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EXPLORING THE HOLE INFORMATION PARADOX

Item Type Electronic Thesis; text

Authors Rathi, Chirag

Citation Rathi, Chirag. (2020). EXPLORING THE INFORMATION PARADOX (Bachelor's thesis, University of Arizona, Tucson, USA).

Publisher The University of Arizona.

Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.

Download date 03/10/2021 21:23:31

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Link to Item http://hdl.handle.net/10150/651341 EXPLORING THE BLACK HOLE INFORMATION PARADOX.

By

CHIRAG RATHI.

A Thesis Submitted to The Honors College In Partial Fulfillment of the Bachelor’s degree With Honors in THE UNIVERSITY OF ARIZONA MAY 2020

Approved by:

Dr. Dimitrios Psaltis

Department of Astronomy Abstract.— Mere weeks after published his theory of , Karl

Schwarzschild came up with the simplest, static solution to Einstein’s field equations. This solu- tion implied the existence of a compact object with an infinite potential well. This implied that even could not crawl out of this potential well and hence the phrase ”even cannot escape a black hole” came into being. Black holes are characterized by the existence of an , a surface that disconnects the interior regions of the black hole from the exterior .

These compact objects existed on paper, without any direct observational evidence, for a century.

However, with the recent shift in interests towards astronomy and the success of projects like the (EHT), we have been able to image the ”shadow” of the M87 black hole. General relativity predicts that the shadow of the black hole is going to be stationary while the weather around the black hole can be turbulent. The EHT project not only has the potential to look for any classical evidence for the quantum structure of black holes but also allows us to investigate further into the Black Hole Information Paradox. So, in this project, we develop a method to analyze the Fourier transforms of the perturbed metric around a black hole and describe the macroscopic consequences of each of the proposed solutions to the information paradox. If we analyze the Fourier transforms of the perturbations, we can test the validity of general relativity’s prediction about the black hole’s shadow.

Version 1.0 - May 7, 2020 Information Paradox Chirag Rathi Contents

1 Introduction 3

1.1 The Information Paradox...... 4

2 Solutions to the Information Paradox6

2.1 The Remnant Solution...... 6

2.2 ...... 9

2.3 Firewalls...... 10

2.4 Fuzzballs...... 11

2.5 ...... 12

2.6 Non-Kerr Black Holes...... 15

2.7 ...... 16

2.8 ...... 17

2.9 Superspinars...... 19

3 Event Horizon Telescope Project 20

3.1 Why are we testing GR with EHT?...... 22

3.2 Properties of Black Hole Shadows:...... 22

3.3 Proposed Tests:...... 23

3.4 Challenges and Implementation:...... 24

4 Empirical evidence of Black Hole Perturbations 25

5 Observations from the code 28

6 Conclusions 29

May 7, 2020 2 Information Paradox Chirag Rathi 1 Introduction

The twentieth century witnessed an exponential growth in the field of theoretical physics. Two of the most groundbreaking contributions that swept the physics community of its feet were the classical theory of general relativity and . These two theories have been the most successful in describing the universe and have withstood the test of time. However, these two theories have been at the throats of each other for the past century. There is no smooth transition between the quantum and the classical realms of reality. When we try to reconcile general relativity with quantum mechanics, it looks like we are trying to fit square pegs into round holes.

One of the major problems we face on astronomical scales is the Black Hole Information Paradox.

Before we begin to understand the crux of the information paradox and what problems it presents to fundamental physics, we need to understand two of the most basic principles–unitarity and equivalence–presented by quantum mechanics and general relativity respectively.

Principle of Unitarity: The principle of unitarity is taken as an axiom or a basic postulate of quantum mechanics. It basically states that a pure quantum state must evolve into a pure quantum state. So, initially if we have a pure quantum state and it obeys unitarity then:

|ψfinali = Smn |ψinitiali . (1)

Here |ψinitiali is the initial (pure) quantum state, |ψfinali is the final quantum state and Smn is the matrix. Smn takes the initial quantum state to the final quantum state [28].

Principle of Equivalence: This principle in general relativity dictates that an observer falling into the black hole experiences nothing extraordinary at the event horizon.

The information paradox puts the principle of unitarity into jeopardy and when one tries to preserve unitarity, the equivalence principle is perilled. Since quantum mechanics is considered simply to be a more fundamental description of reality, theorists are willing to consider general relativity as an approximation to quantum .

May 7, 2020 3 Information Paradox Chirag Rathi

1.1 The Information Paradox

Although there has been a lot of work in the last four decades, it is fair to say that we are nowhere near a self-consistent theory of . Classical general relativity provides the most successful description of gravitational fields and the space-time metric. Although it is reasonable to ignore quantum gravitational effects on large scales, quantization plays an important role in the behavior of matter fields described by Einstein’s equations. Thus, the problem arises when one has to connect classical space-time metric to quantum mechanical matter fields.

