Consequences of Quantum Mechanics in General Relativity

UNIVERSITY OF CINCINNATI

Consequences of quantum mechanics in general relativity

by Souvik Sarkar

A thesis submitted to the graduate school in partial fulﬁllment for the degree of Doctor of Philosophy

in the Department of Physics Cenalo Vaz, L.C.R. Wijewardhana

August 2018 UNIVERSITY OF CINCINNATI

Abstract

Cenalo Vaz, L.C.R. Wijewardhana

Department of Physics

Doctor of Philosophy

by Souvik Sarkar

We have presented a gauge theoretic approach for gravity in 2 + 1-D, starting from the

Chern-Simons action with the SO(2, 2) and ISO(2, 1) Lie group structures. With these groups, we are able to construct some known results from general relativity namely the

Banados, Teteilboim and Zanelli (BTZ) solution and the spinning point particle solution respectively. We obtain the canonical structure in each case. To get a more physical picture we probe the canonical structure by embedding the Arnowitt-Deser-Misner

(ADM) metric into the corresponding spacetime. After several canonical transformations we found in each case that the spacetime is described by a two dimensional phase space with two degrees of freedom, the mass and the angular momentum. These are stationary solutions. In the second part of this thesis, we have extended this approach to non- rotating dynamical collapse. During a spherical collapse the end state does not always lead to a black hole. Instead, due to quantum effects, collapsing shells in the exterior of the apparent horizon are accompanied by outgoing Unruh radiation in its interior. Both collapsing shells and the outgoing Unruh radiation appear to stop at the apparent horizon. This solution is obtained by solving the Wheeler-DeWitt equation (WDW) which is the Hamiltonian constraint, elevated to operator form by applying Dirac’s quantization procedure. As a consequence, we can expect that the collapse process would end in a quasi-stable, static compact object without forming a black hole. We have shown that, using Einstein equations, such a stable configuration is possible where the BTZ horizon radius is 87% of the boundary radius, so the BTZ horizon lies inside the boundary. In the last part, we consider a scalar field model of dark matter(DM) which forms a Bose

Einstein condensate. We coupled a non-relativistic scalar field with φ4 interactions to linearized gravity in the Gravitoelectromagnetic formulation. This model predicts that halo sizes vary significantly depending on the mass of the scalar field particles and the nature of the self-interaction. Dark matter halos ranging from asteroid size to galaxy size are possible within this model. This was one of the major challenges in the Cold Dark

Matter (CDM) paradigm. We examined the stability of the halos by studying small oscillations about the equilibrium using a collective coordinates approach with both types of self-interaction. We found oscillations about the minima of the energy with a time period of 10 billion years for attractive interactions and 1.3 billion years for repulsive interactions.

ii iii Acknowledgements

Firstly, I would like to express my sincere gratitude to my advisors Dr. Cenalo Vaz and

Dr. L. C. R. Wijewardhana for the continuous support during my PhD study and research and for their patience, motivation, enthusiasm and knowledge. Their persistent guidance helped me in every way in my reasearch and writing of this thesis. It has been an honour to be PhD student under them. Without their help, I could not have imagined this day.

I would like to use this opportunity to thank my thesis committee members Dr. Rostislav

A. Serota, Dr. Philip Argyres, Dr. Kay Kinoshita for their time, interest and helpful comments in my research.

I would like to convey my heartiest gratitude to Dr. Paul Esposito, Dr. Peter Suranyi,

Dr. Michael Ma for their Constructive discussions and sharing their thoughts. It has considerably improved the environment to carry on the research during my PhD.

All the group sessions with my other PhD colleagues have added another polished layer in my research journey through my PhD career. I thank all of them from the core for their interest and helping me in every way possible.

Last but not the least, I would mention my parents , Sreetama and my friends for their constant support, love and encouragement to reach my goal.

It has been a wonderful journey through my research career and many more people are responsible for making it memorable. I would like to thank all of them. Their contributions in my life is unforgettable and will be cherished through the rest of my life.

Thank you...

iv Contents

Abstract i

Acknowledgements iv

List of Figures vii

Physical Constants viii

1 Introduction1

2 Chern-Simons theory and quantization of BTZ black hole8 2.1 Introduction...... 8 2.2 The Connection...... 10 2.3 Chern-Simons gravity...... 14 2.4 Equations of motion...... 19 2.4.1 Static case...... 20 2.5 Fall-oﬀ conditions and boundary action...... 23 2.6 Embedding...... 25 2.6.1 Spinning particle...... 25 2.6.2 BTZ black hole...... 27 2.7 New canonical variables...... 28 2.8 The boundary action...... 30 2.9 Summary...... 34

3 Classical dust collapse model 36 3.1 Collapse wave functionals...... 39 3.2 A quasi-classical static conﬁguration...... 46 3.3 Energy extraction...... 48 3.4 Summary...... 49

v Contents

4 Dark matter as Bose-Einstein condensate 51 4.1 Introduction...... 51 4.2 Dark matter and BEC...... 54 4.3 The action...... 56 4.3.1 The GPP Action...... 56 4.3.2 The Gravitational Action...... 59 4.3.3 Equations of Motion...... 61 4.3.4 Stable Conﬁguration...... 64 4.3.4.1 Total energy...... 64 4.3.4.2 Minimum energy conﬁguration...... 67 4.3.4.3 Single vortex ansatz...... 70 4.3.4.4 Size of condensates(halos)...... 72 4.3.5 Vortex Oscillations...... 74 4.4 Summary...... 78

5 Conclusion 81

A Simpliﬁcation of the Hamiltonian constraint 92

B Derivation of unique solution of WDW equation 94

C Calculations of Israel’s junction conditions 97

D The Gross-Pitaevskii equation 99

vi List of Figures

4.1 Gravitational acceleration on the equatorial plane of the BEC for our ansatz 74 4.2 Gravitational acceleration on the equatorial plane of the BEC for our ansatz 77 4.3 Gravitational acceleration on equatorial plane of the BEC for our ansatz 78

vii Physical Constants

Speed of light c = 2.997 924 58 × 108 ms−1

Reduced Planck constant ~ = 1.054 571 800 × 10−34 J.s Gravitational constant G = 6.674 08 × 10−11 Nm2kg−2

Boltzmann constant k = 1.380 650 5 × 10−23 JK−1

−35 Planck length lP = 1.616 229 × 10 m

−8 Planck mass mP = 2.176 470 × 10 kg

−44 Planck time tP = 5.391 16 × 10 s

viii To my parents. . .

ix Chapter 1

Introduction

There are four fundamental forces in nature based on its properties e.g. strong, weak, electromagnetic, and gravitational force. While the description of the first three forces is well-built on the principles of quantum mechanics on the microscopic level with a very high precision, general relativity takes into account the gravitational force on the macroscopic (galactic) scales. So far, quantum mechanics and general relativity are proved to be two of the most successful theories in modern physics to account for these fundamental forces. More specifically, general relativity discards the concept of force and introduces a new way to look at gravity as the curvature of spacetime itself, which in turn, dictates the objects to move in the shortest path in the presence of matter. On the other hand, according to quantum mechanics, the microscopic forces are mediated by quantum particles which are the quanta of a force field which pervades all over the universe. This generation of quanta out of the field is known as ‘quantization’ of force. Since gravity is simply the curvature of spacetime, quantization of gravity means quantizing spacetime itself which is difficult to realize in our usual understanding.

The task of quantizing gravity is one of the most outstanding problems of modern theoretical physics. Many attempts have been made to reconcile these two theories starting in 1930s. Decades of hard work has enabled us to develop insights into quantum ﬁeld theory and quantization of constrained systems. Despite the progress in these areas, there is still 1 Quantization of gravity no self-consistent quantum theory of gravity. There are some unavoidable reasons that make quantum gravity very challenging to formulate. They are the following:

• General relativity is a highly non-linear theory and standard power counting argu- ments indicate that the theory is nonrenormalizable. This was ﬁnally conﬁrmed in 1986 by explicit computations.

• Observables in quantum ﬁeld theory are local, wherein, quantum gravity physical observables are nonlocal.

• Quantum ﬁeld theory respects causality but for quantum gravity, light cones and causal structure are themselves subject to quantum ﬂuctuations.

• It turns out, that quantum gravity is in itself a constrained system. Hamiltonian is identically zero when acting on physical states. Therefore time evolution in quantum gravity does not have any obvious meaning. It has to be handled carefully.

Our aim is to understand the quantum theory of gravity which is far more complicated in 3 + 1-D. In consideration of these problems, it is reasonable to look at a simpler theory namely 2 + 1-D gravity and develop our insight based on it. We might hope to generalize that understanding to 3 + 1 case. We could have looked at 1 + 1 case, but in one spatial dimension, the concept of Bekenstein-Hawking entropy of black holes does not make sense. This is because the entropy is proportional to the area or the circumference of the black hole. In 1 + 1-D, the boundary of a black hole consists of only two points on which the quantum entropy law cannot be applied. Keeping these shortcomings in mind, we expect that 2 + 1-D gravity may serve well as our prototype. In 1992, Banados, Teteilboim, and Zanelli (BTZ) showed in their paper [1] that in 2 + 1-D, it is possible to have a vacuum solution with a negative cosmological constant, which has many of the same properties as the more physical 3 + 1-D black holes. Due to these similarities, their solution has become the primary model to understand “quantum gravity”.

Gravity in 2 + 1-D has two convenient properties which makes it even more interesting. They are : 2 Quantization of gravity

• 2 + 1-D gravity is trivial, meaning it has no propagating modes or gravitational waves. This is because the Riemann curvature is fully determined by the Ricci tensor. The number of degrees of freedom in this system is exactly equal to the number of constraints given by Einstein’s equations. This makes 2 + 1-D gravity a “topological ﬁeld theory” which is much easier to quantize than its 3 + 1-D counterpart.

• Unlike Schwarzschild and Kerr solutions, the BTZ spacetime is asymptotically anti- de Sitter which is a maximally symmetric vacuum solution with negative curvature. This manifold is exactly equal to the “group manifold” SL(2,R). This enables us to translate many diﬃcult analytical questions into simple algebraic computations.

Quantum mechanics is believed to be the underlying theory of all physical processes known in nature. Nevertheless, it cannot be used to predict the outcome precisely but gives the probability for a particular process to happen. Quantum mechanics is based on one main principle, namely unitarity. All the information about a particle state is encoded in the wave function describing it until it collapses. Unitarity means that given a wave function at present time, its time evolution in later time as well as in the past time is uniquely determined by evolution operator(unitary). One consequence of this is that in quantum theory, information is never truly lost, nor is it truly copied, at least in principle. But a paradox arose in 1975 after Stephen Hawking [2] and Bekenstein [3–5] made a remarkable connection between thermodynamics, quantum mechanics, and black holes. They have shown within the field theoretic framework that black holes surrounded by quantum fields emit radiation (Hawking radiation) and slowly shrink in size, eventually evaporating completely leaving nothing behind. According to the quantum field theory in curved spacetime, Hawking radiation involves a pure state of two mutually entangled particles where an outgoing particle escapes as Hawking radiation and the infalling one is swallowed by the black hole. Therefore the exterior is entangled with the interior. For an observer with access only to the exterior, the outgoing particle is in a mixed state and since

3 Quantization of gravity

the quantum numbers of the particle inside the black hole can never escape, there will only be an exterior mixed state if the black hole evaporates completely. This mixed state is quantum mechanically described by a density matrix rather than a wave function. In transforming from a pure state to a mixed state one must lose information. For instance, in our example, we took a state described by a set of eigenvalues and coeﬃcients, a large set of numbers, and transformed it into a state described by temperature, which is a single quantitative number. All the other structures of the state were lost in the transformation. In other words, during the process of falling into the black hole, a non- unitary transformation has been performed on the state of the system. As we recall, non- unitary evolution is not allowed to occur naturally in a quantum theory because it fails to preserve probability; that is, after non-unitary evolution, the sum of the probabilities of all possible outcomes of an experiment may be greater or less than 1. This is the famous ‘information paradox’. The fact that the information is lost is reﬂected in the thermal nature of the emitted radiation. But any thermal system can be assigned to an entropy via the Gibbs law dE = SdT . Since the quantum radiation is thermal, we can calculate the black hole entropy by just calculating the black hole temperature.

While it is true that Gibbs law gives the correct Bekenstein-Hawking entropy from the calculated temperature, no one has been able to explain the entropy directly from quantum mechanical or statistical grounds. In fact, it is proven that semi-classical gravity is insuﬃcient to account for this entropy. This is a profound result since the thermodynamic entropy is obtained at the semi-classical level, thus, leaving us with two possible choices :

• Thermodynamic entropy does not always have a statistical mechanical basis.

• Gravity is not a fundamental interaction, but rather a composite eﬀect of some more fundamental underlying theory.

The second option is from the point of view of String theory. It turns out that by using some known results regarding monopoles in certain types of ﬁeld theories, string theory

4 Quantization of gravity is able to count for states that would contribute to a certain class (unphysical) of black hole of a given mass.

Several theories have been put forward to resolve the information paradox. The most straightforward solution is to assume that the evaporation process leaves behind a remnant by some unknown mechanism. But this is diﬃcult to imagine since it would possess large degeneracy while remaining stable. To contain information from an evaporated black hole, the remnant would have to have an inﬁnite number of internal states which does not seem to be a viable option.

In 1993, Leonard Susskind, working with Larus Thorlacius and John Uglum [6], build- ing on ideas of Gerard‘t Hooft and John Preskill, tried to give a revolutionary idea. They emphasized on the non-local behavior that would be needed in order to resolve the information paradox. Their principles of black hole complementarity [7–9] is based on following postulates :

• Unitarity: A distant observer sees a unitary S-matrix which describes the black hole evolution from infalling matter to outgoing Hawking radiation within the standard quantum theory.

• EFT : Outside the stretched horizon, physics can be described by an eﬀective ﬁeld theory of Einstein gravity and matter.

• No Drama : Equivalence principle holds at the horizon. A freely falling observer experiences nothing out of the ordinary while crossing the horizon.

According to this theory, different observers see the same bit of information at different places. For an external observer, the infinite time dilation at the horizon itself makes it appear as if it takes an infinite amount of time to reach the horizon. However, according to the infalling observer, nothing special happens at the horizon itself, and both the observer and the information will hit the singularity eventually. Though it sounds like cloning of information, there is only one bit of information in the Hilbert space, which is

5 Quantization of gravity not localized. Both the observers are right in their own point of view but since nothing can escape the black hole, there is no scope for them to compare both their stories simultaneously. In a way, the notion of locality is lost, but not the quantum theory. Another aspect of this complementarity theory is the existence of the stretched horizon, which is a hot and physical membrane hovering about a Plank length outside the event horizon. According to an external observer, the infalling information stays on the membrane heat- ing it up which then radiates it as Hawking radiation, with the entire evolution being unitary.

However, in 2012, Almheiri, Marolf, Polchinski and Sully (AMPS) [10] showed (under certain assumptions) that the equivalence principle and unitarity contradict each other during evaporation process of the black hole. Considering that the effective field theory is valid outside the stretched horizon, a Bogoliubov transformation is possible from the frame of a distant observer (for whom the quantum field is in a pure state) to that of freely falling observer such that the freely falling observer will see a violation of equivalence principle as he crosses the horizon. AMPS proposed that the most conservative solution to this contradiction is that there would be drama at the horizon of the black hole in the form of a firewall that would destroy the infalling object. It seems surprising because the curvature is relatively small at the event horizon of a sufficiently large black hole where general relativity should hold. This conflict arises between physics at infinity and physics near the horizon, but far from the singularity.

Hawking recently proposed a resolution [11] to this paradox by suggesting that it may incidentally happen that collapse does not always lead to an event horizon. Matter stops collapsing once each shell reaches its apparent horizon due to some quantum mechanical mechanism. In other words, quantum mechanics is believed to play a major role in the collapsing phase in such a way that the collapse does not end up in a black hole. Therefore, no singularity is formed. And in the absence of event horizon, the paradox of information loss becomes irrelevant.

6 Quantization of gravity

With this in mind, we will attempt to quantize gravity with a view to constructing a quantum description of dust collapse model in 2 + 1 dimensions. Then we will consider an example of a static conﬁguration and demonstrate that such conditions are possible under which the physical boundary of the system does not cross the BTZ horizon radius forming no black hole, hence proving Hawking’s proposal.

Like the quantization of other three forces, quantizing gravity naturally directs us to think of the Lagrangian (Hamiltonian) formalism of some kind of field variables. As Witten shows in his paper [12] that 2 + 1-D gravity has a simple gauge theoretic interpretation, it seems promising to explore more into this description. It is very interesting to see that if we take a matrix-valued gauge field which depends on local frame field, then the 3-D action for gravity can be derived from the boundary term of a 4-D action. This is the Chern-Simons action. General relativity is described by this action with an additional constraint that the gauge fields are invertible. The advantage of this approach is that it admits many different classes of solutions that may have metric interpretations. It also comes with solutions where metric of the spacetime cannot be interpreted in a usual way. However, it is interesting to explore this description in its own right. In the next chapter, we will elaborately describe this action principle and present that it can lead to some well-known solutions to Einstein’s equations.

