A Double-Headed Centaur in a Black de Sitter Universe

(1)

Universiteit van Amsterdam

Institute for Theoretical Physics

A Double-Headed Centaur

1 in a Black de Sitter

Universe

MSc Thesis

Author: Nikolaos Petropoulos

Supervisor: Prof. Dr. Erik Verlinde

Co-Supervisor: Dr. Dionysios Anninos

Examiner: Dr. Jan Pieter van der Schaar

October 25, 2018

1_{Centaurs were creatures in Ancient Greek Mythology. Their upper body was of a human and their lower body and legs}

(2)

(3)

Abstract

We study some aspects of black holes in de Sitter (dS) spacetime with a particular focus on the Nariai and Rotating Nariai geometries. These are maximal black hole geometries on cosmological spacetimes.

We first review spacetime structure of dS. We study different coordinate systems which give rise to different symmetries and causality in dS spacetime.

We then proceed with black hole geometries in dS. We make thermodynamic and rotodynamic analysis and compare them with Minkowski black holes highlighting some essential differences which give rise to potentially special microscopic behaviours. Our focal point are the Nariai and Rotating Nariai geometries which have an Sn−2_{fibration over a dS}

2 base space.

Moreover, we make some analysis on quantum field theory in dS spacetimes. We consider pure dS and the Rotating Nariai geometries. We comment on the structure of the vacua that are mainly used on dS, focusing on the so-called α−vacua and the Euclidean one.

Furthermore, we explore a direction towards examining possible microscopic properties of the cos-mological horizon. This has already been studied for the black hole horizons, mainly via the AdS/CFT correspondence. We employ the Kaluza-Klein decomposition from higher to lower dimensional effective theories. Then we establish a connection with the so-called Centaur geometries.

Finally, we give some insights on constructing more geometrically involved spacetimes, namely the double-headed Centaurs.

(4)

Acknowledgements

There are many people I would like to thank for helping me finish this project and my master’s track in general. Their support was both intellectual and moral and each one was a unique piece to completing the puzzle.

Firstly, I would like to thank my family, for even though they are not familiar with physics, their help all these years made all of this possible.

Furthermore, I would like to thank my supervisor professor Erik Verlinde. I would like to thank him for the guidance he provided me and all the enlightening talks we had during my thesis project. Each talk was instructive, inspiring and intuitive, not to mention extremely fun.

Also, I cannot express in words my gratitude for my co-supervisor Dr. Dionysios Anninos. He guided me through an amazing journey in physics, for either general topics or specific about my thesis. He was always excited to talk physics and answer questions, trivial or not, at any time. The word co-supervisor is not halfway to capture his role in my career; most likely the word mentor would better describe him.

Moreover, I would like to thank all the PhDs and colleagues for instructive talks. Most of all, my colleague Stratos Pateloudis for the endless physics and philosophical conversations, being for fun or not, and for his help with writing my project.

Finally, I would like to thank my friends and especially my partner Sofia, for immense moral support, picking me up when I fell down and helping me find my focus when I lost it.

(5)

(6)

List of Figures

2.1 The embedded hyperboloid depicting d-dimensional de Sitter space into a (d+1)-dimensional flat space. The dotted line represents an Sd−1_{. Figure source: [}₁_]. _{. . . .} ₇ 2.2 Penrose-Carter diagram of global dS. Lines represent Sd−1and points Sd−2. . . 9

2.3 Causal past O− and future O+ of an observer sitting at the North Pole. We can see that he/she can’t receive signals past the diagonal line (null) of O− and send signals past the diagonal of O+_. _{. . . .} ₁₀ 2.4 Penrose diagram for Eddington-Finkelstein coordinate system, with the arrows

represent-ing the time flow which comes from the Killrepresent-ing vector K = ∂/∂t. I± rely at r = ∞, the Southern and Northern causal diamonds in region r ∈ [0, 1] and the cosmological horizons at r = 1. Figure source: [1]. . . 11

2.5 This is the diagram of Kruskal coordinates. The arrows point to the directions of increasing U, V . The poles lie at U V = −1, I± at U V = 1 and the horizons at U = 0 and V = 0. Figure source: [1]. . . 11

2.6 This diagram depicts τ = constant spacelike flat (d-1)-dimensional slices, which extend all the way to I−. . . 13

3.1 Penrose diagram of maximally extended Schwarzschild geometry. K = ∂/∂t is timelike in regions I, IV and spacelike in regions II, III. The gravitational singularity is depicted by wavy lines at r = 0. Null infinities lie at I±, spacelike infinity at i0, future and past timelike infinities at P and Q respectively and the black hole event horizons at r+. . . 15

3.2 Penrose diagram of maximally extended Schwarzschild-de Sitter geometry. It has one more horizon than the classical Schwarzschild geometry. Here also K = ∂/∂t is timelike in regions I and spacelike in regions II and III. Notice that there is an infinite sequence of gravitational singularities and spacelike infinities, so K changes signature an infinite number of times. The dots on the left and right side of the diagram imply repeated copies of it (the "beads" of the string). We can get rid of these "beads on a string" by imposing periodicity conditions on spacetime and keeping one singularity and one spacelike infinity (since as is the case in flat Schwarzschild geometry, there’s probably no physical meaning associated to the copied region). . . 16

(9)

3.3 Penrose diagram of extremal SdS spacetime. In figure (a) we see the black hole spacetime. As the observer comes from r > 3M to cross the horizon, he/she will either end up at the black hole singularity at r = 0 or to one of the asymptotic points P. On figure (b) is the inverse spacetime, namely the white hole one, in which we have an observer emanating from the white hole singularity at r = 0 or asymptotic P and will cross the horizon at r = 3M to either reach r = ∞ at I+ or the asymptotic points Q. The three dots at the right and left side of the diagram mean that the diagram is repeated infinitely in the horizontal direction, just like in non extremal SdS. Figure source: [20]. . . 20

3.4 Penrose diagram for dS2(x S2) geometry. The black hole event horizon lies at ρ = 0 and the cosmological horizon at ρ = β. The intermediate region is where the observer lies at the "sandwiched" area. Each point in this diagram if we include the whole dS2x S2 is an S2, while τ = constant spacelike slices have an S2 x S1topology. . . 21

3.5 Penrose-Carter diagram of Reissner-Nordström-de Sitter (RNdS) spacetime, for M = |Q|. The red dotted lines represent a single (τ, R) chart, where r is the old radial coordinate and R the cosmological one. Figure source: [2]. . . 23

3.6 Time evolution of the cosmological contracting chart (τ−, R). We can see the forming of the singularity at (τ−, R) = (0, ∞), "grinding" the space down to R = 0 as τ− → ∞. Figure source: [2]. . . 24

3.7 On the left, we have the Penrose diagram of the gluing of the two cosmological charts (τ±, R) at the point of the cosmological horizon r = rc. The extended diagram repeats infinitely many times in the horizontal direction just like before. On the right, there is the embedding of the throat-like 3-cylindrical geometry with topology S2 _{x R, with half the} cylinder being covered by the expanding chart and the other half by the contracting one. Figure source: [2]. . . 25

3.8 Penrose-Carter diagram of extended Kerr spacetime along θ = 0. The dashed (red) lines represent the ring singularity, which lies at r = 0 and θ = π₂, but it is usual to represent it in the θ = 0 diagram. We can see the presence of two horizons, namely r− which is the inner (Cauchy) horizon and r+the outer horizon with the area r ∈ (r−, r+) being the ergosphere. Future and past conformal infinities lie at I± and there’s another infinity at r = −∞ if one passes through the ring singularity to the other side, which is a naked singularity for r < 0. Figure source: [2]. . . 26

3.9 Penrose diagram of global maximally extended Kerr-de Sitter (KdS) spacetime. We can see here the presence of another horizon at rc(cosmological) comparing to Kerr geometry. Furthermore, the singularity for r < 0 is no longer naked since there’s the cosmological horizon at −rn. Note that as in the SdS geometry, there’s an infinite extension of repeated spacetimes which with the proper periodic conditions can be identified as one KdS universe. Also the spacelike infinities r = −∞ and r = ∞ are disconnected. Figure source: [2]. . . . 27

