University of Amsterdam
Master of Physics and Astronomy
Theoretical Physics
Master Thesis
A Near-Horizon View on Vacuum Decay
&
The Materialisation of a Braneworld
Author
Damien Cornel
August 28, 2019
Supervisor
Dr. Jan Pieter van der Schaar
Second Reader
Dr. Ben Freivogel
Abstract
Recent development of a braneworld scenario has shed a new light on the notoriously difficult problem of finding metastable de Sitter vacua in the Landscape. Motivated by this braneworld construction and the Weak Gravity Conjecture, we examine the dual description of the instability of extremal black holes and near-horizon anti-de Sitter. In particular, we look at shells tunneling through the horizon of a black hole, splitting the geometry, which will correspond to domain walls in the near-horizon region. The existence of these domain walls is also supported by the conjecture that every non-supersymmetric anti-de Sitter spacetimes is at most metastable. After elaborating on these subjects, we analyse the decay of a non-supersymmetric AdS5 false vacuum to a
super-symmetric AdS5 true vacuum. We show that it is exactly when the domain wall, separating these
vacua, is in coherence with the Weak Gravity Conjecture, the induced cosmology on the domain wall is four-dimensional de Sitter. The cosmological constant in this braneworld can be arbitrarily small with a fine tuning of the brane tension and is guaranteed to be positive. Finally, we discuss some cosmological applications, such as how to implement matter in the braneworld, concluding with prospects for further study.
Acknowledgements
First of all, I would like to thank Jan Pieter, I have enjoyed working with you as my supervisor, in the past year. In moments where I found myself getting lost in all the information, a meeting with Jan Pieter would always inspire me and rejuvenate my motivation, due to his positivity and enthusiasm. Second of all, I would like to thank Ben, my second reader. Thank you for interesting and useful discussions, as well as providing me with some critical questions regarding the setup.
Furthermore, I would like to thank all the other members of the cosmology group for interesting and fun journal clubs. In particular, I want to thank Lars Aalsma and John Stout. Thank you Lars, for always taking the time to answer my questions.
I would to acknowledge Marjorie Schillo, for helpful discussions through email.
Moreover, I want to thank my family for their support. My brothers, for our weekly escapes to other worlds. My mother for her support in various ways. Lijntje, as well. And of course, my father, with an always supportive character. Making the pursuit of this master possible, after having walked some rocky roads. Thank you.
On top of that, I am grateful to Laurens, because bff. Finally, thank you Lulu, for all the love and support.
Contents
1 Introduction 2
2 Vacuum Decay 5
2.1 Decay Without Gravity . . . 6
2.2 Decay With Gravity . . . 9
3 The Swampland 14 3.1 The Weak Gravity Conjecture . . . 15
3.2 Instability of Anti-de Sitter . . . 18
4 The Brane and the Shell 21 4.1 Tunneling of Charged Shells . . . 21
4.2 Extremal Shells and Branes . . . 23
4.3 Near-Horizon AdS . . . 26
4.4 Super-Extremal Shells and Branes . . . 28
4.5 Non-Extremal Near-Horizon Geometries . . . 30
5 Dimensional Reduction Near the Horizon 33 5.1 Reducing the Sphere . . . 33
6 De Sitter Domain Wall Cosmology from Decaying AdS 38 6.1 Randall-Sundrum . . . 39
6.2 Brane Equations of Motion . . . 40
6.3 The Curious Case of n = 4 . . . 42
6.4 Bubbles Near the Horizon . . . 46
7 Braneworld Cosmological Applications 48 7.1 Collisions of Bubbles . . . 48
7.2 Matters of Matter . . . 50
7.3 Matters of Strings . . . 54 8 Conclusion and Outlook 57
Chapter 1
Introduction
When peering into the night sky with a telescope, one is met with wonder and awe. The vastness of space can be humbling but it is filled with astounding beauty. Our gaze into space falls onto planets, stars and nebulae within the galaxy we call home, the Milky Way. If we look further we see that the there are numerous other galaxies, with many of them much larger than the one we are in. Studying galactic dynamics, we see they are governed by the laws of gravity. On the vast scales we use Einstein’s Theory of General Relativity, which tells us that energy deforms the fabric of spacetime, such that the trajectories of objects close to one of a large energy will be affected. We see this happen even to light in the form of gravitational lenses. On smaller scales however, say here on Earth in our everyday life, we do not need to use Einsteins equations to describe gravity. Newton’s laws of gravity will suffice as a great approximation in describing the motion of the apple falling from the tree. It also accounts for why one’s cat and pens remain at rest on ones desk, without floating away. However, say you have a small magnet and a metal pen laying around, you do not need a large magnet to lift the pen from your desk (you would need quite a large magnet to lift your cat). So, the electromagnetic force of this tiny magnet on the pen is stronger than the grav-itational pull of the entire planet. It seems that gravity is a rather weak force, surely, weaker than the electromagnetic force. In physics terms, the gravitational coupling constant is much smaller than the electromagnetic coupling constant. The weakness of gravity is at the core of this thesis, in the form of the Weak Gravity Conjecture (WGC) [1], which will be discussed thoroughly in chapter 3.
Another interesting thing to note about gravity, is that we do not know how it behaves on the smallest of scales, say the quantum world. On the smallest scales we describe dynamics with quantum mechanics, or quantum field theory. If we naively try to combine Einstein gravity with quantum field theory, the theory spits out infinities, it diverges and one quickly finds out that grav-ity is non-renormalisable. If we go back to our telescope, and gaze into space, we can see that most galaxies out there have a super-massive black hole at the centre. Black holes are highly gravitational objects, but quantum at their core. Thus we need a theory of quantum gravity to understand them properly. It is not only this, but also for a complete understanding of the Universe that a theory of quantum gravity is highly coveted. In some sense it feels like a piece of the puzzle is missing if we do not have a theory of quantum gravity. This is where string theory comes into play. String theory is to this day the only consistent theory of quantum gravity. A caveat, is that it follows from string theory that there should be ten spacetime dimensions (eleven in M-theory, but let’s stick with ten
for the sake of argument). We clearly only see four spacetime dimensions, so we have to get rid of six of them. This is done with compactification. As a Gedankenexperiment, imagine having a sheet of paper. This is a two-dimensional manifold. Rolling it up in a cylinder does not change the space, it is just a different embedding in a higher dimensional space. Now imagine someone holds this cylinder shaped paper far away from you, or the radius being very small. The sheet of paper now looks like a line, a one-dimensional object. Even though, it still is two-dimensional. This is how one can think about compactification. What happens is that the degrees of freedom of the extra dimensions get reduced, which results in additional fields in your lower-dimensional theory. This only works when probing energy scales that are low compared to the radius of the compact dimension. We will see an explicit example later. One can do this to all the six extra dimensions to end up with a four-dimensional theory. However, this compactification can be done in numerous ways. As in our Gedankenexperiment, one dimension is chosen to be compactified on a circle, but this is not a limiting case. As such, what we are left with after compactification of the extra dimensions is a vast Landscape of possible vacua. But how do we know where we are in this Landscape? Perhaps more importantly, are we even part of the Landscape? We live in a de Sitter-like Universe, meaning that we measure a positive cosmological constant, very small, but positive. A point of discussion, that has been growing over the past couple of years, is whether the Landscape even admits four-dimensional de Sitter vacua. This is one of the main motivations for this thesis, the notoriously difficult problem of finding de Sitter solutions from compactification of a ten-dimensional string theory. In this thesis, we study a braneworld scenario which naturally inherits an expanding, four-dimensional de Sitter cosmology[2][3]. Namely, we will study the decay of a five-dimensional anti-de Sitter spacetime to one with a lower cosmological constant, through the creation of a charged membrane, a 3-brane. This brane separates the two five-dimensional spacetimes. The decay described here is a consequence of the Weak Gravity Conjecture, of which an extension states that all non-supersymmetric anti-de Sitter vacua are at most metastable, mean-ing they can decay through bubble nucleation[4]. We will see that it is exactly the decay channels that are in coherence with the Weak Gravity Conjecture, which give rise to a four-dimensional de Sitter cosmology on the brane.