7 Chapter 2

Chern-Simons theory and quantization of BTZ black hole

2.1 Introduction

In this chapter, we will explain the gauge-theoretic approach to 2 + 1-D gravity. 2 + 1-D pure gravity has no local, propagating degrees of freedom. In the absence of matter, general relativity describes a spacetime of constant curvature. The absence of local dynamics means that the gravity in three dimensions is totally determined by its global effect, hence it is a topological theory. Therefore, we can expect the problems with quantum gravity to be handled easily in the 2 + 1 case. Although this fact does not make anything easier. Nevertheless, it is far from being trivial [13]. The vacuum solutions of pure gravity are multi-conical spacetimes, obtained by identification of points in flat space [14–16]. In the presence of a cosmological constant, the solutions obtained are maximally symmetric, i.e. the Anti-de Sitter(AdS) and de Sitter(dS) spacetimes. The Bañados, Teitelboim, and Zanelli (BTZ) [1] black hole solution is locally AdS but globally it is characterized by conserved charges at the boundary of the AdS spacetime [17]. Due to similarity with the 3 + 1 case, it provides much simpler ground to study quantum effects of gravity.

8 Quantization of gravity

Classical 2 + 1 dimensional gravity and super-gravity can be viewed as Chern-Simons gauge theories of the Poincare, Anti-de Sitter and de Sitter group and their supersym- metric generalizations [12, 18–20]. One can also show that the spinning black hole solution of Banados, Teitelboim, and Zanelli (BTZ) [1] can be recovered by identiﬁcation of points by a discrete subgroup of SO(2, 2) [21]. We can construct the Chern-Simons action as

1 Z 1 I = γ Aa ∧ dAb + f b Ac ∧ Ad (2.1) C.S. 2 ab 3 cd

a by expanding the gauge super ﬁeld in the bases of generators Tba. Here fbc are the structure constants of Lie algebra and γab = Tr(TbaTbb) plays the role of the metric of the Lie algebra. For the action to contain the kinetic terms of all gauge ﬁelds the metric has to be non- degenerate. By construction, the action is invariant under the gauge transformation

a δgAµ = −DµΛ, where Λ = Λ Ta and Dµ = ∂µ + [Aµ, ]. If the action (2.1) describes the correct gauge theory of (super)gravity, then gauge transformation is equivalent to diffeomorphism. This is true for small diffeomorphism on-shell. In general Chern-Simons action admits many solutions that have no metric interpretation. To have a metric interpretation it is necessary to impose the additional constraint that the gauge field is invertible. With this constraint, solutions to Einstein’s theory in 2+1 dimensions emerge from the Chern-Simons action. In my work, we will limit ourselves to those solutions with metric interpretations. In this case the gauge fields are the dreibein and spin connection, and the field strength torsion vanishes from the classical equations of motion and the curvature is constant. Following Kuchaˇr[22], we will describe the canonical structure of Chern-Simons action for metric compatible solutions. There are other approaches that have been pursued in [23–26]. They focus on solving constraints and using them to derive simplified Hamiltonian in space of finite numbers of degrees of freedom. In case of spacetime having symmetry, one can start with reduced action for the system.

In our approach [27], we will simplify the constraints via a series of canonical transformations. The advantage of this approach is that these transformations can be modiﬁed and thus be useful in a variety of systems i.e. Einstein-Maxwell system [28], Lovelock gravity

9 Quantization of gravity

[29] and the cases when the matter is included [30, 31]. In section (2.2), we will provide basics of first order formalism. In section (2.3), we will discuss the Chern-Simons action with SO(2, 2) group structure. In section (2.4), we will calculate the equations of motion and showed that static cases correspond to two different solutions. In section (2.5), the fall-off conditions will be described which are important to find the boundary variations from the bulk action. In section (2.6), the canonical structure of the two spacetimes will be considered by embedding the ADM metric into the static solutions. In section (2.7) and (2.8), we will perform the quantization of the rotating BTZ black hole. In the end, we will show that the wave function that describes the system stays the same on each spacelike hypersurface and are fully characterized by two global conserved charges.

2.2 The Connection

Three dimensional gravity is topological by nature which can be seen from Einstein- Hilbert action on 3 − D manifold

Z 1 3 √ IEH = d x −g(R + 2Λ) (2.2) 16πG M

Gibbons, Hawking and York proposed an additional boundary term that depends on the boundary geometry [32, 33]

Z √ 1 2 IGHY = d x −hK (2.3) 8πG M where h is the determinant of the induced metric hµν deﬁned by

hµν = gµν + nµnν (2.4)

on the boundary, nµ is the normal vector to the hypersurface at the boundary and K is the trace of the extrinsic curvature of the boundary. This term cancels the terms involving

δ(∂σgµν). Therefore, making δgµν = 0 is sufficient to make the action stationary. We can 10 Quantization of gravity map the diffeomorphism as a gauge transformation that leaves the total action invariant if we consider Einstein’s gravity as a gauge theory. This is called first order formalism

a where vierbeins e µ are the fundamental field variables. This is referred as frame field and a all the frame fields at a particular point makes the frame bundle. e µ can be thought of as a transformation matrix between the tangent space and coordinate frame at a particular point in spacetime. If we take the basis vectors of local tangent space to be orthonormal with respect to the Minkowski metric, then they satisfy the following relations

µν a b ab g e µe ν = η a b ηabe µe ν = gµν (2.5)

Here the Greek indices are spacetime indices and the Latin indices are vierbein indices. Given a vector ﬁeld whose tangent space components are V a, the covariant derivative can

a be deﬁned using the spin connection ω µ

a a a b DµV = ∂µV + ωµbV (2.6)

If V µ are the components of the same vector in the coordinate basis, then the covariant derivative of the vector is deﬁned by

ν ν ν λ DµV = ∂µV + ΓλµV (2.7)

The above two equations are compatible if and only if we demand that the net parallel transport of vierbeins gives a vanishing covariant derivative

a a ρ a abc Dµe ν = ∂µe ν − Γµνe ρ + ωµbeνc = 0 (2.8)

ρ ρ a ρ a b From the last equation (2.8) it follows that Γµν = e a∂µeν +e aωµbeν . The torsion tensor ρ ρ is given by Tµν = 2Γ[µν] and can thus be written as

a a a b Tµν = 2(∂[µeν] + ω[µb eν]) (2.9)

11 Quantization of gravity

. If the connection is torsion free, then the above equation can be written in terms of exterior calculus notation as

a a a b Dωe = de + ω b ∧ e (2.10)

a a where e and ω b are the frame and spin connection one-form. This is Cartan’s ﬁrst structure equation. In that case, the curvature tensor can be deﬁned using usual expression

a a b for gauge ﬁeld strength as [Dµ,Dν]V = RµνbV . Using the above deﬁnition of covariant derivative Riemann tensor takes the form

a µ ν a a c Rµνbdx ∧ dx = dω b + ω c ∧ ω b (2.11)

which is analogous to familiar gauge theory ﬁeld strength F = dA + A ∧ A. To construct the Chern-Simons action in 3-D we will take a four dimensional manifold with the Chern- Pontryagin action of the form

Z Z 1 4 µνλρ a b IP = Tr.(F ∧ F ) = d x F µνF λργab (2.12) 4 M4

We will assume that γab is non-degenerate to make sure that the action contains kinetic terms of all gauge ﬁelds. The integrand can be written in the form

1 1 γ dAa + f a AcAd dAb + f b AeAf ab 2 cd 2 ef 1 ⇒ γ dAadAb + f a AeAf dAb (2.13) ab 2 ef

a where we have used the Jacobi identity of structure constants fbc to eliminate the last term containing four A’s. The Lagrangian now can be readily written as total divergence of a boundary term which is the Chern-Simons action

Z 1 I = dL ,L = γ AadAb + f a AcAeAb (2.14) P CS CS ab 3 ce

12 Quantization of gravity

In 2 + 1-D these ﬁelds can be written as one-forms

1 ea = ea dxµ, ωa = abcω dxµ (2.15) µ 2 µbc where abc is three dimensional Levi-Civita tensor and invariant under SO(3, 1). Einstein Hilbert action in 3 dimensions with cosmological constant can be written as

1 Z 1 Λ I = ea ∧ (dω + ωbωc) + ea ∧ eb ∧ ec (2.16) 8πG a 2 abc 6 abc

a Varying this action with respect to ωa and e will yield the equation of motion

b c Ta = dea + abcω ∧ e = 0 1 Λ R = dω + ωb ∧ ωc = − eb ∧ ec (2.17) a a 2 abc 2 abc

a respectively. The triads e µ are invertible to have any meaningful solution in terms of General relativity.The second equation is the Einstein’s equation in ﬁrst order formalism. It can be shown that the action is invariant under two sets of gauge symmetries

• Local Lorentz Transformation(LLT ): given by

a abc δle = ebτc a a abc δlω = dτ + ωbτc (2.18)

• Local Translations (LT ):

a a abc δte = dρ + ωbρc a abc δtω = −Λ ebρc (2.19)

Here τ a and ρa are local functions and its components are equal to the number of Lorentz transformations and translations respectively.

13 Quantization of gravity

Gravity in 2 + 1-D behaves like a gauge theory in many ways because of the action (2.16) which is of the ﬁrst order form. Starting from the action (2.1) Euler-Lagrange equation gives

F [A] = dA + A ∧ A = 0 (2.20)

Therefore A is a ﬂat connection and its ﬁeld strength vanishes.

In the following section, we show that under certain group structure Chern-Simons action leads to well-known solutions to Einstein equations.

2.3 Chern-Simons gravity

Vacuum three dimensional gravity can be treated as a gauge theory if we consider following three groups

• ISO(2, 1) for pure gravity(Λ = 0)

• SO(2, 2) for anti-de Sitter space(Λ > 0)

• SO(3, 1) for de Sitter space(Λ < 0) on a Chern-Simons action of the form

1 Z 2 ICS = Tr A ∧ (dA + A ∧ A) 2 M 3 Z 1 µνλ 2 = Tr Aµ ∂νAλ + AνAλ 2 M 3 Z 1 a b 1 b c d = γab A ∧ (dA + f cdA ∧ A ) (2.21) 2 M 3

µνλ These A’s are the gauge connections (one-forms), is completely antisymmetric tensor,γab

is the metric of the Lie algebra i.e., γab = Tr(TbaTbb) with Tba’s are the generators of the 14 Quantization of gravity

a Lie algebra and f bc are the structure constants of the group. For ISO(2, 1) group Aµ’s can be expanded in the basis of generators of the corresponding group as

a a Aµ = e µPca + ω µJba (2.22)

a a e µ and ω µ are the dreibeins and the spin connections respectively.Pca and Jba are the translation and Lorentz generators of ISO(2, 1) group satisfying the following commutation rule

h i h i h i c c Pca, Pbb = 0, Pca, Jbb = abcPc, Jba, Jbb = abcJb (2.23)

It is worth mentioning that the above action is invariant under inﬁnitesimal gauge trans-

a formation δAµ = −Dµα with α = α Tba, if we deﬁne the covariant derivative Dµ =

∂µ + [Aµ, ]. The quadratic invariant form (Casimirs), namely P.b Jb+ J.b Pb determines the

γab as

Tr(PcaJbb) = Tr(JbaPbb) = δab (2.24)

Substituting all these in (2.21) we can write the action in the form

Z 1 3 µνλ a b b c d a b ICS = δab d x e µ ∂νω λ + cdω νω λ + ω µ∂νe λ (2.25) 2 M

Chern-Simons action can also be seen as the surface term of the four dimensional gauge theory action of the form R Tr(F ∧ F ) as

Z Z µνρσ 2 Tr(F ∧ F ) = (Bulk term) + ∂σ Tr Aµ∂νAρ + AµAνAρ (2.26) S ∂S 3 where Fµν = [Dµ,Dν] = ∂µAν − ∂νAµ + [Aµ,Aν] is the non-abelian ﬁeld strength tensor. We can generalize it by including Cosmological constant (Λ < or > 0). Spacetime is not ﬂat anymore but locally homogeneous with a constant curvature determined by Λ. It turns out that for Λ > 0, the spacetime is going to be what we call anti-de Sitter spacetime. It is a vacuum solution of Einstein’s equations. This spacetime has a constant

15 Quantization of gravity

negative curvature proportional to −Λ. Anti-de Sitter spacetime has its own symmetry SO(2, 2). It is easy to see that gauge group for this spacetime will be SO(2, 2). On the other hand, for Λ < 0 the spacetime will have a constant positive curvature and it is de Sitter spacetime. It has a SO(3, 1) symmetry.

Moving on to anti-de Sitter spacetime, inclusion of Cosmological constant Λ corresponds to changing the commutation relations as

h i h i h i c c c Pca, Pbb = ΛabcJb , Pca, Jbb = abcPc, Jba, Jbb = abcJb (2.27)

Starting from (2.21) with this commutation relations we will have now one extra term in the action

Z 1 3 µνλ a b b c d Λ c d ICS = δab d x e µ ∂νω λ + cd ω νω λ + e νe λ 2 M 3 a b +ω µ∂νe λ (2.28)

For further calculation it is convenient to separate the time components of the action

Z 1 3 ij a b b c d c d ICS = δab d x e t 2∂iω j + cd ω iω j + Λe ie j 2 M a b b c d a b a b +ω t 2∂ie j + 2 cde iω j − e i∂tω j − ω i∂te j (2.29)

We can immediately see that dreibeins and spin connections are canonically conjugate

a i ij b to each other. In other words, the momentum conjugate to e i is Πa = δab ω j. In a a this ﬁrst order form, if we treat {e i, ω i} on equal footing, then canonical momenta do not depend on time derivatives of ﬁeld variables. This tells us that it is a second class primary constraint. This theory does not have any degrees of freedom since it has twelve

a second class and six ﬁrst class constraints. We choose e i to be the conﬁguration space variables, the Hamiltonian density will take the form

a b a b H = −δab e tF [ω] + ω tF [e] (2.30)

16 Quantization of gravity

where

dm c i Fa[e] ≡ acdδ e iΠm ≈ 0

1 F [ω] ≡ ∂ Π i + δcmδdnΠ kΠ l + Λijec ed ≈ 0 (2.31) a i a 2 acd kl m n i j

These are the six constraints of the theory. First three constraints represent vanishing of torsion tensor and last three enforce constancy of the spacetime curvature. In other words, Einstein’s gravity is torsion free gravity where the aﬃne connections are symmetric in their lower two indices.

To retrieve the canonical structure of the spacetime, we need to write down the constraints in terms of the metric functions which we can do by considering the standard ADM metric

2 ds2 = N dt2 − A2(dr + N rdt)2 − B2(dφ + N φdt)2 − C2(dr + N rdt)(dφ + N φdt) (2.32)

We are interested in axisymmetric solutions. This ADM form implies foliation of three dimensional spacetime labeled by time parameter t such that each leaf (spatial hypersurface) will have the following axisymmetric metric in circular coordinates (r, φ)

ds2 = A2(r)dr2 + B2(r)dφ2 + C2(r)drdφ (2.33)

Now (2.32) can be written as

Q 2 ds2 = N 2dt2 − L2(dr + N rdt)2 − R2 dφ + N φdt + dr (2.34) R

with the following identiﬁcation

r c4 C2 L = A2 − ,R = BQ = 4B2 2B

C2 N = NN r = N r N φ = N φ + N r (2.35) 2B2

17 Quantization of gravity

Now we can apply the ADM formalism on this metric. We can deﬁne this particular kinds of combination of functions A, B, C as our canonical variable L, R, Q and N,N r,N φ are lapse and shift functions respectively which are linear combinations of old lapse and shift functions. We will see that these lapse and shift are nothing but Lagrange multipliers. We can ﬁnd a dreibein which leads to the metric given in (2.34). It is given in lower triangular matrix form

  N 0 0   a  r  e µ = N LL 0  (2.36)   N φRQR

In terms of these variables the six constraints (2.31) will be

r φ r F0[e] ≡ LΠ2 − RΠ1 + QΠ1 ≈ 0 φ r F1[e] ≡ RΠ0 + QΠ0 ≈ 0 r 0 F2[e] ≡ LΠ0 + R ≈ 0 r r φ r φ F0[ω] ≡ ∂rΠ0 + Π1 Π2 − Π2 Π1 + ΛLR ≈ 0 r r φ r φ F1[ω] ≡ ∂rΠ1 + Π0 Π2 − Π2 Π0 ≈ 0 r r φ r φ F2[ω] ≡ ∂Π2 − Π0 Π1 + Π1 Π0 ≈ 0 (2.37)

r 0 φ 0 From the third and the second constraints we have Π0 = −R /L and Π0 = QR /LR. φ Substituting, these values and Π1 from the ﬁrst equation, into the last three constraints will yield

r φ r φ 0 0 F0[ω] ≡ Π1 Π2 − Π2 Π1 + ΛLR − (R /L) ≈ 0 r 0 φ 0 r F1[ω] ≡ ∂rΠ1 − (R /L)Π2 − (QR /LR)Π2 ≈ 0 r 0 r F2[ω] ≡ ∂rΠ2 + (R /R)Π2 (2.38)

r r φ Deﬁning PL = Π1 ,PQ = Π2 and PR = Π2 , we have the simpliﬁed Hamiltonian as

g r φ H = −NH − N Hr − N Hφ (2.39)

18 Quantization of gravity

g with the Hamiltonian constraint H and momentum constraints Hr, Hφ as

L Q R0 0 Hg = P P + ΛLR − P 2 + P P − ≈ 0 L R R Q R Q L L QR0 H = LP 0 − R0P − P ≈ 0 r L R R Q 0 Hφ = (RPQ) ≈ 0 (2.40)

The last constraint tells us that RPQ = α(t). Using the third constraint second constraint can be written as

0 0 0 LPL − R PR + QPQ ≈ 0 (2.41)

Therefore, we construct a six dimensional phase space spanned by L, R, Q and their conjugate momenta. These three constraints will dictate the dynamics of a point in this phase space, since the Hamiltonian is itself combination of these three constraints. As an example, axisymmetric solutions are obtained by taking Q = 0 and circularly symmetric solutions by Q = N φ = 0.

