Phys 4390: General Relativity
|
|
- David Owens
- 6 years ago
- Views:
Transcription
1 Phys 4390: General Relativity Dr. David McNutt, 1 (call me Dave) 1 Department of Physics Saint Mary s University January 9, / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
2 My Background I am a mathematician with an interest in differential geometry, dynamical systems and their applications in physics and engineering. My PhD focused on the classification of space-times in General Relativity (GR), in particular my focus was on certain "toy-models" of gravitational radiation: the pp-wave and Kundt wave solutions. At the moment my work in GR has been limited, however I do have considerable experience with the theory. Figure: A purely electromagnetic plane-wave spacetime causing a null cone from event Q to be focused at a second vertex R 2 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
3 Outline 1 Course Goals Course Texts Academic Policies 2 Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity 3 When is Gr Important Some Context Special Relativity From SR to GR 3 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
4 Course Goals Course Goals Course Texts Academic Policies At completion of this course, students will be able to: Employ tensor analysis to approach problems in general relativity. Understand and explain the underlying physical principles of general relativity (GR) Have a quantitative understanding of the application of GR in modern astrophyics. 4 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
5 Course Goals Course Texts Academic Policies Introduction. Review of Special Relativity (SR), and use of tensor notation. Tensor algebra and Calculus: metrics, curvature, covariant differentiation. Fundamental concepts in GR: Principle of Equivalence, Mach s Principle, Principle of Covariance, Principle of Minimal Coupling. Energy momentum tensor and Einstein s equations. Scwarzchild solutions and black holes. Applications of GR in astrophysics: compact objects, gravitational waves, lensing cosmology. This outline, especially the last point, is subject to change depending on student s interest and time constraints 5 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
6 Course Texts Course Goals Course Texts Academic Policies I will primarily use the textbook "Introducing Einstein s Relativity" by Ray D Inverno, as this is a good introduction to GR with some advanced topics covered. That said, I may also take material from: Robert M. Wald s "General Relativity", John Stewart s "Advanced General Relativity", James Hartle s "Gravity: An Introduction to Einstein s General Relativity". 6 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
7 Approach to Teaching Course Goals Course Texts Academic Policies After this class I will not be using slides. Instead I will work on the board and follow class notes. As I am left handed and calligraphy was never a strong point, I will provide typed class notes. These notes may follow after the class is done, but it is my sincere hope to prepare them prior to class. These notes will be posted on my personal website, and blackboard. 7 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
8 Academic Integrity Course Goals Course Texts Academic Policies I encourage students to discuss work with each other in order to mutually understand the topics taught. Similarly, I see no harm with people working together on assignment questions. I object to solutions being copied word-for-word without any attempt to rehash to reflect the student s understanding. 8 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
9 Marking Scheme Course Goals Course Texts Academic Policies The current marking scheme is: Assignments 60% Final Exam 40% There will be a total of 5 assignments, with one every two weeks. 9 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
10 A Big List Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity 1 Precision gravity in the solar system 2 Relativistic stars 3 Black holes 4 Cosmological Models 5 Gravitational lensing 6 Gravitational waves 7 Quantum gravity 10 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
11 Precision Gravity Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity To measure the gravity field of the earth, NASA and the German Aerospace Center launched GRACE:Gravity Recovery and Climate Experiment. This has become an important tool for studying the ocean, geology and climate of the earth. Combining the data from GRACE with that of LAGEOS:LAser GEOdynamics Satellites, it is hoped that the relativistic effect of frame-dragging could be measured. These satellites are intended to provide an orbiting laser ranging benchmark for geodynamical studies of the Earth. Figure: Gravity Anomaly map from GRACE; taken from Wikipedia Figure: LAGEOS-1 satellite; taken from Wikipedia 11 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
12 Relativistic Stars Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity White dwarfs and neutron stars employ non-thermal pressure sources, i.e., electron and neutron degeneracy respectively, to resist contraction. At low masses white dwarves may be analyzed without relativity, however at higher masses these models are inaccurate. Neutron stars are relativistic. Recent computational models on the ignition of supernovae are considering general relativistic effects. Figure: Chandrasekhar Limit Graph; taken from wikiepdia 12 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