Towards the end of the twentieth century, it was well understood that black holes behave like thermodynamic systems [7][8]. The laws of black hole mechanics were also developed. The first law of black hole mechanics states:

dE = T dSBH + ΩdJ + ΦdQ. (2)

Here T, Ω and Φ are interpreted as the temperature, the angular velocity and electric potential of the black hole [22]. The second law states that the quantity SBH does not decrease with time. This quantity was interpreted as the entropy of the black hole by Bekenstein. The Einstein-Hilbert theory of gravity it was found that SBH depends on the surface area, A, of the black hole horizon [22].

A S = , (3) BH 4¯hG whereh ¯ is the Planck’s constant and G is the universal gravitational constant. These results of black hole dynamics are consequences of the classical theories of gravity. In lieu of a perfect quantum- mechanical description of gravity, applied quantum field theory to curved space- time as a decent approximation to quantum gravity in the classical limit. In 1974, he proposed that quantum mechanical effects cause black holes to emit particles as if they were hot bodies [16].

The temperature of the black hole is expected to be:

hκ M = 10−6 o , (4) 2kπ M

where k is the Boltzmann constant, κ is the surface gravity of the black hole, Mo is the of the

May 7, 2020 4 Information Paradox Chirag Rathi

Sun and M is the mass of the black hole. The emission of radiation confirms the thermodynamic of black holes. Consider the of a into a black hole. In the classical theory of gravity, the black hole will rapidly settle down into a stationary state characterized only by its mass, charge and angular (the no-hair theorem). The Schwarzschild solution, the

Reissner-Nordstrom metric, the and the Kerr-Newman metric are the four solutions that represent the black hole solutions of general relativity [1].

Hawking predicted that the event horizon produces pairs of quanta. One particle of each pair escapes into space towards infinity while its partner falls into the singularity. Hawking traced the wave packets associated with these particles, backwards in time and found that they originated from two distinct waves: one with positive frequency and another with negative frequency. However, the mixing of positive and negative frequencies in the stationary black hole solution does not happen and hence, particle creation should not be expected.

So far, we have not considered superradiance. Zel’dovich showed that low-frequency electro- magnetic (EM) waves scattered by a rotating cylinder, are amplified [10]. Following this proposal,

Misner suggested that rotating black holes would also amplify the scattered EM low frequency waves [25]. Upon quantization, this amplification leads to the emission of particles from the rotat- ing black holes. This emission was a precursor to . Stephen Hawking proposed that in addition to the radiation in the superradiant modes, there is a steady emission at all modes

κ by the black hole with temperature 2π [16]. The thermal emission from the black hole is dubbed the Hawking radiation. If the thermal emissions are highly mixed, it is impossible to reconstruct any information about the black hole that emitted this radiation. This conclusion violates the principle of Unitarity in quantum mechanics and the principle of equivalence in general relativity. This leads to the infamous Black Hole Information

Paradox.

The main idea is that a black hole, initially in a pure quantum state, evaporates out of existence by emitting highly mixed states. A typical initial state of the black hole is consid- ered to be non-degenerate. So, at a later time, the full quantum state of the system must be

|Ψi = |0initiali |ψradiationi. This implies that the final state is an impure product state in the bases of |ψradiationi and has a vanishing entropy. Since SBH ∝ A, this predicts the eventual disappearance

May 7, 2020 5 Information Paradox Chirag Rathi of the black hole.

The first point to note about this paradox is that it is non-trivial. It is a common misconception that higher-order corrections to Hawking radiation will resolve the paradox. Hawking’s calculations have been revisited numerous times and redone in different settings. However, the results turn out to be valid and a thermal emission from the black hole is expected. The resolution to this puzzle is deep rooted in quantum gravity. Earlier it was assumed that the quantum gravitational effects are confined within the Planck scale length (lp). Now one has to consider that may be quantum

α gravity affects the space-time over horizon scale distances N lp for α > 0 [23].

To circumvent this problem, many solutions have been constructed over the past few decades and they remain extremely theoretical. However, with the success of LIGO and the EHT project, we may obtain some insight into the resolution of this paradox. In the next section I am going to summarize the solutions to this paradox and discuss about the observational consequences (if any) of these proposed solutions.

2 Solutions to the Information Paradox

In the following sub-section I discuss some proposed solutions to the information paradox and the potential of gaining insights on them using astrophysical observations. They are tabulated on page

6.

2.1 The Remnant Solution

There is a class of solutions that propose a halt in the evaporation process of the black hole.

The black hole does not evaporate out of existence but stops at the Planck scale, leaving behind remnants. If this solution were to be accepted, there are two possible types of remnants: stable and long-lived (quasi stable).

