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Computational Relativity
-Black Holes
andGravitational
Waveson a Laptop
Ray d’InvernoFaculty of Mathematical Studies
University of Southampton
G T
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)
g
G T
R 0
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 veryTEDIOUS… TIME-CONSUMING… ERROR-PRONE… STRAIGHTFORWARDalgebraic 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
2 1
2 2
1024 16
10 16
10 15
3 5 10 10
5 10
10 5 10
10
64
10 6 4
6
3 6
18
7 11
11
10
( )
( )
( )
( )
(
secs
years
years
Age of Universe)T T
3 107
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
• E mc 2
New Mathematics
Newtonian time
Newtonian space
• Time is absolute
• Euclidean distance is invariant
d dx dy dz 2 2 2 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
ds dt dx dy dz2 2 2 2 2 • 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 MotionNewtonian 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
G T
R 0
dsm
rdt
m
rdr r d r d2 2
1
2 2 2 2 2 212
12
sin
• 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 HolesSchwarzschild (Eddington-Finkelstein coordinates)
Tidal forces in a black hole
Gravitational Waves
Indirect evidence: Binary Pulsar 1913+16 (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
dsm
rdt
m
rdr r d r d2 2
1
2 2 2 2 2 212
12
sin
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 2+2
• 1 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
Summary
Tonight’s Gig