The initial value problem in general relativity
|
|
- Iris Patrick
- 6 years ago
- Views:
Transcription
1 LC Physics Colloquium, Spring 2015
2 Abstract In 1915, Einstein introduced equations describing a theory of gravitation known as general relativity. The Einstein equations, as they are now called, are at once elegant and extremely complicated. Thus it was not until the middle of the 20th century that Yvonne Choquet-Bruhat showed they permit an initial value problem i.e. that if the state of a system is specified at an initial time, then there exists a corresponding solution to the equations specifying the state at a later time. In this talk we first discuss initial value problems in classical physics, before describing important features of the initial value problem in general relativity. We outline some of the challenges in studying the initial value problem, some recent progress, and list some important unsolved problems in this exciting area of research.
3 Shameless advertising Math 490: Curvature of Space & Time Prof. Stavrov MTΘF 11:30-12:20
4 Caveat (emptor) I am a mathematician... mathematical general relativity! Other important topics: Data & observation Numerical simulation Theoretical physics Getting these communities of people talking to one another! Please do not hesitate to ask questions throughout!
5 Initial value problem (IVP) Given state of system now what happens in the future? Ingredients Questions Evolution equations Initial conditions Short-time questions: Existence? Uniqueness, Continual dependence on initial conditions? Global questions: Behavior of solutions?
6 Classical dynamics Dynamical equations d x dt = p Principle of least action Conservation law Minimize tf t i d p dt = V ( x) { } 1 2 p 2 V ( x) dt H = 1 2 p 2 + V ( x) is conserved Free evolution V = 0 d p dt = 0 straight line trajectory
7 Initial value problem theory Fundamental theorem of ODEs For any initial x 0, p 0 there exists a unique short-time solution. Global behavior Determined by conservation of energy H.
8 Example: Simple Harmonic Oscillator Potential p V (x) = 1 2 x 2 x Equations H V dx dt = p dp dt = x x
9 Example: Electromagnetism Equations Energy t E = B 4πJ E = 4πρ t B = E B = 0 H = 1 ( E 2 + B 2) dv 2 Partial differential equations, but linear Constraints: satisfied initially preserved by evolution Energy does not give point-wise control
10 Doing electromagnetism First pass Given a charge configuration, what is E? What is B? Given E and B, what is trajectory of a test particle? Second pass Electromagnetic waves Reformulation using potentials, gauge condition wave equation Initial value problem Given E and B now, how do they evolve? Initial E and B must satisfy constraints Wave equation formulation is mathematically well-behaved
11 General relativity: Spacetime diagrams Particles, fields, etc. all defined in space and time View a space time from vantage point of an observer r me = 0 Dr. S Dr. O t me = 2 t me = 1 t me = 0
12 General relativity: Spacetime diagrams Particles, fields, etc. all defined in space and time View a space time from vantage point of an observer r me = 0 Dr. S Dr. O t K = 0 t me = 2 t me = 1 t me = 0 c = 1 special relativity (Also: G = 1)
13 Classical IVP General relativity IVP in GR General relativity: Geometry, scaling, maps I Which lines are straight? I Metric length scale at each point of map projections/
14 General relativity: Geometry and spacetime Einstein (and Hilbert, Poincare, et al.) Spacetime metric g (spacetime length scale) Gravitational model: particles obey principle of least action, with respect to length determined by g Needs to be same for any observer ( geometric ) Einstein s equation Ric 1 2 R g = 8πT }{{}}{{} Geometry Matter fields Ric, R are notions of curvature, depend on derivatives of g T also involves g
15 General relativity: Simple example a small planet r = 0 Dr. S: crash & burn land on planet t = 2 t = 1 Qualitatively Newtonian dynamics... t = 0
16 General relativity: Simple example a small planet r = 0 Dr. S: crash & burn land on planet Dr. O: angular momentum! t = 2 t = 1 Qualitatively Newtonian dynamics... t = 0
17 General relativity: Large mass black hole region Schwarzchild 1915; r = 2m r = 3m r = 4m region with mass m What about inside?
18 General relativity: Large mass black hole region Schwarzchild 1915; Kruskal 1960 region with mass m m 2m 4m r = m r = 2m r = 3m r = 4m 3m Let s go on an adventure...
