physics of quantum chromodynamics, since as in conventional condensed-matter physics, we seek to map the phase diagram and calculate the properties of the phases.

Quark matter occurs in various forms, depending on the temperature T and quark chemical potential µ, which you can think of as the pressure of the system; it is an ana- logue to electric potential and gravitational potential in which force ﬁelds are thought as being the cause of things moving, be they charges, masses, or, in this case, chemicals. More precisely, in a thermodynamic system containing n particle species, its Helmholtz energy A is a function of its temperature T , the volume V and the number of particles of each species N1,N2, ..., Nn, i.e.

A = A(T,V,N1,N2, ..., Nn).

The chemical potential of the i-th species is then deﬁned by the partial derivative ∂A µi = . ∂Ni

The details of the various phases should not concern us: we are mainly interested in the phase transition from quark gluon plasma to other phases. We note that from the 5.1 An Introduction to Quark Gluon Plasma 72

Figure 5.4: The simplified phase diagram of quark matter. CFL stands for colour-flavour locked phase which is a superfluid. Colour-flavour locking means that the quarks form Cooper pairs, whose colour properties are correlated with their flavor properties in a symmetric pattern. For example, a Cooper pair of an up quark and a down quark must have colours red and green. For details see [52] and [53]. Heavy ion colliders are exploring the physics close to the T -axis.

quark matter phase diagram in Figure 5.5 that QGP exists for high temperature; as the temperature is lowered, if the chemical potential is low, QGP makes transition into ordinary hadronic matter. The curve that marks this transition ends on the upper left away from the T -axis. This is the critical point. For high enough chemical potential, QGP makes transition to non-hadronic matter (non-CFL and CFL) with interesting properties that we will not discuss. Note that there is a triple point where the phase transition curves of QGP intersects: This is the lowest possible temperature allowed for QGP. While this picture is qualitatively well-accepted, we don’t know enough physics of quark matter to know quantitatively where precisely is this triple point. We will later on try to estimate the location of this triple point, but now let us focus on the critical point. 5.2 Estimating The QGP Critical Point 73

5.2 Estimating The QGP Critical Point

In this section, we shall review some facts from particle physics. As is well known, at very high temperature such that the particles have energy much larger than their rest mass, we may describe them using relativistic kinematics and eﬀectively neglect their masses. That means we can treat them as hot relativistic free gas. The number densities of the partons of species i are then described by the quantum distribution

Z 3 d pi 1 ni = (2π)3 eβEi ± 1

1 where β = T is the inverse temperature and the negative sign corresponds to bosons while positive corresponds to fermions. For relativistic case we can approximate Ei with pi, upon integrating we obtain the standard result in terms of the Riemann zeta function  ζ(3) 3  T  π2 ni = 3 ζ(3)  T 3 4 π2 for boson and fermion respectively. Similarly, one can compute the energy density (in the unit c = 1),

π2  T 4 Z 3 30 d pi Ei  ρi = = (2π)3 eβEi ± 1 7 π2  T 4 8 30 for boson and fermion respectively.

Recall from particle physics that these expressions are valid for each spin, ﬂavour, charge and colour of the particle. If the system is a mixture of fermion and boson however, we will need to include degeneracy factors gi for the particle of species i, so that the total energy density is X π2 ρ = g ρ = g T 4 i i ∗ 30 i 7 where g∗ = gb + 8 gf with gb and gf referring to the degeneracy factors for bosons and fermions respectively. The degeneracy factors count the total degrees of freedom, i.e. multiply over spin, ﬂavour, charge and colour. Note that as temperature drops, 5.2 Estimating The QGP Critical Point 74 particles tend to decouple from each other (for example, neutrino decoupled from other particles at around T ∼ 1 MeV). As a particle decouple from the rest, it will no longer contribute to the degeneracy factor, thus g∗ tends to decrease with time as the universe expands and cools.

Much of the details discussed thus far in this section can be found in [54].

In QGP, the gluon have 2 helicity states and 8 possible colours so that makes gb = 16. For a fixed flavour, each quark have 3 colors, 2 spin states and 2 charge states (quark and antiquark). Thus that makes gf = 12. Assuming T < 1 GeV, there are two active 2 quark flavours (up and down), we have

7 7 g = g + g = 16 + (2 × 12) = 37. ∗ b 8 f 8

For phase transition of QGP into hadronic matter, we consider the lightest hadron, namely, pions. A gas of relativistic pions have g = 3 since it is made of 3 types of pions (π+, π− and π0). It follows that the energy density of pion is 3π2 ρ = T 4. π 30 1 Using the equation of state of relativistic particles p = 3 ρ, we have the pressure of the system as 3π2 P = T 4. π 90 Likewise for QGP, we have 37π2 P = T 4. QGP 90

The total pressure of the hadronic phase consisting of pion gas is actually 3π2 P = B + T 4 π 90 where B is the pressure exerted by the QCD vacuum on the quarks and gluons. In other words, the true ground state of the QCD vacuum has a lower energy −B than the perturbative QCD vacuum.

At the phase transition, we set the pressures to be equal 37π2 3π2 T 4 = B + T 4 90 90 which gives the critical temperature

1 45B 4 T = ≈ 144 MeV c 17π2 5.2 Estimating The QGP Critical Point 75

1 since B 4 ≈ 200 MeV (See, for example, a very nice pedagogical introduction to quark matter phases at [56]). That is slightly more than a trillion Kelvin!

Indeed a variety of methods including lattice QCD and experiments have suggested that the critical point we sought lie at a temperature of about 150 MeV and baryonic chemical potential in the range of 350 to 450 MeV [55]. 5.3 Charging Up Black Holes in 5 Dimension 76

5.3 Charging Up Black Holes in 5 Dimension

We are now ready to explore AdS/CFT in the context of quark gluon plasma following [44]. Since we are considering QGP in 4 dimension, the corresponding gravitational dual lives in 5 dimension. The chemical potential in the 4 dimensional field theory corresponds to electrical charges on the 5 dimensional black hole [57]. So we must first look at some more properties of charged black holes in addition to what we have explored in Section 3.3. (However, there is certain fine prints, see Section 5.7).

Recall that for AdS-Schwarzschild black hole with typical S3 horizon, the temperature is bounded away from zero temperature. This corresponds to deconfinement-confinement transition in the field theory defined on R×S3 [37]. That is, AdS black hole corresponds to unconfined phase while AdS without black hole represents the confined phase of the field theory. However, this is no longer the case for charged AdS-Schwarzschild black hole. Indeed, with sufficiently large electric potential on the horizon, it is possible to access the zero-temperature axis while remaining in the de-confined phase.

Figure 5.5: The phase diagram for charged spherical black holes in global AdS. Note that the horizontal axis is the AdS-Schwarzschild case where Tc marks the critical temperature corresponding to the Hawking-Page phase transition. The vertical axis is the extremal charged case. The image is lifted from [37].

However, the black holes used as dual description of QGP must be very diﬀerent from this kind of charged spherical black hole since we should not allow the black hole to attain zero temperature. As we throw more and more electrical charges into the black hole, the black hole will cool down; the corresponding description in the ﬁeld theory 5.3 Charging Up Black Holes in 5 Dimension 77 is that the temperature of the QGP also decreases. However QGP cannot be of zero temperature - as is evident from the phase diagram of quark matter. Thus conversely, some mechanism should also prevent our black hole from reaching zero temperature, i.e. extremal state.

We begin by noticing that spherical black holes would be bad for AdS/CFT correspondence in the case of describing QGP in our universe: In the quark matter phase diagram, the curve corresponds to the transition between QGP and non-hadronic matter rises indeﬁnitely to the right as we increase the chemical potential. In the dual description, a black hole slightly below the curve is unstable, but this point can be as high as one likes as we increase the chemical potential. This means that some highly charged and extremely hot black holes must be unstable. This contradicts the fact that Hawking-Page transition occurs at low temperature, and so extremely hot large AdS black holes should be stable. In other words, they reach thermal equilibrium with their own Hawking radiation as we have discussed before.

The idea is to consider black holes with ﬂat event horizon instead.

We ﬁrst consider the bulk electromagnetic 1-form, expressed in terms of a gauge such that the connection is not singular (See p.416, [37]):

Q 1 1 A = 3 3 2 − 2 dt. 16π K r reh

The chemical potential of the dual ﬁeld theory is proportional to the (negative) magnitude of the asymptotic value of A, where the constant of proportionality must have 1 unit of inverse length, say [59]. γL

A Recall that the entropy of the black hole satisﬁes the usual S = 4 relationship, i.e.

3 3 3 S = 2π K reh.

Therefore,

2 Q 2 3 Q µ = 3 3 2 = 2 . 16π γLK reh 16πKγLS 3

The chemical potential increases with Q, and is bounded by the extremal value µE: 5.3 Charging Up Black Holes in 5 Dimension 78

2 2 3 QE µ ≤ µE = 2 3 16πKγLSE √ 1 2 h 64 3 3 i 2 3 2 2 2 2 9 2π M LK = 1 h 1/3 i 2 4(2) 2 2 16πKγL 9 π ML K

2 q 64 1 3 4 " 3 1 3 # 2 9 2 πM 4 L 2 K 4 = 1 1 1 1 6 2 2 16(2) 3 (2) πKLπM LK γ " 1 # 1 M 4 = 5 3 3 2 4 πγL 2 K 4 1 1 M 4 = 5 3 6 . 2 4 πγ K L where in the second step we have used equations 3.16 and 3.17.

