Astrophysical Wormholes
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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 ................. -
Quantum Detection of Wormholes Carlos Sabín
www.nature.com/scientificreports OPEN Quantum detection of wormholes Carlos Sabín We show how to use quantum metrology to detect a wormhole. A coherent state of the electromagnetic field experiences a phase shift with a slight dependence on the throat radius of a possible distant wormhole. We show that this tiny correction is, in principle, detectable by homodyne measurements Received: 16 February 2017 after long propagation lengths for a wide range of throat radii and distances to the wormhole, even if the detection takes place very far away from the throat, where the spacetime is very close to a flat Accepted: 15 March 2017 geometry. We use realistic parameters from state-of-the-art long-baseline laser interferometry, both Published: xx xx xxxx Earth-based and space-borne. The scheme is, in principle, robust to optical losses and initial mixedness. We have not observed any wormhole in our Universe, although observational-based bounds on their abundance have been established1. The motivation of the search of these objects is twofold. On one hand, the theoretical implications of the existence of topological spacetime shortcuts would entail a challenge to our understanding of deep physical principles such as causality2–5. On the other hand, typical phenomena attributed to black holes can be mimicked by wormholes. Therefore, if wormholes exist the identity of the objects in the center of the galaxies might be questioned6 as well as the origin of the already observed gravitational waves7, 8. For these reasons, there is a renewed interest in the characterization of wormholes9–11 and in their detection by classical means such as gravitational lensing12, 13, among others14. -
Active Galactic Nuclei: a Brief Introduction
Active Galactic Nuclei: a brief introduction Manel Errando Washington University in St. Louis The discovery of quasars 3C 273: The first AGN z=0.158 2 <latexit sha1_base64="4D0JDPO4VKf1BWj0/SwyHGTHSAM=">AAACOXicbVDLSgMxFM34tr6qLt0Ei+BC64wK6kIoPtCNUMU+oNOWTJq2wWRmSO4IZZjfcuNfuBPcuFDErT9g+hC09UC4h3PvJeceLxRcg20/W2PjE5NT0zOzqbn5hcWl9PJKUQeRoqxAAxGoskc0E9xnBeAgWDlUjEhPsJJ3d9rtl+6Z0jzwb6ETsqokLZ83OSVgpHo6754xAQSf111JoK1kTChN8DG+uJI7N6buYRe4ZBo7di129hJ362ew5NJGAFjjfm3V4m0nSerpjJ21e8CjxBmQDBogX08/uY2ARpL5QAXRuuLYIVRjooBTwZKUG2kWEnpHWqxiqE+MmWrcuzzBG0Zp4GagzPMB99TfGzGRWnekZya7rvVwryv+16tE0DysxtwPI2A+7X/UjASGAHdjxA2uGAXRMYRQxY1XTNtEEQom7JQJwRk+eZQUd7POfvboej+TOxnEMYPW0DraRA46QDl0ifKogCh6QC/oDb1bj9ar9WF99kfHrMHOKvoD6+sbuhSrIw==</latexit> <latexit sha1_base64="H7Rv+ZHksM7/70841dw/vasasCQ=">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</latexit> The power source of quasars • The luminosity (L) of quasars, i.e. how bright they are, can be as high as Lquasar ~ 1012 Lsun ~ 1040 W. • The energy source of quasars is accretion power: - Nuclear fusion: 2 11 1 ∆E =0.007 mc =6 10 W s g− -
Searching for Evaporating Primordial Black Holes Using Fermi Gamma Ray Telescope Data
Searching for Evaporating Primordial Black Holes using Fermi Gamma Ray Telescope Data Jagdev Bains CID 00552650 Horns Group, DESY A thesis submitted for the degree of MSci May 2011 Abstract All confirmed photon event data (≥ 100MeV) recorded by the Fermi Gamma Ray Telescope's (FGRT's) Large Area Telescope (LAT) was used to attempt to locate photons emitted by evaporating primordial black holes (PBHs) via Hawking radiation. An 'event' is a single photon recorded by the LAT. HEALPix was used to group photons spatially into pixels with unique iden- tification numbers, and all events within these pixels were sorted in time. The time interval between subsequent photons in each pixel was calculated and a histogram made for each pixel. The expected time distribution of events was compared with the measured for each pixel to identify any can- didates for further analysis; if the expected varied significantly with the measured at the smallest time intervals (26µs − 10s). Several pixels were found that met the criteria for further analysis, however they were all found to include known high energy photon sources. No pixels were found with extra photons at short time intervals that could not be attributed to gamma- ray bursts (GRBs) or known sources. A limit was calculated on the range at which the LAT could detect a PBH burst, between 16.69 and 289.16pc. Alle photonischen (≥ 100MeV) und vom Large Area Telescope (LAT) des Fermi Gamma Ray Telescope's (FGRT's) dokumentierten, bestaetigten Ereignisse wurden benutzt, um Photonen zu lokalisieren, die durch sich verfluechti- gende primordiale schwarze Loecher (PBHs) mit Hilfe der Hawking Strahlung emittiert wurden. -
