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Instabilities of Spinning Higher-Dimensional Black Holes

Steven Lageveen

July 23, 2020

Studentnumber 11965703

Supervisor dr. Jácome Armas

Second Examiner dr. Alejandra Castro Anich

Course Bachelor Project Physics & Astronomy

Size 15 EC

Conducted between 30-03-2020 and 21-07-2020 Submission Date 21-07-2020

Faculteit der Natuurwetenschappen, Wiskunde en Informatica UvA & Faculties Faculteit der Bètawetenschappen VU

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Samenvatting

In het begin van de afgelopen eeuw kwam Einstein met een revolutionaire nieuwe theorie over zwaartekracht genaamd de algemene relativiteitstheorie. Deze theorie weerlegde het eeuwenoude theorie van Newton en stelde dat zwaartekracht helemaal geen kracht was, maar het resultaat van een gekromde ruimtetijd. Dit heeft vergaande gevolgen en zorgt er onder andere voor dat tijd en lengte niet meer universeel zijn, maar afhangen van de waarnemer. Een nog gekkere voorspelling van de theorie is dat er gebieden kunnen zijn met zo’n sterke zwaartekracht dat zelfs licht niet meer kan ontsnappen. Dergelijke gebieden worden zwarte gaten genoemd en zijn nu, bijna honderd jaar later, nog steeds een groot mysterie. Dit betekent dat er op het moment nog veel onderzoek wordt gedaan naar hoe zwarte gaten precies werken. Maar zwarte gaten in de normale vier dimensies, één tijd en drie ruimte, vinden we natuurlijk nog niet interes-sant genoeg, dus in dit project kijken we naar hoe zwarte gaten zich gedragen in meer dan vier dimensies. De methode die we daarvoor gebruiken berust op veel van dezelfde ideeën als algemene relativiteitstheorie zelf en we kunnen een hoop eigenschappen van deze mysterieuze objecten zo bepalen, dat er bijvoorbeeld bepaalde onstabiele staten gevormd worden waarbij het zwarte gat uit elkaar valt.

Abstract

This thesis aims to follow and explain the analysis performed in the paper "Instabilities of Thin Black Rings: Closing the Gap" [1]. To this end we first gain the necessary background by studying the essentials of manifolds and general relativity, before moving onto the intricacies of the blackfold approach and its perturbations. From the study of general relativity we obtain that spacetime is a manifold in which matter has to obey the Einstein equations. In the context of the blackfold approach these equations receive a different form known as the blackfold equations which have their own specific set of solutions. This approach describes black branes as being wrapped around a submanifold that is embedded in a background spacetime and here we examine two such branes that also satisfy the blackfold equations. The first of these solutions is something known as a boosted black string, and the second a spinning black ring. They are both higher dimensional objects where there is a spherical black hole at each point along a line, only in the case of the black ring this line is curved into a closed ring. We analyze these solutions by forming small perturbations in them and then investigate the resulting modes. We find that for both cases there are two stable elastic modes and two hydrodynamic modes. For the black string one of the hydrodynamic modes will always attenuate while the other will disrupt the solution and is associated with the Gregory-Laflamme instability. For the black ring we find the same results, only differing in that we now find that the expression for the hydrodynamic modes changes such that the attenuation and instability only occur for modes beyond a critical point.

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Contents

Contents 2 1 Introduction 3 2 Background Theory 4 2.1 Manifolds . . . 4 2.1.1 Atlases . . . 4 2.1.2 Tensors . . . 5 2.1.3 Curvature . . . 6 2.1.4 Embedding . . . 7 2.2 General Relativity . . . 8 3 Blackfold approach 9 3.1 Perturbations . . . 11

4 Black Rings & Strings 11 4.1 Black Strings . . . 11 4.2 Black Rings . . . 12 5 Discussion 13 6 Conclusion 13 7 Acknowledgements 14 References 15

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1

Introduction

Ever since the faithful day where Isaac Newton was hit by a falling apple, gravity has been a major constituent of physics and is still the subject of much research. Over the centuries there have been many different theories on the nature of gravity, some more plausible than others, but the current settled upon theory is the theory of general relativity that was developed by Einstein in the early 1900’s. This theory of gravity makes many bizarre predictions about our universe, such as time dilation and length contrac-tion or the lensing of light, but the prediccontrac-tion that is by far the strangest, and least understood, is the prediction of the existence of black holes. These black holes are objects in spacetime containing singular-ities from which not even light can escape. Needless to say, black holes have attracted much interest in the scientific community ever since their first prediction and have often been the subject of heated debate. Most research on black holes has been contained inside the typical four dimensional spacetime, but some of the more recent studies have instead looked at the characteristics of higher dimensional black holes. This may sound like a bizarre and maybe even useless thing to study, but it actually does have quite a bit of merit. One reason it is useful is that some modern theories, like string theory, require there to be more than four dimensions to spacetime so it would be useful to know how black holes would be affected by these extra dimensions. It might even soon be possible to create such higher dimensional black holes right here on earth in our super colliders. If we do find them we could learn much about the universe, as the current theoretical predictions expect these black holes to behave differently in higher dimensions [2]. Besides these concrete uses, the study of these objects is useful even if spacetime does not turn out to contain higher dimensions, because their study is an exercise that can award us with a greater depth in our understanding of the theory of gravity in and of itself.