May 7, 2020 6 Information Paradox Chirag Rathi Chaotic shadows Changes in GW signals No observational consequence Observational Consequences GW show extreme pericenter precision Different GW signals in ringdown phase Shadows and different measured QNM of GW Photons from the shadow take longer to reach observer Generation of highly relativistic/high frequency photons Gravitational QNM differs from that of BH in the late stages No No No No No No No Yes Yes Hawking Radiation No No No No No No No Yes Yes Horizon GR GR Classical GR Classical GR Classical GR theory GR modification Semi-classical GR Theory of Gravity Firewalls Fuzzballs Remnants Solutions Gravastars Wormholes Boson stars Superspinars Naked Singularity Non-Kerr black holes Table 1: Table 1:GW A summary = of Gravitational the Waves, proposed BH solutions = to Black the Holes, black GR hole = information General paradox. Relativity. Note that QNM = Quasi Normal Modes,

May 7, 2020 7 Information Paradox Chirag Rathi

Figure 1: A Penrose diagram for a black hole remnant. Here it is assumed that the lifetime of the remnant is infinite. The solid curve outside the Planckian Region denotes the . Outgoing arrows represent the Hawking Radiation while the incoming arrows denote represent the infalling matter. h1 and h2 are the two Cauchy hypersurfaces. For further discussion, look at [12]

A long-lived remnant’s lifetime is longer than that of the black hole. These types of remnants radiate particles extremely slowly and decrease their mass over a long period of time. However, it becomes extremely hard to distinguish between stable and long-lived remnants. All the solutions that point towards the existence of remnants, arise from the generalized uncertainty principle [2] and modifications to the theories of gravity [12].

The overall wavefunction of the black hole maintains unitarity by correlating the interior states of the black hole to the states of the Hawking radiation outside [12].

E X radiation n |ψtotali = Cn ψn ⊗ |φinteriori (5) n

Thus the remnant can hold a large number of quantum states, far beyond its storage capacity as suggested by the Bekenstein-Hawking entropy.

Although this is an interesting proposal, there are two major issues that do not allow us to test it astrophysically:

1. The remnants are too microscopic to be resolved: The current technological limitations cannot

resolve any astrophysical effects on the Planck scale.

May 7, 2020 8 Information Paradox Chirag Rathi

2. The remnants form in astronomical timescales. Even the quasi-stable remnants outlive the

black holes and it is not feasible to look for any macroscopic effects in such timescales.

2.2 Gravastars

The is a short-hand term for a gravitational vacuum star. Gravastars are the final state of gravitational collapsed stars. This class of compact object was originally proposed by Mazur and Mottola as an alternative solution to black holes [24]. The model of a gravastar consists of an interior de Sitter region connected to an exterior Schwarzchild region via a shell of stiff matter

(ρ = p)[30]. Although their formation process remains unclear, gravastars hold the potential to solve the two fundamental problems related to black holes: the singularity and the black hole information paradox.

In black hole , there exist unstable circular of photons that play an important role in generating shadows of black holes. The study of the shapes of these shadows is critical to verifying the existence of black holes. Thus the direct observations of black holes, which the EHT project aims to do, means to observe the geometry in the vicinity of the unstable circular orbits.

However, the feature of unstable circular photon orbits is not unique to the black holes. Ref. [30] worked with a thin-shell spherical model of a gravastar and analyzed the optical images of those photon orbits, assuming an optically transparent surface. They concluded that gravastars also produce these unstable circular orbits and it becomes difficult to distinguish between gravastars and black holes.

May 7, 2020 9 Information Paradox Chirag Rathi

Figure 2: This figure was adopted from [30]. This image shows the shadow cast by an infinite optical plane behind the gravastar. In the yellow domain, the photons emitted by the infinte plane come towards the observer. The orange domain consists of an infinite number of bright and dark rings. The dotted circle corresponds to the unstable circular in Schwarzchild background.

On the other hand, there are other proposals that try to single out the unique observational consequences of a gravastar. For instance, Chirenti and Rezolla studied the axial perturbations on gravastars and found that the quasi normal modes of gravastars is different from that of black holes [13]. Moreover, Broderick and Narayan suggested that if the black hole candidates with matter were indeed gravastars, then they should heat up and emit radiation at IR wavelengths [9].

Figure 3: This figure was adopted from [13]. The left panel shows the behavior of the real part ωR of the QNM eigenfrequencies. The right panel shows the same thing but for the imaginary part. In both panels, the thick horizontal lines represent the QNM frequencies for a Schwarzchild black hole.

2.3 Firewalls

While there are a set of solutions that look for alternative solutions to black holes, there are a few classes of solutions that look forward to preserving the idea of black holes. The Black Hole

May 7, 2020 10 Information Paradox Chirag Rathi

Complementarity is one of the proposals that wants to preserve the black hole solution in general relativity. The Black Hole Complementarity principle is based on four postulates [3]:

• Postulate 1: The process of formation and evaporation of a black hole, as noted by a distant

observer, can be entirely described with the help of quantum theory.

• Postulate 2: Outside the stretched horizon of a massive black hole, physics can be described

by a set of semi-classical field equations.

• Postulate 3: To a distant observer, the black hole appears to be in a quantum system with

discrete energy levels.