13 "Global" Cosmology Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity In order to describe curved spacetimes we must use GR This requires working on scales much larger than galactic clusters; often dust or perfect fluid models are considered for the spacetimes. In the weak field approximation it can be shown that the Newtonian description is sufficient, with great accuracy. The Friedmann equations dictate cosmic expansion and allow for one to study a number of possible scenarios for curvature of the universe. The cosmological constant, which originally frustrated Einstein s vision of a static universe has been shown to be a necessary part of cosmology. Figure: Two dimensional examples of surfaces with negative, vanishing, and positive curvature respectively; taken from Cosmology tutorial 13 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
14 Gravitational Lensing Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity While originally discussed by Chwolson (1924) and Klin (1936), Einstein published a canonical article on the subject in Three classes of gravitational lensing: strong lensing, weak lensing, and microlensing. Figure: Artist conception of gravitational lensing from a galaxy cluster; taken from NASA Figure: Einstein s Cross: An example of strong gravitational lensing, where four images of the same distant quasar appear around a foreground galaxy. 14 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
15 Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity Gravitational Waves GR predicts that ripples in spacetime can propagate at the speed of light, these ripples are called gravitational waves. Mergers of compact objects should produce immense amounts of gravitational radiation All mass produces gravitational waves, in this sense if one could detect gravitational waves, the universe would be very "bright". Unfortunately these are incredibly difficult to detect due to weak coupling F of matter Fgrav Figure: Artist conception of LISA (Laser Interferometer Space Antenna) spacecraft; taken from NASA elec 15 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
16 Planck Scale and Quantum Gravity Precision Gravity In the Solar System Relativistic Stars: white dwarfs, neutron stars, and supernovae (oh my!) Cosmological Modeling Gravitational... Quantum Gravity Using the fundamental constants of nature, it is possible to derive units associated with an era when quantum gravity is relevant on the "Planck" Scale. Taking, G and c, we can produce Planck length, mass and time. l p = G c = m c m p = G = kg t p = lp c = G c 5 = s 16 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
17 Hand-waving argument When is Gr Important Some Context Special Relativity From SR to GR As a simple argument consider the following: In the Newtonian approximation of a test particle in a closed orbit with speed v, radius R around a mass M GM R 2 = v 2 R v 2 = GM R Dividing v 2 by c 2 yields a dimensionless ratio v 2 c 2 = GM Rc 2 17 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
18 Comparison of Gm/Rc 2 values When is Gr Important Some Context Special Relativity From SR to GR Blackholes 1 Neutron stars 10 1 Sun 10 6 Earth 10 9 Figure: Taken from Fig 1.1 of Hartle provides a 18 / 27 Dr. David McNutt, (call me Dave) Phys comparison 4390: General of Relativity masses and distances.
19 When is Gr Important Some Context Special Relativity From SR to GR Successes and Failures of Newtonian Picture Replaced the Aristotelian picture that: Objects move when acted upon by a force, and tend to a stationary state when force is removed. Could not explain the force of gravity, which is a constant force but objects were accelerated. Newton s First Law gave us the first clue about relativity. If the force is such that F = 0 then v = C where C is a constant vector This allows for the concept of inertial frames of reference. Any frame for which v = C is defined to be an inertial frame of reference Newton s Laws cannot impose the constancy of the speed of light, which led to an erroneous believe in absolute simultaneity instead of a relative one. 19 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
20 When is Gr Important Some Context Special Relativity From SR to GR Newtonian Transformations between Inertial Frames of Reference Making the transformation: x n = x vt, y n = y, z n = z, t n = t it is easily shown that the second derivative of x w.r.t. t satisfies d2 x dt 2 = d2 x n dt 2 n and F = F n 20 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
21 The Light Cone in Special Relativity When is Gr Important Some Context Special Relativity From SR to GR Speed of light is the same in all inertial frames Normal matter is restricted to speeds less than c New concept of simultaneity, namely Relative Simultaneity 21 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
22 When is Gr Important Some Context Special Relativity From SR to GR Coordinate Transformations in Special Relativity Consider the Lorentz transformation, x n = x vt, yn = y, zn = z, tn = t vx/c2 1 (v/c) 2 1 (v/c) 2 As before, the frame will be boosted by speed v along x axis relative to the original frame at O 22 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
23 When is Gr Important Some Context Special Relativity From SR to GR Space-time Diagram Under Lorentz Boost 23 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
24 When is Gr Important Some Context Special Relativity From SR to GR Correspondence of electric and (Newtonian) Gravitational Force Newtonian Gravity Electrostatics Forces Between Sources Fg = GMm r 2 ē Mm Fe = qq 4πɛ 0 r 2 ēqq Force Derived From Potential Fg = m Φ g( x m) Fe = q Φ e( x q) Potential Outside a Spherical Source Φ g = GM r Φ e = Q 4πɛ 0 r Field Equation 2 Φ g = 4πGρ m 2 Φ e = ρ e/ɛ 0 Note: if ḡ( x = Φ g( x) then g( x) = 4πρ m( x), which is similar to Ē = ρe( x)/ɛ / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
