Gravitational radiation and angular momentum flux from a slow rotating dynamical black hole
aa r X i v : . [ g r- q c ] D ec Gravitational radiation and angular momentum flux from a slow rotating dynamical black hole
Yu-Huei Wu ∗
1. Center for Mathematics and Theoretical Physics, National Central University, Chungli, 320, Taiwan.2. Department of Physics, National Central University, Chungli, 320, Taiwan.
Chih-Hung Wang †
1. Department of Physics, Tamkang University, Tamsui, Taipei 25137, Taiwan.2. Institute of Physics, Academia Sinica, Taipei 115, Taiwan.3. Department of Physics, National Central University, Chungli, 320, Taiwan.
A four-dimensional asymptotic expansion scheme is used to study the next order effects of the nonlinearitynear a spinning dynamical black hole. The angular momentum flux and energy flux formula are then obtainedby constructing the reference frame in terms of the compatible constant spinors and the compatibility of thecoupling leading order Newman-Penrose equations. By using the slow rotation and small-tide approximationfor a spinning black hole, we chose the horizon cross-section is spherical symmetric. It turns out the flux formulais rather simple and can be compared with the known results. Directly from the energy flux formula of the slowrotating dynamical horizon, we find that the physically reasonable condition on the positivity of the gravitationalenergy flux yields that the shear will monotonically decrease with time. Thus a slow rotating dynamical horizonwill asymptotically approaches an isolated horizon during late time.
PACS numbers: 04.30.Db,04.20.Ha,04.30.Nk,04.70.Bw, 97.60.Lf,95.30.Sf
I. INTRODUCTION
Null infinity and black hole horizon have similar geometricalproperties. They are both three dimensional hypersurfaces andhave the gravitational flux across them. The physical proper-ties of null infinity can be studied in the conformal spacetimewith finite boundary. Thus the conformal method provides analternative way to study Bondi-Sachs gravitational radiationnear null infinity, which was first proposed by Penrose [19].The boundary of a black hole is asymptotically non-flat andone may not be able to apply the conformal method to studythe boundary problem of a dynamical black hole. Rather thanusing the symmetry for the whole space-time to locate theboundary of a black hole, Ashtekar et al. use a rather mildcondition on the symmetry of the three dimensional horizon[1, 2]. This quasi-local definition for the black hole boundarymakes it possible to study the gravitational radiation and thetime evolution of the black hole.In this paper, we use the Bondi-type coordinates to writethe null tetrad for a spinning dynamical horizon (DH). Theboundary conditions for the quasi-local horizons can be ex-pressed in terms of Newman-Penrose (NP) coefficients fromthe Ashtekar’s definition on DH. Unlike Ashtekar et al ’s [1, 2]three dimensional analysis, we adopts a 4-dimensional asymp-totic expansion to study the neighborhoods of generic iso-lated horizons (IHs) and dynamical horizons (DHs). Sincethe asymptotic expansion has been used to study gravitationalradiations near the null infinity [9, 13], it offers a usefulscheme to analyze gravitational radiations approaching an-other boundary of space-time, black hole horizons. We first ∗ Electronic address: [email protected] † Electronic address: [email protected] set up a null frame with the proper gauge choices near quasi-local horizons and then expand Newman-Penrose (NP) coeffi-cients, Weyl, and Ricci curvature with respect to radius. Theirfall-off can be determined from NP equations, Bianchi equa-tions, and exact solutions, e.g., the Vaidya solution. This ap-proach allows one to see the next order contributions from thenonlinearity of the full theory for the quasi-local horizons.We have shown that the quasi-local energy-momentum fluxformula for a non-rotating DH by using asymptotic expansionyields the same result as Ashtekar-Krishman flux [14, 15]. Forslow rotating DH, we have presented our results in [14], how-ever, we use an assumption of vanishing NP coefficient λ onDH. Furthermore, the energy-momentum flux formula has ashear (NP coefficient σ ) and a angular momentum (NP coef-ficient π ) coupling term. Since it is unclear whether the exis-tence of this term carries any physical meaning or it may dueto our assumptions, we thereby extend our previous work onIHs and DHs into a more general case.An algebraically general structure (Petrov type I) of spacetimeis thought to be related with gravitational radiation for an iso-lated source and can tell us more about the inner structure ofthe gravitating source. The Weyl scalars Ψ k , k = 0 , .., canbe expanded in terms of an affine parameter r along each out-going null geodesic based on assumption of compatificationof null infinity [4, 19]. Here Ψ k = O ( r k − ) , k = 0 , .., andone may find that it peels off more and more when movinginward along null ray. From Ashtekar’s definition of isolatedhorizon, it implies that Ψ , Ψ vanishes on horizon. There-fore the space-time is algebraically special on isolated hori-zon. However, spacetime may not be algebraically special foran arbitrary DH. The corresponding peeling theorem for anarbitrary DH is crucial for our gravitational radiation study.Due to the difficulties of knowing the fall-off of Weyl scalars,we use Kerr-Vaidya solution to serve as our basis for choosingthe fall-off of Weyl scalars,which is Ψ , Ψ vanishing on DH,in our previous work on slow-rotating DH [14]. So space-time structure on slow-rotating DH is still assumed to be al-gebraically special. However, according to the gravitationalplane-wave