y.k.lau institute of appl. maths, cas. bondi sachs formulation of the kerr metric gravitational wave...
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Y.K.Lau Institute of Appl. Maths, CAS
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Bondi Sachs Formulation of the Kerr Metric
Gravitational Wave Physics of a Kerr Black Hole
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Joint work with
Shan Bai, Zhoujian Cao, Xuefei Gong, Xiaoning Wu
IAM, AMSS
Yuguang ShiPeking University
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Numerical Relativity
Numerical Simulation Using Large Scale Computation
Compute GravitationalWaveform
Data MiningOf ExperimentalDatabase
Relativistic Two body Einstein field
equations
T8G
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Angular momentum carries away gravitational radiation
Rotating star
Gravitational radiation emitted
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Wavefront Tracking
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Distortion of wavefront by angular momentum of the source ?
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Constant time slice
Wavefront Two sphere as snapshot of wave front
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Describes the distortion of the two sphere
Eigenvectors Two states of polarization of GW
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Why Angular Momentum is Tough?
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Dilemma
• In need of a pragmatic description at the level of application
• Lack of good theoretic foundation
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Kerr Metric of a Prototype of Rotating Source
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Motivations for Studying Kerr
• Waveform generated by Rotating Star
• Extreme Mass Ratio Inspiral Problem (LISA Template)
• Stability of Black Hole
• ………..
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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
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Construction of Wavefronts
Solution of Eikonal Equation
0,, uug
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22
2222222
2
sin)(
aMrr
aaru
r
u
Eikonal Equation
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......)sin
21
4ln2(
222
r
aMrMrtu
Asymptotic Solution as r tends toinfinity
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Moving frame adapted to the null hypersurface
alan
amam
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Conserved Quantity for Gravitational Radiation Field
• Maxwell Theory: Total Charge Conserved
• General Relativity: • Mass, Momentum, Angular Momentum are
not conserved!
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Newman Penrose Constants
Newman-Penrose Constants
Absolutely Conserved at Null Infinity
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NP const defined
NP constant---independent on the wavefront that intersects infinity# of NP constants=10
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0 4
......6
10
5
00
0
rr
NP constants constructed from the multipole expansion of the ingoing radiation field
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# of NP constants=10
Definition of NP constants reads:
0du
dGm
u -- time measured at null infinity
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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
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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?
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NP constant of Kerr spacetime
NP constants of Kerr spacetime all vanish!
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Evolution of Binary Stars
No hair theorem means Ker
r end state
NP constant vanishes for generic initial d
ata set ?
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• NP constants for binary system (Dain and Kroon)
• What happens from binary to Kerr?
• Possible ways out of this dilemma?
212
120 mmrG
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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