However, before we get to the braneworld construction, we will discuss all the ingredients needed to construct it. We will see that there is a beautiful interplay going on between some of these ingre-dients, with the Weak Gravity Conjecture at the core. The Weak Gravity Conjecture tells us about the (super-)extremal decays channels of extremal black holes, as well as the instabilities of anti-de Sitter spacetimes. Apart from being linked through the Weak Gravity Conjecture, these two have a deeper link through the near-horizon geometry of extremal black holes. For a four-dimensional extremal black hole, the near-horizon geometry will turn out to be AdS2× S2. From this
near-horizon view, the link between the instabilities of black holes and anti-de Sitter spaces becomes all the clearer, and it is one of the key motivations to study this braneworld scenario.
The structure of thesis is as follows. We will start in chapter 2 with a study of the process of vacuum decay. This gives us an idea of the instabilities of spacetimes and how one can describe this decay in a semi-classical way with instantons. In chapter 3 we will continue our discussion on the Landscape and will introduce the Swampland. After having introduced the Swampland, we will go into detail about the Weak Gravity Conjecture and discuss its origins and some of its conse-quences for the instability of extremal black holes as well as the instability of (non-)supersymmetric anti-de Sitter spacetimes. Following up on that discussion, we will see in chapter 4 that there is a deeper relation between the instability of extremal black holes and the instability of (non-)supersymmetric anti-de Sitter spacetimes, coming from the analysis of the near-horizon geometry
of a four-dimensional extremal black hole. We study the near-horizon geometry further, by di-mensionally reducing it to two-dimensions in chapter 5. By then we will have gathered all the ingredients needed for this thesis, and we get to the braneworld construction. In chapter 6, we will derive the equations of motion for brane that nucleates into existence via the decay of the ambient five-dimensional anti-de Sitter spacetime. We will show that the brane cosmology is governed by a four-dimensional Friedmann equation describing a de Sitter cosmology. The braneworld will have an automatically positive cosmological constant, which can be tuned to be arbitrarily small (like in our Universe) via a tuning of the brane tension. In chapter 7 we will turn to some cosmological ap-plications like adding matter to the braneworld and discussing possible collisions with other branes. Finally, we end in chapter 8 with our conclusions, discussions and outlook for further study.
Chapter 2
Vacuum Decay
During the discussion of the Weak Gravity Conjecture we have already encountered the subject of vacuum decay. In this chapter we will discuss this further following the paper by Coleman and De Luccia [5][6] in which they computed the effects the inclusion of gravity has on the decay process. We will start by the evaluation of the decay process in a classical field theory with a single scalar field without gravity and will later add gravity to its effects. The theory we will consider is defined by the following action
S = Z d4x 1 2(∂µφ) 2 − V (φ) . (2.1)
We are working in four dimensions here but the concepts we will discuss may be generalised to higher dimensions as well. The shape of the scalar potential V (φ) is depicted in figure 2.1a. Looking at the plot alone, one observes that the potential has two minima, φ+ and φ− of which only one is
a global minimum, namely φ−. In a pure classical field theory case, both of these points, φ = φ+
and φ = φ− correspond to stable equilibrium states. In the quantum version of the field theory
however, only φ = φ− corresponds to a stable state. The state for which φ = φ+ is a metastable
state. It is metastable in the sense that yes, it is a minimum of the potential, however, it is prone to decay via tunneling. The vacuum state of φ = φ+ is called the true vacuum whereas φ = φ+ is
named a false vacuum state. The process of decay is thus aptly named: false vacuum decay. We will mostly refer to it as vacuum decay throughout this thesis, for mere convenience.
Vacuum decay is very much like a first order phase transition in statistical physics. Take for example a super-heated fluid. When super-heating the fluid, one can, imagine a similar situation as our potential in figure 2.1a. Most of the material is still in its fluid state, the false vacuum, but it becomes energetically favourable to enter the gas phase, the true vacuum. It is then through thermal fluctuations that bubbles of the true vacuum start to nucleate within the material. If a bubble nucleates with an insufficient size, the ambient pressure and surface tension are too high and the bubble shrinks again. However, if a bubble nucleates of sufficient size, the vapor pressure inside the bubble is high enough and the bubble can expand and convert the entire fluid into the true vacuum, the gas phase. This is very similar to the idea of vacuum decay we will discuss here. Instead of thermal fluctuations causing the decay in the phase transition, in the field theory case quantum fluctuations are the cause of the tunneling event.
(a) Original (b) Wick rotated
Figure 2.1: General shape of the scalar potential in consideration. Two min-ima can be seen, where the local minimum is called the false vacuum and the global minimum is called the true vacuum. On the left side is the original po-tential and on the right side is the popo-tential after Wick rotation to Euclidean time. We see a flip in sign of the potential.
2.1
Decay Without Gravity
We will work in the semi-classical limit, meaning the limit such that ~ is small. In the semi-classical limit, working in the WKB approximation, the probability for a tunneling event per unit of time per unit volume can be expanded in ~ as follows
Γ V = Ae
−B/~1 + O(~) , (2.2)
where V is the volume, not to be confused with the potential V (φ). For our purposes we do not need to worry about the coefficient A, as it is a constant of proportionality. The main interest lies within the coefficient B. It will be useful for us to go to Euclidean spacetime. The reason for this is that it will change our tunneling problem, which is a quantum mechanical process, into a classical problem. Let us Wick rotate to Euclidean time t −→ iτ such that the Euclidean action for our scalar field is given by SE = Z d4x 1 2(∂µφ) 2+ V (φ) , (2.3)
where the change in sign respective to the action in (2.1) of the potential should be noted. It is exactly this change of sign in the potential that allows us to treat the problem classically. See the potential in figure 2.1b for a graphical depiction. Whereas in the Lorentzian case the field had to tunnel to get over the barrier, in the Euclidean case it can just role there. With the use of the Euler-Lagrange equation ∂µ ∂L ∂(∂µφ) ! −∂L ∂φ = 0 (2.4) we find the Euclidean equation of motion
∂2 ∂τ2 + ∇ 2 ! φ = dV dφ. (2.5)
Now let us take φ to be a solution to the Euclidean equation of motion, an instanton, with the properties that φ approaches the false vacuum φ = φ+at Euclidean infinity, φ is not a constant and
lastly that φ has an action less than or equal to any other solution that also obeys the aforementioned properties. In this situation, the coefficient B in the decay amplitude (2.2) is given by the difference in the actions between the points φ and φ+
B = SE(φ) − SE(φ+). (2.6)
As given by the WKB approximation. The field φ in the theories of our interest has symmetry group O(4) [7] meaning it is only a function of some distance from the centre φ = φ(ρ). As such we can rewrite our action in a simplified manner in terms of spherical coordinates, leaving
