Constructing a transversable wormhole in >2 dimensions

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MSc Physics and Astronomy

Theoretical Physics Master Thesis

Constructing an eternal traversable

wormhole in a four-dimensional Anti-de

Sitter background

by Suzanne Bintanja 11027010 60EC June 2019 - June 2020 Supervisor/Examiner: dr. Ben Freivogel Second Examiner: dr. Diego Hofman July 11, 2020

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Abstract

This thesis gives a construction of an eternal traversable wormhole in a four-dimensional Anti-de Sitter background. The theory we consiAnti-der is that of the four-dimensional Einstein-Hilbert gravitational action with a negative cosmological constant, a U (1) gauge field and massless Dirac fermions coupled to the gauge field. The wormhole is constructed by introducing a non-local coupling, between opposing sides of a Reissner-Nordstr¨om black hole, of massless Dirac fermions that carry no energy in the lowest Landau level. We find the wormhole geometry by considering the backreaction of the coupling to the geometry. The coupling renders the Einstein-Rosen bridge of the black hole traversable. The wormhole geometry consists of an asymptotically Anti-de Sitter black hole geometry at large radii. In the throat of the wormhole the geometry is that of a direct product of deformed two-dimensional Anti-de Sitter space and a two-dimensional sphere.

Title: Constructing an eternal traversable wormhole in a four-dimensional Anti-de Sitter background

Author: Suzanne Bintanja, sbintanja@gmail.com, 11027010 Supervisor/Examiner: dr. Ben Freivogel,

Second Examiner: dr. Diego Hofman, End date: July 11, 2020

Institute of Physics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.iop.uva.nl

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1 Introduction

Wormholes are objects that connect points in the universe that are conventionally not connected, or in other words, they connect spacelike separated points. Crucial questions concerning wormholes are whether they could exist physically, and whether they could be used to travel to distant parts of the universe, or even alternate universes. However, most wormholes are not traversable. This means that in order to travel through one, one has pass through horizons and singularities. In contrast, a traversable wormhole can be traversed without encountering horizons or singularities. In recent years there have been a number of new developments that try to answer questions concerning traversable wormholes, and this work will expand on these developments.

An important research avenue in theoretical physics is to find what geometries are allowed by the Einstein equations, since it is conjectured that all physical spacetimes are solutions to these equations. The existence of traversable wormhole solutions forms an interesting question in the field of theoretical physics. Once one has found and understood these solutions one can for example answer questions on quantum teleportation protocols, and by constructing wormholes in negatively curved spacetimes one can probe the nature of holography. Recently it has been conjectured that wormholes could even be used to explain the nature of spacetime by their role in explaining the island rule in the von Neumann black hole entropy.

In 1988 Morris et al. [1] found a traversable wormhole solution. However, they showed that traversable wormhole solutions can only exist in the presence of so-called exotic matter. Exotic matter is matter that violates the different energy conditions [2]. The energy conditions originate from the notion that energy should not be negative. The existence of traversable wormholes has been connected in particular to violations of the (average) null energy condition [2, 3]1. It has been shown that the (average) null energy condition can be violated by quantum effects [2, 3]. Nonetheless time has proven that controllable violations of the energy conditions that result in traversable wormhole solutions are hard to come by.

A notable development in the search for traversable wormhole solutions was made in 2016 when Gao et al. [4] constructed a traversable wormhole in an Anti-de Sitter background. The setup was that of a BTZ black hole in three dimensions. Gao et al. [4] then coupled the right and left boundary CFTs that are located on either side of the black hole by adding a term

δS = Z

dxdt h(t, x)OR(t, x)OL(−t, x)

1_{The null energy condition states that for any null vector k}a_{we have T}

µνkµkν≥ 0. The average version of

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to the action. For a certain choice of the coupling function h(t, x) this term results in negative energy shockwaves in the interior. With this negative contribution to the stress tensor the backreaction of the geometry is found and it turns out that the non-local coupling renders the black hole traversable. It is important to note that in the setup of Gao et al. [4] the coupling function was set to zero at a certain time. This implies that after this time there is no more negative energy to support the wormhole solution. Therefore, the traversable wormhole has a finite lifetime.

The recipe of Gao et al. [4] for constructing a traversable wormhole in an Anti-de Sitter background by adding a non-local coupling between the holographic CFT’s has been repeated by Maldacena et al. [5] and Maldacena and Qi [6]. In these papers the setup is that of a black hole in a two-dimensional Anti-de Sitter background. Maldacena and Qi [6] even found a wormhole solution that is eternal, or in other words, a traversable wormhole that exists at all times.

Maldacena et al. [7] constructed a traversable wormhole in a flat four-dimensional background. Their construction was based on two oppositely charged near extremal Reissner-Nordstr¨om black holes that were connected by the wormhole throat. They created negative energy by a Casimir-like effect of massless fermions traveling between the mouths of the black holes and through the wormhole throat. In order to stop the two oppositely charged black hole mouths to collapse onto one another the setup is altered so that the black holes are rotating around each other. This rotation however causes radiation to be emitted and gives the traversable wormhole solution a finite lifetime.

Other traversable wormhole solutions can be found by giving up spherical symmetry. When considering only geometries that are not invariant under rotations one evades the need to violate the different energy conditions. Examples of traversable wormhole solutions that do not preserve spherical symmetry can be found in [8, 9].

Up until now, no eternal traversable wormhole solutions have been found that preserve spherical symmetry in more than two spacetime dimensions. This thesis will provide the first known construction of such a solution. The theory we consider is that of the four-dimensional Einstein-Hilbert gravitational action with a negative cosmological constant, a U (1) gauge field and massless Dirac fermions coupled to the gauge field. The solution we find is that of an eternal traversable wormhole in a four-dimensional Anti-de Sitter background. This solution can for example be used to obtain a better understanding of holography and quantum systems that model traversable wormholes. Understanding the solution and its consequences may lead to a more complete comprehension of the interplay between gravitational and quantum physics.

Summary of results

As advertised, we find a solution to the four-dimensional Einstein-Hilbert action with a negative cosmological constant, a U (1) gauge field and massless Dirac fermions coupled to the gauge field that has a traversable wormhole geometry. The wormhole geometry we find consists of

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two regimes. At small radii the geometry is a deformation of AdS2× S2. At larger radii this

AdS2× S2 can be glued to the geometry of a near extremal magnetically charged black hole in

an AdS4 background. An observer would experience passage through the wormhole as follows.

Far away, the geometry is that of AdS4 with a near-extremal magnetically charged black hole.

Once the observer approaches the black hole mouth an approximately AdS2× S2 geometry

develops. The observer could travel through this AdS2× S2 throat where after some time she

sees the geometry as that of the mouth of a near-extremal magnetically charged black hole again. Once she emerges from the black hole mouth and moves away from the wormhole the geometry is that of (a different part of) AdS4 once again. She has travelled to the “other side”!

The construction of the wormhole solution crucially depends on the introduction of a non-local coupling between the two disconnected AdS4 boundaries. The non-local coupling is introduced

between the fermionic fields on both boundaries

Z

d3x√γh ¯ΨR−ΨL++ ¯ΨL+ΨR− ,

where h is the coupling constant, Ψ± are projections of the Dirac spinors onto the eigenspace

of the gamma matrix corresponding to the radial coordinate and the label L, R denotes the field on the left respectively right boundary. A consequence of introducing such a coupling is the changing of the boundary conditions. By solving the equations of motion with these new boundary conditions, and by using point splitting we can determine the contribution to the stress tensor originating from the non-local coupling to linear order in h. By tuning h we can make the contribution negative. With this negative contribution to the stress tensor we then solve the Einstein equations perturbatively and numerically and find the traversable wormhole geometry.