2.4 Equations of motion

The equations of motion are given by Poisson brackets with the Hamiltonian H. These are

˙ r 0 R = −NPL + N R NQ L˙ = −NP − P + (N rL)0 R R Q 0 2L Q QR 0 Q˙ = N P − P + N r + N φ R R Q R L R 0 0 2 N R NP 0 P˙ = NΛR − − Q + N φ R L R2 R 00 0 0 2 N N L NLP Q 0 P˙ = NΛL − + + Q − P P + (N rP )0 − N φ P R L L2 R2 R2 Q L R Q N R0 P˙ = P P + N rP 0 = − P (2.42) Q R Q L Q R Q 19 Quantization of gravity

where we have used the last constraint to get the result in the last equation of motion. This tells us that α(t) must be constant. So far we have made no additional assumptions except isotropy. Therefore, (2.40) and (2.42) must be satisﬁed for any isotropic classical solutions.

2.4.1 Static case

In this section we will discuss the static case and will work out two known examples starting from equations of motion. In static case, the time derivative of all canonical variables must vanish. Using RPQ = α with the ﬁrst two constraints equations of motion will be of the form

(N rL)0 αQ P = − R N R2 N rR0 P = L N 0 2αL Q QR 0 N − P + N r + N φ R = 0 R2 R L R N 0R0 α2N NΛR − − + N rP 0 = 0 L2 R3 L 00 0 0 2 N N L α NL αQ α 0 NΛL − + + − + (N rP )0 − N φ = 0 L L2 R4 R3 R R α2L αQ R0 0 P P + ΛLR − + P − = 0 L R R3 R2 L L R0 αQR0 P 0 − P − = 0 (2.43) L L R LR2

We have eight unknown functions but seven equations. The system is over-determined. So we have freedom to choose one of the unknown functions and we have set N r = 0.

2 This substitution gives PL = 0, PR = −αQ/R . We can readily check the last equation is satisﬁed identically. Now we are left with four remaining equations with ﬁve unknowns, namely

2αNL 0 + N φ R = 0 R2 N 0R0 α2N NΛR − − = 0 L2 R3 20 Quantization of gravity

00 0 0 2 N N L α NL αQ α 0 NΛL − + + − − N φ = 0 L L2 R4 R3 R α2L R0 0 ΛLR − − = 0 (2.44) R3 L

We still have freedom to choose another function and we take Q = 0. We can solve the ﬁrst equation, N φ0 = −2αNL/R3. The remaining equations are

N 0R0 α2N NΛR − − = 0 L2 R3 N 00 N 0L0 3α2NL NΛL − + + = 0 L L2 R4 α2L R0 0 ΛLR − − = 0 (2.45) R3 L

A close inspection reveals that the third equation can be obtained from the other two equations. So, there are actually two independent equations

N 0R0 α2N NΛR − − = 0 L2 R3 α2L R0 0 ΛLR − − = 0 (2.46) R3 L

These are two independent equations with three unknown functions. Therefore, we choose R(r) = r as expected in the static case. At this point we can consider two diﬀerent cases, i)Λ = 0 and ii)Λ 6= 0.

For Λ = 0 case, the corresponding gauge group SO(2, 2) turns into the Poincar´egroup by a Wigner-Inonu contraction. With R(r) = r the equations in (2.46) will yield the following solutions

1/µ L(r) = p1 + α2/µ2r2 p 2 2 2 N(r) = N+ 1 + α /µ r φ φ 2 N (r) = N+ + N+α/µr (2.47)

21 Quantization of gravity

φ where µ,N+ and N+ are constants of integration. To see what this means, we set N+ = 1 φ and N+ = 0, the corresponding line element is given by

N −2 j 2 ds2 = N 2dt2 − dr2 − r2 dφ − dt (2.48) µ2 µr2

If we compare this with spinning point particle metric, µ can be identiﬁed as the mass of the particle and j as the angular momentum.

With Λ 6= 0, solving second equation of (2.46) gives the solution

α2 −1/2 L(r) = Λr2 − M + (2.49) r2

where M is the constant of integration. Substituting this in the ﬁrst equation of (2.46), we have

α2 1/2 N(r) = N Λr2 − M + (2.50) + r2

φ φ 2 φ together with N = N+ + N+α/r . With N+ = 1 and N+ = 0 we recover the BTZ black hole metric

J 2 ds2 = N 2dt2 − N −2dr2 − r2 dφ − dt (2.51) r2

where we have identiﬁed M as the mass of the BTZ black hole and −α as the angular momentum.

22 Quantization of gravity

2.5 Fall-oﬀ conditions and boundary action

The total action on the hypersurface can be written as the sum of a bulk and boundary action

Z Z ˙ ˙ ˙ SΣ = dt dr PLL + PRR + PQQ − H + S∂Σ (2.52)

The boundary terms play an important role by canceling unwanted boundary variations in the action. Its values depend on the boundary conditions that are imposed. And it is reasonable to impose the boundary conditions such that the solutions asymptotically approach stationary spacetime as r → ∞. For the maximally extended point particle and the BTZ black hole, the boundary conditions are described below. The basic idea is to consider the asymptotic expansions of the canonical variables integer power of 1/r.

With this spirit, for spinning point particle, we adopt the boundary conditions such that it matches with (2.48) as r → ∞

R −→ r + O∞(r−2) 2 1 j± −2 ∞ −3 L −→ − 3 r + O (r ) µ± 2µ± Q −→ O∞(r−2)

∞ −1 PR −→ PR0 + O (r ) ∞ −1 PL −→ O (r ) −1 ∞ −2 PQ −→ j±r + O (r ) j± −2 ∞ −3 N −→ 1 + 2 r N± + O (r ) 2µ± N r −→ O∞(r−1)

φ φ j± −2 ∞ −3 N −→ N± + r + O (r ) (2.53) µ±

The higher order terms O∞(r−n) represents the asymptotic behavior that fall as r−n. It is implied that some function of t is multiplied with these terms. The ± sign refers to the quantities at right and left inﬁnities respectively. These fall-oﬀ conditions are consistent with the constraints and the equations of motion (2.43). We will consider all the terms of 23 Quantization of gravity

the Hamiltonian density H whose variation actually contribute to the boundary terms. As it turns out that the variations with respect to L and PQ will contribute to the boundary terms

Z Z 0 φ φ φ dt NR δ (1/L) − N RδPQ |∂M = − dt[N+δµ+ + N−δµ− − N +δj+ − N −δj−] (2.54)

This unwanted boundary variations must be canceled in the boundary action. Therefore, we take the boundary action to be

Z φ φ S∂Σ = dt[N+µ+ + N−µ− − N +j+ − N −j−] (2.55)

This boundary action aﬃrms the role of µ and j as the mass and angular momentum of the spinning particle.

For the BTZ black hole case, we adopt the following boundary conditions

R −→ r + O∞(r−2) r−1 M r−3 L −→ + ± + O∞(r−4) Λ1/2 2Λ3/2 Q −→ O∞(r−6)

∞ −2 PL −→ O (r ) ∞ −4 PR −→ O (r ) −1 ∞ −2 PQ −→ −J±r + O (r ) M N −→ Λ1/2r − ± r−1 N + O∞(r−2) 2Λ1/2 + N r −→ O∞(r−2)

φ φ −2 ∞ −4 N −→ N ± + J±r + O (r ) (2.56)

By explicitly calculating the variations of the contributing terms, we ﬁnd that variations

with respect to L and PQ contribute to the variations as before. So the boundary action

24 Quantization of gravity

will be

Z 1 S = − dt (N M + N M ) + N φ J + N φ J (2.57) ∂Σ 2 + + − − + + − −

Though inclusion of these boundary terms are necessary as it would unfreeze the evolution at the two inﬁnities, it would lead to the problem that the lapse and shift functions may also be varied at the boundary which would give µ± = j± = M± = J± = 0. As Kuchaˇr φ proposed, the solution to this problem is to assume N± and N± as a prescribed functions of t. This procedure is called parameterization at inﬁnity.

2.6 Embedding

The canonical structure of a spacetime can be understood with the help of the embedding procedure. We know the canonical structure of the ADM metric. We will imagine the two metrics describing the spinning particle and the BTZ black hole as leaves of a particular foliation of spacetime and we will embed the hypersurfaces of the ADM metric into these metrics.

2.6.1 Spinning particle

The metric in this case can be cast in terms of the Killing time and area radius as

2 j 2 ds2 = F dT 2 − dR2 − R2 dφ − dT (2.58) µ2F µR2

j2 where F = 1 + µ2R2 , µ and j are the mass and angular momentum respectively. With the rescaling of the Killing time as T = T/µ, the above metric can be conveniently rewritten as

2 2 2 2 2 2 j ds = F1dT − dR − R dφ − 2 dT (2.59) F1 R 25 Quantization of gravity

2 j2 where F1 = µ + R2 . The functions T and R are assumed to be functions of the ADM variables t and r. Comparing corresponding terms of this metric with that of the ADM metric in (2.34), we ﬁnd the following

0 R0T˙ − T R˙ N = L 0 F −1RR˙ 0 − F T˙ T N r = 1 1 L2 jT˙ N φ = − R2 2 −1 02 02 L = F1 R − F1T 0 jT Q = − (2.60) R

Inserting the lapse and shift functions into the ﬁrst equation of (2.42) we get

0 1 ˙ r 0 F1T 0 LPL PL = (−R + N R ) = − ⇒ T = − (2.61) N L F1 which upon inserting into the expression for L2, gives

j2 R02 F = µ2 + = − P 2 (2.62) 1 R2 L2 L

From the last equation of (2.60) we also ﬁnd

02 QR R 2 j = 2 − PL (2.63) LPL L

Thus we are able to write the mass and angular momentum in terms of canonical data.

Furthermore, diﬀerentiating F1 with respect to r, we ﬁnd

R02 0 R0 R0 0 F 0 = − P 2 = −2P P 0 + 2 1 L2 L L L L L 2R0 2P 2R0 = − H − L H − P 2 (2.64) L g L r R Q

26 Quantization of gravity

where Hg and Hr are the Hamiltonian and momentum constraints respectively. Therefore,

j2 0 2R0 2P 2P µ2 + − P 2 = − H − L H − Q H ≈ 0 (2.65) R2 Q L g L r R φ is linear combination of constraints. But µ0 and j0 do not require to be zero separately.

2.6.2 BTZ black hole

For the BTZ black hole the metric is expressed as

1 J 2 ds2 = F (R)dT 2 − dR2 − R2 dφ − dT (2.66) F (R) R2 where F (R) = ΛR2 − M + J 2/R2, M and J being the mass and angular momentum respectively. Embedding (2.34) into the BTZ metric will yield the same result as (2.60), just F1 replaced by F . Inserting the lapse and the shift into ﬁrst equation of (2.42) we obtain T 0 and substituting its value into the expressions for L2 and Q, we have

J 2 R02 F = ΛR2 − M + = − P 2 R2 L2 L 02 QR R 2 J = 2 − PL (2.67) LPL L

Here we again write the mass and angular momentum in terms of canonical data. We also have seen

R02 2 R0 R0 0 (F − ΛR2)0 = − P 2 − ΛR2 = −2P P 0 − 2ΛRR0 + 2 L2 L L L L L 2R0 2P 2R0 = − H − L H − P 2 (2.68) L g L r R Q

Thus we may write

J 2 0 2R0 2P 2P −M + − P 2 = − H − L H − Q H ≈ 0 (2.69) R2 Q L g L r R φ

27 Quantization of gravity

2.7 New canonical variables

The goal for seeking new variables is to cast the constraints in simpliﬁed form. From the expressions for µ(M) and j(J), it is clear that the mass and the angular momentum cannot be part of the same canonical chart since their Poisson brackets do not vanish. In the quantum theory, they are not simultaneously observable. We will work with non- zero cosmological constant as the spinning particle is the limit Λ → 0 taken together with M → −µ2. From the expression for F in (2.67) it is straightforward to see that if we deﬁne two canonical variables Z and PZ in the following way, they are canonically conjugate to each other i.e. {Z,PZ }P.B. = 1, where

R02 LP Z = − P 2 − ΛR2 − P 2 ,P = − L (2.70) L2 L Q Z 2F

The Poisson brackets of Z and PZ with other canonical variables are given below

2R0 0 {Z,R} = 0 {Z,P } = − − 2ΛR P.B. R P.B. L2 {Z,Q}P.B. = 2PQ {Z,PQ}P.B. = 0 R0P 0 {P ,R} = 0 {P ,P } = − L Z P.B. Z R P.B. F 2L {PZ ,Q}P.B. = 0 {PZ ,PQ}P.B. = 0 (2.71)

From now on we will use Z and PZ as our new canonical variables in stead of L and PL.

Looking at the Poisson brackets, the necessity of new canonical variables P R and Q is obvious. However, simple evaluation reveals that the Poisson brackets of

LP P Q = Q + L Q (2.72) F

with Z,PZ and R is zero and is also conjugate to PQ. The remaining problem is to ﬁnd

P R and we will use generating functional procedure to achieve this. The canonical trans-

formation from original chart {L, R, Q, PL,PR,PQ} to new chart {Z,R, Q, PZ , P R,PQ}

28 Quantization of gravity

can be found to be generated by

Z 2 0 PQ 0 −1 R G[L, R, PL,PQ] = dr LPL 1 − − R tanh (2.73) F LPL

and from this P R is determined to be

0 0 ΛRLPL (R /LPL) P R = PR − − 0 2 (2.74) F 1 − (R /LPL)

The fall-oﬀ conditions for the new variables can be easily determined from fall-oﬀ conditions of old canonical variables (2.56). Once we have the new canonical variables, we can write the Hamiltonian constraint in terms of new variables

2FP R0 Hg = Z [QP + RP ] − [2P (RP )0 + RZ0] (2.75) RL Q R 2FRL Q Q

and the momentum constraint now will be

Q − 2P P H = Z0P − R0P + QP 0 − Z Q H (2.76) r Z R Q R φ

Using the expression in (2.65), the Hamiltonian constraint can be greatly simpliﬁed

0 g F R H = [QPQ + RP R] − 2 Hr (2.77) RPL L PL

Now the full Hamiltonian can be written as sum of constraints

g Hf = RP R + QPQ 0 0 0 Hfr = Z PZ − R P R + QPQ 0 Hfφ = (RPQ) (2.78)

g r φ and the Hamiltonian is H = −NeHf − Ne Hfr − Ne Hfφ where the new multipliers are

NF Ne = RPL

29 Quantization of gravity

0 r r NR Ne = N + 2 L PL 0 φ φ r NR Q Ne = N − N + 2 (2.79) L PL R

We also notice that

0 0 R g Q Z PZ = Hf + Hfr − Hfφ (2.80) R R

So we could just as well consider the constrained system

g Hf = RP R + QPQ 0 HfZ = Z PZ 0 Hfφ = (RPQ) (2.81)

In the next section we will absorb the boundary action into the bulk action. This enables us to write the constrained system in more simpliﬁed form.