SE = 2π2 Z dρ " 1 2 dφ dρ 2 + V (φ) # , (2.7) and the resulting equations of motion are
d2φ dρ2 + 3 ρ dφ dρ = ∂V ∂φ. (2.8) In what follows we will drop the second term in (2.8) as it will always be small. It comes down to ρ being large away from the wall, and d
dρφ will be small close to the wall. This will become more
evident later. If we now assume there is a small energy difference between the two vacua we can find an explicit approximation for φ. As such we define the small energy difference by
= V (φ+) − V (φ−). (2.9)
with this definition we can write the potential in term of some function V0(φ) that we choose such
that it obeys the following constraints
V0(φ+) = V0(φ−), and,
dV0
dφ|φ=φ± = 0. (2.10)
The potential in terms of this function can then be written as
V (φ) = V0(φ) + O(). (2.11)
Given all the above we can approximate φ as d2φ dρ2 =
∂V0
∂φ, (2.12) where we have dropped the second term in (2.8) as well as neglected terms proportional to . Equation (2.12) gives us a kind of energy conservation equation
d dρ " 1 2 dφ dρ 2 − V0 # = 0. (2.13) As we have chosen in the beginning that φ at infinity will be φ+, it is this that determines the value
in the brackets of (2.13) 1 2 dφ dρ 2 − V0= −V0(φ±), (2.14)
Which we can rewrite into the following integral equation, determining φ in terms of some integra-tion constant. We choose this integraintegra-tion constant to be the point at which φ is the average of the two minima, hence
Z φ φ++φ− 2 dφ0 p2[V0(φ0) − V0(φ±)] = Z ρ ¯ ρ dρ0. (2.15) This leaves us to find an expression for ρ. We will do this by evaluating B form equation (2.6). We assume that ¯ρ0 is large compared to the scale at which φ varies significantly. The situation is that
of a bubble of true vacuum floating in a sea of false vacuum. the two vacua are then separated by a wall, which corresponds to the transition region. Working in the thin wall approximation, this transition region in our care is small compared to the radius of the bubble. As such we can split the evaluation of B into three integrals: the inside B−, the outside B+and the wall itself Bσ. Starting
out with the outside. Outside of the wall we have φ = φ+, inserting this into (2.6) one simply gets
B+= 0. (2.16)
Inside of the wall where φ = φ−, the true vacuum, we have
B−= 2π2 Z ρ¯ 0 ρ0 3dρV (φ−) − V (φ+) , (2.17) = −π 2 2 ρ¯ 4 0. (2.18)
Where we have used that the derivative of φ at the minima is zero and as given by (2.9). Within the wall itself ρ = ¯ρ0. As such ρ3= ¯ρ30 in the integral in the Euclidean action. We then have
BΣ= 2π ¯ρ30σ, (2.19) in which S1 is defined as σ = Z φ+ φ− dφ0 p2[V0(φ0) − V0(φ+)] . (2.20) Using equation (2.14) it is also possible to write the integral of σ in terms of ρ, such that one gets
σ = 2 Z
dρV0(φ) − V0(φ+)
(2.21) This integral will be useful in the gravitational case. We suggestively use σ for this integral as it represents a surface tension. Note that it explicitly depends on the potential. We then take the sum of these different constituents to get the full expression
B0= 2π ¯ρ30σ −
π2
2 ρ¯
4
0. (2.22)
We can simplify this expression by finding the value of ¯ρ such that B is stationary ∂B0
∂ ¯ρ0
= 0 → ρ¯0=
3σ
this makes sense as ¯ρ is a constant, thus we do not want B to depend on it as a variable. Having solved this derivative equation (2.23) for ¯ρ we get the final expression for B
B0=
27π2σ4
2 . (2.24) We now have an expression for the B coefficient which governs the exponential suppression of the probability of the tunneling event. So what about the growth of the bubble? At the moment the bubble of true vacuum nucleates within the false vacuum, the configuration of the field is given by
φ(x, t = 0) = φ(x, τ = 0), and, ∂φ
∂t(x, t = 0) = 0. (2.25) We will assume the bubble will not be too small and thus the evolution can be described by the classical Lorentzian equations of motion, simply by analytically continuing the Euclidean equations. Since we have been working in the thin wall approximation, it is evident from equation (2.23) that when is small, ¯ρ will be large. The Lorentzian equations of motion are
−∂ 2 ∂t2+ ∇ 2 ! φ = dV dφ. (2.26) The comparison with the Euclidean equations of motion tells us that the solutions to the Lorentzian equations will just be the analytical continuation of the Euclidean ones, therefore
φ(x, t) = φ(x, iτ ) = φ(ρ) = φ(p|x|2− t2). (2.27)
From this it can be seen that the O(4) symmetry of the Euclidean solution is translated to O(3, 1) symmetry for the expansion of the bubble. The position of the bubble in a spacetime diagram is governed by the following function
t = q
|x|2− ¯ρ2
0, (2.28)
such that the bubble nucleates at t = 0 with a radius or ¯ρ. After nucleation the bubble uniformly accelerates and asymptotes the light cone. Due to the O(3, 1) symmetry, the motion of the bubble looks the same to any Lorentz observer.
Thus far, the computations have all been done in a situation without gravity. In the next section we will add gravity to the theory and see that its effects should not be ignored.
2.2
Decay With Gravity
One thing that can already be noted as making a difference when adding gravity, is by looking at the action in equation (2.3). There is of course no absolute zero of energy density, as such one can add and constant of energy to the potential without affecting the physics. When including gravity, this is not the case. Given the action of our gravitational theory
S = Z dx4√−g 1 2g µν∂ µφ∂νφ − V (φ) − 1 2κR , (2.29) where R is the Ricci scalar and κ = 8πG. It is evident that adding a constant to the potential energy in the gravitational theory corresponds to adding a factor of√−g as well. This extra factor
in the gravitational theory would correspond to a cosmological constant. When the false vacuum decay occurs, the cosmological constant changes. As such one of the main differences between the gravitational and the non-gravitational case is that we now need to specify the initial value of the cosmological constant for computations.
In an endeavour to compute the same quantities as before we will have to go to Euclidean spacetime. Wick rotating to Euclidean time, the Euclidean gravitational action takes the form
SE= Z dx4√g 1 2g µν∂ µφ∂νφ + V (φ) + 1 2κR . (2.30) We again begin a search for an instanton solution describing the vacuum decay and we will assume rotational invariance. It thus makes sense to take the most general rotational invariant Euclidean metric, which is of the form
ds2= dη2+ ρ(η)2dΩ23. (2.31)
Here dΩ23 is the line element of a unit 3-sphere, whose radius is determined by the function ρ(η)
and η is a radial coordinate. For this metric we have the following Christoffel symbols and the rest is zero Γ0ij= ρdρ dηδij, Γi0j= 1 ρ dρ dηδ i j = Γ i j0. (2.32)
With these we can compute the Einstein equations as well as the equations of motion coming from the Euler-Lagrange equations given in (2.4). Starting off with the Einstein equations. For our purposes the ηη-components will be of interest. Thus what we want to compute is
Gηη= −κTηη. (2.33)
the energy momentum tensor is given by Tηη= 1 2 dφ dη 2 − V (φ), (2.34) and the Einstein tensor
Gηη= Rηη− 1 2gηηR α α, = 3 " 1 ρ dρ dη 2 − 1 ρ2 # . (2.35)
Plugging these into equation (2.33) we get dρ dη 2 = 1 +1 3κρ 2 " 1 2 dφ dη 2 − V (φ) # . (2.36) Now we will use (2.4) and compute the Euclidean equations of motion. these are given by
d2φ dη2 + 3 ρ dρ dη dφ dη = dV dφ. (2.37)
Comparing this to the equations obtained in the previous section, equation (2.8), we see that the only difference is in the name of the variable and the coefficient multiplying the first order derivative of φ in the second term. there is an additional factor of dηd ρ there. This will not affect our computation, given that as before we will neglect this term in the thin-wall approximation.