19 General relativity: Large mass black hole region Schwarzchild 1915; region with mass m m 2m 4m r = m r = 2m r = 3m r = 4m Dr. S s fate...? 3m
20 General relativity: Large mass black hole region Schwarzchild 1915; region with mass m m 2m 4m r = m r = 2m r = 3m r = 4m 3m
21 Lessons learned from Schwarzchild solution Observations Some coordinate systems behave better than others. Interesting features (e.g. BH) may be regions of space time. Singularities may form; due to non-linearity. Questions Can interesting features form dynamically? Are singularities typically hidden? Weak cosmic censorship conjecture. Need an initial value formulation.
22 Towards an initial value problem Recall initial value problem framework Specify initial conditions ( state at t = 0), satisfying constraints if applicable Use equations to evolve in time (need good formulation/theory) Verify constraints are preserved by evolution General relativity? Which time coordinate should we use?? What if we choose another time coordinate?? Are there constraints?? Are the equations even solvable from an IVP perspective?
23 The short-time initial value problem (I) Local perspective (Choquet-Bruhat et al., 1950 s ): Focus on a small region; choose wave-adapted coordinates Einstein s equation becomes a (non-linear) wave equation, which can be solved Patch together little pieces to form a spacetime Wave-like behavior, including gravity waves Maximally-extended nice spacetime
24 The short-time initial value problem (II) Hamiltonian perspective (A.-D.-M. et al., 1960 s ): Choose an arbitrary time function Decompose equations analogous to E&M Clearly illustrates constraint and evolution equations: T g = Nk 0 = R + k 2 (trk) 2 T k = 2 g = k (trk) Conserved quantity: Energy-momentum Ongoing research: Understanding & constructing solutions to constraint equations Ongoing research: Solutions to evolution equations
25 Beyond short-time existence Lot s of fun questions... Isolated systems Singularity formation Black holes & weak cosmic censorship Stability of black holes Cosmology Stability of symmetric models Structure formation
26 Dynamical formation of singularities Incompleteness (Hawking & Penrose; 1970) Expansion θ satisfies dθ dt < 1 3 θ2 1 Thus θ < 1 3 t + θ 1 0 If θ 0 < 0, paths collide. θ 0 > 0 θ 0 < 0 Curvature singularities Understood in some symmetric situations Lots of work yet to be done
27 Formation of BHs & weak cosmic censorship conjecture Generic scenario (?) θ gravitational collapse Singularity BH region Horizon Singularity formation Hidden inside BH region cosmic censorship Outside observer Many examples... few theorems... Preskill-Thorne Hawking bets
28 Stability problems Stability Compare a symmetric solution to small, nearby configurations Important for theoretical and physical reasons Famous results Minkowski space time (Christodoulou-Klainerman) Rapidly expanding space times (Friedrich) Current hot topic Stability of Schwarzschild space time
29 Structure formation Expectations Small inhomogeneities radiated away Large inhomogeneities large-scale structure Known results Linear approximations Lots of heuristics Lots of good data coming in! Early stages... The math is exceedingly difficult new ideas needed!
30 Concluding remarks General relativity is complicated... and fascinating! We know many things and a lot remains to be done! Thank you for your attention. Questions?
31 Resources Introductory books General Relativity by Woodhouse A First Course in General Relativity by Schutz Also books by Carroll, Hartle, etc. More advanced General Relativity and the Einstein Equations by Choquet-Bruhat Partial Differential Equations in General Relativity by Rendall Also books by Ellis & Hawking, Wald, etc.
An Overview of Mathematical General Relativity
An Overview of Mathematical General Relativity José Natário (Instituto Superior Técnico) Geometria em Lisboa, 8 March 2005 Outline Lorentzian manifolds Einstein s equation The Schwarzschild solution Initial
More informationSingularity formation in black hole interiors
Singularity formation in black hole interiors Grigorios Fournodavlos DPMMS, University of Cambridge Heraklion, Crete, 16 May 2018 Outline The Einstein equations Examples Initial value problem Large time
More informationQuantum Gravity and Black Holes
Quantum Gravity and Black Holes Viqar Husain March 30, 2007 Outline Classical setting Quantum theory Gravitational collapse in quantum gravity Summary/Outlook Role of metrics In conventional theories the
More informationGeometric inequalities for black holes
Geometric inequalities for black holes Sergio Dain FaMAF-Universidad Nacional de Córdoba, CONICET, Argentina. 26 July, 2013 Geometric inequalities Geometric inequalities have an ancient history in Mathematics.
