Gravitational Wave Memory in Cosmological Spacetimes
|
|
- Evan Davis
- 5 years ago
- Views:
Transcription
1 Gravitational Wave Memory in Cosmological Spacetimes Lydia Bieri University of Michigan Department of Mathematics Ann Arbor Black Hole Initiative Conference Harvard University, May 8-9, 2017
2 Overview Spacetimes and Radiation Gravitational Radiation with Memory A Footprint in Spacetime Isolated Systems Cosmological Setting
3 Photos: Courtesy of ETH-Bibliothek Zu rich
4 In 1915, Albert Einstein completed the Theory of General Relativity. In a letter of A. Einstein to A. Sommerfeld from November 1915, he mentions: At present I occupy myself exclusively with the problem of gravitation and now believe that I shall master all difficulties with the help of a friendly mathematician (Marcel Grossmann). But one thing is certain, in all my life I have never labored nearly as hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now. Compared with this problem, the original relativity is child s play.
5 Postcard from Albert Einstein to Hermann Weyl, 1923 The early cosmology of Einstein and de Sitter 83 Courtesy of ETH-Bibliothek Zürich Fig. 6.5 Einstein s postcard to Weyl. Written on Tuesday before Whitsun, which
6 Spacetimes in General Relativity Definition Spacetimes (M, g), where M a 4-dimensional manifold with Lorentzian metric g solving Einstein s equations: G µν := R µν 1 2 g µν R = 8π T µν, where G µν is the Einstein tensor, R µν is the Ricci curvature tensor, R the scalar curvature tensor, g the metric tensor and T µν denotes the energy-momentum tensor. For T µν 0 these equations reduce to the Einstein-Vacuum equations: R µν = 0. (1) Solutions of (1): Spacetimes (M, g), where M is a four-dimensional, oriented, differentiable manifold and g is a Lorentzian metric obeying (1).
7 Asymptotically Flat versus Cosmological Spacetimes In the cosmological case, we add to the original Einstein equations the term containing Λ, the positive cosmological constant: R µν 1 2 g µν R + Λg µν = 8π T µν, (2) Asymptotically Flat Spacetimes: Fall-off (in particular of metric and curvature components) towards Minkowski spacetime at infinity. Natural definition of null infinity understand gravitational radiation. These are solutions of the original Einstein equations with asymptotically flat initial data. Cosmological Spacetimes: Solutions of the cosmological Einstein equations (2). Null infinity is spacelike. no natural way to discuss radiation.
8 H 01203$"1F > < Foliations of the Spacetime O L P O P O P O P O P O L P O O N N L N N L KML Foliation by a time function t spacelike, complete Riemannian hypersurfaces H t. Foliation by a function u null hypersurfaces C u. S t,u = H t C u
9 Foliation of Null Infinity Future null infinity I + is defined to be the endpoints of all future-directed null geodesics along which r. It has the topology of R S 2 with the function u taking values in R. Thus a null hypersurface C u intersects I + at infinity in a 2-sphere S,u. Consider a null hypersurface C u in the spacetime M. Let t and explore limits of local quantities. For instance: Hawking mass tends to Bondi mass along any C u as t.
10 Shears and Expansion Scalars Viewing S as a hypersurface in C, respectively C: 4.6 The Characteristic Initial Value Problem In Section Denote 3.3 the we second discussed fundamental about the form Cauchy of S problem in C by χ, for and the the Einstein equatio ticular, second we saw fundamental that the initial form of data S in set C by consists χ. of the triplet (H 0, g, k), wh three-dimensional Their traceless Riemannian parts are called manifold, the shears g is the andmetric denoted onby H 0 ˆχ, and ˆχ k is a symm tensorrespectively. field on H 0 and such that g, k satisfy the constraint equations. Recall tha be thethe firsttraces and second trχ and fundamental trχ are theforms expansion of Hscalars. 0 in M, respectively. In Null this section, Limits of we the will Shears: discuss lim in detail the Cu,t r 2 ˆχ formulation = Σ(u) and of the characteristic problem, lim i.e. the Cu,t rˆχ case = where Ξ(u). the initial Riemannian (spacelike) Cauchy hypesu replaced by two degenerate (null) hypersurfaces C C intersecting at a twosurface S.