In the theory of general relativity, spacetime is described as being a manifold which are the mathe-matical description of spaces that have a more complicated structure, or curvature, than the Euclidean spaces that we are familiar with. It is this curvature then that is the cause of the effects of gravity in the theory, and it can take rather complicated forms, especially when they concern black holes. We will use an approximation known as the blackfold approach to describe black holes, or black branes as they are known when generalized to higher dimensions, for this allows us to examine their properties analytically. The approach consists of wrapping a black brane onto a submanifold that is embedded in a background spacetime. What all of this actually entails is something that should become clear further on in the paper. Specifically, the focus of this paper will be on examining perturbations of solutions to the blackfold method from which we discover different modes and an instability in these black branes. We use the blackfold approach to first describe a boosted black string and look at its solutions and instabilities, and then we do the same for the case of a spinning black ring which we can then compare. Both of these are higher dimensional black holes and using the blackfold approach to describe black strings will actually be a way of approximating the ring as a series of strings.

As mentioned in the abstract, this paper is based on the paper "Instabilities of Thin Black Rings: Closing the Gap" [1] and will therefore closely follow it when perturbing the black strings and rings. The blackfold approach will in a similar manner closely follow the outline given in the paper "Essentials of Blackfold Dynamics" [3] and much of the explanation of manifolds and general relativity is based on the book "Spacetime and Geometry, An Introduction to General Relativity" [4].

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2

Background Theory

Before we can get into the blackfold approach itself, we must first cover some exciting background theory. As the blackfold approach lies within the theory of general relativity, it is first necessary to get acquainted with the concepts and mechanics of that theory before we approach the blackfold approach. But even before we can begin to understand general relativity, we first have to arm ourselves with the mathematics of manifolds as these can be seen as the foundations of general relativity and they are also crucial to the methods developed later on.

2.1

Manifolds

Manifolds, briefly stated, are how we mathematically describe curved spaces. There are a few caveats to this as we shall see shortly, but for our use cases understanding them will tell us all of what it means for a space to be curved. Moving our laws of physics to such spaces changes them, so it is important to know exactly how they change and how that changes the ways in which they need to be approached.

In concise, mathematical terms a C∞ manifold, also known as a differentiable manifold, is a set M

together with a maximal atlas. In layman’s terms, this statement says that a manifold is a set of points that locally (in a small region) looks like flat space, Rn, and that each of these local sections are smoothly

sewn together so that there are no gaps or sharp edges. The idea of a manifold comes from observing that the effect of more complicated topologies often turn out to be negligible on a small scale. A good example of this is the surface of a sphere: if we zoom in close enough, things like parallel lines will seem to always stay parallel, as we would expect from flat space, and it becomes hard to distinguish the space from a regular plane. But as we zoom out to larger scales, the curvature will start to effect the observer, and coming back to the example of parallel lines, we see that they can cross each other for we see that lines parallel at the equator cross at the poles.

This more intuitive understanding is surely useful, but in order to be able to work with manifolds we need to translate it back to the concise definition so that we understand all of its components. The first step in this process will be to understand exactly what the atlas part of the definition encompasses. 2.1.1 Atlases

As we will see shortly, atlases in mathematics are very similar to the colloquial use of the term and is defined as a collection of charts or coordinate systems with a few specific properties. However, in order to formalize the concept of atlases we first need to define a few basic mathematical tools revolving around maps and sets:

• Map: assigns a set of points from a set A to another set B.

• Injective: a map that assigns a maximum of one element from A per point in B.

• Surjective: a map for which each element of B has at least one point from A mapped into it. • Invertible: a map for which an inverse can be defined that is both injective and surjective. • Open: a set of points in Rn that do not contain their edge (e.g. open interval on a line).

• Smooth: a map for which the functions it consists of are infinitely differentiable, denoted C∞.

The last definition, smoothness, sounds rather abstract, but it is just a concise and general way of stating that a line contains no gaps or sharp edges. It ensures that the manifolds and atlases are well behaved and removes most of the annoying edge cases.

Having now defined these tools we can turn back to formulating the concept of atlases. Atlases are a collection of open sets Ua, that satisfy the following two conditions:

1. The union of the open sets must equal the complete set M.

2. The open sets must be smoothly sewn together, or in other terms: if Ua∪Ub6= ∅, then the (φa◦φ−1b )

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The first of these means that if we take all of the points from the individual open sets and add them together in a new set, then that set is the same as the complete set M. A maximal atlas is then simply an atlas that contains all possible open sets for the given set M. The second requirement is quite subtle mathematically speaking, but intuitively it means that if two open sets overlap, then they have to agree of the space in the overlap so we do not get any discontinuities.

