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Malcolm@60 Computer Algebra: From the Visible To the Invisible Ray d’Inverno School pf Mathematics University of Southampton Why Me and General Relativity? “Is it true that only three people in the world understand Einstein’s theory of General Relativity?” Sir Arthur Eddington “Who is the third?” “... and there are only a few people in the world who understand General Relativity...” The Einstein Theory of General Relativity by Lilian R Lieber and Hugh R Lieber Outline of Lecture Algebraic Computing Special Relativity General Relativity Black Holes Gravitational Waves Exact Solutions Numerical Relativity Einstein’s Field Equations (1915) • Full field equations G T • Vacuum field equations R 0 • Complicated (second order non-linear system of partial differential equations) for determining the curved spacetime metric g • How complicated? SHEEP: 100,000 terms for general metric Algebraic Computing John McCarthy: LISP Symbolic manipulation planned as an application Jean Sammett: “… It has become obvious that there a large number of problems requiring very TEDIOUS… TIME-CONSUMING… ERROR-PRONE… STRAIGHTFORWARD algebraic manipulation, and these characteristics make computer solution both necessary and desirable “ Ray d’Inverno: LAM (LISP Algebraic Manipulator) Why LISP? Lists provide natural representation for algebraic expressions 3+1 (+ 3 1) ADM (* A D M) 2+2 (+ 2 2) DSS (* D (** S (2 1))) Recursive algorithms easily implemented e.g. (defun transfer ... ... (transfer .....)) Automatic garbage collector Example: Tower of Hanoi Problem: Monks are playing this game in Hanoi with 64 disks. When they finish the world will end. Should we worry? Example: Tower of Hanoi Problem: Monks are playing this game in Hanoi with 64 disks. When they finish the world will end. Should we worry? Use: 1 move a second, 1 year 3 107secs Moves 2 1 64 (210 ) 6 2 4 (1024) 6 16 (103 ) 6 16 1018 15 3 5 107 1011 secs 5 1011 years 10 (5 1010 ) years 10T (T Age of Universe) SHEEP FAMILY LAM (Ray d’Inverno) ALAM (Ray d’Inverno) CLAM (Ray d’Inverno & Tony Russell-Clark) ILAM (Ian Cohen & Inge Frick) SHEEP (Inge Frick) CLASSI (Jan Aman) STENSOR (Lars Hornfeldt) Einstein’s Special Relativity (1905) Two basic postulates • Inertial observers are equivalent • Velocity of light c is a constant Train at rest New underlying principle: Relativity of Simultaneity v Einstein train thought experiment Train in motion v New Physics Lorentz-Fitzgerald contraction • length contraction in the direction of motion Time dilation • slowing down of clocks in motion New composition law for velocities • ordinary bodies cannot attain the velocity of light Equivalence of Mass and Energy • E mc 2 New Mathematics Newtonian time Special Relativity: • Time is absolute Newtonian space Minkowski spacetime • Euclidean distance is invariant • Interval between events is an invariant d 2 dx 2 dy 2 dz 2 ds2 dt 2 dx 2 dy 2 dz 2 “Henceforth, space by itself, and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality” Hermann Minkowski Einstein’s General Relativity (1915) A theory of gravitation consistent with Special Relativity Galileo’s Pisa observations: “all bodies fall with the same acceleration irrespective of their mass and composition” Einstein’s General Relativity (1915) A theory of gravitation consistent with Special Relativity Galileo’s Pisa observations: “all bodies fall with the same acceleration irrespective of their mass and composition” Einstein’s Equivalence Principle: “a body in an accelerated frame behaves the same as one in a frame at rest in a gravitational field, and - a body in an unaccelerated frame behaves the same as one in free fall” Leads to the spacetime being curved Einstein’s lift thought experiment A Theory of Curved Spacetime Special Relativity: - Space-time is flat - Free particles/light rays travel on straight lines General Relativity: - Space-time is curved - Free particles/light rays travel on the “straightest lines” available: curved geodesics Example: Planetary Motion Newtonian explanation: combination of • inertial motion (motion in a straight line with constant velocity) • falling under