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 Shi Peking University

Numerical Relativity Numerical Simulation Using Large Scale Computation Compute Gravitational Waveform Data Mining Of Experimental Database Relativistic Two body Einstein field equations

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 ………..

Kerr black hole in Boyer- Lindquist coordinate Dragging effect of star’s angular momentum

Construction of Wavefronts Solution of Eikonal Equation

Eikonal Equation

Asymptotic Solution as r tends to infinity

Moving frame adapted to the null hypersurface

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

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

# of NP constants=10 Definition of NP constants reads: 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 Kerr end state NP constant vanishes for generic initial data set ?

NP constants for binary system (Dain and Kroon) What happens from binary to Kerr? Possible ways out of this dilemma?

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