These open sets have a few different names, such as charts, or coordinate systems, but we will mostly stick with open sets here, yet later switch to coordinate systems when putting them to practice as this terminology is more familiar to non-mathematicians.

With the definition of a maximal atlas now clear, we have yet again encountered a new unknown that needs clarification: open sets. An open set is a subset of a manifold, for which there exists an invertible map to Rn and for which the image can be constructed by a collection of open balls. Open

balls can be seen as an extension of open intervals to higher dimension, so they are all the points inside an n-dimensional sphere while excluding the boundary of that sphere. One characteristic of these open sets is that it is usually not possible to describe an entire manifold using just a single open set, but in practice it is often enough to choose one open set and deal with the exceptions individually when necessary. Nevertheless, this requirement of using multiple open sets gives reason to describe the spaces and their contents in a way that is independent of any particular open set. The way that we will achieve such description, will be by formulating relations using constructs known as tensors.

2.1.2 Tensors

Tensors are the solution to describing things in a coordinate independent way and are therefore a very powerful and useful tool when working with manifolds. Because flat space itself is a manifold and many other spaces can be seen as manifolds, this property means that tensors pop up everywhere in physics, indeed one would be hard-pressed to find a field where they are not used.

A simple definition for tensors, and the one we will use, is that they are a multilinear map of a collection of dual vectors and vectors to R. Breaking this statement down we have that the notion of a collection is given by multiplying the vectors and dual vectors using the Cartesian product and a multilinear map means that the map is linear in each of its argument or for our purposes it suffices to think of it as a map. Because dual vectors are linear maps from vectors to R and vice versa we can see that a tensor itself is a collection of vectors and dual vectors and the number of each determines its rank, (#vectors, #dualvectors).

The most important question however is what dual vectors and vectors are. To answer this we will first lackadaisically define dual vectors in terms of vectors by saying that they are linear maps from vectors to R. This means that all of the care has to be put into the definition of vectors. For anyone at all familiar with physics this might seem an easy task as we are usually told that vectors are simply arrows pointing from one point to another. Sadly this view only works in flat spaces and has to be abandoned when we move to more general contexts. In curved spaces vectors are only well defined at the point where they are located, meaning that each point in a manifold has a vector space associated with it that contains all the possible vectors at that point. These spaces are called tangent spaces and the vectors within them are known as tangent vectors. Vectors then, are defined to be the derivatives of lines passing through a point and the tangent space is formed by doing this for all possible curves through a point. As there are often infinitely many vectors in a tangent space, it is useful to describe them in terms of a combination of basis vectors. A basis for a tangent space is a collection of vectors from which any other vector in the space can be created by combining and scaling them, except for the vectors in the basis themselves. The natural choice for basis vectors are the partial derivatives associated with a chosen coordinate system, and they also give a good indication of how vectors transform under coordinate transformation, namely by the chain rule:

Vµ= ∂x µ ∂xνV ν (Components) ∂µ= ∂xν ∂xµ∂ν (Basis).

Where µ denotes the components in the new coordinate system. The term component can be confusing as it can refer both to the individual elements denoted by the super- or subscripts and to the factors in front of the basis vectors or dual vectors. Most of the time the difference should be clear from the context, but it is unfortunate nonetheless. It is important to note that the matrix given by the derivative factor in the component transformation is the inverse of the matrix given by the factor in the basis transformation,

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meaning that the vector when taken in its entirety remains unchanged under a coordinate transformation. The same turns out to be true for dual vectors, only their constituents transform exactly oppositely to those of the vectors:

ωµ= ∂xν ∂xµων (Components) dxµ=∂x µ ∂xνdxν (Basis).

Here the basis dual vectors are given by dxµand are known as differentials or gradients. Adding these

transformation rules together for each dual vector and vector then gives the transformation rules for a tensor, which also does not change under coordinate transformations. This may not be entirely surpris-ing, as it was the exact feature we required of these tensors in the first place, yet is good to confirm that it is indeed the case.

So tensors are very important because they allow us to write down our laws of physics in curved spaces, but that is not their only use case. There are a few tensors that are related to curvature, and our description of it, that will be useful later on, so we will define them in the next subsection.

2.1.3 Curvature

Metric Tensor

Perhaps the most important tensor when it comes to general relativity is the metric tensor. This is a (0, 2) symmetric tensor that also has an inverse. Symmetric just means that its indices can be swapped while the tensor remains positive. When working with these metrics they are most often used simply to raise or lower indices, or in other terms, to turn vectors into dual vectors and vice versa. Anyone who has worked with special relativity is probably familiar with the Minkowski metric tensor which looks like ηµν = {−1, 1, 1, 1}, but Euclidean space also has a metric, one that looks like {1, 1, 1, 1} in four

dimensions. Actually, a manifold can have many expressions for the metric tensor, as the components depend on the coordinate system being used which also means that they can look arbitrarily complicated by using a sufficiently complex system. More generally however, the metric tensor is tied to the curvature of a given space and one can always find a coordinate system in which the components are constant if the space is flat and vice versa. However, the metric tensor does not entirely describe the curvature and in order to do so we need to introduce a symbol called a connection.