25 When is Gr Important Some Context Special Relativity From SR to GR Moving Charges: Maxwell s Equations and Lorentz Force The Lorentz force describes how a collection of moving charges are affected by a velocity dependent force from magnetic fields. F e = q(ē + v B) The velocity dependent term is missing in Newtonian Gravity. For any frame chosen, acceleration depends on mass alone in Newtonian gravity, this implies it is not relativistic. It is possible to add B g term, however this becomes very complicated. 25 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
26 Measuring Ē and B Fields When is Gr Important Some Context Special Relativity From SR to GR Using neutral charges one can construct an inertial frame. Any particle at rest can be used to measure Ē, since F e = qē Once in motion one can measure B through the expression F e = q(ē + v B) Sadly this approach cannot be used to measure gravity, since there is no form of matter that is "neutral" to gravity. 26 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
27 Steppin Stone: SR to GR When is Gr Important Some Context Special Relativity From SR to GR In the presence of gravity, freely falling frames are locally inertial, we call this the Principle of Equivalence. These particles will follow geodesics, which can be seen as the path of least resistance. Returning to the point of no "neutral" charges. Any particle is a source of gravitational field, so as they move through spacetime they also bend it. We can approach GR by formulating SR in this new frame and taking our physical laws and applying the Principle of Covariance (Physical Laws are preserved under changes of coordinates.) Some people believe we need additional principles. 27 / 27 Dr. David McNutt, (call me Dave) Phys 4390: General Relativity
Classical Field Theory
April 13, 2010 Field Theory : Introduction A classical field theory is a physical theory that describes the study of how one or more physical fields interact with matter. The word classical is used in
More informationRelativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION. Wolfgang Rindler. Professor of Physics The University of Texas at Dallas
Relativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION Wolfgang Rindler Professor of Physics The University of Texas at Dallas OXPORD UNIVERSITY PRESS Contents Introduction l 1 From absolute space
More informationCurved Spacetime III Einstein's field equations
Curved Spacetime III Einstein's field equations Dr. Naylor Note that in this lecture we will work in SI units: namely c 1 Last Week s class: Curved spacetime II Riemann curvature tensor: This is a tensor
More informationAnnouncements. Lecture 6. General Relativity. From before. Space/Time - Energy/Momentum
Announcements 2402 Lab will be started next week Lab manual will be posted on the course web today Lab Scheduling is almost done!! HW: Chapter.2 70, 75, 76, 87, 92, 97*, 99, 104, 111 1 st Quiz: 9/18 (Ch.2)
More informationRelativity, Gravitation, and Cosmology
Relativity, Gravitation, and Cosmology A basic introduction TA-PEI CHENG University of Missouri St. Louis OXFORD UNIVERSITY PRESS Contents Parti RELATIVITY Metric Description of Spacetime 1 Introduction
More informationPedagogical Strategy
Integre Technical Publishing Co., Inc. Hartle November 18, 2002 1:42 p.m. hartlemain19-end page 557 Pedagogical Strategy APPENDIX D...as simple as possible, but not simpler. attributed to A. Einstein The
More informationSpecial Relativity: The laws of physics must be the same in all inertial reference frames.
Special Relativity: The laws of physics must be the same in all inertial reference frames. Inertial Reference Frame: One in which an object is observed to have zero acceleration when no forces act on it
More informationMeasuring the Whirling of Spacetime
Measuring the Whirling of Spacetime Lecture series on Experimental Gravity (revised version) Kostas Glampedakis Prologue: does spin gravitate? M 1 M 2 System I: F = GM 1M 2 r 2 J 1 J 2 System II: M 1?
More information2.5.1 Static tides Tidal dissipation Dynamical tides Bibliographical notes Exercises 118
ii Contents Preface xiii 1 Foundations of Newtonian gravity 1 1.1 Newtonian gravity 2 1.2 Equations of Newtonian gravity 3 1.3 Newtonian field equation 7 1.4 Equations of hydrodynamics 9 1.4.1 Motion of
More informationFundamental Theories of Physics in Flat and Curved Space-Time
Fundamental Theories of Physics in Flat and Curved Space-Time Zdzislaw Musielak and John Fry Department of Physics The University of Texas at Arlington OUTLINE General Relativity Our Main Goals Basic Principles
More informationInertial Frame frame-dragging
Frame Dragging Frame Dragging An Inertial Frame is a frame that is not accelerating (in the sense of proper acceleration that would be detected by an accelerometer). In Einstein s theory of General Relativity
More informationGeneral Relativity. Einstein s Theory of Gravitation. March R. H. Gowdy (VCU) General Relativity 03/06 1 / 26
General Relativity Einstein s Theory of Gravitation Robert H. Gowdy Virginia Commonwealth University March 2007 R. H. Gowdy (VCU) General Relativity 03/06 1 / 26 What is General Relativity? General Relativity
More informationAstronomy 182: Origin and Evolution of the Universe
Astronomy 182: Origin and Evolution of the Universe Prof. Josh Frieman Lecture 6 Oct. 28, 2015 Today Wrap up of Einstein s General Relativity Curved Spacetime Gravitational Waves Black Holes Relativistic
More informationA873: Cosmology Course Notes. II. General Relativity
II. General Relativity Suggested Readings on this Section (All Optional) For a quick mathematical introduction to GR, try Chapter 1 of Peacock. For a brilliant historical treatment of relativity (special