We now deﬁne temperature-normalized chemical potential by µ µ¯ ≡ . T

Since µ is increasing with Q, and T is decreasing with Q, we have overall,µ ¯ increasing with Q. However,µ ¯ is not classically bounded above since T can be arbitrarily small. Thus we have a very lousy bound

0 < µ¯ < ∞ which is hardly useful at all.

We will seek to improve this bound by ﬁnding a ﬁnite upper bound.

For now, we recall the phase diagram for quark matter, and notice that there should be two qualitatively diﬀerent phase transition: for low chemical potential the QGP makes phase transition into hadronic matter, while for high chemical potential it makes transition into non-hadronic (non-CFL) matter. 5.4 Transition to Conﬁnement at Low Chemical Potential 79

5.4 Transition to Conﬁnement at Low Chemical Po- tential

One idea is to consider the phase transition of flat black holes into the Horowitz-Myers AdS Soliton. Recall that we have established the Horowitz-Myers phase transition temperature bound 4.2: 1 T ≥ . 2πKL 1 Once K is fixed, every flat black hole with temperature below 2πKL will be replaced by the appropriate soliton, which has lower free energy.

Assuming that the same mechanism works for charged case, the Horowitz-Myers AdS soliton metric is r2 dr2 r2 A B g(AdSRNSol) = − dt2 + + − + L2dφ2 + r2 dθ2 + dθ2 . L2 r2 A B L2 r2 r4 1 2 L2 − r2 + r4 c.f. the ﬂat charged black hole metric in AdS:

r2 2M Q2 dr2 g(AdSRN ) = − − + dt2 + 0 L2 3π2K3r2 48π5K6r4 r2 2M Q2 L2 − 3π2K3r2 + 48π5K6r4 2 2 2 2 + r dφ + dθ1 + dθ2 .

Again, as with the non-charged case, the parameter A and B of the soliton need not matched that of the black hole.

Similar calculation as the case of Horowitz-Myers AdS soliton without charge gives the same result that 1 T ≥ . 2πKL

Therefore at low temperature, the charged ﬂat AdS black hole makes phase transition into AdS soliton, correspondingly, the QGP makes phase transition to hadronic matter. However this cannot be the whole story since the temperature bound obtained above is independent of chemical potential. Thus it predicts that the plasma phase should make phase transition to hadronic matter as we lower the temperature, at all values of chemical potential, no matter how large. However we know that this is not the case: the plasma eventually makes transition into non-hadronic matter instead at some relatively high chemical potential. Thus this picture is incomplete: something else must account for the phase transition at high chemical potential. 5.5 Stringy Instability at High Chemical Potential 80

5.5 Stringy Instability at High Chemical Potential

We proposed that this something is the Seiberg-Witten (non-perturbative) instability that we discussed in section 4.2. In the Euclidean metric, consider a BPS 3-brane of tension Θ wrapping one of the r = const. sections of that space. The Seiberg-Witten action is then

S = Θ(Brane Area) − µ(Volume Enclosed by Brane). 4Θ Since the brane is assumed BPS, we have µ = . L Recall that to obtain the Euclidean metric, we Wick-rotate so that t now parametrizes a 4-th circle, in addition to the 3 circles that parametrize the flat torus. The Euclidean metric is thus a metric on a manifold which is radially foliated by copies of the 4-torus. t One can think of as an angular coordinate on this 4-th circle with periodicity 2πP L chosen so that the metric is not singular at reh. This gives rise to conformal boundary which is a conformal torus. Since P need not be the same as the torus parameter K, this torus need not be cubic. The field theory is then defined on this space which is the Euclidean version of ordinary flat spacetime.

We have, area and volume given respectively by

Z Z √ 3 A = gττ dτ r dΩk,n−2 and Z r Z Z √ √ 0 3 0 V = gττ gr0r0 (r ) dr dΩk,n−2 dτ reh where the integral over τ is from 0 to the period impose on imaginary time.

For AdS uncharged topological black hole, we have

( 1 ) 2 2 4 4 3 r 16πM r − reh S(r; M, L, Θ,K(k = 0)) = 2πP LΘΓk r 2 + k − 2 − L 3Γkr L where the notation means that S depends on K only when k = 0, and Γk is the area of the event horizon. The action is seen to vanished on the horizon.

This can be re-written as   h 2 16πM i  L kr − 4   3Γk reh  S = 2πP LΘΓk 1 + .  h kL2 16πML2 i 2 L  1 + 1 + 2 − 4  r 3Γkr 5.5 Stringy Instability at High Chemical Potential 81

In the k = +1 case, the action S is always positive and increasing function of r, while in the case of k = −1, the action first increases but then reaches a maximum and decreases, eventually becoming negative at large r [17]. Thus non-perturbative instability arises in the case of negatively curved event horizon, that means such black holes are not stable in the context of string theory. The instability is due to brane nucleation (brane pair-production) as brane with negative energy that is unbounded below approaches the boundary. In the dual field description, this corresponds to negative mass square scalar field.

We note that for the uncharged ﬂat case, the action asymptotes to a ﬁxed value 16 2 2 3 π ΘPML :   h 2 16πM i  L kr − 4   3Γ0 reh  lim 2πP LΘΓ0 1 + r→∞  h kL2 16πML2 i 2 L  1 + 1 + 2 − 4  r 3Γ0r

 16πM   L − 4  3Γ0 reh  = 2πP LΘΓ0   +  2 L   1 4  2 4   2ML   16πM 2 3   L − 3Γ 3π K  = 2πP LΘΓ 0 + 0  2  L     ML = 2πP LΘ(8π3K3) 3π2K3 16 = π2ML2ΘP. 3

For our purpose we need only to focus on the ﬂat case. 5.5 Stringy Instability at High Chemical Potential 82

Figure 5.6: Brane action for negatively curved event horizon (left) and flat event horizon (right) for typical values of parameters. Diagrams are from [17].

Adding in electric charges, ﬂat AdS black hole has Seiberg-Witten action of the form

( 1 ) r2 2M Q2 2 r4 − r4 S(r;Θ, L, M, Q, K) = 16π4ΘP LK3 r3 − + − eh L2 3π2K3r2 48π5K6r4 L    Q2 2M 4  4 2 3  48π5K6r2 − 3π2K3 reh  = 16π ΘPL K 1 + 2 .  h 2ML2 Q2L2 i 2 L  1 + 1 − 3π2K3r4 + 48π5K6r6 

At large r, we have

M r4 lim S = 16π4ΘPL2K3 − + eh . r→∞ 3π2K3 L2

Note that this result is approximate since we ignore all couplings of the probe brane and stringy corrections to the gravitational action. Since we are trying to exclude an entire range of values of Q, not just a single (extremal) value, this should not aﬀect the picture too much.

2 Now recall from section 3.3.4 that if we ley x = r > 0, then the event horizon reh satisﬁes 2ML2x Q2L2 x3 − eh + = 0. eh 3π2K3 48π5K6 5.5 Stringy Instability at High Chemical Potential 83

The cubic 2ML2x Q2L2 G(x) = x3 − + . 3π2K3 48π5K6 has local minimum at r 2ML2 x = . min 9π2K3

Existence of event horizon implies G(xmin) ≤ 0, i.e. √ Q2 64 2π2 3 ≤ ≈ 99.255. L(KM) 2 9

Note that if we ﬁx M, L and K while varying Q, the local minimum xmin remains invariant: increasing Q only lifts the cubic higher.

We can evaluate the brane action at inﬁnity for the extremal hole by letting r → √ eh xmin: S(∞;Θ, L, M, Q, K) 16 lim = − π2ΘL2M < 0. Q→QE P 9 Here we divided by P since otherwise we get −∞ with P diverges in the extremal limit (since P is proportional to inverse T , and T tends to zero as Q → QE).

Thus by the Seiberg-Witten instability, near-extremal black holes dual to strongly coupled ﬁeld theories on ﬂat spacetime is unstable. We shall next quantify this “nearness”.

With hindsight, we deﬁne the critical value (near-extremal value) of Q2 by

2 2 16π 3 3 Q = √ L(KM) 2 ≈ 91.1715L(KM) 2 . NE 3 Recall also that from section 3.3.4, the extremal charge is √ 2 2 64 2π 3 3 Q = L(KM) 2 ≈ 99.2550L(KM) 2 E 9 so that the ratio gives √ 2 QNE 16 9 1 3 3 2 = √ √ = √ ≈ 0.9186. QE 3 64 2 4 2 That is, Q NE ≈ 0.9584. QE This critical charge is about 96% of the extremal charge. 5.5 Stringy Instability at High Chemical Potential 84

The event horizon that corresponds to the black hole having critical charge is reh = p xeh(NE), where, we claim: r ML2 x (NE) = . eh 3π2K3

It is easy algebra to check that indeed G(xeh(NE)) = 0. Also, xeh(NE) > xmin so xeh(NE) is indeed the event horizon instead of the inner horizon.

We also note that M r4 S(∞;Θ, L, M, Q ,K) = 16π4ΘPL2K3 − + eh NE 3π2K3 L2 M ML2 = 16π4ΘPL2K3 − + = 0. 3π2K3 3π2K3L2

This means that the system is critical in the sense that for any Q such that Q > QNE, the system becomes unstable in the Seiberg-Witten sense.