More informationThe cosmic censorship conjectures in classical general relativity
The cosmic censorship conjectures in classical general relativity Mihalis Dafermos University of Cambridge and Princeton University Gravity and black holes Stephen Hawking 75th Birthday conference DAMTP,
More informationGlobal properties of solutions to the Einstein-matter equations
Global properties of solutions to the Einstein-matter equations Makoto Narita Okinawa National College of Technology 12/Nov./2012 @JGRG22, Tokyo Singularity theorems and two conjectures Theorem 1 (Penrose)
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 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 informationA Brief Introduction to Mathematical Relativity
A Brief Introduction to Mathematical Relativity Arick Shao Imperial College London Arick Shao (Imperial College London) Mathematical Relativity 1 / 31 Special Relativity Postulates and Definitions Einstein
More informationQuasi-local mass and isometric embedding
Quasi-local mass and isometric embedding Mu-Tao Wang, Columbia University September 23, 2015, IHP Recent Advances in Mathematical General Relativity Joint work with Po-Ning Chen and Shing-Tung Yau. The
More informationOrbital Motion in Schwarzschild Geometry
Physics 4 Lecture 29 Orbital Motion in Schwarzschild Geometry Lecture 29 Physics 4 Classical Mechanics II November 9th, 2007 We have seen, through the study of the weak field solutions of Einstein s equation
More informationBlack hole instabilities and violation of the weak cosmic censorship in higher dimensions
Black hole instabilities and violation of the weak cosmic censorship in higher dimensions Pau Figueras School of Mathematical Sciences, Queen Mary University of London w/ Markus Kunesch, Luis Lehner and
More informationCosmic Censorship Conjecture and Topological Censorship
Cosmic Censorship Conjecture and Topological Censorship 21 settembre 2009 Cosmic Censorship Conjecture 40 years ago in the Rivista Nuovo Cimento Sir Roger Penrose posed one of most important unsolved problems
More informationEntanglement and the Bekenstein-Hawking entropy
Entanglement and the Bekenstein-Hawking entropy Eugenio Bianchi relativity.phys.lsu.edu/ilqgs International Loop Quantum Gravity Seminar Black hole entropy Bekenstein-Hawking 1974 Process: matter falling
More informationNon-existence of time-periodic dynamics in general relativity
Non-existence of time-periodic dynamics in general relativity Volker Schlue University of Toronto University of Miami, February 2, 2015 Outline 1 General relativity Newtonian mechanics Self-gravitating
More information8.821/8.871 Holographic duality
Lecture 3 8.81/8.871 Holographic duality Fall 014 8.81/8.871 Holographic duality MIT OpenCourseWare Lecture Notes Hong Liu, Fall 014 Lecture 3 Rindler spacetime and causal structure To understand the spacetime
More informationThe stability of Kerr-de Sitter black holes
The stability of Kerr-de Sitter black holes András Vasy (joint work with Peter Hintz) July 2018, Montréal This talk is about the stability of Kerr-de Sitter (KdS) black holes, which are certain Lorentzian
More informationQuantum gravity and aspects of relativity
Quantum gravity and aspects of relativity Branislav Nikolic Institute for Theoretical Physics, University of Cologne Bonn-Cologne Graduate School in Physics and Astronomy who are we??? Gravitation and
More information3 Rindler Space and Hawking Radiation
3 Rindler Space and Hawking Radiation The next couple of lectures are on Hawking radiation. There are many good references to learn this subject, for example: Carroll s GR book Chapter 9; Townsend gr-qc/970702;
More informationFrom Quantum Mechanics to String Theory
From Quantum Mechanics to String Theory Relativity (why it makes sense) Quantum mechanics: measurements and uncertainty Smashing things together: from Rutherford to the LHC Particle Interactions Quarks
More informationToward Binary Black Hole Simulations in Numerical Relativity
Toward Binary Black Hole Simulations in Numerical Relativity Frans Pretorius California Institute of Technology BIRS Workshop on Numerical Relativity Banff, April 19 2005 Outline generalized harmonic coordinates
More informationA Panoramic Tour in Black Holes Physics