More informationLecture 1 General relativity and cosmology. Kerson Huang MIT & IAS, NTU
A Superfluid Universe Lecture 1 General relativity and cosmology Kerson Huang MIT & IAS, NTU Lecture 1. General relativity and cosmology Mathematics and physics Big bang Dark energy Dark matter Robertson-Walker
More informationGeneral Relativity. PHYS-3301 Lecture 6. Chapter 2. Announcement. Sep. 14, Special Relativity
Announcement Course webpage http://www.phys.ttu.edu/~slee/3301/ Textbook PHYS-3301 Lecture 6 HW2 (due 9/21) Chapter 2 63, 65, 70, 75, 76, 87, 92, 97 Sep. 14, 2017 General Relativity Chapter 2 Special Relativity
More informationDynamics of star clusters containing stellar mass black holes: 1. Introduction to Gravitational Waves
Dynamics of star clusters containing stellar mass black holes: 1. Introduction to Gravitational Waves July 25, 2017 Bonn Seoul National University Outline What are the gravitational waves? Generation of
More informationGeneral Relativity and Gravity. Exam 2 Results. Equivalence principle. The Equivalence Principle. Experiment: throw a ball. Now throw some light
General Relativity and Gravity Special Relativity deals with inertial reference frames, frames moving with a constant relative velocity. It has some rather unusual predictions Time dilation Length contraction
More informationGravitation. Adrian Ferent. This is a new quantum gravity theory which breaks the wall of Planck scale. Abstract
Gravitation Adrian Ferent This is a new quantum gravity theory which breaks the wall of Planck scale. My Nobel Prize Idea Abstract The Photon Graviton pair (coupled) has the same speed and frequency, and
More informationSpecial theory of relativity
Announcements l CAPA #9 due Tuesday April 1 l Mastering Physics Chapter 35 due April 1 l Average on exam #2 is 26/40 l For the sum of the first two exams (80 points); l >=67 4.0 l 61-66 3.5 l 50-60 3.0
More informationRELG - General Relativity
Coordinating unit: Teaching unit: Academic year: Degree: ECTS credits: 2017 230 - ETSETB - Barcelona School of Telecommunications Engineering 749 - MAT - Department of Mathematics 748 - FIS - Department
More informationTwo Lectures on Physics
Two Lectures on Physics Cogne 2003 1. The Large Scale Structure of the World 2. The Small Scale Structure of the World The Large Scale Structure of the World The goal of physics is to understand the universe
More informationAstronomy 421. Lecture 24: Black Holes
Astronomy 421 Lecture 24: Black Holes 1 Outline General Relativity Equivalence Principle and its Consequences The Schwarzschild Metric The Kerr Metric for rotating black holes Black holes Black hole candidates
More informationA brain teaser: The anthropic principle! Last lecture I said Is cosmology a science given that we only have one Universe? Weak anthropic principle: "T
Observational cosmology: The Friedman equations 1 Filipe B. Abdalla Kathleen Lonsdale Building G.22 http://zuserver2.star.ucl.ac.uk/~hiranya/phas3136/phas3136 A brain teaser: The anthropic principle! Last
More informationRelativity and Black Holes
Relativity and Black Holes Post-MS Evolution of Very High Mass (>15 M Θ ) Stars similar to high mass except more rapid lives end in Type II supernova explosions main difference: mass of iron core at end
More information2.1 Basics of the Relativistic Cosmology: Global Geometry and the Dynamics of the Universe Part I
1 2.1 Basics of the Relativistic Cosmology: Global Geometry and the Dynamics of the Universe Part I 2 Special Relativity (1905) A fundamental change in viewing the physical space and time, now unified
More informationSPECIAL RELATIVITY! (Einstein 1905)!
SPECIAL RELATIVITY! (Einstein 1905)! Motivations:! Explaining the results of the Michelson-Morley! experiment without invoking a force exerted! on bodies moving through the aether.! Make the equations
More informationBlack Holes. Jan Gutowski. King s College London
Black Holes Jan Gutowski King s College London A Very Brief History John Michell and Pierre Simon de Laplace calculated (1784, 1796) that light emitted radially from a sphere of radius R and mass M would
More informationA5682: Introduction to Cosmology Course Notes. 2. General Relativity
2. General Relativity Reading: Chapter 3 (sections 3.1 and 3.2) Special Relativity Postulates of theory: 1. There is no state of absolute rest. 2. The speed of light in vacuum is constant, independent
More information8. The Expanding Universe, Revisited
8. The Expanding Universe, Revisited A1143: History of the Universe, Autumn 2012 Now that we have learned something about Einstein s theory of gravity, we are ready to revisit what we have learned about
More informationBinary Black Holes, Gravitational Waves, & Numerical Relativity Part 1
1 Binary Black Holes, Gravitational Waves, & Numerical Relativity Part 1 Joan Centrella Chief, Gravitational Astrophysics Laboratory NASA/GSFC Summer School on Nuclear and Particle Astrophysics: Connecting
More informationPHYM432 Relativity and Cosmology fall Introduction. Dr. David K. Sing
PHYM432 Relativity and Cosmology fall 2012 1. Introduction Dr. David K. Sing 1 PHYM432 General Relativity and Cosmology Fall 2012-2013 Instructor: Dr. David K Sing Office: Physics building room 514 Email
More informationTutorial I General Relativity
Tutorial I General Relativity 1 Exercise I: The Metric Tensor To describe distances in a given space for a particular coordinate system, we need a distance recepy. The metric tensor is the translation
More informationBasic Physics. Remaining Topics. Gravitational Potential Energy. PHYS 1403 Introduction to Astronomy. Can We Create Artificial Gravity?