solutions, Ψ and Ψ indicate the ingoing and out-going gravitational waves, respectively. It seems physicallyunsatisfactory to assume Ψ vanishing on DH. Moreover, thealgebraically general space-time allows four roots of the equa-tion, which correspond to the principal null-directions of Weylscalars, and describes the gravitational radiation near the grav-itating source. Therefore it would be more reasonable for oneto consider an algebraically general space-time on an evolvingDH. From the reduction and the decoupling of the equationsgoverning the Weyl scalars, instead of assuming Ψ , Ψ = 0 on DH, we set Ψ , Ψ vanishing on a spinning DH. This is asimilar setting with the perturbation method (Also see Chan-drasekhar [7]).We present the results of asymptotic expansion for a spinningDH in Sec. III. However, it maybe too general to yield someinteresting physical results. By considering the small-tide andslow rotate of DH and using slow rotate Kerr solution as a ba-sis, we use two sphere conditions of DH cross section for ourlater calculation. The NP coefficient λ (shear for the incom-ing null tetrad n ) on DH is no longer assumed to be vanishedwhen calculating the flux formula. The index on NP coef-ficients denotes their values on DH. Directly from non-radialNP equations, we find that σ and π coupling terms can betransformed into π terms only, so the problems of our previ-ous work [14] are resolved.Though the exact solution for a stationary rotating black holehas been found near fifty years, the spacetime with rotationremains its ambiguity and difficulty for quasi-local mass ex-pressions and boundary condition. For example, the existenceof angular momentum will not change the boundary condi-tion for the null infinity, however, it will affect the boundarycondition of a black hole. Among the well-known quasi-localmass expressions named Komar, Brown-York and Dougan-Mason, only Komar integral of the quasi-local mass for anarbitrary closed two surface can go back to the unique New-tonian quasi-local mass [16]. Unfortunately, Brown-York andDougan-Mason mass can return to the unique surface integra-tion of the Newtonian mass in the covariant Newtonian space-time only for the spherically symmetry sources. In GR, quasi-local mass expressions for Kerr solution disagree one another[5]. Different quasi-local expressions give different values ofquasi-local mass for Kerr black hole. At null infinity, there isno generally accepted definition for angular momentum [17].Unfortunately, no explicit expression for Bramson’s angularmomentum in terms of the Kerr parameters m and a is given[6]. We use Komar integral to calculate angular momentumsince it gives exactly ma for Kerr solution. Although differ-ent quasi-local expressions yield the different results for Kerrsolution, our main motivation is to analyze and discuss thecompatibility of the coupling NP equations from asymptoticexpansions. We both calculate quasi-local mass and flux for aspinning DH based on two spinors (Dougan-Mason) and Ko-mar integral. It is found that these two expressions yield thesame result. Bondi and Sachs use no-incoming radiation condition forgravitational wave on null infinity [3, 4]. However, no-incoming radiation condition is only true for linearized the-ories, e.g., electrodynamics and linearized GR, as to excludethe incoming rays. The incoming pulse waves do not destroythe asymptotic conditions for null infinity since they are ad-mitted by formalism. Their existence may play an importantrole in the interpretation of the new conserved quantities (NPconstants) [11, 12]. The interpretation and physical meaningof these constants have been a source of debate and contro-versy until today. Some physical discussions and applicationof them can be found in [8, 10]. Despite the vagueness ofthe physical meaning of these conserved quantities, in the fullnonlinear gravitational theory, the mass and momentum are nolonger absolutely conserved and can be carried away by theoutgoing gravitational wave, so as to give a positive energyflux at infinity. Here we consider a spacetime inner bound-ary, e.g, a spinning DH in this paper. By the aid of usingasymptotic constant spinor to define spin frame as the refer-ence frame for our observation, mass and angular momentumflux can be calculated. According to the coupling NP equa-tions from the asymptotic expansion analysis, such a systemwill gain energy and will cause the radius of the black holeto increase. From similar argument, the outgoing waves donot change the boundary conditions of the quasi-local hori-zons (DHs or IHs) and make no contribution to flux, whileincoming wave will cross into DHs. The existence of incom-ing wave indicates the difference between IHs and DHs. Themass and momentum are carried into black hole by the incom-ing gravitational wave.In the Ashtekar-Kirshnan’s 3-dimensional analysis, it gives noconstraints on σ ’s time evolution. However, through our 4-dimensional asymptotical expansion scheme, we observe thatthe σ will monotonic decrease with time once the positivityof gravitational energy flux is hold on a slow rotating DH. Itmeans that the slow rotating DHs will gradually settle downto IHs as σ approach zero, which is physically reasonable. Itis similar to a physical assumption saying that the mass losscannot be infinite large for null infinity [13]. However, ratherthan assuming that mass loss cannot be infinite large we ob-tain this result directly from asymptotic expansion analysis fora slow rotating DH together with the physical arguments. Fur-ther from the commutation relations, we find that the horizonradius of a slow rotating DH will not accelerate. The