We can turn back to the Euclidean action, and insert R = 6 " 1 ρ d2ρ dη2+ 1 ρ2 dρ dη 2 − 1 ρ2 # , (2.38) while simultaneously making use of the fact that we have rotational symmetry and thus switching to spherical coordinates as
Z
dx4(...) → 2π Z
ρ3dη(...). (2.39) Such that we get the following action
SE= 2π2 Z dη ρ3 1 2 dφ dη 2 + V (φ) ! + 3 κ ρ 2d2ρ dη2 + ρ dρ dη 2 − ρ ! . (2.40) In order to find the instanton solution, in the thin-wall approximation, what is left to do is is actually just copy (2.15) while making the changes
ρ → η, and, ρ → ¯¯ η, (2.41) such that Z φ φ++φ− 2 dφ0 p2[V0(φ0) − V0(φ±)] = η − ¯η. (2.42) Once we find a solution for φ, we can solve equation (2.36) to find ρ. For this we need one integration constant which we will choose to be ¯ρ ≡ ρ(¯η). However, for now we will not focus on finding ρ. We will first try to find ¯ρ by computing the coefficient B and demanding that its derivative vanishes. In order to do this we want to simplify the Euclidean action given in equation (2.40). To simplify we first get rid of the second order derivative using integration by parts. We do not care about the resulting surface term because in the end we are looking for B, which is the difference between actions and thus the surface term will cancel out, leaving us with
SE= 2π2 Z dη ρ3 1 2 dφ dη 2 + V (φ) ! −3 κ ρ dρ dη 2 + ρ ! . (2.43) Subsequently we can use equation (2.36) to eliminate the first order derivative term and get
SE= 2π2 Z dη ρ3V (φ) − 3ρ κ . (2.44) Which will make our life a lot easier. We now continue our endeavour to compute B. Like before, we will split the integral into three parts and put them together at the end. As a reminder B is defined as the difference between Euclidean actions at φ and φ+, as given in (2.6). Outside of the
bubble we thus have again that
because they just cancel each other. In the wall itself, ρ = ¯ρ again and like before we can replace V by V0. Such that we get the following
BΣ= 4π2ρ¯
Z
dηV0(φ) − V0(φ+) ,
= 2π2ρ¯3σ.
(2.46) In which we have used the definition of σ given in (2.21). Inside the bubble we have again that φ = φ−, which is a constant. As such we can use (2.36) to make a change of variable in the action
such that the integral is over ρ, by changing dρ = dη
r 1 −1
3κρ
2V (φ), (2.47)
and inserting it into the the action. We are left with the following expression for B on the inside of the bubble B−= − 12π2 κ Z ρ¯ 0 ρdρ "r 1 − 1 3κρ 2V (φ −) − r 1 −1 3κρ 2V (φ +) # . (2.48) For now, we will leave it in integral form. Noting that the full expressions will again be the sum of the constituents B = B−+ BΣ+ B+. One can then evaluate specific cases by filling in values for
the scalar potentials. Subsequently requiring that B is stationary one can find an expression for the radius of the bubble ¯ρ and a final expression for B. As an example, one can set
V (φ+) = 0, and, V (φ−) = −ε. (2.49)
This situation corresponds to the vacuum decay of some space with zero energy density to one with negative energy density. Plugging these values into the equation for B one gets
B = 2π2ρ¯3σ −12π 2 κ Z ρ¯ 0 ρdρ "r 1 +1 3κρ 2ε − 1 # . (2.50) This integrates to B = 2π2ρ¯3σ +6π 2 κ ρ¯ 2−12π 2 εκ2 " 1 +1 3κ ¯ρ 2ε 3/2 − 1 # . (2.51) We then take the derivative of B with respect to ¯ρ and require it to vanish. From this we get the following expression for the radius of the bubble
¯
ρ = 12σ
4ε − 3κσ2. (2.52)
We can perform a consistency check by turning off gravity. In other words, we set κ = 8πG = 0 and we see that
¯ ρ =3σ
ε ≡ ¯ρ0. (2.53) Which is a nice confirmation. We have worked this out in one example for simplicity and clarity, but it is straightforward to do for different values of the scalar potentials V (φ±). For ease of
computation we chose one of them to be zero, but this is not a restriction. If they are both non-zero the computation quickly becomes an algebraic mess. In the case of decay from a space with zero energy density, to one with negative energy density, we see that the radius of the bubble will be larger when it materialises due to the inclusion of gravity. The nucleation rate will also be different, it follows from filling in the value for ρ that we computed in (2.52)
B = B0 (1 − ¯ρ2
0κε/12)2
. (2.54)
From this it can be concluded that the nucleation rate will also be diminished as a consequence of the inclusion of gravity. It should be noted that the radius of the bubble and the nucleation rate will not always be diminished by gravity. In our example this is the case, however, as is discussed in [5], if one would consider the decay of a space with positive energy density to one with zero energy density, exactly the opposite happens. In that case, gravity will increase B and ρ, hence it will be more likely for a bubble to nucleate, and when it does, it will have a larger radius compared to the case when gravity is not included.
Concluding, some vacua are unstable to decay. Through the mechanism of quantum tunneling, a false vacuum can tunnel through a potential barrier to the true vacuum. In a spacetime, this can be seen as a spherical bubble of true vacuum nucleating in the false vacuum. We have derived an expression for the exponent that governs the nucleation rate of bubbles, as well as an expression for the radius at nucleation. In the next chapter, we introduce the Swampland and discuss the Weak Gravity Conjecture.
Chapter 3
The Swampland
The vastness of the string theory Landscape of vacua is a curse to some and a blessing to others. One could argue that the fact that there are so many possible low energy effective field theories, it surely will include the one unique solution that describes our Universe. Critics of the Landscape would argue that this overprediction of possible theories is a problem of string theory or a lack of our understanding. Criticism of this kind is unfounded. If we look at quantum field theory, this will give an infinite set of consistent theories. It is for experiments to determine the right theory and fitted values of the parameters. A similar argument can be used in favour of string theory. Ideally we would able to cook up an experiment that would pick out the right parameters for us to get an idea of where in the Landscape we are. However, this would include figuring out the geometry and size of the extra dimensions experimentally, which seems impossible to this day. As such, it seems we are still a long way of a string theory description of our Universe.
It used to be the general belief that if a low energy effective field theory1 (LEEFT) looked
consistent, it probably was consistent. Unfortunately, it is not this simple. Many effective field theories that look consistent in the IR (at low energies) turn out to be inconsistent when coupled to gravity in the UV (at high energies). The vast collection of semi-classical consistent effective field theories that are in fact quantum inconsistent, is dubbed the Swampland[8][9][10], see [11] for a nice review. One can picture the situation as a small island of actual consistent theories, the Landscape, surrounded by an extensive Swampland of seemingly consistent theories which are actually not consistent as pictorially depicted in figure 2.1. It speaks for itself then that there is a general desire for some set of rules or principles that allow one to distinguish the Landscape from the Swampland. This is a field of studies that has certainly been growing over the past few years. It is a search for some characteristics that all UV complete theories in the Landscape have but the seemingly consistent theories of the Swampland lack. A search for UV characteristics that leave footprints in the IR.
In fact, there are even arguments that perhaps string theory does not allow for de Sitter vacua in the Landscape. Observing that we live in a de Sitter-like Universe, string cosmologists assume that the Landscape must have de Sitter vacua. However, so far no one has been able to construct a stable de Sitter vacuum in string theory, see [12] and references therein for a nice review on this. The closest to a construction being that of KKLT[13]. In which they add an anti-D 3 brane to a
1For convenience we usually drop the ’low energy’ and just write ’effective field theories’, carrying the same
Figure 3.1: A schematic drawing of all low energy effective field theories of which a vast area consists of theories that seem to be consistent when coupled to gravity, named the Swampland. Lastly, we see that in this Swampland there is an area of LEEFTs that can actually be consistently coupled to gravity, this is called the Landscape.
supersymmetric AdS vacuum such that it uplifts the vacuum to a meta-stable de Sitter vacuum. There are however arguments for why this does not work, also discussed in [12]. With this in mind, the braneworld construction we consider provides interesting prospects.