Figure 1: The ergosphere of Kerr s black hole A Panoramic Tour in Black Holes Physics - A brief history of black holes The milestones of black holes physics Astronomical observations - Exact solutions
More informationFrom An Apple To Black Holes Gravity in General Relativity
From An Apple To Black Holes Gravity in General Relativity Gravity as Geometry Central Idea of General Relativity Gravitational field vs magnetic field Uniqueness of trajectory in space and time Uniqueness
More informationAccelerated Observers
Accelerated Observers In the last few lectures, we ve been discussing the implications that the postulates of special relativity have on the physics of our universe. We ve seen how to compute proper times
More informationHolography Duality (8.821/8.871) Fall 2014 Assignment 2
Holography Duality (8.821/8.871) Fall 2014 Assignment 2 Sept. 27, 2014 Due Thursday, Oct. 9, 2014 Please remember to put your name at the top of your paper. Note: The four laws of black hole mechanics
More informationHawking s genius. L. Sriramkumar. Department of Physics, Indian Institute of Technology Madras, Chennai
Hawking s genius L. Sriramkumar Department of Physics, Indian Institute of Technology Madras, Chennai Institute colloquium Indian Institute of Technology, Palakkad April 4, 2018 Plan of the talk Introduction
More information14. Black Holes 2 1. Conformal Diagrams of Black Holes
14. Black Holes 2 1. Conformal Diagrams of Black Holes from t = of outside observer's clock to t = of outside observer's clock time always up light signals always at 45 time time What is the causal structure
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 informationGravitational wave memory and gauge invariance. David Garfinkle Solvay workshop, Brussels May 18, 2018
Gravitational wave memory and gauge invariance David Garfinkle Solvay workshop, Brussels May 18, 2018 Talk outline Gravitational wave memory Gauge invariance in perturbation theory Perturbative and gauge
More informationOverview and Innerview of Black Holes
Overview and Innerview of Black Holes Kip S. Thorne, Caltech Beyond Einstein: From the Big Bang to Black Holes SLAC, 14 May 2004 1 Black Hole Created by Implosion of a Star Our Focus: quiescent black hole
More informationGravitational Memory and BMS Symmetry in Four and Higher Dimensions
Gravitational Memory and BMS Symmetry in Four and Higher Dimensions S. Hollands based on joint work with A. Ishibashi and R.M. Wald King s College, London 12 January 2017 arxiv:1612.03290 [gr-qc] History
More informationIntroduction to Numerical Relativity I. Erik Schnetter, Pohang, July 2007
Introduction to Numerical Relativity I Erik Schnetter, Pohang, July 2007 Lectures Overview I. The Einstein Equations (Formulations and Gauge Conditions) II. Analysis Methods (Horizons and Gravitational
More informationNon-linear stability of Kerr de Sitter black holes
Non-linear stability of Kerr de Sitter black holes Peter Hintz 1 (joint with András Vasy 2 ) 1 Miller Institute, University of California, Berkeley 2 Stanford University Geometric Analysis and PDE Seminar
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 informationCosmic acceleration from fuzzball evolution. Great Lakes 2012
Cosmic acceleration from fuzzball evolution Great Lakes 2012 Outline (A) Black hole information paradox tells us something new about quantum gravity (B) Early Universe had a high density, so these new
More informationCausality, hyperbolicity, and shock formation in Lovelock theories
Causality, hyperbolicity, and shock formation in Lovelock theories Harvey Reall DAMTP, Cambridge University HSR, N. Tanahashi and B. Way, arxiv:1406.3379, 1409.3874 G. Papallo, HSR arxiv:1508.05303 Lovelock
More informationScott A. Hughes, MIT SSI, 28 July The basic concepts and properties of black holes in general relativity
The basic concepts and properties of black holes in general relativity For the duration of this talk ħ=0 Heuristic idea: object with gravity so strong that light cannot escape Key concepts from general
More informationMyths, Facts and Dreams in General Relativity
Princeton university November, 2010 MYTHS (Common Misconceptions) MYTHS (Common Misconceptions) 1 Analysts prove superfluous existence results. MYTHS (Common Misconceptions) 1 Analysts prove superfluous
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 informationSecond quantization: where quantization and particles come from?