11 Gravitational Waves - Energy Radiated Fluctuation of curvature of the spacetime propagating as a wave. Gravitational waves: Localized disturbances in the geometry propagate at the speed of light, along outgoing null hypersurfaces. I + I + observe gravitational waves source H Picture: Courtesy of NASA. Gravitational radiation: gravitational waves traveling from sour
12 Memory Effect of Gravitational Waves Gravitational waves traveling from their source to our experiment. Three test masses in a plane as follows. The test masses will experience 1 Instantaneous displacements (while the wave packet is traveling through) 2 Permanent displacements (cumulative, stays after wave packet passed). This is called the memory effect of gravitational waves. Two types of this memory. lass. Quantum Grav. 29 (2012) L Bieri et al
13 Memory - Continued - Isolated Systems Ordinary (formerly called linear ) effect => was known for a long time in the slow motion limit [Ya.B. Zel dovich, A.G. Polnarev 1974] Null (formerly called nonlinear ) effect => was found by [D. Christodoulou 1991]. Contribution from electro-magnetic field to nonlinear effect => was found by [L. Bieri, P. Chen, S.-T. Yau 2010 and 2011]. Contribution from neutrino radiation to nonlinear effect => was found by [L. Bieri, D. Garfinkle 2012 and 2013]. See also works by Braginsky, Grishchuk, Thorne, Blanchet, Damour, Wiseman, Will. Recent works on memory include Wald, Tolish, Favata, Flanagan, Strominger, Winicour, Loutrel, Yunes, Hawking, Perry, Zhiboedov, Pasterski and more.
14 Contribution to the null memory (Christodoulou memory) for a fairly general stress-energy tensor with decay r 2 in the outgoing null direction (L. Bieri, D. Garfinkle). 2 Types of Memory Due to: Fields that do and Fields that do not go out to null infinity! We find an electromagnetic analog of gravitational wave memory. [L. Bieri, D. Garfinkle 2013] charged test masses observe a residual kick.
15 Detection Detectors of electromagnetic radiation absorb energy from the wave. Flux of energy in the wave: goes as r 2. Sensitivity of the detector falls off like r 2. Detectors of gravitational waves sensitivity falls off like r 1. Gravitational wave detector works not by measuring power absorbed from the wave but rather by following the motion induced in the detector by the wave. What permanent changes occur? Gravitational: Permanent displacement. Electromagnetic: Residual velocity (kick).
16 Gravitational Wave Experiment For a situation where the geodesics are not too far away from each other, one can replace the geodesic equation for γ 1 and γ 2 by the Jacobi equation (geodesic deviation from γ 0 ). with where k, l = 1, 2, 3. d 2 x k dt 2 = R kt lt x l (3) R kt lt = R (E k, T, E l, T )!!!Information about the curvature and null structures required!!! Analyze the spacetimes!
17 Memory - Permanent Displacement Asymptotically Flat Spacetimes The permanent displacement of test masses is related to the difference (Σ + Σ ) in the asymptotic shears, which themselves depend on the radiated energy in a nonlinear way. x = ( d 0 r ) (Σ+ Σ ). (4) There are the following contributions to the permanent displacement x: The ordinary memory is sourced by P, that is the change in the radial component of the electric part of the Weyl tensor. The null memory is sourced by F, the energy radiated to infinity (including shear and component of energy-momentum tensor).
18 The Christodoulou Memory Effect Christodoulou derived the null memory effect of gravitational waves in a fully nonlinear setting with exact solutions (no approximations used). It is based on the precise description of the null asymptotic behavior of relevant spacetimes, as these was established in the Christodoulou-Klainerman work on the nonlinear stability of the Minkwoski space. Other methods use approximations. New method by Bieri and Garfinkle using a perturbation of the Weyl tensor; this is gauge invariant. A recent paper by P. Lasky, E. Thrane, Y. Levin, J. Blackman and Y. Chen suggests a method for detecting gravitational wave memory with aligo.
19 Cosmology: de Sitter, FLRW and ΛCDM Observations of 1998 of Distant Supernovae Accelerating Expansion of the Universe Most popular cosmological theories: ΛCDM (with cold (i.e. non-relativistic) dark matter) Friedmann-Lemaître-Robertson-Walker (FLRW) (with a perfect fluid) de Sitter (ds) (modeling early inflation period of the Universe) Positive cosmological constant.