2.8 The boundary action

In terms of new variables the action takes the form

Z h i ˙ ˙ ˙ g r φ S = dtdr PZ Z + P RR + PQQ − NeHf − Ne Hfr − Ne Hfφ + S∂Σ (2.82) where the boundary action is given by (2.57). If the lapse and shift functions at the boundary are allowed to be varied freely, it would imply that the mass and the angular momentum both vanish at infinity. To avoid this problem and allow for non-vanishing mass and angular momentum, those functions must be treated as prescribed functions of the ADM time t i.e., lapse and shift must have fixed ends. To determine what these functions are, we compare the asymptotic ADM metric in (2.34) at fixed r

2 2 2 φ2 2 2 φ 2 2 ds = (N± − R N± )dt − 2R N±dtdφ − R dφ (2.83) 30 Quantization of gravity

with the asymptotic metric in the co-moving frame

2 2 2 2 2 ds = t± + 2Ω±R dtdφ − R dφ (2.84)

where t and Ω denote proper time and angular velocity. To match, we must take

q 2 2 ˙ def N± = ± 1 + r Ω±t± = ±τ˙± φ ˙ def N± = ∓Ω±t± = ±ω˙ ± (2.85)

where t± and ω± represent the proper time and the angular velocity as measured along constant r worldlines at the inﬁnities. The surface action now looks

Z 1 S = − dt (M τ˙ − M τ˙ ) + J ω˙ − J ω˙ (2.86) ∂Σ 2 + + − − + + − −

We will consider the Liouville form

Z ∞ 1 Θ1 := drPZ δZ − (M+δτ+ − M−δτ−) (2.87) ∞ 2

According to the fall-oﬀ conditions, limx→±∞ Z(r) = −M±. Therefore there exists a density function Γ(r) such that

Z r 0 0 0 Z(r) = −M− − dr Γ(r ),Z (r) = −Γ(r) (2.88) −∞

We rewrite the Liouville form

Z ∞ Z r 0 0 1 Θ1 := drPZ (r) −δM − dr δΓ(r ) − δ(τ+M+ − τ−M−) −∞ −∞ 2 1 1 + τ δM − τ δM 2 + + 2 − − Z ∞ Z ∞ Z r 1 0 0 = δM− − τ− − drPZ (r) − drPZ (r) dr δΓ(r ) 2 −∞ −∞ −∞ Z ∞ 1 0 0 1 + τ+ δM− + dr δΓ(r ) − δ(τ+M+ − τ−M−) 2 −∞ 2 Z ∞ Z ∞ Z r 1 0 0 = δM− (τ+ − τ−) − drPZ (r) − drPZ (r) dr δΓ(r ) 2 −∞ −∞ −∞ 31 Quantization of gravity

Z ∞ 1 0 0 1 + τ+ dr δΓ(r ) − δ(τ+M+ − τ−M−) (2.89) 2 −∞ 2

From the result above we are able to identify the conjugate variables as

1 Z ∞ m = M−, pm = (τ+ − τ−) − drPZ (r) (2.90) 2 −∞

In terms of these new variables the Liouville form is

Z ∞ Z r 1 0 0 Θ1 := pmδm + dr τ+δΓ(r) − PZ (r) dr δΓ(r ) (2.91) −∞ 2 −∞

Using the identity

Z ∞ Z r Z ∞ Z r 0 0 0 0 drPZ (r) dr δΓ(r ) = − drδΓ(r) dr PZ (r ) (2.92) −∞ −∞ −∞ ∞

we ﬁnd the Liouville form

Z ∞ Z r 1 0 0 Θ1 := pmδm + dr τ+ + dr PZ (r ) δΓ(r) (2.93) −∞ 2 ∞

This form now enables us to identify two other new canonical variables

Z r 0 1 0 0 Γ(r) = −Z (r),PΓ(r) = τ+ + dr PZ (r ) (2.94) 2 ∞

0 0 We notice that PΓ = PZ = T /2. Therefore, the Killing time is identiﬁed with the

momentum PΓ up to a constant. We can choose the constant in such a way that T

matches with τ+ at inﬁnity. Then we can write

Z r 0 0 T = 2PΓ = τ+ + 2 dr PZ (r ) (2.95) ∞

We can check that the momentum conjugate to the Killing time is PT = −Γ/2.

32 Quantization of gravity

Now we will look at the other Liouville form

Z ∞ Θ2 := dr P RδR + PQδQ − (J+δω+ − J−δω−) (2.96) −∞

−1 Again under the fall-oﬀ conditions, limx→±∞ PQ = −J±r . With the same logic, there exists a density function Σ(r) such that

Z r 0 0 R(r)PQ(r) = −J− + dr Σ(r ) (2.97) −∞

But according to the third constraint RPQ = constant. This means Σ(r) has to vanish.

We conclude, then, J+ = J− = J and

Z ∞ J Θ2 := dr P RδR − δQ − Jδ (ω+ − ω−) −∞ R Z ∞ JQ Z ∞ Q = dr P R + 2 δR − Jδ (ω+ − ω−) + dr −∞ R −∞ R Z ∞ JQ Z ∞ Q = dr P R + 2 δR + (ω+ − ω−) + dr δJ (2.98) −∞ R −∞ R

Thus this Liouville form tells us that we can choose

Z ∞ Q pJ = (ω+ − ω−) + dr (2.99) −∞ R as the momentum conjugate to J and on the other hand,

JQ P = P + (2.100) R R R2 conjugate to R. Now the new constraints will look like

HR = RPR 0 HT = T PT (2.101)

33 Quantization of gravity

and may be adjoined to the canonical action by means of new Lagrange multipliers. In reduced form the action will be of the form

Z ∞ h i ˙ ˙ ˙ T R S = pmm˙ + pJ J + dr PT T + PRR − (N PT + N PR) (2.102) −∞

The conﬁguration space of vacuum 2 + 1 − D gravity is spanned by coordinates T , R and two degrees of freedom,m and J. Now we are in a position to quantize the system. The momenta are raised to operator status and constraints act as an operator on a state

wave functional Ψ(m, J, t; T,R). The two constraints PT = 0 and PR = 0 tell us that the wave functionals are independent of T and R. Therefore the spacetime is described by the wave functional Ψ(m, J, t). We can even further say that the wave functional is time independent. Since the Hamiltonian is a combination of constraints , Hamiltonian vanishes,

iΨ(˙ m, J, t) = 0 ⇒ Ψ(m, J, t) = Ψ(m, J) (2.103)

The wave functional is characterized by two global parameters m and J. The time independence arises from the fact that the spacetime is stationary and it has time-like killing vector ﬁeld. Being time independent, the wave functional once prepared stays the same on all space-like hypersurfaces which is expected from the BTZ spacetime.

2.9 Summary

In this chapter, we have used the canonical reduction approach developed by Kuchaˇrto construct the canonical description of 2 + 1-D case of the BTZ black hole. The advantage of this approach is that it can be naturally extended to describe the quantum eﬀects of dynamical canonical collapse. Here, we have presented that starting from the Chern- Simons action, some well-known solutions can be derived if we associate certain Lie group structure with the corresponding gauge ﬁeld. To achieve this goal, we derived the

34 Quantization of gravity

Hamiltonian of the system which turns out to be identically zero since it is derived from a generally covariant system. The whole phase space is six-dimensional and therefore the constraints depend on six phase variables. Once we have the Hamiltonian, we readily found the equations of motion for the phase variables. The static case is given by the fact that the time derivative of the variables vanishes. Residual diffeomorphism constraints allow us to choose one of the variables to remove the under-determinacy of the solutions. These choices lead to two solutions, namely i) the spinning point particle (Λ = 0), ii) the BTZ solution (Λ 6= 0). To understand the underlying canonical structure we embed general axisymmetric metric in the ADM form into one of the solutions. Fall-off conditions are important since the metric has to approach the metric given by (2.48) and (2.51). With these fall-off conditions, it turns out that variations of some terms in the action do not die off at the boundary. To cancel these unwanted variations the boundary action plays a key role. To probe the underlying canonical structure, we embedded the two solutions into the ADM metric whose canonical structure is already known to us. This embedding procedure enables us to write the global parameters m and J in terms of canonical data. Then we sought for new canonical variables that will reduce the constraints into a simpler form. This has been achieved by a series of canonical transformations. The important point is that the information about the global parameters is preserved throughout the calculation. In terms of these new variables, the configuration space turns out to be two dimensional with two degrees of freedom. The constraints removed the dependence of phase space variables and the fact that the Hamiltonian is identically zero makes the wave functional time independent. Therefore the spacetime will be described by a wave functional with only two degrees of freedom, m, and J.

35 Chapter 3

Classical dust collapse model

The canonical formalism mentioned in the previous chapter becomes much simpler for non-rotating black holes and maybe generalized to describe the graviational collapse of matter. This allows us to consider the non-rotating gravitational collapse in 2 + 1-D and determine the collapse wave functional for the system. The earliest study of three- dimensional gravitational collapse without cosmological constant was carried out in [34]. In the case of circularly symmetric and homogeneous dust collapse, it has been shown that the collapse to a black hole sensitively depends on initial data. In the absence of the cosmological constant, the collapse may or may not occur depending on the initial velocity of the collapsing shells. If it collapses then it forms a naked, conical point source singularity [35]. Whereas in AdS spacetime the end state of collapse leads to BTZ black hole provided that the initial density is suﬃciently large. If not, then the end state is again naked conical singularity. These results provide the ground for carrying out numerical studies of critical phenomena associated with the collapse [36]. It has been also conﬁrmed for inhomogeneous dust collapse in [37, 38]. In [39–42] quantization procedure of dust collapse have been carried out that leads to interesting results. In this chapter, we are going to describe the quantum gravitational collapse in 2 + 1-D. Rotating collapse has been already described classically in [43].We will employ canonical quantization on Lemaˆıtre-Tolman-Bondi (LTB) family of solutions [37, 44–46] that describes the collapse

36 Quantization of gravity or expansion of spherically symmetric inhomogeneous mass distribution. Quantizing the LTB metric will enable us to understand the dynamics of the collapsing system under the influence of the quantum effects which will be described by functional solutions of the Wheeler-DeWitt equation(WDW) equation [47, 48]. We will see that the equation admits two sets of solutions. In the first solution, the collapse will form the apparent horizon and matter on both sides will coalesce at the horizon. The exterior infalling waves are collapsing dust shells which are accompanied by interior outgoing Unruh radiation. The second solution represents the case when matter moves away from the apparent horizon. Having found these solutions, now we may attempt to realize Stephen Hawking’s proposal in a quantized model of dust collapse in 2 + 1-D with negative cosmological constant. Hawk- ing’s proposal is in agreement with the first solution only. However, the superposition of these two solutions leads to the picture that both the exterior infalling and interior outgoing wave modes are transmitted through the apparent horizon. The outgoing modes are suppressed by the Boltzmann factor. This corresponds to the traditional picture of black hole evaporation.

For the moment if we take the first solution where continued collapse does not occur, we can expect the collapse process to end up in a spherically symmetric quasi-stable, static configuration of finite size. We will argue that the interior outgoing Unruh radiation during collapse will generate enough conditions to sustain such a quasi-stable configuration. The process of outgoing Unruh radiation will lead to a negative mass singularity which will weaken gravity slowly causing an expansion eventually at a later time[49].

We will consider a dust ball collapsing in 2 + 1-D spacetime with negative cosmological constant. In general, the collapsing solution is described by LTB family of solutions. In a comoving and synchronous coordinates (τ, ρ, φ), the metric takes the following form

Re2 ds2 = dτ 2 − dρ2 − R2dφ2 (3.1) 2(E − F )

37 Quantization of gravity

The physical radius R(τ, ρ)) and energy density (τ, ρ) obey the following

(R∗)2 = 2E − ΛR2 Fe 2πG(τ, ρ) = (3.2) RRe

∗ where Λ is the cosmological constant, G is Newton’s constant in 2 + 1-D, and e are derivatives with respect to comoving time (τ) and shell label (ρ) respectively. We note that without loss of generality we can rescale the shell labels in such a way that at some initial time the physical radius coincides with the shell label, i.e. R(0, ρ) = ρ. Exploiting this freedom gives an idea about the quantities E and F which may be given as

Z ρ F (ρ) = 2πG ρ0(0, ρ0)dρ0 0 2 2 E(ρ) = [∂τ R(0, ρ)] + Λρ (3.3)

The quantity 2F (ρ) represents total gravitational mass content inside the shell labeled by ρ. E(ρ)/2 is the total energy per unit mass in the shell labeled by ρ. They are called mass function and energy function. From the deﬁnition of F (ρ), we would expect that it is a positive and monotonically increasing function of ρ. We will exculde the possibility of shell crossing singularity by assuming Re > 0. In the case, when E is constant, the solution to (3.1) for collapsing dust is given by

r r ! 2E √ Λρ2 R(τ, ρ) = sin − Λτ − sin−1 (3.4) Λ 2E

From this solution, it is evident that shells are inevitably doomed to hit the singularity that is, the phyical radius will shrink to zero at ﬁnite proper time

r 2 1 −1 Λρ τ0(ρ) = √ sin (3.5) Λ 2E

Different shells will collapse to the singularity at different proper times which is indicated by the dependence of ρ. It can be shown that the apparent horizon is defined by ΛR2 −

38 Quantization of gravity

2F = 0 where the null divergence vanishes. A shell labeled by ρ becomes trapped once it crosses this apparent horizon i.e. R < p2F/Λ. Only those shells that satisfy F > 0 will be trapped at proper time

r 2 r ! 1 −1 Λρ −1 F τah(ρ) = √ sin − sin (3.6) Λ 2E E

As τ0(ρ) > τah(ρ), collapsing shells will be trapped ﬁrst before they hit the singularity. In other words, singularity formation is not a necessary condition for the formation of trapped surface.

3.1 Collapse wave functionals

To describe the collapse wave functionals we need to consider the canonical structure of the spacetime which is well described by the Arnowitt-Deser-Misner(ADM) metric. To that goal, we embed cicularly symmetric ADM metric

ds2 = N 2dt2 − L2(dr + N rdt)2 − R2dφ2 (3.7) into LTB family of solution given by (3.1). N and N r are lapse and shift functions. The total action can be written in terms of the metric functions as

Z h i ˙ ˙ g r SEH = dtdr PLL + PRR − NH − N Hr + S∂Σ (3.8)

where the angular dependence has been integrated out, PL and PR are conjugate momenta of L(t, r) and R(t, r) respectively given by (G = 1)

1 h i P = − R˙ − N rR0 L N 1 h i P = − L˙ − (N rL)0 (3.9) R N

39 Quantization of gravity

S∂Σ is the surface term of the action. Since the lapse and shift appear as Lagrange multipliers in the action, the corresponding Hamiltonian and momentum constraints are given by

R0 0 Hg = −P P − ΛLR + ≈ 0 L R L 0 0 Hr = R PR − LPL ≈ 0 (3.10)

The matter content will also contribute to the collpase. Following the paper [50, 51],

k the dust is described by eight spacetime scalars (, τ, Z ,Wk) with decomposition of four velocity in the cobasis as

k Uµ = −τ,µ + WkZ ,µ (3.11) where is the dust proper energy density, τ is the proper time along particle ﬂow lines, Zk are the comoving coordinates of the dust and W k ae the spatial components of four velocity in the dust frame. The dust action is given by

Z 3 p α SD = d x |g| (U Uα + 1) (3.12)

In the case when the particles do not move spatially in the dust frame i.e. W k = 0 the dust action can be simpliﬁed in the form

Z d r d SD = dtdr Pτ τ˙ − NH − N Hr (3.13) where

LR P = − (τ ˙ − N rτ 0) (3.14) τ N and the constraints look

r τ 02 Hd = P 1 + τ L2

40 Quantization of gravity

d 0 Hr = τ Pτ (3.15)

To derive the constraints we have used the fact that is a Lagrange multiplier since the action does not involve any time derivatives of dust proper energy density . The total Hamiltonian and momentum constraints are sum of these two constraints. It is obvious from the action that the Hamiltonian is identically zero. The equations of motion of all variables can readily be found out using Poisson brackets.

