On Uniqueness of Kerr Metric Near Null infinity

On Uniqueness of KerrOn Uniqueness of KerrMetricMetric Near Null infinityNear Null infinity

XiaoningXiaoning Wu WuInstitute of Mathematics, AMSS,Institute of Mathematics, AMSS,Chinese Academy of SciencesChinese Academy of Sciences

1963, Kerr metric, vacuum, stationary,1963, Kerr metric, vacuum, stationary,axial-symmetric, asymptotically flat metric.axial-symmetric, asymptotically flat metric.

Uniqueness theoremUniqueness theorem

Uniqueness of Kerr metric Uniqueness of Kerr metric (Carter, 1971)(Carter, 1971)

The Kerr metric is the only vacuum black hole The Kerr metric is the only vacuum black holesolution with Msolution with M2 2 > a> a22, regular horizon and, regular horizon andstationary and axial-symmetric, asymptotic flatstationary and axial-symmetric, asymptotic flatdomain of outer communicationsdomain of outer communications.

Uniqueness of Kerr-Newman metric.Uniqueness of Kerr-Newman metric.

Recently, people get many more strongerRecently, people get many more strongerresults.results.

( (ChruscielChrusciel and and TodTod, 2007; , 2007; ChruscielChrusciel and Costa, 2008; and Costa, 2008; AlexakisAlexakis, , IonescuIonescu and and KleinermanKleinerman, 2009; , 2009; AmselAmsel, Horowitz, , Horowitz, MarolfMarolf,,

Robert, 2009; Robert, 2009; …………))

Bondi metric Bondi metric (H. Bondi et. al. Proc. Roy. Soc. Lond.(H. Bondi et. al. Proc. Roy. Soc. Lond.A , A , 1962)1962)

Question : What is the Bondi form of KerrQuestion : What is the Bondi form of Kerrmetric ?metric ?

Mapping Space-timeMapping Space-time

Gravitational DetectorGravitational Detector (LIGO, LISA, GEO600, TAMA, (LIGO, LISA, GEO600, TAMA, …………))

Tide force Tide force

Question :Question : Can we get some information about the Can we get some information about the

space-time geometry by using some dataspace-time geometry by using some datawhich are measurable near null infinity ?which are measurable near null infinity ?

Peeling off theorem Peeling off theorem

Characteristic initial value method Characteristic initial value method (E. T. Newman and T. W. J. (E. T. Newman and T. W. J. UntiUnti, J. Math. Phys.1962, J. Math. Phys.1962 H. Friedrich, Proc. Roy. Soc. Lond. A, 1981) H. Friedrich, Proc. Roy. Soc. Lond. A, 1981)

Suppose (M, g) be a vacuum, asymptotic flat, axial-Suppose (M, g) be a vacuum, asymptotic flat, axial-symmetric, stationary space-time, symmetric, stationary space-time, ttaa is the time-like is the time-likeKilling fieldKilling field

Where the null tetrad satisfy Bondi gauge in theWhere the null tetrad satisfy Bondi gauge in theneighborhood of null infinityneighborhood of null infinity

Killing EquationsKilling Equations

Unknown functionsUnknown functions

Lowest order coefficientsLowest order coefficients

Higher order coefficientsHigher order coefficients

All unknown functions can be expressed in terms ofAll unknown functions can be expressed in terms of

algebraic combination of algebraic combination of ψψkkoo and its derivative.and its derivative. HereHere

“……”“……” terms and f are all functions of lower order terms and f are all functions of lower order coefficients which we have known. coefficients which we have known.

General solution of General solution of ψψko

The freedom of space-time are reduced to a set ofConstants {Dk

m}. In fact, {Dkm} are just Jenis-Newman

Multi-pole moments of space-time.

Question : How to fix the value of {Dkm} to get

Kerr Metric ?

Axial-symmetric Axial-symmetric → → m=0m=0

Type-D conditionType-D condition

Submitting the general solution in to above equationSubmitting the general solution in to above equation

It is easy to see that type-D condition force It is easy to see that type-D condition force{{DDkk

mm} to be zero.} to be zero.

Main result : Main result : (X. Wu and S. (X. Wu and S. BaiBai, Phys. Rev. D. 2009), Phys. Rev. D. 2009)

Suppose (M, g) be an asymptotic flat, vacuum, Suppose (M, g) be an asymptotic flat, vacuum,stationary, axial-symmetric, Type-D space-time,stationary, axial-symmetric, Type-D space-time,then it must be Kerr space-time in the then it must be Kerr space-time in the BondiBondicoordinates neighborhood.coordinates neighborhood.

RemarksRemarks Type-D condition can be relaxed to algebraicType-D condition can be relaxed to algebraic

special condition.special condition.

This method also can show Newman-PenroseThis method also can show Newman-Penroseconstants vanish for vacuum, asymptoticconstants vanish for vacuum, asymptoticalgebraic special space-times.algebraic special space-times.

Thank youThank you