The construction is organised as follows. In Chapter 2 we review the vielbein formalism that is used to describe fermions in curved spacetime setups and the AdS/CFT correspondence. In particular we explain the mechanism of the correspondence for spin-1₂ fields. Chapter 3 discusses black holes with a focus on the near-extremal Reissner-Nordstr¨om black hole in four-dimensional Anti-de Sitter space. This is followed by a review of the traversable wormhole construction of Maldacena et al. [7] in Chapter 4. Chapter 4 is included because a lot of the elements of Maldacena’s construction are used in this work. In Chapter 5 we explain the general strategy of splitting the solution into different regions. This is followed by a discussion on the setup we use in Chapter 6. In Chapter 7 we quantise the fermionic fields and calculate its propagators, which we use in Chapter 8 to calculate the different contributions to the stress tensor. Then in Chapter 9 the Einstein equations are solved with these stress tensor contributions and the wormhole geometry is found. In Chapter 10 we present a concluding discussion followed by an outlook in Chapter 11. Throughout this thesis we will use the convention c = ~ = 1.

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2 Preliminaries

This chapter contains a review of some formalism and concepts that will be used throughout the remainder of this thesis. In section 2.1 the description of curved spacetimes in the vielbein formalism will be reviewed, in section 2.2 the description of Dirac fermions in both flat and curved spacetimes is reviewed, and finally in section 2.3 the concept of the AdS/CFT correspondence will be reviewed.

2.1 Vielbein formalism

In this section the description of curved spacetimes in the vielbein formalism will be reviewed. The review will primarily follow Carroll [10], and Lippoldt [11]. Conventionally in quantum field theories with integer spin fields, tensors are described in the coordinate basis defined by _∂x∂µ and dxµ (this can be seen from the fact that all tensors have greek upper and lower

indices). We are however free to choose any basis that we like. A more convenient choice for describing half integer spin fields is the vielbein basis. The vielbein basis is defined by vectors ea and co-vectors ea(note the latin indices), which are such that gµν = eaµebνηab where g is the

metric, η is the Minkowski metric, and eaeb = δab. These requirements do not fix the vielbein

basis uniquely. One can show that the vielbein basis is fixed up to Lorentz symmetries, or equivalently the set of vielbein bases has a local O(p, n − p) symmetry. One can go from the coordinate basis to the vielbein basis in the following manner

Tµ1...µpν1...νn−p = e a1 µ1 . . . e ap µp T b1...bn−p a1...ap e ν1 b1 . . . e νn−p bn−p . (2.1)

One can show that the covariant derivative in the vielbein basis is equal to

D(e)µT

a_{= ∂}

µTa+ ω_(e)_µa_bTb, where ω_(e)_µa_b := eνaDµeνb = eνa∂µeνb+ eνaΓµ ρν e ρ

b. (2.2)

In this formalism ω is called the spin connection1. One can easily see that this implies that D(e)µe

a ν = 0.

1

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2.2 Dirac fermions

In this section the description of Dirac fermions in both flat and curved spacetimes is reviewed. First the flat case is considered which is then used, together with the vielbein formalism, to describe Dirac fermions in curved spacetimes.

2.2.1 In flat spacetimes

First the symmetries of fermions in spacetime will be discussed through which the Spin group will be introduced. After the introduction of the Spin group we define the Clifford algebra. Next we define the Spin group as a subset of the Clifford algebra. Then we will have all the tools to introduce Dirac fermions. The main references we will follow are Posthuma and Vonk [12], and Blaine Lawson and Michelson [13].

In order to introduce Dirac fermions we first need to consider the symmetries of flat Minkowski spacetime. Relativity forces physics to be invariant under the action of elements of the symmetry group SO(m, n) (where m and n depend on the signature and dimension of spacetime). However, different types of particles transform according to different representations of the symmetry group. In particular fermions transform according to representations of the Spin group. The Spin group Spin(m, n) is defined as the double covering of the component of the Lie group SO(m, n) that contains the identity (which is a group itself, and is denoted by SO+(m, n))2. The fact that the Spin group is the double covering of (a component) of the symmetry group connects to the intuitive idea of what fermions are, namely that fermions are particles that obtain a minus sign after a rotation over 2π, and are only invariant under rotations over 4π. At each point in spacetime particles are invariant under actions of the symmetry group. In the case of fermions this group is the Spin group. Now the following question arises: how can one describe an entire spacetime with such a structure? The answer to this question is that one can give spacetime a Spin group structure by considering a spinor bundle. For now it is enough to know that such a spinor bundle does exactly what one wants it to do, namely at every point in spacetime it defines a vector space with a spin structure. A more thorough introduction to spinor bundles is given by Posthuma and Vonk [12].

Now let us define the Clifford algebra in flat spacetimes. Let us denote with Rm,nthe Minkowski

spacetime with n spacelike directions and m timelike directions. Then we can define the Clifford algebra Cliffm,n as the algebra generated by the linear map that sends a vector v to ψ(v)

subject to the following relation

ψ(v1)ψ(v2) + ψ(v2)ψ(v1) = −2ηµνv₁µv2ν. (2.3)

Some examples of Clifford algebras are

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The fact that the spin group is defined as the double covering of SO(m, n) can be understood from the fact that there exists an exact sequence 0 Z2 Spin(m, n) SO(m, n) 1 .

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• Cliff0,0= R;

• Cliff0,1= C.

From equation 2.3 one immediately recognises the anticommutating relation that fermions satisfy. It may now come as no surprise that the group of symmetries that fermions satisfy, Spin(m, n) is a subgroup of the invertible elements of Cliff(m, n) (where the group multiplica-tion is the multiplicamultiplica-tion in the algebra).

Finally we are at a point where we can introduce fermionic (quantum) fields. Fermionic fields are fields that are valued in a representation of their symmetry group, or in this case the Spin group. Now in order to be more concrete as to what such representations look like let us focus on the case m = 1 and n = 3. This is the case we will be considering in this thesis. One can show that Spin(1, 3) ∼= SL(2, C), so fermionic fields are valued in representations of SL(2, C). It turns out that the irreducible representations of SL(2, C) can be labeled by (u, v) where u, v ∈ {0,1₂, 1,3₂, . . . }, here 2u and 2v are the so called highest weights of the representation (the weights are eigenvalues of the representations). In Table 2.1 the physical interpretation of the first few irreducible representations are given. Dirac fermions are now defined as fermions valued in the (1₂, 0) ⊕ (0,1₂) representation of SL(2, C).

Table 2.1: Physical interpretation of the (u, v) irreducible representations of SL(2, C), in parenthesis is the number of components the object has.

u = 0 u = 1₂

v = 0 Scalar (1) Left-handed Weyl spinor (2)

v = 1₂ Right-handed Weyl spinor (2) Vector (4)

Now that we have discussed some representations and their physical interpretations we can look into the explicit form of the (1₂, 0) ⊕ (0,1₂) representation of SL(2, C) that describes Dirac fermions. This representation is generated by a set of four matrices γµ ∈ Mat_4×4(C), µ ∈ {1, 2, 3, 4}, such that

γµγν+ γνγµ= −2ηµν, (2.4)

often called the gamma matrices. Here we immediately recognise the defining relation of the Clifford algebra Cliff1,3. In other words, the gamma matrices generate the Clifford algebra

relation. As noted before we now need to impose this structure at every point in spacetime (now chosen to be R1,3, four dimensional Minkowski space), through a spinor bundle structure.

However, because our underlying manifold is trivial, we can use the same gamma matrices at each point in spacetime. Let us now consider a theory of Dirac fermions coupled to a U (1) gauge field A in four-dimensional Minkowski spacetime. The requirement that the Lagrangian density is invariant under the symmetry group fixes the Lagrangian density to be

L = i ¯χ( /∂ − i /A)χ − m ¯χχ, (2.5)

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m.

