Hunting for Micro Black Holes

Home , Micro black hole, Schwarzschild radius, Trapped surface

University of Amsterdam Institute for Theoretical Physics

Master’s Thesis

Author: Supervisor: Lucas Ellerbroek Prof. dr. Erik Verlinde

October 30, 2009 Alice laughed. “There’s no use trying,” she said: “one can’t believe impossible things.” “I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”

Lewis Carroll, Through the Looking Glass (And What Alice Found There), 1871

Cover illustration: Sir John Tenniel, “Alice’s Adventures in Wonderland”, 1865 Abstract

In braneworld scenarios, the visible universe is pictured as a brane embedded in a higher- dimensional compact space, through which only the gravitational field propagates. In these scenarios, gravity could be the dominant force at TeV-scale energies. The discovery of hidden dimensions would therefore be a major breakthrough in finding a solution to the hierarchy problem. A phenomenological consequence of these scenarios is that black holes can exist at energy scales as low as 1 TeV. This leads to predictions about black hole events at future LHC experiments, that would prove the existence of hidden dimensions. This possibility sparked a lot of discussion among scientists, and a lawsuit was filed by worried citizens, afraid to be accreted by this black hole. However, careful study of Hawking radiation of TeV-scale black holes indicates that they evaporate very rapidly. Also, the signature of the radiation spectrum will probably be obscured by other emission effects. This narrows the chances of observing a black hole in a detector. In this research, a model is proposed of the creation of a black hole within a hot quark-gluon plasma. Under certain conditions, the plasma could be partly absorbed by the black hole, thus extending its lifetime. This could improve the chances of detecting a microscopic black hole. Contents

1 Introduction 5

2 Black holes in 4 dimensions 8 2.1 Formation ...... 8 2.1.1 Gravitational collapse ...... 9 2.1.2 High-energy collisions ...... 10 2.2 Properties ...... 13 2.2.1 The Unruh eﬀect and Hawking radiation ...... 14 2.2.2 Superradiance ...... 20 2.2.3 Schwinger discharge ...... 22 2.3 Evaporation ...... 25

3 Braneworld scenarios 26 3.1 Motivation ...... 26 3.2 The ADD scenario ...... 27

4Blackholesin4+n dimensions 32 4.1 Formation: high-energy collisions ...... 32 4.2 Properties ...... 34 4.2.1 Schwarzschild radius ...... 34 4.2.2 Hawking temperature ...... 35 4.2.3 Greybody factors ...... 36 4.3 Evaporation ...... 38 4.3.1 Balding phase ...... 39 4.3.2 Spindown phase ...... 40 4.3.3 Schwarzschild phase ...... 40 4.3.4 Planck phase ...... 44 4.3.5 Lifetime ...... 44

5 Primordial black holes 46 5.1 Mass limits for PBHs in braneworld scenarios ...... 46 5.2 Primordial high-energy collisions ...... 47

2 6 Black holes in the lab 51 6.1 The quark-gluon plasma ...... 51 6.1.1 Creation ...... 52 6.1.2 Cooling and Expansion ...... 53 6.1.3 Hadronization ...... 54 6.2 The QGP Absorption Model ...... 54 6.2.1 Growth and decay ...... 55 6.2.2 Numerical results ...... 58 6.2.3 Theoretical Issues ...... 64 6.2.4 The Gunﬁre Model ...... 66 6.2.5 Feasibility ...... 66

7 Conclusions 69

A Lorentz Force Lagrangian 72

B Emission coeﬃcient for black holes with ﬁxed mass 73

C Graphs 75 4 CONTENTS

Notation

The equations in this thesis are written in SI units unless speciﬁed otherwise. The reason for this is that the numerical results become more insightful. The following relation between the gravitational constant and the Planck mass is used: c G = 2 (0.1) Mpl

The four-dimensional value of the Planck scale is denoted by Mpl.Foritshigher- dimensional equivalent the symbol M∗ is used. Chapter 1

Introduction

With much anticipation, the scientific world watches the Large Hadron Collider in Geneva, where for the first time TeV-energies will be reached in particle experiments. Some theorists speculated (for example [1, 2, 3]) that one of the products created in these collisions might be a black hole. When these predictions were published, a big hype was spawn among scientists and non-scientists alike. The small black hole originally proposed became blown up to gargantuan proportions by the media. Headlines featured doomsday scenarios, and even a lawsuit was filed to pull financial support out of the project. Fortunately for the European Organization for Nuclear Research (CERN), the judge ruled that the Apocalypse lay ‘outside of her jurisdiction’. In the meanwhile, scientific evidence had been comprised by some of the leading theorists on this topic that there was nothing to worry about [4, 5]. The official safety assessment from CERN was endorsed by numerous leading physicists, including Stephen Hawking:

The world will not come to an end when the LHC turns on. The LHC is absolutely safe. ... Collisions releasing greater energy occur millions of times a day in the earth’s atmosphere and nothing terrible happens [6].

Some questions that immediately come to mind are: how can black holes be fabricated at LHC? What will happen to them upon creation? Can they even be detected? These questions are the backbone of this thesis. To answer the ﬁrst question: black holes are gravitational objects, so they can only exist at TeV-scales if gravity acted on this scale. And according to our current knowledge of nature, it does not. The relative weakness of gravity as opposed to the other three forces is called the hierarchy problem. It is one of the major outstanding problems in physics. Among its proposed solutions are theories called braneworld scenarios. In these scenarios, the scale of gravity, which is the Planck scale, is lowered down. This means that gravity becomes stronger on short distances than we would expect it to be according to Newtonian physics. This is explained by the existence of extra, hidden dimensions where only gravity propagates. Because some of the gravitational ﬂux ‘leaks out’ to the extra dimensions, only a weak gravity force remains in the observable universe. In this way, the

5 6 CHAPTER 1. INTRODUCTION observed Planck scale is merely an effective value of a fundamental scale, which can be as low as the electroweak scale, around 1 TeV. In this line of reasoning, it comes as no surprise that should these scenarios represent reality, and gravity is so strong on such short distances, the creation of black holes may well occur at LHC experiments. Current measurements of gravity do not rule out these braneworld scenarios. Hence the excitement upon this prediction. The detection of a black hole would tell us much about the structure of spacetime, it would prove the existence of extra dimensions, and it might solve the hierarchy problem. Also, for the first time it would be possible to probe the quantum gravity regime. On the other hand, it might mean the end of particle physics as we know it, since if scattering events hide behind event horizons, none of them will be detected. The consequences for physics may be dire, but the Earth will not be obliterated by these microscopic black holes. The argument mentioned by Hawking, that cosmic ray observations tell us that black holes formed in collisions do not pose a threat to the Earth, is central in the safety assessment released by CERN. The expectation is that these black holes are so tiny, that they will evaporate through thermal radiation, originally proposed by and named after Hawking himself. Since the temperature of black holes is inversely proportional to their size, microscopic black holes will be extremely hot. They will decay rapidly in detectors, before they get the chance to accrete matter. It is even a question if we would even measure this radiation, since the spectrum is not exactly thermal. For this reason, it would be interesting to know if black holes produced at accelerators can somehow be made larger, before they get a chance to evaporate. One possibility might be to force the black hole to absorb large quantities of a hot plasma. It is possible that this has already happened in the early radiation-dominated universe. In the future we might even be able to simulate this situation in a laboratory environment. In particular, we could make a black hole inside a very hot quark-gluon plasma, making the hole grow by absorption. This thesis explores the exotic events that TeV-scale Gravity (TeVG) implied by braneworld scenarios makes possible. It follows the storyline laid out in the paragraphs above. In chapter 2, the general properties of black holes in four dimensions are reviewed. The different means of particle emission by black holes, including Hawking radiation, are derived. The differences between large and small black holes are made clear. It is also explained why microscopic black holes have not been produced on Earth up until this moment. In chapter 3, the con- cept of braneworld scenarios is introduced. One scenario in particular, the Arkani-Hamed Dimopoulos Dvali (ADD) model, is highlighted as the modification of gravity can be deduced most elegantly from its geometry. Black hole behavior in the context of this model is investigated in chapter 4. The general properties of black holes are compared between 4 dimensions and scenarios with extra dimensions. Also, more specifically, the evolution of microscopic black holes formed in high-energy collisions is predicted. Their radiation spectrum is affected by the black hole geometry and radiation backreaction; this effect modifies the thermal spectrum. It is described by greybody factors. The chapter ends with an estimate of black hole lifetimes in detectors. The subject of chapter 5 are small black holes in a cosmological context. These primordial black holes are also affected by the braneworld scenarios. A novel creation mechanism is discussed where microscopic black holes grow by absorbing radiation in the primordial 7 plasma. This mechanism is an analogon for the model introduced in chapter 6. This chapter starts with a thermodynamic approach of the quark-gluon plasma. It is then hypothesized that a microscopic black hole created within this plasma will under certain circumstances be able to grow significantly. The feats and limitations of this model are discussed. The chapter concludes with a prediction of its feasibility with current technology. Chapter 2

Black holes in 4 dimensions

This chapter deals with the different stages of the life of a black hole in 3+1 dimensions. Dif- ferent scenarios and conditions for formation are formulated. Black hole properties, such as its Schwarzschild radius and temperature, are discussed. We also derive the main mechanisms for particle emission by a black hole. This leads to a condition for black hole evaporation in four dimensions. A black hole is a massive object, with such a strong gravitational field that nothing, not even light, can escape it once it passes the black hole’s event horizon. The distance from the horizon to the center of the hole is called the Schwarzschild radius, dependent on the black hole mass. It can be interpreted as the distance below which the escape velocity exceeds the speed of light. Apart from its mass, the only quantities that define a black hole are its angular momentum and electric and magnetic charge. This is the No Hair theorem: black holes are ‘bald’ in the sense that there are no irregularities around their edges. At a glance, black holes seem to be eternally-growing objects: they grow by attracting matter, that continues to add up to their mass. However, this does not have to be the case. Thermodynamically, black holes are perfect blackbodies – they absorb all ingoing radiation and emit thermal radiation at the Hawking temperature. In a suitable environment it could completely evaporate. In the following paragraphs we take a closer look at the thermal properties of a black hole and the conditions for its evaporation. Firstly, the processes that lead to the formation of a black hole are described.

2.1 Formation

Generally speaking, a black hole is created by putting a lot of mass within a very small volume. General relativity predicts that this mass will curve spacetime around it so an event horizon is formed outside the object. Black holes come in diﬀerent sizes; these are listed in table 2.1. The focus of this thesis will be on ‘small’ black holes: primordial and microscopic. In this section, two formation scenarios are reviewed: gravitational collapse (leading to large black holes) and high-energy collisions (leading to small black holes).

8 2.1. FORMATION 9

Mass (kg) Type Formation Large ∼ 1035 − 1040kg Supermassive Gravitational collapse of gas clouds ∼ 1033 kg Intermediate size Collisions of smaller black holes ∼ 1031 kg Stellar Gravitational collapse of stars Small ∼ 1011 − 1020 kg Primordial Gravitational collapse in the primordial plasma 10−24 kg Microscopic High-energy collisions of particles

Table 2.1: Classiﬁcation of black holes by size. We will identify primordial and microscopic black holes as ‘small’ throughout this thesis.

2.1.1 Gravitational collapse The pressure keeping a star from collapsing under its own mass is generated by nuclear fusion reactions inside the star. When the nuclear fuel has run out, the star starts shrinking. The release of gravitational radiation results in an explosion (a type II supernova), where the outer ring of matter is blown away. Depending on its mass, three scenarios [7] are possible for the fate of the star:

• M<1.4M (Chandrasekhar limit): The collapse is halted by the Fermi degeneracy pressure of electrons in the star. The stable remainder is called a white dwarf.

• 1.4M protons in the nuclei to form neutrons and neutrino’s. The neutrino’s are blown away, leaving a dense object consisting solely of neutrons: a neutron star.

• M>3−4M: Gravitational collapse goes on until all the mass of the star falls through aradius9GM/4c2. Buchdal’s theorem states that at this point, no static solution of the equation of state is possible. Beyond this point, black hole formation is inevitable.

White dwarfs and neutron stars can accrete matter from nearby stars in what is called a binary system; this process can result in a type Ia supernova, with a black hole as its remainder. Figure 2.1 displays a conformal diagram of black hole formation. The shaded region represents the collapsing body; the exterior region is Schwarzschild (the spherically symmetric solution to Einstein’s equation). The intersection of the dotted line and the boundary of the shaded region represents the collapse through the Schwarzschild radius

2GM r = (2.1) h c2 Beyond this point, all the mass is contained within a region of spacetime separated from inﬁnity. All matter and light falling into it will eventually hit the singularity at r =0.Such an object is called a black hole. 10 CHAPTER 2. BLACK HOLES IN 4 DIMENSIONS

Figure 2.1: Conformal diagram of gravitational collapse.

The aforementioned mass limits imply that stellar black holes cannot be lighter than 3 to 4 times the mass of the Sun. However, theory states that shortly after the Big Bang (at t ∼ 10−15 s), black holes could have been formed in the primordial plasma. Density perturbations from quantum ﬂuctuations, blown up by inﬂation, would have been subject to gravitational collapse. The result are primordial black holes (PBHs). This type of black holes is further discussed in chapter 5. The next section introduces another creation mechanism: particle collisions at high energies.

2.1.2 High-energy collisions From the previous section we gather that a black hole is formed if a collection of massive particles is crammed inside its own Schwarzschild radius. This argument was generalized by Thorne, who formulated the ‘hoop conjecture’ as follows: Black holes with horizons form when and only when a mass M gets compacted into a region whose circumference is in every direction C 2πrh [8]. Note that by this conjecture, the critical condition for black hole formation is its density, which depends on its mass. This critical density is

3c6 ρ = (2.2) c 32πG3M 2 2.1. FORMATION 11

This is the minimum density of an object that fits inside a sphere of its own Schwarzschild radius, as the hoop conjecture prescribes. For stellar-sized objects with a mass greater than the Oppenheimer-Volkoff limit, this density is achieved inevitably by gravitational collapse. Another possibility is creating a dense object through a forced crushing or compression of matter. For instance, a high-energy collision of two relativistic particles might create a black hole. Before the collision, the particles and the metric around them are Lorentz- contracted. Upon collision, the metric becomes (near) Schwarzschild. Theoretically, the maximum compression of a particle is achieved by fitting it inside a volume of its own Compton wavelength. The density is then

M 3 3c3M 4 ρmax = = 2 3 (2.3) λc 32π Since the density of an object cannot be larger than this value, the density ρ of black holes should satisfy ρc <ρ<ρmax. This sets the condition M Mpl for black hole formation. If this condition is violated, the object will be subject to quantum gravitational eﬀects. Following now is a summary of he black hole formation process as analyzed in literature.

Figure 2.2: Collision of two relativistic particles at small enough impact parameter to create a black hole.

The colliding particles can be described by two Aixelburg-Sexl shockwaves (the Lorentz- boosted version of Schwarzschild). Moving at near-light velocity, they can be considered massless so the curvature is zero except on the null trajectory; see figure 2.2.Iftheimpact parameter is small enough, b<2rh, by the hoop conjecture a black hole is formed. Penrose [9] and D’Eath and Payne [10] verified this for the case of two particles colliding at zero impact parameter. Upon collision, a closed trapped surface forms. This is a surface where the outward, future-directed null rays converge everywhere. Every worldline within this surface must have an endpoint at some finite affine parameter, implying a singularity. The closed trapped surface then corresponds with the event horizon. The area theorem suggests that once this surface exists, classically it can only grow. Figure 2.3 illustrates the formation of a closed trapped surface; the lightcones from the uppermost surface are pointing inward. 12 CHAPTER 2. BLACK HOLES IN 4 DIMENSIONS

Figure 2.3: Formation of a closed trapped surface.

The above suggests that particle collisions result in black holes only at transplanckian energies. The Planck scale has the value of 1016 TeV in four dimensions. At LHC, never before achieved collision energies will be reached. Typical experiments have E ∼ 7-11 TeV. This is still a factor 1015 below the Planck scale. Quantum field interactions will dominate the process, while gravity is too weak for the hoop conjecture to be fulfilled. The idea of black hole creation on Earth can be banished to science-fiction literature.