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. -
GURPS4E Ultra-Tech.Qxp
Written by DAVID PULVER, with KENNETH PETERS Additional Material by WILLIAM BARTON, LOYD BLANKENSHIP, and STEVE JACKSON Edited by CHRISTOPHER AYLOTT, STEVE JACKSON, SEAN PUNCH, WIL UPCHURCH, and NIKOLA VRTIS Cover Art by SIMON LISSAMAN, DREW MORROW, BOB STEVLIC, and JOHN ZELEZNIK Illustrated by JESSE DEGRAFF, IGOR FIORENTINI, SIMON LISSAMAN, DREW MORROW, E. JON NETHERLAND, AARON PANAGOS, CHRISTOPHER SHY, BOB STEVLIC, and JOHN ZELEZNIK Stock # 31-0104 Version 1.0 – May 22, 2007 STEVE JACKSON GAMES CONTENTS INTRODUCTION . 4 Adjusting for SM . 16 PERSONAL GEAR AND About the Authors . 4 EQUIPMENT STATISTICS . 16 CONSUMER GOODS . 38 About GURPS . 4 Personal Items . 38 2. CORE TECHNOLOGIES . 18 Clothing . 38 1. ULTRA-TECHNOLOGY . 5 POWER . 18 Entertainment . 40 AGES OF TECHNOLOGY . 6 Power Cells. 18 Recreation and TL9 – The Microtech Age . 6 Generators . 20 Personal Robots. 41 TL10 – The Robotic Age . 6 Energy Collection . 20 TL11 – The Age of Beamed and 3. COMMUNICATIONS, SENSORS, Exotic Matter . 7 Broadcast Power . 21 AND MEDIA . 42 TL12 – The Age of Miracles . 7 Civilization and Power . 21 COMMUNICATION AND INTERFACE . 42 Even Higher TLs. 7 COMPUTERS . 21 Communicators. 43 TECH LEVEL . 8 Hardware . 21 Encryption . 46 Technological Progression . 8 AI: Hardware or Software? . 23 Receive-Only or TECHNOLOGY PATHS . 8 Software . 24 Transmit-Only Comms. 46 Conservative Hard SF. 9 Using a HUD . 24 Translators . 47 Radical Hard SF . 9 Ubiquitous Computing . 25 Neural Interfaces. 48 CyberPunk . 9 ROBOTS AND TOTAL CYBORGS . 26 Networks . 49 Nanotech Revolution . 9 Digital Intelligences. 26 Mail and Freight . 50 Unlimited Technology. 9 Drones . 26 MEDIA AND EDUCATION . 51 Emergent Superscience . -
Life and Adventures of Binary Supermassive Black Holes
Life and adventures of binary supermassive black holes Eugene Vasiliev Lebedev Physical Institute Plan of the talk • Why binary black holes? • Evolutionary stages: – galaxy merger and dynamical friction: the first contact – king of the hill: the binary hardens – the lean years: the final parsec problem – runaway speedup: gravitational waves kick in – anschluss • Life after merger • Perspectives of detection Origin of binary supermassive black holes • Most galaxies are believed to host central massive black holes • In the hierarchical merger paradigm, galaxies in the Universe have typically 1-3 major and multiple minor mergers in their lifetime • Every such merger brings two central black holes from parent galaxies together to form a binary system • We don’t see much evidence for widespread binary SMBH (to say the least) – therefore they need to merge rather efficiently • Merger is a natural way of producing huge black holes from smaller seeds Evolutionary track of binary SMBH • Merger of two galaxies creates a common nucleus; dynamical friction rapidly brings two black holes together to form a binary (a~10 pc) • Three-body interaction of binary with stars of galactic nucleus ejects most stars from the vicinity of the binary by the slingshot effect; a “mass deficit” is created and the binary becomes “hard” (a~1 pc) • The binary further shrinks by scattering off stars that continue to flow into the “loss cone”, due to two-body relaxation or other factors • As the separation reaches ~10–2 pc, gravitational wave emission becomes the dominant mechanism that carries away the energy • Reaching a few Schwarzschild radii (~10–5 pc), the binary finally merges Evolution timescales I. -
A Unifying Theory of Dark Energy and Dark Matter: Negative Masses and Matter Creation Within a Modified ΛCDM Framework J