3.10 Allowed parameter space of (r, α) at extremality T = 0. We see that the parameter curves are bounded by the Extremal Kerr (blue line → r+(α)) and the Rotating Nariai (orange line → r+(α) ∼ rc(α)); the point where the two meet is called the ultracold limit which is the case where r−= r+= rc. The extremal value of α is αmax=

p

7 − 4√3. We can see again that we cannot put arbitrarily large rotating black holes in dS. . . 29

(10)

3.11 This is the plot of J (Ω) and J0(Ω) for the flat Kerr-Newman black hole, with Q = 1 in the domain Ω ∈ h−_√5 2, 5 √ 2 i

. As is shown in the plot the blue curves correspond to the J+(Ω), J+0 (Ω) branch and the orange ones to J−(Ω), J−0 (Ω). The J+ branch is the Kerr-Newman branch which behaves like J+ ∼ Ω close to the origin and is connected to the extremal Reissner-Nordström black hole at Ω = 0. The J− branch is connected to the electrically neutral extremal Kerr black hole. We can see that the first derivatives J_±0(Ω) diverge also at the end of the domain of Ω, thus indicating a second order phase transition (since δ G

δΩ → ∞ at these points.) . . . 35 3.12 These are the plots of the J+( ˜ΩH) for ˜ΩH ∈ [−3, 3] and ˜ΩH ∈ [−13, 13] in order to have

a more broad visualization of the function closer and further away from the origin for the case of the extremally rotating KdS black hole. The left plot is for ˜ΩH ∈ [−3, 3] and the right one for ˜ΩH∈ [−13, 13]. We can see that for ˜ΩH ≥ 1 we have J+∝Ω˜1H

. As is obvious from the plots, the domain of ˜ΩH extends to ˜ΩH∈ R∗ as in flat Kerr; the range though of the function ˜ΩH(α) ∈ (−∞, −1] ∪ [1, ∞) for the extremally rotating branch so we can see that starts rotating from a larger minimum rotation than its flat counterpart; the linear branch near the origin is covered by the Rotating Nariai case as we show below). . . 39

3.13 These are the plots of J+( ˜ΩH) again for the same domain of ˜ΩH as the ones in fig.3.12. The left plot is for the domain ˜ΩH ∈ [−3, 3] and the right one for ˜ΩH ∈ [−13, 13]. We can see that in sharp contrast to the case of the extremal Kerr and extremal flat KN black hole, here we have no divergences J+0 and hence no signs of obvious phase transitions. A first interpretation could be that the presence of the cosmological horizon "smoothens" out any unusual behaviour of the black hole, since its size is restricted by the Rotating Nariai black hole limit.. . . 39

3.14 These are the plots of J+( ˜ΩC) = J+( ˜ΩH) for the Rotating Nariai black hole, with the left one being in the domain ˜ΩC ∈ [−1, 1] which is also the range of the function ˜ΩC(α) and again for a broader perspective on the right the plot for ˜ΩC∈ [−13, 13]. We can see that for ˜ΩC∈ [−1, 1] we have the Rotating Nariai black hole and for ˜Ω ∈ (−∞, −1] ∩ [1, ∞) we have the extremally rotating KdS, as advertised above. At the points | ˜Ω| = 1 we have the Ultracold Geometry. We see that when the black hole becomes extremely large, its values are bounded from above and below as in the flat KN case. . . 41

3.15 Here are the plots of J+0( ˜ΩC) for the Rotating Nariai black hole. Again we see that for ˜ΩC ∈ [−1, 1] we have the Rotating Nariai branch and for ˜Ω ∈ (−∞, −1] ∩ [1, ∞) the extremally rotating KdS hole. Also for this black hole, we see that there are no divergences in the first derivative of J+ in contrast to the flat KN case and so no apparent phase transitions of the black hole or the cosmological horizon. . . 41

5.1 This is a 3D plot of R(2)_(r

c, φ) for values rc∈ [−1, 1] and φ ∈ [−0.5, 0.5] from two different angles, for more clarity. It is obvious that at specific points in parameter space (rc, φ) the curvature changes sign and becomes negative. Of course for the Rotating Nariai we always have positive curvature, but if we have a running dilaton then we can tweak it so we get negative Ricci scalar. Also there is a fine point, where for φ = 0 we have R(2) _{= 2, ∀ r}

(11)

5.2 This is the same plot as before, only in a 2D cut of the 3D plot for specific values of the dilaton φ = 0.5 on the left and φ = −0.5 on the right plot. We can see that the Ricci scalar function is not symmetric with respect to the origin for the two values of the dilaton field. Also, the curvature becomes negative (and singular at a point) in the neighbourhood of rc∼ 0.4 for this specific value of the dilaton. . . 66 5.3 Here is the depiction of AdS space, which is a negatively curved space. It is a maximally

symmetric solution to the Einstein equations with a Λ < 0. holography is understood in AdS space as the duality between a CFT description which lives on the timelike boundary of AdS and quantum gravity description which lives in the bulk (inside the disk), as is seen on the right picture. The left picture is a triangular tesselation of AdS space. The Euclidean picture of global AdS space can be thought of as a (d + 1)−dimensional Poincare disk, which is a higher dimensional realization of the 2D disk on the left.. . . 70

5.4 This is a Euclidean depiction of the A/dS Centaur. On the left side where we have the hemisphere is the cosmological horizon which marks the end of the geometry and on the right side we have the AdS boundary. As we mention later on, the dilaton field evolves in a decreasing fashion from the AdS boundary towards the dS cosmological horizon. Figure source: [52]. . . 71

5.5 An example form of the potential used in [52] for the values of the dilaton φb = 10, φh= −2. We can also see near the origin the IR cutoff scale (l ) which lies at φ = ± ; this is the point of gluing the two geometries together. When the slope of the dilaton potential is decreasing φ ∈ (−2, 0) we have the dS2 part and when it is increasing for φ ∈ (0, 10) we have the AdS2 part. . . 73 5.6 The plots of E(β) (left) and δ S(β) (right) for the dilaton potential of [52] and similar ones

for the domain T ∈ (0, 50). We see that for the cosmological horizon, both the correction to the entropy and the energy are increasing with respect to the temperature of the horizon. 76

5.7 This is the general V (φ) plot for the particular choice made here. s = 1 corresponds to Λ > 0 and s = −1 to Λ < 0. The plot is for the domain φ ∈ (−3, 3); for s = 1 and φ = 1/3 we have the Nariai solution. No such solution exists for s = −1.. . . 77

5.8 Here is the dilatonic potential used in A/dS2 Centaur, but with flipped signs; that is the AdS2/BH Centaur. At φ = φhwe have a black hole horizon instead of a cosmological one. The plot is for the domain φ ∈ (−2, 10). Again we have an IR cutoff scale at φ = ± . . . 78

5.9 On these plots we can see how the dilaton field ˜V (φ) changes with respect to |E | and φ. On the top two plots we have plotted in 3D for two different perspectives for two different plots, with the orange curve corresponding to φb > 0 (A/dS2 Centaur) and the blue one to φb< 0 (AdS2/Black Hole Centaur) and on the bottom one 2D cuts for constant values of E and for φb > 0 (bottom left) and φb < 0 (bottom right). For E = 0 we get the original non-deformed Centaur. We can see that from a value larger than some critical E = Ec we have something like a Centaur spacetime phase transition. This could be probably because since E in d = 2 corresponds to m in d = 4, for large values of angular momentum we have the particle exiting the cosmological horizon and hence this anomaly in the geometry. Analogously, we have similar for the black hole case with the transition probably corresponding to crossing the black hole event horizon. . . 80

(12)

5.10 Here we have the plots of the Ricci scalar R. On the top two 3D plots we see how the spacetime curvature changes with respect to |E | and φ and on the bottom 2D cuts for E = constant values. The orange 3D curves corresponds to the Centaur with a positive bound-ary value for the dilaton (A/dS2 Centaur) and the blue ones for negative (AdS2/Black Hole Centaur) for the Rotating Nariai geometry. Also the bottom left 2D cut is for φb > 0 and the bottom right for φb < 0. We see that the Centaur spacetime phase transition that takes place corresponds to a change to a Centaur which no longer has a change in curvature R, but instead stays constant for both parts of the geometry. . . 81