Or not? What if we could lower the fundamental Planck scale in our theories? If Mpl would be of order 1 TeV, black holes may well be observed at colliders in the near future. This can be true in the presence of large extra dimensions. The observation of mini black holes at the LHC would have major implications. We would learn a great deal about the structure of spacetime; unseen dimensions would be unveiled. Furthermore, the existence of ‘TeV-Gravity’ (TeVG) could solve some of the major outstanding problems in physics, such as the hierarchy problem and — when applied to early universe black holes — the abundance of cosmic dark matter. It may eventually put us on track to ﬁnd a uniﬁed theory of quantum gravity. However, black hole production at TeV scale experiments would present a problem: all hard scattering processes would be cloaked by event horizons. This could mean the end of high energy physics as we know it [11]. In chapter 4 we consider black holes in higher dimensions. We have now laid out the circumstances that can lead to black hole formation in four dimensions. Another important 2.2. PROPERTIES 13

Uncharged (Q =0) Charged(Q =0) Nonrotating (J =0) Schwarzschild Reissner-Nordstr¨om Rotating (J =0) Kerr Kerr-Newman

Table 2.2: Classiﬁcation of black holes by physical properties.

conclusion is that to create black holes through high-energy collisions, the center-of-mass energy must well exceed the Planck mass. The Planck mass therefore is the lower mass limit for the smallest black holes. The remainder of this chapter is devoted to the properties of four-dimensional black holes.

2.2 Properties

This chapter outlines the most important properties of a black hole in four dimensions: the Schwarzschild radius, temperature and means of particle emission. First of all, we turn to the most general deﬁnition of black hole: the No Hair theorem. This was brieﬂy stated before; in full, it reads

Stationary, asymptotically ﬂat black hole solutions to general relativity coupled to electromagnetism that are nonsingular outside the event horizon are fully characterized by the parameters of mass, electric charge1, and angular momentum. All other information (for which ”hair” is a metaphor) about the matter which formed a black hole or is falling into it, ”disappears” behind the black-hole event horizon and is therefore permanently inaccessible to external observers. [7]

This implies that all other properties of a black hole can be deduced from the three quantities M, Q and J. For example, the Hawking temperature follows directly from the mass. By the No Hair theorem, four diﬀerent classes of black holes can be deﬁned, according to the parameters that are zero (table 2.2).Themostsimplecaseisablackholewithno charge Q nor angular momentum J, described by the Schwarzschild metric

2 2 −1 2 2 2 ds = −h(r)dt + h(r) dr + r dΩ2 (2.4) 2GM h(r)=1− (2.5) c2r This is the type of black hole most prominently featured in this thesis, for reasons that will become clear. The Schwarzschild radius was described earlier as the point of no return for objects falling into a black hole. Once a test mass or light beam is at a distance called the Schwarzschild radius from the center of the black hole, it will eventually be absorbed. This follows from

1Theoretically, a black hole could also posses magnetic charge, which we will ignore in this thesis since we focus on realistic situations. 14 CHAPTER 2. BLACK HOLES IN 4 DIMENSIONS general relativity, where the Schwarzschild metric describes the exterior region of a nonrotating, uncharged black hole. Its features are spherical symmetry, a pure singularity at its center and a spherical surface around it called the event horizon, at the Schwarzschild radius. The event horizon is a ‘one-way surface’, in the sense that all timelike and null geodesics passing through this surface will eventually hit the singularity at the center of the black hole. The value of the Schwarzschild radius (2.1) follows from solving the Einstein equations in GR. Another, more intuitive way to arrive at this result is to calculate the maximum distance from a massive body M where a test particle cannot escape Newtonian gravity. This is done by equating the classical escape velocity to c. In other words, we equate the classical kinetic energy of a test particle moving at the speed of light to the potential energy from the gravitational field: 1 Gm mc2 − =0 (2.6) 2 r 2 This immediately yields the value rh =2GM/c . The Schwarzschild radius will turn out to be the crucial quantity for calculating black hole temperature and lifetime. Hawking showed that black holes have a temperature and radiate as perfect blackbodies. The notion of a black hole temperature seems counter-intuitive. After all, how can a completely black object that absorbs everything including light, emit thermal radiation? This can be explained by considering quantum fluctuations in the vicinity of the horizon. Particle emission by a black hole could be conceived as the creation of a virtual particle pair; one particle is absorbed by the black hole, while the other one flies of to infinity. In this chapter we will discuss three such effects: • Hawking radiation: thermal radiation of a black hole; • Schwinger discharge: e−e+ pair creation in the vicinity of a charged black hole, leading to charged particle emission and charge loss of the black hole; • Superradiance: Hawking radiation of spin particles, by stimulated emission of a rotating black hole. These effects will be explained in the next sections in the context of four dimensions. Their higher-dimensional analogues are derived and used further on.

2.2.1 The Unruh effect and Hawking radiation Black holes radiate through a process called Hawking radiation. This is explained by the Unruh effect. The basic principle that underlies the Unruh effect is that observers with different notions of field modes will disagree on the particle content of a state. In particular, what seems to a static observer as a vacuum state, will be a thermal spectrum to a uniformly accelerated observer. Seen in the background of a black hole, this leads to the conclusion that a black hole must emit thermal radiation. The Hawking temperature, associated with thesurfaceofablackhole,isproportionaltoitssurfacegravity. To derive the Unruh effect, the following steps are made. We compare empty flat space as viewed by a static observer (with Minkowski coordinates) and by a uniformly accelerated observer (with Rindler coordinates). They will follow trajectories through spacetime 2.2. PROPERTIES 15 generated by different timelike Killing vectors. By solving the Klein-Gordon equation with respect to these two different cases, two sets of field modes are obtained. By investigating the analyticity properties of these modes, it will become clear that they do not share the same vacuum state. As it turns out, the Minkowski vacuum state will have a thermal character in terms of the Rindler modes. Finally, it is reasoned why this implies the Hawking effect. This derivation is based on [12, 13]. For simplicity, we consider flat spacetime in 1+1-dimensions. In inertial coordinates, the metric is ds2 = −dt2 + dx2 = dud¯ v¯ (2.7) where we have defined null coordinates for convenience:

u¯ = t − x (2.8) v¯ = t + x (2.9)

Paths of constant acceleration α are hyperbola in the (x, t)-plane satisfying the equation 1 x2 − t2 = (2.10) α2 We deﬁne Rindler coordinates u = η − ξ,v = η + ξ that are adapted to accelerated motion. They are related to the original coordinates by the coordinate transformation − 1 −au u¯ = a e (2.11) 1 av v¯ = a e (2.12) These coordinates cover the region x>|t| of Minkowski space (the ‘Rindler wedge’) and have ranges −∞ <η,ξ<∞. Figure 2.4 shows a Kruskal diagram of Rindler space as a subspace of Minkowski space. H+ and H− are Killing horizons for the timelike Killing vector in Rindler space, to which we will come shortly. In terms of the new coordinates, equation (2.10)reads 1 1 = e2aξ = constant (2.13) α2 a2 This means that uniformly accelerated observers follow paths of constant ξ in Rindler spacetime. Rindler coordinates are therefore convenient for our purposes.

We want to know the timelike Killing vector of Rindler space, because it generates the paths of constant acceleration. In Rindler coordinates, the metric (2.7)reads

ds2 = e2aξdudv (2.14)

Because η does not appear in the metric coefficients, ∂η must be a (timelike) Killing vector. Since this is still Minkowski space, it should correspond to a known symmetry of flat space. Expressing ∂η in terms of flat coordinates, we have

∂u¯ ∂v¯ ∂η = ∂η ∂u¯ + ∂η ∂v¯ (2.15)

=¯u∂u¯ − v∂¯ v¯ (2.16) 16 CHAPTER 2. BLACK HOLES IN 4 DIMENSIONS

Figure 2.4: A Kruskal diagram of Rindler space. The coordinates (η, ξ) can be used in region I and region II, where they point the other way. The Killing horizons H± for the vector field ∂η correspond to the past and future infinities as experienced by the Rindler observer. which is the generator for the Lorentz boost - indeed, the generator of a known symmetry. With this result in mind, we take a look at the transformations between field modes in Minkowski and Rindler spacetime. We consider the most simple case of a massless scalar field in 1+1 dimensions. In inertial coordinates (¯u, v¯), the Klein-Gordon equation reads

∂2φ φ = = 0 (2.17) ∂u∂¯ v¯ This equation has the standard orthonormal plane wave solutions † ∗ φ = (aku¯k + aku¯k) (2.18) k

† with (ak,ak) the standard creation and annihilation operators, and ﬁeld modes

− 1 ikx−iωt u¯k =(4πω) 2 e (2.19)

From these solutions, we can construct a Fock space with a vacuum state |ΩM deﬁned as

ak|ΩM = 0 (2.20)

We will refer to this state as the ‘Minkowski vacuum’. Note that the ﬁeld modes are positive- frequency with respect to the timelike Killing vector, i.e. they satisfy

∂tu¯k = −iωu¯k (2.21) 2.2. PROPERTIES 17

For a massless ﬁeld, the dispersion relation is ω = |k| > 0. We can thus separate the ﬁeld modes into left- and right-moving waves:

iωu¯ u¯k ∼ e ,k>0 (2.22) ∼ e−iωv¯,k<0 (2.23)

In Rindler space, the Klein-Gordon equation reads

∂2φ φ = e−2aξ = 0 (2.24) ∂u∂v We can construct a Fock space of solutions deﬁned in the Rindler wedge. The solutions should be positive-frequency with respect to a future-directed timelike Killing-vector. This is not the same vector ∂η throughout the Rindler wedge. In region II, as can be seen from ﬁgure 2.4, the desired Killing vector is ∂−η = −∂η, so the following conditions apply:

∂ηuk = −iωuk in region I (2.25)

−∂ηuk = −iωuk in region II (2.26)

This compells us to define two different sets of field modes to cover the entire Rindler wedge: − 1 − (4πω) 2 eikξ iωη in region I uI = (2.27) k 0 inregionII

II 0 inregionI uk = − 1 + (2.28) (4πω) 2 eikξ iωη in region II

We can, as an alternative to (2.18), expand φ in terms of these modes: I I I† I∗ II II II† II∗ φ = (bkuk + bk uk + bk uk + bk uk ) (2.29) k

I I† II II† Here, bk,bk ,bk ,bk are the creation and annihilation operators associated with the Rindler ﬁeld modes (2.27)and(2.28). The resulting Fock space has vacuum state |ΩR,deﬁnedas I | II| bk ΩR =0,bk ΩR = 0 (2.30)

We claim that this Rindler vacuum is not equivalent to the Minkowski vacuum. For if this were true, the modes uk could be expressed as a supersposition of the modesu ¯k.Thisis, however, not possible because the modes uk are non-analytic at the pointu ¯ =¯v = 0. Here, at the crossover point between regions I and II, there is a sign-change in the exponent of the Rindler modes. That means that the Minkowski vacuum expressed in Rindler modes contains both annihilation and creation operators. This state must therefore also contain negative frequency-modes. Hence the Minkowski and Rindler Fock spaces do not share the 18 CHAPTER 2. BLACK HOLES IN 4 DIMENSIONS same vacuum. As a consequence, a Rindler observer sees the Minkowski vacuum as a state containing particles. The key question now is: how many particles? To obtain an answer, the following steps are made. First, we need new sets of modes that overlap (2.27)and(2.28), and are analytic acrossu ¯ =¯v = 0. We can then make sure that these new modes contain only positive frequencies, and thus must share the vacuum with the Minkowski modes. By ﬁnding the I II relation between the new creation and annihilation operators and the bk,bk of the Rindler modes,itcanbepredictedwhatkindofstate|ΩM is in terms of these Rindler operators. We will see that we are dealing with a thermal state. We can write (2.27)and(2.28)alternativelyas iω iω u¯ a θ(¯u),k>0 v¯ a θ(−u¯),k>0 I ∼ II ∼ uk − iω ,uk − iω (2.31) v¯ a θ(¯u),k<0 u¯ a θ(−u¯),k<0

Here, the Heaviside step function θ(¯u) is used to account for the validity of the modes in their appropriate regions of the Rindler wedge. We are looking for linear combinations of these modes that cover the entire spacetime. Speciﬁcally, taking k>0, we want to analytically I II continue the function uk to region II. For this we need the uk -modes.Itturnsoutthatthe following combinations give the desired result:

− πω ∗ iω I a II ∼|| a uk + e u−k u¯ (2.32) ∗ πω − iω I a II ∼|| a u−k + e uk v¯ (2.33) We have performed a rotation of the ﬁeld modes in the complex plane by a factor e−iπ. This is valid, because the branch cuts of the complex powers in (2.31) lie in the upper half complex plane ofu ¯ andv ¯. These new modes are analytic throughout spacetime. Therefore, we know they only consist of positive frequencies and must share the same vacuum with the Minkowski modes, as was desired. We can decompose φ in terms of either of these new mode expansions. Taking (2.32), we write πω † πω I I − a II∗ I I∗ − a II φ = ck(uk + e u−k )+ck (uk + e u−k) k (2.34) πω † πω II I∗ a II II I a II∗ +ck (u−k + e uk )+ck (u−k + e uk )

The goal is now to write the new operators ck, that annihilate the Minkowski vacuum, in I II terms of the Rindler operators bk,bk to see what this vacuum looks like to a Rindler observer. iω − iω Equating the coeﬃcients of the contribution ∼ u¯ a and ∼ v¯ a from (2.29)and(2.34), and realizing they both annihilate the Minkowski vacuum, we now see that

− πω II† I | I a | 0=ck ΩM =(bk + e b−k ) ΩM (2.35) − πω I† II| II a | 0=ck ΩM =(bk + e bk ) ΩM (2.36) We now have deﬁned the Minkowski vacuum in Rindler coordinates. We want to ﬁnd the particle content of this state. This is done by writing it as a state in Rindler fock space and then calculating the expectation value of the Rindler number operator. 2.2. PROPERTIES 19

The Minkowski vacuum can be expressed as ∞ |ΩM = fn|nI ,nII (2.37) nI ,nII=0

Summation over k is implied, but omitted in notation. Here, |nI ,nII is taken from the Fock space generated by the Rindler operators:

1 † † | √ I nI II nII | nI ,nII = (bk ) (bk ) ΩR (2.38) nI !nII! Combining (2.35)and(2.36) we see that the number operators in both regions are equal, so that nI = nII. Furthermore, we ﬁnd fn from combining (2.35)with(2.37): √ √ − πω nfn|n − 1,n = −e a n +1fn|n, n +1 (2.39) n n − πω fn+1 = −e a fn (2.40) Combining all this, we write the Minkowski vacuum in terms of the Rindler Fock space: ∞ − nπω |ΩM = N −e a |n, n (2.41) n=0 ∞ 1 − nπω I† II† = N − e a (b )n(b )n|Ω (2.42) n! k k R n=0 ∞ − nπω I† II† −e a b b = N e k k |ΩR (2.43) n=0

Taking the trace over nI and nII in the expression ΩM |ΩM yields the normalization factor:

− 2πω − 1 N =(1− e a ) 2 (2.44)

Thislookslikeathermalstate.Wenowwanttoﬁndthenumberofparticlesobservedbya I† I Rindler observer in region I. This comes down to wedging the number operator bk bk between |ΩM and tracing over the states in region II:

− 2πω e a 1 | I† I | ΩM bk bk ΩM = − 2πω = 2πω (2.45) 1 − e a e a − 1 This result is a Planck spectrum with temperature a T = (2.46) 2π We conclude that a uniformly accelerated observer traveling through the Minkowski vacuum observes a thermal spectrum of particles. This is the Unruh effect. We now proceed to show that the Unruh effect directly leads to the Hawking effect. 20 CHAPTER 2. BLACK HOLES IN 4 DIMENSIONS

The Kruskal diagram of Rindler space (figure 2.4) closely resembles that of a Schwarzschild black hole, with the Killing horizons playing the role of the black hole horizon, and regions I and II representing the part of Minkowski space outside the horizon. We claim that the central region of the Schwarzschild diagram resembles flat space to a distant Rindler observer. This is justified by comparing the length scale set by the acceleration of the observer with the length scale set by the curvature: √ r r − 2GM a−1 = (2.47) GM We see that close to the horizon, r − 2GM 2GM, this length scale is much smaller than the curvature scale, 2GM. So, close to the horizon, spacetime looks essentially flat to distant observers. This is consistent with the fact that the horizon is merely a coordinate singularity; an observer crossing it experiences nothing out of the ordinary. A uniformly accelerated (Rindler) observer at late times will thus see a flat region with no particles; he observes the Minkowski vacuum |ΩM . A static, near-horizon observer will thus look like a Rindler observer in flat space. By the Unruh effect, he will observe thermal radiation at T = a/2π. This radiation will be redshifted as it propagates to x →∞.For an observer at r>2GM, the appropriate redshift factor is given by V T = T (2.48) V where 2GM V = 1 − (2.49) r is the redshift factor for static observers in the Schwarzschild metric. As limx→∞ V =1, V a κ T =lim = (2.50) r→2GM 2π 2π Here, κ =lim(Va)isthesurfacegravity[7]; in Schwarzschild, it equals 1/4GM. A Schwarzschild black hole therefore emits thermal radiation at a temperature c3 T = (2.51) 8πkGM where we have put back numerical constants. This is the Hawking effect. A black hole will emit particles of all different species through this effect: photons, fermions, gauge bosons, etc. This radiation will have the spectrum of a blackbody, described by the well-known Stefan-Boltzmann law. The radiative power is proportional to P ∝ T 4. In this thesis, Hawking radiation is a very important effect as it is the only means of particle emission by Schwarzschild black holes.