• Postulate 4: An infalling observer experiences nothing out of the ordinary as he crosses the

event horizon. This is the principle of equivalence from general relativity.

Nearly after two decades the Black Hole Complementarity principle came into the picture, the solution was disputed by the firewall paradox. It states that the equivalence principle from general relativity has to be violated and the infalling observer must encounter a highly energetic wall of particles as he/she crosses the event horizon. The firewall paradox predicates on the monogamy of entanglement. The late radiation has to be maximally entangled with the early radiation and with the infalling counterpart of the late radiation [32]. Since the monogamy of entanglement prevents this, it is proposed that the principle of equivalence is violated as the observer crosses the event horizon.

The obvious observational consequence is the firewall at the event horizon. However, it is not feasible to throw someone into a black hole only to detect a curtain of energetic particles for LIGO observations. Barausse, Cardoso and Pani discussed the observational consequences of a firewall [5].

They modeled the firewall as a thin spherical shell and found out that the ringdown frequencies of massive black holes would be drastically modified due to its existence.

2.4 Fuzzballs

In the classical framework, black holes are a solution to Einstein’s field equations that have an enormous entropy, Sbek. However, in accordance with the no-hair theorem, black hole states are

May 7, 2020 11 Information Paradox Chirag Rathi unique, which suggests an entropy S = ln(1) = 0. However, black holes might not be as classical as was previously assumed. Quantum effects might smear the information in the hole all over the horizon in a formation. There are eSbek fuzzball states and the structure of the black hole can be understood by an ensemble of such fuzzball microstates [17]. This large entropy of a black hole can lead to substantial quantum effects at the horizon.

Black holes, by virture of being macroscopic objects, have a low probability of tunneling from one state to another. However, there are eSbek states available for tunneling. The small probability amplitudes and the availability of a large number of states compensate for each other and allows the collapsing black hole to tunnel to a linear combination of fuzzball states [17]. This fuzzball state has no horizon and radiates in a manner that does not lead to the information paradox.

When it comes to the observational consequences of the fuzzball solution, gravitational wave astronomy can be extremely helpful. The main reason is that the individual fuzzball solutions differ significantly from the black hole solutions. That implies that sufficiently fine-grained observations of the neighborhood region to the horizon should enable one to differentiate between a fuzzball and a black hole [17]. Observations of gravitational wave patterns from mergers and in particular, the details of the ringdown phase differ from that of a black hole merger.

2.5 Naked Singularity

Gravitational collapse of a body is one of the most important processes in astrophysics. If the pressure gradient force is not sufficiently strong compared to the gravitational force, a body can continue to collapse under the influence of its gravity. The general standpoint is that if the body continues to collapse under its own gravitational influence, it ends up as a compact object with a singularity, which is shielded from the by an event horizon [27]. However, it remains possible that the gravitational collapse may also result in the formation of a naked singularity (not shielded by any horizons).

Many studies in gravitational collapse have shown that the end product of the collapse might not neccessarily be a black hole. A naked singularity is also possible by the continual gravitational collapse. In fact, [4], [20] and [19] have shown that a collapsing cloud, subjected to the right physical conditions is capable of forming a static naked singularity.

May 7, 2020 12 Information Paradox Chirag Rathi

At singularities, the smoothness of spacetimes is lost. Since classical physics inherently assumes the continuity of , this branch of physics cannot deal with singularities. This is the limitation of classical physics. However, classical physics can be applied successfully to study the regions neighboring the singularity. In order to study spacetimes within the realm of classical physics, the cosmic censorship hypothesis was conjectured by Roger Penrose.

Cosmic Censorship Hypthesis: Every singularity must posses an event horizon that hides the singularity from the universe outside the event horizon. This conjecture ensures that the spacetime region inside the event horizon is causally disconnected from the exterior regions of the universe. If this conjecture is true, one can predict the evolution of the entire spacetime region outside black holes.

The cosmic censorship hypothesis remains unproven till now even though enormous amount of work has been put into testing it. However, the EHT project provides the opportunity to analyze and test this hypothesis.

The work done by [31] considered thin accretion disks around naked singularities and studied their images. The study found out that the naked singularities having photon spheres do create images and shadows, similar to those in the black hole case. However, when there are no photon spheres, it becomes significantly easy to distinguish between naked singularities and black holes.

Ref. [31] also studied the naked singularity images from Joshi-Malafarina-Narayan-1, with photon spheres, (JMN-1) and Janis-Newman-Winicour (JNW), without photon spheres. Figures 2 and 3 are borrowed from the paper.