Correspondingly,

3 2 3 3 3 3 2 2 2 4 2 2 4 S ≥ SNE = 2π K [xeh(NE)] = 1 π ML K ≈ 0.877383(π ML K) 3 4 2 2 3 c.f. SE ≈ 0.647322(π ML K) 4

Recall from 3.19 that 1 3S 3 2M T = 1 − 2 3 2π2KL2 3S Thus 1 3 3 2 3 1 1 1 1 2M3 4 2 4 2 4 TNE = 1 1 π M L K − 3 3 3 3 2 3 2π2KL2 3 4 3(2π 2 M 4 L 2 K 4 ) 3 3 4 − 3 1 − 3 − 3 − 1 − 3 1 − 3 − 3 = π 2 M 4 L 2 K 4 − 3 4 π 2 M 4 L 2 K 4 2 3 1 ! 1 − 4 3 4 − 3 4 2 M − 3 = π 2 2 K3L6

1 1 M 4 = 1 3 3 6 . (2)3 4 π 2 K L Thus

1 1 1 M 4 M 4 T ≥ TE = 1 3 3 6 ≈ 0.068228 3 6 . (2)3 4 π 2 K L K L 5.5 Stringy Instability at High Chemical Potential 85

This is bounded away from zero. Thus AdS ﬂat black hole cannot be arbitrarily cold. Adding charges to about 96% of the extremal value makes the black hole unstable.

Indeed,  3 2 3 3 3 2 2 2 4 SNE = 2π K (xeh(NE)) = 3 (π ML K)  3 4 

1  1 M 4 T = .  NE 1 3 3 6  (2)3 4 π 2 K L

This leads to a rather surprisingly result M S T = S T = NE NE min min 3 which is independent of geometric parameters K and L.

Therefore, a computation of the minimal black hole entropy from an analysis of say, the microscopic degree of freedom, would allow evaluation of minimal temperature without requiring knowledge of the geometric parameters.

Now recall that the chemical potential satisﬁes

2 Q 2 3 Q µ = 3 3 2 = 2 . 16π γLK reh 16πKγLS 3 Thus,

1 2 16π2 3 2 2 3 √ L(MK) 2 3 µNE = 2 3 3 2 2 2 16πγKL 3 (π ML K) 4 3 4 2 4π 1 3 3 2 3 1 L 2 M 4 K 4 = 3 4 2 − 1 1 1 16πγKL 2 3 3 2 πM 2 LK 2

1 1 3 4 M 4 = . 4πγ K3L6

Also 1 1 M 4 2γ √ TNE = 1 3 3 6 = µNE. (2)3 4 π 2 K L 3π

Seiberg-Witten instability sets in if a point in the phase diagram is below the line 2γ T = √ µ. 3π 5.5 Stringy Instability at High Chemical Potential 86

Equivalently, if √ 3π γ ≤ . 2¯µ

We also note that µNE 2γ µ¯max = = √ ∞ TNE 3π if γ is not too big.

This is indeed the case since recent estimates for QCD critical points give T ∼ 150 MeV and µ ∼ 350-450MeV (Section 5.2) imply thatµ ¯ ∼ 2 or 3, that is, γ / 0.51. So we have a holographic phase diagram (Figure 5.8). We see that if charge is low, then there is phase transition from black hole to Horowitz-Myers soliton; but hot black holes which are highly charged can now become unstable due to Seiberg-Witten instability, and the black hole phase make transition into another phase with as yet unknown properties that we will simply call Seiberg-Witten phase. This picture is, we stress, over-simpliﬁed. Recall that we have made several assumptions: we ignore the coupling of probe brane and stringy correction, we also assumed that phase transition from black hole to AdS soliton continues to hold for charged black holes, which is not the complete story since transition at low chemical potential also requires inclusion of black holes with scalar hair [44].

The point A represents critical black hole which is the holographic dual to the quark Q matter critical point. By modifying the charge over mass ration M of the dual black holem we can deﬁne trajectories in the holographic phase plane. Starting at the critical Q black hole, by increasing M , we obtain a curved trajectory (dotted curve AB) that bends downward towards the right. This curve terminates as Seiberg-Witten instability sets in. The point at which this curve terminates corresponds to the coldest black hole that can be obtained in this way, starting at a black hole with a deﬁnite QGP interpretation. Thus we propose that this termination point is dual to the triple point of quark matter. We will now try to estimate the triple point using what we know about the critical point. 5.5 Stringy Instability at High Chemical Potential 87

Figure 5.7: Holographic phase diagram where AdSRN0 refers to stable black hole phase, HMSol refers to Horowitz-Myers AdS Soliton phase, and SW refers to Seiberg-Witten phase with as yet unknown properties. The diagram is modified from [44]. 5.6 From the Critical Point to the Tripple Point 88

5.6 From the Critical Point to the Tripple Point

From 3.19 1 3S 3 2M T = 1 − 2 3 2π2KL2 3S we have 1 3S 3 K = 4 2M 3 2 2 2 π L T + 3S 3S which upon multiplying numerator and denominator by 2M yields

4 9S 3 K = . (5.1) 1 3TS 3 2 2 2 4π ML 1 + 2M

On the other hand,

" 1 2 # 1 S 3 Q T = 1 2 − 5 2 3 π πKL 24KS 3

 1 2 !2  1 S 3 16πKγLS 3 µ 1 = 1  2 − 2 5  2 3 π πKL 2 3 24KS 3

" 1 2 2 2 2 # 1 S 3 32 π Kγ L µ = 1 2 − 4 1 . 2 3 π πKL (3)2 3 S 3

But 1 3S 3 2M T = 1 − 2 3 2π2KL2 3S so we have

1 " 1 2 2 2 2 # 3S 3 2M 1 S 3 32 π Kγ L µ 1 − = 1 2 − 4 1 2 3 2π2KL2 3S 2 3 π πKL (3)2 3 S 3 1 2 2 2 S 3 32 πKγ L µ = 1 − 2 1 . 2 3 π2KL2 (3)(2)(2) 3 S 3

It follows that 1 2 " 1 !# 2 S 3 2 3 3 2M S 3 µ = 2 2 − 4 . πγ L K 16 3S 2 3 π2KL2 5.6 From the Critical Point to the Tripple Point 89

That is, 2 2 2 2 1 16πγ L µ K 2M S 3 2 1 − K + 4 = 0. 3(2) 3 S 3 3S 2 3 π2L2

This is a quadratic equation in K. Solving this using quadratic formula and after some algebraic manipulations give

2 4 q 2 3 3S 3 12γ2µ2S2 4π2ML2 1 ± 1 − πM 2 K = 12γ2µ2S2 πM 2 That is, 2 4 2 3 3S 3 1 K = 2 2 . 4π ML q 12γ2µ2S2 1 ∓ 1 − πM 2

The sign in front of the square root is chosen to be positive so that K corresponds to the 4 1 parameter of the event horizon (recall that reh ∝ K3 , so larger value of reh corresponds to smallar value of K):

2 4 2 3 3S 3 1 K = 2 2 . 4π ML q 12γ2µ2S2 1 + 1 − πM 2

Equating with 5.1, we have

4 2 4 9S 3 2 3 3S 3 1 = . 1 3TS 2 2 q 2 3 4π2ML2 1 + 4π ML 12γ2µ2S2 2M 1 + 1 − πM 2

Thus r 2 2 2 2 12γ µ¯ T S 1 2 3TS 9 + 9 1 − = 2 3 2 3 3 1 + . πM 2 2M Deﬁne ST σ ≡ M gives r ! 12γ2µ¯2σ2 3σ 9 1 + 1 − = 6 1 + π 2

That is, r ! 12γ2µ¯2σ2 3 1 + 1 − = 2 + 3σ π 5.6 From the Critical Point to the Tripple Point 90

It thus follows that r 12γ2µ¯2σ2 3 1 − = 2 + 3σ − 3 = 3σ − 1. π That is, 12γ2µ¯2σ2 9 1 − = 9σ2 − 6σ + 1 π Thus we obtain a quadratic in σ:

12γ2µ¯2 9 1 + σ2 − 6σ − 8 = 0 (5.2) π

4 Taking the limit γ → 0 gives the positive value of σ as 3 . This is the upper bound for σ: 1 4 ≤ σ < 3 3 where the lower bound is supplied by the previous result that M S T = S T = . NE NE min min 3

So in terms of σ, we have

1 3S 3 2M T = 4 − 2 3 π2KL2 3S 1 3 σM 3 2M = T − 4 2 2 σM 2 3 π KL 3 T

Thus 1 σM 3 2T 3 T T + = 4 3σ 2 3 π2KL2 1 1 1 3σ σ 3 M 3 2T T 1/3 ⇒ σT + = 4 3 2 3 π2KL2 1 1 3σ σ 3 M 3 2 4 3 ⇒ σ + T = 4 3 2 3 π2KL2 3 1 3 4 σM 4 ⇒ T = 3 3 3 3 2 2 4 2 4 2π K L σ + 3 3 1 1 3 4 σ 4 M 4 ⇒ T = 3 . 3 3 2 2 2 4 4 2π (KL ) 1 + 3σ 5.6 From the Critical Point to the Tripple Point 91

Also, 1 1 M 4 TNE = 1 3 3 6 = Ttriple. 2(3) 4 π 2 K L because we argued that the triple point corresponds to the coldest stable black hole. Therefore