5.11 Here we have the plots of C[β] with respect to the value of φh. The orange curve corresponds to the charged A/dS2 horizon and the blue to the AdS2/BH. We can see that the specific heat doesn’t distinguish for different electric fields, in general, but it has some anomalies for E = ±2φh

|φh|, corresponding to the Centaur phase transition.. . . 83 A.1 Here are the plots of S(Ω) for each geometry respectively. We have the extremal

Kerr-Newman where the S+ branch corresponds to the extremally charged Kerr-Newman and the S− the extremal electrically neutral Kerr black hole (top left), the Rotating Nariai which we can see that is the continuation of KdS near the origin for Ω ∈ (−1, 1) (top right) and finally we have the extremally rotating KdS black hole (bottom) where the Scbranch corresponds to the cosmological horizon and the S+ to the black hole event horizon; we can see that the S+curve is the same as the extremal electrically neutral Kerr on the top left S− one. But again we see that in the KdS case we it doesn’t diverge near the origin, since the presence of the cosmological horizon induces the Rotating Nariai geometry when the black hole becomes big enough and smoothens out any divergences that we may had in flat space. . . 88

A.2 Here are the plots of M (Ω) for each geometry on the same order as in fig.A.1. Also for the Kerr-Newman case, each branch M+ and M− corresponds to extremally charged Kerr-Newman and extremal neutral Kerr black holes, respectively. We can see that in the case of the KdS we have only one branch this time, since we can associate entropy to the cosmological horizon but it doesn’t have any mass; we have only the extremally rotating KdS black hole mass curve. Again we can see the difference between a rotating black hole in Minkowski and dS spacetimes. In the former, we can have an arbitrarily massive rotating black hole since there are no bounds on how big it can get; spacetime is infinite. On the other hand, in dS the black hole size, and hence its mass, is restricted by the size of the cosmological horizon with the maximal case being the Rotating Nariai spacetime. This is why we don’t have any divergence near the origin of Ω, which the KdS branch is cut and continued by the Rotating Nariai branch near the origin; in sharp contrast we see that the flat Kerr branch diverges near the origin. . . 89

D.1 This is the two-point function for the spinors as a function of time seperation Sb_{(u). The} blue curve corresponds to the full Sb solution, while the orange one to the one for large u which correspond to IR frequencies ω. We can see that in the IR, Sb

IR(u) is a good approximation of the full two-point function. . . 97

(13)

D.2 This is the correlation function diagram, taken intact from [39]. We can see that for small deformation 0 < |z| < 2γ the low energy effective S_IR1 is in good agreement with the full solution for large u (low ω). As |z| ≥ 2γ we can no longer describe interactions with S_IR1 and we need the full solution again.. . . 101

(14)

1

Introduction

We live in a Universe, in which a constant battle between two forces who strive for domination is taking place. These are the forces who want to bring everything together and the forces who want to tear everything apart. The contraction forces team is comprised of Gravity and the expansion team of Dark Energy. There has been a constant epic fight between these two giants and it will probably be eternal.

Furthermore, if we look into the history and probable fate of our Universe, we see that it teaches us in a physical way that "history repeats itself". Let’s take a quick look at the timeline of our Universe, in order to understand both the battle and the cyclic behaviour of our Universe.

So after the Big Bang, it experienced an unimaginably fast expansion due to cosmic inflation1 _which

is driven by a scalar field called inflaton. On this time, we had the Universe approaching a de Sitter geometry; this is a spacetime geometry with maximal symmetry and positive spacetime curvature and a geometry which no single observer can have access to its whole. But inflation lasted for a fraction of time and then ended, thus the Universe entered a reheating phase where gravity would dominate and that would be the start of the complexity, beauty and order we see in the large scale structure today (e.g. planets, galaxies, superclusters, etc). Had inflation still persisted, we probably wouldn’t be able to be here since everything would tear apart and wouldn’t interact to form complex structures that life needs to evolve or even exist for that matter. Now we are in a post-reheating phase, where due to the exit of the inflationary phase we can observe the "imprints" of the quantum fluctuations of the inflaton field.

Current cosmological observations indicate that now we live in a dark energy (produced by the cos-mological constant Λ ∼ 10−52m−2) dominated Universe and at the far future it will have a dark and cold fate. All matter and energy will dilute, except for the dark energy, and nothing will be able to interact

1_{At least this is the most widely accepted interpretation in Cosmology which agrees with the data we observe on the}

(15)

with anything thus resulting in what is called the thermal death of the Universe. If there could be, in this far future an observer, he/she would be surrounded by complete darkness; he/she would only be able to "see" some uniform thermal radiation and some thermal fluctuations of about the temperature T ∼ 10−29K. This makes it at least worthwhile to study spacetimes with positive curvature and the simplest and most symmetric one being de Sitter. The very difficult part, which has also been a very big obstacle in the study of de Sitter spacetime, is the presence of this observer-dependent cosmic horizon; this what we call the Static Patch, rendering the study of is of tremendous physical importance. Naively, our Universe when we look at it through the eyes of an observer seems to be, in some sense, inside a black hole.

Now we know that the Holy Grail of High Energy Theoretical Physics is the quest to a theory of quantum gravity. This theory is needed to explain what happens when quantum effects start being non negligible in gravity; this happens in the center of a black hole and at the Big Bang.

For the black holes we have a pretty concrete understanding through the magnificent duality named the holographic AdS/CFT correspondence. The theory states that we can describe a quantum gravity theory living on the (d + 1)-dimensional bulk of AdS spacetime is equivalent to a Conformal Field Theory living on the d-dimensional boundary; concretely is AdSd+1/CFTd duality. This is a specific type of a

Gauge/Gravity duality, which came from string theory and enabled physicists to probe to the nature of the black holes; it shed light to the darkest objects of the Universe. We know that we can associate an entropy to the black hole horizons through the Bekenstein-Hawking formula:

S = c

3_A

4 G ~ (1.1)

where A : area of the black hole event horizon, c : speed of light, ~ : Planck’s constant, G : Newton’s gravitational constant.

The duality allowed a precise microscopic derivation of the entropy formula from string theory. This allowed us to interpret entropy as a counting of microstates of the black hole, which has the form:

S ∼ ln Ω (1.2)

Since this Gauge/Gravity duality could be a more universal underlying truth of Nature, theorists have tried to apply it also in dS in order to seek for a quantum theory of gravity in our Universe [1], [17], [27], [29], [33], [28], [34], [32], [35] and others. But it is a more tricky business, since dS spacetime has a spacelike boundary at future/past infinity I± and the CFT which is dual to dS, is not that well understood. Furthermore, as mentioned above there is a non-localized cosmic horizon which follows the observer wherever he goes. Attempts to apply holography in the Static Patch and have a Worldline Holography have been made in [24], [57].

Moreover, there have been some attempts lately in [52] realize the dS cosmological horizon as some infrared (IR) limit of a microscopic theory living in AdS space. This attempt was made in order to be able to use the well known holographic duality of AdS/CFT, to probe for the cosmological horizon. Although it may be unphysical, since physical spacetime has no AdS part, it can be a good toy model to understand a little better aspects of dS with the powerful tools AdS/CFT has provided us.

(16)

As Hawking and Gibbons observed [5], the cosmological horizon could have some similar behaviour to a black hole horizon, but different in some other aspects. For example, when things enter the cosmological horizon it becomes smaller, in sharp contrast to what happens to the black hole horizon. Nevertheless, he observed that we could also associate an entropy to the cosmological horizon as well and one way to interpret it as the amount of ignorance of what lies outside of it; larger horizon is more empty inside which means larger ignorance. Another way to interpret it is the dimensionality of the Hilbert space H of quantum dS spacetime. Generally though, there’s no clear understanding on what entropy counts in dS. This is one of the challenges a putative holographic theory for dS should overcome.

All of the above have led me to do my thesis on dS spacetime. We will make a general review of dS spacetime and what black holes in dS could look like. Furthermore, we will make comparisons with black holes in Minkowski and dS spacetime and point out crucial differences. Finally, we will give some future directions that we find interesting towards exploring the nature of the cosmological horizon and black holes in general.