slowrotating dynamical horizon increases with a constant speed.There is one more interesting point about the peeling proper-ties for a slow rotating DH. It is known that the peeling proper-ties refer to different physical asymptotic boundary conditionsof a slow rotating black hole. By comparing our current workto previous one [15], which have different peeling properties,and also due to the monotonic decrease of σ , we propose thatthe setting of Weyl scalars in this work makes it excludes thepossibility to absorb the gravitational radiation from near bygravitating source.The plan of this paper is as follows. In Sec. II, we re-view the definition of DH and express Ashtekar-Krishnan’s3-dimensional analysis of DH in terms of NP coefficients. Thegauge choices and boundary conditions of a spinning DH areapplied to the asymptotic expansion in Sec. III. In Sec. IV,we first examine the gauge conditions of slow rotating Kerrsolution in Subsec. IV A. Later we use the two sphere condi-tion for a slow rotating DH with small tide in Subsec. IV B.The results of asymptotic expansion are largely simplified byconsidering DH’s cross section as a two sphere. Angular mo-mentum and its flux for a slow rotating DH are calculated byusing Komar integral in Sec. V. Energy-momentum and itsflux of a slow rotating DH are obtained in Sec. VI. We firstcalculate mass and mass flux by using Komar integral in Sub-sec. VI A. Then, mass and mass flux of a slow rotating DH iscalculated by using two spinor method in Subsec. VI B. Thetime evolution of shear flux and its monotonic decrease is dis-cussed here. We find that either Komar integral or two spinormethod yields the same result.In this paper, we adopt the same notation as in [1, 2] for de-scribing generic IHs and DHs. However, we choose the differ-ent convention (+ − −− ) , which is a standard convention forthe NP formalism [18]. The necessary equations, i.e., com-mutation relations, NP equations and Bianchi identities, forasymptotical expansion analysis can be found in p. 45-p. 51of [7]. We use " ˆ= " to represent quantities on a dynamicalhorizon (ignore O ( r ′ ) ) and use " ∼ = " to represent quantities ona slow rotating horizon (ignore O ( a ) ). II. ASHTEKAR DYNAMICAL HORIZONA. The dynamical horizon
The generic IHs are taken as the equilibrium state of the DHs.The DHs can be foliated by marginally trapped surface S.Therefore, the expansion of the outgoing tetrad vanishes onDHs. a. Definition
A smooth, 3 dimensional, space-like sub-manifold H of space-time is said to be a dynamical horizon (DH) if it can be foliated by a family of closed 2-manifoldsuch that: (1) on each leaf, S , the expansion Θ ( ℓ ) of one nullnormal ℓ a vanishes, (2) the expansion Θ ( n ) of the other nullnormal n a is negative.From this definition, it basically tells us that a dynamical hori-zon is a space-like hypersurface, which is foliated by closed,marginally trapped two surface. The requirement of the ex-pansion of the incoming null normal is strictly negative sincewe want to study a black hole (future horizon) rather than awhite hole. Also, it implies Re[ ρ ] ˆ=0 , Re[ µ ] < . (1) B. Dynamical horizon in terms of Newman-Penrosecoefficients
If we contract the stress energy tensor with a time-like vector,then in components T k represents the energy flux of matterfield. Therefore we can use a time-like vector T a and contractit with the stress energy tensor to define the flux of the mat-ter energy. Here we are more interested in the energy of thematter field associated with a null direction. One can thus cal-culate the flux of energy associated with ξ a = N ℓ a . The fluxof matter energy across H along the direction of ℓ is given by F matter := Z H T ab T a ξ b d V. (2)The dynamical horizon is a space-like surface, the Cauchydata ( (3) q ab , K ab ) on the dynamical horizon must satisfy thescalar and vector constraints H S : = (3) R + K − K ab K ab = 16 πT ab T a T b , (3) H aV : = D b ( K ab − K (3) q ab )= D b P ab = 8 πT bc T c (3) q a b (4)where P ab := K ab − K (3) q ab .If the dominate energy condition is satisfied, it turns out that H has to be a space-like hypersurface [2]. The unit time-like vector that is normal to H is denoted by T a and the unitspace-like vector that orthogonal to the two sphere and tangentto H is denoted by R a . In order to study them in terms ofNewman-Penrose quantities, they can be defined by using thenull normals ℓ a and n a . Therefore, T a = 1 √ ℓ a + n a ) , R a = 1 √ ℓ a − n a ) (5)where T a T a = 1 , R a R a = − . The four metric has the form g ab = n a ℓ b + ℓ a n b + (2) q ab (6) = T a T b − R a R b + (2) q ab . (7)The three metric (3) q ab that is intrinsic to dynamical horizon H is (3) q ab = g ab − T a T b = (2) q ab − R a R b . (8)The two metric (2) q ab that is intrinsic to the cross section twosphere S is (2) q ab = (3) q ab + R a R b = − ( m a m b + m a m b ) . (9)The induced covariant derivative on H can be defined in termsof 4-dimensional covariant derivative ∇ a by D b V a := (3) q b c (3) q a d ∇ c V d , (10)so the three dimensional Ricci identity is then given by (3) R abc d w d = − [ D a , D b ] w c . (11)The induced covariant derivative on cross section S can bedefined in terms of 4-dimensional covariant derivative ∇ a by (2) D b V a := (2) q b c (2) q a d ∇ c V d . (12)The extrinsic three curvature K ab on H is K ab = (3) q ( a c (3) q b ) d ∇ c T d = ∇ a T b − T a (3) a b where (3) a b = T c ∇ c T b . One can also introduce the extrinsictwo curvature (2) K ab on S by (2) K ab = (2) q ( a c (2) q b ) d D c R d = D a R b + R a (2) a b , where (2) a b = R c D c R b = R ( c (2) q b d ) ∇ c R d . After a