Binary systems as sources of Binary systems as sources of gravitational wavesgravitational waves

Gideon KoekoekGideon KoekoekMarch 5March 5thth 2008 2008

Theoretical research done at Nikhef/VuTheoretical research done at Nikhef/Vu

Overview of presentationOverview of presentation

Introduction: the binary systemIntroduction: the binary system The Martel-Poisson methodThe Martel-Poisson method Methods used: current researchMethods used: current research Conclusions & outlookConclusions & outlook

Introduction Introduction

Our prime target: Our prime target: the binary system!the binary system!

A A sure sourcesure source of GW’s: of GW’s: Hulse & TaylorHulse & Taylor

Their GWs are about Their GWs are about to be measured (LIGO, to be measured (LIGO, VIRGO experiments)VIRGO experiments)

Test-bedTest-bed for GR and for GR and extensionsextensions

General Relativity: General Relativity: any system with a any system with a quadrupole moment produces GWsquadrupole moment produces GWs

Introduction Introduction How to calculate gravitational waves?How to calculate gravitational waves?

TRgRcG4

8

2

1

Take background metric Take background metric ggBBμνμν

add a disturbance add a disturbance hhμνμν and and

plug into the Einstein plug into the Einstein EquationsEquations

hgg B

Source of the Source of the GWsGWs

General schemeGeneral scheme::

Differential equations that couple Differential equations that couple the gravitational wave the gravitational wave hhμμνν to to

the energy momentum the energy momentum TTμνμν

Curvature of Curvature of spacetimespacetime

Introduction Introduction In the case of a In the case of a Minkowski Minkowski

backgroundbackground, this is an easy , this is an easy exercise.exercise.

1000

0100

0010

0001

Bg

• Waves move with speed of lightWaves move with speed of light

• Waves have two (!) polarisationsWaves have two (!) polarisations

• Are transversalAre transversal

T

c

Gh

dt

d4

22

2 8

The wave The wave equation!equation!

This is all well-known…This is all well-known…

All constantsAll constants

IntroductionIntroductionHow about the binary system?How about the binary system?

Mm

??Bg

ApproximationApproximation: one of the stars is very much heavier than the other : one of the stars is very much heavier than the other (EMRI). The metric of the system is then that of the heavy star: (EMRI). The metric of the system is then that of the heavy star: Schwarzschild metric. Schwarzschild metric.

mM

sin000

000

00)2

1(0

000)2

1(

2

2

1

r

rr

Mr

M

g SVariablesVariables!!

IntroductionIntroduction

10 coupled, 10 coupled, partial partial differential differential equations in equations in 10 variables10 variables

Difficult! How to solve this system?Difficult! How to solve this system?

AlsoAlso, the source , the source TTμνμν will be will be

difficult, as the stars difficult, as the stars orbit each other in a orbit each other in a non-trivial way.non-trivial way.

(Epicycle method; see last year’s (Epicycle method; see last year’s presentation)presentation)

TRgRcG4

8

2

1

Non-constant Non-constant background metricbackground metric

Non-trivial energy-Non-trivial energy-momentum tensormomentum tensor

hgg S T

The challenge: The challenge:

IntroductionIntroductionExpert groups are working on Expert groups are working on

this by numerical methods:this by numerical methods:

(Frans Pretorius, 2006)(Frans Pretorius, 2006)

..but this is a ..but this is a veryvery demanding computation!demanding computation!

Our goal:Our goal:

Try to find the gravitational waves for the binary Try to find the gravitational waves for the binary system in an system in an analytical wayanalytical way. . We do this by using a We do this by using a formalism developed recently by Martel & Poisson, formalism developed recently by Martel & Poisson, and work our way from there.and work our way from there.

The Martel-Poisson The Martel-Poisson methodmethod

The Martel-Poisson methodThe Martel-Poisson methodFollowing a program started by John Wheeler (1957), Following a program started by John Wheeler (1957),

Martel & Poisson devised a formalism (2004)Martel & Poisson devised a formalism (2004)

1.1. Linearize the Einstein Equations in Linearize the Einstein Equations in hhμνμν

2.2. Decompose the waves into Decompose the waves into tensorial spherical tensorial spherical harmonicsharmonics....

3.3. Plug into Einstein Equations and find the EOMs for Plug into Einstein Equations and find the EOMs for the coefficients..the coefficients..

4.4. Do some very clever gauging to eliminate all Do some very clever gauging to eliminate all unphysical degrees of freedom..unphysical degrees of freedom..

5.5. Think very hard..Think very hard..

The Martel-Poisson methodThe Martel-Poisson method

The Martel-Poisson methodThe Martel-Poisson method..in the end: ..in the end: only two linear, uncoupled only two linear, uncoupled

Klein-Gordon equations Klein-Gordon equations remain. remain.