On Chapter 1, we will make an introduction of what is dS spacetime according to the Einstein equations with a positive cosmological constant Λ > 0. We will start by deriving dS as a solution to the equations and then examining the general topological2and causal structure of dS spacetime. This we will achieve by exploring different coordinate patches on dS spacetime, which come from different embeddings of dS, and the so-called Penrose Diagrams, on which we will elaborate later on. Different patches, can have significantly different spacetime topology and symmetries. We will mostly follow [1], [3], [13].

On Chapter 2, we make a general review on black hole with a cosmological horizon spacetimes, that is black holes which have been embedded in an expanding spacetime3_{. As we will show, the presence of}

a cosmological horizon puts severe restrains on the black holes’ size, rotation and mass. This can have a dramatic impact on any microscopic theory that tries to describe black holes in dS. We also mention briefly the black hole geometries in Minkowski spacetime, just to make a comparison. Furthermore, we focus on the Nariai and Rotating Nariai spacetimes, which will be the key point spacetimes of my whole thesis, since they have some very interesting characteristics. These are the largest non-rotating and rotating black holes we can have in an expanding spacetime, respectively. The study of the latter is of particular interest, since the generalization to any dimensional spacetime, has a dS2 part which

remains intact. In the near-horizon limit, they have SL(2, R) x SO(3) and SL(2, R) x U (1) isometries, respectively. Just for reference, most astrophysical black holes we observe in space have SL(2, R) x U (1) near-horizon limit, thus making the study of these symmetries important and physically relevant.

Moreover, we calculate some macroscopical thermo- and roto-dynamical properties and made proper comparisons of (un)charged rotating black holes (Kerr-Newman black holes in different limits) in Minkowski and extremal (rotating) black holes in dS. These will prove to have some very interesting differences, which we make more manifest with plots that encode them. We compare our calculations with the ones made in [39] and draw proper conlcusions. Here we mostly follow [39], [4], [24].

On Chapter 3, we give a brief picture on how a quantum field theory (QFT) might look in dS spacetimes. We make a brief review of QFT in Minkowski space and then extend to curved backgrounds. More specifically, we focus on pure dS and Rotating Nariai spacetimes. In the pure dS analysis, we

2_{By topological, we mean the geometry of the constant time hypersurfaces of each coordinate patch.}

3_{Most often we study the Static Patch description of these black hole spacetimes, and sometimes the Global form may}

(17)

review the different vacua that are chosen in dS spacetime and mostly about the widely used α-vacua. Furthermore, we make a connection between the d = 4 Rotating Nariai geometry and d = 2 dS2with an

E field, as was noted in [3]. As we will also see, in curved backgrounds QFTs are remarkably different, since the choice of a vacuum is very ambiguous. On this chapter, we mainly follow [38],[54], [53] and [37], [25].

On Chapter 4, we use the Kaluza-Klein (KK) decomposition mechanism, to reduce from a 4D theory of gravity to an effective 2D dilaton gravity for the Nariai and dilaton gravity with a gauge field for the Rotating Nariai spacetimes4_{. This is due to the fact that the near-horizon limit of the Nariai is dS}

2 x S2

and the Rotating Nariai is an S2 fibration over dS2, so we can reduce our theory on the spheres. The

dilaton will control the size of the spheres, which tells us how far away we are from the cosmological and/or black hole horizons. It will also play the role of the radial direction in the lower dimensional theory, if we are in the near-extremal limit. Using this and something called the A/dS Centaur geometry in [52], which is having dS2 inside AdS2 spacetime, as an IR realization. We show that by building this

we can probe for the cosmological horizon and black hole horizon in dS, and being able at the same time to use the tools of AdS/CFT. In order to have both horizons at the same Centaur geometry, we need to glue two seperate Centaurs together. One of them will have the cosmological horizon and the other one the black hole one. It will be left as a future direction, to construct this Double-Headed Centaur, which may be AdS2in the UV and encode both the cosmological and black hole event horizons in the IR.

Finally, we draw my conclusions and give further directions to probe and apply AdS/CFT for the nature of the cosmological horizon black hole horizons in dS. To do this for each horizon seperately, is not such a hard task. On the contrary, as will be manifest in the last chapter the task of gluing the two Centaurs to create the double-headed Centaur geometry, will be very involved and far from trivial.

Afterwards, we provide some Appendices in which we include some extra plots and calculations which we didn’t include in the main corpus. Furthermore, we include a toy quantum mechanical model which was derived in [39] and we review inD, just for a reference on what a possible toy model for some features of a black hole might look like.

4_{For the Rotating Nariai case, the rotation in the higher dimensional theory is encoded in a gauge field in the lower one,}

(18)

2

Pure de Sitter spacetime from the Einstein Equations

On this chapter, we make an introduction on pure de Sitter space, the different coordinate patches which makes different symmetries more apparent and their corresponding Penrose Diagrams which make more manifest the causal structure of each coordinate patch. Firstly, we describe the transformations used in the embedding space ((d + 1)−dimensional flat space) in order to get each coordinate system on the embedded one ( d−dimensional de Sitter) and at the same time present the Penrose diagram which cor-responds to each one; the notion of the Penrose diagram will be disambiguated below. On this chapter, we mostly follow [3], [36] and [1].

2.1 A maximally symmetric solution with Λ > 0

Considering the Einstein-Hilbert action endowed with a positive cosmological constant Λ > 0 and no matter fields in d = 4-dimensions, we have (G = c = 1):

SEH = 1 16π Z M d4xp|g| (R − 2Λ) (2.1)

where M is the 4-dimensional manifold over which we integrate our action. Varying this action with respect to the metric:

δSEH = 0 =

δSEH

(19)

we end up with the equations of motion namely the Einstein equations devoid of any matter content (T_µνmatter = 0): Gµν ≡ Rµν− 1 2gµνR = − Λ 8πgµν (2.3)

where we can identify the stress-energy tensor of the vacuum as:

T_µνvac= −Λ

8πgµν (2.4)

By multiplying both sides of the Einstein equation with gµν we get the form of the Ricci scalar R:

R = 4Λ > 0, ∀ Λ > 0 (2.5)

Furthermore, the Riemann tensor Rµνρσ has the form in this case:

Rµνρσ=

1

12(gµρgνσ− gµσ) R (2.6)

This maximally symmetric solution to the Einstein’s equations with Λ > 0 is called the pure de Sitter spacetime solution (or just de Sitter space). It is worth mentioning that the cosmological constant that we observe in our universe through astronomical obervations is found to be of order Λ ∼ 10−52m−2, thus rendering the study of spacetimes with positive cosmological constant of physical importance. Moreover, we have:

R = constant, ∀ x ∈ R4 K ≡ RµνρσRµνρσ=

R2

6 = constant (2.7) where K is the Kretschmann scalar

This tells us that pure de Sitter space is free of any gravitational singularities.

Embedding the geometry

In order to have a Euclidean picture of pure de Sitter spacetime we can start by embedding a d-dimensional hyperboloid to an (d + 1)-dimensional Minkowski space. Then de Sitter can be viewed as the induced metric on the hyperboloid:

− X02+ n

X

i=1

(20)

Figure 2.1: The embedded hyperboloid depicting d-dimensional de Sitter space into a (d + 1)-dimensional flat space. The dotted line represents an Sd−1. Figure source: [1].

The geometry can be visualized as follows: The induced metric is: ds2 = −dX2

0 +

Pn

i=1dX 2

i and the various coordinate systems of de Sitter

spacetime, cover partially or globally this n-dimensional hyperboloid. We mostly study the d = 4-dimensional de Sitter (dS) spacetime Λ = +_l32 and variations thereof. Also we use l = 1 from now on

and reinstate when needed.

In order to use the coordinates on the 3-sphere S3 it is convenient to use the parametrization:

ω1= cos ψ (2.9)

ω2= sin ψ cos θ (2.10) ω3= sin ψ sin θ cos φ (2.11) ω4= sin ψ sin θ sin φ (2.12)

where: θ, ψ ∈ (0, π), φ ∈ [0, 2π), 4 X i=1 (ωi)2= 1 (2.13)

Thus, the metric on S3 becomes:

dΩ23= 4

X

i=1

(21)

2.2 Coordinate Patches and Penrose Diagrams of pure de Sitter

On this section we initially review different coordinate patches on dS [1] with their respected symmetries and the Penrose Diagrams for each patch respectively. Different coordinate systems on dS make different symmetries more manifest.

