Computer Algebra in General Relativity

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Computer Algebra in General Relativity Powered By Docstoc

      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
• 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
    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

                                          (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

Einstein train thought experiment             Train in motion
                     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

                         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
         2m 2  2m
   ds  1 
                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

 (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
           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
Tonight’s Gig

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