2.2.2 In curved spacetimes

In this subsection we will generalise the construction that allows one to describe Dirac fermions in flat spacetimes to curved spacetimes. The construction in general spacetimes is more involved compared to the case of flat spacetime. This is a consequence of the fact that the manifold over which the spinor bundle was constructed is Minkowski space (which is trivial). This implies that the spinor bundle reduces to the direct product of the flat spacetime with the Spin group. However, in general this is not the case. For more complicated manifolds there is no basis for the gamma matrices that leaves the metric invariant at each point in spacetime, so the gamma matrices have to depend on the spacetime coordinates. However, there is a way around this. We can use the construction of flat spacetime by employing the vielbein formalism discussed in section 2.1. The vielbein basis is not uniquely defined and has (in four dimensions) an additional O(1, 3) invariance, which we can use to “store” the information on the coordinate dependence of the gamma matrices. We can think of the coordinate dependent gamma matrices as being equal to γ_(e)µ = eµaγa, with γa the coordinate independent (flat)

gamma matrices. With this construction it seems as if we can fully describe fermions on any manifold. However, because the invariance under the Spin group has to be a local symmetry we need to adjust the covariant derivative in the vielbein formalism, given in equation 2.2. This can be accomplished by defining

Γ_(e)µ = −1

8ηacηbdω

ab (e)µ[γ

c_{, γ}d_]. _(2.6)

The covariant derivative then becomes

D(e)µ = ∂µ+ Γ(e)µ. (2.7)

Now we can finally write the Lagrangian density for Dirac fermions with mass m coupled to a U (1) gauge field A in general spacetimes as

L = i ¯χ( /D(e)− i /A)χ − m ¯χχ. (2.8)

Note that one can straightforwardly generalise the construction of this section to general dimensions, by studying the representations of the Spin group, and generalise by considering different gauge symmetries that are coupled to the fermions.

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2.3 The AdS/CFT correspondence

This section will be used to review the concept of the AdS/CFT correspondence. The corre-spondence was first proposed in [14], and some of its important aspects were first discussed in [15, 16, 17]. This review is organised as follows. First Anti-de Sitter space (AdS) is constructed as a solution to the vacuum Einstein equations with a negative cosmological constant. Fur-thermore some different coordinate systems that describe (patches) of the Anti-de Sitter space will be discussed. Finally the symmetry group of AdS will be discussed. Hereafter a review of conformal field theory (CFT) is given, and the symmetry group of a CFT is discussed. After AdS and CFT are introduced the conjecture of the AdS/CFT correspondence is discussed. Finally the most important concept for this thesis of the correspondence is discussed, namely the holographic dictionary for fermions. The main reference we follow is Nastase [18], which we recommend for a more thorough introduction of the AdS/CFT correspondence. For the introduction on CFT in section 2.3.2 we will also follow Qualls [19]. Finally for the review of the dictionary for fermions we use [20, 21, 22].

2.3.1 Anti-de Sitter space

Let us first recall the Einstein equations

Rµν−

1

2gµν(R − 2Λ) = 8πGNTµν. (2.9)

Here Λ is the cosmological constant, GN denotes Newton’s gravitational constant, and Tµν is

the stress tensor. Furthermore, remember that the vacuum (Tµν = 0) Einstein equations can

be derived from the Einstein-Hilbert action

SEH =

1 16πGN

Z

dd+1x√g (R − 2Λ) , (2.10)

by varying the action with respect to the metric g and requiring the variation to be equal to zero. We will see in a moment that Anti-de Sitter space is a solution to the vacuum Einstein equations with negative cosmological constant Λ. In order to introduce Anti-de Sitter space (in d + 1 dimensions), let us first write the negative cosmological constant as

Λ = −d(d − 1)

2`2_AdS , (2.11)

where `AdS is known as the AdS length. The physical significance of the AdS length will

become clear after we have defined AdS spacetime. We start by considering R2,d with the

following metric3

3_{Note that for just this once our coordinate indices run from -1 to d, since we will define AdS as an embedding}

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ds2= −dX−12 − dX02+ d

X

i=1

dX_i2. (2.12)

Anti-de Sitter spacetime is now defined as the hypersurface embedded in the geometry of equation 2.12 defined by − X₋₁2 − X₀2+ d X i=1 X_i2 = −`! 2_AdS. (2.13)

Now we can see why `AdS is called the AdS length. It is because the hypersurface that defines

the AdS spacetime is just a sphere (of signature (2, d)), with radius `AdS, so `AdS is a measure

for the size of the AdS spacetime. From equations 2.12 and 2.13 one immediately sees that the symmetry group of AdS equals SO(2, d). This will be important when we introduce the correspondence conjecture. We have now formally defined AdS spacetime, however we would like to have a form of the metric for AdS (in d + 1 dimensions) in which it is not just an embedding in R2,d. One way to achieve this is to consider the following coordinate

transformation

X−1= `AdScosh(ρ) sin(τ )

X0= `AdScosh(ρ) cos(τ )

Xi= `AdSsinh(ρ)Ωi for i ∈ {1, . . . , d} (2.14)

where Ωi are coordinates for the d-dimensional unit sphere with positive signature for all i.

These coordinates are called global coordinates. The metric in these coordinate becomes

ds2= `2_AdS − cosh2(ρ)dτ2+ dρ2+ sinh2(ρ)d~Ω2_d−1 , (2.15) with d~Ω2

d−1 the metric on the d − 1-dimensional unit sphere. This metric can be written in a

more familiar form by the coordinate transformation r = sinh(ρ)`AdS, and t = τ `AdS

ds2= − 1 + r 2 `2 AdS dt2+ dr 2 1 +_`2r2 AdS + r2d~Ω2_d−1. (2.16)

These global coordinates cover the entire AdS spacetime. We can find the Penrose diagram of d + 1-dimensional AdS space by considering tan(x) = sinh(ρ) in equation 2.14. The metric then becomes ds2 = ` 2 AdS cos2_(x) −dτ2+ dx2+ x2d~Ω2_d−1 . (2.17)

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Here x ∈ [0,π₂] for d = 1, and x ∈ [−π₂,π₂] otherwise. From this we see that the boundary geometry is a flat strip for d = 1 and R × Sd−1 otherwise. With this knowledge the Penrose diagram can be drawn. The Penrose diagram is given in Figure 2.1.

Figure 2.1: Penrose diagram of d + 1 dimensional AdS spacetime. The Penrose dia-gram for d = 1 is given by the shaded area, while the circle corresponds to a d − 1 dimensional sphere.

It remains to show that AdS spacetime actually is a solution to the vacuum Einstein equations. The Ricci tensor of AdS equals Rµν = −_`2d

AdS

gµν. From this we see that the curvature is

R = −d(d+1)_`2 AdS , so that Rµν− 1 2gµν(R − 2Λ) = gµν − d `2_AdS + 1 2 d(d + 1) `2_AdS − d(d − 1) 2`2_AdS = 0,

and we can conclude that the Anti-de Sitter space is indeed a solution to the Einstein equations.

2.3.2 Conformal field theory

In order to understand what a conformal field theory is one first needs to understand what conformal transformations are. A conformal transformation is an extension of a scale transfor-mation xµ7→ λxµ_{. Conformal transformations generalise scale transformations in the following}

sense. Scale transformations preserve angles in a metric. However there are more coordinate transformations that leave angles invariant, and those are precisely the conformal transforma-tions. Note that even though the angles are left invariant under such transformations distances are not necessarily preserved. On flat spacetimes conformal transformations can be defined as precisely the coordinate transformations that cause the metric to change in the following manner

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η0 = Ω−2(x)η, as xµ7→ x0µ(x). (2.18)

From this it follows that one can see a conformal transformation as a local rescaling of the metric. In d 6= 2 the most general form of a conformal transformation is the following

x0µ= xµ+ νµ(x), with νµ(x) = aµ+ ωµρxρ+ λxµ+ bµxρxρ− 2xµbρxρ. (2.19)

Here ω is antisymmetric and corresponds to rotations, while λ, aµ, and bµ correspond to scale transformations, translations, and special conformal transformations respectively. Altogether this gives us a total (d+1)(d+2)/2 components for the parameters of conformal transformations. The conformal transformations form a symmetry group. From the above it follows immediately that this group should be of dimension (d + 1)(d + 2)/2. It turns out that when considering R1,d the symmetry group equals SO(2, d). A first check on this claim is that the dimension

of the group matches the intuitive degrees of freedom in a conformal transformation. From this we see that when considering a flat spacetime with signature (1, d − 1) that is invariant under conformal transformations, the symmetry group equals SO(2, d). Theories with this background and these symmetries are called d-dimensional conformal field theories.