straight-forward but tedious calculation, we can write the extrinsic cur-vature in terms of NP spin coefficients. Here we present thegeneral extrinsic three curvature and two curvature withoutany assumption of gauge conditions. The extrinsic three cur-vature K ab is K ab = (3) q ( a c ∇ c T b ) = (2) q ( a c ∇ c T b ) − R ( a R c ∇ c T b ) = A (2) q ab + S ab + 2 W ( a R b ) + BR a R b (13)where S ab = 1 √ σ − λ ) m a m b + C.C. ] ,A = − Re ρ − Re µ √ ,W a := − (2) q a c K cb R b = 14 [ κ + ν − τ − π − α + β )] m a + C.C.,B = −√ ǫ − Re γ ) , and C.C. denotes the complex conjugate terms. The extrinsictwo curvature (2) K ab is (2) K ab = (2) q ( a c D c R b ) = (2) q ( a c (3) q c d (3) q b ) e ∇ d R e = (2) q a d (2) q b e ∇ d R e = 12 (2) K (2) q ab + (2) S ab where (2) K = −√ ρ + Re µ ) , (14) (2) S ab = 1 √ σ + λ ) m a m b + C.C. (15)The calculation of two acceleration (2) a a yields (2) a a = R b D b R a = R ( c (2) q a d ) ∇ c R d = Cm a + Cm a , where C = −
14 ( κ − ν + π − τ ) , (16)so the two acceleration is tangent to S . We now perform 2+1 decomposition to study the variousquantities on H . The curvature tensor intrinsic to S is givenby − (2) R abc d = − (2) q a f (2) q b g (2) q c k (2) q j d (3) R fgk j − (2) K ac (2) K b d + (2) K bc (2) K a d , which is the Gauss-Codacci equation. This leads to the rela-tion between the scalar three curvature (3) R and the scalar twocurvature (2) R − (3) R = − (2) R − (2) K + (2) K (2) ab K ab − D a α a (17)where α a := R b D b R a − R a D b R b = (2) a a − R a (2) K .From (17), we obtain the Einstein tensor on H − (3) G ab R a R b = − (2) R + (2) K − (2) K ab (2) K ab . (18)The expansion of the out going tetrad ℓ a can be calculated toyield Θ ( ℓ ) := −
12 ( ρ + ρ ) = 12 √ K + (2) K + B ] . (19)Now we use the following relations (20)-(23) to calculate H S + 2 R a H aV , where H S and H aV are the scalar and vectorconstraints defined in (3) and (4). K = 2 A − B, (20) (2) K = − K − B + 2 √ ( ℓ ) = − A + 2 √ ( ℓ ) , (21) K ab K ab = 2 A + S ab S ab − W a W a + B , (22) (2) K ab (2) K ab = 12 (2) K + (2) S ab (2) S ab . (23)From the momentum constraint (4) and use integration byparts, we get R b D a P ab = D a β a − P ab D a R b (24)where β a := K ab R b − KR a . Thus, γ a := α a + β a = R b D b R a − W a − √ ( l ) R a . (25)For a general space-time, the matter energy flux can be calcu-lated as following H s + 2 R a H aV = (3) R + K − K ab K ab + 2 R a D b P ab = (2) R + (2) K − (2) K ab (2) K ab + K − K ab K ab − P ab D a R b + 2 D a γ a (Use (17)) = (2) R − σ ab σ ab + 2 W a W a − W a (2) a a +4Θ ( ℓ ) (Θ ( ℓ ) − √ B ) + 2 D a γ a . By applying the 2+1 decomposition on the covariant deriva-tive D and using integration by parts, we have D a γ a = 2 D a ( (2) a a − W a − √ ( ℓ ) R a )= 2( (2) a a (2) a a − W a (2) a a − D a Θ ( ℓ ) R a + 12 (2) K Θ ( ℓ ) ) , where the term (2) D a ( (2) a a − W a ) has been discarded since itwill vanish due to the integration over a compact two surface S , and then W a W a − W a (2) a a + D a γ a )= 2( W a − (2) a a )( W a − (2) a a ) − D a Θ ( ℓ ) R a − (2) K Θ ( ℓ ) . Here we can define ζ a := W a − (2) a a = −√ (2) q ( da R c ) ∇ c ℓ d = 12 [ κ − τ − ( α + β )] m a + C.C. (26)Finally, we get H s + 2 R a H aV = (2) R − σ ab σ ab + 2 ζ a ζ a − D a Θ ( ℓ ) R a + Θ ( ℓ ) ( − (2) K + 4Θ ( ℓ ) − √ B ) where σ ab = 1 √ S ab + (2) S ab ) = √ σm a m b + C.C. is the shear of null normal ℓ a . This equation is completelygeneral. On the dynamical horizon, the outgoing expansion Θ ( ℓ ) vanishes. It then becomes F matter = 116 π Z ∆ H N ( (2) R − σ ab σ ab + 2 ζ a ζ a ) d V. (27)If the gauge condition κ ˆ= π − τ ˆ= π − ( α + β ) ˆ=0 (28)is satisfying, where ˆ= denotes the equating on DH, then ζ a = − πm a + C.C. and ζ a ζ a = − ππ . So the flux formula interms of NP in this gauge is F matter = 116 π Z ∆ H N ( (2) R − σσ − ππ ) d V. (29) C. Angular momentum flux and energy fluxes
By contracting the vector constraint H aV with the rotationalvector field ψ a , which is tangential to S , we can obtain angu-lar momentum of a black hole. Then we integrate the result-ing equation over the region of ∆ H and use the integration byparts together with the identity L ψ (3) q ab = 2 D ( a ψ b ) . It leadsto − dJ = J S − J S = 18 π I S K ab ψ a R b dS − π I S K ab ψ a R b dS = Z ∆ H ( T ab T a ψ b + 116 π P ab L ψ (3) q ab ) d V. (30) Our expression has some minus sign different from Ashtekar’s expressionbecause of convention.
The angular momentum associated with cross-section S is J ψS = − π H S K ab ψ a R b dS where ψ a need not be an axialKilling field. The flux of angular momentum due to matterfields F ψ matter and gravitational waves F ψ grav are F ψ matter = − Z ∆ H T ab T a ψ b d V, (31) F ψ grav = − π Z ∆ H P ab L ψ (3) q ab d V, (32)and the balance equation J ψS − J ψS = F ψ matter + F ψ grav , whichdescribes the difference of angular momentum between twocross section, is due to the matter radiation and gravitationalradiation.Each time evolution vector t a defines a horizon energy E t ∆ .From equation (27), we find the total energy flux is the com-bination of the matter flux and gravitational flux F matter + F grav = 116 π Z ∆ H N (2) R d V (33)where the matter flux is equation (2) and the gravitational fluxis F grav = 116 π Z ∆ H N ( σ ab σ ab − ζ a ζ a ) d V. (34)If we use the gauge conditions in (29), we then have F grav = 14 π Z ∆ H N ( | σ | + | π | ) d V. (35)The matter flux expression (2) of Vaidya solution would be F matter : = Z H T ab T a ℓ b N d V = 14 π Z Φ N d V (36)where we use πT ab ℓ a ℓ b = Φ . The total flux of Ashtekar-Krishnan then becomes F total = 14 π Z [ | σ | + | π | + Φ ] N d