PHYS 1403 Introduction to Astronomy Basic Physics Chapter 5 Remaining Topics Gravitational Potential Energy Escape Velocity Artificial Gravity Gravity Assist An Alternate Theory of Gravity Gravitational
More informationAn introduction to gravitational waves. Enrico Barausse (Institut d'astrophysique de Paris/CNRS, France)
An introduction to gravitational waves Enrico Barausse (Institut d'astrophysique de Paris/CNRS, France) Outline of lectures (1/2) The world's shortest introduction to General Relativity The linearized
More informationBlack Holes. Robert M. Wald
Black Holes Robert M. Wald Black Holes Black Holes: A black hole is a region of spacetime where gravity is so strong that nothing not even light that enters that region can ever escape from it. Michell
More informationRelativistic Astrophysics Neutron Stars, Black Holes & Grav. W. ... A brief description of the course
Relativistic Astrophysics Neutron Stars, Black Holes & Grav. Waves... A brief description of the course May 2, 2009 Structure of the Course Introduction to General Theory of Relativity (2-3 weeks) Gravitational
More informationBasic Physics. What We Covered Last Class. Remaining Topics. Center of Gravity and Mass. Sun Earth System. PHYS 1411 Introduction to Astronomy
PHYS 1411 Introduction to Astronomy Basic Physics Chapter 5 What We Covered Last Class Recap of Newton s Laws Mass and Weight Work, Energy and Conservation of Energy Rotation, Angular velocity and acceleration
More informationENTER RELATIVITY THE HELIOCENTRISM VS GEOCENTRISM DEBATE ARISES FROM MATTER OF CHOOSING THE BEST REFERENCE POINT. GALILEAN TRANSFORMATION 8/19/2016
ENTER RELATIVITY RVBAUTISTA THE HELIOCENTRISM VS GEOCENTRISM DEBATE ARISES FROM MATTER OF CHOOSING THE BEST REFERENCE POINT. GALILEAN TRANSFORMATION The laws of mechanics must be the same in all inertial
More informationCover Page. The handle holds various files of this Leiden University dissertation.
Cover Page The handle http://hdl.handle.net/1887/42442 holds various files of this Leiden University dissertation. Author: Saravanan, S. Title: Spin dynamics in general relativity Issue Date: 2016-07-07
More informationGeneral Relativity and Cosmology. The End of Absolute Space Cosmological Principle Black Holes CBMR and Big Bang
General Relativity and Cosmology The End of Absolute Space Cosmological Principle Black Holes CBMR and Big Bang The End of Absolute Space (AS) Special Relativity (SR) abolished AS only for the special
More informationGravity: What s the big attraction? Dan Wilkins Institute of Astronomy
Gravity: What s the big attraction? Dan Wilkins Institute of Astronomy Overview What is gravity? Newton and Einstein What does gravity do? Extreme gravity The true power of gravity Getting things moving
More informationTheoretical Aspects of Black Hole Physics
Les Chercheurs Luxembourgeois à l Etranger, Luxembourg-Ville, October 24, 2011 Hawking & Ellis Theoretical Aspects of Black Hole Physics Glenn Barnich Physique théorique et mathématique Université Libre
More informationGeneral Relativity: Einstein s Theory of Gravitation. Arien Crellin-Quick and Tony Miller SPRING 2009 PHYS43, SRJC
General Relativity: Einstein s Theory of Gravitation Presented By Arien Crellin-Quick and Tony Miller SPRING 2009 PHYS43, SRJC The Motivations of General Relativity General Relativity, or GR, was created
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Exploring Black Holes General Relativity and Astrophysics Spring 2003
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics 8.224 Exploring Black Holes General Relativity and Astrophysics Spring 2003 ASSIGNMENT WEEK 5 NOTE: Exercises 6 through 8 are to be carried out using the GRorbits
More informationEinstein Toolkit Workshop. Joshua Faber Apr
Einstein Toolkit Workshop Joshua Faber Apr 05 2012 Outline Space, time, and special relativity The metric tensor and geometry Curvature Geodesics Einstein s equations The Stress-energy tensor 3+1 formalisms
More informationAstr 2320 Tues. May 2, 2017 Today s Topics Chapter 23: Cosmology: The Big Bang and Beyond Introduction Newtonian Cosmology Solutions to Einstein s
Astr 0 Tues. May, 07 Today s Topics Chapter : Cosmology: The Big Bang and Beyond Introduction Newtonian Cosmology Solutions to Einstein s Field Equations The Primeval Fireball Standard Big Bang Model Chapter
More informationASTR 200 : Lecture 21. Stellar mass Black Holes
1 ASTR 200 : Lecture 21 Stellar mass Black Holes High-mass core collapse Just as there is an upper limit to the mass of a white dwarf (the Chandrasekhar limit), there is an upper limit to the mass of a
More informationSurvey of Astrophysics A110
Black Holes Goals: Understand Special Relativity General Relativity How do we observe black holes. Black Holes A consequence of gravity Massive neutron (>3M ) cannot be supported by degenerate neutron
More informationModern Physics notes Paul Fendley Lecture 35. Born, chapter III (most of which should be review for you), chapter VII