" 1 # 1 M 4 Ttriple = 1 3 3 6 2(3) 4 π 2 K 4 L 4 1 ! 1 M 4 = 3 3 1 2π 2 (KL2) 4 3 4   1 3 1 3 1 M 4 3 4 σ 4 2 4 1 =  critical  1 +  3  1 3 3 3 2 4 3σcritical 4 2π 2 (KL2) 4 1 + σcrtical 3σcritical   3 1 1 3 1 3 4 M 4 σ 4 2 4 =  critical  + σ .  3  critical 3σcritical 3 3 2 4 3 2π 2 (KL2) 4 1 + 3σcritical

Thus we have 3 1 2 4 Ttriple = + σcritical Tcritical. (5.3) 3σcritical 3

Also √ 3π µ = T . triple 2γ triple

The energy density of the ﬁeld theory is proportional to the mass per unit horizon area of the dual black hole. Since the entropy is a quarter of the horizon area, this means M T that the relevant quantity is 4S or equivalently 4σ . The energy densities at the quark matter critical point and triple point satisfy

" 3 # 4 ρtriple σcriticalTtriple σcritical 1 2 = = + σcritical Tcritical ρcritical Tcriticalσtriple Tcriticalσtriple 3σcritical 3

1 and since σ = , we have triple 3

3 4 ρtriple 2 = + σcritical . ρcritical 3 5.6 From the Critical Point to the Tripple Point 92

1 4 With the bound that 3 ≤ σ < 3 , we have 3 3 1 2 4 2 4 Ttriple = + σcritical Tcritical ≥ Tcritical. 3σcritical 3 4

Thus, with Tcritical ∼ 150 MeV, we have

Ttriple ' 63 MeV. and ρtriple 3 ≤ 2 4 . ρcritical

The energy density at the critical point is estimated to be not far below the maximal 3 density attained in the RHIC experiment [55]. We thus take ρcritical ∼ 1000 MeV/fm .

For comparison, 1 GeV/fm3 = 1.78 × 1018 kg/m3; while nuclear density is around 2.8 × 1017 kg/m3.

3 We thus obtain the upper bound on ρtriple / 1680 MeV/fm . Numerically, we can vary γ and compute σ using 5.2 and then compute T from √ triple 3π 5.3. Then µcritical can be computed from the relation µtriple = 2γ Ttriple and ρtriple from 3 ρtriple 2 4 = + σcritical . We then obtain the table shown on next page [44] (Figure 5.9). ρcritical 3 Using Shuryak’s upper bound on the triple point temperature [60]

Ttriple ≤ 70 MeV we have 63 MeV ≤ Ttriple ≤ 70 MeV and 3 3 1530 MeV/fm ≤ ρtriple ≤ 1680 MeV/fm

From the table we see that, the upper bound Ttriple ≤ 70 MeV means that 0.10 / γ / 0.15.

For γ = 0.10. This gives

3 Ttriple ≈ 70 MeV; µtriple ≈ 1100 MeV; ρtriple ≈ 1500 MeV/fm corresponding to µ¯triple ≈ 15.7.

For γ = 0.15, we have similarlyµ ¯triple ≈ 10.2.

This is a huge improvement from the bound that does not take into account the Seiberg- Witten instability: 0 < µ¯ < ∞. 5.7 Caveat: QCD Dual in AdS/CFT 93

Figure 5.8: Some numerical data by varying γ. Temperature and chemical potential are measured in MeV while energy density is in MeV/fm3. The first row is the assumed values at the critical point for comparison purpose only. I.e. the first row is set such that critical point is the same as the triple point.

5.7 Caveat: QCD Dual in AdS/CFT

Despite what we have been doing so far trying to use AdS/CFT to understand quark gluon plasma, we must point out that the exact dual of QCD is not known: placing a black hole in a 5-dimensional spacetime does not endow its dual 4-dimensional finite- temperature Yang-Mills theory with the specific content of finite temperature QCD. In particular, the dual theory has no true quarks. Furthermore, the gauge symmetry of the Yang-Mills field is SU(N) with large N, not 3 as in QCD (corresponding to the 3 colours of quarks). In addition, QCD is also not supersymmetric. See also [61] for difference between the computed shear viscosity in weakly coupled N = 4 super Yang-Mills theory as compared to QCD.

Nevertheless, the theory analyzed does bear similarity to QCD, which may be due to the important factor that controls the behavior of the system is temperature, while microscopic details of the physics are less important[15]. Chapter 6

Dilaton Black Holes in Anti-de Sitter Space

6.1 Asymptotically Flat Spherically Symmetric Dila- ton Black Holes

For simplicty we ﬁrst consider the (3 + 1)-dimensional case. For the sake of comparison we shall recall the Reissner-Nordstr¨omsolution:

2M Q2 2M Q2 −1 g(RN) = − 1 − + dt2 + 1 − + dr2 + r2dΩ2 r r2 r r2

Q where the Maxwell ﬁeld is given by F = . As with previous chapters, we assume rt r2 Q > 0 for simplicity.

The global structure of Reissner-Nordstr¨omblack hole is quite diﬀerent from the Schwarzschild black hole (See Appendix 1 for details). For 0 < Q < M, we have two horizons

p 2 2 r± = M ± M − Q where r = r+ is the event horizon and r = r− is the Cauchy horizon or simply, the inner horizon.

In low energy limit of string theory, scalar field called dilaton cam couple to the Maxwell field, giving the action Z √ S = d4x −g R − 2(∇φ)2 − e−2φF 2 6.1 Asymptotically Flat Spherically Symmetric Dilaton Black Holes 95 where φ denotes the dilaton scalar field. We assume that the dilaton decays and vanishes at infinity. Because of the coupling between dilaton field and Maxwell field, the dilaton is not an independent “hair” of the black hole.

The corresponding black hole solution is remarkably simple, known as the Garﬁnkle- Horowitz-Strominger or GHS black hole [62]:

2M 2M −1 Q2 g(GHS) = − 1 − dt2 + 1 − dr2 + r r − dΩ2. r r M

In the general case in which the dilaton does not vanish at inﬁnity but equals to some value φ0, then the last term takes the form

Q2e−2φ0 r r − dΩ2. M

As mentioned before, the dilaton is coupled to the electric ﬁeld and hence is not an independent parameter of the black hole. The precise relation between the dilaton and the electric charge Q is given by

Q2e−2φ0 e−2φ = e−2φ0 1 − Mr or in the case of vanishing dilaton at inﬁnity,

Q2 e−2φ = 1 − . Mr

Note the absence of dependence of electrical charge in the gtt and grr terms. The r-t plane is thus similar to the Schwarzschild black hole, but the sphere is smaller in area since Q2 r r − < r2 M for any nonzero electrical charge.

Interestingly the GHS black hole behaves differently compared to the Reissner-Nordström black hole when electrical charge is increased. In the latter case, the event horizon moves inward while the Cauchy horizon moves outward, finally the two horizon coincide when extremality is reached. For the GHS black hole however, the event horizon stays fixed at r+ = 2M and it has no inner horizon. This may be due to the instability of the inner horizon, and when dilaton is included the inner horizon vanishes. However, there are examples of solutions with dilaton which possess a nonsingular inner horizon [63]. The effect of decreasing electrical charge on the GHS black hole is to decrease its area, which 6.1 Asymptotically Flat Spherically Symmetric Dilaton Black Holes 96

2 goes to zero at Q = rM. This√ gives the extremal limit: the event horizon becomes singular at Q2 = 2M 2, i.e. Q = 2M, unlike the extremal limit of Reissner-Nordstr¨om black hole that satisﬁes Q = M.

One can understand that the dilaton black hole has larger charge over mass ratio in the extremal limit because the scalar ﬁeld contributes an extra attractive force, and so for any ﬁxed M, we need a larger Q to balance it.

String Metric: As a remark, strings do not couple directly to the physical 2φ metric gGHS, but rather to the conformally related string metric e gGHS:

φ0 1 − 2Me dρ2 g(string) = − ρ dt2 + + ρ2dΩ2. 2 −φ0 φ 2 −φ 1 − Q e 2Me 0 Q ρ 0 Mρ 1 − ρ 1 − Mρ See for example, [63] and [64]. Interestingly, as pointed out in [63], when r is small,

Q2 1 e−2φ = 1 − ⇒ e2φ = ∼ 0 Mr Q2 1 − Mr and so the string coupling is becoming very weak near the singularity. Of course the solution cannot be trusted that close to the singularity, but nevertheless it is tempting to imagine what it might mean if the full string theoretic solution has a similar behavior. Could it mean that contrary to the common folklore that large quantum ﬂuctuation plagues the near singularity region, the quantum eﬀects are actually suppressed there?

In general, we can introduce a free parameter α ≥ 0, that governs the strength of coupling between the dilaton field and the Maxwell field. This yields the Garfinkle- Maeda or GM black hole solution [65]:

1−α2 α2−1 2α2 r r 2 r r 2 r 2 g(GM) = − 1 − + 1 − − 1+α dt2+ 1 − + 1 − − α +1 dr2+r2 1 − − 1+α dΩ2. r r r r r where 2α r− 1+α2 Q e−2φ = e−2φ0 1 − ,F = dt ∧ dr r r2 where the asymptotic value of the dilaton ﬁeld φ0 will be taken to be zero in the following discussion. and the horizons are at 1 + α2 h i r = M ± pM 2 − (1 − α2)Q2 , α 6= 1 for r . ± 1 ± α2 −

When α = 1 (the coupling strength that appears in the low energy string action), the 6.2 Topological Dilaton Black Holes 97

GM solution reduces to the GHS solution, and there ceases to be an inner horizon, while α = 0 case reduces to the Reissner-Nordstr¨omsolution.