Global Coordinates (t, ψ, θ, φ)

The coordinate system that covers the whole hyperboloid is called the Global de Sitter Patch. It is obtained by setting:

X0= sinh t (2.15)

Xi= ωicosh t, i ∈ {1, 2, 3, 4} (2.16) where t ∈ R. The metric is then expressed as:

ds2= −dt2+ cosh2t dΩ2₃ (2.17) We can see that global coordinates foliate (constant-t spacelike slices) dS with a S3 _(positive

cur-vature) that starts with radius r = ∞ at t = −∞, shrinks to r = 0 at time t = 0 and again grows exponentially to infinite radius at t = +∞. Furthermore, it is a time-dependent background with a topology of R1_{x S}3_{so there is no time translation symmetry since t is not a Killing vector and we have}

only rotation symmetries. The global patch although it gives an overall picture of dS, it is not of so much physical relevance since there’s no observer that has causal access to the whole patch, as we will see below.

Conformal Coordinates (T, ψ, θ, φ)

Now an elegant depiction of the causal structure of the spacetime, is obtained through the

Carter-Penrose Diagram, by transforming the coordinates, we can picture the infinite spacetime into a finite diagram.

More explicitly, we use:

cosh t = 1

cos T (2.18)

The metric then takes the form: ds2= 1

cos2_T(−dT

2_{+ dΩ}2

3), T ∈ (−π/2, π/2) (2.19)

The null rays travel at angles = 45o _{, timelike curves at > 45}o _{(more vertical) and spacelike at < 45}o

(more horizontal). The spacelike slice on I+ (future null infinity) relies at t = +∞ and I− (past null infinity) at t = −∞ and these are the only spacelike boundaries that the de Sitter geometry enjoys.

The left vertical boundary of the Penrose diagram is called South pole and the right North pole, which is a matter of choice. The causal structure of Global dS in these coordinates is manifest on fig. 2.2.

As we can see from the diagram in fig.2.3, null (light) rays in dS who emanate from the infinite past from the one pole, will reach the other pole at the infinite future. So an observer who’s sitting at e.g.

(22)

Figure 2.2: Penrose-Carter diagram of global dS. Lines represent Sd−1_{and points S}d−2_.

the North Pole can only observe past events happening on the one side of the diagonal being shaped by the null rays O− and is causally connected to the observer. On the other hand, the observer can only affect events happening to the future (communicate with) on the restricted region O+_{. This peculiarity is}

only seen in dS spacetime (neither Minkowski nor Anti-de Sitter (AdS) spacetimes); it comes due to the expansion of spacetime as time progresses, rendering some regions for one observer "unphysical" since he/she will never be able to probe experiments, receive signals from or communicate with, in any causally consistent way. In that sense, there is no notion of a "global observer" in a dS spacetime and that’s why the global patch does not provide any physical intuition about dS whatsoever.

Eddington-Finkelstein Coordinates (x+_{, r, θ, φ)}

These coordinates are most commonly used in Schwarzschild black hole geometry. Nevertheless, they can also be used in empty de Sitter case. Solving by integration:

dt = dx+ dr

1 − r2 (2.20)

we get the solution for x+_:

x+= t + 1 2ln 1 + r 1 − r (2.21) The metric then takes the form:

ds2= −(1 − r2)(dx+)2− 2dx+_{dr + r}2_dΩ2

2 (2.22)

where: dΩ2is the metric on the S2.

Furthermore we can define:

x−= t − 1 2ln 1 + r 1 − r (2.23)

(23)

Figure 2.3: Causal past O− and future O+ _{of an observer sitting at the North Pole. We can see that}

he/she can’t receive signals past the diagonal line (null) of O− and send signals past the diagonal of O+.

with the metric being:

ds2= −1 − r2_(x +, x−) dx+dx−+ r2dΩ22 (2.24) with: r(x+, x−) = tanh x+_−x− 2

We can see that in this coordinate system, at r = constant hypersurfaces we have a timelike killing vector ∂/∂x+ _{which is the same as ∂/∂t, for fixed r. The Penrose diagram for this coordinate system is}

in fig.2.4.

Kruskal Coordinates (U, V, θ, φ)

Using the Eddington-Finkelstein coordinates and making the transformation:

x+ = − ln (−V ), x−= ln U (2.25) then we get the metric:

ds2= 1 (1 − U V )2

h

−4dU dV + (1 + U V )2dΩ2₂i (2.26) This is also a universal covering of de Sitter, with the horizons lying at U V = 0, the poles at U V = −1 and I+−_{at U V = 1. The U and V are increasing towards the upper diagonal directions (right and left}

respectively) and the Southern (Northern) causal diamond is the region with U > 0, V < 0 (U < 0, V > 0). This coordinate system is visualized in fig.2.5.

Static Patch Coordinates (t, r, θ, φ)

This is a particularly useful coordinate patch, is obtained through the intersection of the causal past and future O+∩ O− _{and is called the Southern (Northern) Causal Diamond or Southern (Northern)}

Static Patch (although we will just call it the Static Patch). As will be made clear later on, this patch is of physical interest and we will focus mainly on it. It is obtained by writing:

(24)

Figure 2.4: Penrose diagram for Eddington-Finkelstein coordinate system, with the arrows representing the time flow which comes from the Killing vector K = ∂/∂t. I± _{rely at r = ∞, the Southern and}

Northern causal diamonds in region r ∈ [0, 1] and the cosmological horizons at r = 1. Figure source: [1].

Figure 2.5: This is the diagram of Kruskal coordinates. The arrows point to the directions of increasing U, V . The poles lie at U V = −1, I± at U V = 1 and the horizons at U = 0 and V = 0. Figure source: [1].

(25)

X0=p1 − r2 _{sinh t} _(2.27)

Xk = rωk, k ∈ {1, 2, 3} (2.28) X4=p1 − r2 _{cosh t} _(2.29)

then the metric takes the form:

ds2= −(1 − r2) dt2+ (1 − r2)−1 dr2+ r2dΩ2₂, |r| ∈ [0, 1] (2.30) As we can see from the metric, each observer in the static patch is surrounded by a Cosmological Horizon, which is also the case in our Universe, relying at r2 _{= 1. This is the only region where an}

observer can send signals to and receive a response from, at a time before I+_{is reached and it covers one}

fourth (left or right triangle) of global dS.

Also the importance of this coordinate patch is made manifest through the existence of a timelike Killing vector ∂/∂t (hence the name ’Static’). This is because the static patch has a time-independent background (∂tgµν = 0). This is of physical importance since we have time translation and reversal

symmetry, hence we can define a time evolution (Hamiltonian H) for r < 1. The symmetry group of the Static Patch is SO(4) x R in d = 4.

Planar Patch (τ, xi), i ∈ {1, 2, 3}

These coordinates are defined by taking:

X0= sinh τ −1 2xix i_e−τ _(2.31) Xi= xie−τ (2.32) X4= cosh τ −1 2xix i_e−τ _(2.33)

Note that in the notation used in these coordinates we have:

{x1, x2, x3} ≡ {x, y, z} ∈ R3 (2.34)

Planar coordinates cover half of dS (upper or lower triangle) and are useful since they have flat constant time R3 _{foliations (zero curvature). These spacelike slices have infinite volume and peculiarly}

(only in dS) are extended all the way to the infinite past, as we can see from fig.2.6. The metric in this patch takes the form:

ds2= −dτ2+ e2τd~x2, _{τ ∈ R,} ~_{x ∈ R}3 (2.35) Also a more commonly used form planar patch metric is related to the notion of conformal time η, which is defined as:

ds2= 1 η2(−dη

2_{+ d~}_x2_), _{η = −e}−τ_, _~

x ∈ R3 (2.36)

(26)

Figure 2.6: This diagram depicts τ = constant spacelike flat (d-1)-dimensional slices, which extend all the way to I−.

a "big bang" type singularity at the infinite past and covers half of global dS, while the other half can be obtained by taking τ → −τ and it shrinks to a point at future infinity I+. The last one is referred to as "big crunch" type singularity [17]. In this patch it is obvious that de Sitter space has SO(4, 1) symmetries. These are [13]:

T ranslations : ~x → ~x + ~α (2.37) Rotations (ISO(3)) : ~x → R(~x) ~x (2.38) Special Conf ormal : ~x → ~x + ~α |~x|

1 + 2~α~x − 2|~α|2_|~_x|2 (2.39)

Reparametrization : η → λη, ~x → λ~x (2.40) The dimensionality of the de Sitter group is: dim(SO(4, 1)) = 5∗4₂ = 10 which are indeed the num-ber of generators in our case 3 Translations +3 Rotations +3 Special Conformal transformations +1 Reparametrization = 10 symmetry transformations.