V. (37)Further, the integral of N (2) R can be written as Z ∆ H N (2) Rd V = Z R R dR I (2) Rd V = 8 π ( R − R ) where R and R are the radii of the horizon at the bound-ary cross-sections. For a rotating non-spherical symmetricdynamical horizon , we find the relation of the change in thehorizon area in the dynamical processes can be written as Z dR R − R )2 = 116 π Z ∆ H N (2) R d V = F matter + F grav (38) = Z ∆ H T ab T a ξ b d V + 116 π Z ∆ H N ( σ ab σ ab − ζ a ζ a ) d V. Hence, from this equation we can relate the black hole areachange with energy and angular momentum change. Thisgives a more general black hole first law in a dynamical space-time. If we now define the effective surface gravity [2] as κ R := 12 R , (39)then the area of horizon is A = 4 πR and the differential ofthe area is dA = 8 πRdR , therefore κ R π dA = 12 dR. (40)For the time evolute vector t a = N ℓ a − Ω ψ a = ξ a − Ω ψ a ,the difference of the horizon energy E tS can be expressed as[2] dE t = E tS − E tS = Z T ab T a t b d V + 116 π Z N ( σ ab σ ab − ζ a ζ a ) d V − π Z ∆ H Ω P ab L ψ q ab d V. (41)By using (30) together with the linear combination of Z dR Z ∆ H T ab T a ξ b d V (42) + 116 π Z ∆ H N ( σ ab σ ab − ζ a ζ a ) d V, we can obtain a generalized black hole first law for dynamicalhorizon κ R π dA + Ω dJ = dE t . (43) III. ASYMPTOTIC EXPANSION FOR A SPINNINGDYNAMICAL HORIZONA. Frame setting and gauge choice
We choose the incoming null tetrad n a = ∇ a v to be thegradient of the null hypersurface v = const. We then have g ab v ,a v ,a = 0 . It gives us the gauge conditions ν = µ − µ = γ + γ = α + β − π = 0 . Then we further choose n a flagplane parallel, it implies γ = 0 . For the setting of outgoingnull tetrad ℓ , we first choose ℓ to be a geodesic and use null ro-tation type III to make ǫ − ǫ = 0 . We choose m, m tangent tothe cross section S , and thus ρ ˆ= ρ, π ˆ= τ . From the boundaryconditions of a spinning DH ( see eq (1)), recall ρ ˆ=0 , π ˆ =0 , σ ˆ =0 . (44) This also implies ω ˆ=0 . Kerr solution preferred gauge on horizon is µ ˆ= µ, π ˆ= α + β ˆ= τ . on DH. We summarize our gauge choices and boundary con-ditions κ = ǫ − ǫ = ν = µ − µ = γ = π − α − β = 0 ,ρ ˆ=0 , π ˆ= τ . (45)In order to preserve orthonormal relations, we can choose thetetrad as ℓ a = (1 , U, X , X ) , n a = (0 , − , , ,m a = (0 , , ξ , ξ ) . in the Bondi coordinate ( v, r, x , x ) .Now we make a coordinate transformation to a new comovingcoordinate ( v, r ′ , x , x ) where r ′ = r − R ∆ ( v ) and R ∆ ( v ) is radius of a spinning DH. Here ℓ a = (1 , U − ˙ R ∆ , X , X ) , n a = (0 , − , , ,m a = (0 , , ξ , ξ ) , where ˙ R ∆ ( v ) is the rate of changing effective radius of DH.From this coordinate, we may see that the dynamical horizonis a spacelike or null hypersurface. Here, tangent vector ofDH is R a = ℓ a − ˙ R ∆ n a ˆ= ∂∂v where ˙ R ∆ ≥ . Therefore, itimplies R a R a ≤ and U, X k = O ( r ′ ) . B. The peeling properties and falloff of the Weyl scalars
Since we use κ = ν = 0 , σ = 0 , λ = 0 , then we have ( δ − α + π )Ψ − ( D − ǫ )Ψ ˆ=0 , (46) (∆ + µ )Ψ − ( δ − π − β )Ψ ˆ=3 σ Ψ , (47) ( D + 2 ǫ ) σ ˆ=Ψ , (48) ( D + 4 ǫ )Ψ − ( δ + 4 π + 2 α )Ψ ˆ= − λ Ψ , (49) ( δ + 4 β − τ )Ψ − (∆ + 4 µ )Ψ ˆ=0 , (50) (∆ + 2 µ ) λ ˆ= − Ψ . (51)in vacuum. Therefore, one can set Ψ ˆ=Ψ ˆ=0 as peeling prop-erties for a spinning DH. This is a similar with perturbationmethod and one may refer to p. 175 and p. 180 in [7].The falloff of the Weyl scalars is algebraically general (this isa more general setting than [14] and [15]) on DH where Ψ = Ψ = O ( r ′ ) , Ψ = Ψ = Ψ = O (1) . (52)By considering Vaidya solution as our compared basis formatter field part, the falloff of the Ricci spinor componentsare Φ = O (1) , Φ = Φ = Φ = Φ = Φ = O ( r ′ ) . (53) C. From the radial equations µ = µ + ( µ + λ λ ) r ′ + O ( r ′ ) ,λ = λ + (2 µ λ + Ψ ) r ′ + O ( r ′ ) ,α = α + [ λ ( π + β ) + α µ ] r ′ + O ( r ′ ) ,β = β + [ µ π + β µ + α λ ] r ′ + O ( r ′ ) ,ρ = [Ψ − ð π + π π + σ λ ] r ′ + O ( r ′ ) ,σ = σ + ( − ð π − π + µ σ ) r ′ + O ( r ′ ) ,π = π + [2 µ π + 2 λ π ] r ′ + O ( r ′ ) ,ǫ = ǫ + [2 α π + 2 β π + π π ] r ′ + O ( r ′ ) ,ξ k = ξ k + [ λ ξ k + µ ξ k ] r ′ + O ( r ′ ) ,U = 2 ǫ r ′ + O ( r ′ ) ,X k = 2( π ξ k + π ξ k ) r ′ + O ( r ′ ) . Ψ = Ψ + ( µ Ψ − σ Ψ + λ Φ ) r ′ + O ( r ′ ) , Ψ = ( − δ Ψ + 3 π Ψ ) r ′ + O ( r ′ ) , Ψ = Ψ + (3 µ Ψ − σ Ψ ) r ′ + O ( r ′ ) , Ψ = [ − δ Ψ + ( π − β )Ψ ] r ′ + O ( r ′ ) . Φ = Φ + 2 µ Φ r ′ + O ( r ′ ) , Φ = 12 ˙ R ∆ Φ r ′ + O ( r ′ ) , Φ = − ˙ R ∆ Φ r ′ + O ( r ′ ) , Φ = Φ r ′ + O ( r ′ ) , Φ = Φ r ′ + O ( r ′ ) , Φ = Φ r ′ + O ( r ′ ) . D. From the non-radial equations
The following equations refer to the equation numbers fromp. 45-p. 47 in [7]. We re-label (304)-(306) as (NC1), (NC2),(NC3). (NC1) δ ǫ = 0 , (NC2) ˙ P = ˙ R ∆ [ µ P + λ P ] + σ P , (NC3) P c ∇ ln P = α − β , (54)where P ( v, x k ) := ξ = − iξ , P c ∇ := δ , c ∇ := ∂∂x + i ∂∂x . We relabel (a), (b), (c), (g), (d), (e), (h), (k), (m), (l) as (NR1),(NR2), (NR3), (NR4), (NR5), (NR6), (NR7), (NR8), (NR9), (NR10). (NR1) − ˙ R ∆ (Ψ − ð π + π π + σ λ ) = σ σ + Φ , (NR2) ˙ σ = ˙ R ∆ [ − ð π − π + σ µ ] + 2 ǫ σ + Ψ , (NR3) ˙ π = ˙ R ∆ [2 µ π + 2 λ π ] + 2 σ π , (NR4) ˙ λ = ˙ R ∆ [2 µ λ + Ψ ] + ð π + π − λ ǫ + µ σ , (NR5) ˙ α = ˙ R ∆ [ α µ + λ ( π + β )] + σ β , (NR6) ˙ β = ˙ R ∆ [ α λ + µ ( π + β )] + σ ( α + π ) , (NR7) ˙ µ = ˙ R ∆ ( µ + λ λ ) + ð π + π π + σ λ − µ ǫ + Ψ , (NR8) ð σ = 0 , (NR9) δ λ − δ µ = µ π + λ ( α − β ) , (NR10) ð π − ð π = 2Im ð π = − λ σ ) − , (NR10) ReΨ = − Re[ δ ( α − β )] − Re( λ σ )+( α − β )( α − β ) . The following equations refer to the equation numbers (a), (b),(c), (d) on p. 49 of [7]. Here, we relabel (a), (b), (c), (d) as(NB1), (NB2), (NB3), (NB4). (NB1) − δ Ψ + (4 α − π )Ψ − ˙ R ∆ [ − δ Ψ + 3 π Ψ ]= − δ Φ + π Φ , (NB2) ˙Ψ = ˙ R ∆ (3 µ Ψ − σ Ψ ) − λ Ψ + µ Φ , (NB3) − δ Ψ − π Ψ = ˙ R ∆ ( − δ Ψ + ( π − β )Ψ ) , (NB4) ˙Ψ = ˙ R ∆ Ψ − λ Ψ − ǫ Ψ . E. Compatible constant spinor conditions for a rotatingdynamical horizon