6.2 Topological Dilaton Black Holes

Gao and Zhang [66] generalized the GM solution to include dilatonic topological black hole in asymptotically AdS spacetime in n-dimension,

2 2 2 2 2 ds = −U(r)dt + W (r)dr + [f(r)] dΩk,n−2 where k = −1, 0, +1 and

r n−3 r n−31−γ(n−3) 1 r n−3r U(r) = k − + 1 − − − Λr2 1 − − r r 3 r

( )−1 r n−3 r n−31−γ(n−3) 1 r n−3r W (r) = k − + 1 − − − Λr2 1 − − r r 3 r

r n−3−γ(n−4) × 1 − − r

where

r n−3γ 2α2 [f(r)]2 = r2 1 − − ; γ = r (n − 3)(n − 3 + α2)

3 Note that in this notation Λ is the eﬀective cosmological constant |Λ| = L2 where L is the curvature scale of de Sitter or Anti-de Sitter space, independent of dimensionality, c.f. the convention in Section 3.1.

We also note that for n ≥ 5, α 6= 0, we have U(r)W (r) 6= 1 in general. This is not surprising since the presence of scalar ﬁeld contributes to the stress energy tensor and thus aﬀects the geometry of spacetime leading to gttgrr 6= −1 in general [67].

The mass of the black hole, and the charge parameter q, are, with L = 1,

Γ n − 3 − α2 M = n−2 (n − 2) rn−3 + k rn−2 16π + n − 3 + α2 − 6.2 Topological Dilaton Black Holes 98 and (n − 2)(n − 3)2 q2 = rn−3rn−3. 2(n − 3 + α2) + − respectively. The charge parameter q is directly proportional to the black hole electrical charge Q [68].

Let us consider k = 0, L = 1, n = 5, α = 0 = γ which should reduce to the case of ﬂat charged black hole. Using the above formula, we compute that 8π3K3 3 M = 3r2 = π2K3r2 16π + 2 + and r 2 r 2 1 U(r) = − + 1 − − − Λr2 r r 3 r 2 r r 2 1 = − + + + − − Λr2 r r2 3 r2 r2 r2 r2 = − + + + − + = W (r)−1 r2 r4 L2

If we compare this with the explicit form of metric in Chapter 3, r2 2M Q2 U(r) = − + L2 3π2K3r2 48π5K6r4 we see that 2M Q2 r2 = , r2 r2 = . + 3π2K3 + − 48π5K6

The event horizon is the solution of r2 2M Q2 U(r) = − + = 0 L2 3π2K3r2 48π5K6r4 2M which is not 3π2K3 . Thus the notation of Gao and Zhang is somewhat misleading: one should treat r+ and r− as merely parameters that relate to the horizons instead of the horizons themselves. The authors in [68] for example, use the symbols c and b in place of r+ and r− and refer to them as “integration constants”.

6.2.1 Seiberg-Witten Action for Flat AdS Dilaton Black Holes

Consider the 5-dimensional ﬂat dilaton black hole in AdS with L = 1. As above, 2M r2 = . + 3π2K3 6.2 Topological Dilaton Black Holes 99

Since Q2 is proportional to q2, we have n − 3 2 Q2 ≡ Q2(α) = Q2(α = 0) = Q2(α = 0) . n − 3 + α2 2 + α2

That is,

2 Q2 = (48π5K6r2 r2 ). 2 + α2 + −

I.e.

Q2(2 + α2) Q2(2 + α2) r2 = = . − 96π5K6r2 5 6 2M + 96π K 3π2K3

We ﬁrst explore the comparatively easy case of α = 1 in which we have

1 3Q2 2(1) 1 r 2 2 Q2 = 32π5K6r2 r2 , r2 = , γ = = , f(r)3 = r3 1 − − . + − − 64Mπ3K3 2(2 + 1) 3 r

Thus the Euclidean metric satisﬁes

1− 2 1 r 2 r 2 3 r 2 3 g = − + 1 − − + r2 1 − − ττ r r r 1 r 2 r 2 3 = − + + r2 1 − − . r r and

( 1 )−1 − 1 r 2 r 2 3 r 2 3 g = − + + r2 1 − − 1 − − . rr r r r 6.2 Topological Dilaton Black Holes 100

The Seiberg-Witten action takes the following form Z Z Z Z Z √ 3 √ √ S = Θ gττ dτ f(r) dΩk,n−2 − 4Θ dr gττ grr dΩk,n−2 dτ

1 1 1 r 2 2 r 2 2 r 2 6 = 2πΘPLΓ r3 1 − − r2 − + 1 − − k r r r ( − 1 1 ) Z r 2 6 2 2 0 r− 0 3 r− − 2πΘPLΓk 4 dr 1 − 0 (r ) 1 − 0 reh r r ( 1 2 1 ) 2 2 2 3 Z r 2 3 3 2 r+ r− 0 0 3 r− = 2πΘP LAk r r − 1 − − 4 dr (r ) 1 − 0 r r reh r ( 2 1 1 ) 2 3 2 Z r 2 3 3 3Q 2 2M 0 0 3 3Q = 2πΘP LAk r 1 − 3 3 2 r − 2 3 2 − 4 dr (r ) 1 − 3 3 0 2 . 64π K Mr 3π K r reh 64π K M(r )

For general α > 0 in n = 5, similar calculation shows that the Seiberg-Witten action satisﬁes

2 s S 3α 2M 2−α2 α2 3 2(2+α2) 2+α2 2 2+α2 =r (J) − 2 3 2 (J) + r (J) 2πΘP LAk 3π K r Z r 2 0 0 3 α − 4 dr f(r ) (J) 2+α2 . reh where Q2(2 + α2) J := 1 − . 64π3K3Mr2

We will only focus on the α = 1 case. Note that for α = 0 the general action reduces to that of ﬂat Reissner-Nordstr¨omcase. We begin with

( 1 )−1 − 1 r 2 r 2 3 r 2 3 g = − + + r2 1 − − 1 − − . rr r r r The horizon is at 1 4 2 1 2M r = (r ) 4 = eh + 3π2K3 which is ﬁxed independent of the electrical charge, just like its asymptotically ﬂat GHS cousin.

For any ﬁxed dilaton coupling α, varying the electrical charge means equivalently, varying the parameter r−, via the relationship 3Q2 r2 = . − 64Mπ3K3 6.2 Topological Dilaton Black Holes 101

1 2 At extremal limit, the horizon becomes singular with reh = r+ = r−, i.e.

1 1 2M 4 3Q2 2 = 3π2K3 64Mπ3K3

So the extremal charge is

1 8 × 2 4 3 3 3 3 4 4 4 4 QE = 3 πM K ≈ 13.11 M K . 3 4

3 Again this is greater than QE ≈ 9.96(KM) 4 for flat AdS Reissner-Nordströmblack hole, a similar behavior as asymptotically flat counterpart.

The action vanishes at the horizon. Taking typical values of the parameters give the plot of the action as function of r as shown in Figure 6.1 below.

Figure 6.1: The action of flat AdS dilaton black hole.

Indeed, for Q = 0 the action reduces to that of uncharged flat AdS black hole (the dilaton, being a secondary hair coupled to the Maxwell field, also vanishes when electrical charge is zero), which asymptotes to a positive value. Unlike flat AdS Reissner- Nordströmblack hole with action increases to a maximal before plunging to negative, the action of flat AdS dilaton black hole is always positive. In particular, lim S(r, QE) = +∞. For any fixed charge Q, increasing the charge makes the ac- r→∞ tion started out with smaller value than the one with charge Q, but subsequently take over at some finite value of r. The value of r in which this take over occurs decreases 6.3 Holography of Dilaton Black Holes in AdS 102 with increasing charge. Thus the presence of dilaton stabilizes the black hole (at least in this special case with α = 1) against non-perturbative instability in the Seiberg-Witten sense.

Thermodynamically, dilaton black holes in AdS is stable for small coupling α but possess unstable phase for large α [68].

6.3 Holography of Dilaton Black Holes in AdS

We remark that dilaton black holes in Anti-de Sitter space have been explored for its holography [70] [71] and application in AdS/CFT, notably, Gubser and Rocha [69] argued that dilatonic black hole in AdS5 or a relative of it with similar behavior might be dual to Fermi liquid. Since there are black holes in Einstein gravity which are thermodynamically unstable while dynamically stable (c.f. the conjecture of Gubser and Mitra [58]), more works are needed to establish the dynamical (in)stability for dilaton black holes in Anti-de Sitter space for large α. A generic phase diagram involving Fermi liquid allows the Fermi liquid to attain zero temperature (Figure 6.2). In the dual description therefore, we should expect that the black hole should be allowed to reach extremal charge without subjected to the Seiberg-Witten instability, which we have shown to be the case at least for α = 1 coupling. More works need to be done to study the stability in the general case.

Figure 6.2: The phase diagram of heavy-fermion systems, where δ is an adjustable parameter such as pressure or chemical doping. The diagram is from [72]. Conclusion

The main theme of this thesis is the inter-connection between various fields of physics − notably between gravity (general relativity and string theory), particle physics (Yang- Mills field, quark-gluon plasma), condensed matter (CFL quark matter, Fermi liquid and color-superconductor, although we did not discuss much about them), cosmology (dense state of matter in the early universe) and astrophysics (compact stars). This unifying scheme is very exciting, at least for me on a personal level. I like differential geometry when I first learned about it for the same reason − it inter-relates various themes of mathematics. To quote John Opera in his book Differential Geometry and Its Applications,

It is a subject which allows students to see mathematics for what it is − not the compartmentalized courses of a standard university curriculum, but a uniﬁed whole mixing together geometry, calculus, linear algebra, diﬀerential equations, complex variables, the calculus of variations, and various notions from the sciences.