An important example of a CFT in the view of the AdS/CFT correspondence is a four-dimensional CFT called N = 4 super Yang-Mills (SYM). This is the simplest example of a four-dimensional CFT that is also supersymmetric4. The most famous example of the AdS/CFT correspondence that has been proven (in a specific limit) involves the N = 4 SYM CFT. We will comment on this more in the next section when we introduce the correspondence.

2.3.3 The correspondence

Finally we are ready to introduce the conjectured correspondence between negatively curved AdS spacetimes in d + 1 dimensions and conformal field theories in d dimensions. As mentioned before the most famous example of this correspondence involves the conformal field theory N = 4 SYM. One can show that in a specific limit the partition function of this theory coincides with the partition function of string theory on AdS5× S5. We will not discuss the

specifics of this limit here but a review can be found in Chapter 10 of Nastase [18]. The fact that these theories have the same partition function has major consequences. It implies that every result in one theory can be “translated” to a result in the other theory trough the so-called dictionary. One thing to note is that the two theories have the same symmetry groups. The symmetry group of AdS5 is SO(2, 4), the same as the symmetry group of a four

dimensional conformal field theory. Furthermore the symmetry group of the sphere S5 equals SO(6) ∼= SU(4), which is precisely the symmetry group of the R-Symmetry of N = 4 SYM

4

We will not explain supersymmetry in its full glory in this thesis. However, we will briefly discuss the supersymmetric stability of black holes in chapter 3. For now it is enough to know that supersymmetry is a symmetry that forces fermions to have bosonic super partners and vice versa, into which they transform under a supersymmetric transformation.

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(which is the symmetry group having to do with supersymmetry). Another important fact about the correspondence is that it is a duality. This means that when one theory is strongly coupled, the other one is weakly coupled. Because of this we can use the dictionary to do calculations in the limit that we desire (and in which the calculation is easier).

It is believed that the AdS/CFT correspondence holds in more general cases. It is conjectured that any string theory on a d + 1-dimensional spacetime that is asymptotically AdS (AAdS)5 that preserves supersymmetry has a d-dimensional CFT dual. A first indication that this conjecture may be true comes from the fact that the symmetry groups of the theories coincide. Furthermore the d-dimensional boundary of an asymptotically AdS spacetime is conformally flat, which implies that one could in principle define a CFT on the boundary of an AAdS spacetime. In the correspondence the higher dimensional theory is often referred to as the bulk theory.

The AdS/CFT correspondence is the most well-known example of the holographic principle. This principle states that a theory of quantum gravity on d + 1 dimensions can be stored on a d-dimensional surface much like a holograph [23]. A hint for this principle comes from black holes. The entropy of a black hole scales with the area of its horizon, as was famously shown by Hawking. This means that the information that resides inside the black hole can be thought of as stored on the horizon of the black hole.

2.3.4 The dictionary for spin-1₂ fields

In order to start using the AdS/CFT correspondence it is important to understand the dictionary, which tells us how to go back and forth from one theory to its dual. For the scope of this thesis it is enough to understand what the spin-1₂ objects in both theories are, and how they are connected through the correspondence. We will also constrict ourselves to the case where the AdS theory has an even dimension, since this is the setup we will be using. However it is important to note that the case we will discuss is qualitatively different from the case where the AdS theory has an odd dimension.

In a CFT with odd (2n + 1 with n ∈ Z) dimensions, massless spin-1₂ objects are described by spinor operators of scaling dimension ∆ = n + 1₂. These operators have 2n components. However in the bulk theory massless spin-1₂ objects are spinor fields with 2n+1 components [22]6_{. This implies that the dictionary must give us a rule that ensures that the boundary}

objects have half the components compared to the bulk objects in such a manner that one can reconstruct one from the other.

Let us now discuss concretely how the dictionary works. Let the bulk dimension be equal to four. We will label the time coordinate with 1 and the holographic coordinate (the coordinate through which the boundary of the AdS space is defined; in the different coordinate patches: ρ in equation 2.15, r in equation 2.16 and x in equation 2.17) with 2. Furthermore we denote

5

A spacetime is asymptotically AdS when it solves the Einstein equations and preserves the conformal structure of “pure” AdS near the boundary.

6_{When the CFT has even dimensions (2n with n ∈ Z) both the bulk and boundary theory spin-}1

2 objects

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the spin-1₂ field in the bulk with χ. In order to construct the boundary spinors in the CFT we decompose χ into projections on the eigenspaces of γ2 (the gamma matrix corresponding to the holographic coordinate) by [21]7

χ = Ψ++ Ψ−, where Ψ±:=

1 2(1 ± γ

2_)χ. _(2.20)

Let the gamma matrices be defined as followed

γ1 := iσx⊗ 1, γ2 = σy⊗ 1, γ3 := σz⊗ σx, γ4:= σz⊗ σy, (2.21)

where σi for i ∈ {x, y, z} are the Pauli matrices given by

σx := 0 1 1 0 , σy := 0 −i i 0 , σz:= 1 0 0 −1 . (2.22)

The projections of χ onto the eigenspaces of γ2 will now have the following property

Ψ+= ξ+ iξ+ , and Ψ−= ξ− −iξ− , (2.23)

where ξ+ and ξ− are bispinors given (in components of χ) by

ξ+= 1 2 χ1− iχ3 χ2− iχ4 , and ξ−=χ1+ iχ3 χ2+ iχ4 . (2.24)

Now the bispinors ξ± evaluated at the holographic boundary are spin-1₂ operators in the three

dimensional CFT with scaling dimension ∆ = 3₂.

7

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3 Black holes

This chapter contains a brief review of black holes. In particular our focus will be on the near-extremal Reissner-Nordström black hole in a four-dimensional AdS background. First the history behind the idea of black holes will be discussed in section 3.1. Section 3.2 will consist of a brief review of different black hole solutions, after which the near-extremal Reissner-Nordström black hole in a four-dimensional AdS background will be studied in detail. Then in Section 3.3 the thermodynamics of black holes, and specifically the four-dimensional near-extremal Reissner-Nordström in AdS, will be discussed. In order to construct a static traversable wormhole it is important that the “original” black hole solution that will be rendered traversable is thermodynamically stable. If the black hole is not thermodynamically stable, how can one expect the traversable wormhole to be stable. Finally the supersymmetric stability of the our dimensional near-extremal Reissner-Nordström in AdS will be discussed in Section 3.4. The supersymmetric stability of the black hole solution we use to construct a traversable wormhole is important because in order to use the AdS/CFT correspondence the bulk theory needs to preserve at least some supersymmetry. More detailed discussions on black hole solutions and their thermodynamics can be found in [10, 24, 25, 26, 27, 28], among others.

3.1 History

When Einstein developed the theory of general relativity in 1915, it became clear that the manner in which light travels through space can be influenced by gravity. An interesting question that naturally arises is then: can there be regions of spacetime from which light cannot escape. The answer to this question is well-known, and these regions in spacetime are known as black holes. When one considers a point-like, or spherical mass, there is the Schwarzschild solution to the Einstein field equations, that becomes singular at the Schwarzschild radius [29]. Light originating from within this radius cannot be observed from any location outside of the Schwarzschild radius [30].

Now that we have defined what it is to have a black hole, the next question that arises is how one can describe different black holes. The answer to this question came in a number of publications around the year 1970. Israel [31] proposed for the first time the no-hair theorem. In the years following the first publication of this theorem more general proposals for the no-hair theorem were published, and they were summarised in the famous book Gravitation [28]. The no-hair theorem states that a stationary black hole (a thermodynamically stable black hole) can be described uniquely by its mass, electromagnetic charge and angular momentum. The name no-hair for the theorem comes from the fact that according to the theorem a black

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hole can have no other characteristics (no hair) besides mass, charge and angular momentum.