20 de Sitter Spacetime The de Sitter metric reads ds 2 = dt 2 + a 2 (t)dω 2 where Ω denotes the three-dimensional Euclidean space and a(t) is the expansion factor. This metric is conformal to the Minkowski metric diag( 1, +1, +1, +1). We introduce the conformal time η = dt/a. ds 2 = a 2 ( dη 2 + dω 2 ). (5) Thus we have the conformal behavior g ij = a 2 m ij with i, j = 0, 1, 2, 3. Here, m ij denotes the Minkowski metric with Cartesian coordinates (η, x, y, z).
21 de Sitter Spacetime de Sitter spacetime, a solution of the Einstein equations with a positive cosmological constant, models the early inflation period of the universe. (L. Bieri, D. Garinkle, S.-T. Yau) We find in de Sitter spacetime, that there is a factor of (1 + rh 0 ) multiplying F, the energy per unit solid angle radiated to infinity. Thus, the null memory is enhanced.
22 FLRW and ΛCDM FLRW: Friedmann - Lemaître - Robertson - Walker The FLRW metric reads ds 2 = dt 2 + a 2 (t) ( dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) ) Universe started as a small perturbation from FLRW. by now: these perturbations have grown waves propagate through highly inhomogeneous medium. Consider gravitational waves in ΛCDM cosmology.
23 For FLRW (A. Tolish, R. Wald): For sources at the same luminosity distance, the memory effect in a spatially flat FLRW spacetime is enhanced over the Minkowski case by a factor of (1 + z).
24 ΛCDM ΛCDM Our inhomogeneous spacetime two zones: wave zone and cosmological zone. (L. Bieri, D. Garfinkle, N. Yunes [forthcoming preprint]) For gravitational wave memory we find that in the wave zone the memory is similar to the one with Minkowski as a background, whereas in the cosmological zone the memory is given by the memory in the wave zone multiplied by a factor including the redshift and a magnification factor due to lensing.
25 Thank you!
Gravitational Wave Memory and an Electromagnetic Analog
Gravitational Wave Memory and an Electromagnetic Analog Lydia Bieri University of Michigan Department of Mathematics Ann Arbor Infrared Physics: Asymptotic and BMS symmetry, soft theorems, memory, information
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 informationGravitational Wave Memories and Asymptotic Charges in General Relativity
Gravitational Wave Memories and Asymptotic Charges in General Relativity Éanna Flanagan, Cornell General Relativity and Gravitation: A Centennial Perspective Penn State 8 June 2015 EF, D. Nichols, arxiv:1411.4599;
More informationStationarity of non-radiating spacetimes
University of Warwick April 4th, 2016 Motivation Theorem Motivation Newtonian gravity: Periodic solutions for two-body system. Einstein gravity: Periodic solutions? At first Post-Newtonian order, Yes!
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 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 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 in General Relativity
Quasi-local Mass in General Relativity Shing-Tung Yau Harvard University For the 60th birthday of Gary Horowtiz U. C. Santa Barbara, May. 1, 2015 This talk is based on joint work with Po-Ning Chen and
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 informationNull Cones to Infinity, Curvature Flux, and Bondi Mass
Null Cones to Infinity, Curvature Flux, and Bondi Mass Arick Shao (joint work with Spyros Alexakis) University of Toronto May 22, 2013 Arick Shao (University of Toronto) Null Cones to Infinity May 22,
More information3 The Friedmann-Robertson-Walker metric
3 The Friedmann-Robertson-Walker metric 3.1 Three dimensions The most general isotropic and homogeneous metric in three dimensions is similar to the two dimensional result of eq. (43): ( ) dr ds 2 = a
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 informationGravitation: Cosmology
An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013
More informationCosmology: An Introduction. Eung Jin Chun
Cosmology: An Introduction Eung Jin Chun Cosmology Hot Big Bang + Inflation. Theory of the evolution of the Universe described by General relativity (spacetime) Thermodynamics, Particle/nuclear physics
More informationIntroduction to the Current Events Bulletin
Introduction to the Current Events Bulletin Will the Riemann Hypothesis be proved this week? What is the Geometric Langlands Conjecture about? How could you best exploit a stream of data flowing by too
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 informationNon-existence of time-periodic vacuum spacetimes
Non-existence of time-periodic vacuum spacetimes Volker Schlue (joint work with Spyros Alexakis and Arick Shao) Université Pierre et Marie Curie (Paris 6) Dynamics of self-gravitating matter workshop,
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 informationTHE DARK SIDE OF THE COSMOLOGICAL CONSTANT
THE DARK SIDE OF THE COSMOLOGICAL CONSTANT CAMILO POSADA AGUIRRE University of South Carolina Department of Physics and Astronomy 09/23/11 Outline 1 Einstein s Greatest Blunder 2 The FLRW Universe 3 A
More informationAn 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 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 informationA simple estimate of gravitational wave memory in binary black hole systems
Classical and Quantum Gravity NOTE A simple estimate of gravitational wave memory in binary black hole systems To cite this article: David Garfinkle 0 Class. Quantum Grav. 00 Manuscript version: Accepted
More informationÜbungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor.