Astronomy & Astrophysics manuscript no. theory_dark_universe_arxiv c ESO 2018 November 5, 2018 A unifying theory of dark energy and dark matter: Negative masses and matter creation within a modified ΛCDM framework J. S. Farnes1; 2 1 Oxford e-Research Centre (OeRC), Department of Engineering Science, University of Oxford, Oxford, OX1 3QG, UK. e-mail: [email protected]? 2 Department of Astrophysics/IMAPP, Radboud University, PO Box 9010, NL-6500 GL Nijmegen, the Netherlands. Received February 23, 2018 ABSTRACT Dark energy and dark matter constitute 95% of the observable Universe. Yet the physical nature of these two phenomena remains a mystery. Einstein suggested a long-forgotten solution: gravitationally repulsive negative masses, which drive cosmic expansion and cannot coalesce into light-emitting structures. However, contemporary cosmological results are derived upon the reasonable assumption that the Universe only contains positive masses. By reconsidering this assumption, I have constructed a toy model which suggests that both dark phenomena can be unified into a single negative mass fluid. The model is a modified ΛCDM cosmology, and indicates that continuously-created negative masses can resemble the cosmological constant and can flatten the rotation curves of galaxies. The model leads to a cyclic universe with a time-variable Hubble parameter, potentially providing compatibility with the current tension that is emerging in cosmological measurements. In the first three-dimensional N-body simulations of negative mass matter in the scientific literature, this exotic material naturally forms haloes around galaxies that extend to several galactic radii. These haloes are not cuspy. The proposed cosmological model is therefore able to predict the observed distribution of dark matter in galaxies from first principles. -
Engineering the Quantum Foam
Engineering the Quantum Foam Reginald T. Cahill School of Chemistry, Physics and Earth Sciences, Flinders University, GPO Box 2100, Adelaide 5001, Australia [email protected] _____________________________________________________ ABSTRACT In 1990 Alcubierre, within the General Relativity model for space-time, proposed a scenario for ‘warp drive’ faster than light travel, in which objects would achieve such speeds by actually being stationary within a bubble of space which itself was moving through space, the idea being that the speed of the bubble was not itself limited by the speed of light. However that scenario required exotic matter to stabilise the boundary of the bubble. Here that proposal is re-examined within the context of the new modelling of space in which space is a quantum system, viz a quantum foam, with on-going classicalisation. This model has lead to the resolution of a number of longstanding problems, including a dynamical explanation for the so-called `dark matter’ effect. It has also given the first evidence of quantum gravity effects, as experimental data has shown that a new dimensionless constant characterising the self-interaction of space is the fine structure constant. The studies here begin the task of examining to what extent the new spatial self-interaction dynamics can play a role in stabilising the boundary without exotic matter, and whether the boundary stabilisation dynamics can be engineered; this would amount to quantum gravity engineering. 1 Introduction The modelling of space within physics has been an enormously challenging task dating back in the modern era to Galileo, mainly because it has proven very difficult, both conceptually and experimentally, to get a ‘handle’ on the phenomenon of space. -
4. Kruskal Coordinates and Penrose Diagrams
4. Kruskal Coordinates and Penrose Diagrams. 4.1. Removing a coordinate Singularity at the Schwarzschild Radius. The Schwarzschild metric has a singularity at r = rS where g 00 → 0 and g11 → ∞ . However, we have already seen that a free falling observer acknowledges a smooth motion without any peculiarity when he passes the horizon. This suggests that the behaviour at the Schwarzschild radius is only a coordinate singularity which can be removed by using another more appropriate coordinate system. This is in GR always possible provided the transformation is smooth and differentiable, a consequence of the diffeomorphism of the spacetime manifold. Instead of the 4-dimensional Schwarzschild metric we study a 2-dimensional t,r-version. The spherical symmetry of the Schwarzschild BH guaranties that we do not loose generality. −1 ⎛ r ⎞ ⎛ r ⎞ ds 2 = ⎜1− S ⎟ dt 2 − ⎜1− S ⎟ dr 