As mentioned before, given the lack of experimental aid to determine our location in the Land-scape, we turn to conjectures and principles that can help us navigate the Swampland. The idea here being to try and motivate general properties that a UV-complete theory of quantum gravity should have, that would be manifest in the IR. These properties can then be used to formulate criteria for the Swampland. Many of these are conjectural to this day and should not be taken as proven, yet most of them stand on firm grounds. The one we will discuss the most, and also strongly believe a theory of quantum gravity should adhere to, is the Weak Gravity Conjecture[1]. This happens to be one of the Swampland conjectures that has been proved in certain situations, see for example [14][15]. The WGC is at the core of this thesis and we will discuss it and some of its implications more thoroughly in the following sections.
3.1
The Weak Gravity Conjecture
In words, the Weak Gravity Conjecture simply states that gravity should be the weakest force. This is surely true in our own Universe and there are reasons to believe this to be true in any consistent theory of quantum gravity. Hence, it is one of our landmarks in charting the Swampland. The original arguments for this conjecture came mostly from black hole physics[1]. In particular it was proposed by requiring that black holes should be able to decay. We will review these arguments in this section, restricting ourselves first to a four-dimensional theory and later stating the conjecture for general dimensions.
Let us consider a theory that is coupled to gravity and has a U (1) gauge symmetry with a coupling g. Such that we consider the following action
S = Z d4X√−h " Mp2 2 R − 1 4g2F 2 # . (3.1) Here, R is the Ricci scalar with respect to the metric h, Mpthe four-dimensional Planck mass and
F the field tensor corresponding to the gauge field. The WGC then has an electric and a magnetic form. Let us first state them both and then elaborate on them.
• The electric WGC states that there should exist some particle in the theory with a mass m and charge q that satisfies the following inequality
m ≤ gqMp. (3.2)
• The magnetic WGC states that the cutoff scale Λ of the effective field theory is approximately bounded from above by the gauge coupling g, as in the following inequality
Λ . gMp (3.3)
We will not go into detail about the magnetic version of the WGC as it is the statement of the electric WGC that we will use throughout this thesis. We just give the magnetic statement for completion. Some comments about these definitions will be at place. In the electric WGC we have used the word particle. This is perhaps not the right formulation as what we mean by it is some state with a mass below the Planck mass m < Mp. This is not an absolute as often the considered
state is that of a charged black hole, for which the mass is larger than the Planck mass. We should make clear that this is not a statement that should hold for all particles. It is just the statement that there should exists at least some particles for which the inequality holds. This statement, that there should be some particle for which this holds, is also known as the mild form of the WGC. A more constraining version of the WGC, which is known as the strong WGC, states that the inequality should hold for a light charged particle. Now let us turn to the arguments for this conjecture coming from black hole physics. Starting with the metric of a charged black hole, named a Reissner-Nordström black hole. The Reissner-Nordström black hole solution is the following
ds2= −f (r)dt2+ f (r)−1dr2+ r2dΩ22, f (r) = 1 −2M r + g2Q2 r2 . (3.4) We will evaluate this metric in more detail in a later chapter but for now let us just observe that it has a horizon at
r±= M ±
p
M2− g2Q2. (3.5)
Evaluating the metric of (3.4) one sees that charged black holes have two horizons. Such charged black holes satisfy an extremality bound between the mass and charge
M ≥ gQMp. (3.6)
Here we left in the coupling and the Planck mass, however throughout most of this thesis we will work in natural units. When this extremality bound is saturated, the event horizons coincide2and
2Actually, there is always a finite proper distance between the inner and outer horizon, see [17] [18], and we will
the black hole is called extremal. From evaluating the horizons in (3.5) it is evident that this bound should hold. If we consider M < Q, the square root becomes an imaginary number, consequently the black hole gets an imaginary horizon. This gives rise to a naked singularity and violates the Weak Cosmic Censorship Conjecture, something we rather not have happening. The Weak Cosmic Censorship Conjecture asserts that there can be no naked spacetime singularities, they have to be hidden from an observer at infinity by an event horizon.
Let us think about the decay channels of an extremal black hole, for which the bound (3.6) is saturated. and thus
M = Q, r±= Q. (3.7)
We consider some object tunneling through the horizon of mass m and charge q. We consider spherically symmetric shells, which we will go through in more detail in chapter 4. There are the following possibilities for the charge to mass ratio of this tunneling shell
q m > 1 super-extremal, = 1 extremal, < 1 sub-extremal. (3.8)
Starting with the sub-extremal shell. For an extremal black hole, this decay channel is not allowed, precisely for the reasons we discussed before. Namely, emission of a sub-extremal shell from an extremal black hole will result in a naked singularity. For a non-extremal black hole, with M > Q, sub-extremal decay channels are allowed.
Extremal shells are allowed, they keep the bound saturated. Actually, some interesting physics come to play when considering the tunneling of extremal shells, which is the subject of chapter 4. In the next sections we will also discuss an extension to the WGC that asserts that extremal decay channels are only allowed in a supersymmetric theory. As such we see that a super-extremal decay channel with q/m > 1 has to exist. Otherwise extremal black holes will not be able to radiate away their charge without leaving behind a naked singularity.
Another interesting argument comes from the problem of black hole remnants. Imagine we have a gauge coupling of g ∼ 10−100 and a black hole of a mass of M ∼ 10Mp. Our extremality bound
would look as follows
10 ≥ 10−100Q. (3.9) Due to this very small gauge coupling, the extremality bound is still easily satisfied for any charge in the range of 0 ≤ Q ≤ 10100. If there would not exist a very light particle for which the charge to mass ratio is bigger than one, thus satisfying the WGC, the black hole will never be able to radiate away all its charge. What would then be left is a Planck scale black hole remnant labelled by a charge anywhere in the range of 0 ≤ Q ≤ 10100. This could then lead to a family of 10100 Planck
scale black hole remnants. This argument, as given in the original WGC paper, seems closely related to the conjecture that a theory of quantum gravity should not have global symmetries. Namely in the limit when the gauge coupling goes to zero g −→ 0, the gauge symmetry becomes indistinguishable from a global symmetry and it causes this dilemma of remnants. There have been different arguments against remnants as well. For example in [19], Susskind argued that while the probability for a black hole to materialise in a given thermal atmosphere is highly suppressed due to their large entropy, remnants however would be quite favourable. If one were to consider a thermodynamic system like the Sun. It will be almost infinitely more favourable for it to convert its
energy into zero temperature stable black hole remnants. One can imagine why this is problematic. We now have an idea of why gravity should indeed be the weakest force in a consistent theory of quantum gravity. It is now time to discuss some of its implications. In this section we have already seen the implications it has for the decay of black holes. In other words, the WGC tells us something about the stability of black holes. To sketch an idea of where we are going, we note that the near-horizon geometry of an extremal black hole, is anti-de Sitter. Given the implications for black hole decay coming from the WGC, it is not difficult to imagine that the WGC has implications for the stability of AdS as well. We will make this connection more rigorous in chapter 4 where we will see that the stability of an extremal black hole is kind of a dual description of the stability of the near-horizon AdS region. In the next section we will discuss the implications of the WGC on the stability of AdS spacetimes and a sharpened version of the WGC.
3.2
Instability of Anti-de Sitter
Since the original paper of the Weak Gravity Conjecture, there have been many extensions. In the previous section we stated the WGC for some particle state. Given some charged brane, the tension σ of the brane is bounded to be less than or equal to the tension if the brane were an extremal black brane. This is just another way of stating the WGC. It also tells us that when the WGC bound is saturated, the attractive gravitational force and the repulsive electric force between these kind of branes cancels out. States that saturate this bound are well known in string theory, they are BPS states3 in a supersymmetric theory. As written in the introduction, what we will be
working towards is the decay of a non-supersymmetric AdS5 to a supersymmetric AdS5 vacuum.