Hyperbolic Patch (τ, ψ, θ, φ)

We can foliate dS with hyperbolic (negative curvature) slices, using the hyperbolic coordinates:

X0= sinh τ cosh ψ (2.41) Xi= ωi sinh τ sinh ψ i ∈ {1, 2, 3} (2.42)

X4= cosh τ (2.43)

The metric here takes the form:

ds2= −dτ2+ sinh2t (dψ2+ sinh2ψ dΩ2₂) (2.44) The constant time slices are H3 (constant negative curvature hyperbolic slices). This is the least

(27)

3

Black Holes in de Sitter

In this chapter we examine geometries that involve black holes in dS, mostly following [4],[24], [2] and [5]. As will become apparent below, it is qualitatively different than black holes in Minkowski spce since there’s one more horizon appearing in every solution, namely the Cosmological Horizon. This means that we can’t put an arbitrarily large black hole in dS, since it is confined by the cosmological horizon that contains the black hole. With this come further constrains apart from the mass (which is related to the radius of the event horizon of the black hole) like the angular momentum of the black hole, if it’s rotating. Furthermore, the different limits of the horizons (black hole and cosmological) will be examined in some geometries, which give rise to different topologies and thermodynamical limits. Moreover, since the asymptotic boundary in dS is spacelike (the boundaries lie at I±which are temporal instead of spatial infinities) there’s a different intuition when defining angular velocity Ω, entropy S or mass M when we follow an approximate procedure as in AdS or Minkowski.

Finally, in every black hole geometry we first make a reference to the analog in Minkowski space and then go to dS. In any case if we start from a black hole in dS and take the limit Λ → 0, we should always get the corresponding black hole in flat space.

3.1 Schwarzschild-de Sitter (SdS) spacetime

The Schwarzschild black hole geometry is a spherically symmetric solution to the Einstein equations, described by the one-parameter M metric:

(28)

Figure 3.1: Penrose diagram of maximally extended Schwarzschild geometry. K = ∂/∂t is timelike in regions I, IV and spacelike in regions II, III. The gravitational singularity is depicted by wavy lines at r = 0. Null infinities lie at I±, spacelike infinity at i0, future and past timelike infinities at P and Q

respectively and the black hole event horizons at r+_.

ds2= − 1 −2M r dt2+ 1 − 2M r −1 dr2+ r2dΩ2₂ (3.1)

where M : Schwarzschild black hole mass, dΩ2₂: the metric on the S2

There is one horizon at the Schwarzschild metric which lies at gtt = 0 ⇒ r = 2M . Furthermore, we

can see that this geometry enjoys an SO(3) symmetry (rotational symmetry on a S2_{), apart from}

time-translation symmetry. We can also observer that the Killing vector ∂/∂t is timelike future directed in region I, timelike past directed in region IV, spacelike in regions II, III and null on the horizons. The causal structure of this geometry is visualized in fig.3.1.

Now the simplest step towards a non-pure dS solution, involves putting a non rotating black hole in empty de Sitter, with a causal structure as in fig.3.2. In static coordinates, the metric has the two parameter (M, Λ) form in d = 4: ds2= − 1 − 2M r − Λr2 3 dt2+ 1 − 2M r − Λr2 3 −1 dr2+ r2dΩ2₂ (3.2)

Here we can see that gtt = 0 has three complex solutions and by imposing reality conditions [10], [11]

which are constraints on the black hole mass 0 < 9M2Λ < 1, we get two positive roots (physical) namely rh, rc where 0 < rh ≤ rc and one negative rn < 0 (unphysical). These indicate the locations of the

horizons (since ∂/∂t becomes null).The reality condition comes from the upper boundary on the mass of any black hole in dS, which is Mmax =

3√Λ

−1

. Out of these two radii, the smaller one which is located at r = rH corresponds to the black hole event horizon and the larger one at r = rC corresponds

(29)

Figure 3.2: Penrose diagram of maximally extended Schwarzschild-de Sitter geometry. It has one more horizon than the classical Schwarzschild geometry. Here also K = ∂/∂t is timelike in regions I and spacelike in regions II and III. Notice that there is an infinite sequence of gravitational singularities and spacelike infinities, so K changes signature an infinite number of times. The dots on the left and right side of the diagram imply repeated copies of it (the "beads" of the string). We can get rid of these "beads on a string" by imposing periodicity conditions on spacetime and keeping one singularity and one spacelike infinity (since as is the case in flat Schwarzschild geometry, there’s probably no physical meaning associated to the copied region).

rh= 2 √ Λcos θ 3+ 4π 3 (3.3) rn= 2 √ Λcos θ 3+ 2π 3 (3.4) rc= 2 √ Λcos θ 3 (3.5)

where cos θ = −3√ΛM and imposing the condition for all the roots to be real 0 < 9M2Λ < 1 we have cos θ ∈ (−1, 0] ⇒ θ ∈ [π/2, π). Furthermore, for M = Mmax we get the degenerate case where

rc = rh=

√ Λ

−1

which in the near horizon limit gives rise to another solution to the Einstein equations, namely the Nariai limit, which we will analyze later on. For M > Mmaxwe get a spacetime with a naked

singularity and no event horizon. Also note that for M = 0, we have the pure (empty) dS geometry. Note the very different conformal structre of SdS spacetime, as opposed to the ordinary Schwarzschild in Minkowski. Firstly, we can describe the maximally extended Schwarzschild geomtry with a single coordinate system (e.g. Kruskal-Szekeres coordinates) containing one physical (gravitational) singularity at r = 0. On the other hand, in SdS we see that we need two different coordinate patches to describe the two different regions; one for the region near the event horizon and one for the region near the cosmological. Furthermore, here we see that ∂/∂t changes signature an infinite number of times, since there’s a infinite sequence of gravitational singularities r = 0 and spacelike infinities r = ∞. As we can see in the Penrose diagram of SdS, we can interpret the geometry in an "exotic" way as in [2], where there

(30)

is an infinitely long ’string of beads’ of (acasually) connected universes, where each bead corresponds to a ’sandwiched’ area of the obsever between the two horizons. Here, the way we make our spacetime slicing of the region rh< r < rc corresponds to one of the four possible ’colour’ combinations of the holes

(black-black, black-white, white-black, white-white) and cuts the infinite number of beads at some line. Nevertheless by imposing periodicity conditions on the Penrose diagram, the beads are identified as one Schwarzschild-de Sitter universe.

In addition, the mass of the black hole is bounded from above, since as M increases (for Λ = constant), rhand rccome closer (rhincreases and rcdecreases), with the upper bound being rh= rc. As mentioned

above, we can’t put an arbitrarily large (heavy) black hole in dS.

We can also study the surface gravity of each horizon [5]. Defining the Killing vector:

K = ∂/∂t, |K| → r

Λ

3r as r → ∞ (3.6)

then we can define the black hole and cosmological surface gravity (κH, κC) and areas (AH, AC) of the

horizons from: Kb∇bKa= κ Ka (3.7) which yield: κC= Λ 6rc (rc− rh) (rc− rn) (3.8) κH = Λ 6rh (rc− rh) (rh− rn) (3.9) with: AC= 4πrc2 (3.10) AH= 4πrh2 (3.11)

The temperatures that corresponds to the event and the cosmological horizons are, respectively:

TH = κH 2π (3.12) TC = κC 2π (3.13)

In addition, if we take into account quantum fluctuations (and TH > TC) then the black hole evaporates drh

dt < 0 by emission of Hawking radiation. It is then reabsorbed by the cosmological horizon drc

dt > 0

and we can see that the thermal evolution of the geometry flows towards pure de Sitter, which is the most entropic configuration. This comes from [5], since cosmological horizons have a thermodynamic

(31)

interpretation just as black hole event horizons. The entropy in that case corresponds to the ’ignorance’ of the observer for the region outside his/her horizon. Since there’s thermal radiation emitted from the cosmological horizon, as the observer collects with a detector thermal particles he/she gains entropy and for the overall entropy to remain constant, the horizon shrinks down since according to the Bekenstein-Hawking formula:

Shorizon∝ (Area), Stot= Shorizon+ Sobserver (3.14)

. From this we can also understand why as we increase M , rh increases while rc decreases until they

coincide at Mmax, if we replace Sobserver with Sblack hole.