May 7, 2020 13 Information Paradox Chirag Rathi

Figure 4: The images of a Schwarzchild black hole and a JMN-1 naked singularity with accretion disks [(a)-(f)]. For M0 ≥ 2/3, the JMN-1 naked singularity has a with a single in [(b)]. For M0 < 2/3 does not have a photon sphere and for 1/3 < M0 < 2/3 has two accretion disks [(c) and (d)] and for M0 < 1/3 has a single accretion disk extending up to the singularity [(e)]. For further details, please refer to [31]

Figure 5: The images of a Schwarzchild black hole and a JNW-1 naked singularity with accretion disks [(a)-(f)]. For γ ≥ 1/2, the JNW singularity has a photon sphere and a singe accretion disk√ [(b)-(c)]. For γ <√1/2 does not have any photon spheres and two accretion disks for 1/2 < γ < 1/ 5 [(d)]. For γ < 1/ 5 we see a single accretion disk extending up to the singularity [(e)]. For further details, please refer to [31]

May 7, 2020 14 Information Paradox Chirag Rathi

2.6 Non-Kerr Black Holes

In four dimensional spacetime, the no hair theorem dictates that any black hole can be charac- terized by three properties: mass, charge and . In the static case, where the angular momentum vanishes and there is spherical symmetry, one obtains the Schwarzschild and

Reissner-Nordstr¨omsolutions to Einstein’s field equations [1]. The spinning generalization of the

Schwarzschild solution results in the Kerr metric and the charged spinning generalization of the

Reissner-Nordstr¨omsolution results in the Kerr-Newman metric. These four metrics, together form the black hole solutions of general relativity.

However, there is little direct observational evidence that astrophysical black holes are described by Kerr spacetime metric and thus one cannot exclude the possibility of observing a geometry that significantly deviates from the Kerr metric. Thus, studying the details of a non-Kerr spacetime metric becomes very important.

Refs. [18] and [35] studied the shadows of a non-Kerr black hole numerically and found out that the shadows of a non-Kerr depended on a quadrupole-deviation parameter

(q) and a spin parameter (S). For the case with negative q, the shadow of the non-Kerr rotating black hole becomes prolate and it splits into two disconnected main shadows. These shadows are symmetric about the equatorial plane. For the case with positive q, the shadow of the non-Kerr rotating black hole becomes oblate and the main shadows are joined together in the equitorial plane. The numerical analysis also found that the Einstein ring is broken for a value more than the critical value of q, |qc|.

Figure 6: The shadow of a Non-Kerr rotating black hole with different q parameters and a fixed spin parameter, S = 0.2M 2. For further details please refer to [35]

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Figure 7: The shadow of a Non-Kerr rotating black hole with different q parameters and a fixed spin parameter, S = 0.98M 2. For further details please refer to [35]

Figure 8: The change in the shape of Einstein rings (white rings in the figures) for different values of q. For further details, please refer to [35]

2.7 Wormholes

While a solution to general relativity predicts the existence of a singularity shielded by an event horizon, there are other solutions that prohibit the singularity formation. One of such solutions is the formation of an Einstein-Rosen bridge, commonly known as a . The wormhole solution suggests that the gravitational collapse of a body results in the formation of a compact object that potentially connects two different points in spacetime. These bridges are consistent solutions to

Einstein’s field equations, but their existence still remains on paper. However, the advent of gravitational wave astrophysics has the potential to address the issue of observing wormholes.

The detected gravitational wave signal is characterized by three phases: the inspiral stage; the merger phase where the two objects coalesce; and the ringdown phase when the merger produce settles to a stationary equilibrium solution. The analysis of the ringdown phase becomes extremely important to detect wormholes. The ringdown phase is comprised of Quasi-Normal Modes (QNMs) and the QNM spectrum can characterize a Kerr black hole’s mass and angular momentum. This

May 7, 2020 16 Information Paradox Chirag Rathi reasoning suggests that the ringdown signal from the gravitational waves can help us identify dark, compact objects. If the compact object is horizonless, then the GW signal should consist of the usual light-ring ringdown, followed by proper modes of vibration of the the object itself [11].

The former signals mimic the black hole case but the latter (which is the QNM spectrum) differ significantly from their black hole counterpart. [11] plots the gravitational waveforms of a black hole and a wormhole and it becomes evident that the later stages of the post merger phase becomes very important to the study of compact objects.

Figure 9: The gravitational waveforms of a wormhole compared to that of a black hole. For further details, please refer [11]

In the post merger phase, the initial ringdown signal depends on the light ring of the final object. However, the QNM spectrum in the later stages of the post merger phase can be the key to differentiating between a black hole and a wormhole.

2.8 Boson stars

The EHT project has enormous potential to image a black hole and for a long time it was theorized that images of a so-called shadow can be interpreted as a conclusive proof of the existence of an event horizon of the black hole. However, there exists a compact object with no event horizon and no hard surface. This is known as a boson star. While the shadow-like and photon ring structures could prove the existence of black holes, boson stars have been shown to produce images that very successfully mimic the features in the image of a Kerr black hole [34].

Boson stars are localized bundles of energy formed by an assembly of spin-0 . The boson star is a solution to the Einstein-Klein-Gordon set of equations that describe the gravitational field

May 7, 2020 17 Information Paradox Chirag Rathi created by an ensemble of spin-0 bosons [34]. These boson stars are macroscopic quantum objects subject to the Heisenberg’s uncertainity principle. This principle dictates the fact that boson stars may not undergo complete gravitational collapse to form a black hole.