Connection

A manifold can have many different connections, but luckily the existence of a metric is enough to define a unique connection of which the curvature can be considered to be that of the metric. These connections are what we use in general relativity and are called Christoffel connections:

Γλµν = 1 2g

λσ(∂

µgνσ+ ∂νgσµ− ∂σgµν).

Connections look like tensors, but they are in fact different as they do not transform in the same way so we call them symbols instead. The connection is a way of relating vectors between tangent spaces to each other, but their fundamental use is to define the covariant derivative, ∇µ:

∇µVν= ∂µVν+ ΓνµσVσ.

Which can be seen as the generalization of the divergence vector to curved spaces. These derivatives are important because "normal" derivatives do not generalize nicely to curved spaces for they depend on a specific coordinate system, as we can see with the transformation of basis vectors. This means that the operation of a normal derivative on a tensor does not necessarily return a tensor, which is what the term with the connection corrects for, in fact we define the properties of the connection from this. The covariant derivatives are used in various places, one of particular note is in the definition of a geodesic which is the notion of a shortest path in a curved space, and another will be in the blackfold approach to denote a subset of the Einstein equations.

Riemann tensor

This tensor is of particular note as it determines everything about the curvature of a space. We will not go into the technicalities of its definition, but will note that it is given by the expression:

σµν= ∂µΓρνσ− ∂νΓρµσ+ Γ ρ µλΓ λ νσ− Γ ρ νλΓ λ µσ.

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For our uses the Riemann tensor’s most important use is to define the Ricci tensor. This tensor is given by a contraction (a summation over two indices) of the Riemann tensor:

Rµν= Rλµλν,

and the contraction of this tensor in turn is known as the Ricci scalar: R = gµνRµν.

The Ricci tensor and scalar will both be used in the definition of the Einstein equations that we will see later on, but the Ricci scalar has its own special use in the blackfold approach. The scalar is inversely proportional to something called the radius of curvature, and therefore defines the scale of the curvature of a manifold which is useful when employing approximations. The radius of curvature of a manifold is roughly speaking how curved a space is so if we again examine the surface of a sphere as a manifold, then this radius of curvature would be the radius of the sphere on which the surface lies. In more complicated manifolds we can imagine there to be "bumpy" regions with strong curvature that smoothen out when looking at larger scales.

2.1.4 Embedding

One of the key components of the blackfold approach consists of the notion of embedding a smaller man-ifold into a larger background one. This can be thought of as assigning points inside the larger manman-ifold to points of another manifold, that often consists of fewer dimensions. Let us make this a bit more intuitive by examining the surface of a sphere which is a two dimensional manifold, called S2, that in

essence "lives" inside another three dimensional manifold R3. In general it is always possible to embed

a given manifold into Euclidean space R2n that is twice the dimension of the manifold, but it is often

possible to embed manifolds in smaller Euclidean spaces. These embeddings can help with visualizing curved spaces, but can also be very misleading as the process of embedding a manifold can actually lead to novel curvature in the manifold. To see exactly how this works we first have to take a look at how mapping between manifolds works in the general case.

Given two arbitrary manifolds, M and N with coordinates xµand ya, we can define a map φ : M 7→ N

together with a random function in N, f : N 7→ R. We can then move f to M, so to speak, by composing φ and f, φ∗f = (f ◦ φ)(f ◦ φ). This is quite useful and common, and is known as the pullback of f. The reverse of this would be a push forward from M to N, but this is not in general possible, but it is however possible for vectors, which cannot be pulled back, and is given by: (φ∗V )f = V (φ∗f ). Writing

this out in terms of components and basis vectors in N gives: (φ∗V )a∂af = Vµ

∂ya

∂xµ∂af,

where we make use of the chain rule for the partial derivative. This in fact the generalization of the coordinate transformation of vectors as those can be seen as maps from M 7→ M.

Similar but oppositely, dual vectors can only be pulled back and have the same matrix but are con-tracted over a different index. This means that a tensor consisting purely of vectors can be pushed forward, and one consisting solely of dual vectors can be pulled back. We will use this feature later on for a few different things, but a general useful tensor to pull back is the background metric tensor, which is then known as the induced metric tensor γab.

Getting back to the idea of embedding submanifolds we know that this process consists of a map from N to M with N being a subset of M and thus the points mapping to the "same" points on a larger manifold. In order for this to be an embedding instead of an immersion this map has to be smooth and invertible, meaning that N is a nice connected set of points in M. In the context of an embedded manifold the induced metric will receive the meaning of describing the added geometry induced on the manifold by the embedding. This metric also defines its own Christoffel symbol, which in turn defines a covariant derivative, denoted ∇a (note the latin subscript), which is used within the submanifold.

