G T Computational Relativity Black Holes and Gravitational Waves on a Laptop Ray d’Inverno Faculty of Mathematical Studies University of Southampton.
Download ReportTranscript G T Computational Relativity Black Holes and Gravitational Waves on a Laptop Ray d’Inverno Faculty of Mathematical Studies University of Southampton.
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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?
“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?
7
Use: 1 move a second, 1 year secs Moves
2 64 1 ( 2 ) 2 4 ( 1024 ) 6 16 ( 10 3 6 ) 16 10 18 15 7 10 11 secs 11 years 10 ( 10 ( 10 ) years 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
v 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
•
Time is absolute
Newtonian space
•
Euclidean distance is invariant d
2
dx
2
dy
2
dz
2
Special Relativity: Minkowski spacetime
•
Interval between events is an invariant ds
2
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
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”
Galileo’s Pisa observations:
“all bodies fall with the same acceleration irrespective of their mass and composition”
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
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• Vacuum field equations
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0 • Einstein originally: too complicated to solve • Schwarzschild (spherically symmetric, static, vacuum) solution
ds
2 1 2
r m
dt
2 2
m r
1
dr
2 2
r d
2
r
2 sin 2
d
2
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
ds
2 2
m r
dt
2 2
m r
1
dr
2 2
r d
2
r
2 sin 2
d
2
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