2 (4.1) ⎝ r ⎠ ⎝ r ⎠ To describe outgoing and ingoing null geodesics we divide through dλ2 and set ds 2 = 0. −1 ⎛ rS ⎞ 2 ⎛ rS ⎞ 2 ⎜1− ⎟t& − ⎜1− ⎟ r& = 0 (4.2) ⎝ r ⎠ ⎝ r ⎠ or rewritten 2 −2 ⎛ dt ⎞ ⎛ r ⎞ ⎜ ⎟ = ⎜1− S ⎟ (4.3) ⎝ dr ⎠ ⎝ r ⎠ Note that the angle of the light cone in t,r-coordinate.decreases when r approaches rS After integration the outgoing and ingoing null geodesics of Schwarzschild satisfy t = ± r * +const. (4.4) r * is called “tortoise coordinate” and defined by ⎛ r ⎞ ⎜ ⎟ r* = r + rS ln⎜ −1⎟ (4.5) ⎝ rS ⎠ −1 dr * ⎛ r ⎞ so that = ⎜1− S ⎟ . (4.6) dr ⎝ r ⎠ As r ranges from rS to ∞, r* goes from -∞ to +∞. We introduce the null coordinates u,υ which have the direction of null geodesics by υ = t + r * and u = t − r * (4.7) From (4.7) we obtain 1 dt = ()dυ + du (4.8) 2 and from (4.6) 28 ⎛ r ⎞ 1 ⎛ r ⎞ dr = ⎜1− S ⎟dr* = ⎜1− S ⎟()dυ − du (4.9) ⎝ r ⎠ 2 ⎝ r ⎠ Inserting (4.8) and (4.9) in (4.1) we find ⎛ r ⎞ 2 ⎜ S ⎟ ds = ⎜1− ⎟ dudυ (4.10) ⎝ r ⎠ Fig. -
Physics 200 Problem Set 7 Solution Quick Overview: Although Relativity Can Be a Little Bewildering, This Problem Set Uses Just A
Physics 200 Problem Set 7 Solution Quick overview: Although relativity can be a little bewildering, this problem set uses just a few ideas over and over again, namely 1. Coordinates (x; t) in one frame are related to coordinates (x0; t0) in another frame by the Lorentz transformation formulas. 2. Similarly, space and time intervals (¢x; ¢t) in one frame are related to inter- vals (¢x0; ¢t0) in another frame by the same Lorentz transformation formu- las. Note that time dilation and length contraction are just special cases: it is time-dilation if ¢x = 0 and length contraction if ¢t = 0. 3. The spacetime interval (¢s)2 = (c¢t)2 ¡ (¢x)2 between two events is the same in every frame. 4. Energy and momentum are always conserved, and we can make e±cient use of this fact by writing them together in an energy-momentum vector P = (E=c; p) with the property P 2 = m2c2. In particular, if the mass is zero then P 2 = 0. 1. The earth and sun are 8.3 light-minutes apart. Ignore their relative motion for this problem and assume they live in a single inertial frame, the Earth-Sun frame. Events A and B occur at t = 0 on the earth and at 2 minutes on the sun respectively. Find the time di®erence between the events according to an observer moving at u = 0:8c from Earth to Sun. Repeat if observer is moving in the opposite direction at u = 0:8c. Answer: According to the formula for a Lorentz transformation, ³ u ´ 1 ¢tobserver = γ ¢tEarth-Sun ¡ ¢xEarth-Sun ; γ = p : c2 1 ¡ (u=c)2 Plugging in the numbers gives (notice that the c implicit in \light-minute" cancels the extra factor of c, which is why it's nice to measure distances in terms of the speed of light) 2 min ¡ 0:8(8:3 min) ¢tobserver = p = ¡7:7 min; 1 ¡ 0:82 which means that according to the observer, event B happened before event A! If we reverse the sign of u then 2 min + 0:8(8:3 min) ¢tobserver 2 = p = 14 min: 1 ¡ 0:82 2. -
Spacetimes with Singularities Ovidiu Cristinel Stoica
Spacetimes with Singularities Ovidiu Cristinel Stoica To cite this version: Ovidiu Cristinel Stoica. Spacetimes with Singularities. An. Stiint. Univ. Ovidius Constanta, Ser. Mat., 2012, pp.26. hal-00617027v2 HAL Id: hal-00617027 https://hal.archives-ouvertes.fr/hal-00617027v2 Submitted on 30 Nov 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Spacetimes with Singularities ∗†‡ Ovidiu-Cristinel Stoica Abstract We report on some advances made in the problem of singularities in general relativity. First is introduced the singular semi-Riemannian geometry for met- rics which can change their signature (in particular be degenerate). The standard operations like covariant contraction, covariant derivative, and constructions like the Riemann curvature are usually prohibited by the fact that the metric is not invertible. The things become even worse at the points where the signature changes. We show that we can still do many of these operations, in a different framework which we propose. This allows the writing of an equivalent form of Einstein’s equation, which works for degenerate metric too. Once we make the singularities manageable from mathematical view- point, we can extend analytically the black hole solutions and then choose from the maximal extensions globally hyperbolic regions.