Embedding of the ADM metric (3.7) in the metric given by (3.1) is useful in order to ﬁnd a suitable canonical variable. This procedure enables us to write the mass function F in terms of the canonical variables as

1 R02 1 F = P 2 − + ΛR2 = ΛR2 − F (3.16) 2 L L2 2

It has already been shown in [22, 52] that the mass function can be elevated to a canonical variable by making proper canonical transformation. If we take momentum conjugate to

the mass function as PF = LPL/F, by directly taking Poisson bracket with F one can show that they satisfy the commutation relation

{F,PF }P.B. = 1 (3.17)

It is to be noted that F = 0 deﬁnes the apparent horizon and plays an important role

in the collapse. F does not commute with PR. Therefore they do not form a canonical

chart. We need to make a canonical transformation from {R, L, PR,PL} to new canoni-

cal variables {R,F, PR,PF }. This transformation can be shown to be generated by the generating function G[R, L, PL]

Z 0 −1 0 G[R, L, PL] = dr LPL − R tanh (LPL/R ) (3.18)

41 Quantization of gravity

And PR is given by

0 0 ΛRLPL (LPL/R ) PR = PR + − 0 (3.19) F 1 − (LPL/R )

In terms of these new variables, the constraints can be simpliﬁed to the form

1 Hg = − FP P + F −1R0F 0 L F R 0 0 Hr = R PR + F PF (3.20)

The fall-off conditions of the variables are important as they are needed to asymptotically match the flat metric at infinity. The fall-off conditions are given in [40]. These fall-off conditions lead to

Z Z S∂Σ = − dtM+(t)τ˙+ + dtM0(t)τ˙0 (3.21)

This is called parameterization at infinity. This removes the necessity of fixing the lapse function at the boundary. We are interested in finding new variables that absorb the boundary terms of the action. To this end, we will aim to cast the homogeneous part of the action into Liouville form. This is done by introducing a a density function Γ(r) as

Z r 0 0 0 F (r) = M0 + dr Γ(r ), Γ(r) = F (r) (3.22) 0

Consider the Liouville form

Z ∞ Θ = PF δF + τ+δM+ − τ0δM0 (3.23) 0

Using the above deﬁnition of F we can write

R ∞ 0 0 R ∞ Θ = 0 dr PF (r ) − τ0 δM0 + (δM+ − δM0) 0 drPF (r) R ∞ R r 0 0 − 0 drδΓ(r) 0 dr PF (r ) + τ+M+ R ∞ = p0δM0 + p+δM+ + 0 PΓ(r)δΓ(r) (3.24)

42 Quantization of gravity

where

p0 = −τ0 Z ∞ p+ = τ+ + drPF (r) 0 Z r 0 0 PΓ(r) = − dr PF (r ) (3.25) 0

To get this result we have made use of the identity

Z ∞ Z r Z ∞ Z ∞ 0 0 0 0 drPF (r) dr δΓ(r ) = drδΓ(r) dr PF (r ) (3.26) 0 0 0 r

The new action will be of the form

Z Z h i ˙ ˙ ˙ ˙ g r SEH = dt p0M0 + p+M+ + dr Pτ τ˙ + PRR + PΓΓ − NH − N Hr (3.27)

with constraints given by

r 1 τ 02 Hg = − −FP 0 P + F −1R0Γ + P 1 + ≈ 0 L Γ R τ L2 0 0 0 Hr = R PR − ΓPΓ + τ Pτ ≈ 0 (3.28)

We see that after the canonical transformations the whole system is described by a six dimensional phase space spanned by the dust proper time τ(t, r), the physical shell radius

0 R(t, r), the mass density Γ(r) = F (r) and their conjugate momenta Pτ (t, r),PR(t, r) and

PΓ(t, r). The Hamiltonian constraint can further be simpliﬁed using the momentum

constraint to eliminate PΓ. The constraints can now be replaced by (Appendix A)

2 2 Γ P 2 + FP − = 0 τ R F 0 0 0 R PR − ΓPΓ + τ Pτ = 0 (3.29)

where the prime denotes derivatives with respect to the ADM radial variable r and the mass function F ≡ ΛR2 − 2F . To study the quantum mechanical behavior of the collapsing shells, we will apply standard Dirac’s quantization procedure on the constraints. We

43 Quantization of gravity

ˆ will replace X by X and PX by −∂/∂X into the constraints. With this substitution, the Hamiltonian constraint becomes the Wheeler-DeWitt equation. Noting the momentum constraint, we choose the wave functional on which quantum mechanical constraints will act, as

Z Ψ[τ, R, Γ] = exp −i drΓ(r)W(τ(r),R(r), Γ(r)) (3.30)

It automatically satisﬁes the momentum constraint provided W does not have any explicit r dependence. To solve this we will consider a one-dimensional discrete lattice system where the discrete points ri’s are distance σ apart. Taking our ansatz for wave functional (3.30), the WDW equation will yield

" 2 2 # 2 2 2 ∂Wi ∂Wi 1 ∂ W ∂ W ∂Wi ωi + Fi + + ωi 2 + Fi 2 + Ai + Bi = 0 ∂τi ∂Ri Fi ∂τi ∂Ri ∂Ri (3.31)

With the substitution Wi = −iWi and equating each co-eﬃcient to zero as it is true for arbitrary ωi, we have three independent equations to be solved

2 2 ∂Wi(τi, Γi,Ri) ∂Wi(τi, Γi,Ri) 1 + Fi = ∂τi ∂Ri Fi 2 2 ∂ Wi(τi, Γi,Ri) ∂ Wi(τi, Γi,Ri) ∂Wi(τi, Γi,Ri) 2 + Fi 2 + Ai(Ri, Γi) = 0 ∂τi ∂Ri ∂Ri Bi(Ri, Γi) = 0 (3.32)

The solution to these equations is (Appendix B)

Z p 2 1 − ai Fi Wi = aiτi + dRi (3.33) Fi

p where ai = 1/ 2(Ei − Fi). The wave functional will be then

Y Ψ = lim Ψi(τi,Ri, Γi) σ→0 i

44 Quantization of gravity

( " #) Z p1 − a2F Y ωibi i i = lim e × exp −iωi aiτi ± dRi (3.34) σ→0 F i i where ωi = σΓi. The wave functional is deﬁned except at the apparent horizon (Fi = 0). Therefore, two diﬀerent solutions exist, one for exterior and one for interior to the apparent horizon. One way to get rid of this singularity is to match the solutions on both sides at the apparent horizon. The standard technique is to analytically continue the solutions to the complex plane. This procedure proved its importance by showing Hawking radiation as a tunneling process through the apparent horizon. The two sets of solutions to the WDW equation are

 √ 1−a2F  ωibi R i i e × exp −iωi aiτi + dRi F Fi > 0 (1)  i Ψi = πω √ (3.35) − i 1−a2F g ωibi R i i e i,h × e × exp −iωi aiτi + dRi Fi < 0  Fi and √  πωi 2 − 1−a Fi  gi,h ωibi R i e × e × exp −iωi aiτi − dRi F Fi > 0 (2)  i Ψi = √ (3.36) 1−a2F ωibi R i i e × exp −iωi aiτi − dRi Fi < 0  Fi where

gi,h = ∂RFi(Ri,h)/2 (3.37) is the surface gravity of the i-th shell at the apparent horizon. If we look at the phase velocity of (3.35), we see that the flow on both sides of the apparent horizon would reach it and stop there. On the other hand, (3.36) represents a flow away from the apparent horizon on both sides. The interior infalling wave will continue collapsing to the central singularity. We note that in the first case, the interior outgoing shells flow towards the apparent horizon with a relative probability determined by Boltzmann factor at Hawking temperature of the shells. In the second case, the outgoing shells have the same suppression factor. This represents thermal (Unruh) radiation. These two solutions 45 Quantization of gravity

are two bases for the dynamics of the quantum system. Superposition of these two solutions leads to the picture of continued collapse to the singularity with an outgoing thermal radiation in the exterior. In the case of (3.35), there will be Unruh radiation inside but no radiation outside and the continued collapse will stop at the apparent horizon before ending up in the singularity. However more physics is needed to provide the selection rule for this particular choice of the wave functional.

3.2 A quasi-classical static conﬁguration

In this section, we will present a toy model of quasi-classical dust configuration in 2 + 1- D in support of our analysis in the previous section [53]. Since (3.35) tells us that the collapse stops at the apparent horizon in the form of a quasi-stable compact object, we will look for solutions to Einstein equations with finite boundary radius that encapsulates the effect of the Unruh radiation. Two criteria have to be satisfied

• The matter should condense on the apparent horizon.

• The solution should match smoothly with external BTZ vacuum at the boundary.

We will consider a spherically symmetric dust ball with metric

ds2 = e2A(r)dt2 − e2B(r)dr2 − r2dφ2 (3.38) where A(r) and B(r) are smooth inﬁnitely diﬀerentiable functions of r. The stress-energy

µ tensor for this dust ball will be of the form T ν = diag{−(r), pr(r), pφ(r)} where (r) is

the energy density of the dust ball, pr(r) is the radial and pφ(r) is the tangential pressure of the dust ball. The ﬁeld equations will be

A0 − Λe2B = 4πGp (r) r r e2(A−B)B0 + Λe2A = 4πGe2A(r) r 46 Quantization of gravity

−2B 2 00 0 0 02 2 e r (A − A B + A ) − Λr = 4πGpφ(r) (3.39)

from these three equations it is right away clear that the three components of stress energy tensor are not all independent. We can arbitrarily choose any two stress energy components. Those will determine the unknown metric functions A(r) and B(r) from the corresponding two equations. The third will be determined by the remaining field equation. From the previous section, we know that the radius of the apparent horizon is determined by vanishing F, i.e.F = ΛR2 − 2F = 0. Here R is the physical radius which is, in this case, nothing but r since we are dealing with a quasi-stable stable configuration. After the collapse has stopped at the apparent horizon, the mass function at a radius r is F (r) = Λr2/2. We can immediately find out the energy density (r) as

Λ (r) = (3.40) 2πG

a constant. This determines tt-component of stress energy tensor. For spherical symmetry, we will choose the tangential pressure to be zero. From the second equation we can solve for B(r)

2B 2 −1 e = (C1 − Λr ) (3.41)

The meaning of the constant C1 will be clear later in this chapter. With the solution for B(r), φφ-component of ﬁeld equations yields

√ ! ! Λr e2A = cosh2 arctan √ + C (3.42) 2 2 C1 − Λr

This metric function indicates the existence of a singularity at r = 0. Similarly, from the rr-component of ﬁeld equations we have

√ √ Λ tanh C + arctan √ Λr 2 2 Λ C1−Λr p = + √ (3.43) r 2 2 Λr − C1 r C1 − Λr

47 Quantization of gravity

We also want to match the interior geometry smoothly with the outer BTZ vacuum

geometry at the boundary rb. The vacuum BTZ metric is given by

ds2 = f(R)dT 2 − f(R)−1dR2 − R2dφ2 (3.44)

2 where f(R) = ΛR − GMs and Ms is the BTZ mass of the dust ball. The junction

conditions require that, at rb = Rb, (Appendix C)

s e2A(rb) −B(rb) p 0 0 dT = dt e = f(Rb) 2A (rb) = (lnf) |Rb (3.45) f(Rb)

Solving the last two equations, we can determine the integration constants

2 2 C1 = Λ(2r − r ) b s ! ! −1 rb −1 rb C2 = − tan + tanh (3.46) p 2 2 p 2 2 rb − rs rb − rs

2 where we deﬁne GMs = Λrs .

3.3 Energy extraction

As we saw that our solutions depend on two constants, namely the BTZ radius rs and

the boundary radius rb. We can ﬁnd the relationship between these two constants from (3.46) as

Ms = Mb − (C1/G − Mb) (3.47)

where the quantity (C1/G − Mb) = M0 is interpreted as the mass-energy extracted from the center of the dust ball by the outgoing Unruh radiation. To estimate the value of

M0, we will consider each shell as a quantum harmonic oscillator with frequency ωi. We already know the mean energy for a single quantum harmonic oscillator, located at lattice

48 Quantization of gravity

point i,

ω hE i = i coth(β ω /2) (3.48) i 2 i i

−1 −1 where βi = (kTi) = (Λri) . For this particular collapse that we consider here, ωi =

σΛri/G and therefore βiωi = σ/G. We should be careful about the size of σ since it cannot P be arbitrarily small in the continuum limit as the total energy, E = ihEii, would go unbounded. Instead, it will be such that it is microscopically large but macroscopically small so that each lattice spacing has the size of many Planck lengths. Having said that, the mean energy of the Unruh radiation inside the apparent horizon will be

N N 1 X X Λiσ2 ΛN(N + 1)σ2 Λr2 hEi ∼ ω = = = b (3.49) 2 i 2G 4G 4G i=1 i=1

where N is the total number of shells within the boundary, i.e., rb = Nσ. Substituting 2 this mean energy expression back in (3.47), the constant C1 comes out to be C1 = 5Λrb /4 and therefore we have √ 3r r = b (3.50) s 2

So we see that the BTZ radius rs lies within the boundary of the gravitational body (rb) implying that the collapse does not end up forming an event horizon.

3.4 Summary

In this chapter, we have shown that within the quantum description it is possible that a spherical collapse does not lead to a black hole. Instead the collapse process stops when all shells arrive at the apparent horizon. During the collapse, infalling shells of matter do not cross the apparent horizon and are accompanied by outgoing Unruh radiation from the center which also stops at the apparent horizon. No Unruh radiation escapes beyond the apparent horizon. The outgoing Unruh radiation leaves behind a negative 49 Quantization of gravity mass singularity at the center as the system settles into a quasi-stable static configuration. We were able to find such a static solution which smoothly matches to the BTZ vacuum at the boundary of the dust ball and the energy density describing the dust has coalesced on the apparent horizon. The solutions are uniquely determined by the boundary radius and BTZ horizon radius of the dust ball. They differ by the magnitude of the mass of the central singularity. The boundary lies outside the BTZ horizon radius and no Unruh radiation escapes. The quantum effects prevent the shells from crossing their corresponding apparent horizon leading to the formation of no black hole. This result is in accordance with Stephen Hawking’s proposal.

50 Chapter 4

Dark matter as Bose-Einstein condensate

4.1 Introduction

With the advancement of observational astronomy, we already have enough evidence that the universe contains much more matter content than the visible part. The reason for not detecting this missing matter is that it does not interact with the ordinary baryonic matter with suﬃcient strength that constitutes the visible universe. This missing matter is called dark matter. In fact, baryonic matter only constitutes 4% of the total mass-energy of the universe. Rest of the mass-energy is composed of dark matter and dark energy out of which 23% is dark matter. Several astrophysical studies of distant supernovae [54, 55] have also conﬁrmed the excess matter content of the universe. The dark energy provides a negative pressure on the universe that made the universe expand and is well described by a positive cosmological constant [56–58].

Though the dark matter cannot be detected using ordinary observational techniques, it shows the evidences of its existence and its huge impact on the universe through its gravitational eﬀect. In 1932 J.H. Oort [59] found that missing matter in the galactic

51 Quantization of gravity

disk which was not explained by any kind of experiments. In 1933, Swiss astronomer F. Zwicky [60] became the first one to apply the virial theorem to predict the total mass of a large structure, particularly of Coma cluster. Zwicky measured the variance of peculiar velocities of visible matter in Coma cluster via red shift and showed that the relative speed of the galaxies in Coma cluster is far too great to be held together by the gravitational attraction of its visible matter only. So there must have been something else that holds them together. His studies suggest that the total gravitational mass of Coma cluster is about two orders of magnitude larger than the visible mass. However, at present, it is believed that the potential in which the stars are moving is not only because of the disk but the total matter in the galaxy that exists in the form of a halo, called dark matter halo. These two observations mark the beginning of the search for the dark matter. In 1970, Rubin studied the galaxy rotation curve for M31 [61] and showed that unlike the theoretical prediction that tells that the velocity of stars and other objects should fall as 1/r1/2 from the center of the galaxy, velocity remains almost constant in the outer edge of the galaxy. Several studies [62–70] confirmed that the majority of the spiral galaxies exhibit flat galaxy rotation curve in the outer part of the galaxy. The rotation curves of neutral hydrogen clouds in spiral galaxies measured by Doppler

effect are found to be roughly flat with tangential velocity typically v∞ ∼ 200 km up to maximum observed radius of 50 kPc. This flat curve can be explained if we consider that the gravitational mass of the galaxy increases linearly with the distance from the center. This would imply that the spiral galaxies are surrounded by massive non-luminous dark matter halo whose gravitational pull helps to hold everything together over cosmological distances. Other evidence like CMB anisotropies [71, 72] combined with large-scale data and type Ia supernova luminosity data show 4% contribution of visible matter to the total energy of the universe. Gravitational lensing [73], X-ray spectra in galaxies [74] and the high-velocity dispersion and gas temperature in a cluster of galaxies [75] all these results point to the fact that galaxies are composed of a luminous galactic disk surrounded by the dark matter halo. To explain the observational evidence of dark matter such as Bullet cluster, physicists have proposed various particles that could reasonably explain

52 Quantization of gravity it. These theories claim that dark matter is composed of weakly interacting particles instead of modiﬁcation of gravity. In spite of all this, the true nature of dark matter is still unknown to us. Observations seem to support the idea that dark matter is a non-baryonic, non-relativistic and weakly interacting particle. Therefore it is necessary to entertain other theories that could give some new insights into the dark matter. One such explanation is scalar ﬁeld model of Dark matter SFDM.

In the Standard Model, dark matter is generally modeled as a system of collisionless particles which is called Λ Cold Dark Matter model(ΛCDM). It claims that the universe is mostly dominated by cold, neutral and weakly interacting massive particles (WIMP) which are non-baryonic [76], pressureless and behaves like a cold gas. Though they have not been detected yet. To explain the observational data, ΛCDM model was proposed [77, 78]. In the Standard Model of cosmology the matter component of the universe decomposes itself into baryons, neutrinos etc. and cold dark matter which would be responsible for structure formation in the universe. This model then considers a flat universe with 96% of unknown matter which is of great importance in the cosmological context. It also supposes a homogeneous and isotropic universe which is best described by Friedmann’s equations today. We now know that the universe is not homogeneous and isotropic on the scale of megaparsecs but Standard Model gives us a framework to study the evolution of structures such as galaxies or clusters starting from small fluctuations in the density of the early universe. As of now, ΛCDM model seems to be the most successful model in fitting observational data [79]. The ΛCDM model successfully describes the accelerated expansion of the universe. It also explains the Cosmic Microwave Background radiation with high precision. It also provides tools to study large-scale isotropy of the universe. But this model faces several challenges to explain observations on smaller scales such as central densities of dark halos (Cusp-core problem) and low surface brightness galaxies, an excess of satellite galaxies predicted by N-body simulations (Missing satellite problem) etc. [80–82]. Another problem occurs in case of pure disk galaxies which do not appear naturally in numerical simulations of structure formation within Standard Model

53 Quantization of gravity cosmology. Some anomalies between the CMB mass power spectrum, obtained by the Sloan Digital Sky Survey (SDSS) and that predicted by the CDM paradigm has also been found in [83].