Ref. [34] produced images of accretion tori surrounding a black hole and a boson star using the open source GYOTO code. In the paper, they performed ray-tracing computations of accretion tori around a black hole and a boson star and produced 1.3 mm images (the working waveband of the EHT project).

Figure 10: The reference images of a Kerr black hole as computed by the GYOTO code (left) in Boyer-Lindquist coordinates. In the right the same image is computed by the GYOTO code in quasi-isotropic (QI) coordinates. For further details please refer [34].

Figure 11: Boson star image: Specific intensity distribution for a frequency of ω = 0.7 and an azimuthal number k = 1. For further details, please refer to [34].

While the images of the shadow of a compact object might not offer conclusive evidence of the nature of that object, gravitational wave astronomy might provide clues to this puzzle. The absence of an event horizon will allow differentiating between a boson star and a black hole. Ref. [21] analyzed the inspiral of a compact object into supermassive objects. This paper

May 7, 2020 18 Information Paradox Chirag Rathi compared the models of a with models in which the central compact object is a supermassive boson star. It found that geodesics within the supermassive boson star model show extreme pericenter precision making the emitted gravitational waves distinguishable from those of a supermassive black hole.

2.9 Superspinars

A superspinar is a rapidly rotating compact object whose exterior geometry is described by the over-spinning Kerr geometry.

The Kerr spacetime is an exact solution to the Einstein field equations in vacuum and is char- acterized by two parameters: the gravitational mass (M) and the Kerr parameter (a), which is defined as the angular momentum divided by the gravitational mass. The solution describes a rotating Kerr black hole if a2 ≤ M 2 while it describes a naked singularity if a2 > M 2 [26]. In

4D general relativity, the angular momentum is limited by the Kerr bound. However, ref. [15] has conjectured that in string theory, this bound can be violated and the compact object can spin faster. Hence the name, superspinars. Thus, observational evidence of superspinars will not only confirm the existence of such exotic compact objects but also provide evidence in support of string theory.

Ref.[33] studied the observational consequences of primordial Kerr superspinars and demon- strated that these exotic compact objects can act as ultra-high energy accelerators. It also claimed that in the field of near-extreme superspinars, extremely high energy processes can occur frequently with energy amplifications on the order of 106 and higher. The generated particles must be highly relativistic.

The figure below, taken from [33] plots the radiation lines from the vicinity of a Kerr black hole and a superspinar.

May 7, 2020 19 Information Paradox Chirag Rathi

Figure 12: The iron line profiles from the Keplerian discs in the vicinity of a Kerr black hole (a = 0.998) and a superspinar (a = 3.0). The inclination of the observer in the left panel is 30◦ and 80◦ in the middle panel. The influence of different superspinar radii on the profiled lines of radiation is shown in the right panel. For further details, please refer to [33].

3 Event Horizon Telescope Project

The information paradox introduced as a consequence of black hole thermodynamics, has led to proposals that look forward to preserving the principle of unitarity. This need for a consistent quantum evolution requires a modification to the semi-classical description not just at the singu- larity but at the horizon or larger scales. If these modifications extend beyond the horizon, they can have observable consequences like metric fluctuations [14].

The small size of black holes makes the task of imaging them extremely challenging. The imaging requires a horizon-scale resolution of tens of microarcseconds. The aim of the Event Horizon

Telescope (EHT) is to capture the first images of the black holes in the centers of the

Galaxy and the M87 . Measurement of the shape and size of the black hole’s shadow can allow us to test the cosmic censorship hypothesis and the no-hair theorem and deduce the classical effects on the quantum structure of black holes by studying its shadow [29]. Observations of the accretion flows can aid in the measurement of the spins of the black holes and other properties of spacetimes around them.

Measurements at optical wavelengths are unable to achieve this feat at the moment but advances in millimeter wavelength astronomy have enabled the different telescopes around the world to combine into the EHT project. Imaging the shadow of the black hole will allow us to:

1. Test general relativity.

2. Observe the interactions of matter and magnetic fields with black hole horizons.

Certain conditions need to be met in order to image the black hole (especially the one in the center

May 7, 2020 20 Information Paradox Chirag Rathi of our galaxy):

1. The ’s atmosphere needs to be transparent (mm wavelength).

2. The Galaxy needs to be transparent (mm wavelength).

3. The accretion disks around the BH need to be transparent (mm-IR wavelength).

The image of the accretion flow becomes fully transparent at the mm-to-IR wavelengths and this is just pure coincidence that all three potential barriers to observation are transparent in this waveband. The next question is if the accretion flow is transparent at this waveband range, then what would be the signature of the presence of a horizon? Von Laue calculated the cross section of √ a Schwarzchild black hole to a parallel beam of photons and found a radius equal to 27GM/c2.