If we push the induced metric forward out of the submanifold, we get something called the first fundamental form hµν, which is different from the background metric as some information was lost during

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and it also acts as a projector that projects tensors from the background space to the submanifold. The process of pulling back the induced metric to produce hµν has the form

hµν = ∂aXµ∂bXνγab.

As hµν projects tensors onto the submanifold, we might expect that there exists a complementary tensor

⊥µν that projects tensors onto the remaining directions which lie orthogonal to the submanifold. This

tensor does indeed exist and the combination of the two gives back the background tensor: gµν =

hµν+ ⊥µν, in fact this will be how we define ⊥µν.

Taking covariant derivatives of tensors inside the submanifold with ∇µ is only well defined along

directions tangent to the submanifold, after pulling these tensors back of course, so it is useful to project the derivative onto the submanifold using h ν

µ or:

∇µ= hµν∇ν.

This derivative still has both tangential and orthogonal components, but for the tangential tensors it is functionally the same as taking the derivative ∇a of the tensor in the submanifold before pulling it back.

This concludes all the mathematical tools that we will need for the blackfold approach, so all that remains is the small detail of what the theory of general relativity, on which all of this is based, entails and how to use it.

2.2

General Relativity

The theory of general relativity is a very interesting and beautiful theory that deserves to be explained in great depth and detail, however, we will not do so in this paper and instead will take the quicker route of only covering what we need to in order to use the blackfold approach. That being said, it is worthwhile to take a quick look at the principles and considerations that led Albert Einstein to this theory in the first place.

The main idea that led to the development of general relativity is called the principle of equivalence. This principle comes in a few different forms, the original form now known as the weak equivalence prin-ciple, the Einstein equivalence principle on which general relativity is based, and the strong equivalence principle which is designed to also include areas of physics besides gravitation. The weak form of the principle states that the inertial mass and the gravitational mass are equal, or in other words that the constant in Newton’s F = mia is the same as the one in his law of gravity Fg = mg∇Φ. The more

popular example of the implications of this principle is that it is impossible for an observer in a closed box to tell the difference between gravity and constant acceleration by observing free falling particles. This only works for small enough regions though, due to the direction of the force for a point source of gravity being different than that of constant acceleration. After the development of the theory of special relativity where mass is a manifestation of energy and momentum, it became logical to broaden the prin-ciple to state that in the box it would be impossible to tell the difference between constant acceleration and gravity by any means. Lastly, the strong equivalence principle extends this to non-gravitational laws of physics. An example of this last principle being broken but not the others would be if gravitational binding energy was unevenly distributed among the inertial and gravitational mass of an object.

All of this brings forth the idea that we should consider unaccelerated objects to be freely moving in a gravitational field, which means that gravity is not a force as forces are things that cause acceleration. This means that we can no longer define a universal inertial frame, and must instead use locally inertial frames that only encompass small enough regions. This is very reminiscent of manifolds in which locally the space is seemingly flat, where here locally laws reduce to those of special relativity. Spacetime then indeed turns out to be a manifold according to general relativity with special relativity being the laws in a flat spacetime, and the main result of the theory is a set of equations relating the curvature of spacetime to the distribution of energy and momentum. These equations are known as the Einstein equations and take the form:

Rµν−1

2Rgµν = 8πGTµν.

The terms on the left consist of the Ricci tensor and scalar, and the metric tensor, and constitute the curvature part of the equations. This side of the equation is also often denoted as the Einstein tensor Gµν. The right hand side of the equation consist of some constants and a momentum energy tensor,

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or stress tensor. This stress tensor is often assumed to be that of a perfect fluid, but adding correction terms to it can significantly influence the resulting solutions, as it can be shown that dark matter could be explained by such corrections [5]. In this paper the stress tensor, when not zero in vacuum, is taken to be that of a viscous fluid, which is given by:

Tµν = uµuν+ P Pµν− 2ησµν− ζθPµν.

3

Blackfold approach

As noted before, the main focus of this paper is to use the blackfold approach to describe black branes and then to perturb the solutions. For this purpose it is then useful to explain what exactly this blackfold approach encompasses and how to use it.

In general relativity it is possible to split the degrees of freedom into long and short wavelength com-ponents, or gµν = {g

(long)

µν , gµν(short)}. The long degrees of freedom live in a ’far zone’, at a distance r  r0,

and describe the background geometry in which the black branes live. The short degrees of freedom then live in the near zone where r  R with R a length scale that is much larger then the size scale of the blackfold, r0.