Übungen zu RT2 SS 2010 (1) Show that the tensor field g µν (x) = η µν is invariant under Poincaré transformations, i.e. x µ x µ = L µ νx ν + c µ, where L µ ν is a constant matrix subject to L µ ρl ν ση
More informationarxiv: v1 [gr-qc] 6 Jun 2017
Gravitational wave memory in ΛCDM cosmology Lydia Bieri, 1, David Garfinkle, 2,3, and Nicolás Yunes 4, 1 Department of Mathematics, University of Michigan, Ann Arbor, MI 4819-112, USA 2 Department of Physics,
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 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 informationQuasi-local Mass and Momentum in General Relativity
Quasi-local Mass and Momentum in General Relativity Shing-Tung Yau Harvard University Stephen Hawking s 70th Birthday University of Cambridge, Jan. 7, 2012 I met Stephen Hawking first time in 1978 when
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 informationOn the Hawking Wormhole Horizon Entropy
ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria On the Hawking Wormhole Horizon Entropy Hristu Culetu Vienna, Preprint ESI 1760 (2005) December
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 informationTHE UNIVERSITY OF CHICAGO THE GRAVITATIONAL WAVE MEMORY EFFECT AND CLASSICAL SCATTERING PROBLEMS A DISSERTATION SUBMITTED TO
THE UNIVERSITY OF CHICAGO THE GRAVITATIONAL WAVE MEMORY EFFECT AND CLASSICAL SCATTERING PROBLEMS A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE
More informationGeometric inequalities for black holes
Geometric inequalities for black holes Sergio Dain FaMAF-Universidad Nacional de Córdoba, CONICET, Argentina. 3 August, 2012 Einstein equations (vacuum) The spacetime is a four dimensional manifold M with
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 informationGeneral Relativity and Cosmology Mock exam
Physikalisches Institut Mock Exam Universität Bonn 29. June 2011 Theoretische Physik SS 2011 General Relativity and Cosmology Mock exam Priv. Doz. Dr. S. Förste Exercise 1: Overview Give short answers
More informationTHE BONDI-SACHS FORMALISM JEFF WINICOUR UNIVERSITY OF PITTSBURGH. Scholarpedia 11(12):33528 (2016) with Thomas Mädler
THE BONDI-SACHS FORMALISM JEFF WINICOUR UNIVERSITY OF PITTSBURGH Scholarpedia 11(12):33528 (2016) with Thomas Mädler NULL HYPERSURFACES u = const Normal co-vector @ u is null g @ u @ u =0 Normal vector
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 informationElectromagnetic spikes
Electromagnetic spikes Ernesto Nungesser (joint work with Woei Chet Lim) Trinity College Dublin ANZAMP, 29th of November, 2013 Overview Heuristic picture of initial singularity What is a Bianchi spacetime?