4.2 Dark matter and BEC

These discrepancies give motivations to explore new alternative models that can explain structure formations at galactic scales. The idea of modeling dark matter as a scalar ﬁeld parameterized by its mass and self-interaction was ﬁrst proposed in [84, 85]. Later it was implemented independently in [86, 87] to describe galactic halo as bosonic dark matter. According to this model, under non-relativistic regime, the dark matter consists of ultra-light particle (m ≈ 10−22eV) [88]. With such small mass two properties of the particles can be deduced:

• The de Broglie wavelength associated with these particles are of the order of kpcs which is the typical size of galaxies. Therefore quantum eﬀects cannot be ignored on large scales. Instead of looking at the individual particles, dark matter behaves as a quantum system [89] as a whole. Such a large Compton wavelength makes it diﬃcult for any structure formation on small scales, which was one of the problems with traditional dark matter models.

• The critical temperature below which they form condensates exceed the temperature of the universe at all times. So there is a possibility that they would form such a condensate at very early epochs in which small fraction of bosons fell into the ground state with no momentum.

The above two considerations indicate that dark matter scalar field can form a Bose- Einstein condensate with quanta of the field which leads to the cosmic structure formation [90–92]. Halos are nothing but spherically condensed Bose gas with all quanta of the scalar field in a single ground state. These halos were formed at the time when the universe 54 Quantization of gravity fell below its critical temperature. Such a low mass (m ≈ 10−22eV) of the particle is not hard to understand as they are also predicted by higher dimensional cosmology and String theory [93–96]. With a larger mass of the particles and couplings, it can form asteroid-sized stable BEC structures. At small scales, it can significantly deviate from the model predicted by CDM due to its superfluidity nature exhibited by BEC. As in [97, 98], it has been shown that self-interacting BECs has the ability to form galactic halos with constant density core. The rotation curve can also be modified to agree with the observational data [99]. The inner structure is very much dependent on the angular momentum distribution of dark matter particles. In the laboratory, we have evidence of a BEC to form a vortex structure when rotated with sufficient angular velocity. Certain fine structure in our Milky Way galaxy can only be explained if the dark matter particles have a net rotational motion [100]. Considering axionic dark matter, vortices may form due to rotation [101]. With these evidence in mind, rotating condensates attract special attention as described in [102–104]. In the following sections, we will provide a rigorous formulation of gravitationally bound non-relativistic rotating BECs [105].

The structure of thesis goes as follows. In the section (4.3), we will define the total action that describes the whole system. The section (4.3.1) and (4.3.2) concern about two assumptions, namely, weak-field limit and non-relativistic limit. Under these assumptions, we will calculate the action up to the leading order. In section (4.3.3), we will derive the equations of motion and incorporate the hydrodynamic analogy to find the equation of state. In section (4.3.4), we will use the variational approach to find the stable configuration of the system where all the particles are in lowest angular momentum state and describe different conditions for having a stable or meta-stable state in case of both types of self-interaction. In particular, we will use a single vortex ansatz to check the critical conditions. In section (4.3.5) we will extend this model to a dynamical problem where small oscillations of the halo about the local minimum have been considered to find approximate time scale of the oscillations. In section (4.4), we summarize all the results.

55 Quantization of gravity

4.3 The action

The field for BEC halos, as mentioned before, is taken to be a light, but not massless, scalar field. In our case, we will begin with the action for a complex scalar field in a curved spacetime background (weak-field),

S = Sφ + SG (4.1)

where the scalar ﬁeld action Sφ and the gravitational action SG are given by

Z 4 √ αβ ∗ Sφ = − d x −g g (∇αφ) (∇βφ) + V (|φ|) c3 Z √ S = d4x −gR (4.2) G 16πG and V (|φ|) is the arbitrary trapping potential. For our purpose we take it to be of the form

2 2 m c 2 2m V (|φ|) = |φ| + Ve1(|φ|) (4.3) ~2 ~2 where Ve1(|φ|) is the self-interaction of the scalar ﬁeld. For our analysis, we will take the action in the metric that is linearized about the ﬂat spacetime.

4.3.1 The GPP Action

In the weak ﬁeld approximation, gµν = ηµν +hµν where |hµν| |gµν| , the determinant of √ µν the metric can be expressed in terms of the perturbation as −g ≈ (1+h/2). h = η hµν is the trace of hµν. The scalar ﬁeld action can be written as

Z 1 S ≈ S (φ) − d4x h ηαβ(∇ φ)∗(∇ φ) + V (|φ|) − hαβ(∇ φ)∗(∇ φ) (4.4) φ 0 2 α β α β

56 Quantization of gravity

where we kept terms up to linear order of hµν. The zeroth order action S0(φ) is the action just in the ﬂat background. We will denote the perturbation terms in the action as S1(h, φ) and S2(hαβ, φ). In the non-relativistic limit, when the temperature is low enough the φ ﬁeld can be described by a complex wave function ψ as

2 φ = √~ e−imc t/~ψ (4.5) 2m

The wave function describes a condensed N particle system and is normalized to the particle number

Z d3r |ψ|2 = N (4.6)

Assuming Ve1 is homogeneous function of order 1, the trapping potential, in terms of ψ will be

2 mc 2 V (|ψ|) = |ψ| + Ve(|ψ|) (4.7) 2

Here Ve is the self interaction of the non-relativistic ﬁeld. We will expand the zeroeth order action in the non-relativistic limit

Z Z ↔ 2 3 i~ ∗ ~ 2 S0(φ) ≈ dt d r ψ ∇tψ − |∇ψ| − Ve(|ψ|) (4.8) 2 2m

To get this result, we have dropped terms that are quadratic in ψ/c˙ since in the non- relativistic limit the diﬀerence between the total energy and the rest mass energy becomes so small that we can ignore higher order terms. Noting that the gravitational ﬁeld is weak,

S1(h, φ) is proportional to the trace of the metric perturbation times the zeroeth order action and therefore it is of higher order than other retaining terms. With this consid-

erations the trace action S1(h, φ) can also be dropped without changing the resulting dynamics much and the last term can be expanded as

Z 4 αβ ∗ S2(hαβ, φ) = d xh (∇αφ) (∇βφ)

57 Quantization of gravity

Z 4 h tt ˙ 2 ti ˙∗ ∗ ˙ ij ∗ i = d x h |φ| + h φ ∇iφ + ∇iφ φ + h ∇(iφ ∇j)φ Z Z 1 i c2 ≈ dt d3r mc4htt |ψ|2 + ~ hti(ψ∗∇ ψ − ∇ ψ∗ψ) (4.9) 2 2 i i

ti i 2 tt 4 Deﬁning h = −A /c and h = −2ΦG/c , this correction term of the action will look like

Z Z 3 2 S2(hαβ, φ) ≈ dt d r −mΦG |ψ| − J.A (4.10)

where ΦG is the gravitational potential energy and

i ↔ J = ~ψ∗∇ ψ (4.11) i 2 i is the scalar current. Putting all terms together, the non-relativistic weak ﬁeld GPP action is

Z ↔ 2 4 i~ ∗ ~ 2 2 Sψ ≈ d x ψ ∇tψ − |∇ψ| − Ve(|ψ|) − mΦG |ψ| − J.A (4.12) 2 2m

The fourth term represents interactions between the gravitational field and the scalar field, the last term is what we call gravitomagnetic term which incorporates the frame dragging effect. If we define Aµ = (−ΦG,Ai) and covariant derivative as

im Dµψ = ∇µ − Aµ ψ (4.13) ~ then we can identify the following

i ↔ i ↔ i ↔ ~ψ∗D ψ = ~ψ∗∇ ψ + mA |ψ|2 = ~ψ∗∇ ψ − mΦ |ψ|2 2 t 2 t t 2 t G 2 2 − ~ (D ψ)∗(Diψ) = − ~ |∇ψ|2 − J.A (4.14) 2m i 2m

Thus the scalar ﬁeld action is of the form

Z ↔ 2 4 i~ ∗ ~ 2 Sψ ≈ d x ψ Dtψ − |Diψ| − Ve(|ψ|) (4.15) 2 2m 58 Quantization of gravity

In the weak field approximation, the interaction of scalar field with gravitational field is of the same form as electromagnetic coupling.

4.3.2 The Gravitational Action

In this section we will focus on the gravitational part of the action. We will determine the general form of the metric from the Einstein equations. It is convenient to work in

1 the Harmonic-transverse gauge, defined by hµν = hµν − 2 ηµνh and imposing the gauge ,ν 1 condition hµν = 0. In the weak field limit, the Einstein tensor is Gµν = 2 hµν and the field equation will be

16πG h = T (4.16) µν c4 µν

The gauge condition above tells us the known result that the stress energy tensor is conserved on flat background. Therefore, on flat background the stress energy tensor for the scalar field will be

∗ Tµν = ∇(µφ ∇ν)φ + ηµνL (4.17)

In the non-relativistic limit, keeping only leading order terms, we ﬁnd

4 2 Ttt ≈ mc |ψ| 2 Tti ≈ c Ji

Tij ≈ 0 (4.18)

Referring to the Einstein equations, this tells that, without loss of generality, we can take

2 2 hij = 0 . In this case, h = −h = −htt/c . If we take htt = −4ΦG then, h = 4ΦG/c and htt = −2ΦG. Other metric components are written as

1 2Φ h = h − η h = − G δ , h = h = −A (4.19) ij ij 2 ij c2 ij ti ti i

59 Quantization of gravity

The line element will be

2 2 2 2 i 2 i j ds = c (1 + 2ΦG/c )dt + 2Aidtdx − (1 − 2ΦG/c )δijdx dx (4.20)

In this metric the cross term appears because of the fact that there is a gravitoelectromagnetic interaction between the scalar ﬁeld and gravitational ﬁeld. The line element is subject to the gauge conditions

,µ 4Φ˙ ,µ h = G − ∇.A = 0, h = A˙ /c2 = 0 (4.21) tµ c2 iµ i

˙ 2 In the non-relativistic limit the term proportional to ΦG is of the order v ( 1), so we ˙ can ignore it and the new gauge conditions are now Ai = 0 = ∇.A.

To ﬁnd the bulk Lagrangian for gravitational part, we will match the above line element with the standard line element of Arnowitt, Deser and Misner (ADM)

2 2 2 i i j ds = N dt − Nidtdx − γijdx dx (4.22)

Then the bulk action in terms of the lapse function N, the spatial metric γij, intrinsic (3) scalar curvature R and extrinsic curvature of spatial hypersurface, Kij will be

Nc3 L = γ1/2 (3)R + K Kij − K2 (4.23) G 16πG ij

where K is the trace of Kij.(4.20) allows us to write the lapse, shift function and ﬁrst fundamental form up to linear order

2Φ N ≈ c 1 + G c2 Ni = −Ai 2Φ γ = δ 1 − G (4.24) ij ij c2

60 Quantization of gravity

Then upto the second order, the intrinsic curvature for spatial hypersurface is

4 2 (3)R = ∇2Φ + 3(∇Φ )2 + 8Φ ∇2Φ (4.25) c2 G c4 G G G and upto the ﬁrst order, the extrinsic curvature will be

1 1 f K = γ˙ − ∇ N ≈ ∇ A = ij (4.26) ij 2N ij (i j) 2c (i j) 2c

We can immediately conclude that the trace of the extrinsic curvature K vanishes by virtue of the gauge condition. The gravitational action, after little rearrangement, is

Z Z c2 S = β dt d3r −(∇Φ )2 + f f ij (4.27) G G 8 ij where β = (8πG)−1. The eﬀective total action is therefore

Z Z ↔ 2 3 i~ ∗ ~ 2 2 S = dt d r ψ ∇tψ − |∇ψ| − Ve(|ψ|) − mΦG |ψ| − J.A− 2 2m βc2 β(∇Φ )2 + f f ij (4.28) G 8 ij

Having found the action now we can use this to describe large scale structure as long as ψ(t, ~r) is interpreted as a Schr¨odinger ﬁeld and normalized to the total particle number.

4.3.3 Equations of Motion

Extremizing the action in (4.28) with respect to ψ and making use of the gauge condition we have the Schrodinger equation for ψ

2 ~ 2 0 i Dtψ = − ∇ ψ + mΦGψ + Ve (|ψ|)ψ (4.29) ~ 2m

61 Quantization of gravity

i there Dt = (∇t − A ∇i) is the transport derivative which incorporates the frame dragging due to rotation. Similarly, varying Ai and ΦG give

16πG ∇ f ji = ∇2Ai = − J i j c2 2 2 ∇ ΦG = 4πG |ψ| (4.30) upto the ﬁrst order respectively. We can readily see the solutions

Z ∗ ~0 ~0 3 0 ψ (t, r )ψ(t, r ) ΦG(~r) = −4πGm d r ~0 ~r − r ↔ Z ∗ ~0 ~0 8πiG~ 3 0 ψ (t, r )∇iψ(t, r ) Ai(~r) = 2 d r (4.31) c ~0 ~r − r respectively. it is convenient to treat the BEC as a superﬂuid. Following [99, 106–108]

ψ(t, ~r) = p%(t, ~r)eiS(t,~r) (4.32) we will bring the hydrodynamic analogy together with Madelung transformation [109]. Here % is the particle density and S is the phase of the wave function. Substituting this into Schrodinger equation and comparing the real and imaginary part we get

D % + ~ (∇%.∇S + %∇2S) = 0 t m ~ 2 m 1 0 DtS + (∇S) + ΦG + Ve (%) + Q(%) = 0 (4.33) 2m ~ ~ where Q is the quantum pressure given by

" # ∇2% 1 ∇%2 Q = − ~ − (4.34) 4m % 2 %

The velocity field is given by u = ~∇S/m. Being gradient of a scalar quantity, the velocity field is, in this case , irrotational. The first equation in (4.33) is same as the continuity

62 Quantization of gravity equation

Dt% + ∇.(%u) = 0 (4.35)

We see that it resembles the continuity equation. We also write the second equation in (4.33) as

1 2 1 0 ~∇Q ∇tu − ∇(A.u) + ∇(u) + ∇ΦG + ∇ Ve (%) + = 0 (4.36) 2 m m u being irrotational, follows the identity ∇(u2) = 2(u.∇)u. Similarly, we can write

∇i(A.u) = (A.∇)ui + uj∇iAj and we have

1 D u + (u.∇)u = − ∇ P − ∇ Φ + u ∇ A − ~ ∇ Q (4.37) t i i % i i G j i j m i where we write

% 0 ∇iP = ∇iVe (%) (4.38) m

The reason for deﬁning (4.38) is that it represents general form of the equation of state.

λ 2 For instance, If the interaction is quartic in nature Ve(%) = 4 % , then we ﬁnd

λ P = %2 (4.39) 4m

It is expected that a repulsive interaction (λ > 0) leads to positive pressure and an attractive interaction (λ < 0) to a negative pressure. The way we defined the transport derivative, it automatically takes the frame dragging effect into account. In terms of particle density and velocity field the gravitational potential becomes

Z ~0 3 0 %(t, r ) ΦG(~r) = −4πGm d r ~0 ~r − r Z ~0 ~0 16πGm 3 0 %(t, r )ui(t, r ) Ai(~r) = − 2 d r (4.40) c ~0 ~r − r

63 Quantization of gravity

4.3.4 Stable Conﬁguration

(Meta)stable conﬁguration of rotating BEC corresponds to (local) minimum energy of the system. The Hamiltonian of the system as we get from (4.28), is given by

Z 2 m 1 H = d3r ~ |∇ ψ|2 + V (|ψ|) + Φ |ψ|2 + J.A (4.41) 2m i 2 G 2

We will deﬁne a length parameter R that would give an estimate of average size of the BEC. A general ansatz for ψ(~r) representing a rotating condensate in spherical coordinates would be of the form

X iµt ψ(t, ~r) = w Fkl(r/R)Ykl(θ, φ)e (4.42) k≥|l|=0,1,..

w is the normalization constant, µ is the chemical potential related to the BEC halo, k,

l are integers such that −k ≤ l ≤ k, Fkl(r/R) are real function of r and Ykl(θ, φ) are

the spherical harmonics. For this approach we will treat Fkl(r/R) and R as variational function and parameter respectively holding the total number of particles and angular momentum to be ﬁxed.

4.3.4.1 Total energy

It is needless to say that k and l can take any integer values such that the above relation is satisﬁed. This following procedure can be extended to arbitrary angular momentum state. To understand the model with simplest possible case, we will assume the situation where all particles are in k = 1 = l eigenstate of angular momentum. So in our case, the wave function is

ψ(t, ~r) = wF (r/R) sin θeiϕeiµt (4.43)

64 Quantization of gravity

The normalization constant is

3N 1/2 w = 3 (4.44) 8πR C22

R ∞ 2 2 where N is the total number of bosons in the BEC and C22 is defined as C22 = 0 dξξ F (ξ). Once we define the function , we can get the expression for the gravitational potential by making use of addition theorem of spherical harmonics with vanishing boundary conditions at the origin and at infinity which is given below

l 1 X 1 r< ∗ 0 0 = 4π Y (θ , φ )Ylm(θ, φ) (4.45) |~r − ~r0| 2l + 1 rl+1 lm l,m >

and we ﬁnd ΦG to be

( " Z r/R Z ∞ # 3GmN 2 R 2 2 2 ΦG(r, θ) = − dηη F (η) + dηηF (η) 2C22R 3 r 0 r/R " 3 Z r/R 2 Z ∞ #) 1 2 R 4 2 r −1 2 + (1 − cos θ) 3 dηη F (η) + 2 dηη F (η) (4.46) 15 r 0 R r/R

From the expression of Ai in (4.31) we see that all terms will vanish except Aφ. This result is particular to this speciﬁc wave function we chose.