In general, black p-branes have a metric in D = 3 + p + n spacetime that looks like ds2= −  1 − r n 0 rn  dt2+ dr 2 1 −rn0 rn + r2dΩ2n+1+ p X i=0 dzi2. (1) If we use the coordinates σa = (t, zi) and we have a velocity field ua with uaubη

ab = −1, we can use a

Lorentz transformation to boost the above solution to get: ds2=  ηab+ rn 0 rnuaub  dσadσb+ dr 2 1 −rn0 rn + r2dΩ2n+1.

The remaining directions not covered by the p-brane are collectively denoted as X⊥, which means that

collective coordinates of the blackfold are given by

φ(σa) = {X⊥(σa), r0(σa), ui(σa)}. (2)

For the long wavelength effective theory it is necessary to allow slow variations in ∂X⊥, ln r

0,and ui

along the p-brane over the length scale R. It is important to note that if R → ∞ then the metric above would be exact, but if R is finite then we need to include correction terms that are generally proportional to the derivatives of ln r0, ua or γab, but here we’ll only be working with this first order approximation

and will ignore those correction terms. The γabterm here is the induced metric obtained from embedding

the smaller manifold into the larger one and it replaces ηab in the metric denoted earlier. The smaller,

submanifold will from here on out be referred to as the worldvolume for the sake of clarity. We will also take the background metric to be the Minkowski metric which is given by:

ds2= −dt2+ D−1 X i=1 dzi2 , (3)

and for the induced metric we have:

γab= g(long)µν ∂aXµ∂bXν.

All of this sets up the blackfolds and their geometry, but it is important to remember that they have to obey the Einstein equations. So circling back we get that the effective equations for our blackfolds in the far region look like:

R(long)µν − 1 2R (long) gµν(long)= 8πGT eff µν, where Teff

µν is the effective stress tensor for which, according to [6], we can use the quasilocal stress tensor

as defined in that paper.

The Einstein equations may seem like they will be messy to work with, but luckily we can use that according to [6] the Einstein equations with an index orthogonal to the boundary of the black brane are

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equivalent to the of conservation of the quasilocal stress tensor. This means that conserving the stress tensor is the same as solving a subset of the Einstein equations intrinsic to the worldvolume. The stress tensor for a boosted black brane is given by:

Tab= Ω(n+1) 16πGr n 0(σ) nu aub − ηab ,

which we can identify with a perfect fluid by taking:  = Ω(n+1) 16πG(n + 1)r n 0, P = − 1 n + 1 .

Which simplifies the equations somewhat and makes them more comprehensible. As mentioned before we will instead be using the stress tensor of a viscous fluid which we get by adding one of the correction terms, but the general idea is the same. With this stress tensor we see that we will have a generalization of equation ?? to the larger manifold which splits up into this equation paired with an extrinsic one. The first step in doing so is to push forward the stress tensor so that it is now written as:

Tµν = ∂aXµ∂bXνTab,

then it can be shown [3] that the stress tensor satisfies the relation hρν∇µTµν = ∂bXρ∇aTab.

Where hρ

νis the first fundamental form as we defined earlier, and ∇µis the projected covariant derivative

that we also defined in section 2.1.4.

Lastly, we will need one other important tensor which is called the extrinsic curvature tensor, given by:

Kµνρ = hµσ∇νhσρ.

This tensor measures how the vectors orthogonal to the worldvolume transform along its surface, and its trace gives the mean curvature vector Kρ.

Now, having all the tensors and derivatives in their appropriate forms, we can continue with finding the blackfold equations. First we have that the general extrinsic dynamics of a brane can be worked out in terms of a stress tensor [7] obeying:

⊥ρ µT

µν = 0,

but because the effective stress tensor for a blackfold is derived from general relativity and assuming that spacetime diffeomorphism invariance hold, the stress tensor has to obey:

∇µTµρ = 0,

which is the same as equation ?? only now written in global coordinates. This can be decomposed along parallel and orthogonal directions, or intrinsic and extrinsic components, by doing:

∇µTµρ = ∇µ(Tµνhνρ) = Tµν∇µhνρ+ h ρ ν ∇µTµν = Tµνhνρ∇µhσρ+ h ρ ν ∇µTµν = TµνKµνρ+ ∂bXρ∇aTab.

Because the entire thing has to equal zero, as noted earlier, we get that both terms themselves have to equal zero resulting in the blackfold equations:

TµνKµνρ= 0 (extrinsic) (4) ∇aTab= 0 (intrinsic). (5)

These will be the important equations governing our black brane and will be our focus in the remainder of the paper.

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3.1

Perturbations

Having set up the equations used in the blackfold approach it now follows to perturb some of the variables in the general case before we examine the effects that these perturbations will have in individual cases of a boosted black string and a spinning black ring.

The variables to be perturbed together with their perturbations are:  →  + δ ua → ua+ δua Xµ ⊥(σ) → X µ ⊥(σ) + δX µ ⊥(σ). (6)

Filling these perturbations into the stress tensor given in equation 2.2 then gives a perturbation in it of the form: δTab=  n + 1  (nuaub− γab)δ  + 2nu (aδub)− δγab  − 2ηδσab− ζ(∇ aδua− ua∇a(KρδXρ))Pab.