More informationGravity and action at a distance
Gravitational waves Gravity and action at a distance Newtonian gravity: instantaneous action at a distance Maxwell's theory of electromagnetism: E and B fields at distance D from charge/current distribution:
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 informationInitial-Value Problems in General Relativity
Initial-Value Problems in General Relativity Michael Horbatsch March 30, 2006 1 Introduction In this paper the initial-value formulation of general relativity is reviewed. In section (2) domains of dependence,
More informationPart two by Lydia Bieri, David Garfinkle, and Nicolás Yunes
Part two by Lydia Bieri, David Garfinkle, and Nicolás Yunes Gravitational Waves and Their Mathematics Introduction In 2015 gravitational waves were detected for the first time by the LIGO team [1]. This
More informationSelf trapped gravitational waves (geons) with anti-de Sitter asymptotics
Self trapped gravitational waves (geons) with anti-de Sitter asymptotics Gyula Fodor Wigner Research Centre for Physics, Budapest ELTE, 20 March 2017 in collaboration with Péter Forgács (Wigner Research
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 informationA UNIFIED TREATMENT OF GRAVITATIONAL COLLAPSE IN GENERAL RELATIVITY
A UNIFIED TREATMENT OF GRAVITATIONAL COLLAPSE IN GENERAL RELATIVITY & Anthony Lun Fourth Aegean Summer School on Black Holes Mytilene, Island of Lesvos 17/9/2007 CONTENTS Junction Conditions Standard approach
More informationPhysics 133: Extragalactic Astronomy ad Cosmology
Physics 133: Extragalactic Astronomy ad Cosmology Lecture 4; January 15 2014 Previously The dominant force on the scale of the Universe is gravity Gravity is accurately described by the theory of general
More informationGravitation: Tensor Calculus
An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013
More informationThe Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant
The Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant Jared Speck & Igor Rodnianski jspeck@math.princeton.edu University of Cambridge & Princeton University October
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 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 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 informationEmergent Universe by Tunneling. Pedro Labraña, ICC, Universidad de Barcelona and Facultad de Ciencias, Universidad del Bío-Bío, Chile.
Emergent Universe by Tunneling Pedro Labraña, ICC, Universidad de Barcelona and Facultad de Ciencias, Universidad del Bío-Bío, Chile. The Emergent Universe scenario Is Eternal Inflation, past eternal?
More informationPHY 475/375. Lecture 5. (April 9, 2012)
PHY 475/375 Lecture 5 (April 9, 2012) Describing Curvature (contd.) So far, we have studied homogenous and isotropic surfaces in 2-dimensions. The results can be extended easily to three dimensions. As
More informationLevel sets of the lapse function in static GR
Level sets of the lapse function in static GR Carla Cederbaum Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 72076 Tübingen, Germany September 4, 2014 Abstract We present a novel
More informationMemory effect, supertranslations and symmetries at null infinity
Memory effect, supertranslations and symmetries at null infinity K. Kajantie asymptotic Helsinki Institute of Physics Helsinki 27 March 2018 Project (Jokela-Kajantie-Sarkkinen): How do you measure supertranslations
More informationLecture 05. Cosmology. Part I
Cosmology Part I What is Cosmology Cosmology is the study of the universe as a whole It asks the biggest questions in nature What is the content of the universe: Today? Long ago? In the far future? How
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 informationCHAPTER 3 THE INFLATIONARY PARADIGM. 3.1 The hot Big Bang paradise Homogeneity and isotropy
CHAPTER 3 THE INFLATIONARY PARADIGM Ubi materia, ibi geometria. Johannes Kepler 3.1 The hot Big Bang paradise In General Relativity, the Universe as a whole becomes a dynamical entity that can be modeled
More information4 Evolution of density perturbations
Spring term 2014: Dark Matter lecture 3/9 Torsten Bringmann (torsten.bringmann@fys.uio.no) reading: Weinberg, chapters 5-8 4 Evolution of density perturbations 4.1 Statistical description The cosmological
More informationAdding Light to the Gravitational Waves on the Null Cone
Marshall University Marshall Digital Scholar Physics Faculty Research Physics Spring 4-2014 Adding Light to the Gravitational Waves on the Null Cone Maria Babiuc-Hamilton Marshall University, babiuc@marshall.edu
More informationSupplement to Lesson 9: The Petrov classification and the Weyl tensor
Supplement to Lesson 9: The Petrov classification and the Weyl tensor Mario Diaz November 1, 2015 As we have pointed out one of unsolved problems of General Relativity (and one that might be impossible
More informationDerivation of relativistic Burgers equation on de Sitter background
Derivation of relativistic Burgers equation on de Sitter background BAVER OKUTMUSTUR Middle East Technical University Department of Mathematics 06800 Ankara TURKEY baver@metu.edu.tr TUBA CEYLAN Middle