3.2 Black hole solutions

In this section we will discuss some black hole geometries that are solutions to the Einstein field equations. We will consider four-dimensional flat and negatively curved spacetimes. Also we will restrict ourselves to solutions that preserve spherical symmetry, which means that the black hole solutions we discuss will have no angular momentum. The action we will consider is the Einstein-Maxwell action given by

SEM = Z d4x√g 1 16πGN (R − 2Λ) − 1 4g2F 2 . (3.1)

Here GN is Newton’s gravitational constant, Λ denotes the cosmological constant, and F is the

field strength of a U(1) gauge field with coupling constant g. In this action the cosmological constant is given by Λ = −_`23

AdS

in accordance with equation 2.11. Flat spacetime is given by the `2_AdS→ ∞ limit. The equations of motion for this action are given by

Rµν−

1

2gµν(R − 2Λ) = 8πGNTµν, (3.2)

and

DµFµν = 0. (3.3)

Here we recognise equation 3.2 as the Einstein field equations. It is obtained by requiring the variation of the Einstein-Maxwell equation with respect to the metric to vanish, while we recognise equation 3.3 as the Maxwell equations, that follow from requiring that the variation of the action with respect to the gauge field vanishes. Consider now the following metric

ds2= −f (r)dτ2+ dr 2 f (r) + r 2_(dθ2_{+ sin(θ)}2 dφ2), (3.4) with f (r) = 1 −2GNM r + r2_e r2 + r2 `2, (3.5) and

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r2_e = πq 2_G N g2 and A = q 2sin(θ)dφ. (3.6)

In the above equations M denotes the mass of the black hole, while q is an integer that denotes the magnetic charge of the black hole. The Einstein-Maxwell stress tensor is then given by

T_µνEM = FµλFνλ− 1 4gµνF 2₌ q2 8g2_r4diag f (r), 1 f (r), r 2_{, r}2_sin2_(θ) . (3.7)

The Ricci tensor of the geometry is given by

Rµν = diag (3r4_{+ r}2 e`2)h r6_`4 , − 3r4+ r_e2`2 r2_h , r2_e r2 − 3r3 `2 , r2 e r2 − 3r3 `2 sin2(θ) , (3.8) where h := r4− 2G_NM r`2+ r2`2+ r2_e`2,

and the Ricci scalar is given by

R = −12

`2. (3.9)

Combining all the above shows that the metric given in equation 3.4, with field strength as given in equation 3.6 is a solution to the equations of motion of the Einstein-Maxwell action. The black hole described above is called the Reissner-Nordstr¨om black hole and was discovered by Reissner [32], Weyl [33], Nordstr¨om [34], and Jeffery [35]. Note that in the solution we described above, the black hole only carries magnetic charge. To obtain an electrically charged black hole the metric and gauge field have to be chosen slightly different to include the electrical field and charges. One can obtain the uncharged Schwarzschild solution, first discovered by Schwarzschild [29], by setting q = 0. Furthermore, as noted above the black hole will be imbedded in a flat spacetime by setting `2_AdS → ∞. This would imply that the solution is modified by setting fflat(r) = 1 − 2GNM r + r2_e r2. (3.10)

The Penrose diagram of the Reissner-Nordstr¨om black hole embedded in Anti-de Sitter space looks similar to the Penrose diagram of ‘vacuum’ Anti-de Sitter space as shown in Figure 2.1, with the addition of the horizon. The Penrose diagram for the Reissner-Nordstr¨om black hole in AdS is shown in Figure 3.1.

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Figure 3.1: Penrose diagram of Reisnner-Nordstr¨om black hole in AdS. The shaded surfaces are depicting the black hole horizon.

3.3 Thermodynamic stability

From the second law of thermodynamics it follows immediately that black holes must have entropy, since the total entropy of a system must increase, and objects with positive entropy can fall into a black hole. This realisation is one of the hints towards the existence of laws of thermodynamics for black holes. Bekenstein [36] and Bardeen et al. [27] where the first to introduce the laws of black hole thermodynamics. The zeroth law of black hole thermodynamics states that the surface gravity is constant on the black hole horizon (given by r+, the largest r

such that f (r) = 0). Bekenstein [36] and Bardeen et al. [27] realised that through this constant surface gravity on the horizon a temperature and entropy associated with the black hole can be defined by1 β = 1 T = 4π ∂rf (r)|r=r+ , (3.11) and S = 4πr 2 + GN . (3.12)

The first law of black hole thermodynamics mirrors the first law of “ordinary” thermodynamics, and states the following

dM = T dS + µdq. (3.13)

1

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In this equation M is the black hole mass, while T and S are the black hole temperature and entropy as given in equations 3.11 and 3.12. q is the charge of the black hole while µ is the electromagnetic potential. If the black hole has nonzero angular momentum, the first law should be modified to contain a term proportional to the exterior derivative of the angular momentum. However, since we consider only spherically symmetric solutions this term vanishes. The second law of black hole thermodynamics states (similarly to the second law of ordinary thermodynamics) that the entropy cannot decrease. Lastly the third law states that the entropy approaches zero as the temperature decreases to zero. Applied to black holes this law implies that there is no way to create an extremal black hole2 in finite time. A more detailed overview of the laws of black hole thermodynamics is given by Wall [26].

By using the laws of black hole thermodynamics, one can study the thermodynamic stability of black hole solutions. Next, we will consider the thermodynamical stability of the Reissner-Nordstr¨om black hole in AdS as given in equation 3.4. By Callen [37] we can determine the thermodynamical stability of the solution by determining whether the Hessian matrix of the potential energy (or mass) is positive definite with respect to the extensive variables (that are not fixed at the boundary): entropy and charge3_{. First let us express the mass M in the}

extensive variables S and q by using equations 3.6, 3.12 and the definition r+

M (S, q) = 1 4 r S πGN + 1 16`2 r S3_G N π3 + q2 g2 s π3 SGN . (3.14)

The Hessian of M is defined as

∂2M ∂q2 ∂ 2_M ∂S∂q ∂2_M ∂S∂q ∂2_M ∂S2 ! . (3.15)

By Sylvester’s criterion a two by two matrix is positive definite when both the upper left component and the determinant are positive [38]. A straightforward calculation shows that

∂2_M

∂q2 > 0 and that the determinant of the Hessian of M is positive whenever

q2 g2 > S 4π2 − 3S2_G N 16`2_π3. (3.16)

It follows that the black hole solution as given in equation 3.4 is thermodynamically stable whenever equation 3.16 is satisfied.

2_{An extremal black hole is a black hole with zero temperature, or equivalently, a solution to the Einstein field}

equations such that the function f (r) in the metric as given in equation 3.4 has a double zero.

3_{An extensive variable is a variable that scales with the size of the system, or scales with the quantity of}

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Extremal black holes

In order to construct a traversable wormhole we would like to consider the four-dimensional Reissner-Nordstr¨om black hole in AdS in the near-extremal limit. However we can only expect to end up with a stable traversable wormhole if the metric that we start with (as given in equation 3.4) is a thermodynamically stable solution. From the first law as given in equation 3.13 we see that the temperature is related to the mass of the black hole by T = ∂M_∂S. From this we gather that an extremal black hole corresponds to

q2 g2 = S 4π2 + 3S2GN 16`2_π3.

By comparing this relation to the requirement for thermodynamic stability given in equation 3.16, we can conclude that (near) extremal black holes are thermodynamically stable. For a more complete picture of the stability of black holes, see for example Section 10 of Bellucci [39].

3.4 Supersymmetric stability

In order to be able to use the AdS/CFT correspondence in a setup with a black hole in an AdS background, we need this setup to preserve some supersymmetry. Supersymmetric configurations are also called BPS states, and are defined as states that are massive such that their mass is equal to the supersymmetric central charge. Generalising one can have

1

2-BPS states, 1

4-BPS states etc. where the mass equals half, respectively a quarter of the

supersymmetric central charge. It has been shown that the Reisnner-Nordstr¨om black hole in a four-dimensional AdS background (in the near-extremal limit) has a N = 2 supersymmetric limit. First Romans [40] showed that one of the necessary conditions for a geometry to preserve N = 2 supersymmetry is satisfied by the Reisnner-Nordstr¨om black hole, after which Klemm and Nozawa [41] showed that the condition that is satisfied is sufficient for the setup the preserve N = 2 supersymmetry. In a N = 2 supersymmetric theory there are by definition two charges. Even though the setup we discuss in this thesis contains only one charge we believe that it can be extended to a supersymmetric setup. For a more extensive review on the supersymmetry of black hole geometries in curved spacetimes see Klemm and Nozawa [41].