Plugging the perturbations into the blackfold equations, 4 and 5, then give: ∇aδTab− Tcb∇c(KρδXρ) − 2Tac∇a(KbcρδX ρ ⊥) + T acb K ρ ac ⊥ρλδX⊥λ = 0 (7) δTabKabi+ Tabniµ∇a∇bδXµ = 0, (8)

Which are quite messy but luckily they clean up a bit in the specific cases we will consider.

4

Black Rings & Strings

Having covered all the necessary theory, it is now useful to apply the blackfold approach to a boosted black string, and also to a black ring. A black string can be seen as a line where at each point there exists an n-sphere that is a black hole and so it has a geometry of R × Sn+1. A black ring is quite similar and

is a ring with at each point an n-sphere, or S1× Sn+1. In fact we can see that a black ring can locally be

considered to be a black string in the limit where the radius R of the ring goes to infinity. It is useful to first examine the case of the black string as it is almost exact, whereas our approach for the black ring will consist of zooming in on small sections where the curvature is manageable and we can describe them as boosted black strings, before proceeding to perturb them.

4.1

Black Strings

In the case of a boosted black string we have the background Minkowski metric given by equation 3 in which we embed a black string with the metric given in equation 1 with p = 1, σa= {t, z}, and with u

a

being boosted in the z direction. The string is embedded via the map: Xµ=      t µ = t z µ = 1 0 µ = 2, . . . , D − 1 .

This embedding then leads to an induced metric of the form: ds2= −dτ2+ dz2.

The black string is just a straight line so it does not have any extrinsic curvature and K i

ab = 0. The

solution has to conform to the blackfold equations and we can perturb it just as in equations 6. We then assume that the solution will take the shape of plane waves:

δ(σ) = δei(−ωτ +kz) δuz(σ) = δuzei(−ωτ +kz) δXµ(σ) = δXµei(−ωτ +kz). (9) Because there is no extrinsic curvature, the perturbed blackfold equations, 7 and 8, reduce to:

∇aδTab= 0

Tabniµ∇a∇bδX µ ⊥= 0.

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If we then demand that the determinant of these equations equations equals zero, we get two hydrody-namic solutions, due to the extrinsic equations, and two elastic solutions, due to the intrinsic equations. At ideal order the solutions for ω are:

ω1,2(k) = nβ ±√n + 1(1 − β2) n + 1 − β2 k (elastic) (10) ω3,4(k) = β(n + 2) ± i√n + 1(1 − β2) n + 1 + β2 k (hydrodynamic). (11)

It is noteworthy that the two elastic modes will always be positive and real, and thus represent stable solutions that oscillate. It is however possible that higher order corrections would produce instabilities in these modes by having a small positive complex component that would still increase exponentially, or they could attenuate the modes by adding a small negative complex component. The fourth mode however, has a negative complex component and will therefore dampen over time, but it is otherwise stable. This negative part means that this solution will always dampen out and be stable regardless of any higher order corrections. Lastly the third mode has a positive complex component and will increase exponentially disrupting the state. This instability is associated with the well known Gregory-Laflamme instability of black strings [8].

4.2

Black Rings

With the simpler case of the black string covered, it is time to take a look at the spinning black rings. In the ultra spinning regime the black ring will have a thin donut-like shape where the radius of the of the S1 geometry is much larger than the Sn+1 geometry. In this limit we can zoom in on a local section of the black ring and find that it looks like a black string that is boosted along its z direction. This allows for the blackfold approach to work in a way where we take the exact solution of the black string at a local point and then add to it perturbatively to describe the entire ring.

Using this approach we again consider a Minkowski background method, only this time it is more useful to write three of the spatial coordinates in polar terms:

ds2= −dt2+ dr2+ r2dφ2+

D−3

X

i=1

dxi2.

We then again have the metric 1, only now we have that σa = {t, ϕ}and a R22instead of a dz for the

black ring and we can proceed to embed it using: Xµ=      τ µ = t R µ = r ϕ µ = φ .

This leads us to an induced metric of the form:

ds2= −dτ2+ R2dϕ2,

with ϕ ranging between 0 and 2π. We can see here that if we define z = Rϕ and take the limit R → ∞, we will get back the geometry of the boosted black string. This is to be expected as increasing the radius of the sphere causes a line segment to resemble a straight line and so taking this to infinity flattens out all of the curvature.