The AdS/CFT correspondence has the same role in physics for connecting ideas in various concepts in physics. Nevertheless the correspondence is far from being proved ﬁrmly in stone. Ultimately physics has to be experimentally veriﬁed, and more work has to be done to develop this correspondence, as well as to establish whether it is indeed a correct theory.

Regardless, we note the interesting fact that string theory was ﬁrst formulated as an attempt to understand quarks, but was then replaced by the better theory of Quantum Chromodynamics; it has now come to a full circle by its application to understand quark matter via AdS/CFT. However, AdS/CFT is not equivalent to string theory, even if this principle is correct, it says very little about the prospect of string theory as a theory of quantum gravity, which remains elusive. 6.3 Holography of Dilaton Black Holes in AdS 104

We shall end by a quote by Leon Lederman and David Schramm:

At ﬁrst glance, all of this sounds like medieval mystics discussing the music of the spheres, angels on the head of a pin, or some similar early approach to cosmology. Is it just a mathematical game we are playing, is it just semantics, or is it reality?

Figure 6.3: Many angels dancing on the head of a pin (artist unknown). Appendix A

Penrose Diagram

To see a world in a grain of sand And a heaven in a wild ﬂower, Hold inﬁnity in the palm of your hand And eternity in an hour.

A Penrose diagram, named after mathematical physicist Roger Penrose, is a two- dimensional diagram that captures the causal relations between different points in spacetime. A penrose diagram is also called conformal diagram or Carter-Penrose diagram. It is similar to spacetime diagram where the vertical axis represents time, and the horizontal axis represents space, and slanted lines at an angle of 45◦ correspond to light rays. However, unlike spacetime diagram, a Penrose diagram uses conformal transformation to map infinities into a finite diagram, thereby allowing us to study the causal structure of spacetime. Here we look at Penrose diagram of Minkowski space, and various black hole solutions.

The most basic question we can ask about two points x and y in a given spacetime is about their causal relation. That is, is y in, or outside of the past or future light cone of x. Because of the highly distorted geometry near a black hole, causal structure becomes important aspect for black hole research, but in order to understand Penrose diagram, we started oﬀ with the simplest spacetime - Minkowski spacetime.

Penrose Diagram for Minkowski Spacetime

Recall that Minkowski spacetime has the ﬂat metric ds2 = −dt2 + dx2 + dy2 + dz2, or when expressed in the polar coordinates, ds2 = −dt2 + dr2 + r2(dθ2 + sin2 θdφ2), where 106

−∞ < t < +∞, and 0 ≤ r < ∞. Of course technically speaking the polar coordinate has coordinate singularity at r = 0 and we should have covered the neighborhood by another coordinate patch. But we will not be concerned by this technicality here.

Our objective now is to represent the infinite Minkowski spacetime in a finite diagram. So one might try to, for example, re-scale the timelike and radial coordinates so that they cover a finite range. One possible function for re-scaling is the arctan function π π 0 since it is bounded between − 2 and 2 . By introducing new coordinates t = arctan(t) and r0 = arctan(r), we can convert the Minkowski metric into the form

1 1 ds2 = − dt02 + dr02 + tan2 r0(dθ2 + sin2 θdφ2), cos4 t0 cos4 r0

π 0 π 0 π with 2 < t < 2 , and 0 ≤ r < 2 . Now the new coordinates have ﬁnite ranges. dt0 cos2 t0 However the slope of the light cones given by dr0 = ± cos2 r0 , which is not equal to ±1. This is not particularly useful as ideally we want to preserve light cone structures and null geodesics should be mapped into null geodesics. In other words, we want a transformation which preserves angles - a conformal transformation. We discuss this in a more rigorous manner:

Suppose M is a manifold endowed with metric gµν. If Ω is a smooth, strictly positive 2 function, then the metricg ˜µν = Ω gµν is said to arise from gµν due to conformal transformation. And we say that gµν andg ˜µν are conformally related or conformal to each other. The angles on manifolds are measured using the generalised cosine law: If Xµ ν and Y are 2 vectors on a manifold M with metric gµν, then the angle θ between the vectors is given by:

g XµY ν cos θ = µν . p α βp γ δ gαβX X gδγX X

Clearly the angles between two vectors are the same regardless of whether it is measured 2 with respect to the original metric gµν or the conformally related metricg ˜µν as the Ω terms cancels out. Also, the ratio of the length of any two vectors measured by the two metrics remain the same for the same reason, and clearly null curves with respect to one metric is also null with respect to the other.

This is all good, but it tells us nothing about how to construct a suitable conformal metric. It may take sometime to realize that even for the simple case of Minkowski spacetime, we need to change coordinate systems before re-scaling. Indeed, by introducing new coordinates deﬁned by u := t − r and v := t + r, we can transform this metric into the following form: 107

1 ds2 = −dudv + (u − v)2(dθ2 + sin2 θdφ2). 4

It is clear that the (u, v) axes are rotated with respect to the (t, r) axes by 45◦, so that radial light rays travel on lines of constant u or constant v. This construction is useful in many settings, for example, when scaled by a factor √1 , it is called the 2 light-cone coordinates: x+ = √1 (t + r), x− = √1 (t − r), which is useful in, for example, 2 2 quantization of relativistic string (See, for example, [73]).

With the (u, v) coordinates, we now perform re-scaling using arctan function as before and introduce new coordinates (U, V ) with the corresponding (T,R) coordinates such 1 1 π π that U = arctan(u) = 2 (T − R) and V = arctan(v) = 2 (T + R), with − 2 < U < 2 , π π − 2 < V < 2 , and U ≤ V (since u ≤ v). This transforms the Minkowski metric to the form:

1 ds2 = −4dUdV + sin2(V − U)(dθ2 + sin2 θdφ2) . 4 cos2 U cos2 V

If we were to convert back to the associated (T,R) coordinate, using T = V + U and R = V − U, we obtain

ds2 = Ω−2(T,R) −dT 2 + dR2 + sin2 R(dθ2 + sin2 θdφ2) , where Ω = 2 cos U cos V = cos T + cos R. So we see that Minkowski metric ds2 is conformally related to ds˜2 by ds˜ 2 = Ω2(T,R)ds2 = −dT 2 +dR2 +sin2 R(dθ2 +sin2 θdφ2), with ranges given by 0 ≤ R < π and −π < T < π. The spatial part of this metric is a three-sphere with constant curvature. This is ﬁne although the original Minkowski spacetime is ﬂat. The conformally related metric is consider “unphysical” and its only purpose is to help us to understand the causal structures in the original metric.

Looking back at the Minkowski metric, it has three extremal points, namely t = ∞, t = −∞ and r = ∞. The point r = −∞ is excluded by the definition of polar coordinates which demand r ≥ 0. In (T,R) coordinates, these correspond to the points (0, π), (0, −π) and (π, 0) respectively, which if you look carelly, lies outside the range we mention before: 0 ≤ R < π and −π < T < π. This is fine because these points are not part of Minkowski space, but the conformal infinity. The union of the original spacetime and its conformal infinity is called conformal compactification. In other words, conformal infinities are 3-dimensional boundaries of 4-dimensional regions of the original spacetime manifold, defined by Ω = 0. We introduce the following infinities: 108

I+: Future timelike infinity (T = π, R = 0); I0: Spatial infinity (T = 0,R = π); I−: Past timelike infinity (T = −π, R = 0); J +: Future null infinity (T = π−R, 0 < R < π); J −: Past null infinity (T = −π + R, 0 < R < π).

These are shown in the following Penrose diagram (Figure A.1).

Figure A.1: Penrose diagram for Minkowski Spacetime.

Now we shall spend some time understanding the Penrose diagram, namely, we are interested to ﬁnd out how do geodesics look like on the diagram.

Analyzing the Penrose Diagram

Note that a generic point on the Penrose diagram is a 2-sphere (but certainly not of the same size throughout the diagram). However I+,I0,I− are points since R = 0 and R = π corresponds to the poles of S3. On the other hand, J + and J − are hypersurfaces with topology R × S2. We start with spacelike geodesics. The timelike geodesics are then curves orthogonal to the spacelike ones. A spacelike geodesic in Minkowski spacetime have constant time coordinate and extend to spatial infinity. Note that spatial infinity is like “point at 109 infinity” in complex analysis of which we imagine as follows: given any fixed time t, we can extend a curve to any direction of space off to infinity, and then we identify all those “infinities” into one which we call I0. In complex analysis, this point corresponds to the pole of the Riemann sphere. So all spacelike geodesics converge to I0. For T = 0 in the Penrose diagram, the geodesic is represented by the line segment joining the origin to I0. Indeed the spacelike geodesics are those curves which are orthogonal to the T -axis and which converge to I0. The timelike geodesics, on the other hand, start at I− and converge to I+. All these are shown in Figure A.2.

Figure A.2: Penrose diagram for Minkowski Spacetime with some timelike geodesics in red, spacelike geodesics in blue and null geodesics in orange. 110

For those who prefer more symmetry, we can draw the Penrose diagram in the following way, with topological identiﬁcations on the boundaries, and hence the diagram on the left half plane is identiﬁed with the diagram on the right half plane as in Figure A.3 below.