2.2.2 Superradiance Since Hawking radiation includes all particle species, rotating black holes will typically emit spin particles. In a certain energy regime, this radiation will be amplified by a mechanism 2.2. PROPERTIES 21 called superradiance. The basic idea is that a physical body (in our case, a black hole) rotating in vacuum, is ‘tickled’ by virtual vacuum fluctuations. In return, it amplifies these fluctuations by transferring energy and angular momentum to them. This coupling process converts them into real radiation. By this superradiance mechanism, the black hole loses angular momentum along with its mass [14, 15]. In this section, we show how this phenomenon comes about. The spacetime around a rotating, uncharged black hole is described by the Kerr metric:

2GMr 2GMra sin2 θ ρ2 ds2 = − 1 − dt2 − 2dtdφ + dr2 + ρ2dθ2 ρ2 ρ2 ∆ (2.52) 2GMra sin2 θ + r2 + a2 + sin2 θdφ2 ρ2 with ∆=r2 + a2 − 2GMr, ρ2 = r2 + a2 cos2 θ (2.53) Here, a = J/M√ is the angular momentum per unit mass. The hole has event horizons at 2 2 2 r± = GM ± G M − a . From the metric, we derive the angular velocity of the event horizon: dφ a Ω= = (2.54) dt r=r+ 2GMr+ Now consider a quantum ﬁeld theory with massless scalar ﬁeld Φ in the absence of sources. This obeys the Klein-Gordon equation

µ ∇µ∇ Φ = 0 (2.55)

We can solve this equation using (2.52) and separation of variables. This is a straightforward calculation, of which we present the solutions for the scalar ﬁeld at the outer horizon r+ and at inﬁnity:

∼ 1 −iωt±iωr∗ →∞ Φ r e ,r (2.56) −iωt±i(ω−mΩ)r∗ Φ ∼ e ,r→ r+ (2.57)

Thetortoisecoordinateisdeﬁnedas 2 2 dr∗ r + a = (2.58) dr ∆ Physically, we can only have ingoing waves at the horizon. This means the group velocity dω/dk should be negative, so we choose the negative sign in (2.57). This means the phase velocity will be positive, corresponding to outgoing waves, if

0 <ω

This means that ingoing waves at the horizon are observed as outgoing waves at infinity. Energy and angular momentum are thus extracted from the black hole. The rotation of 22 CHAPTER 2. BLACK HOLES IN 4 DIMENSIONS the hole amplifies the ingoing waves. Note that m must be positive, so the waves should be co-rotating with the black hole. This amplification of Hawking radiation in the energy regime (2.59) is called superradiance. Superradiance is an important effect to be considered when calculating particle emission by rotating black holes. In this thesis, however, it is not that important because the focus in later chapters will be on nonrotating black holes.

2.2.3 Schwinger discharge Electromagnetic charge could end up beyond an event horizon if charged particles are involved in the black hole formation process. The question addressed in this section is: what happens with electric charge carried by a black hole? Our analysis is inspired by [16], one of the earliest sources on this subject available. Charged black holes are characterized by the Reissner-Nordström-metric. They can be classified according to their charge-to-mass ratio Q/Q 1 Q Q∗ ≡ pl = √ (2.60) M/Mpl 4π 0G M This ratio is an important indicator of what the black hole spacetime will look like. When Q∗ = 1, the black hole is said to be ‘extremal’ and will have one event horizon. In the case Q∗ > 1, there will be a singularity at r = 0 without a horizon, a so-called ‘naked singularity’. Such black holes will not be observed in nature. A non-extremal black hole (Q∗ < 1) has two event horizons, but for observers outside the outer horizon it will effectively behave as a Schwarzschild black hole. Charged black holes attract both matter (through gravity) and charge (through the electromagnetic force). The quantity Qe (2.61) Mm (in Planck units) compares these two forces. If it is larger than 1, it means that the black hole will not accrete any charged matter gravitationally. In this case it is energetically favorable for a black hole to neutralize through the Schwinger process described in this section. Moreover, a black hole is not capable of retaining any charge within a reasonable amount of time if its mass is smaller than m/e2 ∼ 1012kg. Microscopic black holes would then immediately disqualify as a subject for charged black hole analysis. However, the values of the constraints will turn out to change in higher dimensional scenarios. In this section, we deduce the origin and rate of the neutralization process. A charged black hole will emit charged particles of opposite charge, eventually becoming neutral. This is achieved through Hawking radiation of charged particles and a different process called Schwinger discharge. Due to the strong electric field of the black hole, Schwinger pair production of charged (anti)particles occurs in a region outside the horizon. The particle carrying opposite charge to the black hole will be absorbed, while the other (anti)particle propagates away from the black hole. We now derive the amplitude for Schwinger pair production in the presence of a strong, homogeneous E-field. We do this via the instanton-action in QED, while a canonical method is also available [17]. 2.2. PROPERTIES 23

2 Our derivation is valid in the semiclassical regime Mm Mpl, where the Compton wavelength of the emitted particles is small as compared to the horizon radius of the black hole. This corresponds to a large black hole of M 1016 kg in four dimensions (again, this constraint will change if extra dimensions are introduced). The Klein-Gordon equation in 1+1 dimensions, with space-dependent gauge field A(x), reduces to a one-dimensional tunneling problem for which we can use the instanton method. The pair production rate will thus correspond to the tunneling probability: − 1 P = e SE (2.62) where SE is the Euclidean action for barrier penetration, also called the instanton action. To calculate it, we transform to Euclidean time. We consider two created particles with mass m and opposite charge q, uniformly accelerated by a homogeneous electric field E. The acceleration of each particle due to the electric field will be ±Eq/m.Dueto(2.10), the trajectory of the particles will be the hyperboloid 2 m x2 − t2 = (2.63) Eq Since the energy needed to create the particles is drawn from the electric field, the following energy condition must be satisfied when the particles are a distance 2r apart: 2rqE =2mc2 (2.64) To compute the probability for pair creation, we make a transformation to Euclidean time: t → it. The trajectory of the particles then becomes a circle in Euclidean spacetime. Phys- ically, this makes sense because the electric field transforms into a magnetic field — and charged particles move along circular orbits in a uniform magnetic field. So, in Euclidean time, the particles will be experiencing a Lorentz force. Furthermore, because of the transformation there is a sign change in the potential term in the action. The appropriate instanton action is (see appendix A): 1 S = dτ( mv2 − qAv) (2.65) 2 Here, v is the orbital velocity of the particles. The equations of motion, in combination with the energy condition (2.64)givev = c. Performing integration over an orbit with radius r, the two terms add up to the exponent in the formula for pair production rate, in agreement with [18]: 3 2 − c m P = e π qE (2.66) where we have put back physical constants in their place. Taking for the field strength the valueoftheCoulombfieldofablackholewithchargeQ at the horizon, we see that if (in Planck units) Qe > (Mm)2, the discharge rate is rapid. The exact rate can be obtained by relating the probability P to dN/dt and integrating over a volume around the black hole. Two mechanisms for charge loss by a black hole compete: Schwinger discharge and Hawk- ing radiation. If 2 2 Qe (Mm/Mpl) 2 2 (2.67) Qpl log (Mm/Mpl) 24 CHAPTER 2. BLACK HOLES IN 4 DIMENSIONS

Figure 2.5: A diagram of charged black hole behavior in (Q, M) parameter space. Above curve I solutions are superextremal so no black holes can exist. Above curve IV Schwinger discharge occurs; above curve II this is a rapid process. Schwinger discharge dominates over discharge via Hawking radiation above curve III, while curve V denotes the minimum black hole charge. The shaded region is the part of parameter space where black holes lose their charge through rapid Schwinger discharge. The ratio e/m is actually a much larger number than in this diagram.

the Schwinger discharge rate exceeds the rate of charge loss through Hawking radiation [19].

The results of this section are displayed in ﬁgure 2.5. We conclude that in four dimensions, only large black holes (M>e2/m ∼ 1012 kg) can hold on to their charge for a signiﬁcant time period. Black holes larger than 1016 kg qualify for our semi-classical analysis. This is the regime where the compton wavelength of the electron exceeds the Schwarzschild radius. For values Q∗ Mm2/e (in Planck units), these black holes will rapidly discharge, either through Hawking radiation or the Schwinger process. Any black hole that holds on to its 5 35 charge smaller than 10 M ∼ 10 kg will be nonextremal. It would be interesting to give a estimate of how the timescale for discharge compares with the timescale for evaporation.

It should be emphasized that the results of this section apply mainly to large black holes. Their validity in higher-dimensional scenarios, and the characteristics of charged microscopic black holes, are discussed in section 4.3.1. 2.3. EVAPORATION 25

2.3 Evaporation

The results from the previous section show that black holes emit particles. If the emission rate is greater than the absorption rate, the black hole will reduce in mass and size. This process cancontinueuntiltheblackholehasshrunktoMpl; in this regime, quantum gravitational eﬀects become important. Whether or not black holes will evaporate completely is a matter unknown, but let us assume they do. As M M∗,whichisthecaseduringthelargerpart of the process, the quantum gravity regime does not come into play. The major contributor to particle emission is Hawking radiation. In this section, we formulate the conditions for complete evaporation of a black hole. Consider a nonrotating, uncharged Schwarzschild black hole of mass M.Weassume that no matter will cross the horizon by gravitational attraction. The black hole is in thermal equilibrium with its surroundings if the Hawking temperature is exactly equal to the temperature of the environment. But even the slightest change in temperature will tip the balance. If the Hawking temperature drops a little bit, absorption through the horizon will exceed emission and the black hole will grow. Since temperature and mass are inversely related, the temperature will drop even further. And vice versa; a black hole slightly hotter than its surroundings will have an increasing emission rate. In other words, a black hole has a negative heat coeﬃcient. This makes it a thermally unstable object; in a thermal bath, it will either grow eternally or evaporate completely. When the temperature of a black hole exceeds the cosmic microwave background (CMB) of 2.73 K, it will ultimately evaporate. This corresponds to an upper mass limit

22 M0 < 4.494 · 10 kg = 0.00752 · Mearth (2.68) for evaporation. All heavier black holes are colder than the CMB and will never evaporate. From this we conclude that stellar black holes will always keep growing, by thermal absorption of radiation and gravitational accretion of matter. Smaller black holes will generally evaporate; they are the main focus of this thesis. The two types that are discussed are primordial black holes (formed in the early universe) and microscopic black holes (formed in high energy collisions). The evaporation of (small) black holes is a combination of the three effects discussed in the previous section: Hawking radiation, superradiance and Schwinger discharge. Furthermore, black holes are not perfect blackbodies; radiation spectra are multiplied by a greybody factor dependent on the species and energy of the emitted particle. The cumulative effect of these properties leads to an evaporation process in different phases. This is the main subject of chapter 4. The introduction to black holes is now complete. We have reviewed the types of black holes that can be created through different mechanisms. We have shown the impossibility of creating black holes with accelerators on Earth in four dimensions. A black hole’s means of particle emission were reviewed; they will also be relevant in higher dimensional scenarios. We have shown that only small black holes can eventually evaporate. In the next chapters we will see the consequences of higher-dimensional scenarios on the creation, behavior and evaporation of black holes. Chapter 3

Braneworld scenarios

Current measurements of gravity probe only up to distances no shorter than 1 cm. This allows for hidden, extra dimensions to exist on smaller scales. In this chapter, we start with an explanation of what prompted the proposal of these ‘braneworld scenarios’1.Wethen describe more in detail one particular model, the ADD model, that will be used throughout the remainder of the thesis.

3.1 Motivation

In this section we motivate why braneworld scenarios are currently being developed. We also explain how these scenarios are relevant to the subject of this thesis: black hole creation. One of the most important unresolved issues in physics is the hierarchy problem: the large discrepancy between the electroweak scale (MEW ∼ 300 GeV) and the Planck scale 19 (Mpl ∼ 10 GeV). Standard Model interactions occur at the electroweak scale, while gravity becomes important above the Planck scale. Their ratio is radiatively stable, yet in a unified field theory they should be related to some common scale. Several solutions are proposed to the hierarchy problem, among which the most popular candidates are supersymmetry and braneworld scenarios. These will be the focus of this thesis. Braneworld scenarios picture the SM-universe as a 4-dimensional brane embedded in a compact higher dimensional bulk space. As we will see, gravity becomes stronger at short distances as it extends into the higher dimensions. Physically, this means that the gravity is weak because there is a loss of flux to the extra dimensions. This leads to a fundamental Planck scale M∗ much lower than the effective Mpl that we measure at large distances. These scenarios could provide a solution to the hierarchy problem and have therefore been studied in recent years. Theories with a low value for the Planck scale could have significant side-effects. Giddings and Thomas [11] among others, predicted that if the Planck scale would be as low as 1

1The terms ‘braneworld scenario’ and ‘higher dimensional scenario’ are used interchangeably throughout the text.

26 3.2. THE ADD SCENARIO 27

TeV, we could observe gravitational effects in future particle experiments at the LHC. In particular, black holes could be created in these experiments. This process will be explained in the next chapters. The microscopic black holes would evaporate almost instantly through Hawking radiation and therefore pose no danger to the Earth. If observed, however, they would give evidence of the existence of extra dimensions. Their radiation spectrum could give information about the number and size of the extra dimensions, and subsequently the value of the Planck scale. It comes as no surprise that TeV-scale gravity (TeVG) is a hot item in physics today. In recent articles on black hole production, the Planck scale is usually assumed to have a value close to the electroweak scale, around 1 TeV. This is certainly the lowest possible value; otherwise, quantum gravitational effects would have already been observed in particle experiments. It is also a logical choice, as it could provide a solution to the hierarchy problem. But it could also be viewed as a consequence of ‘wishful thinking’: we would never observe black hole production (and solve the hierarchy problem) if the Planck scale would have a higher value. Having made this critical note, we will assume a value of 1 TeV throughout this thesis, and comment qualitatively on the consequences of a higher value. In the next section, an overview of experimental bounds on the Planck scale is given. The possibility of 6 or 7 extra dimensions is convenient, as it is the same amount of dimensions predicted by superstring theory and M-theory, respectively. Although the ADD model can be studied in the context of these theories, this thesis does not cover that subject. Our main interest will be the behavior of gravity in this model, that can be studied independently of string theories. Due to the change in the gravitational force law, the size and Hawking temperature of a black hole will be modified. This leads to a different lifetime for black holes from what would be expected with just four dimensions. Furthermore, the radiation spectrum will differ greatly from the four-dimensional case. These properties are discussed in chapter 4.

3.2 The ADD scenario

This section introduces the braneworld scenario that is considered in this thesis: the ADD scenario. The relation between extra dimensions and the Planck scale is explained. Braneworld scenarios could help explain the hierarchy problem because the value of the Planck scale is lowered. This can be achieved in two ways: by introducing either multiple large extra dimensions or only one extra dimension that is warped. These two diﬀerent models are called the Arkani-Dimopoulos-Dvali (ADD) model [2] and the Randall-Sundrum (RS) model [3] after their discoverers. We choose the ADD model as the context for this thesis, because the lowering of the Planck scale can be elegantly explained by its characteristics. In this model, the Planck value is determined by the number and size of the extra dimensions2. The ADD model is set up as follows: our visible, 4-dimensional universe is picture as a brane embedded in a ﬂat, compact n-dimensional space of volume ∼ Rn (the ‘bulk’). For simplicity, we assume the extra dimensions all have size R. While the the SM matter and

2In the RS model, this value would be determined by the warp factor of the extra dimension. In principle, the results of this research could alternatively be obtained by use of this scenario. 28 CHAPTER 3. BRANEWORLD SCENARIOS degrees of freedom live on the brane, gravity propagates both on the brane and through the bulk. The bulk therefore only has graviton degrees of freedom. In this set-up, gravity is modiﬁed, while the other three forces of nature behave normally as they only exist on the brane.

Figure 3.1: Gravitational ﬁeld lines extend through the extra dimensions. The four-brane is visualized as a one-dimensional line, around which the n extra dimensions with size R are wrapped (only one extra dimension is drawn in this picture). The altered behavior of the ﬁeld due to the extra dimensions can only be measured at distances R(below).