4 A (r, θ) = ~ Φ (r, θ) (4.47) φ mc2 G

Since in the non-relativistic regime, the galactic halos are well described as Newtonian systems, the Einstein-Klein-Gordon equations for complex scalar ﬁeld ψ are minimally

λ 4 coupled to gravity and provided with a self-interaction potential as V (|ψ|) = 4 |ψ| . From these two results in (4.41) we found the four terms of the Hamiltonian as

2 Z ∞ N~ 2 02 2 HK (R) = 2 dξ ξ F (ξ) + 2F (ξ) 2mR C22 0 2 Z ∞ 3λN 2 4 HV (R) = 3 2 ξ F (ξ) 40πR C22 0 2 2 Z ∞ Z ξ Z ∞ Z ∞ Gm N 2 2 2 2 2 2 HΦ(R) = − 2 dξξF (ξ) dηη F (η) + dξξ F (ξ) dηηF (η) 2RC22 0 0 0 ξ 65 Quantization of gravity

1 Z ∞ Z ξ 1 Z ∞ Z ∞ + dξξ−1F 2(η) dηη4F 2(eta) + dξξ4F 2(ξ) dηη−1F 2(η) 25 0 0 25 0 ξ 2 2 Z ∞ Z ξ Z ∞ Z ∞ 3~ GN −1 2 2 2 2 2 HA(R) = 2 2 3 dξξ F (ξ) dηη F (η) + dξF (ξ) dηηF (η)(4.48) c C22R 0 0 0 ξ where ξ = r/R is a dimensionless [110] parameter. The second term in HK comes from the contribution of the azimuthal motion of the condensate, HV is the effect of self interaction and HΦ and HA are the contributions of the gravitational field. To write the Hamiltonian in a simpler form we will define the following

Z ∞ m n Cmn = dξξ F (ξ) 0 Z ∞ m 0n Bmn = dξξ F (ξ) 0 Z ∞ Z ξ m 2 n 2 Dmn = dξξ F (ξ) dηη F (η) 0 0 Z ∞ Z ∞ m 2 n 2 Amn = dξξ F (ξ) dηη F (η) (4.49) 0 ξ

The Hamiltonian can now be cast in the form

H(R) = HK (R) + HV (R) + HΦ(R) + HA(R) Gm2N 2 1 = − 2 D12 + A21 + (D−14 + A4,−1) 2RC22 25 2 2 2 2 N~ (B22 + 2C02) 3λN C24 3G~ N 2 + 3 2 + 2 3 2 (D−12 + A01) (4.50) 2mR C22 40πR C22 c R C22

The Hamiltonian H(R) does not include the rest mass energy, Nmc2 of the condensate,

2 therefore total energy is given by E(R) ≈ Nmc + H(R). If HB(R) is the binding 2 energy per particle, then E(R) = N |mc − HB(R)|. Right away we identify |HB(R)| = |H(R)| /N.

Following the results in [111] let us deﬁne dimensionless parameters ρ and n as

r 3/2 M |λ| nM c R = p ρ, N = p ~ (4.51) 2 p m ~c m c |λ|

66 Quantization of gravity

p where Mp = ~c/G is the Planck mass. Then the binding energy per unit mass is

3 H ~ c A B C H(ρ) = = 2 + 2 + 3 (4.52) mN |λ| Mp ρ ρ ρ

where

n 1 A = − 2 D12 + A21 + (D−14 + A4,−1) 2C22 25 (B + 2C ) B = 22 02 2C22 3 3C24 3~ C = n sgn(λ) 2 + 2 2 (D−12 + A01) (4.53) 40πC22 |λ| cMp C22

We can readily see that A is negative and represents contribution of gravitational potential energy. The C02 term in B captures the effect of rotation while B represents the kinetic energy of the bosons. The last term in C refers to the frame dragging effect. Defining

2 2 3 a dimensionless parameter b = |λ| cMp /~ , the strength of the self-interaction can be characterized by the constant 3n/b2.

4.3.4.2 Minimum energy conﬁguration

From (4.52) the minimum of H can be found to be

s B B2 3C ρeq = ∓ + (4.54) |A| |A|2 |A|

We have two conditions for these two roots to be real and those are

• C > 0

• C < 0 & B2 > 3 |A| |C|

To make expressions look even simpler we deﬁne further

1 1 a = − 2 D12 + A21 + (D−14 + A4,−1) 2C22 25 67 Quantization of gravity

3C24 q = 2 40πC22 3(D−12 + A01) d = 2 (4.55) C22

With these substitution (4.53) will be

A = an d C = qn + sgn(λ) (4.56) qb2 where b2 is deﬁned before (4.3.4.1). The parameter a, q and d characterize the strength of the gravitational interaction, self-interaction and the eﬀect of rotation respectively. In terms of these the dimensionless equilibrium radius will be

s ! B 3 |a| q d ρ = 1 + 1 + + sgn(λ) n2 (4.57) eq |a| n B2 qb2

The existence of equilibrium radius is ensured by the fact that the term in the square root has to be non-negative. In Thomas-Fermi approximation, kinetic energy of the bosons is ignored. Therefore B approaches zero and equilibrium radius approaches a constant value independent of the number of the number of the particles. In absence of rotation equilibrium is possible for only repulsive interactions. If one does not include Thomas- Fermi approximation equilibrium radius decreases with increasing n.

1. C > 0

If C > 0 then either λ > 0 or

λ < 0 and b < pd/q (4.58)

We see that d = 0 is the case without rotation. In that case, C will be positive only for repulsive interactions. As long as C > 0, the number of particles n and the interaction strength b are subject to non-relativistic approximations. The validity of non- relativistic

68 Quantization of gravity

approximations can be estimated from the requirement that the de Broglie wavelength

of bosons λdB is much larger than the Compton wavelength λC . In general de Broglie wavelength is approximately the same size of the condensate which is of the order of the scale factor R for BEC. From (4.51) it follows that the condition for non-relativistic approximations to hold is

bρeq 1 (4.59)

We can put stronger condition on number of particles n by justifying not to include higher order general relativistic corrections . In other words, the equilibrium radius of the stable

2 conﬁguration has to be much larger than the Schwarzchild radius, ρS = 2n/b , therefore

ρS ρeq s b 3q 4B + 3d 1/2 ⇒ n + b2sgn(λ) def= n (4.60) 2 |a| 3q S

This set an upper bound on the particle number for any given value of b. For attractive interactions, the non-relativistic approximation ρeq given by (4.57) holds for small n since b is bounded from above. For repulsive interaction, there is no such restriction on b which can grow unboundedly. In fact within the framework of non-relativistic approximations, (4.57) admits a global minimum of the energy. Increasing the particle number n also increases the strength of gravitational attraction causing the radius of the condensate to decrease. In the large n limit the system approaches the equilibrium radius

s 3q d ρ = + 1 (4.61) n→∞ |a| qb2

2. C < 0

69 Quantization of gravity

C < 0 follows that the only possible case is

λ < 0 and b > pd/q (4.62)

The validity of non-relativistic approximation still is given by (4.59). As b can be arbitrarily large, this condition can always be satisﬁed. The energy is unbounded from below.

A local minimum can exist if n is smaller than certain critical value nc.

B q −1/2 def n ≤ 1 − = nc (4.63) p3q |a| db2

Above this critical value no metastable conﬁguration would exist.The equilibrium radius in terms of nc is

s ! B n2 ρeq = 1 + 1 − 2 (4.64) |a| n nc

The radius now continues to decrease with increasing n. Smallest metastable condensate corresponds to n = nc. In attractive case, the existence of minimum size comes from the fact that the inward gravitational and self interaction pressure is balanced by the outward quantum pressure. This condition has been employed in [104] to estimate the mass and radius of a non rotating axion drop. They found an upper bound in the axion drop mass. The requirement that we ignore general relativistic eﬀects leads to

p def nc b B/2 |a| = nS (4.65)

and the critical radius corresponding to critical number nc is

ρc = B/(|a| nc) (4.66)

4.3.4.3 Single vortex ansatz

[106, 107, 112] focus on using the variational approach to get an estimate of the size of 70 Quantization of gravity

the condensate. And these approaches have shown a good agreement with more precise results [106, 111, 113–117] For further analysis, we have to choose an ansatz for F (ξ). We ask for a trial wave function that behaves as a vortex of width R and it’s also required to be continuous. Since we are following the variational approach, this trial wave function does not need to be a solution to the equation of motion. Continuty of wave function at the origin suggests that as ξ approaches zero, F (ξ) should behave as ξl where l ≥ 1. Keeping this in mind, our ansatz will be,

F (ξ) = ξe−ξ2/2 (4.67)

We can right away write down the wave function in cylindrical coordinate as

ψ(ζ, φ) = ζe−ζ2−z2 eiφ (4.68)

ζ is the cylindrical radius. From (4.31) the gravitational potential becomes

GmN R2 Φ (r, θ) = 1 + (1 − 3 cos2 θ) erf(r/R) G r 4r2 2 2 # re−r /R 1 3R2 1 R2 + + cos 2θ − + (4.69) π1/2R 2 4r2 2 4r2

The necessary constants for this particular ansatz are

23 a = − √ ,B = 5/4 30 2π 1 5 q = , d = p2/π (4.70) 16(2π3)1/2 3

In case of C > 0, we can write the condition (4.59) and (4.60) as

" s # 4.09b 168 bρ = 1 + 1 + 4.66 × 10−3 + sgn(λ)n2 eq n b2 p 2 n nS = 0.14b 378 + b sgn(λ) (4.71)

These conditions can be met by small condensates with attractive interactions provided

71 Quantization of gravity

b is chosen suitably(b < 12.9). For repulsive interactions there is no upper limit for b and large condensates are possible. On the other hand, for C < 0 metastable state is posible with the critical number of bosons

168−1/2 n = 14.6 1 − (4.72) c b2

The non-relativistic condition is given by nc NS = 1.43b. This holds true as long

as b & 16.5. As the number of bosons reaches critical value nc, the equilibrium radius approaches

168 ρ = ρ (n → n ) = 0.279 1 − (4.73) c eq c b2

The critical number and radius of the BEC are now seen to be uniquely determined by the strength of self-interaction. It is to say that length scale R and the total mass M = mN depend on self-interaction and the mass of bosons.

4.3.4.4 Size of condensates(halos)

To test our model, we can consider the condensates formed by particles of mass and interaction strength typical of QCD axion, m ∼ 10−5 and b ∼ 2 × 107. the axion decay

−8 3/2 constant is taken to be roughly fa/Mp ∼ 5 × 10 (c/~) . From (4.51) we ﬁnd that

n M 2 N = c p ≈ 1.1 × 1060 (4.74) c b m

19 with a total mass of Mc = mNc = 1.95 × 10 kg. We also get the critical length scale to be

R = bρ ~ ≈ 5.6 × 106 ~ (4.75) c c mc mc

72 Quantization of gravity

The Compton wavelength for QCD axion is about 0.02 m, which gives the critical length

to be about Rc ≈ 110 km. Another way to deﬁne the size of a condensate is to ﬁnd the

radius within which 99.9% of mass is conﬁned, denoted by R99. It is roughly 3.5 times

the critical length, that is R99 = 385 km. When λ < 0, assuming b 102,(4.63) suggests that the maximum size of the condensate

is approximately independent of b. Then ρc ∼ 1/nc is also asymptotically independent of b. Making use of (4.51) one can show that the particle mass and interaction strength are

required to obtain the desired value of Rc and Mc,

2 1/2 2 1/2 B~Mp BcMp Rc m ≈ , b ≈ (4.76) |a| cMcRc 3q~Mc

42 Assuming R99 ∼ 50 kpc and the mass roughly three times that of visible galaxy Mc ∼ 10 kg, we ﬁnd mc2 ∼ 10−24 eV and b ∼ 104. We explained before that all non-relativistic

approximation hold for the condensate as long as bρc 1 is satisﬁed. From our ansatz for vortex wave function (4.67), the gravitational force on equatorial plane, given by

FG = −∇ΦG|θ=π/2 is outward directed up to a distance of about 0.609R as seen from the ﬁgure (4.1). This repulsive nature is revealed as a consequence of rotation. For stable orbits to exist within this distance, the region must be dominated by ordinary (baryonic) matter. This fact can be further used to estimate the length scale of the wave function. Several papers [118–120] have already provided us with good evidences, via near-infrared and optical photometry, that suggests that Milky Way is dominated by ordinary baryonic matter up to around 6 − 8 kpc from the center, which is roughly the location of our solar

system. This in turn implies that Rc ≈ 10 − 13 kpc giving the estimate for R99 ∼ 35 − 50 kpc. It is worth noting that (4.69) admits inverse square law for the gravitational force when r R. An analysis of circular orbits within the BEC on the equatorial plane in the outer region r > 0.609R reveals the tangential speed can be written as

Φ vφ = ±prΦ0 + ~ G (4.77) G mc2r

73 Quantization of gravity

Figure 4.1: Gravitational acceleration on the equatorial plane of the BEC for our ansatz where the prime denotes derivative with respect to r. We can identify the first term due to regular gravitational force and the second term as the effect of frame dragging. The second term is however negligible compared to the first term. In the interior region the tangential speed remains the same except the first term which will be significantly modified by the presence of baryonic matter (assuming it dominates). We want to make one point here that frame dragging term gets significant near the center as it depends on the depth of the potential well not it’s gradient.

4.3.5 Vortex Oscillations

In this section we want to describe the dynamics of imploding BEC. To this goal we will employ collective coordinates in terms of condensate radius [106][121–123] with

1/2 2 mH(t)r2 N r − r i +φ ψ(t, ~r) = e 2R(t)2 sin θe 2~ (4.78) π3/2R(t)3 R(t)

Here R(t) and H(t) are two independent variables characterizing the dynamics of BEC cloud. It is more convenient to obtain the action (4.28) in terms of these two variables.

74 Quantization of gravity

Then we will vary the action to get the equations of motion for R(t) and H(t). With the help of (4.31) and applying integration by parts the action can be expressed as

Z ↔ 2 4 i~ ∗ ~ 2 1 2 1 S = d x ψ ∇tψ − |∇ψ| − Ve(|ψ|) − mΦG |ψ| − J.A 2 2m 2 2 " ( √ ) Z 5 5 2 1 5 2G 2 λ = − dt N m(R2H˙ + H2R2) + ~ + N 2 ~ + √ 4 4mR2 R3 3c2 16 2π3/2 23Gm2 288Gm2H2R 499Gm2H2R − √ + √ − √ (4.79) 30 2πR 25c2 π 50c2 2π

Unlike the stationary case, the time dependence in (4.78) implies that the radial current

Ar is no longer vanishing and therefore contributes to the equation of motion. Variation of the action with respect to H(t) gives

√ 125 πc2R˙ (t) ρ˙(t) H(t) = √ √ = √ (4.80) 2 (1152−499 2)n (1152 − 499 2)GmN + 125 πc R(t) √ ρ(t) + 125 πb2

We have written it in terms of dimensionless parameters deﬁned in (4.51). Subject to this condition, the equation of motion for the condensate, ∂L/∂R(t) = 0, yields

ρ˙2 ρ¨ + = F (ρ) (4.81) 2ρ(1 + µρ) where

m2c4 P P P P F (ρ) = − 1 + 2 + 3 + 4 (4.82) ~2 ρ2 ρ3 ρ4 ρ5

The coeﬃcients are given by √ 125 πb2 µ = √ (1152 − 499 2)n 23n P1 = √ 75 2πb√4 23(576 2 − 499)n2 − 9375πb2 P2 = 9375√πb6 √ 3n(−sgn(λ)25 2b2 − 16(384 − 83 2)π) P = 3 2000π3/2b6

75 Quantization of gravity √ (sgn(λ)3b2 + 160π)(576 2 − 499)n2 P = − (4.83) 4 5000π2b8

Integrating (4.81) once, one gets the ﬁrst integral of motion

1 µρ Z ρ ρ0F (ρ0) ρ˙2 − µ dρ0 = E (4.84) 2 1 + µρ 1 + µρ0

If we identify E as the total energy of the BEC, the eﬀective mass of the system is then given by

µρ m (ρ) = (4.85) eff 1 + µρ and the eﬀective potential energy is given by

Z ρ 0 0 2 4 0 ρ F (ρ ) 2m c A B C Veff (ρ) = −µ dρ = + + (4.86) 1 + µρ0 5~2b4 ρ ρ2 ρ3 where A, B and C are given by (4.53),(4.81) and (4.70). If the initial radius of rotating BEC is larger than the equilibrium radius and starts collapsing, we would expect it to bounce back (or at least slow down) once it crosses the equilibrium radius.

