A Hole in the Black Hole
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A Mathematical Derivation of the General Relativistic Schwarzschild
A Mathematical Derivation of the General Relativistic Schwarzschild Metric An Honors thesis presented to the faculty of the Departments of Physics and Mathematics East Tennessee State University In partial fulfillment of the requirements for the Honors Scholar and Honors-in-Discipline Programs for a Bachelor of Science in Physics and Mathematics by David Simpson April 2007 Robert Gardner, Ph.D. Mark Giroux, Ph.D. Keywords: differential geometry, general relativity, Schwarzschild metric, black holes ABSTRACT The Mathematical Derivation of the General Relativistic Schwarzschild Metric by David Simpson We briefly discuss some underlying principles of special and general relativity with the focus on a more geometric interpretation. We outline Einstein’s Equations which describes the geometry of spacetime due to the influence of mass, and from there derive the Schwarzschild metric. The metric relies on the curvature of spacetime to provide a means of measuring invariant spacetime intervals around an isolated, static, and spherically symmetric mass M, which could represent a star or a black hole. In the derivation, we suggest a concise mathematical line of reasoning to evaluate the large number of cumbersome equations involved which was not found elsewhere in our survey of the literature. 2 CONTENTS ABSTRACT ................................. 2 1 Introduction to Relativity ...................... 4 1.1 Minkowski Space ....................... 6 1.2 What is a black hole? ..................... 11 1.3 Geodesics and Christoffel Symbols ............. 14 2 Einstein’s Field Equations and Requirements for a Solution .17 2.1 Einstein’s Field Equations .................. 20 3 Derivation of the Schwarzschild Metric .............. 21 3.1 Evaluation of the Christoffel Symbols .......... 25 3.2 Ricci Tensor Components ................. -
Stephen Hawking: 'There Are No Black Holes' Notion of an 'Event Horizon', from Which Nothing Can Escape, Is Incompatible with Quantum Theory, Physicist Claims
NATURE | NEWS Stephen Hawking: 'There are no black holes' Notion of an 'event horizon', from which nothing can escape, is incompatible with quantum theory, physicist claims. Zeeya Merali 24 January 2014 Artist's impression VICTOR HABBICK VISIONS/SPL/Getty The defining characteristic of a black hole may have to give, if the two pillars of modern physics — general relativity and quantum theory — are both correct. Most physicists foolhardy enough to write a paper claiming that “there are no black holes” — at least not in the sense we usually imagine — would probably be dismissed as cranks. But when the call to redefine these cosmic crunchers comes from Stephen Hawking, it’s worth taking notice. In a paper posted online, the physicist, based at the University of Cambridge, UK, and one of the creators of modern black-hole theory, does away with the notion of an event horizon, the invisible boundary thought to shroud every black hole, beyond which nothing, not even light, can escape. In its stead, Hawking’s radical proposal is a much more benign “apparent horizon”, “There is no escape from which only temporarily holds matter and energy prisoner before eventually a black hole in classical releasing them, albeit in a more garbled form. theory, but quantum theory enables energy “There is no escape from a black hole in classical theory,” Hawking told Nature. Peter van den Berg/Photoshot and information to Quantum theory, however, “enables energy and information to escape from a escape.” black hole”. A full explanation of the process, the physicist admits, would require a theory that successfully merges gravity with the other fundamental forces of nature. -
Black Holes. the Universe. Today’S Lecture
Physics 311 General Relativity Lecture 18: Black holes. The Universe. Today’s lecture: • Schwarzschild metric: discontinuity and singularity • Discontinuity: the event horizon • Singularity: where all matter falls • Spinning black holes •The Universe – its origin, history and fate Schwarzschild metric – a vacuum solution • Recall that we got Schwarzschild metric as a solution of Einstein field equation in vacuum – outside a spherically-symmetric, non-rotating massive body. This metric does not apply inside the mass. • Take the case of the Sun: radius = 695980 km. Thus, Schwarzschild metric will describe spacetime from r = 695980 km outwards. The whole region inside the Sun is unreachable. • Matter can take more compact forms: - white dwarf of the same mass as Sun would have r = 5000 km - neutron star of the same mass as Sun would be only r = 10km • We can explore more spacetime with such compact objects! White dwarf Black hole – the limit of Schwarzschild metric • As the massive object keeps getting more and more compact, it collapses into a black hole. It is not just a denser star, it is something completely different! • In a black hole, Schwarzschild metric applies all the way to r = 0, the black hole is vacuum all the way through! • The entire mass of a black hole is concentrated in the center, in the place called the singularity. Event horizon • Let’s look at the functional form of Schwarzschild metric again: ds2 = [1-(2m/r)]dt2 – [1-(2m/r)]-1dr2 - r2dθ2 -r2sin2θdφ2 • We want to study the radial dependence only, and at fixed time, i.e. we set dφ = dθ = dt = 0. -