We will now determine exactly how gravity is modified. In the ADD model, two test masses m1 and m2 placed apart at a distance r R will be attracted through Newton’s force law. See figure 3.1. Because the field lines extend on the brane as well as through the n extra dimensions, the gravitational potential is

n+1 m1m2 1 ∼ V (r) n−1 n+2 n+1 ,r R (3.1) c M∗ r

Here, M∗ is the generalized Planck mass scale, which diﬀers from the known 4-dimensional 3.2. THE ADD SCENARIO 29 scale. However, when we place the masses further away at r R, the gravitational ﬁeld lines cannot penetrate the extra dimensions further than R. The usual 1/r potential is obtained:

n+1 m1m2 1 ∼ V (r) n−1 n+2 ,r R (3.2) c M∗ Rn r

This must match the known behavior of gravity in four dimensions, V (r)=Gm1m2/r (see ﬁgure 3.2). From this ﬁgure it is clear that on short distances, gravity is stronger in braneworld scenarios than in four-dimensional scenarios. Since experiments have measured the strength of gravity only up to 1 cm, braneworld scenarios might well be realistic. Match- ing (3.1)and(3.2)atr = R we can relate the four- and higher-dimensional Planck scales to each other by the size and number of the extra dimensions:

n n+2 lpl M∗ ∼ M (3.3) pl R

For clarity, we emphasize that throughout this text, the subscript pl refers to the four- dimensional value, while ∗ denotes the 4 + n-dimensional equivalent. This relation reflects that the larger the number and size of the extra dimensions, the lower the Planck mass will be. This is visualized in figure 3.3. There are some constraints on R and n. First of all, the case of n = 1 is ruled out because at a Planck scale of a few TeVs, the dimension size would exceed the size of the universe. On the other hand, if n>7, the dimension size will be shorter than the modified Planck scale. Quantum gravity would be needed to describe the physics of the bulk space. Therefore our analysis is restriced to models with 2 up to 7 extra dimensions. In these cases, the dimension size is less than a mm for M∗ ≥ 1 TeV, which is in agreement with experiments. IntheprevioussectionthechoiceforM∗ = 1 TeV was motivated. As can be seen from table 3.1, this possibility is not ruled out by experiment. The modified Planck quantities are listed in table 3.2 while the values of R arelistedintable3.3. The relevance of the modified value of Tp∗ is debatable, since the temperature corresponds to SM particles confined to the brane. We therefore assume that the effective Planck temperature has the value Tpl.This allows us to apply thermodynamics at transplanckian energies without a breakdown of the semiclassical approach. The modified Planck scale has its effect on all gravitational objects. In particular, black hole solutions, which are governed by GR, will have different properties. These will be described in the next section. 30 CHAPTER 3. BRANEWORLD SCENARIOS

2 2 Type of Experiment/Analysis M∗c ≥ M∗c ≥ Collider limits on the production of real or virtual KK gravitons 1.45 TeV (n =2) 0.6 TeV (n =6) Torsion-balance Experiments 3.5 TeV (n =2) Overclosure of the Universe 8TeV(n =2) Supernovae cooling rate 30 TeV (n =2) 2.5 TeV (n =3) Non-thermal production of KK modes 35 TeV (n =2) 3TeV(n =6) Diﬀuse gamma-ray background 110 TeV (n =2) 5TeV(n =3) ThermalproductionofKKmodes 167 TeV (n =2) 1.5 TeV (n =5) Neutron star core halo 500 TeV (n =2) 30 TeV (n =3) Neutron star surface temperature 1700 TeV (n =2) 60 TeV (n =3) BH absence in neutrino cosmic rays 1-1.4 TeV (n ≥ 5)

Table 3.1: Current limits on the size of extra dimensions and the subsequent Planck scale. Table taken from [20]; see full text for references.

Quantity Formula Original Modiﬁed 15 2 2 M∗ M∗ 10 TeV/c 1TeV/c 2 −35 −19 lp∗ c/M∗c 10 m 10 m 2 −43 −28 τp∗ /M∗c 10 s 10 s 2 38 16 Tp∗ M∗c /k 10 K 10 K

Table 3.2: Modiﬁed fundamental scales in TeVG. Modiﬁcation to TeVG results in lower Planck mass and temperature and greater Planck length and time.

n Rmax (m) 1 9.9 × 1011 2 4.3 × 10−4 3 3.3 × 10−9 4 9.0 × 10−12 5 2.6 × 10−13 6 2.5 × 10−14 7 4.6 × 10−15

Table3.3:ValuesofR for which Mpl ∼ 1TeV 3.2. THE ADD SCENARIO 31

Figure 3.2: In the ADD scenario, on scales larger than R, the force of gravity goes as ∼ 1/r2, just like in four dimensions. As the distance decreases beyond R,theinverse-squarelaw changes to ∼ 1/r2+n, making gravity stronger than expected in four dimensions.

1014

1010

106

102

10-2

10 -35 10 -29 10 -23 10 -17 10 -11 10-5

Figure 3.3: Diagram representing the modiﬁed Planck scale as a function of number n and size R of the extra dimensions. In the case M∗ =1TeV,n = 1 is ruled out as a possibility, while for n ≥ 2thevaluesofR range up to 0.1 mm. Chapter 4

Black holes in 4+n dimensions

The main result from the previous chapter is that by solving the hierarchy problem with braneworld scenarios, the Planck mass could be as low as 1 TeV. As a consequence, black holes could be generated at TeV-scale particle experiments. In this chapter, we will calculate some generalized properties of black holes in braneworld scenarios. From the modified horizon radius, we derive the modified Hawking radiation law and evaporation timescale. The creation of PBHs at the dawn of the universe will also be affected — retroactively — by higher dimensional scenarios. This is the subject of chapter 5.

4.1 Formation: high-energy collisions

Assuming the Planck scale would indeed be 1 TeV due to the ADD scenario, black hole 1 creation√ events would possibly occur at LHC. Two particles collide at transplanckian energy s>M∗. According to the hoop conjecture in four dimensions, they would form a black hole if b<2rh (see figure 2.2). This limit is modified in braneworld scenarios; the maximal impact parameter for black hole creation increases with the number of extra dimensions [22]. This modifies the cross section for black hole production, which is proportional to the ‘target area’. It is obtained by summing over the possible parton interactions in a collision. The calculation of cross sections dependent on energies and extra dimensions is a subject of extensive study (see for example [23]). Since we are only interested in an order-of-magnitude estimate in this thesis, we will use the approximation ∼ 2 ∼ 2 σpp→BH πrh πbmax (4.1) for an estimate of the production rate for black holes. For black holes of a few TeVs, the cross 2 ∼ −38 2 section will be of order lp∗ 10 m . An agreeable estimate for the integrated luminosity for typical experiments at LHC is L ∼ 100 fb−1 yr−1. This leads to a production rate of dN = Lσ ∼ 107 yr−1 (4.2) dt 1The original analysis viewed these as point particles. In [21], the particles are more accurately viewed as colliding wavepackets. The hoop conjecture applies in both approaches.

32 4.1. FORMATION: HIGH-ENERGY COLLISIONS 33

This roughly equals one event per second2. The LHC might turn out to be a productive black hole factory. Not all of the collision energy ends up as mass beyond the event horizon. Some of it is lost in the process through radiation. Using fn to denote the remaining mass fraction, we write √ s M ∼ f (4.3) n c2 √ for the mass of a black hole created in a collision with center-of-mass energy s. The factor fn is less than 1 and dependent on the number of extra dimensions. Calculations in [10, 1] putthisfactoratapproximately.84 for a head-on collision, while it drops to .45 when the impact parameter is maximal. These fractions decrease with the number of extra dimensions [20]. Black holes are created on the brane, because that is where all SM-matter is confined to. The brane is subject to a tension force, that keeps the black hole fixed on the brane for reasons that will follow momentarily. The brane tension, described by the stress-energy tensor, is proportional to the induced metric on the brane. This implies the tension force originates from nothing else than the curvature of the brane. The constant of proportionality is the curvature, being the inverse of the dimension size: 1 T ∼ g (4.4) µν Rn µν Large extra dimensions imply small curvature, which is in accordance with our approximation to neglect quantum gravity effects. If the curvature is small compared to the energy scales involved in the high-energy processes studied, the quantum gravity regime is avoided. This is precisely the case in our situation. The gravity induced on the brane through tension is negligible compared to the gravity that governs the black hole formation process. This allows a semiclassical approach to Hawking radiation; the quantum fields can be seen as propagating on a gravitational background. We would like to make sure that once created, the black hole does not ‘slip off’ into the bulk space. An argument for this is presented in [24] and follows from the condition that the black hole is stationary. If this is the case, the dot product of the stress energy tensor with a generator lµ of a null-geodesic at the event horizon must be zero. This is equivalent to the statement that no energy crosses the horizon. We know from (4.4) that the stress energy tensor is proportional to the induced metric on the brane, leading to

µ µ ν 0=lµl ∼ Tµν l l (4.5)

This means that the vector pointing outward the black hole, orthogonal to the horizon, is tangent to the brane. So the event horizon is always perpendicular to the brane; if it slips oﬀ there is a restoring force due to brane tension. In this way black holes stay on the brane once created. To summarize the results from this section, most importantly we note that the lowered Planck mass in the ADD scenario makes black hole creation at the LHC possible. Estimates

2Surprisingly, larger energies lead to smaller cross sections and lower production rates [20]. 34 CHAPTER 4. BLACK HOLES IN 4+N DIMENSIONS for the cross section leads to a production rate of one per second at typical experiments. The black hole consists of most of the collision energy; some of it is lost through radiation. The brane tension keeps it ﬁxed onto the brane; this does not interfere with the formation process. Hawking radiation is able to propagate through the bulk as well as on the brane; their relative intensities are commented on in the next section.

4.2 Properties

This section gives an overview of the properties of black holes that are aﬀected by the presence of extra dimensions. The most important property is the Hawking temperature, that will under the right circumstances lead to evaporation of the black hole. It directly depends on the Schwarzschild radius, which will again be the starting point of our analysis. Another property dependent on the extra dimensions is the amount of radiation that backscatters into the black hole, the so-called greybody factor. This will also be discussed in this section. Plots of black hole properties are displayed in appendix C.

4.2.1 Schwarzschild radius As mentioned above, the single feature of a black aﬀected by higher dimensional scenarios that will play a major role in this thesis, is the Schwarzschild radius. For from this property, the Hawking temperature and subsequent lifetime are derived. This analysis is inspired by [25]. A non-rotating, uncharged, spherically symmetric black hole is described by the generalized Schwarzschild metric

2 − 2 −1 2 2 2 ds = h(r)dt + h(r) dr + r dΩn+2 (4.6) where r (4+ ) n+1 h(r)=1− h, n (4.7) r

We can ﬁnd the higher-dimensional horizon radius rh,(4+n) as a function of the black hole mass M by equating the classical kinetic energy of a particle travelling at the speed of light to the gravitational binding energy. This yields the desired result up to a numerical factor an. This geometrical factor was found in [26] by considering higher-dimensional general relativity. The result is:

1 an M n+1 rh,(4+n) = (4.8) c M∗ M∗ 1 n+3 n+1 8Γ( 2 ) an = n+1 (4.9) (n +2)π 2

Comparing with the 4-dimensional case, n = 0, we see from (3.3)and(4.8)thatforasmall enough black hole, rh,(4)

1+ 2 Mpl n R Mc ∼ Mpl ∼ Mpl (4.11) M∗ lpl

A hole with M>Mc has a horizon radius larger than the dimension size, and will behave as a four-dimensional black hole. A diagram of a black hole smaller than the dimension size is displayed in ﬁgure 4.1.Table5.1 in chapter 5 lists the values of Mc for the case M∗ =1 TeV. Black holes produced at particle colliders will be below this limit, so the relation (4.10) will apply.

Figure 4.1: A black hole with radius smaller than the dimension size, conﬁned to the 4-brane.

4.2.2 Hawking temperature The Hawking temperature of a black hole is inversely related to its Schwarzschild radius: c h(r) c (n +1) T(4+n) = = (4.12) k 4π r=rh k 4πrh From this we infer that a black hole with mass M is colder in higher-dimensional scenarios than it would be in four dimensions:

This can be learned intuitively from black hole thermodynamics. In higher-dimensional scenarios, a black hole with the same mass (proportional to energy) has a larger area (proportional to entropy). The fraction dM/dA ∼ dE/dS, proportional to the Hawking temperature, is then smaller for higher-dimensional black holes. Concluding, small black holes in braneworld scenarios are larger and colder than in four dimensions. Note that by this we mean ‘colder than a black hole of the same mass in four dimensions’. The microscopic black holes created at particle scattering processes are 36 CHAPTER 4. BLACK HOLES IN 4+N DIMENSIONS tremendously hot because of their small size. The Hawking temperature is the most important feature for black hole evaporation. Due to its high temperature, a small black hole will radiate away all its energy. Before discussing evaporation, we will discuss in the next section the fact that the radiation spectrum is not exactly thermal.

4.2.3 Greybody factors Before studying evaporation, we have to take a closer look at how black holes radiate with extra dimensions present. The black hole emits SM-particles3 on the brane, and gravitons in the bulk. The spectrum will turn out not to be exactly thermal. This has consequences for the radiative power of a ‘realistic’ black hole versus a purely thermal one. The deviation from the thermal spectrum is described by greybody factors. Their origin, general form and numerical evaluation will be the topic of this section. Radiation in the bulk via non-detectable gravitons will be observed as an energy defect. This portion, relative to radiation on the brane is called ‘bulk-to-brane ratio’. In the next section, we will see how this aﬀects the radiative power. We begin with describing on-brane radiation and will incorporate the bulk later. From the results of section 2.2.1 we get an expression for the amount of energy emitted by the black hole per unit time at energy ω. We have assumed the emitted particles are massless, so that |k| = ω and the integration variable is changed from momentum to energy. Theresultiscalledtheﬂuxspectrum:

(s) 3 dE (ω) ( ) (ω) dω = σ s (ω) (4.14) dt ,n eω/kT ∓ 1 2π2 To obtain the total radiative power we then have to perform an integration over particle energies and sum over all spin and angular momentum states. The minus and plus sign correspond to bosons and fermions, respectively. The flux spectrum deviates from a thermal (s) 2 one by the quantity σ,n(ω) with dimension [m ], called the greybody factor. It is dependent on the particle’s angular momentum and spin quantum numbers , s and energy, and also on the number of extra dimensions. We will now explain how this factor comes about. Throughout this thesis, a semiclassical approach to Hawking radiation is made. In the semiclassical picture, the tension of the brane is small compared to the black hole mass, such that the presence of the brane does not affect the gravitational background. The only things that are affected by the extra dimensions are the black hole’s Schwarzschild radius and particle emission. Blackbody radiation emitted from the modified horizon travels through the background metric, the hole’s gravitational potential. Some of it will scatter back into the black hole because of this potential. This portion is equal to the absorption probability; the greybody factor is proportional to this. Counter-intuitive as it may seem, the greybody factor for particle emission from a black hole is equal to the absorption cross section for the same particles incident on the black hole. This ensures, however, that the black hole can still be in thermal equilibrium with

3These should be thought of as degrees of freedom (scalars, fermions and gauge bosons) rather than elementary particles. This applies whenever we talk about ‘particles’. 4.2. PROPERTIES 37 its environment. As we explained in chapter 2, this is an unstable equilibrium, leading eventually to either evaporation or eternal growth. The thermal spectrum peaks roughly at ωrh/c =1,thethermalenergyrelatedtothe Hawking temperature (we will omit the factor 1/c from now on). Greybody factors modify the spectrum diﬀerently at both sides of this value, causing a shift in the peak energy. In the high-energy regime, the greybody factor approaches the geometric optics value for the total absorption cross section [20]:

2 n+1 (s) n +3 n +3 2 lim σ,n(ω)= πrh (4.15) ωrh1 2 n +1

This value is constant for ﬁxed M and decreases with the dimension. The dependence on n entersviathemetricofthespacetime(4.6) in which the particles move. This means that an external observer will see a black hole as a thermal blackbody with an eﬀective radius rc that is larger than the Schwarzschild radius:

1 n +3 n+1 n +3 r = r (4.16) c 2 n +1 h

This is also the effective value of r where radiation will backscatter off the potential. Because the black hole is effectively larger, the flux will be integrated over a larger surface leading to a greater radiative power in the high-energy regime. This may seem strange, but can be explained by an analogous example of a stove. The radiation from one single piece of coal inside the stove reflects off the walls. In thermal equilibrium, the temperature of the walls will equal the temperature of the coal. Consequently, the power radiated by the entire stove is much larger than the power of one single piece of coal. In the low energy regime ωrh 1 the greybody factors are strongly spin-dependent. In general, the spectrum will be suppressed in this regime, leading to a shift of the peak to a higher energy level (see figure 4.2). The general form of the greybody factor for emission on the brane is

(s) π | (s) |2 Ah | (s) |2 σ,n(ω)= 2 (2 +1)A,n = 2 (2 +1)A,n (4.17) ω (2ωrh)

2 | (s) |2 where Ah =4πrh is the horizon surface and A,n is the absorption probability, which is a function of ωrh. Analytical and numerical eﬀorts have been made to calculate the low- energy greybody factors. The review [20] provides useful results for scalar particles (s =0), fermions (s =1/2) and gauge bosons (s = 1). These are the three ‘spin channels’ through which radiation is emitted. If we were ever to detect black hole radiation, we could ‘read oﬀ’ the number of extra dimensions by the characteristic signature of the greybody factors. In this thesis we are only interested in the total radiative power of the black hole. We will incorporate numerical results from [20]toaccountforgreybodyfactors.Theresultsare presented in section 4.3.5. Before we come to this, the evaporation process is described step by step in the next section. 38 CHAPTER 4. BLACK HOLES IN 4+N DIMENSIONS

1 2 3 4

Figure 4.2: Enhancement of the radiation spectrum by greybody factors. The blue curve represents purely thermal radiation, the purple curve includes a sample greybody factor. The spectrum is suppressed in the low energy regime, while at higher energies the greybody factor reduces to a constant times the horizon surface area. The peak shifts to the right.

