Y.K.Lau Institute of Appl. Maths, CAS. Bondi Sachs Formulation of the Kerr Metric Gravitational Wave...

Y.K.Lau Institute of Appl. Maths, CAS

Bondi Sachs Formulation of the Kerr Metric

Gravitational Wave Physics of a Kerr Black Hole

Joint work with

Shan Bai, Zhoujian Cao, Xuefei Gong, Xiaoning Wu

IAM, AMSS

Yuguang ShiPeking University

Numerical Relativity

Numerical Simulation Using Large Scale Computation

Compute GravitationalWaveform

Data MiningOf ExperimentalDatabase

Relativistic Two body Einstein field

equations

T8G

Angular momentum carries away gravitational radiation

Rotating star

Gravitational radiation emitted

Wavefront Tracking

Distortion of wavefront by angular momentum of the source ?

Constant time slice

Wavefront Two sphere as snapshot of wave front

Describes the distortion of the two sphere

Eigenvectors Two states of polarization of GW

Why Angular Momentum is Tough?

Dilemma

• In need of a pragmatic description at the level of application

• Lack of good theoretic foundation

Kerr Metric of a Prototype of Rotating Source

Motivations for Studying Kerr

• Waveform generated by Rotating Star

• Extreme Mass Ratio Inspiral Problem (LISA Template)

• Stability of Black Hole

• ………..

222222

222222

2

2

222

2222

22

22

sin2)(

2,cos

sin4

sin)2

1(

Mraar

aMrrar

dtdaMr

dddrdtMr

ds

Kerr black hole in Boyer-Lindquist coordinate

Dragging effect of star’s angular momentum

Construction of Wavefronts

Solution of Eikonal Equation

0,, uug

22

2222222

2

sin)(

aMrr

aaru

r

u

Eikonal Equation

......)sin

21

4ln2(

222

r

aMrMrtu

Asymptotic Solution as r tends toinfinity

Moving frame adapted to the null hypersurface

alan

amam

Conserved Quantity for Gravitational Radiation Field

• Maxwell Theory: Total Charge Conserved

• General Relativity: • Mass, Momentum, Angular Momentum are

not conserved!

Newman Penrose Constants

Newman-Penrose Constants

Absolutely Conserved at Null Infinity

NP const defined

NP constant---independent on the wavefront that intersects infinity# of NP constants=10

0 4

......6

10

5

00

0

rr

NP constants constructed from the multipole expansion of the ingoing radiation field

# of NP constants=10

Definition of NP constants reads:

0du

dGm

u -- time measured at null infinity

Newman Penrose Constants

…..Another problem is to understand the significance (either mathematical or physical) of a certain curious set of ten exactly conserved asymptotically defined quantities …..

Roger Penrose

“Unsolved Problems in General Relativity”

in “Seminars in Differential Geometry” edited by S.T.Yau, Princeton Univ. Press, 1982

Mathematical

Physical

Constraint of NP constants to initial data?

What role will the NP constants play in the evolution of field eqs?

Are there deeper physical meaning for these absolute conserved quantities?

NP constant of Kerr spacetime

NP constants of Kerr spacetime all vanish!

Evolution of Binary Stars

No hair theorem means Ker

r end state

NP constant vanishes for generic initial d

ata set ?

• NP constants for binary system (Dain and Kroon)

• What happens from binary to Kerr?

• Possible ways out of this dilemma?

212

120 mmrG

Journey to the Promised Land

• Numerical Simulation of the angular momentum of gravitational waves generated by tidal distortion of Kerr

• Understand better the Newman Penrose Constant in Binary Black Hole Spacetime