Black Hole Weather Forcasting
NTU Physics Department Chih-Hung Wu 吳智弘 I. Spacetime Locality and ER=EPR Conjecture II. Construction of the Counter-example III. Debate with Professor J. Maldacena J. Maldacena and L.Susskind, “Cool horizon for entangled black holes”, Fortsch. Phys. 61 (2013) 781-811 arXiv:1306.0533 Pisin Chen, Chih-Hung Wu, Dong-hanYeom. ”Broken bridges: A counter-example of the ER=EPR conjecture”, arXiv:1608.08695, Aug 30, 2016 Locality(Impossibility of superluminal signal) 1.Quantum Theory-EPR entanglement 2.General Relativity-ER bridge Violation of Locality?---NO! ER bridge should remain un-traversable even in the quantum theory. En /2 e nLR m n In AdS/CFT framework, an eternal AdS- Schwarzschild BH and the Penrose diagram: En /2 e nLR m n Maximally entangled J. Maldacena, “Eternal black hole in AdS” JHEP 0304 (2003) 021 ,arXiv:0106112 Consider such a scenario: A large number of particles, entangled into separate Bell pairs, and separate them when we collapse each side to form two distant black holes, the two black holes will be entangled. Now they make a conjecture that they will also be connected by an Einstein-Rosen bridge. ER=EPR Known: EREPR Conjecture: EPRER How to realize in the usual Hawking radiation scenario? Second black hole= early half of HR. Step 1: Generate an entangled system Step 2: Formation of the ER bridge Step 3: Collapsing of the bubble Step 4: Evaporation of the black hole Step 5: Communication via ER bridge Assumption: GR with N massless scalar fields 11N S gd4 x ()()() 2 U f 2 i 22i1 The potential has two minima: (AdS space) Background (false vacuum) True vacuum 0 Prepared in advance (e.g. -
Evolution of the Cosmological Horizons in a Concordance Universe
Evolution of the Cosmological Horizons in a Concordance Universe Berta Margalef–Bentabol 1 Juan Margalef–Bentabol 2;3 Jordi Cepa 1;4 [email protected] [email protected] [email protected] 1Departamento de Astrofísica, Universidad de la Laguna, E-38205 La Laguna, Tenerife, Spain: 2Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, E-28040 Madrid, Spain. 3Facultad de Ciencias Físicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain. 4Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain. Abstract The particle and event horizons are widely known and studied concepts, but the study of their properties, in particular their evolution, have only been done so far considering a single state equation in a deceler- ating universe. This paper is the first of two where we study this problem from a general point of view. Specifically, this paper is devoted to the study of the evolution of these cosmological horizons in an accel- erated universe with two state equations, cosmological constant and dust. We have obtained closed-form expressions for the horizons, which have allowed us to compute their velocities in terms of their respective recession velocities that generalize the previous results for one state equation only. With the equations of state considered, it is proved that both velocities remain always positive. Keywords: Physics of the early universe – Dark energy theory – Cosmological simulations This is an author-created, un-copyedited version of an article accepted for publication in Journal of Cosmology and Astroparticle Physics. IOP Publishing Ltd/SISSA Medialab srl is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. -
NIDUS IDEARUM. Scilogs, IV: Vinculum Vinculorum
University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2019 NIDUS IDEARUM. Scilogs, IV: vinculum vinculorum Florentin Smarandache University of New Mexico, [email protected] Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Celtic Studies Commons, Digital Humanities Commons, European Languages and Societies Commons, German Language and Literature Commons, Mathematics Commons, and the Modern Literature Commons Recommended Citation Smarandache, Florentin. "NIDUS IDEARUM. Scilogs, IV: vinculum vinculorum." (2019). https://digitalrepository.unm.edu/math_fsp/310 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. scilogs, IV nidus idearum vinculum vinculorum Bipolar Neutrosophic OffSet Refined Neutrosophic Hypergraph Neutrosophic Triplet Structures Hyperspherical Neutrosophic Numbers Neutrosophic Probability Distributions Refined Neutrosophic Sentiment Classes of Neutrosophic Operators n-Valued Refined Neutrosophic Notions Theory of Possibility, Indeterminacy, and Impossibility Theory of Neutrosophic Evolution Neutrosophic World Florentin Smarandache NIDUS IDEARUM. Scilogs, IV: vinculum vinculorum Brussels, 2019 Exchanging ideas with Mohamed Abdel-Basset, Akeem Adesina A. Agboola, Mumtaz Ali, Saima Anis, Octavian Blaga, Arsham Borumand Saeid, Said Broumi, Stephen Buggie, Victor Chang, Vic Christianto, Mihaela Colhon, Cuờng Bùi Công, Aurel Conțu, S. Crothers, Otene Echewofun, Hoda Esmail, Hojjat Farahani, Erick Gonzalez, Muhammad Gulistan, Yanhui Guo, Mohammad Hamidi, Kul Hur, Tèmítópé Gbóláhàn Jaíyéolá, Young Bae Jun, Mustapha Kachchouh, W. -