Great simplification!Great simplification! ZMZMZMbaS

ab SVDDg

RWRWRWbaS

ab SVDDg )(

The two The two ΨΨ((t,rt,r) ’s (roughly) correspond to the two ) ’s (roughly) correspond to the two polarisationspolarisations of the GWs. From these of the GWs. From these, we can , we can directly calculate the GWs and the emitted energy!directly calculate the GWs and the emitted energy!

10 coupled, partial 10 coupled, partial differential differential equations in 10 equations in 10 variablesvariables

The Martel-Poisson methodThe Martel-Poisson methodA closer look at these two differential equations A closer look at these two differential equations

ZMZMZMbaS

ab SVDDg

RWRWRWbaS

ab SVDDg )(

In which:In which:• DDaa is the generally covariant derivative, determined by the is the generally covariant derivative, determined by the

Schwarzschild metricSchwarzschild metric

• VVZMZM((rr) ) and and VVRWRW((rr) ) are ‘potentials’, fully determined by the are ‘potentials’, fully determined by the

Schwarzschild metricSchwarzschild metric

• SSZMZM((t,rt,r) ) and and SSRWRW((t,rt,r) ) are sources, fully determined by the motion of are sources, fully determined by the motion of

the small star around the bigger onethe small star around the bigger one

These two equations can now be solved!These two equations can now be solved!

Methods used: Methods used: current researchcurrent research

Methods used: current researchMethods used: current researchHow to solve these two equations?How to solve these two equations?

ZMZMZMbaS

ab SVDDg

RWRWRWbaS

ab SVDDg )(

As always in such matters, there are two optionsAs always in such matters, there are two options::

We are doing them both!We are doing them both!

1.1. Numerical integrationNumerical integration: write a clever C++ program : write a clever C++ program that takes the initial conditions and extrapolates from that takes the initial conditions and extrapolates from therethere

2.2. Approximation techniquesApproximation techniques: throw out some terms, do : throw out some terms, do integral transforms, and try to find an approximate integral transforms, and try to find an approximate analytical solutionanalytical solution

Methods used: current researchMethods used: current researchAnalytical scheme:Analytical scheme:

• Includes some approximations and is based on Laplace transforms.• Solutions found are integrals in which the orbit of the smaller star can be freely specified, i.e. solutions work for general orbits• status: implementing initial conditions; results expected shortly.

4M4M> 6M> 6M

Numerical scheme:Numerical scheme:• Spacetime is divided up into grid cells, worldline of the smaller star is plotted • C++ code integrates the EOMs over the worldline• Status: a first version of the code is available, and is ondergoing testing.

Work in Work in progress; progress; results results expected expected soonsoon

Conclusions & Conclusions & outlookoutlook

Conclusions & outlookConclusions & outlook

Solve Martel & Poisson’s two EOMs both numerically and Solve Martel & Poisson’s two EOMs both numerically and via an analytical scheme; compare the results.via an analytical scheme; compare the results.

Immediate future Immediate future (i.e. working on this right now)(i.e. working on this right now)

The greater plan:The greater plan:

Martel & Poisson’s method only works Martel & Poisson’s method only works when one of the masses is small, when one of the masses is small, because otherwise the curvature because otherwise the curvature deviates from Schwarzschilddeviates from Schwarzschild

We know a formalism to calculate We know a formalism to calculate deviations of the metricdeviations of the metric

Blend these two Blend these two formalisms, so the formalisms, so the EMRI-condition can EMRI-condition can be relaxed..even for be relaxed..even for alternative gravities!alternative gravities!

Conclusions & outlookConclusions & outlook Problem:Problem: the gravitational waves in a binary system are very the gravitational waves in a binary system are very

challenging to calculatechallenging to calculate

EMRI approximation: EMRI approximation: by assuming one of the stars to be by assuming one of the stars to be much heavier than the other much heavier than the other, , the the

background metric can be taken as background metric can be taken as SchwarzschildSchwarzschild

Martel & Poisson method: Martel & Poisson method: enables us to find the GWs in a enables us to find the GWs in a Schwarzschild spacetime for any Schwarzschild spacetime for any

source, by solving two scalar EOMs source, by solving two scalar EOMs

Status: Status: Solving the two EOMs, by an analytical scheme using Solving the two EOMs, by an analytical scheme using Laplace Transforms; making progress but no results Laplace Transforms; making progress but no results yet. A yet. A numerical code is being developed: results will numerical code is being developed: results will be compared. be compared.

Work in progress..but results expected soon!Work in progress..but results expected soon!

Any questions?Any questions?

gkoekoek@nikhef.nlgkoekoek@nikhef.nl