4.3 Evaporation

The evaporation process described qualitatively in section 2.3 will now be studied quantitatively in the context of extra dimensions. The goal is to give an estimate of a black hole’s lifetime dependent on its initial mass. The entire evaporation process will be described; the focus will be on the most important part of this: Hawking radiation. We showed in chapter 2 that the smaller a black hole, the hotter. This is still the case in higher-dimensional scenarios. The upper mass bound for black hole evaporation (2.68) is modiﬁed due to the enhanced Hawking temperature law (4.12). Although black holes are colder with extra dimensions present, all black holes smaller than the dimension size R (4.11) are still hotter than the CMB. Their emission rate will exceed the growth, and they will unavoidably evaporate. The evaporation process can be roughly divided into four stages [27]:

• The Balding phase, where the black hole sheds its electric charge and ‘hair’;

• The Spindown phase, where angular momentum (along with a large portion of mass) is lost;

• The Schwarzschild phase, where the remaining mass is lost through thermal radiation; 4.3. EVAPORATION 39

• The Planck phase, where the Planck-sized remainder is subject to quantum gravitational eﬀects.

In the next sections, these phases are described in more detail. Charge loss is considered in the balding phase. Of the next three phases, the Schwarzschild phase will be the most important and the only one quantitatively described in this thesis. As mentioned before, this thesis makes a semiclassical description of black hole decay. This description has the following requirements:

1. The decay is quasi-stationary: the black hole has time to reach thermal equilibrium in between particle emissions.

2. The black hole lives long enough to be a well-deﬁned resonance.

These requirements will be tested at the end of our analysis, to see if the results are consistent with the semiclassical approach in which they were calculated.

4.3.1 Balding phase The No Hair Theorem states that a black hole can be characterized by the quantities M, Q and J. However, the process of black hole formation is a highly asymmetrical process. Thus, upon creation, a black hole will typically have ‘hair’: its gravitational and electromagnetic fields have high multipole moments. The black hole first sheds these multipole moments through a process aptly called balding.IntheM Mpl limit, this is a classical process. The timescale on which it occurs is in order of the crossing time: r τ ∼ h (4.18) balding c This timescale will turn out to be very short compared to black hole lifetime for large enough black holes. At the same time, because of the tremendous electric field, the black hole will sheditschargebyemittingparticleswithalargecharge-to-massratio.Thisprocessisdue to Schwinger pair creation, described in section 2.2.3. We end up with a spinning, uncharged Kerr black hole. We assume that τbalding is negligible compared to lifetime. This will be another consistency- check for our analysis. We also want to know what happens to the charge of the black hole. Particles that create the hole will generally have charge. Will this charge end up in the black hole and if so, how quickly will it be shed? In section 2.2.3 we summarized the mass bounds associated with the different behavior of charged black holes in four dimensions. These bounds will be altered for braneworld scenarios, because the Planck mass is reduced. First of all, black holes with a charge larger than Q M > (4.19) Q∗ M∗ will be extremal, which means they are not to be found in nature because of the naked singularity featured in their metric. Black holes formed in high-energy collisions will typically 40 CHAPTER 4. BLACK HOLES IN 4+N DIMENSIONS

have a charge no larger than 2e and a minimum mass of M∗. This means they will always be nonextremal. This does not apply to primordial black holes, as their charge-to-mass ratio can be larger. For these black holes a lower mass bound will exist given their charge. According to our analysis and [16], a black hole cannot carry even one electron charge for a significant period of time if the Coulomb force due to the charge e exceeds the gravitational force, that is if 2 M (e/Q∗) < (4.20) M∗ m/M∗ This implies that only black holes larger than 10−4 kg can carry charge. This rules out the black holes created in high-energy collisions as suitable candidates. The charge carried by the colliding particles must somehow have vanished (perhaps through a Schwinger-like process) before we can call the system a black hole. Our analysis of microscopic black holes will therefore neglect the effects of charge. In the case of primordial black holes, charge is relevant. The diagram displayed in figure 2.5 will apply with different values of M. We will comment on this in chapter 5.Fornow, we conclude that the timescale of the balding phase can be neglected, and the product is a rotating, uncharged black hole.

4.3.2 Spindown phase Two particles colliding at nonzero impact parameter will produce a spinning black hole, because the angular momentum is conserved. During the spindown phase, the black hole loses mass and angular momentum through Hawking radiation. Superradiant modes stimulate the emission. Although this is a very important phase in the evaporation process — it is estimated [20] that the black hole loses up to 30 % of its mass in this phase — we will consider only the Schwarzschild phase. This will produce an error, because superradiance and modified greybody factors (leading to a higher emission rate) are neglected. These effects have not been fully studied in literature. Because we are interested in order-of-magnitude estimates, neglecting the spindown phase is justified.

4.3.3 Schwarzschild phase The starting point for the Schwarzschild phase is a nonrotating, uncharged black hole of mass M0. As mentioned before, we are interested in the black hole’s total radiative energy loss per unit time. Energy is lost on the brane and in the bulk. On the brane, the black hole emits radiation through the three ‘spin channels’ mentioned in section 4.2.3:scalars, fermions and gauge bosons. In the bulk, only gravitons are emitted. The emissivity in the bulk compared to emissivity on the brane is described by the number Bn, the bulk-to-brane ratio. These ratios are calculated in [20]; the ratio increases with n, as could be expected intuitively. Table 4.2 lists the values of these ratios. (s) The total power is the sum over the radiative power P˜n in each channel times the number of degrees of freedom g(s) in channel s, additionally including the bulk-to-brane ratio: (s) ˜(s) Pn =(1+Bn) g Pn (4.21) s 4.3. EVAPORATION 41

Channel Particle type g(0) g(1/2) g(1) Quarks 0 72 0 Gluons 0 0 16 Charged leptons 0 12 0 Neutrinos4 0 6 0 Photon 0 0 2 Z0 1 0 2 W+ and W− 2 0 4 Higgs boson 1 0 0 Total 4 90 24

Table 4.1: Degrees of freedom for diﬀerent particle types emitted from a black hole.

We will ﬁrst calculate the power for radiation on the brane and comment on the bulk-to- brane ratios afterwards. The values of g(s) for SM-particles on the brane are listed in table 4.1. The expression for the radiative power per channel is obtained by integrating the ﬂux spectrum (4.14) over the entire energy domain:

∞ 3 ( ) ω dω P˜(s) = σ s (ω) (4.22) n 4 ,n ω/kT 2 c 0 e ∓ 1 2π

(s) In the case of blackbody radiation in n =0,wehaveσ,n = Ah and this expression reduces to 2 ˜(s) c Pn = 2 (4.23) 15360πrh This factor is equal for channels s =0, 1 (bosons) and is multiplied by a factor .87 in channel 1 s = 2 (fermions). The total power is obtained by summing over these three channels, yielding

2 c bb Pn = 2 (4.24) rh with emissivity factor bb ≈ .00220. We will use this factor for blackbody radiation with n = 0 to compare with our results for greybody radiation. To obtain the radiative power for greybodies, we use the expression for the greybody factor (4.17) and changing to the dimensionless integration variable x = ωrh. The radiative power per channel can be written as

2 (s) ˜(s) c ˜n Pn = 2 (4.25) rh 4If right-handed neutrinos exist, this number should be changed to 12 42 CHAPTER 4. BLACK HOLES IN 4+N DIMENSIONS with ( ) (n +1)4 ∞ Γ s (x)x ˜(s) = n dx (4.26) n (4π)5 ex ∓ 1 0 (s) | (s) |2 Γn (x)= (2 +1)A,n (4.27)

(s) The dimensionless quantity ˜n is the emissivity factor, having a constant value determined by the greybody factor. It is evaluated numerically in [20, 28]. The values of these emissivities are listed in table 4.2. We see that at no extra dimensions, a black hole emits particles mainly through the the scalar channel. For higher dimensions, the gauge boson channel becomes equally important, while the fermion channel lags behind by a factor ∼ .85. Also, the emissivity grows with dimension n up to four factors of magnitude with respect to a blackbody with n =0.Thisisduetothegreybodyfactors. To obtain the total radiative power of a black hole with mass M in n + 4 dimensions, we rewrite equation (4.21) as follows:

2 4 c M∗ n+1 P (M)= n M 2 (4.28) n 2 ∗ an M We can rewrite this in terms of black hole temperature, for purposes that will become clear in the next chapters. The expression is equivalent to the Stefan-Boltzmann law for a body with emissivity coefficient n: k4 (4π)3 P (M)= A T 4 (4.29) n 3c2 (n +1)4 n h with T the black hole temperature given in (4.12). The emissivities per spin channel and for bulk gravitons are combined in the constant n (the well-known Stefan-Boltzmann constant σ is absorbed into this value): (s) (s) n =(1+Bn) g ñ (4.30) s The values of these total emissivity factors are listed in table 4.3. Note that these values are obtained in literature for a black hole with fixed radius. We claim that these are the same for a black hole with fixed mass; this is shown in appendix B.Wecanseethatthegreybody factors indeed lead to higher emissivities as compared to a blackbody, contrary to intuition. Notethattheradiativepowerinthebulkalsogoeswiththeinversesquareofthehorizon radius: ∝ 4+n ∝ −2 Pn,bulk AhT rh (4.31) because the extra factors of n in the surface area and temperature cancel. Therefore we can simply account for it by the proportionality factor Bn. Neglecting all other phases of evaporation with respect to the Schwarzschild phase, we assume that a black hole evaporates solely by the obtained expression for radiative power (4.28). In the next section we will use this formula to calculate the lifetime of a black hole. 4.3. EVAPORATION 43

(0) (1/2) (1) n Scalars: ñ Fermions: ñ Gauge Bosons: ñ B-t-b ratio Bn BB 2.07 × 10−5 1.80 × 10−5 2.07 × 10−5 N/A 0 2.98 × 10−4 1.64 × 10−4 6.72 × 10−5 N/A 1 2.66 × 10−3 2.32 × 10−3 1.82 × 10−3 .40 2 1.07 × 10−2 9.73 × 10−3 9.71 × 10−3 .24 3 2.97 × 10−2 2.65 × 10−2 2.96 × 10−2 .22 4 6.61 × 10−2 5.75 × 10−2 6.86 × 10−2 .24 5 1.28 × 10−1 1.09 × 10−1 1.35 × 10−1 .33 6 2.23 × 10−1 1.87 × 10−1 2.37 × 10−1 .52 7 3.62 × 10−1 2.99 × 10−1 3.86 × 10−1 .93

(s) Table 4.2: Emissivities ñ for different values of s and n (and the case of a black body ‘BB’ in n = 0); bulk-to-brane ratios Bn. They apply to a black hole with fixed radius. Taken from [28].

n n n/ bb 0 .0176 7.99 1 .370 1.68 × 102 2 1.43 6.50 × 102 3 3.93 1.79 × 103 4 8.81 4.01 × 103 5 18.0 8.19 × 103 6 35.6 1.62 × 104 7 72.8 3.31 × 104

Table 4.3: Total emissivity constants n in all three spin channels combined for diﬀerent values of n. These emissivities are larger compared to those of a blackbody up to four orders of magnitude, as the values in the third column show. 44 CHAPTER 4. BLACK HOLES IN 4+N DIMENSIONS

4.3.4 Planck phase

When the black hole has shrunk to M ∼ M∗, the semiclassical picture is no longer valid, and all known physics breaks down. The radiative power (4.28)blowsupasM → 0. To predict what will happen in this regime, we need a quantum gravitational theory. This might be provided by string theory. The black hole will either vanish, emitting some highly energetic modes, or a Planck-sized stable object will remain. If the Planck phase is ever observed, it will be the first we learn from nature about quantum gravity. The first three phases encompass the larger part of black hole evaporation, as M M∗. When calculating the lifetime in the next section we will therefore neglect any effects that might occur in the Planck phase, assuming total evaporation through Hawking radiation.

4.3.5 Lifetime We calculate the lifetime of a nonrotating, neutral Schwarzschild black hole of mass M in 4+n dimensions. We assume it decays only through Hawking radiation on the brane and in the bulk. The total emitted power is given by (4.28). We use it to obtain a diﬀerential equation for the black hole mass:

2 2 dM c M∗ n+1 = − n M 2 (4.32) 2 ∗ dt an M

Solving for M(t) and putting the initial mass M(0) ≡ M0 we have

n+1 n+3 n+3 2 n+1 M∗c n n +3 M∗ M(t)=M0 1 − t (4.33) 2 an n +1 M0 Putting this equal to zero and solving for t yields for the lifetime of a black hole:

n+3 2 n+1 an n +1 M0 τ = 2 (4.34) M∗c n n +3 M∗ For black holes of a few TeVs, such as the ones that might be produced at LHC, this corresponds to a very short lifetime (τ ∼ 10−26 s). These black holes barely exist before they evaporate. For a black hole with ﬁxed mass, the lifetime reduces with increasing n. The increasing emissivities due to greybody factors enhances this eﬀect. Plots of black hole properties are displayed in appendix C. Taking M0

Black holes heavier than Mc will decay according to the four-dimensional law, which can be obtained from our analysis had we set the Planck scales to their four-dimensional values and n =0: 1 2 3 3 Mplc 0 Mpl M(t)=M0 1 − t for M0 >Mc (4.36) 4 M0

When the mass, reducing according to this law, has reached Mc, the behavior becomes (4 + n)-dimensional again. This is only relevant for heavy black holes. The next chapter discusses a speciﬁc kind of these: primordial black holes. Chapter 5

Primordial black holes

In view of the braneworld scenarios analyzed in the previous chapters, we set out to investigate their effect on a specific class of black holes: primordial black holes (PBHs). These are black holes formed during the early, radiation-dominated plasma-phase of the early universe. The most viable theory is that density perturbations in the primordial plasma collapsed gravitationally and formed horizons. When the universe cooled off and expanded, these remained as heavy black holes. The aim of this chapter is to investigate how the existence of large extra dimensions alters the constraints of black hole formation in the primordial plasma. Firstly, the longer lifetime of black holes in scenarios with extra dimensions, derived in section 4.3.5, has a direct consequence on PBH mass bounds. This will be the topic of the section 5.1.This section is a ‘cosmological intermezzo’ that deviates from the common context of this research — the microscopic regime. Yet it is useful, as it serves as an example of the impact that the discovery of extra dimensions could have on physics. Besides collapse of density perturbations, another process that can lead to PBHs is made possible by the presence of extra dimensions. This is the rapid growth of microscopic black holes by thermal absorption of the primordial plasma. In a cosmological context, this mech- anismfurtheraltersthemassboundsonPBHs.Inthisthesis,itservesasananaloguefor the model presented in chapter 6, where this situation is simulated in a laboratory. The mechanism is explained in section 5.2.

