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

1. Computational Relativity-Black Holes andGravitationalWaveson a LaptopRay d’InvernoFaculty of Mathematical StudiesUniversity of Southampton

2. 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?” 1958 “... 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

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

5. Einstein’s Field Equations (1915) • Full field equations • Vacuum field equations • Complicated (second order non-linear system of partial differential equations) for determining the curved spacetime metric • How complicated? SHEEP: 100,000 terms for general metric

6. Algebraic Computing John McCarthy: LISP Symbolic manipulation planned as an application 1960 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 “ 1966 1967 Ray d’Inverno: LAM (LISP Algebraic Manipulator)

7. Automatic garbage collector 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 .....))

8. 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?

9. Use: 1 move a second, 1 year secs Moves 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?

10. SHEEP (Inge Frick) SHEEP FAMILY LAM (Ray d’Inverno) 1967 1968 ALAM (Ray d’Inverno) 1971 CLAM (Ray d’Inverno & Tony Russell-Clark) 1973 ILAM (Ian Cohen & Inge Frick) 1976 1979 CLASSI (Jan Aman) 1980 STENSOR (Lars Hornfeldt)

11. Train at rest v Train in motion v Einstein’s Special Relativity (1905) Two basic postulates • Inertial observers are equivalent • Velocity of light c is a constant New underlying principle: Relativity of Simultaneity Einstein train thought experiment

12. 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

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

17. 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”

19. 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” Einstein’s lift thought experiment 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” Leads to the spacetime being curved

20. 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

21. 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” Example: Planetary Motion • Newtonian explanation: combination of • inertial motion (motion in a straight line with constant velocity) • falling under gravity • Intuitive idea: rubber sheet geometry

22. Schwarzschild (spherically symmetric, static, vacuum) solution Schwarzschild Solution • Full field equations • Vacuum field equations • Einstein originally: too complicated to solve

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

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

25. Indirect evidence: Binary Pulsar 1913+16 (Hulse-Taylor 1993 Nobel prize) 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

26. 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

27. 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?

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

29. EquivalenceProblem Given two metrics: is there a coordinate transformation which converts one into the other? 1946 Cartan : found a method for deciding, but it is too complicated to use in practise 1965 Brans: new idea 1980 Karlhede: provides an invariant method for classifying metrics 1986 Aman: implemented Karlhede method in CLASSI 1990 Skea, MacCallum, ...: Computer Database of Exact Solutions

30. Limitations Of Exact Solutions No exact solutions for • 2 body problem • n body problem • Gravitational waves from a source e.g. binary black hole system e.g. planetary system e.g. radiating star

31. 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 2+2 • 1 dimensional (spherical/cylindrical) • 2 dimensional (axial) • 3 dimensional (general) • E.g. 100x100x100 grid points = 1 GB memory

32. 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

33. Cylindrical Gravitational Waves Colliding waves Waves from Cosmic Strings

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

35. 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

36. Summary General Relativity Algebraic Computing Black Holes Gravitational Waves Exact Solutions Numerical Relativity Southampton CCM Project EU Network

37. Summary

38. Tonight’s Gig John Arlott Bar Staff Club 8.30pm Bossa Rio