In this section, we adopt a similar idea of Bramson’s asymp-totic frame alignment for null infinity [6] and apply it to setup spinor frames on the quasi-local horizons. We define thespinor frames as Z A A = ( λ A , µ A ) (55)where λ A = λ o A − λ ι A , µ A = µ o A − µ ι A . We expand λ , λ as λ = λ ( v, θ, φ ) + λ ( v, θ, φ ) r ′ + O ( r ′ ) , (56) λ = λ ( v, θ, φ ) + λ ( v, θ, φ ) r ′ + O ( r ′ ) . (57)Here λ is type ( − , and λ is type (1 , .Firstly,we require the frame to be parallely transported alongthe outgoing null normal ℓ a . lim r ′ → DZ A A = 0 . ⇒ ℓ a ∇ a ( λ o A − λ ι A ) = 0 . (58)Then it gives the condition þ λ = 0 on DH. The compatibleconditions are: þ λ = 0 , ⇒ ˙ λ − ǫ λ = 0 (59)ð λ + σ λ = 0 , (60)ð λ − µ λ = 0 , (61)þ λ = − ð λ . (62) IV. SLOW ROTATING BLACK HOLE AND SETTINGS ONA TWO SPHEREA. Slow rotating Kerr horizon in Bondi coordinate
Kerr metric in the Eddington-Finkelstein coordinate ( v, r, θ, χ ) is ds = ∆ − a sin θ Σ dv − dvdr + 2 a sin θ ( r + a − ∆)Σ dvdχ + 2 a sin θdχdr − Σ dθ − ( r + a ) − ∆ a sin θ Σ sin θdχ . (63)By changing the coordinate from ( v, θ, χ ) to ( v, θ, χ ′ ) dχ = Ω ∆ dv + dχ ′ (64)where Ω ∆ = ar + a is the angular velocity on horizon and r ∆ is horizon radius of Kerr solution, we can make the term g vχ dvdχ vanished in the 3-D metric. The 3-D metric in thenew coordinate ( v, θ, χ ′ ) will be ds ˆ= − a sin θ Σ ∆ dv + 2 a sin θ Σ ∆ Ω − dvdχ − Σ ∆ dθ − Ω − a sin θ Σ ∆ dχ (65) ˆ= 0 · dv − Σ ∆ dθ − Ω − a sin θ Σ ∆ dχ ′ (66) ∼ = 0 · dv − r ( dθ + sin θ dχ ′ ) . (67)Here the surface area of slow rotating Kerr is A Kerr ∼ =16 πr .Now, we consider the case of slow rotation so that a is smalland we ignore the a terms. Thus the tetrad components inthe Bondi coordinate (˜ v, ˜ r ′ , ˜ θ, ˜ φ ) are: ℓ a = (1 , U r ′ , , ar ∆ 2 + Dr ′ ) (68) n a = (0 , − , , (69) m a = 1 √ η ∆ (0 , , − r ′ η ∆ , − i sin θ (1 − r ′ η ∆ )) (70)where U := r ∆ − Mr and D := a (2 r ∆ − M ) r . The NP coefficients and Weyl tensors are: κ = σ = λ = ν = 0 , (71) ρ = U ( − r ∆ + r ′ ) r ′ ( η ∆ − r ′ )( η ∆ − r ′ ) ˆ=0 , (72) µ = − r ∆ + r ′ ( η ∆ − r ′ )( η ∆ − r ′ ) ˆ= − r ∆ Σ ∆ ∼ = − r ∆ , (73) π = τ = i √ Dη sin θ η ∆ − r ′ ) ˆ= i √ Dη ∆ sin θ ∼ = i √ a r , (74) ǫ = U [( r ′ − r ∆ ) + a cos θ + ia cos θr ′ ]2[( r ′ − r ∆ ) + a cos θ ]ˆ= U ∼ = 14 r ∆ , (75) γ ˆ= − ia cos θ ∆ , γ + γ ˆ=0 , (76) π = α + β, (77) Ψ = 0 , Ψ = O ( r ′ ) , (78) ImΨ ˆ= − iD cos θ Σ ∆ ( r + a cos θ − a sin θ ) , (79) Ψ ˆ= i √ θr ∆ η ∆ [ D Σ + 2 ia cos θ ] , (80) Ψ = 0 . (81) Remark.
In this approximate Kerr tetrad in Bondi coordinate,the NP coefficients satisfy ν = µ − µ ˆ= π − α − β ˆ= γ + γ ˆ= ǫ − ǫ ˆ=0 ,π = τ , ρ ˆ= ρ, µ < . (82)By examining the approximate Kerr tetrad in Bondi coordi-nates, we found it is compatible with our frame setting for theasymptotic expansions. B. Setting on a two sphere: on horizon cross section
To solve the coupling equations from non-radial NP equationswould be rather complicated and maybe too general to yieldsome interesting physical results. By considering the small-tide and slow rotate of DH and consider slow rotate Kerr so-lution as a basis from pervious subsection, we use two sphereconditions of DH cross section for our later calculation. On asphere with horizon radius R ∆ ( v ) , one can set µ = − R ∆ . (83)Let P, µ on a sphere with radius R ∆ , then P ∝ R ∆ . From(NC2), ˙ ξ k = ˙ R ∆ ( µ ξ k + λ ξ k ) + σ ξ k which dependson the next order nonlinear effect off horizon, we obtain λ = − σ ˙ R ∆ , (84)and ˙ P = ˙ R ∆ µ P = − ˙ R ∆ PR ∆ . (85)Moreover, the effective surface gravity is ˜ κ = 2 ǫ = R ∆ ,and then µ = − ǫ (Recall eq. (39)).Check the commutation relation [ δ , D ] λ and [ δ , D ] σ , itimplies ¨ R ∆ = 0 . (86)This means that the horizon radius will not accelerate (no in-flation). The dynamical horizon will increase with a constantspeed. We note here that if the two sphere condition does nothold, then this result is no longer true.After applying these conditions, we list the main equationsthat will be used in the later section (NR1’) ˙ R ∆ [ − (Ψ + Ψ )+ ( ð π + ð π ) − π π ] = Φ , (NR2’) ˙ σ = ˙ R ∆ [ − ð π − π + σ µ ]+2 ǫ σ + Ψ , (NR3’) ˙ π = 2 ˙ R ∆ µ π , (NR4’) σ ¨ R − ˙ R ∆ ˙ σ ( ˙ R ∆ ) = ˙ R ∆ Ψ + ð π + π + 2 σ ǫ ˙ R ∆ − µ σ , (NR5’) ˙ α = ˙ R ∆ α µ − σ π , (NR6’) ˙ β = ˙ R ∆ µ ( π + β ) + σ π , (NR7’) ReΨ = 2 µ ǫ − π π − Re ð π , ImΨ = − Im ð π , (NR8’) ð σ = 0 , (NR9’) − σ π = ˙ R ∆ µ π ,σ ð π = − ˙ R ∆ µ ð π , (NB2’) ˙Ψ = ˙ R ∆ [3 µ − σ Ψ ] + σ Ψ + µ Φ , (NR1’)+(NR7’) ð π = Φ ˙ R ∆ + 2 µ ǫ . V. ANGULAR MOMENTUM AND ANGULARMOMENTUM FLUX OF A SLOW ROTATING DH
Here we use an asymptotically rotating Killing vector φ a fora spinning DH. It coincides with a rotating vector φ α ˆ= ψ a on a DH and is divergent free. It implies ∆ a φ a := S aa S b a ∇ b φ a = 0 . Therefore, m a δφ a = − m a δφ a . (87)Let φ a = Am a + Bm a , we get A = − B . Therefore, it existsa function f such that φ a = δf m a − δf m a , (88)which is type (0 , . Since f is type (0 , , therefore δf = ð f . By using Komar integral, the quasi-local angular momentumon a slow rotating DH is J ( R ∆ ) = 18 π [ I S ∇ a φ b dS ab ] | ∆ = 18 π I S Im( π ð f ) dS ∆ (use integration by part) = − π I S f Im ð π dS ∆ = − π I S f ImΨ dS ∆ . (89)From (NB2’), we get Im ˙Ψ = 3 ˙ R ∆ R ∆ Im ð π = − ˙ R ∆ R ∆ ImΨ .Together with ∂∂v dS ∆ = 2 ˙ R ∆ R ∆ dS ∆ , the angular momentumflux for a slow rotating DH is ˙ J ( R ∆ ) = − π I S ( ˙ f − ˙ R ∆ R ∆ f )ImΨ dS ∆ = 14 π I S Im[( ˙ f − ˙ R ∆ R ∆ f ) ð π ] dS ∆ . (90)We note that from ddv (NR7’), it yields the same result. Here if π = 0 and f ( v, θ, φ ) = G ( θ, φ ) R ∆ ( v ) , then ˙ J ( R ∆ ) = 0 . Itthen returns to the stationary case. If π = 0 , i.e., ImΨ = 0 ,then J and ˙ J = 0 . It then returns to the non-rotating blackhole.For IH, by using integration by parts and Penrose volume I, p.281, we have I ω a φ a = I f Im ð π = I f ImΨ . (91)For DH, we have K ab φ a R b = 2 W ( a R b ) φ a R b = − W a φ a (92) = − πm a φ a + C.C. = − π ð f + π ð f. (93) VI. THE QUASI-LOCAL ENERGY-MOMENTUM ANDFLUX OF A SLOW ROTATING DHA. Mass and mass flux from Komar integral