Modern Physics notes Paul Fendley fendley@virginia.edu Lecture 35 Curved spacetime black holes Born, chapter III (most of which should be review for you), chapter VII Fowler, Remarks on General Relativity
More informationThe Theory of Relativity
The Theory of Relativity Lee Chul Hoon chulhoon@hanyang.ac.kr Copyright 2001 by Lee Chul Hoon Table of Contents 1. Introduction 2. The Special Theory of Relativity The Galilean Transformation and the Newtonian
More informationLecture 18 : Black holes. Astronomy 111
Lecture 18 : Black holes Astronomy 111 Gravity's final victory A star more massive than about 18 M sun would leave behind a post-supernova core this is larger than 2-3 M sun :Neutron degeneracy pressure
More informationUnderstanding and Testing Relativity
Understanding and Testing Relativity From Einstein s formulations to the tests of today www. library.thinkquest.org www.csep10.phys.utk.edu www.arcive.ncsa.uiuc.edu Boston University - April 25, 2006 1
More informationPHYM432 Relativity and Cosmology 17. Cosmology Robertson Walker Metric
PHYM432 Relativity and Cosmology 17. Cosmology Robertson Walker Metric Cosmology applies physics to the universe as a whole, describing it s origin, nature evolution and ultimate fate. While these questions
More informationCracking the Mysteries of the Universe. Dr Janie K. Hoormann University of Queensland
Cracking the Mysteries of the Universe Dr Janie K. Hoormann University of Queensland Timeline of Cosmological Discoveries 16c BCE: flat earth 5-11c CE: Sun at the centre 1837: Bessel et al. measure distance
More informationHOMEWORK 10. Applications: special relativity, Newtonian limit, gravitational waves, gravitational lensing, cosmology, 1 black holes
General Relativity 8.96 (Petters, spring 003) HOMEWORK 10. Applications: special relativity, Newtonian limit, gravitational waves, gravitational lensing, cosmology, 1 black holes 1. Special Relativity
More informationThe Science Missions of Columbia
The Science Missions of Columbia Tools for Viewing The Universe Tools for Viewing The Universe & Columbia Shuttle Added Corrective Optics to the Hubble Space Telescope Hubble Discovers a New View of The
More informationDo You Need to Understand General Relativity to Understand Gravitation?
Do You Need to Understand General Relativity to Understand? Institute of Mathematical Sciences, Chennai IIAP-Bangalore 13 June 2006 Newton s Three Laws Figure: Newton s Laws. Newton The fundamental law
More information12:40-2:40 3:00-4:00 PM
Physics 294H l Professor: Joey Huston l email:huston@msu.edu l office: BPS3230 l Homework will be with Mastering Physics (and an average of 1 hand-written problem per week) Help-room hours: 12:40-2:40
More informationLimitations of Newtonian Physics
Limitations of Newtonian Physics 18 th and 19 th Centuries Newtonian Physics was accepted as an ultimate truth Science is never absolute Hundreds of experiments can t prove my theory right but only one
More informationCharles Keeton. Principles of Astrophysics. Using Gravity and Stellar Physics. to Explore the Cosmos. ^ Springer
Charles Keeton Principles of Astrophysics Using Gravity and Stellar Physics to Explore the Cosmos ^ Springer Contents 1 Introduction: Tools of the Trade 1 1.1 What Is Gravity? 1 1.2 Dimensions and Units
More informationLaser Interferometer Space Antenna Listening to the Universe with Gravitational Waves
Laser Interferometer Space Antenna Listening to the Universe with Gravitational Waves Scott E Pollack for the LISA team UW General Relativity Labs AAPT Workshop GSFC - JPL 5 January 2007 Outline LISA Overview
More informationTest #3 Next Tuesday, Nov. 8 Bring your UNM ID! Bring two number 2 pencils. Announcements. Review for test on Monday, Nov 7 at 3:25pm
Test #3 Next Tuesday, Nov. 8 Bring your UNM ID! Bring two number 2 pencils Announcements Review for test on Monday, Nov 7 at 3:25pm Neutron Star - Black Hole merger Review for Test #3 Nov 8 Topics: Stars
More informationName Final Exam December 7, 2015
Name Final Exam December 7, 015 This test consists of five parts. Please note that in parts II through V, you can skip one question of those offered. Part I: Multiple Choice (mixed new and review questions)
More informationOutline. General Relativity. Black Holes as a consequence of GR. Gravitational redshift/blueshift and time dilation Curvature Gravitational Lensing
Outline General Relativity Gravitational redshift/blueshift and time dilation Curvature Gravitational Lensing Black Holes as a consequence of GR Waste Disposal It is decided that Earth will get rid of
More informationOuter space: A matter of gravity
1997 2009, Millennium Mathematics Project, University of Cambridge. Permission is granted to print and copy this page on paper for non commercial use. For other uses, including electronic redistribution,
More informationDark Energy vs. Dark Matter: Towards a unifying scalar field?