Novel Shadows from the Asymmetric Thin-Shell Wormhole
Physics Letters B 811 (2020) 135930 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Novel shadows from the asymmetric thin-shell wormhole ∗ Xiaobao Wang a, Peng-Cheng Li b,c, Cheng-Yong Zhang d, Minyong Guo b, a School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, PR China b Center for High Energy Physics, Peking University, No. 5 Yiheyuan Rd, Beijing 100871, PR China c Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, No. 5 Yiheyuan Rd, Beijing 100871, PR China d Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, PR China a r t i c l e i n f o a b s t r a c t Article history: For dark compact objects such as black holes or wormholes, the shadow size has long been thought to Received 29 September 2020 be determined by the unstable photon sphere (region). However, by considering the asymmetric thin- Accepted 2 November 2020 shell wormhole (ATSW) model, we find that the impact parameter of the null geodesics is discontinuous Available online 6 November 2020 through the wormhole in general and hence we identify novel shadows whose sizes are dependent of Editor: N. Lambert the photon sphere in the other side of the spacetime. The novel shadows appear in three cases: (A2) The observer’s spacetime contains a photon sphere and the mass parameter is smaller than that of the opposite side; (B1, B2) there’ s no photon sphere no matter which mass parameter is bigger. -
Naked Firewalls Phys
Naked Firewalls Phys. Rev. Lett. 116, 161304 (2016) Pisin Chen, Yen Chin Ong, Don N. Page, Misao Sasaki, and Dong-han Yeom 2016 May 19 Introduction In the firewall proposal, it is assumed that the firewall lies near the event horizon and should not be observable except by infalling observers, who are presumably terminated at the firewall. However, if the firewall is located near where the horizon would have been, based on the spacetime evolution up to that time, later quantum fluctuations of the Hawking emission rate can cause the `teleological' event horizon to have migrated to the inside of the firewall location, rendering the firewall naked. In principle, the firewall can be arbitrarily far outside the horizon. This casts doubt about the notion that a firewall is the `most conservative' solution to the information loss paradox. The AMPS Argument Almheiri, Marolf, Polchinski and Sully (AMPS) argued that local quantum field theory, unitarity, and no-drama (the assumption that infalling observers should not experience anything unusual at the event horizon if the black hole is sufficiently large) cannot all be consistent with each other for the Hawking evaporation of a black hole with a finite number of quantum states given by the Bekenstein-Hawking entropy. AMPS suggested that the `most conservative' resolution to this inherent inconsistency between the various assumptions is to give up no-drama. Instead, an infalling observer would be terminated once he or she hits the so-called firewall. This seems rather surprising, because the curvature is negligibly small at the event horizon of a sufficiently large black hole, and thus one would expect nothing special but low energy physics. -
Firewalls and the Quantum Properties of Black Holes
Firewalls and the Quantum Properties of Black Holes A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science degree in Physics from the College of William and Mary by Dylan Louis Veyrat Advisor: Marc Sher Senior Research Coordinator: Gina Hoatson Date: May 10, 2015 1 Abstract With the proposal of black hole complementarity as a solution to the information paradox resulting from the existence of black holes, a new problem has become apparent. Complementarity requires a vio- lation of monogamy of entanglement that can be avoided in one of two ways: a violation of Einstein’s equivalence principle, or a reworking of Quantum Field Theory [1]. The existence of a barrier of high-energy quanta - or “firewall” - at the event horizon is the first of these two resolutions, and this paper aims to discuss it, for Schwarzschild as well as Kerr and Reissner-Nordstr¨omblack holes, and to compare it to alternate proposals. 1 Introduction, Hawking Radiation While black holes continue to present problems for the physical theories of today, quite a few steps have been made in the direction of understanding the physics describing them, and, consequently, in the direction of a consistent theory of quantum gravity. Two of the most central concepts in the effort to understand black holes are the black hole information paradox and the existence of Hawking radiation [2]. Perhaps the most apparent result of black holes (which are a consequence of general relativity) that disagrees with quantum principles is the possibility of information loss. Since the only possible direction in which to pass through the event horizon is in, toward the singularity, it would seem that information 2 entering a black hole could never be retrieved. -
Explained in 60 Seconds: the Event Horizon and the Fate of Fish