We can perturb the blackfold equations and assume the solution consist of plane waves again so that the perturbations are given by:

δ(σ) = δei(−ωτ +kRϕ) δuz(σ) = δuzei(−ωτ +kRϕ) δR(σ) = δRei(−ωτ +kRϕ). (12) Due to the spatial topology of the ring being closed, k has to be quantized as m = kR with a discrete m and the extrinsic curvature does not vanish anymore, instead it is now given by: K = −1

R. The elastic

modes will then remain unchanged from the boosted black string, but the hydrodynamic modes now become: ω3,4= √ n + 1 (n2+ 2n + 2) R  (n + 2)m ±p2 (n2+ 2n + 2) − n2m2. (13)

We can then see that they both only have a complex part if m > √2 n

n2+ 2n + 2, which will again

always be stable for mode ω4 and unstable for mode ω3. Here the third mode is again a description of

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5

Discussion

As mentioned earlier, we used some approximation to obtain these results, so there are some inherent inaccuracies to the entire process. In the blackfold approach the near zone metric as listed in 1 is of itself not a solution to the Einstein equations and requires corrections containing terms with gradients of ln r0,

ua, and γ

ab. Some of these are contained in the viscosity term of the stress tensor but it remains the case

that the metric is an approximation. The use of perturbation theory to examine the instability of the blackfolds is itself an approximation that is only accurate for small wavelengths and require more terms as the perturbations get larger. These approximations mean that the blackfold approach is only applicable to black rings in the thin limit where R  r0, but solutions for the boosted black strings are a bit more

exact. However, for both cases we have that the elastic modes obtain an imaginary component at higher orders, but for the hydrodynamic modes the higher order corrections will not change the qualitative results and can only differ by the exact growth. When comparing with different analysis methods [1] we can also see that our blackfold approach for the black ring breaks down for m ≤ n+2

2 but this region is also

where the growth rate of the Gregory-Laflamme instability increases and it is this region that interests us. To this end then the blackfold method proves to be an accurate and useful option for analyzing black holes.

6

Conclusion

At the start of the paper we set out to first understand the blackfold solutions for black holes, and then proceed to perturb these solutions to examine any potential instabilities they could contain. The blackfold approach is based on the idea of splitting up black holes into a short and a long wavelength component and then describing the latter region as a submanifold embedded in a background spacetime. In this scenario, the stress tensor of the black branes would have to comply to the conservation equation ∇µTµρ= 0, which can be split up into an intrinsic component, ∇aTab= 0, and an extrinsic component

TµνK ρ

µν = 0, which together are known as the blackfold equations. We considered two known solutions

to these equations, that of a boosted black string and that of a spinning black ring. We first perturbed the boosted black string solution which led to a description of two elastic modes, given in 10, and two hydrodynamic modes, given in 11. The elastic modes represent stable solutions in this order, as they are alway positive and real, yet may receive complex components from higher order corrections that would grow exponentially. For the hydrodynamic modes, the fourth mode will always attenuate as its complex component is negative, but the third mode describes the Gregory-Laflamme instability that increases exponentially due to the positive complex component.

The process for the black rings consisted of the same steps and the elastic modes found before remain unchanged, but the hydrodynamic modes are now described by the equations given in 13. This still produces an instability the third mode and attenuates in the fourth, only now the complex component only arises when m > √2

n

n2+ 2n + 2.

With this approach then we have found a novel way of describing the growth and onset of the Gregory-Laflamme instability, which for the black ring is the first analytical description of the growth rate of this instability as a function of m, and also the first for the onset of the instability [1].

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7

Acknowledgements

This project has been a tremendously interesting and fun journey to take, and I would therefore sincerely like to thank my supervisor Jacome Armas for his guidance throughout. I may not have reached out very often, but when I did, your patience and help was always really useful, and it is very appreciated.

Aside from thanking Jacome it would be a great amiss if I were to finish this paper without also thanking my friend Matthijs Pool, who has patiently put up with my many small questions regarding the nature of vectors, tensors, and all things manifold. It has been very helpful to me, and has saved Jacome much headache I am sure, so thank you.

Lastly then, I would like to thank another friend Annika van den Brink, as we both performed our projects at the ITFA under Jacome and she has been vital to getting me through the management side of the project. Without you I would probably still be stuck sending that first email.

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References

[1] J. Armas and E. Parisini, “Instabilities of thin black rings: closing the gap,” Journal of High Energy Physics, vol. 2019, Apr 2019.

[2] R. Emparan and H. S. Reall, “Black holes in higher dimensions,” Living Reviews in Relativity, vol. 11, Sep 2008.

[3] R. Emparan, T. Harmark, V. Niarchose, and N. A. Obers, “Essentials of blackfold dynamics,” Journal of High Energy Physics, vol. 2010, Mar 2010.

[4] S. Carroll, Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley, 2004. [5] R. Baier, S. Lahiri, and P. Romatschke, “Ricci cosmology,” 2019.

[6] J. D. Brown and J. W. York, “Quasilocal energy and conserved charges derived from the gravitational action,” Physical Review D, vol. 47, p. 1407–1419, Feb 1993.

[7] B. Carter, “Essentials of classical brane dynamics,” 2000.

[8] R. Gregory and R. Laflamme, “Black strings and p-branes are unstable,” Physical Review Letters, vol. 70, p. 2837–2840, May 1993.

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