Computational Relativity - Black Holes and Gravitational Waves on a Laptop Ray d’Inverno Faculty of Mathematical Studies University of Southampton

Why Me and General Relativity? The Einstein Theory of General Relativity by Lilian R Lieber and Hugh R Lieber “Is it true that only three people in the world understand Einstein’s theory of General Relativity?” “... and there are only a few people in the world who understand General Relativity...” Sir Arthur Eddington “Who is the third?”

Outline of Lecture Algebraic Computing Special Relativity General Relativity Black Holes Gravitational Waves Exact Solutions Numerical Relativity

Einstein’s Field Equations (1915) SHEEP: 100,000 terms for general metric How complicated? Full field equations Vacuum field equations Complicated (second order non-linear system of partial differential equations) for determining the curved spacetime 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))) Automatic garbage collector Recursive algorithms easily implemented e.g. (defun transfer (transfer.....))

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 secs Moves

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) New underlying principle: Relativity of Simultaneity Einstein train thought experiment Inertial observers are equivalent Velocity of light c is a constant Two basic postulates v v Train at rest Train in motion

New Physics Lorentz-Fitzgerald contraction Time dilation New composition law for velocities Equivalence of Mass and Energy length contraction in the direction of motion slowing down of clocks in motion ordinary bodies cannot attain the velocity of light

New Mathematics Newtonian time Newtonian space Time is absolute Euclidean distance is invariant “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 Interval between events is an invariant Special Relativity: Minkowski spacetime

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: Einstein’s Equivalence Principle: Einstein’s lift thought experiment “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” “all bodies fall with the same acceleration irrespective of their mass and composition” Leads to the spacetime being curved

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

Einsteinian explanation: Sun curves up spacetime in its vicinity Planet moves on a curved geodesic of the spacetime Example: Planetary Motion Newtonian explanation: combination of inertial motion (motion in a straight line with constant velocity) falling under gravity 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 Vacuum field equations Einstein originally: too complicated to solve Schwarzschild (spherically symmetric, static, vacuum) solution

Spacetime Diagrams Flat space of Special Relativity Schwarzschild (original coordinates) Gravity tips and distorts the local light cones

Black Holes Schwarzschild (Eddington-Finkelstein coordinates) Tidal forces in a black hole

Gravitational Waves Indirect evidence: Binary Pulsar (Hulse-Taylor 1993 Nobel prize) 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 Ripples in the curvature travelling with speed c

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) Gravitational waves (idealised cases abstracted away from sources) Hundreds of other exact solutions Schwarzschild Reissner-Nordstrom (charged black hole) Kerr (rotating black hole) Kerr-Newman (charged rotating black hole) Plane fronted waves Cylindrical waves But are they all different?

What Metric Is This? Schwarzschild - in Cartesians coordinates Recall: Schwarzschild in spherical polar coordinates

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 2 body problem n body problem Gravitational waves from a source No exact solutions for e.g. binary black hole system e.g. radiating star e.g. planetary system

Numerical Relativity Numerical solution of Einstein’s equations using computers Mathematical formalisms Simulations Need for large scale computers Standard: finite difference on a finite grid ADM 3+1 DSS dimensional (spherical/cylindrical) 2 dimensional (axial) 3 dimensional (general) 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 CCM (Cauchy-Characteristic Matching) Advantages spurious numerical reflections at the boundary central 3+1 exterior null-timelike 2+2 timelike vacuum interface generates global solution transparent interface exact asymptotic wave forms

Cylindrical Gravitational Waves Waves from Cosmic Strings Colliding waves

US Binary Black Hole Grand Challenge NASA Neutron Star Grand Challenge Albert Einstein Institute Numerical Relativity Group Large Scale Simulations

European Union Network 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 Theoretical Foundations of Sources for Gravitational Wave Astronomy of the Next Century: Synergy between Supercomputer Simulations and Approximation Techniques

Summary

Tonight’s Gig