The asymptotic time Killing vector on a DH can be expressedas t a = ∂∂v = [ ℓ a + ( U − ˙ R ∆ ) n a ] | ∆ in corotating coordinate.The Komar mass on a DH is then M ∆ = 18 π I S ∇ a t b N dS ab = 14 π I ǫ N dS ∆ = 14 π I R ∆ N dS ∆ , (94)where ǫ = − µ / . This yield the same with eq. (i) from twospinor calculation when one chose N = λ λ ′ .0We then obtain the mass flux on a DH from Komar integral is ˙ M ∆ = 14 π I ˙ R ∆ R N dS ∆ , (95)and later we shall see that it agrees with eq. (96) from twospinor method. B. Mass and mass flux from two spinor method
By using the compatible constant spinor conditions for aspinning dynamical horizon (60), (61) and the results ofthe asymptotic expansion, we get the quasi-local energy-momentum integral on a slow rotating dynamical horizon I ( R ∆ ) = − π H µ λ λ ′ dS ∆ (i) = − π H Re2 ǫ [Ψ + δ π + 2 β π ] λ λ ′ dS ∆ . (ii)In order to calculate flux we need the time related condition(59) of constant spinor of dynamical horizon in Section III Eand re-scale it. Then ˙ λ = 0 . It’s tedious but straightfor-ward to calculate the flux expression. It largely depends onthe non-radial NP equations and the second order NP coeffi-cients. By using Sec.IV B, we substitute them back into theenergy-momentum flux formula to simplify our expression. From (i)
Apply time derivative to (i), and then we obtain the quasi-local energy momentum flux for dynamical horizon ˙ I ( R ∆ ) = 14 π I ˙ µ λ λ ′ dS ∆ . (96)where it is always positive . Here ˙ µ is the news function ofDH that always has mass gain.From the choice of µ = − R ∆ , we have ˙ µ = ˙ R ∆ R = ˙ R ∆ R where the two scalar curvature is (2) R = R (Themetric of a two sphere with radius R ∆ is d ˜ s = − R ( dθ +sin θdφ ) .). Integrate the above equation with respect to v and use ˙ µ = ˙ R ∆(2) R/ , we then have [14] dI ( R ∆ ) = 18 π Z (2) Rλ ′ λ dS ∆ dR ∆ . (97) From (ii)
We first apply ∂/∂v on (NR7) to get ˙ µ = ˙Ψ + ˙ δ π + δ ˙ π + 2 ˙ β π + 2 β ˙ π ǫ − µ ˙ ǫ ǫ (98)Now, apply time derivative on (ii) and use Sec. IV B yields ˙ I ( R ∆ ) = 14 π I ǫ { R ∆ Φ − σ σ ˙ R ∆ ( ∂∂v ln( R σ σ ))+3 ˙ R ∆ π π R ∆ } λ λ ′ dS ∆ (99)where the total energy momentum flux F total is LHS of (99)and is equal to matter flux plus gravitational flux F total = F matter + F grav . We can write the gravitational flux equal tothe shear flux plus angular momentum flux. F grav = F σ + F J (100)where the shear flux F σ is second term of RHS of (99) andthe angular momentum flux F J is third term of RHS of (99).The coupling of the shear σ and π can be transform into π terms by using (NR9’). Then integrate the above equationwith respect to v , we have dI ( R ∆ ) = 18 π Z R ∆ ˙ R ∆ { R ∆ Φ − σ σ ˙ R ∆ ( ∂∂v ln( R σ σ ))+3 ˙ R ∆ π π R ∆ } λ λ ′ dS ∆ dR ∆ . (101)where dv = dR ∆ ˙ R ∆ . Here we note that if one wants to observepositive shear flux − ∂∂v ln( R σ σ ) ≥ , it implies that ˙ σ ≤ , (102)where ˙ R ∆ , R ∆ > have been considered. So the shear on aspinning DH is monotonically decreasing with respect to v .Recall that the total flux of Ashtekar-Krishnan (37), we com-pare our expression with Ashtekar’s expression. If we choose N = λ λ ′ , then (101) together with (97) gives dI ( R ∆ ) = 18 π Z (2) RN dS ∆ dR ∆ = 18 π Z R ∆ ˙ R ∆ { R ∆ Φ + 2 k σ σ ˙ R ∆ + 3 ˙ R ∆ π π R ∆ } N dS ∆ dR ∆ (103)where we define ∂∂v ln( R σ σ ) := − k for the convenience.This is the relation between the change in DH area (Recall(38)) and total flux with including the matter flux and gravita-tional flux. Shear flux:
In the special case ∂∂v ln( R σ σ ) := − k where k is a constant, we then have R σ σ = Ae − kv . (104)1If k > , σ ց . If k < , σ ր . Therefore, if we want toget positive gravitational flux, the shear σ must decrease withtime v and k > . On the contrary, the negative gravitationalflux implies the shear must grow with time. The negative massloss from shear flux will make the dynamical horizon growwith time is physically unreasonable. Therefore, the secondterm of RHS in eq. (99) should be positive. This says that theshear on a spinning DH will decay to zero when time v goesto infinity and the amount of shear flux F σ is finite. σ → , | v | → ∞ , (105) F σ < ∞ . (106)Hence a slow rotating dynamical horizon will settle down toan equilibrium state, i.e., isolated horizon at late time. Discussion
1. If π = 0 and the shear does not vanishes σ = 0 wehave ˙ I ( R ∆ ) = 14 π I R ∆ ˙ R ∆ [Φ + 2 kσ σ ] N dS ∆ . This goes back to the result of the flux of the non-rotating dynamical horizon. When k = , it goes backto the result of non-rotating DH in [15].2. If both shear and π vanishes, we have ˙ I ( R ∆ ) = 14 π I R ∆ Φ ˙ R ∆ N dS ∆ . This result can be compared with dynamical horizon ofVaidya solution.3. Though we chose the cross section of DH to be twosphere, however, it still imply that the shear term can-not make into zero. This is because the contribution ofshear comes from the next order nonlinear effect of theequations.