3.5 Asymptotics of the Reissner-Nordstr¨

om solution

Before we turn to the topic of wormholes, it will be convenient to explore some limits of the Reissner-Nordstr¨om black hole solution in AdS. More specifically we will look for the near-extremal, near-horizon limit of equation 3.4. First recall that there is a horizon at radius r when f (r) = 0. From equation 3.4 we see that this is equivalent to

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2M GN = r +

r2_e r +

r3

`2. (3.17)

Secondly recall that the black hole is extremal when f (r) has a double zero on the horizon. Therefore the horizon, denoted with r+, satisfies

0= ∂! rf (r)|r=r+ = r2₊− r2 e+ 3r4+/`2 r3 + . (3.18)

It follows that when we consider an extremal black hole, f has a double zero at

r2₊= ` 2 6 −1 + r 1 + 12r 2 e `2 ! . (3.19)

However we would like to know the zeros of f in the near-extremal limit. In the near-extremal limit f has two zeros given by r1 = ¯r(1 + ) and r2= ¯r(1 − ), with ¯r equal to r+ as given in

equation 3.19. We can now approximate the two remaining zeros by setting the constant and cubic term of

r2f (r) − (r − ¯r)2(r − r3)(r − r4) (3.20)

to zero and solving for r3,4. This leads to the following zeros

r3 = −¯r + r −1 − 2r¯2 `2, and r4= −¯r − r −1 − 2r¯2 `2. (3.21)

Using the above we can approximate f in the near-extremal limit by

f (r) = 1 r2_`2(r − r1)(r − r2)(r − r3)(r − r4) = 1 `2 r − ¯r r 2 −r¯ 2 r (r + ¯r)2+ `2+ 2¯r2. (3.22)

The temperature in the near-extremal limit is given by

1 +¯r_`22

2π¯r2 + O(

2_), _(3.23)

where we have used 3.11 and applied it to equation 3.22. In the near-extremal case the near-horizon metric is to leading order in ¯r−r_¯_r given by

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ds2= r¯ 2 C(¯r) − (ρ2− 1)dt2+ dρ 2 ρ2_{− 1} + ¯r2(dθ2+ sin2(θ)dφ2) , (3.24) where C(r) = 1 `2(r − r3)(r − r4) . = 6r ` 2 + 1, (3.25) and ρ = r − ¯r ¯r , t = C(¯r) τ ¯ r = 2π¯rT τ, (3.26)

This geometry is can be matched to the Reissner-Nordstr¨om geometry as given in equation 3.4 in the limit

r − ¯r ¯

r 1.

The geometry of the near-extremal near-horizon limit can be recognised as AdS2× S2. This can

be understood by the fact that as the black hole approaches extremality the throat becomes infinitely long. It follows that in the near-extremal limit the black hole geometry can be described as follows. Far away from the black hole the geometry will be approximately that of AdS4. Closer to the black hole the geometry is given by equation 3.4, while the throat of the

black hole develops an AdS2× S2 geometry. The different limits of the geometry are shown in

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AdS2× S2

AdS4

Figure 3.2: Schematic form of the different limits of the near-extremal Reissner-Nordstr¨om black hole in an AdS4background. At large radii there is an AdS4 geometry, while in the throat an AdS2× S2 geometry develops.

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4 Wormholes in flat spacetime

In this Chapter we will review Maldacena et al. [7], on which the idea of this thesis is based. Maldacena et al. [7] presents a wormhole solution in a four-dimensional Minkowski background. The wormhole geometry is a solution to the Einstein-Maxwell action as given in equation 3.1 with the cosmological constant Λ set to zero (to consider asymptotically flat spacetimes) and a matter term in the action containing a massless spin-1₂ field given by

SM =

Z

d4x√gi¯χ( // ∇ − i /A)χ, (4.1)

which is identical to the matter action we will consider in the rest of this thesis. The solution Maldacena et al. [7] finds is a so-called long traversable wormhole, which means that it takes longer to travel through the wormhole than it takes to travel through the ambient space. This ensures that there is no violation of causality. The solution is a four-dimensional Minkowski spacetime with two oppositely charged near-extremal Reissner-Nordstr¨om black holes, whose throats are connected by a geometry that is a deformation of AdS2× S2, with the AdS in

global coordinates. This setup has no horizons or singularities, and thus corresponds to a traversable wormhole.

The wormhole geometry is made up of three approximations that have overlapping regions of validity, in which they can be patched together by matching conditions. The three ap-proximations are flat space far away from the black holes, the near-horizon geometry of the near-extremal black holes in the wormhole mouths, and the wormhole geometry in the throat. The different regions are shown in Figure 4.1.

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A magnetically charged black hole permits an electromagnetic field in the spherical directions, as was shown in Section 3.2. The massless charged fermions (with charge one) moving along the field lines then give rise to a series of Landau levels. For a fermionic field on the sphere the energy of the lowest Landau level is zero and has degeneracy q (we assume the fermionic field to carry unit charge). Each of the spherical modes with zero energy gives rise to an effective massless two-dimensional field in the time and radial direction. Effectively we thus have two-dimensional massless fermions traveling along the magnetic field lines. The setup consists of two of these black holes with opposite charge q at distance d of one another. In this setup there are electromagnetic field lines connecting the two black holes such that the effectively massless two dimensional travel between the two black holes. The dynamics of the gauge field will give the fermions a subleading mass for large q, which is the case that is considered.

Let us now imagine that there is a throat connecting the black hole mouths, as pictured in Figure 4.1, and let us approximate the geometry in the throat to be as follows

ds2 = r_e2 −(ρ2+ 1)dt2+ dρ 2 ρ2_{+ 1}+ dθ 2_{+ sin}2_(θ)dφ2 . (4.2)

For large ρ this geometry can be matched onto the black hole geometry given by equation 3.4 (with ` → ∞) by the following identifications1

t = τ α, ρ = α(r − re) r2 e , for 1 ρ, r − re re 1, 1 α re , (4.3)

where α is an integration constant that determines the ratio between the time scales in the throat and in the black hole geometry. Moreover α gives an estimation of the value of ρ for which the throat geometry starts to divert from the wormhole solution. This cutoff is given by ρcutoff∼ _rα_e. The effectively massless fermions then travel along a circle between the black

holes and through the wormhole. The length L of the circle in the limit α d is equal to L = πα. The Casimir effect then gives rise to a negative energy

ECasimir= −

1 6 q

α. (4.4)

Furthermore, because the geometry in the throat is not flat, we have to take into account the contribution of the conformal anomaly to the total energy of the configuration. The conformal anomaly contribution to the energy equals

Econformal=

q 24

π

L, (4.5)

1_{In this setup the radius of the extremal black hole is equal to r}

e, and the black hole mouth opens up at

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so that the total energy of the configuration is E = q 24 π L − 1 6 q α = − q 8α. (4.6)

This energy could be thought of as though it is generated by the following stress tensor

Ttt = − q 8π 1 4πr2 e , and Tρρ= Ttt (ρ2_{+ 1)}2. (4.7)

With this negative contribution to the stress tensor we can solve the Einstein equations, with the following metric ansatz in the throat

ds2= r_e2 −(ρ2+ 1 + γ(ρ))dt2+ dρ 2 ρ2_{+ 1 + γ(ρ)} + (1 + ψ(ρ))(dθ 2_{+ sin}2_(θ)dφ2₎ . (4.8)

The ρρ component of the Einstein equation then leads to the following differential equation for ψ ρψ0− ψ + ζ 1 + ρ2 = 0, (4.9) with solution ψ(ρ) = ζ(1 + ρ arctan(ρ)), where ζ := q 8π 8πGN 4πr2 e . (4.10)

It turns out that the solution for γ is of higher order and will not be needed in order to match the geometry in the throat to the black hole geometry to leading order. The matching will be given by

r − re

re

= πζ

4 ρ. (4.11)

Furthermore the value of the constant α can be found by noting that

r − re r2 e dτ = ρdt, so that α = dτ dt = 4re πζ = 16 r3_e qGN . (4.12)

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From the matching we can see that one can connect the blue and red regions in Figure 4.1, and we see that the red wormhole region is valid up to ρ ∼ _rα

e. Therefore through this matching

we have found the wormhole solution!