5.1 Mass limits for PBHs in braneworld scenarios

Consider the question: what is the minimal initial mass of a black hole created during the early stages of the universe, surviving until now and decaying exclusively through Hawking radiation? We will answer the question for the speciﬁc case of M∗ ∼ 1 TeV, which is assumed in the larger part of our previous analysis. The minimum mass for PBHs surviving up until present time is found by substituting

46 5.2. PRIMORDIAL HIGH-ENERGY COLLISIONS 47

n Mmin (kg) Mc (kg) 0 1.3 × 1010 N/A 2 1.4 × 101 2.5 × 1022 3 2.2 × 104 1.9 × 1017 4 4.6 × 106 5.1 × 1014 5 2.6 × 108 1.5 × 1013 6 5.6 × 109 1.4 × 1012 7 6.1 × 1010 2.6 × 1011

Table 5.1: Minimal mass and crossover mass for PBHs in scenarios with M∗ ∼ 1TeV.

the age of the universe τuniv ∼ 13.7 Gyr into (4.33) and solving for M0:

2 n+1 M∗c n n +3 n+3 M0 >M = M∗ τ (5.1) min 2 univ an n +1

For the TeVG-scenarios relevant in this thesis, the values for Mmin and Mc are listed in table 5.1.WecanseethattheconditionM0

5.2 Primordial high-energy collisions

The common creation mechanism for primordial black holes is gravitational collapse of density perturbations [25]. This process will be influenced when viewed in a braneworld context, altering the mass bounds for resulting PBHs. We will not delve into this cosmological subject. Instead, we view another mechanism of PBH creation that is only made possible by the extra dimensions: rapid growth of microscopic black holes in the primordial plasma. The consequences of black holes created by high-energy collisions in the primordial plasma have long been neglected in literature. This is due to the assumption that they are too short-lived to have any significant influence. However, it was suggested in [29]thatinhigher dimensional scenarios, these microscopic black holes can grow rapidly by absorbing matter 48 CHAPTER 5. PRIMORDIAL BLACK HOLES from the primordial plasma surrounding them. In this chapter, we briefly review this theory, without commenting too much about its influence on PBH mass bounds. It merely serves as a basis for the ‘QGP-black hole’ model proposed in chapter 6. The basic idea in [29] is as follows. Since the temperature of a black hole is inversely related to its radius, small black holes are very hot. However, when submerged in a substance with an even higher temperature, there will be a net absorption of radiation across its horizon. In this way, microscopic primordial black holes can rapidly grow into massive ones during the early hot stages of the universe. That is, if describing the process with thermodynamics, which we will comment on in the next chapter. In [29], two cases are considered; an empty bulk space and a bulk space in thermal equilibrium with the 4-brane. In this thesis, these two cases are unified by using the expression for radiative power (4.29). The emission coefficients n account for brane-confined radiation as well as radiation in the bulk by the relation (4.30). It is important to bear in mind that these coefficients will be the same for emission as they are for absorption, as explained in section 4.2.3. Consider a black hole with rh R created at t = 0 within the primordial plasma. The rate of change in mass of the black hole can be expressed in terms of the black hole temperature TBH (4.12) and the temperature of the universe T as

dM k4 (4π)3 = A (T 4 − T 4 ) (5.2) dt 3c4 (n +1)4 n h BH

We see from this differential equation that there is a mass Mthresh corresponding to T = TBH above which a black hole will absorb more radiation than it emits: 2 n+1 n +1M∗c Mthresh = M∗ (5.3) 4πan kT Figure 5.1 displays a graph of these values for increasing T . If a black hole is created with a mass larger than this value, there will be a net absorption of radiation resulting in an increase in mass. If this happens to be the case, the second term in (5.2) becomes smaller and smaller with t, allowing us to neglect it when solving the differential equation. This leaves only the first term in the equation, describing an increase in mass. We will now solve (5.2) for the case M0 >Mthresh. Since we assume that PBH creation takes place in the radiation dominated era, we incorporate the relation between time and temperature given by the Friedmann equations:

c2 M t ∝ pl (5.4) k2 T 2 We use this relation to obtain an expression for M at a given temperature T . We take M0 M∗ and T0 to be the values of M and T at t = 0 (the moment of black hole creation). The solution is n+1 n−1 2 n−1 M0 n+1 k M pl 2 − 2 M(T )=M∗ + Cn 4 3 (T0 T ) (5.5) M∗ c M∗ 5.2. PRIMORDIAL HIGH-ENERGY COLLISIONS 49

where all dimension-dependent numerical factors are absorbed into Cn:

(4π)4(n − 1) C = a2 (5.6) n (n +1)5 n n

1 The values of Cn have a value of order 10 for the range 2 ≤ n ≤ 7. Equation (5.5)takesinto account the cooling of the universe, which has a restraining eﬀect on black hole growth. But because Mpl M∗, the second term dominates almost immediately as the universe cools. The mass will therefore almost instantaneously approach the maximum value

n+1 2 n−1 Cnk Mpl 2 Mmax = 4 3 T0 M∗ (5.7) c M∗

This is a very large number compared to the initial mass M0; see ﬁgure 5.2.In[30], an argument was shown why this does not happen in four dimensions. In that case, the diﬀerential equation for growth reads 4 dM k 3 4 = (4π) 0A T (5.8) dt 3c4 h with the four-dimensional expressions for Schwarzschild radius (2.1) and Hawking temperature (2.51). This has the solution

−1 3 2 256π k 0 M0 →∞ ∼ − M(t ) M0 1 4 3 T0 (5.9) c Mpl

3 Because of the very small factor M0/Mpl, the mass will not get much larger than M0 for initially small (M0 M∗) black holes. Therefore, the growth of a black hole by rapid absorption of the primordial plasma exclusively occurs in braneworld scenarios. Given the temperature of the primordial plasma as a function of t, we could have solved for M(t). This will be the case in the next chapter. The most important feature outlined in this section is the rapid growth of a black hole submerged in a hotter plasma, due to the extra dimensions. In next chapter, a model is proposed to imitate this situation in the lab. 50 CHAPTER 5. PRIMORDIAL BLACK HOLES

(TeV/ 2 ) Mthresh c

1011

108

105

100

T (TeV/ k ) 200 400 600 800 1000

Figure 5.1: Values of Mthresh for increasing T (temperature of the universe). From bottom to top: n =2, 3, 4, 5, 6, 7. If a black hole is created with M0 above this value, the mass grows almost instantaneously to Mmax.

2 Mmax (TeV/c )

1049

1043

1037

1031

1025

T (TeV/ k) 200 400 600 800 1000 0

Figure 5.2: Values of Mmax for increasing T0 (the temperature of the universe at black hole creation). From top to bottom: n =2, 3, 4, 5, 6, 7. This is the maximum mass attained by a black hole with M0 >Mthresh, taking into account the cooling of the universe. Chapter 6

Black holes in the lab

Supposing TeV-scale gravity exists, the creation of black holes at the LHC is possible. How- ever, as the results of section 4 show, any black hole created from a ∼ 10 TeV collision will 2 −26 not last much longer than 10 tp∗ ∼ 10 s. This is too short to experimentally separate the black hole ‘creation’ and ‘decay’ effects. The black hole is too short-lived to observe typical black-hole behavior. The black hole will evaporate before even having the chance of accreting nearby matter. In this chapter, we explore the possibility of creating a heavier, stable black hole. In this scenario, a black hole is created within a quark-gluon plasma. If this plasma is hotter than the black hole, the black hole will start absorbing it thermally. A great portion of the plasma could be ‘eaten’ before it cools down. If we keep firing nuclei into the plasma, we could in theory make the black hole larger and larger, until its lifetime reaches macroscopic scales. The resulting ‘heavy’ black hole could be preserved by adding charge to it and capturing it in a magnetic field. Our analysis is structured as follows. First, we shortly review quark-gluon plasma creation, cooling and hadronization. Next, we describe our experimental setup: the simulta- neous creation of a quark-gluon plasma and a black hole inside it. We compare absorption and emission rates of the black hole to establish conditions for growth. We then assess the feasibility of this model.

6.1 The quark-gluon plasma

At particle colliders such as the LHC (the ALICE experiment) and RHIC, heavy nuclei are collided to imitate the circumstances shortly after the Big Bang. Upon collision, for a very short while before scattering, the nuclei form one very hot ball of matter. If the temperature of this fireball is sufficiently high, the confinement of the quarks inside the nucleons is broken. In the process, gluons are released. Because of the asymptotic freedom of QCD, the deconfined quarks and gluons can move as free particles within the volume of the nucleus. This ‘quark-gluon plasma’ (QGP) will thus behave as an ideal gas of ultrarelativistic

51 52 CHAPTER 6. BLACK HOLES IN THE LAB degenerate fermions. The gas will reach thermodynamic equilibrium, expand and cool, and eventually the quarks and gluons will form hadrons again (hadronization). This process is depicted in ﬁgure 6.1. We are mainly interested in the expansion and cooling phase. What follows now is a brief description of the QGP [31].

Figure 6.1: Formation of the quark-gluon plasma: (a) Collision of relativistic, Lorentz- contracted nuclei; (b) Fireball of extremely dense nuclear matter created; (c) Deconﬁnement of color-charged quarks and gluons: the QGP (d) Expansion and hadronization. Taken from [32].

6.1.1 Creation When at rest, the atomic nucleus is cold: the nucleons within move non-relativistically. If the energy per nucleon E is raised above the binding energy, quarks and gluons may be released. But if the energy surplus is only small, the free quark-gluon density is low and the 6.1. THE QUARK-GLUON PLASMA 53 plasma may reach equilibrium over a long time. If we want to release as many quarks and gluons as possible, the QGP should be as hot and dense as possible. This is called ignition of the QGP. The threshold energy for this to happen corresponds to the Fermi energy of a degenerate fermion gas with density N/V ,withN the total number of nucleons in the two nuclei forming the plasma and V the volume of the nucleus:

2 3 2 3π2N / c = ∼ ∼ 125 MeV (6.1) thr 2m V a

. WehaveapproximatedN/V ∼ 1/a3,wherea ∼ 1.5 fm is the radius of one nucleon. We used the deﬁnition of the compton wavelength a = c/m for the relation between the size and 3 9 mass of a nucleon. The average energy per quark is then 4 thr and per nucleon it is 4 thr ∼ 180 MeV. The gluons in the plasma have an energy density spectrum similar to a thermal blackbody. Consequently, their number increases with temperature. Because the chemical potential of quarks vanishes at high energies, we assume the number of ultrarelativistic quarks is more or less ﬁxed. A hot QGP (Ep/N thr) will then be dominated by the gluons, whose energy spectrum approaches that of a thermal blackbody:

σ4 E ∼ VT4 (6.2) p c with σ4 the Stefan-Boltzmann constant. These hot plasmas are the ones studied in this thesis, as they exist in the same TeV-regime where black hole creation takes place in braneworld scenarios. The plasma will reach equilibrium in a very short time τeq ∼ /kT . Before equilibrium, the energy spectrum will diﬀer from the blackbody spectrum. We will neglect this fact in our analysis, just as we neglected the time a black hole reaches equilibrium in our previous analysis. This is the quasi-stationary approach, whose validity will be commented on in the discussion. In TeV-units, the relation between energy per nucleon and temperature can be expressed as 1 4 E (TeV) / kT = 10−12 p TeV (6.3) N The approximation of the QGP as an ultrarelativistic fermi gas is in adequate agreement with more accurate calculations of the energy density such as in [33]. At ALICE, PbPb- collisions will have energies up to 5.5 TeV per nucleon; this corresponds to a temperature of a few GeV’s1. To double this temperature, the energy density should be enhanced by a factor 16. This makes hot QGP’s diﬃcult to create: the required energy density goes with the fourth power of the temperature.

6.1.2 Cooling and Expansion Once formed, the QGP starts expanding immediately, with near-light speed. Viewing the 1/3 plasma as a sphere with initial radius R0 ∼ N a, and assuming spherical expansion with

1Experimental data shows that the QGP is not a gas, but more likely a perfect liquid. We will stick with the Fermi-gas description. 54 CHAPTER 6. BLACK HOLES IN THE LAB near-light speed, this expansion goes as

R(t)=R0(1 + αt) (6.4) with α = c/aN 1/3.Using(6.2), the temperature decreases as

−3/4 T (t)=T0(1 + αt) (6.5)

1/4 1/3 3/4 The initial temperature equals kT0 = Ep (c/N a) . During the expansion, the number of gluons increases with a factor (1+αt)3/4 and thus the entropy increases during this process. The cooling of the QGP is therefore an irreversible, non-adiabatic process.

6.1.3 Hadronization

When the temperature drops below thr, the conﬁnement phase is reached again, and the partons in the QGP form into hadrons again. Because they have all been ‘jumbled up’ in the plasma, the partons will generally not combine with their pre-collision partners. In- stead,amixtureofmesonsandbaryonsistobeexpected,emerginginjetsfromtheQGP. The hadronization process is expected to start around the edges. The timescale at which hadronization takes place is obtained from (6.5): 4/3 1 T0 τhad ∼ −1+ (6.6) α 9Tthr/4 where Tthr ≡ thr/k or, numerically: −22 −3 4/3 τhad ∼ 10 −1+10 T0(MeV) s (6.7)

This can be interpreted as the brief lifetime of the Fermi-gas approximated QGP.

6.2 The QGP Absorption Model

We consider the possibility that two nucleons collide inside the pre-QGP at a small enough impact parameter to form a black hole. This black hole could in theory absorb nearby partons in the QGP. If the plasma is hot and dense enough, the absorption rate could exceed the evaporation rate, causing the black hole to grow thermally. This is a runaway process: the black hole cools while growing, making the net absorption rate even larger. The absorption will stop if the plasma cools oﬀ below black hole temperature, or if the entire plasma is ‘eaten’ by the black hole and there’s nothing left to absorb. In [33], this scenario is applied to PbPb-collisions at the LHC. The conclusion is that the plasma at LHC is not hot enough (by a factor 108) to establish absorption, and the black hole will evaporate in a fraction of a fm/c. However, this analysis does not take into account that the black hole temperature decreases with its mass. In cases where the black hole and QGP masses are correlated, this lowers the minimum energy scale at which absorption can take place. 6.2. THE QGP ABSORPTION MODEL 55

In this section, we review the conditions for black hole growth. We solve the differential equations for absorption and evaporation, and introduce relevant timescales. We consider two possibilities: the black hole is created by a collision of two of the pre-plasma nucleons, or by two nucleons from a different beam, collided simultaneously at a lower center-of-mass energy inside the QGP. In the first case, QGP and black hole masses are correlated; this important distinction is not made by [33].

6.2.1 Growth and decay We approximate the QGP as a perfect Fermi gas at temperature T . We have a black hole with mass M submerged in this QGP. Based on the model in [29], described in section 5.2, therateofchangeinM is

dM k4 (4π)3 = A (M)(T 4 − T (M)4) (6.8) dt 3c4 (n +1)4 n h BH

Solving this diﬀerential equation will be the aim of this section. From this point on, we will 2 express M and Ep/N in Planck units , making these quantities dimensionless. In this way, the numerical results will be more insightful. Rewriting the diﬀerential equation in terms of M and incorporating results from the previous section, we have 4 2 2 2 dM 4π a n Ep 2 −3 M∗c n − 2 = n M n+1 (1 + αt) − M n+1 (6.9) 2 3 2 dt n +1 M∗ ca N an

We will maintain the quasi-stationary approximation for the QGP, meaning that the plasma has time to reach equilibrium in between particle emissions. The plasma is taken to behave thermodynamically during the whole absorption process. This implies that effects such as the fast cooling of the edges of the plasma, will be neglected. Solving equation (6.9) will be the topic of this section. To do this analytically proves a tough job, because the equation is non-linear. It can be solved numerically. However, the equation can also be decoupled in two separate ones, solved in different regimes of t.Thisis justified because depending on the initial value of M, one term or the other will dominate and continue to do so until either the black hole has evaporated or the plasma has cooled off. It will be shown that growth and decay are such rapid processes that even when the difference in temperature is only slight, one of them can be neglected immediately. If growth dominates initially, the black hole will grow until the plasma has cooled of; then it will decay. This is clarified by figure 6.2.ThevalueofMthresh (see (5.3))canthusbeseenasa‘critical’ value for M0, marking a transition between different mass evolution laws. Considering first the case M0

2 2 The SI values of M and Ep/N can then be obtained by multiplying them with M∗ and M∗c , respectively. The reason not all quantities are expressed in Planck units, is that we want to keep track of the numerical factors. 56 CHAPTER 6. BLACK HOLES IN THE LAB

Figure 6.2: Mass evolution of a black hole within a QGP. The bottom curve corresponds to a hole with M0 Mthresh,and will grow until the plasma has cooled at t ∼ 1/α, after which it decays. The lifetime of the second (heavier) black hole is greatly extended with respect to the ﬁrst one, even though its initial mass was only slightly above Mthresh. Note that the graph is not drawn to scale. in (6.9) yields the solution (4.33):

n+1 n+3 n+3 − n+1 Mdecay(t)=M0 1 − AnM0 t ,M0

2 M∗c n +3 A = n (6.11) n 2 an n +1 The black hole rapidly decays through this evaporation law in a time τ (4.34). This is a lot shorter than the hadronization time τhad (6.6), so the black hole will dissolve within the plasma before detection is possible. If,however,theconditionM0 >Mthresh is satisfied, the first term in (6.9) will dominate while αt 1. The differential equation is solved with only the first term:

n+1 − n−1 n−1 Ep n+1 M0 >Mthresh Mgrowth(t)=M0 1+Bn M0 f(t) , (6.12) N αt 1 4 − 2 2 (4π) (n 1) an n Bn = 5 2 2 3 (6.13) 2(n +1) M∗ c a α 1 f(t)=1− (6.14) (1 + αt)2 6.2. THE QGP ABSORPTION MODEL 57

The function f(t) approaches the value 1 after a time ∼ 1/α ∼ 10−22 s. At this point the plasma will have cooled oﬀ below the temperature of the black hole, which has attained a maximum mass Mmax. The hole will evaporate through (6.10), making the entire equation for the mass evolution of the black hole with M0 >Mthresh: 1 1 1 M(t)=θ(−t + )Mgrowth(t) + θ(t − )Mdecay(t − ) (1) (6.15) α M0 α α Mmax ∼ M (t) (1) (6.16) decay Mmax for in cases of our interest 1/α is far less than the total lifetime. The maximum value of M(t)isapproximatedby 4 M n+1 (1) − thresh (0) Mmax = 1 (0) Mmax (6.17) Mmax n+1 n−1 E n−1 M (0) = M n+1 + B p (6.18) max 0 n N

This is the maximum mass that could theoretically be absorbed by the black hole. The correction factor appearing in the ﬁrst equation is due to neglecting the second term in (6.9) when solving for the case M0 >Mthresh. Finally, the black hole could not absorb more than 2 2 theentireplasmaEp/c and the limits only make sense if M0 Mthresh)is

(1) 2 M1 = max[M0, min(Mmax,Ep/c )] (6.19) Using these results, we arrive at an expression for the lifetime of a black hole within a QGP: τ ,M0 M M1 thresh where τ is the lifetime of a black hole as given by (4.34). It is evident that the lifetime is longer in the second case, as M1 M0. Another interesting quantity to compute is the average power. If it is isolated, a microscopic black hole created in a high-energy process will decay in such a short time, that we would call it ‘explosion’ rather than ‘evaporation’. The question is whether the larger mass and lifetime of the plasma black holes described in this section lead to lower rates of evaporation. It turns out they do; as P decreases with M theincreaseinmassleadstoa lower average radiative power:

2 M1 n +3 P¯ = M∗c = P (M1),M0 >M (6.21) τ n +1 n thresh

Note that this value is an average; the formula for Pn approaches an asymptotic value as M → 0. This is of course due to the shortcoming that the physics of the Planck phase are ill-deﬁned. 58 CHAPTER 6. BLACK HOLES IN THE LAB

Summarizing this section, we solved the diﬀerential equation governing growth and decay of a black hole within a quark-gluon-plasma. It turned out the mass evolution depends strongly on the initial conditions. At M0 = Mthresh, there is a transition between rapid decay (a hotter black hole) and growth (a hotter plasma). In the case of growth, the lifetime of the black hole is extended and the average radiative power decreased. In the next section these results are evaluated numerically, in the two separate cases of non-correlated and correlated initial masses of the black hole and the QGP.