Dark Energy vs. Dark Matter: Towards a unifying scalar field? Alexandre ARBEY Centre de Recherche Astrophysique de Lyon Institut de Physique Nucléaire de Lyon, March 2nd, 2007. Introduction The Dark Stuff
More informationThe structure of spacetime. Eli Hawkins Walter D. van Suijlekom
The structure of spacetime Eli Hawkins Walter D. van Suijlekom Einstein's happiest thought After Einstein formulated Special Relativity, there were two problems: Relativity of accelerated motion The monstrous
More informationA Theory of Gravitation in Flat Space-Time. Walter Petry
A Theory of Gravitation in Flat Space-Time Walter Petry Science Publishing Group 548 Fashion Avenue New York, NY 10018 Published by Science Publishing Group 2014 Copyright Walter Petry 2014 All rights
More informationASTR 1040 Recitation: Relativity
ASTR 1040 Recitation: Relativity Ryan Orvedahl Department of Astrophysical and Planetary Sciences February 17 & 19, 2014 This Week Fiske Planetarium: Thurs Feb 20 (9:30 am) Observing Session: Thurs Feb
More informationCOSMOLOGY The Universe what is its age and origin?
COSMOLOGY The Universe what is its age and origin? REVIEW (SUMMARY) Oppenheimer Volkhoff limit: upper limit to mass of neutron star remnant more than 1.4 M à neutron degeneracy Supernova à extremely dense
More informationIntroduction. Classical vs Modern Physics. Classical Physics: High speeds Small (or very large) distances
Introduction Classical vs Modern Physics High speeds Small (or very large) distances Classical Physics: Conservation laws: energy, momentum (linear & angular), charge Mechanics Newton s laws Electromagnetism
More informationGRAVITATIONAL COLLAPSE
GRAVITATIONAL COLLAPSE Landau and Chandrasekhar first realised the importance of General Relativity for Stars (1930). If we increase their mass and/or density, the effects of gravitation become increasingly
More informationScott Hughes 12 May Massachusetts Institute of Technology Department of Physics Spring 2005
Scott Hughes 12 May 2005 24.1 Gravity? Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 24: A (very) brief introduction to general relativity. The Coulomb interaction
More informationSyllabus: Physics 241 Introduction to Modern Physics Professor Marshall Onellion (office)
1 Syllabus: Physics 241 Introduction to Modern Physics Professor Marshall Onellion (office) 263-6829 Office hours: onellion@wisc.edu MW: 10am- 1pm, F: 10am- noon, or by appointment Text: Kenneth Krane,
More informationGeneral relativity and the Einstein equations
April 23, 2013 Special relativity 1905 Let S and S be two observers moving with velocity v relative to each other along the x-axis and let (t, x) and (t, x ) be the coordinate systems used by these observers.
More informationPhysics 435 and 535 Fall 2016 Gravitational Physics
Physics 435 and 535 Fall 2016 Gravitational Physics Instructor: Prof. Leopoldo A. Pando Zayas Office: Randall 3421, 764-5236, lpandoz@umich.edu Lectures:10:00 11:30 TTh in 335 WH Office Hours: Monday and
More informationFURTHER COSMOLOGY Book page T H E M A K E U P O F T H E U N I V E R S E
FURTHER COSMOLOGY Book page 675-683 T H E M A K E U P O F T H E U N I V E R S E COSMOLOGICAL PRINCIPLE Is the Universe isotropic or homogeneous? There is no place in the Universe that would be considered
More informationTransformation of velocities
Announcements l Help room hours (1248 BPS) Ian La Valley(TA) Mon 4-6 PM Tues 12-3 PM Wed 6-9 PM Fri 10 AM-noon l LON-CAPA #9 due on Thurs Nov 15 l Third hour exam Thursday Dec 6 l Final Exam Tuesday Dec
More informationAstronomy 182: Origin and Evolution of the Universe
Astronomy 182: Origin and Evolution of the Universe Prof. Josh Frieman Lecture 7 Oct. 30, 2015 Today Relativistic Cosmology Dark Side of the Universe I: Dark Matter Assignments This week: read Hawley and
More informationTitle. Author(s)Greve, Ralf. Issue Date Doc URL. Type. Note. File Information. A material called spacetime
Title A material called spacetime Author(s)Greve, Ralf Issue Date 2017-08-21 Doc URL http://hdl.handle.net/2115/67121 Type lecture Note Colloquium of Mechanics, Study Center Mechanics, Dar File Information
More informationStability of Stellar Filaments in Modified Gravity Speaker: Dr. Zeeshan Yousaf Assistant Professor Department of Mathematics University of the Punjab
Stability of Stellar Filaments in Modified Gravity Speaker: Dr. Zeeshan Yousaf Assistant Professor Department of Mathematics University of the Punjab Lahore-Pakistan Hot Topics in Modern Cosmology, XIIth
More informationASTR 200 : Lecture 31. More Gravity: Tides, GR, and Gravitational Waves
ASTR 200 : Lecture 31 More Gravity: Tides, GR, and Gravitational Waves 1 Topic One : Tides Differential tidal forces on the Earth. 2 How do tides work???? Think about 3 billiard balls sitting in space
More informationTa-Pei Cheng PCNY 9/16/2011
PCNY 9/16/2011 Ta-Pei Cheng For a more quantitative discussion, see Relativity, Gravitation & Cosmology: A Basic Introduction (Oxford Univ Press) 2 nd ed. (2010) dark matter & dark energy Astronomical
More informationMassachusetts Institute of Technology Physics Black Holes and Astrophysics Spring 2003 MIDTERM EXAMINATION
Massachusetts Institute of Technology Physics 8.224. Black Holes and Astrophysics Spring 2003 MIDTERM EXAMINATION This exam is CLOSED BOOK; no printed materials are allowed. You may consult ONE 8.5 by
More informationIndex. Cambridge University Press A First Course in General Relativity: Second Edition Bernard F. Schutz. Index.