Explained in 60 Seconds: Seconds 60 Explained in Explained in The event horizon and the fate of fish Clementine Cheetham Keywords Pioneer Productions, UK Event horizon, black holes, analogy, spacetime [email protected] Every time a physicist says the words “event If spacetime is like a river, spacetime at a kip to swim faster than the speed of flow, it horizon” a fish dies. It’s not nice and it’s not black hole is like that river flowing over a will swim merrily away. However, once the fair, but there we are. waterfall. Everything moves through space water flows over that crest and plummets time, wriggling through the spatial ele down towards the base of the falls, our lit We should perhaps expect a certain maso ments and following traditionally straight tle fishy is beyond redemption. It will never chism in the type of person who chooses to paths through time. That includes light, be able to swim fast enough through the dedicate their life to studying something so our precious bringer of information about flow to get back up. impenetrable as black holes and the fact is the Universe. Like a fish swimming down a that no physicist has ever explained why a river, light travels in a straight line through That’s the event horizon. Outside, light black hole is black without using the same spacetime, oblivious to the larger pattern can escape the black hole’s pull — flying fish-killing analogy. An analogy that I will, that guides its journey. faster than spacetime flows into the hole. -
Wormholes Untangle a Black Hole Paradox
Quanta Magazine Wormholes Untangle a Black Hole Paradox A bold new idea aims to link two famously discordant descriptions of nature. In doing so, it may also reveal how space-time owes its existence to the spooky connections of quantum information. By K.C. Cole One hundred years after Albert Einstein developed his general theory of relativity, physicists are still stuck with perhaps the biggest incompatibility problem in the universe. The smoothly warped space- time landscape that Einstein described is like a painting by Salvador Dalí — seamless, unbroken, geometric. But the quantum particles that occupy this space are more like something from Georges Seurat: pointillist, discrete, described by probabilities. At their core, the two descriptions contradict each other. Yet a bold new strain of thinking suggests that quantum correlations between specks of impressionist paint actually create not just Dalí’s landscape, but the canvases that both sit on, as well as the three-dimensional space around them. And Einstein, as he so often does, sits right in the center of it all, still turning things upside-down from beyond the grave. Like initials carved in a tree, ER = EPR, as the new idea is known, is a shorthand that joins two ideas proposed by Einstein in 1935. One involved the paradox implied by what he called “spooky action at a distance” between quantum particles (the EPR paradox, named for its authors, Einstein, Boris Podolsky and Nathan Rosen). The other showed how two black holes could be connected through far reaches of space through “wormholes” (ER, for Einstein-Rosen bridges). At the time that Einstein put forth these ideas — and for most of the eight decades since — they were thought to be entirely unrelated. -
Radiation from an Inertial Mirror Horizon
universe Communication Radiation from an Inertial Mirror Horizon Michael Good 1,2,* and Ernazar Abdikamalov 1,2 1 Department of Physics, Nazarbayev University, Nur-Sultan 010000, Kazakhstan; [email protected] 2 Enegetic Cosmos Laboratory, Nazarbayev University, Nur-Sultan 010000, Kazakhstan * Correspondence: [email protected] Received: 27 July 2020; Accepted: 19 August 2020; Published: 20 August 2020 Abstract: The purpose of this study is to investigate radiation from asymptotic zero acceleration motion where a horizon is formed and subsequently detected by an outside witness. A perfectly reflecting moving mirror is used to model such a system and compute the energy and spectrum. The trajectory is asymptotically inertial (zero proper acceleration)—ensuring negative energy flux (NEF), yet approaches light-speed with a null ray horizon at a finite advanced time. We compute the spectrum and energy analytically. Keywords: acceleration radiation; moving mirrors; QFT in curved spacetime; horizons; dynamical Casimir effect; black holes; Hawking radiation; negative energy flux 1. Introduction Recent studies have utilized the simplicity of the established moving mirror model [1–6] by applying accelerating boundary correspondences (ABC’s) to novel situations, including the Schwarzschild [7], Reissner-Nordström (RN) [8], Kerr [9], and de Sitter [10] geometries, whose mirrors have asymptotic infinite accelerations: lim a(v) = ¥, (1) v!vH where v = t + x is the advanced time light-cone coordinate, vH is the horizon and a is the proper acceleration of the moving mirror. These moving mirrors with horizons do not emit negative energy flux (NEF [11–16]). ABC’s also exist for extremal black holes, including extremal RN [17,18], extremal Kerr [9,19], and extremal Kerr–Newman [20] geometries, whose mirrors have asymptotic uniform accelerations: lim a(v) = constant.