110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian
More informationA Summary of the Black Hole Perturbation Theory. Steven Hochman
A Summary of the Black Hole Perturbation Theory Steven Hochman Introduction Many frameworks for doing perturbation theory The two most popular ones Direct examination of the Einstein equations -> Zerilli-Regge-Wheeler
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 informationGlobal stability problems in General Relativity
Global stability problems in General Relativity Peter Hintz with András Vasy Murramarang March 21, 2018 Einstein vacuum equations Ric(g) + Λg = 0. g: Lorentzian metric (+ ) on 4-manifold M Λ R: cosmological
More informationExcluding Black Hole Firewalls with Extreme Cosmic Censorship
Excluding Black Hole Firewalls with Extreme Cosmic Censorship arxiv:1306.0562 Don N. Page University of Alberta February 14, 2014 Introduction A goal of theoretical cosmology is to find a quantum state
More informationClassical 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 informationEscape velocity and Schwarzschild s Solution for Black Holes
Escape velocity and Schwarzschild s Solution for Black Holes Amir Ali Tavajoh 1 1 Amir_ali3640@yahoo.com Introduction Scape velocity for every star or planet is different. As a star starts to collapse
More informationBlack-Holes in AdS: Hawking-Page Phase Transition
Black-Holes in AdS: Hawking-Page Phase Transition Guilherme Franzmann December 4, 2014 1 / 14 References Thermodynamics of Black Holes in Anti-de Sitter space, S.W. Hawking and Don. N Page (1983); Black
More informationAre naked singularities forbidden by the second law of thermodynamics?
Are naked singularities forbidden by the second law of thermodynamics? Sukratu Barve and T. P. Singh Theoretical Astrophysics Group Tata Institute of Fundamental Research Homi Bhabha Road, Bombay 400 005,
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 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 informationEMERGENT GRAVITY AND COSMOLOGY: THERMODYNAMIC PERSPECTIVE
EMERGENT GRAVITY AND COSMOLOGY: THERMODYNAMIC PERSPECTIVE Master Colloquium Pranjal Dhole University of Bonn Supervisors: Prof. Dr. Claus Kiefer Prof. Dr. Pavel Kroupa May 22, 2015 Work done at: Institute
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.286: The Early Universe October 27, 2013 Prof. Alan Guth PROBLEM SET 6
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.86: The Early Universe October 7, 013 Prof. Alan Guth PROBLEM SET 6 DUE DATE: Monday, November 4, 013 READING ASSIGNMENT: Steven Weinberg,
More informationHolography and Unitarity in Gravitational Physics
Holography and Unitarity in Gravitational Physics Don Marolf 01/13/09 UCSB ILQG Seminar arxiv: 0808.2842 & 0808.2845 This talk is about: Diffeomorphism Invariance and observables in quantum gravity The
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 informationHolography and (Lorentzian) black holes
Holography and (Lorentzian) black holes Simon Ross Centre for Particle Theory The State of the Universe, Cambridge, January 2012 Simon Ross (Durham) Holography and black holes Cambridge 7 January 2012
More informationAsk class: what is the Minkowski spacetime in spherical coordinates? ds 2 = dt 2 +dr 2 +r 2 (dθ 2 +sin 2 θdφ 2 ). (1)
1 Tensor manipulations One final thing to learn about tensor manipulation is that the metric tensor is what allows you to raise and lower indices. That is, for example, v α = g αβ v β, where again we use
More informationPhysics 311 General Relativity. Lecture 18: Black holes. The Universe.
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
More informationOLIVIA MILOJ March 27, 2006 ON THE PENROSE INEQUALITY
OLIVIA MILOJ March 27, 2006 ON THE PENROSE INEQUALITY Abstract Penrose presented back in 1973 an argument that any part of the spacetime which contains black holes with event horizons of area A has total
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 informationI wish to further comment on these issues. Not much technical details; just to raise awareness
Yen Chin Ong Unruh: Can information fall off the edge of spacetime (singularity)? No evidence except prejudice that this is not the case AdS/CFT: Can boundary probe BH interior? Bag-of-gold? ER-bridge?