Figure A.3: Alternative presentation of Penrose diagram for Minkowski Spacetime.

One can even rotate the diagram around to produce a three-dimensional double cone, with I0 now becomes a circle, but still identiﬁed as a point, similarly J + and J − become the surface of the double cone, but are still identiﬁed as points.

Penrose Diagram for Schwarzschild Black Hole

We play the same game with Schwarzschild metric which describes a non-rotating non- charged spherical black hole:

2m 2m−1 ds2 = − 1 − dt2 + 1 − dt2 + r2(dθ2 + sin2 θdφ2). r r

Firstly, transform this metric using Kruskal-Szekeres coordinates (U, V, θ, φ). If r > 2m, the new coordinates U and V are related to the t and r coordinates by the following 111

1 r 2 t U = − 1 er/4m cosh , 2m 4m 1 r 2 t V = − 1 er/4m sinh . 2m 4m

On the other hand, if r < 2m, we have the relations

1 r 2 t U = 1 − er/4m sinh , 2m 4m 1 r 2 t V = 1 − er/4m cosh . 2m 4m

These give Schwarzchild metric in the following form:

32m3 ds2 = e−r/2m(−dV 2 + dU 2) + r2(dθ2 + sin2 θdφ2). r

In order to construct the Penrose diagram, we introduce new coordinates u and v v−u v+u defined by U = 2 and V = 2 . Similar to the light-cone coordinates of Minkowski spacetime, we see that the (u, v) coordinate is obtained from rotating (U, V ) coordinates by 45◦ so that light rays move on curves of constant u or v. Now bringing infinities into finite range by using arctan function: Introduce (yet another!) coordinates (u0, v0) and (U 0,V 0) by u0 := arctan(u) := V 0 − U 0 v0 := arctan(v) := V 0 + U 0. Light rays move on curves of constant u0 and v0, i.e. the 45◦ lines in the U 0V 0-plane. 0 0 π 0 0 π The ranges for u and v are − 2 < u , v < 2 . Now from the defined relations of U and V in terms of r and t, we see that

r − 1 er/2m = U 2 − V 2 = (U + V )(U − V ). 2m

On the horizon r = 2m, it follows that U = ±V . Then since u = V −U and v = V +U, we must have u = v = 0, and so u0 = arctan u = 0 and v0 = arctan v = 0. That is V 0 − U 0 = 0 and V 0 + U 0 = 0. Thus the horizon is represented by the lines V 0 = ±U 0.

At the singularity r = 0, and V > 0, we see by the deﬁning relations that −1 = U 2 − V 2 = uv. 112

Consider the equation u0 + v0 = 2V 0 = arctan u + arctan v. Then

u + v 1 tan(u0 + v0) = tan(arctan u + arctan v) = = (u + v) = V. 1 − uv 2

π 0 0 π So − 2 < u + v = arctan V < 2 . At the boundary of spacetime when V = ∞, this π 0 1 0 0 1 π corresponds to tan V = 2 , and V = 2 (u + v ) = 2 arctan V = 4 . At the same time, 0 1 0 0 π π U = 2 (v − u ) is bounded between − 4 and 4 . Similarly, for r = 0 and V < 0, 0 π the singularity maps into the line V = − 4 . For r 2m, as Schwarzchild metric is asymptotically Minkowski, the Penrose diagram for Schwarzschild spacetime and that of Minkowski spacetime should eﬀectively be the same. We will use curvy lines to represent the singularity. These lines are orthogonal to timelike curves, and so the singularity is called spacelike singularity. Note also that I+ and I− are distinct from r = 0, as there are, in fact a lot, timelike paths that do not hit the singularity.

The Penrose diagram for maximally extended Schwarzschild spacetime is then obtained by time reversal symmetry t → −t, where we have a region which can be interpreted as white hole. In fact there are 4 regions of spacetime as shown in the next ﬁgure: Region I corresponds to our assumed asymptotically Minkowski universe, region II is a black hole, region III is another asymptotically Minkowski universe and Region IV, as mentioned, can be thought as whitehole region.

Figure A.4: Penrose diagram for Schwarzschild spacetime. Any timelike curve, such as the one in green, that passes through the horizon, colored gold, has no choice but to hit the singularity. 113

The central point with coordinate (U 0,V 0) = (0, 0) is called the Einstein-Rosen bridge. π Taking a surface of constant t and consider the equatorial plane θ = 2 , we can reduce Schwarzschild metric to the 2-dimensional surface with Euclidean metric

2m−1 ds2 = 1 − dr2 + r2dφ2. r

We can see the “throat” shape using Embedding diagram. Indeed the metric of the 3-dimensional Euclidean ambient space is

dl2 = dz2 + dr2 + r2dφ2, which on z = z(r), becomes

dl2 = (1 + (z0)2)dr2 + r2dφ2.

2 2 0 2 2m −1 p Setting ds = dl yields (1+(z ) ) = 1 − r , i.e. z(r) = ±2 2m(r − 2m). Plotting this function gives us a “throat”.

Figure A.5: MAXIMA plot of upper half of the throat. 114

We now fill in the geodesics of the spacetime. The region outside the horizon of both asymptotically flat universe is similar to that of Minkowski diagram. However, note the switch of spacelike and timelike coordinates inside the black hole. The Einstein- Rosen bridge is a non-traversable wormhole as only spacelike curves pass through the “throat”, which is clear from the Penrose diagram. Therefore the two asymptotically flat universes cannot communicate with each other short of using tachyons, which as we know, violate causality.

Figure A.6: Penrose diagram for maximally extended Schwarzschild spacetime with curves in red denoting constant r surfaces, while blue curves are constant t surfaces. 115

Gravitational Collapse of a Star into Black Hole

The maximally extended Schwazschild spacetime describes an eternal Schwarzschild black hole, one that always has been and always will be. This is not a good description for astrophysical black holes. In astrophysical context, black holes are the results of gravitational collapse of stars (or possibly clusters of stars or interstellar gas), the exterior gravitational ﬁeld can be modelled by the Schwarzschild metric provided the star rotation is negligible. The interior of the star is of course not vacuum, and hence not described by Schwarzschild metric. Therefore there is no wormhole connecting to another universe, not even a non-traversable one. Similarly, the past of this collapsing star spacetime is not the same as that of the maximally extended Schwarzschild metric. Notably, there is no white hole. The Penrose diagram of such black hole is simply as shown in Figure A.7, where the shaded region contains matter of the star interior:

Figure A.7: Penrose diagram for a black hole formed from a collapsing star.

A construction of this Penrose diagram is described in [11] of which we reproduce here. Basically we describe the spacetime by gluing together Minkowski spacetime and the Schwarzschild spacetime. To be specific, we begin with Penrose diagram for Minkowski spacetime. Consider incoming null line represent infalling massless shell with energy M, separating the Penrose diagram into two parts: region A and B. Region A represents spacetime interior to the infalling spherical shell, and thus is flat. The region B must now be modified since by Birkhoff’s theorem the geometry outside the shell is Schwarzschild. We consider the Penrose diagram of Schwarzschild spacetime with the same infalling massless shell of energy M. Now we have again regions A0 and B0, with 116

B0 being the relevant part that describes geometry outside such shell. So we take region A and B0 and glue them together, hence obtaining the Penrose diagram above.

Figure A.8: Gluing Penrose diagrams. The diagrams are from [11]. 117

Penrose Diagram for Reissner-Nordstr¨omBlack Holes.

Throwing electric charges into a Schwarzschild black hole turned it into a Reissner- Nordstr¨omblack hole, with metric

2m Q2 2m Q2 −1 ds2 = − 1 − + dt2 + 1 − + dt2 + r2(dθ2 + sin2 θdφ2). r r2 r r2

Not unlike Schwarzschild black hole, at any given time t, the Reissner-Nordstr¨omblack hole is a 2-sphere with singularity at r = 0. However we now have, in general, two event horizons given by the roots of the quadratic equations

r2 − 2mr + Q2 = 0,

p 2 2 p 2 2 namely, r+ = m + m − Q and r− = m − m − Q , where r+ ≥ r− ≥ 0 with equality if and only if m2 = Q2. The signature changes twice: When one passes the outer horizon, r+, r-coordinate becomes timelike while t becomes spacelike, which is similar to Schwarzschild black hole. As such, one is compelled to move towards the inner horizon r− (also called the Cauchy horizon), just like one is compelled to move towards Schwarzschild singularity. But once the inner horizon is crossed, signature of spacetime dimensions swicth once more. One can therefore move freely inside the space bounded by the inner horizon, and avoiding the singularity. All these we get from nothing but high school mathematics of the quadratic function r2 − 2mr + Q2. The Penrose diagram for Reissner-Nordstr¨omblack hole must capture these information.

There are in fact 3 cases to consider:

CASE 1: m2 > Q2

As mentioned above, this gives us two event horizons, one within the other. If you were to fall into such a black hole, you would pass through the outer and then the inner horizon. If you have suicidal intend, you must try hard to hit the singularity - because it can be shown that Reissner-Nordstr¨omblack hole admit repulsive gravity near the singularity - certainly not what ordinary people will expect of a black hole!