Since the black holes have opposite magnetic charge there is an attractive electromagnetic force between them. In order to keep the setup stable one needs to consider the wormhole mouths to rotate to prevent them from collapsing into one another. This rotation however produces a positive energy contribution which needs to be small compared to the wormhole energy in order for the wormhole to remain a solution to the Einstein equations. This implies that d should be sufficiently large. At large charge d q2, this gives the following bound

p GNq

5 3 d.

Furthermore the rotation has effects on the trajectory of the fermions, causes gravitational radiation and causes radiation of the gauge field. Fortunately, the effects that cause the displacement of the trajectory of the fermions are of order 1_q, and are small at large q. It turns out that of the radiation effects, the radiation due to the gauge field dominates. Because of the radiation, the lifetime of the setup is finite, and can be calculated to be of order _q2d_G3

N.

Also one could wonder about the thermodynamic stability of the wormhole in the rotating setup. It turns out that for d <√GNq3, the wormhole state is metastable. For larger values of

d the quantum gravity effects in the throat have to be taken into account in order to study the thermodynamic stability of the setup. Finally it is noted that the setup can be embedded into the standard model.

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5 Strategy for constructing a traversable

wormhole

The remainder of this thesis will focus on the construction of an eternal traversable wormhole in a four-dimensional AdS background. The goal is to find a wormhole geometry with no horizons or singularities that is asymptotically AdS4. The geometry will be split into two regions:

far away from the wormhole the geometry will be that of a four-dimensional near-extremal Reissner-Nordstr¨om black hole in AdS4, while the throat will be described by a deformed

AdS2 × S2 geometry. The action that the wormhole geometry is a solution to consists of

the Einstein-Hilbert action with a negative cosmological constant, a U (1) gauge field, and a massless fermionic field charged under the gauge field. By coupling the asymptotic boundaries on either side of the wormhole through a non-local coupling of the massless fermions a negative stress tensor is created. We have to consider three contributions to the stress tensor, the electromagnetic contribution coming from the gauge field, the negative contribution originating from the non-local coupling of the fermionic field at the boundaries, and the vacuum contribution of the fermionic field. We can argue that by choosing suitable boundary conditions this vacuum contribution to the stress tensor vanishes. With these contributions to the stress tensor we can solve the semiclassical Einstein equations. To do so we use the fact that the electromagnetic contribution to the stress tensor will dominate at large radii, so that the geometry in that region is that of the near-extremal Reissner-Nordstr¨om black hole, while at smaller radii the contribution from the non-local coupling starts to contribute to the geometry. In this region the geometry will be that of the deformed AdS2× S2. There exists a

limit where we can match the two geometries. Once we have the matching we have the entire wormhole solution.

The construction is organised as follows. In Chapter 6 we discuss the geometry in which we will calculate the stress tensor contributions, and find suitable boundary conditions for the fermionic field. Furthermore the non-local coupling between the asymptotic boundary is introduced, and the solutions to the equations of motion for the fermionic field in the presence of this non-local coupling is given. In Chapter 7 we use the solutions of the equations of motion for the fermionic field to quantise, and find its propagators. Hereafter the different contributions to the stress tensor are discussed in Chapter 8, where the propagators are used to calculate the negative contribution to the stress tensor originating from the non-local coupling. Finally in Chapter 9 we solve the Einstein equations with the stress tensor contributions found in Chapter 8, and we discuss the matching between the solutions in the different regions.

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6 Curved spacetime setup

Finally we are ready to start constructing our traversable wormhole in four-dimensional Anti-de Sitter. During this construction we will be using various ideas introduced in the earlier chapters. Moreover we will be following Maldacena et al. [7] for many of the elements of the construction. The action we will be considering consists of an Einstein-Hilbert term with a cosmological constant that we will choose to be negative, the kinetic term for a U (1) gauge field with coupling constant g, and a massless Dirac field coupled to the gauge field

S = Z d4x√g 1 16πGN (R − 2Λ) − 1 4g2F 2_{+ i¯} / χ /∇χ) , (6.1)

where ∇ := D(e)− iA. As in Section 2.3 the cosmological constant is given by Λ = −_`32 (since

d = 3). Furthermore we consider the gauge field to be given by A = q₂cos(θ)dφ. The metric that we will use for calculations is the following

ds2 = e2σ(x)(−dt2+ dx2) + R2(x)(dθ2+ sin2(θ)dφ2). (6.2)

Furthermore we define a vielbein basis

e1 = eσdt, e2 = eσdx, e3= Rdθ, e4= R sin(θ)dφ; (6.3)

with corresponding spin connection (components not connected to the ones given below by symmetry are equal to zero)

ω12= σ0dt, ω32= R0e−σdθ, ω42= R0sin(θ)e−σdφ, ω43= cos(θ)dφ. (6.4)

The prime corresponds to taking a derivative with respect to x. This metric is connected to the near-extremal, near-horizon metric of the Reissner-Nordstr¨om black hole as given in equation 3.24 by identifying ρ = tan(x)1, and by choosing

σ(x) = 1 2log ¯ r2 C(¯r) 1 cos2_(x) , and R2(x) = ¯r. (6.5) 1

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We will write the four dimensional spinor χ as a tensor product of two bispinors in the following way χ = e σ(x) 2 R(x)ψ(x, t) ⊗ η(θ, φ). (6.6)

In this representation of the Spin(1, 3) group, the gamma matrices are given by

γ1= iσx⊗ 1, γ2= σy⊗ 1, γ3= σz⊗ σx, γ4 = σz⊗ σy, (6.7)

similar to equation 2.21.

6.1 Equations of motion

From the action it is clear that the equation of motion is given by

/

∇χ = 0. (6.8)

We would however like to know the equation of motion explicitly in terms of the components of the spinor fields. One can show that

γ1eµ₁Dµ= e−σ iσx∂t+ σy σ0 2 ⊗ 1, γ2eµ₂Dµ= e−σ∂x σy⊗ 1, γ3eµ₃Dµ= ∂θ σz R ⊗ σx+ e −σ1 2 R0 Rσy ⊗ 1, γ4eµ₄Dµ= σz R ⊗ σy ∂φ sin(θ) + σx 2 cot(θ) + e−σ1 2 R0 R σy⊗ 1, so that / ∇ =e−σ iσx∂t+ σy ∂x+ 1 2σ 0 +R 0 R ⊗ 1 + σz R ⊗ σy ∂φ− iAφ sin(θ) + σx ∂θ+ 1 2cot(θ) . (6.9)

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0 = /∇χ = e −3 2σ R (iσx∂t+ σy∂x) ψ ⊗ η + e −σ 2 R2 σzψ ⊗ σy ∂φ− iAφ sin(θ) + σx ∂θ+ 1 2cot(θ) η.

We can split the equation as follows

e−32σ R (iσx∂t+ σy∂x) ψ ⊗ η = − λ, e−σ2 R2 σzψ ⊗ σy ∂φ− iAφ sin(θ) + σx ∂θ+ 1 2cot(θ) η =λ.

Here λ is the energy corresponding to the energy eigenstates on the sphere. These energy eigenstates are called Landau levels [42]. For a fermionic field on the sphere the energy of the lowest Landau level is zero and has degeneracy q (we assume the fermionic field to carry charge 1). The equations of motion corresponding to the lowest Landau level (so λ = 0) are

(iσx∂t+ σy∂x) ψ = 0, (6.10) σy ∂φ− iAφ sin(θ) + σx ∂θ+ 1 2cot(θ) η = 0. (6.11)

Equation 6.11 can be solved exactly. Let q > 0, η =η+ η−

and define η±:=P_m∈Zη±m with

η₋m:= cm sin θ 2 j−m cos θ 2 j+m eimφ, and η₊m= 0, (6.12)

where j = q−1₂ , and cm a constant given by

cm = Γ(1 + j + m)Γ(1 + j − m) Γ(2 + 2j) −1 2 .