Later Bardeen generalized the solution to spinning black holes [6]. Subsequent developments in technology and algorithms made it possible to simulate the horizon of Sgr A* and the supermassive black hole (SMBH) in M87.

Observations with a small subset of telescopes in the EHT revealed that:

1. Horizon-scale structures at 1.3 mm for Sgr A* and M87.

2. Source substructure and variability.

3. Highly polarized emissions (large B fields in the vicinity of the black hole).

The Event Horizon Telescope (EHT) is a globe-sized array of radio telescopes operating at 1.3 mm. It aims to capture some of the highest resolution images. The figure below taken from [29] shows the prime targets of the EHT project.

May 7, 2020 21 Information Paradox Chirag Rathi

Figure 13: targets of the EHT project. The two primary targets of this project, M87 and Sgr A*, are indicated in the figure.

3.1 Why are we testing GR with EHT?

There are empirical and theoretical reasons for testing GR with EHT.

1. Black holes’ spacetimes are qualitatively different from those of other astrophysical objects.

It is possible that the quantum structure of black holes will leave classical signatures that are

detectable with gravitational tests.

2. The EHT will probe the gravitational fields of SMBH that are vastly different from those of

other astrophysical objects.

3. All the GR tests that have been up until now are within the . We need to

expose GR to different and difficult situations. EHT will probe 5 orders of higher

gravitational potentials and 15-20 orders of magnitude higher gravitational curvatures [29].

4. EHT observations have the potential to test the no-hair theorem with black holes (which the

LIGO/ probes cannot do).

3.2 Properties of Black Hole Shadows:

The outline of a black hole’s shadow is completely determined by the location of the photon orbits

(with respect to the black hole spin axis) and gravitational lensing. For a non-spinning black hole, the radius of the photon orbit is: r = 3GM/c2. Due to the effects of gravitational lensing the

May 7, 2020 22 Information Paradox Chirag Rathi √ photon orbit’s radius appears at R = 27r. The radius of the photon orbit depends on the relative orientation of the orbital angular momentum with respect to the black hole’s spin axis.

Figure 14: Predicted 1.3 mm images of Sgr A* using three different GRMHD simulations in [29]. Prominent features of the black hole shadow can be seen marked by a red circle in the rightmost panel.

Moreover, because of the accretion flow most of the emission takes place outside the Innermost

Stable Circular Orbit (ISCO). The ISCO lies outside the radius of the photon orbit. This implies that most of the photons that cross the photon orbit will eventually disappear behind the black hole. As a result, the black hole casts a shadow on the surrounding emission. The size and shape of the shadow is determined by the location of the various photon orbits at different orientations with respect to the black hole’s spin [29].

3.3 Proposed Tests:

1. Cosmic Censorship Test: The detection of a black hole shadow is not a proof for the

existence of an event horizon. Naked Singularities can also project shadows (with the right

parameters).

2. Null Hypothesis Test: Sgr A* is described a Kerr metric. This prediction has no free

parameters.

3. No-hair theorem and non-Kerr metrics: No-hair theorem predicts that the shadow of

black holes are nearly circular if and only if q = −a2. Violation of the no-hair theorem makes

the shadows highly asymmetric.

4. Quantum Structure: It is plausible that the spacetime of black holes can have classical

dynamics due to quantum fluctuations at horizon scales. These fluctuations of spacetime lead

to fluctuations of the shape and size (of order unity) of the black hole’s shadow.

May 7, 2020 23 Information Paradox Chirag Rathi

3.4 Challenges and Implementation:

The presence of an accretion flow is necessary around a black hole as it acts as a source of radiation upon which the black hole can cast its shadow. Though the physics of the accretion disks do not affect the shadows of the black holes in any way, these accretion disks can partially or fully obscure the shadow. Moreover, understanding the physics of the accretion disk is key to understanding the surrounding of black hole spacetime. The sheer complexity of the accretion physics makes predictions very difficult. Properties of the black hole’s shadow and the underlying spacetime can be inferred indirectly from models of accretion flows. However, these models run the risk of being biased and flawed. To overcome this limitation, focus should be redirected towards measuring the properties of the shadow and not of the accretion flow. In principle the narrow width of the photon ring around the black hole allows for a precise measurement of its shape and size. The outline of a black hole’s shadow is the set of all points in the image with the highest brightness gradient.

Figure 15: Two snapshots of time-dependent black hole shadows as calculated in [29].

The test for general relativity will be measuring the shapes and sizes of black hole’s shadow.

If the same shapes and sizes of the black hole’s shadow are measured in successive measurements, then it would confirm the predictions of Einstein’s .

Ref. [14] works with non-rotating black holes in Eddington-Finkelstein coordinates because this coordinate system is continuous beyond the horizon and is well behaved at the singularity. This paper introduces perturbations to the spacetime metric around the black hole and the metric does not fluctuate througout the simulation (test of general relativity).