This section will substantiate why this is the setup to consider. We will review two conjectures, which are interlinked and were proposed around the same time by Freivogel, Kleban[4] and Ooguri, Vafa[20]. Their conjectures follow up on the WGC and can be stated as follows
• The Weak Gravity Conjecture bound is saturated if and only if the considered state is a BPS state in the underlying supersymmetric theory.
• Every non-supersymmetric anti-de Sitter vacuum is at most metastable. Every supersymmet-ric anti-de Sitter vacuum is marginally unstable.
Let us touch upon the content of these statements and discuss their nature, in order of presentation. The bound discussed in the previous section (3.2) can actually only be saturated by BPS states. It turns out that allowing saturation of the bound in a non-supersymmetric theory is actually cause for some trouble. This can be understood as follows. Going back to the original WGC, one of its motivations was to allow extremal black holes to decay. We saw that decay channels of sub-extremal particles is not allowed as this would cause a naked singularity, we did allow for extremal particles. However, in a non-supersymmetric theory there is nothing stopping a quantum perturbation from tipping over the balance at extremality. Thus, a perturbation could tip it over to the sub-extremal, resulting in a naked singularity. That is why this extension is needed. In a supersymmetric theory, the extremality bound corresponds to the BPS bound, as such, this bound is protected by super-symmetry and the tipping over of the balance will not happen.
3In a supersymmetric theory, a BPS state is a state which preserves some supersymmetry and saturates the
BPS bound. We will not go into detail about supersymmetric theories as this is beyond the scope of this thesis. However, it is important to understand that this extra symmetry allows for the stable extremal states. The aspects of supersymmetry of importance to this thesis will be explained.
The conjecture about the (in)stability of AdS vacua comes from thinking about a natural ana-logue to demanding that all black holes should be able to decay. Namely, that all vacua should be able to decay. Taking into our minds again the process of vacuum decay discussed in the previous chapter, such that a false vacuum can decay to a vacuum of lower energy, which could be the true vacuum, which is depicted in figure 3.2. These domain walls can be seen as charged branes. It is in
Figure 3.2: A vacuum with cosmological constant Λ+connected via a domain
wall to some lower vacuum of cosmological constant Λ−.
this view that the WGC has been extended and [4] made a conjecture about the instability of AdS spacetimes.
For domain walls separating two supersymmetric vacua, the tension σ obeys a BPS bound. From considering supergravity domain walls separating two vacua V1and V2, with
|V1| > |V2|, (3.10)
the BPS bound is given by the following triangle inequality [21] 2 √ 3 p |V1| −p|V2| ≤ 8πσ ≤ √2 3 p |V1| +p|V2| . (3.11) In the situation we will consider, we mostly care about the minimum value on the left of (3.11). Namely, it is this scenario that corresponds to some domain wall being nucleated with a lower vacuum as interior. Such that in the direction away from the boundary, in the IR, you find the lower vacuum. While going towards the boundary, in the UV, you will find the higher vacuum. As is the situation when considering vacuum decay. The higher bound comes from a situation where on either side of the domain wall you find an IR region. We will not go into detail about the latter but it will come back when evaluating the differences between the braneworld scenario we will be considering as opposed to that of Randall and Sundrum [23].
From considering vacuum decay [5], the false vacuum can decay to a lower one through the nucleation of a domain wall if the tension of the wall obeys
8πσ < √2 3 p |V1| −p|V2| . (3.12) As such the combined result of BPS domain walls and non-BPS domain walls gives the following bound 0 ≤ 8πσ ≤ √2 3 p |V1| −p|V2| , (3.13)
such that every non-supersymmetric AdS vacuum is connected to at least one lower energy vac-uum by a domain wall that obeys inequality in (3.13). These domain walls separating non-supersymmetric AdS vacua nucleate at rest but will have a constant proper acceleration. Thus they expand and their worldline asymptotes the lightcone, as discussed in the previous chapter. On the other hand, every supersymmetric vacuum is connected to at least some other vacuum by a BPS domain wall that saturates the bound (3.13). These BPS domain walls are static and have a flat geometry.
The bound (3.13) is the AdS analogue of the WGC for extremal black holes from the previous section. We have seen how the WGC has implications for both the decay of black holes as well as the decay of AdS spacetimes. There is actually a deeper connection between these two instabilities, via the WGC, which is the topic of the next chapter.
Chapter 4
The Brane and the Shell
In this chapter we will discuss the relation between the instability of extremal black holes and anti-de Sitter spacetimes. In a way, completing the circle of the matters discussed in the foregoing chapters. We start with a brief discussion of the tunneling integral for spherical shells through the horizon of a charged black hole. We study the case of extremal shells, after which we compare the tension of said shells to the tension of domain walls (branes) in AdS. Concluding with the relation between these two decay channels from a near-horizon perspective and some words on the non-extremal decay channels.
4.1
Tunneling of Charged Shells
Throughout this chapter we consider the spherically symmetric sector of Hawking radiation. Most of this chapter is based on a paper by Aalsma and van der Schaar [18]. In their paper they show the equivalence of two descriptions of emission for charged black holes. One from constructing an effective action for a spherical shell, including backreaction, by Kraus and Wilczek[24][25], the other from a tunneling perspective by Parikh and Wilczek[26]. We will not derive the probability for the emission of a spherical shell from a black hole, we will just use the results of Aalsma and van der Schaar. However, let us first examine the extremal limit of a charged black hole.
The metric of a four-dimensional Reissner-Nordström (RN ) black hole of mass M and charge Q is given by ds2= −f (r)dt2+ f (r)−1dr2+ r2dΩ22, f (r) = 1 −2M r + Q2 r2. (4.1) For which, as stated in chapter 3, the inner- and outer horizon are given by
r±= M ±
p
M2− Q2. (4.2)
Naively taking the extremal limit M → Q would result in the conclusion presented in chapter 3, namely, that the horizons overlap as r± = Q. If one considers a spherical shell tunneling through the
horizon, this overlap of the horizons seems to present a problem. If the horizons coincide, it seems there cannot be a tunneling barrier in between them, which is needed for the tunneling description.
Thus it is necessary to carefully examine the extremal limit and study the proper distance between the horizons. As was already mentioned in chapter 3, we will see that there actually is a finite proper distance between the horizons, even in the extremal limit.
To appropriately take the extremal limit, we introduce the non-extremality parameter ε = p
M2− Q2 such that ε Q. In the extremal limit, the inner and outer horizon approach each
other. We define the horizons as
r+ = λ + ε,
r− = λ − ε,
(4.3) such that λ = Q is the value of r to which the horizons evolve. Extremality is thus defined as the limit ε → 0 and r = λ = Q. Writing the metric function of (4.1) in terms of the locations of the horizons (4.3), the metric function takes the following form
f (r) = (r − λ)
2
r2 −
ε2
r2. (4.4)
In order to study the region between the horizons, we make a coordinate transformation, following [17], such that r = λ − ε cos η, t = λ 2 ε ρ (4.5) In these coordinates the inner horizon corresponds to η = 0 and the outer horizon is at η = π. Here, η is a new time-like coordinate and ρ a new spacelike coordinate. Transforming the metric accordingly results in ds2= λ2−h(η)2dη2+ h(η)−2sin2ηdρ2+ h(η)2dΩ22 , h(η) = 1 −ε λcos η. (4.6)
What we then want to consider is the spacetime volume of the region between the inner- and outer horizon as the black hole approaches extremality. The distance between the horizons will be equal to the proper time on a trajectory of constant ρ. The proper time (and distance) will thus be
∆τ = λ Z π 0 dη 1 − ε λcos η = πλ. (4.7) A curious observation is that this is independent of the non-extremality parameter ε. As such there will always be a region of spacetime between the horizons, even at extremality. When ε = 0 one has λ = Q from which follows that the proper time(distance) between the horizons at extremality is
∆τ = πQ. (4.8) Since we now know that there is a tunneling barrier for the shell, we can continue its description. In the following we will borrow some results and state the appropriate references as the full com-putation is beyond the scope of this thesis.