More informationStability and Instability of Black Holes
Stability and Instability of Black Holes Stefanos Aretakis September 24, 2013 General relativity is a successful theory of gravitation. Objects of study: (4-dimensional) Lorentzian manifolds (M, g) which
More informationBLACK HOLES (ADVANCED GENERAL RELATIV- ITY)
Imperial College London MSc EXAMINATION May 2015 BLACK HOLES (ADVANCED GENERAL RELATIV- ITY) For MSc students, including QFFF students Wednesday, 13th May 2015: 14:00 17:00 Answer Question 1 (40%) and
More informationGeneral Relativity (2nd part)
General Relativity (2nd part) Electromagnetism Remember Maxwell equations Conservation Electromagnetism Can collect E and B in a tensor given by And the charge density Can be constructed from and current
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 informationIntroductory Course on Black Hole Physics and AdS/CFT Duality Lecturer: M.M. Sheikh-Jabbari
Introductory Course on Black Hole Physics and AdS/CFT Duality Lecturer: M.M. Sheikh-Jabbari This is a PhD level course, designed for second year PhD students in Theoretical High Energy Physics (HEP-TH)
More informationSo the question remains how does the blackhole still display information on mass?
THE ZERO POINT NON-LOCAL FRAME AND BLACKHOLES By: Doctor Paul Karl Hoiland Abstract: I will show that my own zero point Model supports not only the no-hair proposals, but also the Bekenstein bound on information
More informationOutline. Covers chapter 2 + half of chapter 3 in Ryden
Outline Covers chapter + half of chapter 3 in Ryden The Cosmological Principle I The cosmological principle The Cosmological Principle II Voids typically 70 Mpc across The Perfect Cosmological Principle
More informationASTR 200 : Lecture 30. More Gravity: Tides, GR, and Gravitational Waves
ASTR 200 : Lecture 30 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 informationInside the horizon 2GM. The Schw. Metric cannot be extended inside the horizon.
G. Srinivasan Schwarzschild metric Schwarzschild s solution of Einstein s equations for the gravitational field describes the curvature of space and time near a spherically symmetric massive body. 2GM
More informationInstability of extreme black holes
Instability of extreme black holes James Lucietti University of Edinburgh EMPG seminar, 31 Oct 2012 Based on: J.L., H. Reall arxiv:1208.1437 Extreme black holes Extreme black holes do not emit Hawking
More informationJohn A. Wheeler & Astrophysical Relativity. Charles W Misner University of Maryland
John A. Wheeler & Astrophysical Relativity Charles W Misner University of Maryland or John A Wheeler and the recertification of GR as true physics J A Wheeler phases Particles/nuclear 1933-1955 1955 Fields/general
More information2 Canonical quantization
Phys540.nb 7 Canonical quantization.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system?.1.1.lagrangian Lagrangian mechanics is a reformulation of classical mechanics.
More informationGravitational waves, solitons, and causality in modified gravity
Gravitational waves, solitons, and causality in modified gravity Arthur Suvorov University of Melbourne December 14, 2017 1 of 14 General ideas of causality Causality as a hand wave Two events are causally
More informationBLACK HOLE ENTROPY ENTANGLEMENT AND HOLOGRAPHIC SPACETIME. Ted Jacobson University of Maryland
BLACK HOLE ENTROPY ENTANGLEMENT AND HOLOGRAPHIC SPACETIME Ted Jacobson University of Maryland Goddard Scientific Colloquium, Feb. 7, 2018 Holographic principle Information paradox geometry from entanglement
More informationSpeed limits in general relativity
Class. Quantum Grav. 16 (1999) 543 549. Printed in the UK PII: S0264-9381(99)97448-8 Speed limits in general relativity Robert J Low Mathematics Division, School of Mathematical and Information Sciences,
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 informationWe are used to ponder the information loss paradox from the point of view of external observers. [Also, black hole complementarity principle]
Yen Chin Ong We are used to ponder the information loss paradox from the point of view of external observers. [Also, black hole complementarity principle] Notwithstanding the firewall, horizon should not
More informationEffect of Monopole Field on the Non-Spherical Gravitational Collapse of Radiating Dyon Solution.
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn:2319-765x. Volume 10, Issue 1 Ver. III. (Feb. 2014), PP 46-52 Effect of Monopole Field on the Non-Spherical Gravitational Collapse of Radiating
More informationBlack Holes and Thermodynamics I: Classical Black Holes
Black Holes and Thermodynamics I: Classical Black Holes Robert M. Wald General references: R.M. Wald General Relativity University of Chicago Press (Chicago, 1984); R.M. Wald Living Rev. Rel. 4, 6 (2001).