More informationUniformity of the Universe
Outline Universe is homogenous and isotropic Spacetime metrics Friedmann-Walker-Robertson metric Number of numbers needed to specify a physical quantity. Energy-momentum tensor Energy-momentum tensor of
More informationFormation of Higher-dimensional Topological Black Holes
Formation of Higher-dimensional Topological Black Holes José Natário (based on arxiv:0906.3216 with Filipe Mena and Paul Tod) CAMGSD, Department of Mathematics Instituto Superior Técnico Talk at Granada,
More informationKinetic Theory of Dark Energy within General Relativity
Kinetic Theory of Dark Energy within General Relativity Author: Nikola Perkovic* percestyler@gmail.com University of Novi Sad, Faculty of Sciences, Institute of Physics and Mathematics Abstract: This paper
More informationThe Apparent Universe
The Apparent Universe Alexis HELOU APC - AstroParticule et Cosmologie, Paris, France alexis.helou@apc.univ-paris7.fr 11 th June 2014 Reference This presentation is based on a work by P. Binétruy & A. Helou:
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 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 informationSCIENTIFIC UNDERSTANDING OF THE ANISOTROPIC UNIVERSE IN THE WARPED PRODUCTS SPACETIME FOR AEROSPACE POWER. Jaedong Choi
Korean J. Math. 23 (2015) No. 3 pp. 479 489 http://dx.doi.org/10.11568/kjm.2015.23.3.479 SCIENTIFIC UNDERSTANDING OF THE ANISOTROPIC UNIVERSE IN THE WARPED PRODUCTS SPACETIME FOR AEROSPACE POWER Jaedong
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Friday 8 June 2001 1.30 to 4.30 PAPER 41 PHYSICAL COSMOLOGY Answer any THREE questions. The questions carry equal weight. You may not start to read the questions printed on
More informationCOSMIC INFLATION AND THE REHEATING OF THE UNIVERSE
COSMIC INFLATION AND THE REHEATING OF THE UNIVERSE Francisco Torrentí - IFT/UAM Valencia Students Seminars - December 2014 Contents 1. The Friedmann equations 2. Inflation 2.1. The problems of hot Big
More informationIntroduction to Inflation
Introduction to Inflation Miguel Campos MPI für Kernphysik & Heidelberg Universität September 23, 2014 Index (Brief) historic background The Cosmological Principle Big-bang puzzles Flatness Horizons Monopoles
More informationArvind Borde / MTH 675, Unit 20: Cosmology
Arvind Borde / MTH 675, Unit 20: Cosmology 1. Review (1) What do we do when we do GR? We try to solve Einstein s equation. (2) What is Einstein s equation? and R ab = e[ 1 2 ged ( a g bd + b g ad d g ab
More informationRigidity of Black Holes
Rigidity of Black Holes Sergiu Klainerman Princeton University February 24, 2011 Rigidity of Black Holes PREAMBLES I, II PREAMBLE I General setting Assume S B two different connected, open, domains and
More informationLaw of Gravity and Gravitational Radiation
Law of Gravity and Gravitational Radiation Tian Ma, Shouhong Wang Supported in part by NSF and ONR http://www.indiana.edu/ fluid Blog: https://physicalprinciples.wordpress.com I. Laws of Gravity, Dark
More information3 Spacetime metrics. 3.1 Introduction. 3.2 Flat spacetime
3 Spacetime metrics 3.1 Introduction The efforts to generalize physical laws under different coordinate transformations would probably not have been very successful without differential calculus. Riemann
More informationThe nonlinear gravitational-wave memory in binary black hole mergers
The nonlinear gravitational-wave memory in binary black hole mergers Marc Favata Kavli Institute for Theoretical Physics University of California, Santa Barbara What is memory? Generally think of GW s
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 informationGraceful exit from inflation for minimally coupled Bianchi A scalar field models
Graceful exit from inflation for minimally coupled Bianchi A scalar field models Florian Beyer Reference: F.B. and Leon Escobar (2013), CQG, 30(19), p.195020. University of Otago, Dunedin, New Zealand
More informationClassification theorem for the static and asymptotically flat Einstein-Maxwell-dilaton spacetimes possessing a photon sphere
Classification theorem for the static and asymptotically flat Einstein-Maxwell-dilaton spacetimes possessing a photon sphere Boian Lazov and Stoytcho Yazadjiev Varna, 2017 Outline 1 Motivation 2 Preliminaries
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 informationPAPER 71 COSMOLOGY. Attempt THREE questions There are seven questions in total The questions carry equal weight
MATHEMATICAL TRIPOS Part III Friday 31 May 00 9 to 1 PAPER 71 COSMOLOGY Attempt THREE questions There are seven questions in total The questions carry equal weight You may make free use of the information
More informationPAPER 311 BLACK HOLES
MATHEMATICAL TRIPOS Part III Friday, 8 June, 018 9:00 am to 1:00 pm PAPER 311 BLACK HOLES Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY
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 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 (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 informationAre spacetime horizons higher dimensional sources of energy fields? (The black hole case).