Parikh and Wilczek started their tunneling description with the fact that in the WKB approxi-mation, the tunneling probability is given by the exponential of the classical action. This tunneling
probability is then P ∝ exp −2 Im Z rf ri drpc ! . (4.9) Where ri, rf are the respective initial and final position of the tunneling shell and pc its canonical
momentum. From Aalsma and van der Schaar we then get the following tunneling probability for a spherically symmetric, extremal shell of mass ω = q and charge q
P ∝ eπ((Q−q)2−Q2). (4.10) Here we recognise (Q − q) as the charge, and mass, of the extremal black hole after emission of the extremal shell. The full exponent can be recognised as the difference in entropy between the final and initial state, such that
P ∝ e∆S. (4.11) Which is the expected result of the probability that is suppressed by the negative entropy difference. Later on, we will see that this expression also holds for super-extremal shells within a certain parameter window.
4.2
Extremal Shells and Branes
As mentioned at the start of the chapter, we are working towards the linking between the instabilities of charged black holes and AdS spacetimes. With this in mind, we start by computing the energy density of a tunneling extremal shell.
The metric of a four-dimensional extremal Reissner-Nordström (RN ) black hole is of mass M = Q and charge Q is given by
ds2= −f (r)dt2+ f (r)−1dr2+ r2dΩ22, f (r) = (r − Q)
2
r2 .
(4.12) We consider the emission of a spherically symmetric, extremal shell Σ with charge q and energy ω = q, and we place the shell at some fixed point a. After emission, the black hole is still extremal, however, now with a charge Q − q in the metric (4.12). As such, the shell splits up the geometry of the spacetime. There is the exterior region, the shell itself, and the interior. For an object or function on the interior and exterior, we will use the subscripts - and + respectively. Using this notation we can grasp the geometries of interior and exterior with the following replacement for the metric function in (4.12) f (r) = (r − Q±) 2 r2 , (4.13) where we use Q±= Q (r < a) (+), Q − q (r > a) (−). (4.14)
In order to consistently glue the interior and exterior regions together, the Israel junction conditions, coming from the Einstein field equations4, have to be satisfied[27]. In other words, the
energy-momentum tensor that describes the matter content of the shell, has to source a jump in extrinsic curvature. In the thin-wall approximation, the energy momentum tensor is
8πSij= ∆Kρρδij− ∆Kij,
∆Kij= Kij+− K − ij,
(4.15) where Kij± denotes the extrinsic curvature on either side of the shell. We are working towards an expression for the energy density ρ of the shell, as such we evaluate the time component of the energy-momentum tensor S0
0 and get the following equation
ρ = 1 8π(∆K ρ ρ− ∆K t t) = 1 8π(∆K θ θ+ ∆K φ φ) (4.16) Let us now derive an expression for the extrinsic curvature for a general case and then go to the thin-wall approximation to compute the energy density of the tunneling extremal shell.
We have the two geometries ds2
±, defined by (4.13), joined together by a spherical surface Σ,
which is parametrized by Φ = r − a(τ ) = 0. The induced metric on Σ is then given by γab= gµν
xµ
ξa
xν
ξb, (4.17)
where ξa are coordinates on Σ and {ξa} = {τ, θ, ϕ}. The invariant line element on Σ is then
ds2Σ= −dτ2+ a(τ )2dΩ22. (4.18)
With τ the proper time on the shell. The extrinsic curvature on either side of the shell is defined as Kab= −nµ ∂2xµ ∂ξa∂ξb + Γ µ ρσ ∂xρ ∂ξa ∂xσ ∂ξb ! (4.19) In which nµ is a spacelike unit-normal vector to Σ. Thus, to get an expression for the extrinsic
curvature we need to define nµ. We do this by demanding that it satisfies the following relations
nµnµ = 1, and nµuµ= 0, (4.20)
with
uµ= ∂x
µ
∂τ , (4.21) the four-velocity for a curve that is parametrised by τ . To find this four-velocity we first compute the relation between the global time-coordinate t and the proper time τ on the shell. This we can get directly from computing γtt in equation (4.17). For simplicity we will write f (r) = f in the
following, but it remains a function of r, we will put it back in the end result. The relation between the time-coordinates is then
∂t ∂τ =
p ˙a2+ f
f . (4.22)
Such that the components of the four-velocity are ut= ∂t
∂τ, u
r= ˙a, uθ= uφ= 0. (4.23)
In which the dot denotes the derivative with respect to τ . With the four-velocity we can now define a unit-normal to Σ, using the properties in (4.20). Such that its components are
nt= ∂t ∂τ, n
r= ˙a, nθ= nφ= 0. (4.24)
We now have all the ingredients to compute the extrinsic curvature defined in (4.19). Looking back at equation (4.16) for the energy density ρ, note that we only have to compute Kθ
θ and Kϕϕ, using (4.19) we obtain Kθθ= −nrΓrθθ, = −1 2nrg rr(−∂rg θθ), = rpf + ˙a2, Kϕϕ= −nrΓrϕϕ, = −1 2nrg rr(−∂rg ϕϕ), = r sin2θpf + ˙a2. (4.25)
Then contracting the Kii with the appropriate inverse metric gii we obtain
Kθθ=1 r p f + ˙a2, Kϕϕ=1 r p f + ˙a2. (4.26)
We are considering an extremal shell at a fixed point a. From the extensions to the WGC we discussed in chapter 3, such a shell has to be a BPS state in the underlying supersymmetric theory. Later we will also look at non-extremal expanding shells. However, in this case it is a static shell, meaning the derivative in the square root is zero, and we find the following energy density for the shell ρ = 1 4πr p f−(r) − p f+(r) . (4.27) Inserting the appropriate metric functions, the energy density of the extremal shell is equal to
ρext=
q
4πr2. (4.28)
As mentioned at the start of this section we want to relate the tunnelled spherical shell to a domain wall, a brane, in AdS. So, let us compare this energy density to the tension of a BPS brane in AdS. From the previous chapter, the tension of an extremal (BPS) domain wall separating two supersymmetric AdS vacua is given by [21][22]
8πσext= 2 √ 3 p |V−| −p|V+| , (4.29)
with V− and V+ the respective vacuum energy of the interior and exterior bulk regions, and |V−| >
|V+|. We have just interchanged the subscripts to (±) instead of (1, 2) in the original equation. For
an AdS2 spacetime of charge Q we have
|V | = 3
Q2. (4.30)
We can use the values Q = Q±, as defined in (4.14), to find the respective values of the potential.
This leads to the following equation for the tension σext= 1 4π s 1 (Q − q)2 − r 1 Q2 , (4.31) such that the tension of the extremal brane considered here is
σext= q 4πQ(Q − q) = q 4πQ2 + O q2 Q2 ! , (4.32) where we assumed q Q. Comparing this tension of the BPS brane in AdS2to the energy density
of the extremal (BPS) shell, given by equation (4.28), we find that they are equal to each other in the near-horizon limit r → Q and under the condition that q Q. We have thus related the instability of a four-dimensional extremal black hole to the instability of AdS2, by relating the
emission for an extremal shell to the nucleation of a BPS wall in AdS2. This relation becomes more
clear when one evaluates the near-horizon geometry of an extremal black hole, which we will do in the next section. There we will also discuss an interesting interpretation of these shells as splitting the throat of the black hole, which in the AdS description corresponds to fragmentation. After that we will study super-extremal decay channels.