This constant is chosen such that

X

m∈Z

Z

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We see that σy ∂φ− iAφ sin(θ) + σx ∂θ+ 1 2cot(x) η = X m∈Z   −i∂φ−iAφ sin(θ) + ∂θ+ 1 2cot(θ) ηm− i∂φ−iAφ sin(θ) + ∂θ+ 1 2cot(θ) η₊m  .

Inserting equation 6.12 then gives

−i∂φ− iAφ sin(θ) + ∂θ+ 1 2cot(x) η−m = m sin(θ) − q − 1 2 cot(θ) + j − m 2 cot θ 2 − j + m 2 tan θ 2 η₋m= 0,

so that equation 6.12 is indeed a solution to the equation of motion for η. Next we consider the equations of motion for ψ =ψ+

ψ−

. From equation 6.10 we see that the equation of motion is

(∂t− ∂x) ψ− (∂t+ ∂x) ψ+ =0 0 . (6.13)

In order to solve the equation of motion for ψ, we first need to understand what boundary conditions are appropriate for holographic spacetimes.

6.2 Boundary conditions

In order to find suitable boundary conditions for the matter fields we will first write the matter part of the action in terms of ψ+ and ψ−. After this we will vary the action to find a

suitable boundary action that imposes physical boundary conditions, and write it in terms of the projection of the fermionic field onto the eigenspace of the gamma matrix corresponding to the holographic coordinate γ2. This is necessary because these projections are the physical objects that “live” on the boundary. Finally we will check that these boundary conditions are compatible with conservation of energy and charge of the spacetime. Recall the matter part of the action in equation 6.1

SM =

Z

d4x√gi ¯χ /∇χ. (6.14)

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¯ χ = χ†γ1 = (ψ†⊗ η†)(iσx⊗ 1) = i ψ−† ψ † + ⊗ η†.

Therefore, by taking η to be on-shell we obtain

SM = Z d4x√gi ¯χ( /D_(e)− i /A)χ = − Z d4x√g ψ†₋ ψ₊† ⊗ η†( /D(e)− i /A) ψ+ ψ− ⊗ η = − Z d4x√gψ₊†∂xψ++ ψ+†∂tψ+− ψ†−∂xψ−+ ψ−†∂tψ− η†η. (6.15)

So that by varying ψ+ and ψ−, and using the equations of motion for ψ− and ψ+ we get

δSM = − Z d4x√g δψ₊†∂xψ++ δψ+†∂tψ+− δψ†−∂xψ−+ δψ†−∂tψ− η†η − Z d4x√g ψ†₊∂xδψ++ ψ†+∂tδψ+− ψ−†∂xδψ−+ ψ−†∂tδψ− η†η = − Z d4x√gψ₊†∂xδψ++ ψ+†∂tδψ+− ψ†−∂xδψ−+ ψ†−∂tδψ− η†η = − Z d4x√g ∂x(ψ+†δψ+− ψ†−δψ−) + ∂t(ψ+†δψ++ ψ−†δψ−) η†η + Z d4x√g(∂xψ+†δψ+− ∂xψ−†δψ−+ ∂tψ†+δψ++ ∂tψ−†δψ−)η†η = − Z d4x√g∂x(ψ+†δψ+− ψ−†δψ−) + ∂t(ψ+†δψ++ ψ−†δψ−) η†η. (6.16)

Now considering only the part proportional to ∂x, and using Gauss’s theorem, we see that

δSM ⊃ − Z d4x√g∂x(ψ†+δψ+− ψ−†δψ−)η†η = − Z ∂ d3x√γ(ψ₊†δψ+− ψ†−δψ−)η†η = Z ∂ d3x√γ ¯Ψ−δΨ+− ¯Ψ+δΨ− . (6.17)

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In the final two lines of the calculation the integration is over the boundary surface of the spacetime, γ is the induced metric on the boundary, and in the final line we switched from component notation to a notation in terms of the projections of χ onto the eigenspaces of γ2. Let us now consider some possible contributions to the boundary action and their variation. First we consider Sbdy±= Z ∂ d3x√γ ¯Ψ+Ψ−, (6.18) with variation δSbdy±= Z ∂ d3x√γ δ ¯Ψ+Ψ−+ ¯Ψ+δΨ− .

Note that if we add this boundary term to the matter action, the variation of the action will be only be proportional to δ ¯Ψ+ and δΨ+, so that setting δΨ+|∂ = 0 makes the variation of

the action vanish. Similarly, if we consider

Sbdy∓= −

Z

∂

d3x√γ ¯Ψ−Ψ+, (6.19)

setting δΨ−|∂ = 0 makes the variation of the action vanish.

Note that when one considers a black hole in an Anti-de Sitter background one can only access the AdS boundary on one side of the Einstein-Rosen bridge, as can be seen from the Penrose diagram given in Figure 3.1. In order to access the “other side” of the black hole, we consider the so-called two-sided case in which instead of x ∈ [0,π₂], with the AdS boundary at x = π₂ we have x ∈ [−π₂,π₂] and boundaries at x = ±π₂. We will be denoting the boundaries with left (L) and right (R), where the right (left) boundary is located at x = π₂ (x = −π₂). Another argument for considering the two-sided setup is that we expect that the throat of the wormhole will have a geometry that is approximately AdS2× S2, which has two disconnected

boundaries. Since we have two disconnected boundaries we are allowed to choose boundary conditions on each of the boundaries independently. For reasons discussed in Section 8.2 we will be considering the case where we choose the boundary action as given in equation 6.18 on one side and the “opposite” boundary action given in equation 6.19 on the other side. We will call such boundary conditions mixed. Let us focus on the boundary action given by

Sbdy =

Z

∂

d3x√γ ¯ΨR₊Ψ₋R+ ¯ΨL₋ΨL₊ . (6.20)

Here one might have expected a relative minus sign. However, there is a second relative minus sign from the fact that the outward normal on the left is oriented precisely opposite from the

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one on the right, so that overall there is a no sign in between the right and left terms. This leads to the boundary condition δΨR₊= 0, which is equivalent to

ψR₊− iψR₋= cR; (6.21)

and δΨL−= 0, which leads to

ψ₊L+ iψ−L= cL, (6.22)

where for simplicity we pick cL,R= 0. As a check let us see that these boundary conditions are such that no charge or energy leaks out at the boundary. First we consider the current. The global U (1) symmetry we consider is χ 7→ eiαχ, with α ∈ R. Under this transformation δχ = eiαχ − χ = (1 + iα + O(α2))χ − χ = iαχ + O(α2). Therefore δχ_δα = iχ. The current equals

Jµ= ∂L ∂(∂µχ)

δχ

δα = − ¯χγ

µ_χ,

where L is the Lagrangian density in equation 6.14. Requiring that the x component of the current equals zero implies

0= J! 2= −χ†γ1γ2χ = ψ†σzψ ⊗ η†η ⇐⇒ |ψ+|2 = |ψ−|2. (6.23)

Taking a derivative with respect to time implies that

0= ∂! tψ+†ψ++ ψ+†∂tψ+− ∂tψ†−ψ−− ψ−†∂tψ−. (6.24)

The stress tensor in curved spacetime equals [43]

Tµν = i 2 χγ¯ (µ∇ν)χ − (∇(µχ)γ¯ ν)χ =iηab 2 ¯ χea_(µγb∇_ν)χ − (∇(µχ)e¯ aν)γ b_χ_. _(6.25)

The energy flux on the boundary is given by the tx component of the energy momentum tensor. We see that T12 equals

T12= − ieσ 2 χ†γ1(γ1∇2− γ2∇1)χ + ∇1χ†γ1γ2χ − ∇2χ†γ1γ1χ

Constructing a transversable wormhole in &gt;2 dimensions