gravity Einsteinian explanation: • Sun curves up spacetime in its vicinity • Planet moves on a curved geodesic of the spacetime John Archibald Wheeler: “space tells matter how to move and matter tells space how to curve” • Intuitive idea: rubber sheet geometry Schwarzschild Solution • Full field equations G T • Vacuum field equations R 0 • Einstein originally: too complicated to solve • Schwarzschild (spherically symmetric, static, vacuum) solution 1 2m 2 2m ds 1 2 dt 1 dr 2 r 2 d 2 r 2 sin2 d 2 r r Spacetime Diagrams Flat space of Special Relativity Gravity tips and distorts the local light cones Schwarzschild (original coordinates) Black Holes Schwarzschild (Eddington-Finkelstein coordinates) Tidal forces in a black hole Gravitational Waves Ripples in the curvature travelling with speed c A gravitational wave has 2 polarisation states A long way from the source (asymptotically) the states are called “plus” and “cross” The effect on a ring of tests particles Indirect evidence: Binary Pulsar 1913+16 (Hulse-Taylor 1993 Nobel prize) Gravitational Wave Detection • Weber bars • Ground based laser interferometers • Space based laser interferometers Low signal to noise ratio problem (duke box analogy) Method of matched filtering requires exact templates of the signal New window onto the universe: Gravitational Astronomy Exact Solutions • Black holes (limiting solutions) • Schwarzschild • Reissner-Nordstrom (charged black hole) • Kerr (rotating black hole) • Kerr-Newman (charged rotating black hole) • Gravitational waves (idealised cases abstracted away from sources) • Plane fronted waves • Cylindrical waves • Hundreds of other exact solutions • But are they all different? What Metric Is This? Schwarzschild - in Cartesians coordinates Recall: Schwarzschild in spherical polar coordinates 1 2m 2 2m ds2 1 dt 1 dr 2 r 2 d 2 r 2 sin2 d 2 r r Equivalence Problem Given two metrics: is there a coordinate transformation which converts one into the other? Cartan : found a method for deciding, but it is too complicated to use in practise Brans: new idea Karlhede: provides an invariant method for classifying metrics Aman: implemented Karlhede method in CLASSI Skea, MacCallum, ...: Computer Database of Exact Solutions Limitations Of Exact Solutions No exact solutions for • 2 body problem e.g. binary black hole system • n body problem e.g. planetary system • Gravitational waves from a source e.g. radiating star Numerical Relativity • Numerical solution of Einstein’s equations using computers • Standard: finite difference on a finite grid • Mathematical formalisms • ADM 3+1 • DSS 2+2 • Simulations • 1 dimensional (spherical/cylindrical) • 2 dimensional (axial) • 3 dimensional (general) • Need for large scale computers • E.g. 100x100x100 grid points = 1 GB memory The Southampton CCM Project • Gravitational waves cannot be characterised exactly locally • Gravitational waves can be characterised exactly asymptotically • Standard 3+1 code on a finite grid leads to spurious numerical reflections at the boundary •CCM (Cauchy-Characteristic Matching) central 3+1 exterior null-timelike 2+2 timelike vacuum interface • Advantages generates global solution transparent interface exact asymptotic wave forms Cylindrical Gravitational Waves Colliding waves Waves from Cosmic Strings Large Scale Simulations • US Binary Black Hole Grand Challenge • NASA Neutron Star Grand Challenge • Albert Einstein Institute Numerical Relativity Group European Union Network Theoretical Foundations of Sources for Gravitational Wave Astronomy of the Next Century: Synergy between Supercomputer Simulations and Approximation Techniques • 10 European Research Groups in France, Germany, Greece, Italy, Spain and UK • Need for large scale collaborative projects • Common computational platform: Cactus • Southampton’s role pivotal, team leader in: - 3 dimensional CCM thorn - Development of asymptotic gravitational wave codes - Relativistic stellar perturbation theory - Neutron Star modelling Summary Summary Tonight’s Gig

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posted: | 8/28/2012 |

language: | English |

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