Laws of black hole dynamics
LHS of eq. (103) can be written as dI ( R ∆ )2 = ˜ κ π dA = dR ∆ (107)where A = 4 πR . For a time evolute vector t a = N ℓ a − Ω φ a , the difference of horizon energy dE t can be calculatedas follow dE t = 116 π Z R ∆ ˙ R ∆ { R ∆ Φ + 2 k σ σ ˙ R ∆ } N +[3 N π π − R ∆ Im[( ð ˙ f − ˙ R ∆ R ∆ ð f ) π ] dV and the generalized black hole first law for a slow rotatingdynamical horizon is ˜ κ π dA + Ω dJ = dE t . (108) VII. CONCLUSIONS
Since Ψ , Ψ are gauge invariant quantities in a linear pertur-bation theory, it allows us to chose a gauge, in which Ψ , Ψ vanish on DH. This choice of gauge is crucial for the couplingof the NP equations and the consequence physical interpreta-tion. In this paper, we use a different peeling property fromour earlier work [15][14]. This leads to a physical picture thatcaptures a collapsing slow rotating star and formation of a dy-namical horizon that finally settle down to an isolated horizonat late time. Further from the peeling property, if the shearflux is positive, it excludes the possibility for a slow rotatingDH to absorb the gravitational radiation from nearby gravita-tional sources. The mass and momentum are carried in by theincoming gravitational wave and cross into dynamical hori-zon. We shall see that though it may exist outgoing wave onhorizon, however, it will not change the boundary condition ormake the contribution to the energy flux. A dynamical horizonforms inside the star and eat up all the incoming wave when itreaches the equilibrium state, i.e., isolated horizon.The NP equations are simplified by using two-sphere condi-tions for a slow rotating DH with small tide. By using thecompatibility of the coupling NP equations and the asymp-totic constant spinors, the energy flux that cross into a slowrotating DH should be positive. The mass gain of a slow ro-tating DH can be quantitatively written as matter flux, shearflux and angular momentum flux. Further, a result comes outthat the shear flux must be positive implies the shear mustmonotonically decay with respect to time. This is physicallyreasonable since black hole cannot eat infinite amount of grav-itational energy when there is no other gravitational sourcenear a slow rotating DH. We further found that the mass andmass flux based on Komar integral can yield the same result.Therefore, our results are unlikely expression dependent. Forother quasi-local expressions remain the open question for thefuture study. It would be interesting if one can free the two-sphere conditions, then obtain the metric distorted by gravita-tional wave. Acknowledgments
YHW would like to thank Prof James M. Nester for helpfuldiscussion. YHW would like to thank Top University Projectof NCU supported by Ministry of Education, Taiwan. YHWwas supported by Center for Mathematics and TheoreticalPhysics, National Central University, Taiwan. CHW was sup-ported by the National Science Council of Taiwan under thegrants NSC 96-2112-M-032-006-MY3 and 98-2811-M-032-014.2 [1] A. Ashtekar, C. Beetle, and S. Fairhurst, Class. Quantum Grav. A269
21 (1962).[4] R. K. Sachs, Gravtiational waves in general relativity,
VI.
Theoutoging radiation condition, Proc. Roy. Soc (London), A264,309-338 (1961).[5] G. Bergqvist, Quasilocal mass for event horizons, Class. Quan-tum Grav. 9 1753-1768 (1992).[6] B. D. Bramson, Proc. Roy. Soc. London
A341 , 451-461 (1975).[7] S. Chandrasekhar, The Mathematical Theory of Black Hole,Oxford University Press (1983).[8] S. Dain and J. A. Valiente Kroon, Conserved quantities ina black hole collision, Class. Quantum Grav.
811 (2002).arXiv:gr-qc/0105109v1[9] A. R. Exton, E. T. Newman, and R. Penrose, Conserved Quan-tities in the Einstein-Maxwell Theory, J. Math. Phys.
231 ˛aV233 (1965) [12] E. T. Newman and R. Penrose, New Conservation Laws forZero Rest-Mass Fields in Asymptotically Flat Space-Time,Proc Roy Soc Lond
A305
No.1481 pp. 175-204 (1968)[13] E. T. Newman and T. W. J. Unti, J. Math. Phys. , 891-901(1962).[14] Y. H. Wu, PhD thesis, University of Southampton (2007).[15] Y. H. Wu and C. H. Wang, Gravitational radiations of genericisolated horizons and nonrotating dynamical horizons fromasymptotic expansions, Phys. Rev. D 80, 063002 (2009).arXiv:0906.1551v1 [gr-qc][16] Y. H. Wu and C. H. Wang, Quasi-local mass in the covariantNewtonian spacetime, Class. Quantum Grav.25