6.2.2 Numerical results In this section we present numerical plots of maximum mass, lifetime and average power of a black hole created within a quark-gluon-plasma. The input parameters are the initial black hole mass M0, and the energy density of the plasma, Ep/N (both in Planck units). Although current technology can achieve energies around ∼ 10 TeV at most, we have explored 20 the (M0,Ep/N )-regime up to 10 TeV, because the most interesting results occur at these high energies.

Figure 6.3: The black hole inside the QGP can either be created by a separate beam with an energy independent of the plasma energy (left) or by two nucleons inside the (pre-)QGP with energy Ep/N (right).

On the next three pages, numerical plots are displayed of three quantities:

• Figure 6.4: Maximum mass as a fraction of initial mass M1/M0 (6.19)

• Figure 6.5: Lifetime: τ (6.20)

• Figure 6.6:Average radiative power: P¯ (6.21)

The value used for the number of nucleons in the QGP was N = 400, typical for a PbPb- collision. The plot of M1/M0 differs from the rest because it shows a relative quantity. It can be seen form this that the effect discussed in this section, black hole growth due to a hot QGP, is most significant in the limit Ep/N M0. Thisisnotsurprising,asinthislimitthe temperature difference is the greatest. 6.2. THE QGP ABSORPTION MODEL 59

The values of Ep/N needed to achieve growth of black holes with a mass of a few TeV’s (such as the LHC black holes) are very large, around 107 TeV. This is in accordance with the values calculated in [33]. The phenomenon of growth can however be observed already at lower energies, for black hole temperature decreases with mass. In other words, Mthresh is lower at higher plasma densities. For instance, in an n = 2 scenario, a black hole of M0 ∼ 100 2 7 TeV/c couldabsorbaportionofaplasmaofEp/N ∼ 10 TeVtoreachamaximumsize of 105 TeV/c2. So, growth can occur already at lower energies than the limits mentioned in [33]. The growth factor enhances the lifetime of the black hole; a larger black hole will live n+3 longer. The black hole lifetime is a power law in M with exponent n+1 . Therefore the ratio of the lifetime of a black hole with mass M0 in a QGP as opposed to an isolated black hole is n+3 τ(M1) M1 n+1 = (6.22) τ(M0) M0 Following the same reasoning, the average power will be lowered by a factor

−2 P¯(M1) M1 n+1 = (6.23) P¯(M0) M0

The absolute values of lifetime and average power are plotted in figures 6.5 and 6.6.Inthe last figure, the color red corresponds to a decay rate that exceeds the modified Planck power 20 P∗ ∼ 10 W. This means that the decay is a process that cannot properly be described by our semiclassical picture. Moreover, a theory of quantum gravity is needed to describe it. For high values of n, this applies to a very large portion of the (M0,Ep/N ) parameter space. This symptom will be discussed in the conclusion. We will now address a matter mentioned in the previous section. Two different situations are imaginable, see figure 6.3. The first case is a black hole created within the QGP by a separate particle beam. This implies that M0 is independent of Ep/N . On the other hand, there could be two partons of the QGP colliding to form a black hole. In that special case M0 and Ep/N are correlated via the relation (4.3) between center-of-mass energy and resulting black hole mass: M0 ∼ fnEp/N (6.24) In this case, two of the partons forming the QGP collide with sufficiently small impact parameter to produce a black hole. If the black hole is heavy enough, it will be colder than the QGP and absorb a portion of it, according to the results we have just discussed. This scenario will probably resemble reality, because in such a hot QGP it is likely that black holes are formed. The values of Mmax, τ and P can be read off from the plots on the previous pages if we imagine a linear curve at M0 ∼ fnEp/N for some value of fn. A black hole made and grown in a detector, if we can ever produce one, will be very short-lived. That is, under the conditions specified: the black hole can only attain so much energy as is available in the QGP formed by two nucleons. There will be a value of the energy per nucleon Ep/N where the entire plasma is absorbed. In practice, this will probably not be the case. Cooling will start around the edges of the plasma, that will therefore not be 60 CHAPTER 6. BLACK HOLES IN THE LAB absorbed by the black hole. This effect was neglected in our analysis. Only a fraction of the QGP will be absorbed. It is highly unlikely that we will ever be able to produce a hot enough QGP on Earth to be absorbed by a micro black hole. We will comment on the feasibility of our scenario in section 6.2.5. A typical QGP-grown black hole will live longer than a TeV-scale one; maybe long enough to distinguish its typical black hole properties (such as the greybody characteristics of its radiation spectrum). But it will still be unstable; all small black holes will evaporate. If we were able to somehow put more energy into the plasma, we could fabricate black holes with a more significant impact on measurements. This scenario is discussed in the section 6.2.4. In the next section, the assumptions made at the outset of our analysis are checked consistency with our results. 6.2. THE QGP ABSORPTION MODEL 61

0 5 10 15 20 10log(M1/M0)

n = 2 n = 3

10log(M0) M0 > Ep

QGP entirely absorbed QGP partly absorbed

M < Mthresh

10log(Ep/N)

n = 4 n = 5

n = 6 n = 7

Figure 6.4: The ratio of maximum and initial masses M1/M0 of a QGP black hole plotted 20 on a logarithmic scale for 1 ≤ M0,Ep/N ≤ 10 . 62 CHAPTER 6. BLACK HOLES IN THE LAB

0 8 16 24 >32 10log(τ’/ t ) *

n = 2 n = 3

10log(M0)

10log(Ep/N)

n = 4 n = 5

n = 6 n = 7

Figure 6.5: Lifetime τ of a QGP black hole in units of t∗ plotted on a logarithmic scale for 20 1 ≤ M0,Ep/N ≤ 10 . 6.2. THE QGP ABSORPTION MODEL 63

<15 11 14 17 >0 10log(P / P ) *

n = 2 n = 3

10log(M0)

10log(Ep/N)

n = 4 n = 5

n = 6 n = 7

Figure 6.6: Average power P¯ of a QGP black hole in units of P∗ plotted on a logarithmic 20 scale for 1 ≤ M0,Ep/N ≤ 10 . 64 CHAPTER 6. BLACK HOLES IN THE LAB

6.2.3 Theoretical Issues Aside from the experimental outlook, there are some problems with the theory of microscopic black holes that deserve attention. In this section, the results will be checked for consistency the assumptions made at the end of the introduction of section 4.3. First of all, the assumption that black hole evaporation can be approximated by considering only the Schwarzschild phase does not hold for small black holes, such as the ones that would be produced at LHC. For example, consider the balding time rh/c as a fraction of the total lifetime τ (4.34) of a black hole. According to our results, this fraction would be

− n+2 τ n +3 M n+1 balding = n (6.25) τ an n +1 M∗ Plottedinﬁgure6.7, it can be seen that only above a certain mass limit, this fraction is small. For increasing values of n, this lower bound is higher. In order for the balding time to be 1% of the lifetime, the mass of the black hole needs to be at least 72 TeV/c2 in n =2, and as large as 4000 TeV/c2 in n =7. balding

3.0

2.5

2.0

1.5

1.0

0.5

(TeV/c2 ) 20 40 60 80 100 M

Figure 6.7: Values of τbalding/τ for increasing M. From bottom to top: n =2, 3, 4, 5, 6, 7. At low masses, balding eﬀects become signiﬁcant. They also increase with n.

This implies that the effects of balding cannot be neglected for small black holes of order 10 TeV/c2. Furthermore, the effects of the spindown phase and Planck phase, the latter of which is completely unpredictable, cannot be neglected either. This means that the typical greybody radiation from black holes at LHC will be clouded by these other types of radiation. This will make it difficult to recognize a black hole as such. Furthermore, the small mass of these black holes will make the decay rates unphysically high. This is one problem that the 6.2. THE QGP ABSORPTION MODEL 65

QGP Absorption Model could solve. It will make black holes live long enough to discern the diﬀerent stages of decay. To assess the validity of the semiclassical picture, the two conditions formulated at the start of section 4.3 will be checked [28]. Firstly, a black hole is a well-deﬁned resonance only if its lifetime is longer than the timescale associated with its mass:

τ (6.26) Mc2

Only then does the Quantum Field Theory required to derive Hawking radiation make sense. It turns out that for black holes with M 5TeV/c2, this condition is satisﬁed. For M =5TeV/c2, it is not satisﬁed in n =6, 7. This is again an argument for the case that heavier black holes should be produced in order to observe their thermodynamics. An important point is the condition that the time between emissions should be large enough to allow quasi-stationary decay. The black hole should be allowed to reach equilibrium between every emissions, otherwise the notion of a black hole temperature would make no sense. This condition can be formulated as

1 r h (6.27) F c where F is the total flux of particles, which can be obtained by integrating the formula for total radiative power (4.22)withω2 in stead of ω3 in the integrand. Contrary to the previous condition, this condition is mass-independent. As it turns out, it is not fulfilled for the cases n ≥ 4, indicating a breakdown of thermodynamics. This is due to the large values of the greybody factors, absorbed in n, for higher values of n. A violation of thermodynamics adds to the problem observed in figure 6.6,thatinalarge domain of the energy scale, the decay rates exceed the Planck level. It looks like quantum gravitational effects cannot be neglected, especially in higher dimensions. Summarizing, the conditions that are violated in the thermal model of the ’LHC black hole’ situation, can partly be solved by the QGP Absorption model. However, the breakdown of thermodynamics at n ≥ 4 indicates an intrinsic problem with applying thermodynamics to small braneworld black holes. Lastly, we come back to the bounds on the fundamental Planck scale introduced in chapter 3,intable3.1.Itturnsoutthatthescenariowheren = 2, that seems to be the most ‘black hole-friendly’ scenario, is largely disfavored. Experimental bounds seem to rule out that the Planck scale would be as low as 1 TeV in cases of low n, thereby making the bounds on black hole production more stringent. Qualitatively compared to a Planck scale value of 1 TeV, a higher Planck scale will correspond to a smaller Schwarzschild radius (4.8), a higher temperature (4.12) and a shorter lifetime (4.34). This means that in the plasma-black hole model, the maximum mass will be lower. This implies that the values of the maximum mass (growth factor) computed numerically in figure 6.4 overestimate reality. 66 CHAPTER 6. BLACK HOLES IN THE LAB

6.2.4 The Gunfire Model We want to control the absorption of mass by the black hole and make it grow to a certain size before letting it evaporate at a macroscopic timescale. We propose the so-called ‘gunfire- model’, where nuclei are fired at a high frequency into the plasma; thereby keeping the plasma at a certain temperature. The black hole can then absorb these nuclei, and if it has become large enough for our purposes, we stop the firing. Because the heavy ions are charged, we can imagine an experimental setup where the black hole is levitated by a magnetic field. In this way, a ‘table-top’ black hole may be created. If we would manage to keep the plasma at a constant temperature T0 after its creation, the differential equation (6.9) with only the growth term would have the solution

n+1 − n−1 n−1 Ep n+1 M(t)=M0 1+2A M αt (6.28) n N 0 This would imply infinite growth until a desired mass level is reached. Some idea of numbers involved:ablackholewithlifetime1sisinthemassrange10−8kg (for n =2)upto10−2 kg (for n = 7). For the first one, we would need 105 lead nuclei fired at very high frequency. Current collider technology falls well short of achieving this. It will be very hard to keep the plasma in place, and to aim the nuclei at it. We have not incorporated the effect of the black hole cross section. The most interesting case would be the creation of a stable, charged black hole that can be levitated in a magnetic field. Preliminary calculations that could be made include an estimate of the charge that could be added to the black hole, calculation of Schwinger discharge bounds, and lifetime of the black hole. It would also be interesting to estimate the power output by the black hole. Some problems with the ‘black hole Levitron’ model [34] could be the very short lifetimes of realistic laboratory black holes, and the instability of the thermal black hole system. It would be interesting to incorporate Hawking radiation into this analysis.

6.2.5 Feasibility Theresultsfromsection6.2 imply that in order to establish in-plasma black hole growth in higher dimensional scenarios, energies of at least 105 TeV per nucleon (in the most favorable case) must be reached in particle detectors. Apart from this, a lot of practical problems with this experiment are listed at the end of this section. But first we want to make an estimate of the maximal energy per nucleon that could theoretically be reached in future colliders. There are two types of colliders: the cyclotron and the linear accelerator. The advantage of the first is that particle beams can have multiple crossings per revolution. With this ‘recycling’ of the beams, the number of events in the detectors is largely increased. The disadvantage is that a lot of energy has to be put in generating the centripetal force needed to keep the beam in orbit. The linear accelerator does not have this disadvantage. However, this detector has the problem that a particle beam can only be used once. This is the reason that linear accelerators are often used for electron-positron experiments, that make ‘clean’ collisions. For protons, and also for a quark-gluon-plasma, the cyclotron is the better 6.2. THE QGP ABSORPTION MODEL 67 option. This being as it may, we will consider both types of accelerator as candidates for our experiment. There is a certain limit to the energy per nucleon in both types of accelerator. In the cyclotron, energies are bounded by the magnetic field and accelerator radius. The linear accelerator has an electric field, that is bounded above by the ‘Schwinger limit’, which will be explained shortly. Other effects, such as bremsstrahlung and wearing out, also limit the particle energies. Since we are merely interested in crude estimates, we will leave these out. The main ring of the cyclotron (such as the LHC) consists of an array of very strong magnets, cooled by liquid helium. The particles experience a Lorentz force, that accelerates the particles, increasing their orbital velocity and accounting for centripetal acceleration. After a certain time, the Lorentz force is solely used for the centripetal acceleration; at this point, the maximal velocity is reached. The force balance then reads

γmv2 = Bqv (6.29) r At ultrarelativistic velocity, v ∼ c, and the energy γmc2 per unit of charge equals E = Bcr (6.30) q At the LHC, the field strength of these magnets is about 10 T. The effective bending radius is 3.1 km, making the energy per unit of charge around 7 × 1012V, corresponding to 7 TeV. This is indeed the energy per nucleon at LHC. Let’s make an estimate of the maximal field and radius feasible on Earth. The maximal strength of an electromagnet is 100 T. Even at this level, the material that makes up the magnet will be destroyed by a single pulse3.A durable magnet would have a field strength of 50 T at most. The radius of a detector could at most be 40.000 km – the circumference of the Earth – and even this is practically very unlikely. With these mighty proportions, the energy per nucleon would be multiplied by a factor 105. So the energy per nucleon could at the very most be 106 TeV. This means that cyclotrons are not a very good candidate to experiment with plasma black holes. This leaves linear accelerators as our plan B. A linear accelerator is a long, straight array of cathodes, generating an electric field E. Particles with mass m and charge q released within the tube will experience an acceleration a = Eq/m due to the electric force. Their velocity after a time t due to this acceleration will be at v = (6.31) 1+(at/c)2