Next, we will consider the dynamics of collapse. The equilibrium energy Eeq can be determined from equilibrium radius Eeq = V (ρeq). In terms of rescaled time and energy

2m2c4 1/2 2m2c4 −1 τ = t, E = E (4.87) 5~2b4 5~2b4 the ﬁrst integral of motion will be

1 m (ρ)ρ∗2 + V (ρ) = E (4.88) 2 eff eff

∗ denotes derivative with respect to τ and it has a general solution

Z ρ 0 1/2 0 meff (ρ ) τ − τ0 = − dρ (4.89) 0 2(E − V eff (ρ ))

76 Quantization of gravity

As we mentioned before, with a total energy E > V eff (ρeq) the system will start to collapse

and will oscillate about the local minimum of V eff (ρ) with the condition E < V max ( when λ < 0). As seen from first diagram of figure (4.2) with repulsive interaction, BEC cloud always admits a global minimum of the energy functional. Repulsive force together with the quantum pressure ensures that the potential energy grows without bound as ρ → 0. This ensures that there is no restrictions on the size of condensate except the requirement that it has to be larger than the Schwarzchild radius. However if we consider a halo of mass M ∼ 1042 kg made of bosons of mass mc2 ∼ 10−24 eV and b ∼ 104, we find

n ∼ 37.5 and ρeq ∼ 0.409. We can see that this satisﬁes the condition for non-relativistic 4 regime i.e. bρeq ∼ 4 × 10 1. If the halo is assumed to begin with zero initial velocity

at ρ(0) = 3ρeq, the second diagram in (4.2) shows that the collapse will rebounce and will go through subsequent oscillations about equilibrium. The solution is obtained from numerical integration of (4.81). The time cloud will take to cross the equilibrium radius starting from the initial state is about τ ∼ 0.76 or equivalently 2.5 billion years. On the

Figure 4.2: Gravitational acceleration on the equatorial plane of the BEC for our ansatz contrary, no local minimum for attractive self-interaction would exist unless the number of bosons is below some critical value nc, determined by the strength of interaction. One can argue this phenomenon by understanding that beyond the critical number of bosons the attractive inter-particle force is large enough to outgrow the quantum pressure and the condensate will implode entirely. When the local minimum exists, the equilibrium radius ρeq depends on the actual number of bosons n present as in the ﬁrst diagram of 77 Quantization of gravity

ﬁgure (4.3). The equilibrium energy Eeq = Veq also depends on the number of bosons. To 4 illustrate, we take b ∼ 10 and n/nc ∼ 0.8 and integrate (4.81) to get the collapse process beginning at ρ(0) = 3ρeq withρ ˙(0) = 0. This is shown in the second diagram of ﬁgure (4.3). The negative quantum pressure and weakened gravity make the collapse process of the halo very slow taking approximately 10 billion years, as expected.

The ﬁrst integral of motion allows us to determine the frequency of small oscillations about equilibrium. The characteristic period is

!1/2 r5 b2 m (ρ ) T = 2π ~ eff eq (4.90) 2 mc2 00 V eff (ρeq)

Figure 4.3: Gravitational acceleration on equatorial plane of the BEC for our ansatz

This gives the period to be around 1.3 billion years for repulsive interactions whereas the period is four times larger in the attractive case. This discrepancy comes as a contribution to the BEC pressure from the repulsive interactions strengthening the gravitational attraction and weakening it in the case of negative interactions.

4.4 Summary

The standard CDM paradigm is successful in explaining dark matter on large scales but it deviates signiﬁcantly on scales less than ∼ 50 kpc. The absence of DM cusps at the center

78 Quantization of gravity of the galaxies and lack of low mass and massive sub-halos predicted by this model are two of the inconsistencies that can not be explained by this existing model. With these in mind, searching for alternative models became necessary for observations on smaller scales. One such model is scalar ﬁeld theory of dark matter.

In this chapter, our work is based on axisymmetric background perturbation differing weakly from flat metric. We modeled gravitationally bound BEC dark matter vortices based on Gross-Pitaevskii equation with quartic self-interaction (Appendix D). First, we construct the non-relativistic action for combined matter and gravitational field under the weak-field approximation in harmonic gauge. From this action, the equations of motion for the dynamical variables are readily found which subsequently help to analyze the stability criteria for the system. The equations of motion are complicated and difficult to solve. One way around it is to follow the variational approach. We started with the standard Gaussian Vortex ansatz for our trial wave function assuming it has a radial dependence. We were able to estimate for small and large sized condensates depending on the mass of the scalar field particles and their self-interaction. In attractive interaction case, the effective potential curve admits a local minimum as long as the number of particles is below certain critical value. The effective potential also has one local maximum. If the number of particles is above the critical value or the total energy is larger than the maximum of effective potential, the collapse of the dark matter halo will lead to a black hole. Harko [124], Levkov et. al. [125] and Eby [126] have proposed, in a different scenario that as the central density grows beyond a critical value a fraction of the bosons will get expelled from the condensate to stabilize the system. However in the non-relativistic limit examined here, there is no such upper limit when the self-interaction is repulsive. However, the global minimum of effective potential energy still exists.

Time dependence can be introduced by a collective coordinate description in terms of the condensate radius which now depends on t. Equations for the evolution of R(t) were obtained from a combined effective action. The choice of the trial wave function is not unique. In our ansatz, there is just one free parameter. The Poisson equation can be solved exactly and the gravitational potential of the halo can be evaluated in analytical 79 Quantization of gravity form. Then the motion of the condensate can be described as the motion of a single particle of variable mass in an effective potential. In a different ansatz with multiple parameters, one may expect to get a set of coupled equations which are needed to be solved. In both types of interactions, collapse from a diffused state to an equilibrium state takes billions of years. The time period of small oscillations about the equilibrium turns out to be of the same order of magnitude.

80 Chapter 5

Conclusion

In this report, two seemingly different topics have been studied with a view to achieving two different goals in the end. The first part mainly focuses on quantizing gravity in 2 + 1-D, since it provides a simpler framework to study the quantum effects of gravity than 3 + 1-D case. The second part is devoted to a (relatively) new concept of describing dark matter as Bose-Einstein condensate, its gravitational interaction and how different sized (from galactic scale to extra-galactic scale) dark matter halo can be modeled based on the mass of scalar field particles and their nature of self-interaction.

Focusing on the first part, the main aim was to construct the canonical description of an axisymmetric vacuum solution to Einstein’s gravity in 2 + 1-D. The technique that has been employed to achieve the desired result, can easily be extended to the description of dynamical collapse. This procedure of canonical calculation has been borrowed from work by Kuchaˇr[22] where he has shown the canonical structure of Schwartzchild metric and spherically symmetric dynamical collapse. Our work differs from Kuchaˇr’s work in the sense that we have considered 2 + 1-D case where we have incorporated the effect of rotational motion. As we have shown, the mass and angular momentum can be recovered from the canonical data. The axisymmetric vacuum solution turns out to be governed by a very simple set of constraints that can be obtained from a series of canonical transformations. In doing so, the boundary terms are absorbed in the hypersurface 81 Quantization of gravity

action. The quantum mechanics of the system is described by a time-independent wave function that depends only on the mass and the angular momentum. Hawking radiation arises naturally in the approach employed by Kuchaˇrto describe the spherical quantum dust collapse [127]. But the collapse need not lead to the formation of a black hole. An infalling shell of matter outside the apparent horizon is accompanied by expanding shells emanating from the center inside the apparent horizon. The advantage of this canonical approach is that all the necessary information are incorporated in the canonical data. If the spacetime is foliated in careful manner parameterized by a parameter and each leaf covers the whole spacetime then in principle, the whole collapse dynamics can be determined from the initial data both without and within the horizon until the singularity, if it is formed. Such a foliation is provided by the slices of constant proper time.

Subsequently, 2 + 1-D quantum collapse with a cosmological constant has been probed. Our choice of lower dimensions comes from the fact that it has served as a potential prototype model to study black hole thermodynamics and is expected to give an idea of what might happen in 3 + 1 dimensional case. The asymptotic symmetry group of 2 + 1-D gravity with a cosmological constant is generated by two copies of the Virasoro algebra and its degrees of freedom can be described by a two-dimensional conformal

ﬁeld theory at inﬁnity with central charges cR = cL = 3l/2G [128]. Since then the AdS/CFT correspondence became a tool for describing the BTZ black hole [129–132]. The connection between the AdS/CFT approach and canonical counterpart of the BTZ black hole has been studied in [41]. We have shown, by explicitly solving the Einstein equations that a non-singular quasi-stable spherical state is possible with the boundary radius greater than the BTZ horizon radius with some radial pressure. This is in agreement with Stephen Hawking’s proposal[11].

In the second part, a model is constructed based on the variational approach to analyze the description of the dark matter as a scalar field condensate. As an example if we take the mass of the particle to be m ∼ 10−24 eV and interaction strength b ∼ 104, we get a condensate of the outer radius of the order ∼ 50 kpc. Near the core region, the gravitational field is an outward directed up to a distance of about 17% of the outer 82 References radius because of the density profile of the vortex. In our example, the core is around 6−8 kpc in radius and are dominated by baryonic matter while on a larger scale the BEC dictates the effective gravitational dynamics. Several other aspects of this model can be considered which has not been done here. Our model does not include damping, therefore the BEC will keep oscillating indefinitely. This model does not include the microscopic mechanism for varying the number of particles in the condensate. An interesting case will be when several types of BECs or a single boson with multiple accessible states share a single gravitational potential.

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91 Appendix A

Simpliﬁcation of the Hamiltonian constraint

The Hamiltonian constraint can be simpliﬁed by eliminating PF using the momentum constraint. From the deﬁnition of F in (3.16) we can write

R02 R02 L2 = − FP 2 = − FP 02 (A.1) F F F Γ

and solving for PR from the momentum constraint we have

ΓP 0 − τ 0P P = Γ τ (A.2) R R0

Substitution of these expressions in the Hamiltonian constraint will lead to

√ R0Γ ΓP 0 − τ 0P P L2 + τ 02 = − FP 0 Γ τ τ F Γ R0 Γ R02 Fτ 0P 0 P = − FP 02 + Γ τ R0 F Γ R0 ΓL2 Fτ 0P 0 P = + Γ τ (A.3) R0 R0

92 References

Squaring both sides

(R02−FL2)

2 02 2 02 2 4 2 0 0 02 2 z 2}| 0{2 Pτ R (L + τ ) = Γ L + 2FL ΓPΓτ Pτ + τ Pτ F PΓ 2 02 2 2 0 0 02 2 ⇒ Pτ R = Γ L + 2FΓPΓτ Pτ − Fτ Pτ (A.4)

Substituting L2 from (A.1), we get

R02 P 2R02 + Fτ 02P 2 = Γ2 − FP 02 + 2FΓP 0 τ 0P τ τ F Γ Γ τ 02 2 02 2 R 0 0 2 ⇒ Pτ R = Γ − F (ΓPΓ − τ Pτ ) (A.5) F | {z } 02 2 R PR

This yields the desired result

2 2 Γ P 2 + FP − = 0 (A.6) τ R F

93 Appendix B

Derivation of unique solution of WDW equation

Here we will derive the solution to (3.32). This can be solved by the parameterization

√ ∂τ W = cos η/ F, ∂RW = sin η/F (B.1)

The function η has to satisfy the integrability condition

sin η cos η cos η − √ ∂Rη − ∂RF = ∂τ η (B.2) F 2F 3/2 F which leads to

√ √ ∂τ η = − F tan η∂Rη − ∂R F (B.3)

Inserting (B.1) into the second equation of (3.32), another equation involving η can be obtained

1 A −√ ∂τ η + cot η∂Rη − ∂RlnF + = 0 (B.4) F F

94 References

Therefore the ansatz A = F∂R ln(µF) yields

√ √ ∂τ η = F cot η∂Rη + F∂R ln µ (B.5)

For the integrability condition to be consistent, equation (B.2) and (B.5) can be equated to simplify further. This gives

√ ∂R ln Fµ tan η = 0 (B.6)

Thus we can write

α(τ) tan η = √ (B.7) µ F

sin η and cos η can be found accordingly. Having found these, the expressions for ∂Rη and

∂τ η look

αµ2F 1 ∂Rη = 2 2 ∂R α +√µ F µF µ F ∂ η = ∂ α (B.8) τ α2 + µ2F τ

Inserting these in the integrability leads to

α2 µ ∂ α = ∂ µ − ∂ F (B.9) τ µ2 R 2 R

Since, α is a function of τ, and µ and F are functions of R only, this requires α to be a constant. Therefore we have

α2 1 ∂ µ = ∂ F (B.10) µ3 R 2 R

yielding

1 µ = √ (B.11) 1 − a2F

95 References where β and a = β/α are constants and the solution to (3.32) can readily be found to be √ Z 1 − a2F W = aτ + dR (B.12) F

96 Appendix C

Calculations of Israel’s junction conditions

Within the dust ball, the metric is of the form

ds2 = −e2A(r)dt2 + e2B(r)dr2 + r2dφ2 (C.1)

The dust ball has a boundary deﬁned by r = rb. The extrinsic curvature of this hypersurface at the boundary should match smoothly with external BTZ vacuum, given by

ds2 = −f(R)dT 2 + f(R)−1dR2 + R2dφ2 (C.2) where f(R) = ΛR2 − GM. In the vacuum BTZ spacetime the boundary of this dust ball is given by R = Rb. The extrinsic curvature or the second fundamental form of the both boundaries should match. The extrinsic curvature Kµν of a hypersurface is given by

Kµν = ∇µnν + nµ (n.∇) nν (C.3)

97 References

where nµ is the unit normal vector at the boundary hypersurface. With respect to the metric (C.1) the normal vector to the boundary surface r = rb is given by

B µ −B nµ = (0, e , 0) n = (0, e , 0) (C.4)

Using (C.3) we ﬁnd the non-zero components of the extrinsic curvature as

2A−B 0 0 B 0 K00 = −e A ,K01 = −A ,K11 = (3e − 1)B −B K21 = 1/r, K22 = e r (C.5)

These values should be evaluated at the boundary r = rb for matching. The normal vector at the boundary with respect to the BTZ metric (C.2) is

−1/2 µ 1/2 nµ = (0, f(R) , 0), n = (0, f(R) , 0) (C.6)

They have same non-zero components as before √ 1 f 0 f − 3 K = − pff, K = − ,K = f 0 00 2 01 2f 11 2f 3/2 p K21 = 1/r, K22 = fr (C.7)

All the corresponding second fundamental forms will match if we have (3.45) as our

junction conditions at the boundary rb = Rb.

98 Appendix D

The Gross-Pitaevskii equation

In the presence of interactions, the structure and the dynamics of the Bose-Einstein condensate are governed by non-linear Schrodinger equation, The Gross-Pitaevskii equation. This equation describes the zero-temperature properties of uniform and non-uniform Bose gas when the scattering length a is much smaller than mean interparticle distance. A low energy, the eﬀective interactions between two particles are constant in momentum

2 representations, U0 = 4π~ a/m where a is s-wave scattering length. Incoordinate repre-

sentations, it’s just delta function interaction, U0δ(r − r’) with r and r’ as the positions of two particles. In the fully condensed state, all bosons are same single particle state φ(r), which is normalized. For N particles, te system can be described by

N Y Ψ(r1, r2, ..., rN ) = φ(ri) (D.1) 1

In the mean ﬁed treatment, the interactions among degrees of freedom can be neglected since it is negligible in the length scales smaller than interparticle spacing. Therefore ,the total Hamiltonian is combination of kinetic, potential and interaction energy as follows

N X p2 X H = i + V (r ) + U δ(r − r ) (D.2) 2m i 0 i j 1 i

99 References

V (r) being the trapping potential of the system. With (D.1) the expected value of the Hamiltonian can be calculated in terms of single particle state

Z 2 N − 1 E = N dr ~ |∇φ(r)|2 + V (r) |φ(r)|2 + U |φ(r)|4 (D.3) 2m 2 0

It is convenient to introduce condensed state wave function,

ψ(r) = N 1/2φ(r) (D.4)

The density of particles is given by n(r) = |ψ(r)|2. In the large N approximation, we ignore the terms of the order of 1/N leading to the full energy functional for N particle wave functions as

Z 2 1 E = N dr ~ |∇ψ(r)|2 + V (r) |ψ(r)|2 + U |ψ(r)|4 (D.5) 2m 2 0

To ﬁnd the optimal form of ψ, we minimize the energy functional (D.5) with respect to ψ(r) and ψ∗(r) subject to the condition that the total number of particles

Z N = dr |ψ(r)|2 (D.6) has to be constant. To ensure this constancy, a lagrange multiplier µ is introduce, usually called chemical potential , to the variation such that total variation δE−µδN = 0. Solving the variation with respect to ψ∗(r) and equating to zero gives

2 − ~ ∇2ψ(r) + V (r)ψ(r) + U |ψ(r)|2 ψ(r) = µψ(r) (D.7) 2m 0

100