accelerated particle, 41 acceleration, 46 48 absolute, 2 of the universe, 351 353 accretion disk, 317 active gravitational mass, 197, 202, 355 adiabatic, 103 affine parameter, 161, 166, 175 angular diameter
More informationPreparation of the data analysis of the gravitational wave space antenna.
Preparation of the data analysis of the gravitational wave space antenna. 1) LISA (Laser Interferometer Space Antenna) Why? 2)How? 1 Frequency Limitation Seismic noise cannot be cancelled at low-frequency
More informationHigh-Energy Astrophysics Lecture 6: Black holes in galaxies and the fundamentals of accretion. Overview
High-Energy Astrophysics Lecture 6: Black holes in galaxies and the fundamentals of accretion Robert Laing Overview Evidence for black holes in galaxies and techniques for estimating their mass Simple
More informationASTR Astrophysics 1 - Stellar and Interstellar. Phil Armitage. office: JILA tower A909
ASTR 3730 Astrophysics 1 - Stellar and Interstellar Phil Armitage office: JILA tower A909 email: pja@jilau1.colorado.edu Part one of a year-long introduction to astrophysics: Aim - develop physical understanding
More informationExperimental Tests and Alternative Theories of Gravity
Experimental Tests and Alternative Theories of Gravity Gonzalo J. Olmo Alba gonzalo.olmo@uv.es University of Valencia (Spain) & UW-Milwaukee Experimental Tests and Alternative Theories of Gravity p. 1/2
More informationGravity. Newtonian gravity: F = G M1 M2/r 2
Gravity Einstein s General theory of relativity : Gravity is a manifestation of curvature of 4- dimensional (3 space + 1 time) space-time produced by matter (metric equation? g μν = η μν ) If the curvature
More informationAdvanced Higher Physics
Wallace Hall Academy Physics Department Advanced Higher Physics Astrophysics Problems Data Common Physical Quantities QUANTITY SYMBOL VALUE Gravitational acceleration g 9.8 m s -2 Radius of Earth R E 6.4
More informationSpecial & General Relativity
Special & General Relativity ASTR/PHYS 4080: Intro to Cosmology Week 2 1 Special Relativity: no ether Presumes absolute space and time, light is a vibration of some medium: the ether 2 Equivalence Principle(s)
More informationSyllabus for online Relativity and Cosmology (Instructor: Ta-Pei Cheng)
Syllabus for online Relativity and Cosmology (Instructor: Ta-Pei Cheng) Welcome to this online course on Relativity and Cosmology. The two pillars of modern physics are quantum theory and relativity. Students
More informationLecture Notes on General Relativity
Lecture Notes on General Relativity Matthias Blau Albert Einstein Center for Fundamental Physics Institut für Theoretische Physik Universität Bern CH-3012 Bern, Switzerland The latest version of these
More informationLecture IX: Field equations, cosmological constant, and tides
Lecture IX: Field equations, cosmological constant, and tides Christopher M. Hirata Caltech M/C 350-17, Pasadena CA 91125, USA (Dated: October 28, 2011) I. OVERVIEW We are now ready to construct Einstein
More information5/7/2018. Black Holes. Type II.
Black Holes Type II https://www.youtube.com/watch?v=ctnkk7tnkq8 1 Scientific American 22, 82 (2013) Scientific American 22, 82 (2013) 2 First detection of gravitational waves Recommended reading Physics
More informationUNIVERSITY OF NORTHERN COLORADO
UNIVERSITY OF NORTHERN COLORADO Course Title: Prefix: SCI109 CRN+Section: The Cosmos The Cosmos - 40842 - SCI 109-943 Prerequisites: No Credit: 3 Term: Summer 2012 (Online) Instructors Physics Professor:
More informationGravitational Wave Astronomy the sound of spacetime. Marc Favata Kavli Institute for Theoretical Physics
Gravitational Wave Astronomy the sound of spacetime Marc Favata Kavli Institute for Theoretical Physics What are gravitational waves? Oscillations in the gravitational field ripples in the curvature of
More informationIntroduction to Cosmology
Introduction to Cosmology João G. Rosa joao.rosa@ua.pt http://gravitation.web.ua.pt/cosmo LECTURE 2 - Newtonian cosmology I As a first approach to the Hot Big Bang model, in this lecture we will consider
More information