We can assume that, upon injection, the particles are pre-accelerated up to near-light velocity. A detector with ﬁeld strength E and length l will give particles an energy of 2 γmc2 mc2 Eql = 1+ (6.32) q q mc2

3There is a nice calculation to show this, which I could include. 68 CHAPTER 6. BLACK HOLES IN THE LAB

At a certain field strength, the Schwinger pair production rate will be such that the created pairs generate a field in the opposite direction that cancels the original field. This ‘Schwinger limit’ then sets a maximum electric field in vacuum. Via formula (2.66), this value is

π2c3 E = (6.33) max hqE Supposing we would have cathodes that could produce a ﬁeld in an accelerator of length l, the maximal energy per unit charge would be 1012 TeV. QGP’s produced at these energies would allow us to witness black hole creation and growth in TeVG-scenarios. However, there exists the problem that a QGP is very hard to make in a linear accelerator, as it is produced by a ‘dirty’ collision. It is possible that other devices to conduct particle experiments will be invented, that are not subject to the constraints suggested above. In any case, it is interesting to explore thesegrowthmodels,mightweeverbeabletotestthemexperimentally. Chapter 7

Conclusions

A remarkable feature of the study of microscopic black holes is that its predictions will be put to the test in the near future, at LHC. Modern physics theories not often yield these testable predictions. This thesis outlines the constraints and free parameters of these predictions, and features some recommendations for future research and experiments. One such free parameter is the value of the fundamental Planck scale. As the hierarchy problem would be conveniently solved if it would be as low as 1 TeV, there is no experimental evidence either verifying or falsifying such a claim. Current measurements (table 3.1) seem to indicate that if the Planck scale should be of TeV-order, the number of extra dimensions must be large. Since there is a large possible range of values of the Planck scale, the determination of the actual value will be very difficult. If the value of the fundamental Planck scale is actually pinpointed, there is still one free parameter: the combination of dimension size R and number n. A suitable combination of these two will produce any desired Planck value, while R is also constrained by experiments not to exceed ∼ .1 mm, while n is expected to be a number between 2 and 7 inclusive. This follows from the analysis of the ADD model in chapter 3. The observation of microscopic black holes at detectors would be strong evidence that braneworld scenarios represent reality. The hard part will be to recognize their signature in the experimental data. From chapter 4 it becomes clear that while braneworld micro black holes are larger and colder than their equally heavy counterparts in four dimensions, their tiny mass implies they will be extremely hot. Furthermore, they have a negative heat coefficient, which means that evaporation will be a self-amplifying, runaway process. LHC black holes will therefore instantaneously decay upon creation through radiation, the spectrum of which will not be thermal. This non-thermality is due to two reasons. First of all, the black hole will evaporate in four different phases. In the balding phase, multipole moments and charge are shed. From our analysis it has become clear that it is unlikely that a micro black hole can hold on to its charge for any significant period of time; it will be shed through a Schwinger-type process. Next, in the spindown phase angular momentum and mass are simultaneously shed through Hawking radiation, which is amplified in the superradiant regime. The phase that is thought

69 70 CHAPTER 7. CONCLUSIONS to be most important and featured most prominently in this research is the Schwarzschild phase. The radiation in this phase will be of the most simplified kind. The final fate of a black hole cannot be described in the semiclassical combination of General Relativity and Quantum Field Theory used in this thesis. Whatever happens in this Planck phase, a theory of Quantum Gravity will be needed to understand it. How the transitions between these phases take place — smoothly, or abruptly — is not known. We know from the numerical results that the effects of balding, spindown and Planck phases certainly cannot be neglected for small LHC black holes. The second effect clouding the thermal spectrum are greybody factors. These typically suppress the low-energy Hawking radiation spectrum and amplify the high-energy spectrum. The number and species of particles emitted are highly dependent on the number and size of the extra dimensions. This gives an opportunity to ‘read off’ the number of dimensions from the characteristic radiation spectrum of black holes. However, the enhancement of total emissivity also poses the problem that thermodynamic analysis breaks down for values of n ≥ 4. Two conclusions can be drawn from the results in chapter 5. Due to the lowered value of the Hawking temperature, primordial black holes will live longer in braneworld scenarios than in four dimensions. This implies that the current cosmological density of PBHs is greater than would otherwise be expected. This amplifies the importance of PBHs as dark matter candidates. Furthermore, a novel creation mechanism is made possible by the braneworld scenarios. Microscopic black holes made in high-energy collisions in the primordial plasma can grow to enormous proportions — up to 30 orders of magnitude larger than their original size. While this will further add to the current density of PBHs in the universe, it also inspires a model to simulate this creation mechanism on Earth. This proposed model, called the QGP Absorption Model, supposes a quark-gluon plasma can be approximated as an ultrarelativistic Fermi gas. If we succeed in producing a black hole within such a plasma, two things may happen. If the black hole is initially hotter, it will evaporate instantaneously and its radiation will be insignificant among the fallout of the QGP. If, on the other hand, the plasma is initially hotter, the black hole can absorb a significant proportion of it. If the resulting black hole is large enough, its decay rate would be lowered significantly. This results in a longer lifetime, making it possible to discern its different decay phases and characteristic radiation spectrum. If the lifetime of the black hole is several orders of magnitude larger than the lifetime of the plasma, black hole radiation may be observed separately. In order to produce such a hot QGP, tremendous energies are needed, as these behave as a power law in temperature. No wonder that Earth-bound detectors will be barely capable of reaching such energies with the currently available resources and technology. Aside from practical difficulties, in stead of creating a black hole in a QGP with a separate beam, black holes will already be created by the QGP-partons themselves. While black hole growth might still happen in this situation, the energy scale at which it becomes significant will be higher. Theoretical and practical issues still plague these models of microscopic black hole growth and decay. Also, if braneworld scenarios and TeV-scale gravity are not real, we will probably never observe any of it. However, in view of the forthcoming experiments at LHC, the study 71 of microscopic black holes remains an exciting field of research with copious amounts of experimental data to analyze in the next decade. Appendix A

Lorentz Force Lagrangian

We start with the Lagrangian 1 L = mx˙ i2 + qx˙ jA (xi) − qV (xi) (A.1) 2 j We will show that the Euler-Lagrange equations yield the equations of motion for a charged particle in a magnetic ﬁeld experiencing a Lorentz force. The Euler-Lagrange equations are

j ˙ mx¨i = qx˙ (∂iAj − ∂jAi)+q(−∂iV − Ai) (A.2) j ˙ = qx˙ (δilδjm − δimδjl)∂lAm + q(−∂iV − Ai) (A.3) j j ˙ = qx˙ kij klmx˙ ∂lAm + q(−∂iV − Ai) (A.4) ˙ = q[v × (∇×A)+(−∇V − A)]i (A.5)

Substituting the electromagnetic ﬁelds

E = −∇V − A˙ , B = ∇×A (A.6) yields the familiar Lorentz force law from classical electrodynamics:

fL = q[(v × B)+E] (A.7)

In the special case of no E-ﬁeld and a one-dimensional arbitrary gauge ﬁeld A(x)wehave the Lagrangian 1 L = mx˙ 2 + qxA˙ (x) (A.8) 2

72 Appendix B

Emission coeﬃcient for black holeswithﬁxedmass

We will demonstrate that the emission coefficients n due to greybody factors for black holes with fixed radius, are the same for black holes with fixed mass. For clarity, the subscript n is added to quantities to indicate the number of extra dimensions. Given the Stefan-Boltzmann law for radiative power, the total power radiated by a black hole of mass M is proportional to the inverse square of the Schwarzschild radius:

∝ 4 ∝ γn ∝ γn Pn(M) AhT 2 2 (B.1) rn(M) M n+1

The dimensionless quantity γn is an emissivity coefficient depending solely on n. Setting γ0 = 1, it can be expressed as 2n P (M) − n+1 n γn = M (B.2) P0(M) The result is thought to be independent of M. We now set out to find these coefficients. In literature [20], the emissivity coefficients are given for black holes with equal radius: Pn(M ) n = (B.3) P0(M) with condition M = M n+1 (B.4) to ensure that the radius of black hole M with no extra dimensions is equal to the radius of M in n dimensions. The relation for γn cannowbeexpressedintermsof n:

2n P (M) − n+1 n γn = M n (B.5) Pn(M ) Using (B.1)and(B.4), this leads to γn = n (B.6)

73 74APPENDIX B. EMISSION COEFFICIENT FOR BLACK HOLES WITH FIXED MASS

Indeed, the coeﬃcients γn are mass-independent. In the main text, the notation n is main- tained. Appendix C

Graphs

()m rh 1.5 ´ 10- 18

1.0 ´ 10- 18

7.0 ´ 10- 19

5.0 ´ 10- 19

3.0 ´ 10- 19

2.0 ´ 10- 19

M (TeV/c2 ) 200 400 600 800 1000

Figure C.1: Schwarzschild radius in n+4 dimensions. From top to bottom, n =2, 3, 4, 5, 6, 7.

75 76 APPENDIX C. GRAPHS

T (TeV/k ) 1.00

0.50

0.20

0.10

0.05

M (TeV/c2 ) 200 400 600 800 1000

Figure C.2: Hawking temperature in n+4 dimensions. From bottom to top, n =2, 3, 4, 5, 6, 7.

()s

10- 23

10- 24

10- 25

M (TeV/c2 ) 200 400 600 800 1000

Figure C.3: Black hole lifetime in n + 4 dimensions. From top to bottom, n =2, 3, 4, 5, 6, 7. Bibliography

[1] D.M. Eardley and S.B. Giddings. Classical Black Hole Production in High Energy Collisions. Phys.Rev., D66(044011), 2002.

[2] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali. The Hierarchy Problem and New Dimensions at a Millimeter. Phys.Lett., B429:263–272, 1998.

[3] L. Randall and R. Sundrum. A Large Mass Hierarchy from a Small Extra Dimension. Phys.Rev.Lett., 83:3370–3373, 1999.

[4] John Ellis, Gian Giudice, Michelangelo Mangano, Igor Tkachev, and Urs Wiedemann. Review of the safety of LHC collisions. J. Phys. G: Nucl. Part. Phys. 35 115004, 2008.

[5] S.B. Giddings and M.L. Mangano. Astrophysical Implications of Hypothetical Stable TeV-scale Black Holes. hep-ph/08063381, 2008.

[6] S. Hawking. as quoted on CERN website. url: http://public.web.cern.ch/Public/en/LHC/Safety-en.html, 2008.

[7] S.M. Carroll. Spacetime and Geometry. Addison Wesley, N.Y., 2003.

[8] K.S. Thorne. Nonspherical Gravitational Collapse, A Short Review. In J. Klauder et al., Magic Without Magic: John Archibald Wheeler, Freeman, San Fransisco, pages 231–258, 1972.

[9] R. Penrose. Unpublished. 1974.

[10] P.N. D’Eath. Gravitational Radiation in High Speed Black Hole Collisions. Phys. Rev. D, 46:658–701, 1992.

[11] S.B. Giddings and S. Thomas. High Energy Colliders as Black Hole Factories: The End of Short Distance Physics. Phys.Rev., D65(056010), 2002.

[12] N.D. Birrell and P.C.W. Davies. Quantum Fields in Curved Space. Cambridge University Press, 1982.

[13] G. ’t Hooft et al., editor. Nonperturbative Quantum Field Theory, volume 185 of NATO ASI Series B: Physics. Plenum Press, N.Y., 1988.

77 78 BIBLIOGRAPHY

[14] Y.B. Zel’dovich. Generation of Waves by a Rotating Body. JETP Lett., 14:180, 1971.

[15] W.H. Press and S.A. Teukolsky. Floating Orbits, Superradiant Scattering and the Black- Hole Bomb. Nature, 238:211, 1972.

[16] G.W. Gibbons. Vacuum Polarization and the Spontaneous Loss of Charge by Black Holes. Commun. Math. Phys., 44:245–264, 1975.

[17] S.P. Kim and D.N. Page. Remarks on Schwinger Pair Production by Charged Black Holes. Nuovo Cim., B120:1193–1208, 2004.

[18] R. Parentani and S. Massar. The Schwinger Mechanism, the Unruh Eﬀect and the production of Accelerated Black Holes. hep-th/9603057v2, 1996.

[19] I.B. Kchriplovich. Radiation of Charged Particles by Charged Black Hole. gr- qc/9812060, 1998.

[20] P. Kanti. Black Holes in Theories with Large Extra Dimensions: a Review. Int.J.Mod.Phys., A19:4899–4951, 2004.

[21] S.B. Giddings and V.S. Rychkov. Black Holes from Colliding Wavepackets. Phys.Rev., D70(104026), 2004.

[22] H. Yoshino and Y. Nambu. Black Hole Formation in the Grazing Collision of High- Energy Particles. Phys. Rev., D67(024009), 2003.

[23] T.G. Rizzo. Black Hole Production Rates at the LHC: Still Large. hep-ph/0111230, 2001.

[24] R. Emparan, G.T. Horowitz, and R.C. Myers. Black Holes Radiate Mainly on the Brane. Phys.Rev.Lett., 85:499–502, 2000.

[25] P. Argyres, S. Dimopoulos, and J. March-Russell. Black Holes and Sub-millimeter Dimensions. Phys.Lett., B441:9–104, 1998.

[26] R.C. Myers and J.C. Perry. Black Holes in Higher Dimensional Space-Times. Annals of Physics, 172:304–347, 1986.

[27] S.B. Giddings. High-Energy Black Hole Production. hep-ph/07091107, 2007.

[28] C.M. Harris. Physics Beyond the Standard Model: Exotic Leptons and Black Holes at Future Colliders. Ph.D. Thesis, University of Cambridge, 2005.

[29] J. Conley and T. Wizansky. Microscopic Primordial Black Holes and Extra Dimensions. Phys.Rev., D75(044006), 2006.

[30] B.J. Carr and S.W. Hawking. Black holes in the early Universe. Mon. Not. Roy. Astron. Soc., 168(399), 1974. BIBLIOGRAPHY 79

[31] M. Apostol. Hadronization of the Quark-Gluon Plasma. Romanian Reports in Physics, 52(2):249–260, 2007.

[32] S. Scherer. Dynamics of Coloured Quarks: The Quark Gluon Plasma in the Computer. CSC seminar, Johann Wolfgang Goethe Universit¨at, Frankfurt am Main, 2004.

[33] A. Chamblin, F. Cooper, and G.C. Nayak. Interaction of a TeV Scale Black Hole with the Quark-Gluon Plasma at LHC. Phys.Rev., D69(065010), 2003.

[34] X.D. Arsiwalla and E.P. Verlinde. A Black Hole Levitron. hep-th/09020002, 2009. Dankwoord

Het was een geweldige luxe om een jaar lang met zoveel vrijheid bezig te zijn aan een natuurkundig onderzoek. De mensen die mij daarbij hebben geholpen en ondersteund ben ik veel dank verschuldigd. Als eerste wil ik mijn begeleider Erik Verlinde bedanken. Na een start met een ander onderwerp waar ik minder gelukkig mee was, heb jij mij op de micro-zwarte gaten gebracht. Ik ben erg blij dat ik in alle vrijheid heb kunnen grasduinen in dit enorme onderwerp. Ook ben ik blij dat je me hebt ondersteund in mijn eigen benaderingswijze van de theorie. Ik dank je voor alle keren dat ik bij je heb aangeklopt en jij de tijd hebt genomen om met veel geduld de zaken op te helderen. In de beginfase van mijn scriptie kwam Steven Giddings naar Amsterdam voor een lezing. Ik had het geluk dat hij in de kamer naast de mijne zijn voorbereidingen trof. Omdat hij een van de autoriteiten op het onderwerp van micro-zwarte gaten is kon ik het niet laten hem aantesprekenenevenlaterstondenwevoorhetschoolbordterekenen.Hijheeftmijonder andere op het hart gedrukt om niet al te sensationele voorspellingen over zwarte gaten te doen zonder ze grondig te controleren. Ik wil hem bij dezen bedanken voor de inspiratie die hij in deze sessie heeft gegeven. Verder dank ik ook de mensen met wie ik het afgelopen jaar in hetzelfde schuitje heb gezeten: mijn medestudenten Evert, Shanna, Wolf, Kris, Sebas (2x), Pawel, Panagiota, Dimitrios en Milos. Het is ﬁjn dat de ivoren toren door meer mensen wordt bewoond. Ik bedank ook iedereen die mij buiten de faculteit een jaar lang heeft moeten aanhoren over zwarte gaten: Beets 2003, mijn huisgenoten en andere vrienden. En in het bijzonder mijn ouders en Brechje, omdat jullie altijd achter mij staan.