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Exploring the Black Hole Information Paradox EXPLORING THE BLACK HOLE INFORMATION PARADOX Item Type Electronic Thesis; text Authors Rathi, Chirag Citation Rathi, Chirag. (2020). EXPLORING THE BLACK HOLE 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 Item License http://rightsstatements.org/vocab/InC/1.0/ 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 Astronomy THE UNIVERSITY OF ARIZONA MAY 2020 Approved by: Dr. Dimitrios Psaltis Department of Astronomy Abstract.| Mere weeks after Albert Einstein published his theory of general relativity, 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 photons could not crawl out of this potential well and hence the phrase "even light cannot escape a black hole" came into being. Black holes are characterized by the existence of an event horizon, a surface that disconnects the interior regions of the black hole from the exterior universe. These compact objects existed on paper, without any direct observational evidence, for a century. However, with the recent shift in interests towards gravitational wave astronomy and the success of projects like the Event Horizon Telescope (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 Gravastars..........................................9 2.3 Firewalls........................................... 10 2.4 Fuzzballs.......................................... 11 2.5 Naked Singularity...................................... 12 2.6 Non-Kerr Black Holes................................... 15 2.7 Wormholes......................................... 16 2.8 Boson stars......................................... 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 quantum mechanics. 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: j finali = Smn j initiali : (1) Here j initiali is the initial (pure) quantum state, j finali is the final quantum state and Smn is the density 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 gravity. 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 quantum gravity. 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, Stephen Hawking 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 mass 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 nature of black holes. Consider the gravitational collapse of a star 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 momentum (the no-hair theorem). The Schwarzschild solution, the Reissner-Nordstrom metric, the Kerr metric 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 Hawking radiation. 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 jΨi = j0initiali j radiationi.
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