Now, you may also, upon ﬁnding the interior of the black hole boring, decided to enter the inner horizon again, but now from the interior. Then r will again become timelike coordinate but with reversed orientation, so that you are forced to move in the direction of increasing r, and eventually be spit out from the outer horizon. How can a black hole spit anything out? We interpret this time-reversal version of black hole as again, a white hole. But there is no telling whether this universe that you now emerged in is the same as the original one! Now this journey in and out of black hole (and white hole) can be repeated ad inﬁnitum. 118

Figure A.9: Penrose diagram for Reissner-Nordstr¨omblack hole. This diagram is adapted from [9].

One notices the singularity is orthogonal to spacelike curve, and is therefore called timelike singularity, and as mentioned before, is avoidable. Although we have an infinite tower of asymptotically flat universes with their black holes and white holes connected via traversable wormholes, it is possible to make topological identification along the red lines in the Penrose diagram as shown so that spacetime contains closed loop. But as widely believe, closed timelike curves are bad for physics and therefore such spacetime may well be unphysical. Even the geometry of Reissner-Nordströmblack hole as mentioned above may be unphysical. The inner horizon may be perturbatively unstable due to huge amount of blue-shifted radiation that enters the black hole. Thus bringing charges into a Scwharzschild black hole may not save you after all. 119

CASE 2: m2 = Q2

This is a very interesting case where the mass is in some sense balanced by the charge. It is known as the extremal Reissner-Nordstr¨omblack hole. Now from the quadratic function r2 − 2mr + Q2, it is readily seen that r = m gives the only event horizon, corresponding to the repeated roots of the quadratic function. As such this quadratic function is always nonnegative, which means that r coordinate never become timelike, i.e. r is spacelike on both sides of the horizon deﬁned by r < m and r > m. Thus the singularity is timelike, and so avoidable. The Penrose diagram is given in Figure A.10.

Figure A.10: Penrose diagram for extremal Reissner-Nordstr¨om black hole. This diagram is adapted from [9].

Note that the causal structure is totally different from that of a regular Reissner- Nordströmblack hole no matter how close to the extreme limit it is. We therefore expect physical properties of the family of Reissner-Nordströmblack holes to be discontinuous at the extreme limit. This case is also unstable since a tiny amount of infalling matter will reduce it to the first case.

CASE 3: m2 < Q2

In this case the quadratic r2 − 2mr + Q2 > 0 for all r. There is no event horizon. As with Schwarzschild solution, Reissner-Nordstr¨omis asymptotically ﬂat. So one expects the Penrose diagram to look like that of Minkowski spacetime, except for the pesky 120 singularity at r = 0 (Figure A.11). Here the singularity is not shrouded behind event horizon, and is called a naked one. The Cosmic Censorship Conjecture holds that such situation is not allowed.

How then can the singularity inside a regular or extremal Reissner-Nordstr¨omblack hole be timelike and thus observable by someone who entered the black hole? Assuming that the Cosmic Censorship Conjecture hold, this is another reason to postulate that timelike singularity is unstable.

Figure A.11: Penrose diagram for naked Reissner-Nordstr¨om black hole.

The Reissner-Nordstr¨omblack hole can be generalized to include magnetic charge P , despite magnetic monopole has yet to be discovered. This turns the metric into:

2m Q2 + P 2 2m Q2 + P 2 −1 ds2 = − 1 − + dt2 + 1 − + dt2 + r2(dθ2 + sin2 θdφ2). r r2 r r2

The 3 cases then carry through with minimal modiﬁcations. 121

Penrose Diagram for Kerr Black Holes

A rotating non-charged black hole is described by the Kerr metric:

2mr 4mar sin2 θ ρ2 sin2 θ ds2 = − 1 − dt2− (dtdφ)+ dr2+ρ2dθ2+ (r2 + a2)2 − a2∆ sin2 θ dφ2 ρ2 ρ2 ∆ ρ2

where ∆(r) := r2 − 2mr + a2 and ρ2(r, θ) := r2 + a2 cos2 θ, with a being the angular momentum per unit mass. Not for the faint-hearted!

The coordinates (t, r, θ, φ) are known√ as Boyer-Lindquist√ coordinates, and are related to Cartesian coordinates by x = r2 + a2 sin θ cos φ, y = r2 + a2 sin θ sin φ, and z = r cos θ. The event horizons are given by the roots of the quadratic equation ∆(r) := 2 2 r − 2mr + a = 0. So,√ similar to the Reissner-Nordstr¨omblack√ hole, we have the 2 2 2 2 outer horizon r+ = m + m − a and r− = m − m − a . Note that both r+ and r− are positive if m > a. There are again 3 cases to consider: regular, extremal, and naked. Since they are similar to Reissner-Nordstr¨omcase, we only discuss the regular one, which is also physically most interesting.

Another interesting feature is the singularity, which occurs at ρ := r2 + a2 cos2 θ = 0. This means that both r and a cos θ vanish.√ So the singularity√ corresponds to r = 0 π 2 2 2 2 and θ = 2 . From the equations x = r + a sin θ cos φ, y = r + a sin θ sin φ, and z = r cos θ, one sees that this corresponds to z = 0, and thus a ring x2 + y2 = a2 on the z = 0 plane in Euclidean 3-space. This singularity is timelike and therefore avoidable. There are other features of interest such as the ergosphere, but technically they are not part of black hole, and so we will not consider them.

As with Reissner-Nordströmblack hole, we do not derive the Penrose diagram but simply discuss it. It turns out that the Penrose diagram for Kerr black hole is very similar to that of Reissner-Nordströmblack hole, except for the yellow regions which are absent in Reissner-Nordströmcase (Figure A.13.). These yellow regions indicate that it is possible to pass through the ring singularity to arrive at a asymptotically region with r < 0. In other words, going pass the ring singularity, one exits to another universe, which is not an identical copies of the original. This can be seen by again 2 2 considering the quadratic ∆ := r − 2mr + a . Since this only vanishes at r+, r− > 0, the new spacetime that one exited to does not admit any event horizon! This universe is sometimes called the negative universe.

One can actually make Reissner-Nordstr¨omblack hole to have even more similar looking Penrose diagram to Kerr’s one by considering imaginary charge, i.e. Q2 < 0. But in 122 such case the singularity becomes spacelike, and one had no choice but to hit it, and reappear in another universe, where r− now lies, and continue ad inﬁnitum through an endless collection of black holes and white holes.

Finally, one should also note that generic points in the Penrose diagram are not (met- rically) 2-spheres. Since the black hole is rotating, the horizon actually squashed along the rotation axis, and diﬀerent θ values give diﬀerent geometry. In other words, constant radius r = r± in the Boyer-Lindquist coordinate does not correspond to spherically symmetry, which can be readily checked by substituting r = r± into the Kerr metric and analyze the geometry of the spatial part.

Figure A.12: Kerr and me at Noyori Conference Hall, Higashiyama Campus, Nagoya University, Japan during the 17th Workshop on General Relativity and Gravitation in Japan, December 2007. 123

Figure A.13: Penrose diagram for Kerr black hole. Note the event horizons are now labelled by r± instead of wavy lines. This diagram is adapted from [9]. Yellow regions are dubbed ‘negative universes’. Appendix B

Black Hole Temperature: A Primer

We assume knowledge in quantum mechanics at the level of path integral formulation. Recall that the usual path integral at ﬁnite temperature has the exponential term i Z +∞ exp Ldt ~ −∞ in the propagator, where L is the Lagrangian.

It can be shown using path integral formulation that the usual recipe to obtain the partition function can be summarized as:

(1) Carry out Wick rotation by deﬁning imaginary time τ = it.

(2) Introduce the Euclidean version of the Lagrangian: LE = −LM (τ = it)

(3) Impose τ ∈ [0, β~) and periodicity over τ.

Then i Z +∞ exp Ldt ~ −∞ which governs propagators of field theory at finite temperature in Minkowski space is Wick rotated to the Euclidean version 1 Z β~ exp − LEdτ ~ 0 which governs propagators of Euclidean quantum field theory with periodic time. See for example page 262 of [74]. 125

In other words, Euclidean quantum ﬁeld theory in (n + 1)-dimensional spacetime with 0 ≤ τ < β corresponds to quantum statistical mechanics in n-dimensional space. To quote [74],

Surely you would hit it big with mystical types if you were to tell them that temperature is equivalent to cyclic imaginary time. At the arithmetic level this connection comes merely from the fact that the central objects in quantum physics e−iHT and in thermal physics e−βH are formally related by analytic continuation. Some physicists, myself included, feel that there may be something profound here that we have not quite understood.

Now we consider a typical n-dimensional spherical black hole metric of the form after Wick-rotation: ds2 = V dτ 2 + V −1dr2 + r2dΩ2 where V is a function of r.

Now the neighbourhood near the horizon is trying to look like R2 × Sn−2. This can be achieved by having (r, τ) to behave like polar coordinates. This is where the periodic condition imposed on τ comes in helpful. However, one must make sure that the periodicity is nice and does not cause conical singularity. In other words, we have to make sure that the inﬁnitesimal ratio of the circumference (going around in τ) to the radius (moving in r) is in fact 2π as we approach the origin of R2 (p.412 of [37]) which is at the horizon r = reh. This procedure goes by the name ensuring regularity of the Euclidean section.

For constant r and constant angles, we have ds = V 1/2dτ, and so the circumference C is given by β Z 1 1 C = V 2 dτ = V 2 β. 0 Now requiring that the said ratio to be 2π implies that

1 β d(V 2 ) 2π = lim . − 1 r→reh V 2 dr This simpliﬁes to 4π = V 0| . β r=reh Consequently for Reissner-Nordstr¨omblack hole, with

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