3.2 Near-extremal SdS black hole geometry and the Nariai limit

As mentioned before the generic SdS metric is of the form: ds2= − 1 − 2M r − Λr2 3 dt2+ 1 − 2M r − Λr2 3 −1 dr2+ r2dΩ2₂

with 0 < 9M2_{Λ ≤ 1. Now if we take the extremal case where 9M}2_{Λ = 1 we get the extremal SdS black}

hole. The cosmological and black hole event horizons start to coincide at the point where there is only one degenerate horizon at radius r = 3M with κ = 0 surface gravity. We can easily see that from:

1 −2M r − Λr2 3 = − 1 27M2_r(r + 6M )(r − 3M ) (3.15)

We can also see that due to the change of signs of gtt and grr, radial coordinate r becomes a time

coordinate and time coordinate t a spatial coordinate.

It is also instructive to see the global structure of this geometry. In order to do this need to introduce Kruskal-like null coordinates ˆu, ˆv [20] as:

ˆ u = cot−1u δ (3.16) ˆ v = tan−1v δ (3.17) where: δ = −M (3 − 2 ln 2) < 0, u = t − r∗, v = t + r∗, and the tortoise coordinate r∗ defined by:

r∗= Z _dr 1 − 2M_r −Λr2 3 ! = 9M r − 3M + 2M ln r + 6M r − 3M (3.18)

With these coordinates, the extremal SdS metric can be written as: ds2= − δ

2

27M2_r

(r + 6M )(r − 3M )2

sin2u cosˆ 2_ˆ_v dˆudˆv + r 2_dΩ2

2 (3.19)

We can see that this metric can also be defined on the horizon r = 3M smoothly: lim r→3M r − 3M sin ˆu = lim r→3M r − 3M cos ˆv = −18M 2 δ (3.20)

(32)

The causal structure of this spacetime is evident in fig.3.3 [20]. Any observer coming from r > 3M which is falling in the black hole spacetime will either cross the horizon and end up in the singularity at r = 0 or escape to one of the asymptotic points P which are at u = −∞, v = ∞. On the other hand, in the white hole spacetime observers emanating from r = 0 or P points, will cross the horizon and either reach I+ or the asymptotic points Q at u = ∞ and v = −∞.

Moreover, if we want to see what happens as the two horizons approach each other but don’t exactly coincide, we can "zoom in" the infinitesimal region between the cosmological and black hole horizons by going to the "near extremal" limit, which can be obtained if we take the mass of the black hole to be near extremality 9M2Λ = 1 − 32_{, with 0 < 1 with a proper rescaling of the clock [}₂₁_{], [}₂₂_{] by defining}

new time and radial coordinates (ψ, χ) as: t = 1 √Λψ (3.21) r = √1 Λ 1 − cos χ − 2 6 (3.22)

In these new coordinates, the metric near extremality becomes:

ds2= 1 Λ − 1 + 2 3 cos χ sin2χdψ2+ 1 −2 3 cos χ dχ2+ (1 − 2 cos χ)dΩ2₂ (3.23)

We can see that the S2 _{have varying size with the radius varying over the new coordinate on the}

S1_{, namely χ which is minimal on the black hole horizon and maximal on the cosmological one. In the}

degenerate case (as we will mention in the Nariai case below) the size of the S2 becomes constant. Each horizon has a surface gravity that differs from the other one by O(), so the temperature of each horizon is [21]: κC= √ Λ 1 −2 3 + O(2) (3.24) κH = √ Λ 1 +2 3 + O(2) (3.25)

We can see that κC = κH as → 0. The black hole horizon and the cosmological horizons lie at

χ = 0 and χ = π, respectively. The ψ = constant slices have an S1 _{x S}2 _{topology, with the radius of}

the S2 depending on χ which is the coordinate varying on the S1 with the extreme values (minimum, maximum) corresponding to each horizon respectively.

Furthermore, we can see that the degenerate case studied above corresponds to taking → 0, with the spheres having the same size; the near horizon geometry of the degenerate case is called the Nariai limit and it has the topology of dS2x S2. It is interesting to mention that the topology of the Nariai geometry

has generalization to an arbitrary number of dimensions; in d-dimensions the near horizon geometry of an extremal SdSd black hole after a rescaling of the clock would be dS2x Sd−2.

A more enlightening form of the near horizon metric for the degenerate case, namely the Nariai limit, would be by taking rc → rh and zooming in the intermediate region by introducing new coordinates as

(33)

Figure 3.3: Penrose diagram of extremal SdS spacetime. In figure (a) we see the black hole spacetime. As the observer comes from r > 3M to cross the horizon, he/she will either end up at the black hole singularity at r = 0 or to one of the asymptotic points P. On figure (b) is the inverse spacetime, namely the white hole one, in which we have an observer emanating from the white hole singularity at r = 0 or asymptotic P and will cross the horizon at r = 3M to either reach r = ∞ at I+ or the asymptotic points Q. The three dots at the right and left side of the diagram mean that the diagram is repeated infinitely in the horizontal direction, just like in non extremal SdS. Figure source: [20].

(34)

Figure 3.4: Penrose diagram for dS2(x S2) geometry. The black hole event horizon lies at ρ = 0 and the

cosmological horizon at ρ = β. The intermediate region is where the observer lies at the "sandwiched" area. Each point in this diagram if we include the whole dS2x S2is an S2, while τ = constant spacelike

slices have an S2x S1 topology.

[3]: τ = t rc (3.26) ρ =r − rh rc (3.27) β = rc− rh rc (3.28) Now taking the limit → 0, while keeping β fixed we get the Nariai metric in the form:

ds2= 1 Λ −ρ(β − ρ)dτ2₊ 1 ρ(β − ρ)dρ 2_{+ dΩ}2 2 (3.29)

The Penrose diagram for this dS2x S2 geometry is depicted in fig.3.4.

3.3 Reissner-Nordström-de Sitter (RNdS) spacetime

In Minkowski space we can have an electrically charged non-rotating black hole solution to the Einstein equations, namely the Reissner-Nordström spacetime. The metric solution has a two parameter form (M, Q) for 0= 1: ds2= − 1 −2M r + Q2 r2 dt2+ 1 − 2M r + Q2 r2 −1 dr2+ r2dΩ2₂ (3.30) where: Q is the electric charge.

Solving: gtt= 0 we find two solutions which indicate the horizons: r+, r− which are the outer and inner

(35)

Now there’s an analog of this geometry in dS which is a three parameter solution (Q, M, Λ) with a metric of the form:

ds2= − 1 −2M r − Λr2 3 + Q2 r2 dt2+ 1 −2M r − Λr2 3 + Q2 r2 −1 dr2+ r2dΩ2₂ (3.31)

This solution is called Reissner-Nordström-de Sitter spacetime and we can see it has three horizons (one more than the flat RN case), namely r+, r−, rc corresponding to the outer, inner and cosmological

horizons respectively. The parameter space of this solution is restricted by [40]:

9M2Λ ≤ 1 2 h (1 − 4Q2Λ)32 _{+ 12Q}2_{Λ + 1} i (3.32)

The charge is bounded by the so called Bogomol’nyi bound which in [40] is given as:

Q2≤ M2 1 + 1 3(M 2_{Λ) + O (M}4_Λ2₎ (3.33)

Now in order to cover all three horizons of the geometry, it is useful to change coordinate system to the cosmological coordinates [14], [19] which smoothly cover the region r ∈ (0, ∞) (for |Q| = M case). The coordinate transformation from static coordinates (t, r, θ, φ) to cosmological (τ, R, θ, φ) follows from [2]:

r = τ HR + M (3.34) t = (H)−1ln (τ H) − h(r) (3.35) dh(r) dr = − Hr2 (r − M )f (r) (3.36) f (r) = 1 − 2M r 2 −Λr 2 3 (3.37) where: H = ± q Λ

3 is the Hubble constant.

