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