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 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 informationcarroll/notes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general
http://pancake.uchicago.edu/ carroll/notes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity. As with any major theory in physics, GR has been
More informationQuantum black holes at the LHC
Quantum black holes at the LHC Xavier Calmet Physics and Astronomy University of Sussex Frameworks for Quantum Black Holes (QBHs) at 1 TeV Large extra-dimensions Large hidden sector (and 4 dimensions)
More informationNovember 24, Energy Extraction from Black Holes. T. Daniel Brennan. Special Relativity. General Relativity. Black Holes.
from November 24, 2014 1 2 3 4 5 Problem with Electricity and Magnetism In the late 1800 s physicists realized there was a problem with electromagnetism: the speed of light was given in terms of fundamental
More informationSolutions of Einstein s Equations & Black Holes 2
Solutions of Einstein s Equations & Black Holes 2 Kostas Kokkotas December 19, 2011 2 S.L.Shapiro & S. Teukolsky Black Holes, Neutron Stars and White Dwarfs Kostas Kokkotas Solutions of Einstein s Equations
More informationGeneral Relativity and Important Physical Quantities
General Relativity and Important Physical Quantities Shing-Tung Yau Harvard University 2nd LeCosPA Symposium December 14, 2015 This talk is based on joint work with Po-Ning Chen and Mu-Tao Wang. Exactly
More informationAn introduction to General Relativity and the positive mass theorem
An introduction to General Relativity and the positive mass theorem National Center for Theoretical Sciences, Mathematics Division March 2 nd, 2007 Wen-ling Huang Department of Mathematics University of
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 informationGeneral Relativity in AdS
General Relativity in AdS Akihiro Ishibashi 3 July 2013 KIAS-YITP joint workshop 1-5 July 2013, Kyoto Based on work 2012 w/ Kengo Maeda w/ Norihiro Iizuka, Kengo Maeda - work in progress Plan 1. Classical
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 Role of Black Holes in the AdS/CFT Correspondence
The Role of Black Holes in the AdS/CFT Correspondence Mario Flory 23.07.2013 Mario Flory BHs in AdS/CFT 1 / 30 GR and BHs Part I: General Relativity and Black Holes Einstein Field Equations Lightcones
More informationcarroll/notes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general
http://pancake.uchicago.edu/ carroll/notes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity. As with any major theory in physics, GR has been
More informationBachelor Thesis. General Relativity: An alternative derivation of the Kruskal-Schwarzschild solution
Bachelor Thesis General Relativity: An alternative derivation of the Kruskal-Schwarzschild solution Author: Samo Jordan Supervisor: Prof. Markus Heusler Institute of Theoretical Physics, University of
More informationQuantum Gravity Inside and Outside Black Holes. Hal Haggard International Loop Quantum Gravity Seminar
Quantum Gravity Inside and Outside Black Holes Hal Haggard International Loop Quantum Gravity Seminar April 3rd, 2018 1 If spacetime is quantum then it fluctuates, and a Schwarzschild black hole is an
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 informationhas a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity.
http://preposterousuniverse.com/grnotes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity. As with any major theory in physics, GR has been framed
More informationNew Blackhole Theorem and its Applications to Cosmology and Astrophysics
New Blackhole Theorem and its Applications to Cosmology and Astrophysics I. New Blackhole Theorem II. Structure of the Universe III. New Law of Gravity IV. PID-Cosmological Model Tian Ma, Shouhong Wang
More informationConstrained BF theory as gravity
Constrained BF theory as gravity (Remigiusz Durka) XXIX Max Born Symposium (June 2010) 1 / 23 Content of the talk 1 MacDowell-Mansouri gravity 2 BF theory reformulation 3 Supergravity 4 Canonical analysis
More informationMy Personal Journey on the Geometric Aspect of General Relativity
My Personal Journey on the Geometric Aspect of General Relativity Shing-Tung Yau Harvard University The first annual meeting of ICCM 2017 Sun Yat-sen University, Guangzhou December 28, 2017 This talk is
More informationFormation and Evaporation of Regular Black Holes in New 2d Gravity BIRS, 2016
Formation and Evaporation of Regular Black Holes in New 2d Gravity BIRS, 2016 G. Kunstatter University of Winnipeg Based on PRD90,2014 and CQG-102342.R1, 2016 Collaborators: Hideki Maeda (Hokkai-Gakuen
More information