Now for the H > 0 solution, we get the metric in the form:

ds2= −U−2dτ2+ U2 dR2+ R2dΩ2₂

(3.38) where:

U = Hτ +M

R (3.39)

Following [15], [2], [19] we have the gravitational singularity lying at U = 0 ⇒ Hτ R = −M , the cosmological horizon rc at (τ, R) = (0, ∞), the black hole outer horizon r+ at (τ, R) = (∞, 0) and the

inner horizon r− at (τ, R) = (−∞, 0).

We see that the cosmological chart (τ, R) covers four single charts (t, r) from r = 0 to r = ∞ as depicted in fig.3.5. The boundaries of the chart are the gravitational singularity, the past inner horizon r−, the future outer horizon r+, the past cosmological horizon rc and the future conformal infinity at

(36)

Figure 3.5: Penrose-Carter diagram of Reissner-Nordström-de Sitter (RNdS) spacetime, for M = |Q|. The red dotted lines represent a single (τ, R) chart, where r is the old radial coordinate and R the cosmological one. Figure source: [2].

r = ∞. Moreover, we can see that the singularity U = 0 exists only for τ < 0 (since H, M > 0) while for τ > 0 the geometry is smooth.

Now if we see the τ = constant hyperslices, they are asymptotically flat for R → ∞ and if we see the metric in cosmological coordinates near R = 0 we see that it becomes like a throat-like 3-cylinder (∼= S2 x R), described by: ds2_cyl=M 2 R2dR 2_{+ M}2_dΩ2 2 (3.40)

We see that for τ = 0 the geometry is smooth everywhere, but as we start taking τ → 0− we have a singularity at R = ∞ and as τ → −∞ and R decreases the singularity truncates this 3-cylinder even more, until the whole spacetime shrinks to a point in time τ = −∞. On the other hand, if we go forward in time we see a spacetime that expands out of a singular point. This is why the chart we chose with H > 0 (we picked H = +

q

Λ

3) is called the expanding chart and is labeled by (τ+, R). There’s also the

contracting chart labelled by (τ−, R), which comes if we pick H = −

q

Λ

3 < 0 value and in the metric (60)

we use instead: U = Hτ−+ M R = −|H|τ−+ M R (3.41)

This chart has a similar behaviour, only this time the singularity starts forming at R = ∞ as τ → 0+ and as τ → ∞ and R decreases, the whole spacetime collapses to a singularity when R = 0. We can see the time evolution of the contracting chart in fig.3.6. These cosmological chats were constructed at [16]. In conclusion, the black hole event horizons lie at (τ±, R) = (±∞, 0) and the cosmological ones at

(37)

Figure 3.6: Time evolution of the cosmological contracting chart (τ−, R). We can see the forming of the

singularity at (τ−, R) = (0, ∞), "grinding" the space down to R = 0 as τ−→ ∞. Figure source: [2].

shown in fig.3.7.

3.4 Rotating Black Hole Geometries in de Sitter

Here we study the case of a rotating black hole in de Sitter space. First we make an introduction in the geometry of a black hole in Minkowski spacetime which is a solution to the Einstein-Maxwell theory and has an electric charge Q and angular velocity Ω; that is the Kerr-Newman (KN) spacetime geometry. This will prove to be interesting in its near horizon geometry, since in the extremal charge and non-rotating case in a limit has an AdS2 x S2 geometry, as was studied in [39]. This configuration apart

from its astrophysical relevance (since astrophysical black holes, although are electrically neutral, are rotating near extremality which in the near horizon region have also an AdS2 x S2 geometry) has the

same symmetries as dS2 x S2; this is the near horizon geometry of the Nariai geometry. Furthermore, as

we turn on rotation in the KN black hole, the SO(3) symmetry from the S2 _{piece of the RN black hole}

breaks into the U (1) subgroup. This is also the case when we turn on rotation in the Nariai black hole, going to the Rotating Nariai geometry which we study below. All of this will become more clear when we study some rotodynamic and thermodynamic properties of flat Kerr and KdS and compare whenever possible.

(38)

Figure 3.7: On the left, we have the Penrose diagram of the gluing of the two cosmological charts (τ±, R)

at the point of the cosmological horizon r = rc. The extended diagram repeats infinitely many times

in the horizontal direction just like before. On the right, there is the embedding of the throat-like 3-cylindrical geometry with topology S2

x R, with half the cylinder being covered by the expanding chart and the other half by the contracting one. Figure source: [2].

3.4.1 Kerr-de Sitter spacetime (KdS)

Just to make a brief introduction, the flat Kerr geometry corresponds to a rotating black hole solution to the Einstein equations with rotational velocity parametrized by the spin parameter α and is described by a two parameter (M, α) metric, namely:

ds2= − 1 − 2M r ρ2 dt2+ ρ2 dr 2 ∆ + dθ 2 + r2+ α2+2M rα 2 ρ2 sin 2_θ sin2θdφ2−2M rα sin 2_θ ρ2 dt dφ where: ρ2= r2+ α2cos2θ, ∆ = r2− 2M r + α2

We can see that due to the rotation we have a breaking of Schwarzschild’s geometry symmetry group SO(3) down to a U (1) subgroup (spherical breaks down to axial symmetry). Furthermore, this geometry has one more horizon than Schwarzschild geometry, namely r−which is the inner (Cauchy) horizon and r+

which is the outer (ergosphere) horizon of the rotating black hole. The causal structure of this spacetime is visualized in fig.3.8.

We can also define the angular velocity of the black hole, as the measure of frame dragging effect which will be explained below (comes from the non diagonal elements of the metric), as:

ΩH= − gtφ gφφ _r=r+ (3.42) Normally, we define: e ΩH = ΩH− Ω∞ (3.43)

(39)

Figure 3.8: Penrose-Carter diagram of extended Kerr spacetime along θ = 0. The dashed (red) lines represent the ring singularity, which lies at r = 0 and θ = π₂, but it is usual to represent it in the θ = 0 diagram. We can see the presence of two horizons, namely r−which is the inner (Cauchy) horizon and r+

the outer horizon with the area r ∈ (r−, r+) being the ergosphere. Future and past conformal infinities

lie at I± and there’s another infinity at r = −∞ if one passes through the ring singularity to the other side, which is a naked singularity for r < 0. Figure source: [2].

where: Ω∞= − lim r→∞ gtφ gφφ (3.44)

We see that eΩHis the angular velocity defined in a frame of reference of an observer which is non-rotating

at spatial infinity. On the contrary, ΩH is the angular velocity with respect to a frame of reference that

is rotating at infinity with angular velocity Ω∞; in Minkowski eΩH = ΩH since Ω∞ = 0, while in AdS

Ω∞6= 0.

The de Sitter space analog of flat Kerr, namely a rotating black hole, on a time dependent, expo-nentially growing background, is called Kerr-de Sitter geometry. It is the most astrophysically relevant, since our Universe is in a good approximation dS and the observed black holes are rotating, uncharged and near extremality. This renders the study of these black holes of great importance. It is a solution to the Einstein equations with a positive cosmological constant Λ > 0, giving rise to the three parameter (α, M, Λ) metric written in Boyer-Lindquist like coordinates:

ds2= r2+ α2cos2θ dr 2 ∆r + dθ 2 1 + Λ₃α2_cos2_θ ! + sin2θ 1 + Λ 3α 2_cos2_θ r2_{+ α}2_cos2_θ ! " αdt − r2+ α2 dφ 1 + Λ₃α2 #2 − − ∆r (r2_{+ α}2_cos2_θ) (3.45) where: ∆r= r2+ α2 1 − Λr 2 3 − 2M r

A Double-Headed Centaur in a Black de Sitter Universe

Universiteit van Amsterdam

Institute for Theoretical Physics