Are spacetime horizons higher dimensional sources of energy fields? (The black hole case). Manasse R. Mbonye Michigan Center for Theoretical Physics Physics Department, University of Michigan, Ann Arbor,
More informationNotes on General Relativity Linearized Gravity and Gravitational waves
Notes on General Relativity Linearized Gravity and Gravitational waves August Geelmuyden Universitetet i Oslo I. Perturbation theory Solving the Einstein equation for the spacetime metric is tremendously
More informationThird Year: General Relativity and Cosmology. 1 Problem Sheet 1 - Newtonian Gravity and the Equivalence Principle
Third Year: General Relativity and Cosmology 2011/2012 Problem Sheets (Version 2) Prof. Pedro Ferreira: p.ferreira1@physics.ox.ac.uk 1 Problem Sheet 1 - Newtonian Gravity and the Equivalence Principle
More informationApproaches to Quantum Gravity A conceptual overview
Approaches to Quantum Gravity A conceptual overview Robert Oeckl Instituto de Matemáticas UNAM, Morelia Centro de Radioastronomía y Astrofísica UNAM, Morelia 14 February 2008 Outline 1 Introduction 2 Different
More informationGeneral Relativity Lecture 20
General Relativity Lecture 20 1 General relativity General relativity is the classical (not quantum mechanical) theory of gravitation. As the gravitational interaction is a result of the structure of space-time,
More informationThe early and late time acceleration of the Universe
The early and late time acceleration of the Universe Tomo Takahashi (Saga University) March 7, 2016 New Generation Quantum Theory -Particle Physics, Cosmology, and Chemistry- @Kyoto University The early
More informationConserved Quantities in Lemaître-Tolman-Bondi Cosmology
1/15 Section 1 Section 2 Section 3 Conserved Quantities in Lemaître-Tolman-Bondi Cosmology Alex Leithes - Blackboard Talk Outline ζ SMTP Evolution Equation: ζ SMTP = H X + 2H Y 3 ρ Valid on all scales.
More informationTHE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 2913 ASTROPHYSICS AND RELATIVITY (ADVANCED) ALL QUESTIONS HAVE THE VALUE SHOWN
CC0937 THE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 2913 ASTROPHYSICS AND RELATIVITY (ADVANCED) SEMESTER 2, 2014 TIME ALLOWED: 2 HOURS ALL QUESTIONS HAVE THE VALUE SHOWN INSTRUCTIONS:
More informationThe State of the Universe [2010] There is only data and the interpretation of data (green text = assumptions)
The State of the Universe [2010] There is only data and the interpretation of data (green text = assumptions) Current thinking in cosmology says that the universe is filled with dark matter and dark energy.
More informationCosmology (Cont.) Lecture 19
Cosmology (Cont.) Lecture 19 1 General relativity General relativity is the classical theory of gravitation, and as the gravitational interaction is due to the structure of space-time, the mathematical
More informationQGP, Hydrodynamics and the AdS/CFT correspondence
QGP, Hydrodynamics and the AdS/CFT correspondence Adrián Soto Stony Brook University October 25th 2010 Adrián Soto (Stony Brook University) QGP, Hydrodynamics and AdS/CFT October 25th 2010 1 / 18 Outline
More informationPHY326/426:Lecture 19
PHY326/426:Lecture 19 Dark Energy Finish WIMP signals Evidence for Dark Energy Type Ia Supernovae What is Dark Energy The fate of the Universe The Distance-Redshift relation Recall from lecture 2: The
More informationClosed Universes, de Sitter Space and Inflation
Closed Universes, de Sitter Space and Inflation Chris Doran Cavendish Laboratory Based on astro-ph/0307311 by Lasenby and Doran The Cosmological Constant Dark energy responsible for around 70% of the total
More informationAn exact solution for 2+1 dimensional critical collapse
An exact solution for + dimensional critical collapse David Garfinkle Department of Physics, Oakland University, Rochester, Michigan 839 We find an exact solution in closed form for the critical collapse
More informationIntroduction to General Relativity and Gravitational Waves
Introduction to General Relativity and Gravitational Waves Patrick J. Sutton Cardiff University International School of Physics Enrico Fermi Varenna, 2017/07/03-04 Suggested reading James B. Hartle, Gravity:
More information