4.3
Near-Horizon AdS
In the last section we have linked the spherical decay channel of the four-dimensional extremal black hole to the creation of a domain wall in AdS2. First, we will explicitly derive the
near-horizon geometry of a four dimensional extremal black hole. We will see that in the near-near-horizon limit of an extremal black hole, the geometry is AdS2× S2. Which solidifies the correspondence
between tunneling shells and domain walls in the near-horizon AdS2. To evaluate the near-horizon
geometry, we start with the Euclidean metric of an extremal Reissner-Nordström black hole. We go to Euclidean time because one describes the tunneling shell in terms of a Euclidean action. Thus, in this way we can make the correspondence. We Wick rotate the time coordinate t → it and have to appropriately continue the electric charge as well Q → iQ, such that we Euclidean metric is
ds2= f (r)dt2+ f−1(r)dr2+ r2dΩ22, f (r) = (r − Q)
2
r2 ,
(4.33) where we use the same definition for Q as before
Q = Q (r < a), Q − q (r > a), (4.34)
in which a denotes the position of the extremal shell again. Since we have that both the geometry on the inside and outside are Reissner-Nordström geometries, we can take the near-horizon limit on either side, thus we will not make a distinction here between ± as before, the result will be the same just with a different value of Q. As we have seen that there remains a finite proper distance between the horizons, even in the extremal limit. To formally take the near-horizon limit, we define the following
r = Q + χ, (4.35) where χ is a kind of non-extremality parameter. As can be seen, sending χ → 0 corresponds to the extremal limit, such that r = Q. We then insert the definition (4.35) for r into our Euclidean metric (4.33), such that we get the metric in terms of χ
ds2= f (χ)dt2+ f−1(χ)dr2+ (Q + χ)2dΩ22,
f (χ) = χ
2
(Q + χ)2.
(4.36)
Then expanding around χ = 0 to get the near-horizon geometry, provides ds2= χ 2 Q2dt 2 +Q 2 χ2dχ 2 + Q2dΩ22. (4.37)
Introducing a set of new coordinates
χ = Qe−τcosh z, t = Qeτtanh z, (4.38) and transforming the metric appropriately to find
ds2= Q2(cosh2zdτ2+ dz2+ dΩ22). (4.39) One can clearly see that this metric is the direct product of two spaces of constant curvature. Namely the two-sphere S2(θ, ϕ) and the hyperbolic space H2(z, τ ). The Lorentzian version of
this geometry is known as the Bertotti-Robinson geometry and corresponds to the direct product AdS2× S2. Now the relations between tunneling shells and domain walls is clear. They are two
descriptions, or interpretations, of the the same thing. Extremal shells tunneling through the horizon of a four-dimensional extremal black hole are extremal domain walls (branes) in the near-horizon AdS2 geometry.
As is shown above, when the tunneling shell is extremal itself, e.g. ω = q, the domain wall will separate two near-horizon AdS2 vacua with different cosmological constant. On the outside of the
bubble there will be a vacuum with |V1| = Q32 and on the inside a vacuum with |V2| = (Q−q)3 2.
Because the domain wall itself is extremal the parent black holes of the near-horizon geometries are still extremal too. The relation between these shells and branes seems to be in correspondence with a conjecture by Brill [28]. He showed that there should be an instanton in the throat region describing the creation of the extremal domain wall. This causes a splitting of the black hole near-horizon geometry, such that there is a throat region on either side of the domain wall. To apply this to the case we studied above, of extremal shells and extremal black holes, we start with an extremal black hole of charge Q = Q1+ Q2. In the near-horizon limit of this black hole, the Brill instanton
describes the splitting of the AdS2× S2 space of charge Q, into two AdS2× S2 spaces, of charge
labelled by the charges Q1and Q2. This can be generalised to N splittings of the throat, described
by an N -centered black hole solution. From the AdS point of view, a similar computation has been done in [29] in the context of string theory. There, they constructed some instantons in AdS2 and
showed that when the mass of a nucleated brane is equal to its charge, the corresponding instanton is the same as the one found by Brill. This process described in [29] is called AdS fragmentation. It describes the splitting of an AdS2 throat, labelled by Q into multiple AdS2 spaces, labelled by
Qi, separated by static, flat branes. Where
Q =X
i
Qi. (4.40)
4.4
Super-Extremal Shells and Branes
We have studied extremal shells, but coming from the WGC we know there should exist super-extremal shells too. We will study these decay channel next. In this case the black hole emits a spherical shell such that the energy ω of the tunneling shell is not equal to its charge q. In this case the remaining black hole will not be extremal and the near horizon geometry has to be evaluated separately. Which we will do later in this section. But let us first start by looking at the tunneling probability again
P ∝ e∆S. (4.41) Instead of extremal shells, we now consider super-extremal shells. When an extremal black hole emits such a shell, we can define the remaining black hole, after emission, by the following param-eters
M = Q − ω,
Q = Q − q, (4.42) such that the metric functions in this scenario are given by
f+= (r − Q)2 r2 , f−(r) = 1 − 2(Q − ω) r + (Q − q)2 r2 , (4.43)
where now, in accord with the WGC, the charge-to-mass ratio for the tunneling shell is q/ω > 1. It turns out that when considering sub-extremal shells, the tunneling integral vanishes, due to the horizon becoming imaginary. Which supports the conjecture that these decay channels are forbidden. It was noticed in [18] that by including backreaction, a parameter window opens for super-extremal shells, such that the entropy difference in the exponent is still negative and thus we still have a suppressed tunneling amplitude. They found that when the shell satisfies
q 1 − q 2Q < ω ≤ q, (4.44) the entropy difference is still negative. Following from the discussion of vacuum decay and the WGC, these super-extremal shells should correspond to expanding, super-extremal branes in the near-horizon AdS2 whose tension should be bounded by σ < σext. They are expanding because
ones for which the forces are in balance. To connect these super-extremal shells to super-extremal branes, we need to define an expression for the near-horizon AdS energy, as follows
U = r2 Z
dΩ2ρ, (4.45)
where ρ is defined in (4.27) and dΩ2is the volume element of the unit two-sphere. Integrating over
the local energy density, we get
U = rpf−(r) −
p f+(r)
. (4.46) Using (4.43) for the metric function and subsequently going to the near-horizon limit, we get the following expression
U2= q2− 2Q(q − ω). (4.47) From this one can see that ω < q implies that U < q. Noting that in the extremal case of ω = q, our expression (4.47) tells us that U = q. As such it can be recognised that U < q means σ < σext,
which confirms that these super-extremal shells can also be related to super-extremal branes in the near-horizon AdS region. An interesting thing to note is that, when (4.47) is inverted to an expression for the asymptotic energy ω, such that
ω = q 1 − q 2Q +U 2 2Q, (4.48) one can derive that this near-horizon energy parameter U vanishes when ω = q1 −2Qq . This is exactly the transition point of (4.44) for which the entropy difference in the tunneling amplitude turns positive. This positiveness of the entropy difference corresponds to the superradiant regime, in which a black hole would radiate away its charge very quickly. As such one can conclude that when including backreaction, this superradiant regime is decoupled in the near-horizon limit. Meaning that for all U ≥ 0, the decay is always described by a suppressed tunneling amplitude, with negative entropy difference[18]. The entropy of a black hole is given by
S =A
4, (4.49)
with A the area of the horizon. In [30] the gravitational instanton related to Hawking modes was studied. With the inclusion of backreaction, they found that the decay rate to be as (4.51)
P ∼ e∆S, (4.50) just like (4.11). Aalsma and van der Schaar extended this result to extremal black holes. We consider the decay of an extremal black hole, via emission of a super-extremal shell with ω < q, using the metrics of (4.43). In the near-horizon limit, we can then write the entropy difference in terms of the AdS energy parameter U , as follows
∆S = π Q − ω +p(Q − ω)2− (Q − q)22− Q2 , = π Q − ω +pω2− U22− Q2 , = −2πQhω +pω2− U2i+ O ω 2 Q ! + O U 2 Q ! . (4.51)