May 7, 2020 24 Information Paradox Chirag Rathi

Figure 16: Figure drawn from [14]. Top-left is an unperturbed non-spinning Schwarzchild black hole while the other frames introduce spherically symmetric perturbations to the space-time around the black hole. Blue lines are the photon trajectories that make it into the black hole while the red lines are the trajectories of photons that escape to infinity.

4 Empirical evidence of Black Hole Perturbations

For the honors thesis, we worked on an extension of the EHT project. Since the EHT can measure the visibility amplitudes of the accretion disk around a supermassive black hole, we developed a method to perform a similar task. This method, called the cross-section method, is capable of producing models that map the intensity of the accretion disk. The goal is to compare the visibility amplitude measured by the EHT to that produced by implementing the cross-section method.

To develop the code for this method, we first reproduced the shadow images of the black hole

(akin to Figure 15) at various snapshots of time. To do this, the python code was instructed to read the data file and produce an image of the black hole’s shadow. Some of the reproduced images are shown below.

May 7, 2020 25 Information Paradox Chirag Rathi

(a) Black hole shadow at snap- (b) Black hole shadow at snap- shot ωov = 9π/16 shot ωov = 11π/16

(c) Black hole shadow at snap- (d) Black hole shadow at snap- shot ωov = 13π/16 shot ωov = 15π/16

Figure 17: Four snapshots of black hole shadow images taken at the mentioned time intervals.

Then we manipulated the code to generate the Fourier transforms of shadow snapshots using the Fast Fourier Transform (FFT) package in Python. The shadow images and their corresponding

Fourier transforms are shown below.

May 7, 2020 26 Information Paradox Chirag Rathi

(a) Fourier transform of black hole shadow (b) Fourier transform of black hole shadow at snapshot 0. at snapshot ωov = 2π/16

(c) Fourier transform of black hole shadow (d) Fourier transform of black hole shadow at snapshot ωov = 4π/16 at snapshot ωov = 6π/16

Figure 18: Fourier transforms of the shadow at various snapshots.

The next step was to plot the cross sections of the Fourier transforms along parallel and per- pendicular axis to the black hole spin. Since most of the accretion around a supermassive black hole takes place along the equatorial plane, one should expect more variability in the visibility amplitudes along the horizontal cross sections (parallel to the spin axis of the black hole).

First we found the pixel size in the Fourier transform image of the black hole shadow. Then we mapped the visibility intensity of the Fourier transforms along the horizontal and vertical axis.

May 7, 2020 27 Information Paradox Chirag Rathi

(b) Cross section of the Fourier transform of (a) Cross section of the Fourier transform the black hole’s shadow at snapshot ωov = of the black hole’s shadow at snapshot 0. 2π/16.

(c) Cross section of the Fourier transform of (d) Cross section of the Fourier transform of the black hole’s shadow at snapshot ωov = the black hole’s shadow at snapshot ωov = 4π/16. 6π/16.

Figure 19: Cross sections of Fourier transforms parallel (red curve) and perpendicular (blue curve) to the black hole’s spin axis at various snapshots. The horizontal axis of the cross section plot denotes the length of the baseline array (in gigalambda) while the vertical axis denotes the visibility amplitude of the Fourier transforms.

5 Observations from the code

From the snapshots of the cross sections depicted in Figure 19, we indeed found that there is more variability of visibility amplitude along the horizontal cross sections parallel to the black hole’s spin.

The method developed in this thesis is a very useful tool to test general relativity through the EHT observations. The EHT can measure the visibility amplitudes of the accretion disk around a black hole. The cross-section method can be used to produce different models of visibility amplitudes. Then by varying the parameters in the code, these models can be manipulated to mimic the actual visibility amplitudes measured by the EHT. The cross section method developed in this thesis can applied on data extracted from the EHT project. By analyzing the variation in visibility amplitudes, general relativity’s prediction of stationary shadows of black holes can be tested.

May 7, 2020 28 Information Paradox Chirag Rathi 6 Conclusions

This Honors thesis explored one of the most active areas of research in astrophysics: The Black Hole

Information Paradox. After the prediction of black holes’ existence almost a century ago, these astrophysical compact objects have proven to be the ideal testing ground for general relativity and quantum physics. However, the existence of black holes fundamentally challenges the two most successful theories of physics in the form of the Information Paradox. Section1 explains the arguments that led to this paradox and section2 recaps the potential solutions presented to solve this paradox along with their observational consequences (if any). Section3 summarizes the work of the EHT project and explains how this project has the potential to test general relativity’s prediction static black hole shadow. Section4 and section5 present the work done to develop the cross-section method and the observations made by taking the horizontal and vertical cross sections of the Fourier transforms.

This thesis focused its attention on developing a new method that will be important to inves- tigating the data from the EHT. The next step in the study will be to compare and adjust the models produced by the cross-section method to match the visibility amplitudes derived from the

EHT data.

May 7, 2020 29 Information Paradox Chirag Rathi References

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