Relating the Newman-Penrose constants to the Geroch-Hansen multipole moments
aa r X i v : . [ g r- q c ] A ug Relating the Newman–Penrose constants to theGeroch–Hansen multipole moments
Thomas B¨ackdahl
School of Mathematical Sciences, Queen Mary, University of London,Mile End Road, London E1 4NS, [email protected]
Abstract
In this paper, we express the Newman–Penrose constants in terms ofthe Geroch–Hansen multipole moments for stationary spacetimes. Theseexpressions are translation-invariant combinations of the multipole mo-ments up to quadrupole order, which do not normally vanish.
The Newman–Penrose (NP) constants were defined by Newman and Penrosein [12]. They are quantities defined on the null-infinities, and turn out to beconserved under time translations. Even though they have been studied fora long time, their meaning is still not fully understood. Lately, it has beendisputed whether the NP constants are zero for stationary spacetimes or not.For the Kerr solution they are zero [3]. In fact, it has been shown that they arezero for all algebraically special stationary spacetimes [15]. The NP constantshave also been calculated for a wide set of examples [4, 7, 11]. The originalpaper [12] by Newman and Penrose gives expressions of the NP constants interms of multipole moments. It is unclear, however, how these moments weredefined, if they are coordinate independent and if different moments can bespecified independently. The Geroch–Hansen multipole moments have theseproperties, but were defined later [8, 9]. Therefore, this paper is intended toclearly settle the matter by expressing the NP constants in terms of the Geroch–Hansen multipole moments. These multipole moments also give a possibility ofphysical interpretation.The Geroch–Hansen multipole moments can be freely specified under a sim-ple convergence condition. That is, for any given choice of multipoles, satisfyingthe convergence condition, there is a unique stationay spacetime with these mul-tipole moments. This was shown in [2] for the stationary axisymmetric case.Recently, Herberthson [10] showed this for the general static case using resultsof Friedrich [6]. The result of Frierdich states that for static spacetimes onecan freely specify null data under a convergence condition. These null data arerelated to the multipole moments, but the relation is fairly complicated. Theresults by Friedrich have been extended to the stationary case by Ace˜na [1].Hopefully, the results by Herberthson can also be extended to the stationarycase, but for now it is still an open problem.1or the static case, one could establish the relation between the NP constantsand the Geroch–Hansen multipole moments, using the results of Friedrich andK´ann´ar [7], but it will not give the general stationary case. One would also needto be careful with the translation between formalisms. Therefore, the originaldefinition of the multipole moments, and the asymptotic expansions of Wu andShang [15] are used in this paper.Throughout this paper we use abstract index notation. For coordinateexpressions we sometimes omit the indices, and use the short hand notation dxdy = ( dx ) ( a ( dy ) b ) . In this paper, we will use series expansions of stationary spacetimes in Bondi–Sachs coordinates ( u, r, ζ, ¯ ζ ). Expressed in standard angular coordinates, thecomplex angle ζ = e iφ cot θ . The differential operators ð , ¯ ð are defined as inequation (4.15.117) in [13], for the complex stereographic coordinates ζ, ¯ ζ , i.e. ð f = 1 + ζ ¯ ζ √ ∂f∂ ¯ ζ + s ζ √ f, ¯ ð f = 1 + ζ ¯ ζ √ ∂f∂ζ − s ¯ ζ √ f (1)where s is the spin-weight of f . Observe that this differs slightly from theoperator usually used for the θ, φ coordinates, due to a different choice of spin-frame. The corresponding spin-weighted spherical harmonics are then givenby s Y j,m = p (2 j + 1)( j + s )!( j − s )!( j + m )!( j − m )!¯ ζ j − m ζ j + s ( − m √ π (1 + ζ ¯ ζ ) j × min( j − m,j + s ) X r =max(0 ,s − m ) ( − ζ ¯ ζ ) − r r !( j − m − r )!( j + s − r )!( r + m − s )! (2)where − j ≤ s ≤ j , − j ≤ m ≤ j .We take the following expansion of the null tetrad from [15], using Ψ = ¯Ψ . l a = ∂∂r ,n a = ∂∂u + (cid:18) − − Ψ r + ¯ ð Ψ + ð ¯Ψ r − ¯ ð Ψ + ð ¯Ψ r − (cid:16) | Ψ |
12 + ¯ ð Ψ + ð ¯Ψ (cid:17) r − + O ( r − ) (cid:19) ∂∂r + (cid:18) ζ ¯ ζ √ r Ψ − ζ ¯ ζ √ r ¯ ð Ψ + O ( r − ) (cid:19) ∂∂ζ + (cid:18) ζ ¯ ζ √ r ¯Ψ − ζ ¯ ζ √ r ð ¯Ψ + O ( r − ) (cid:19) ∂∂ ¯ ζ ,m a = (cid:18) − Ψ r + ¯ ð Ψ r + ¯ ð Ψ r + O ( r − ) (cid:19) ∂∂r + (cid:18) ζ ¯ ζ √ r Ψ + O ( r − ) (cid:19) ∂∂ζ + (cid:18) ζ ¯ ζ √ r + O ( r − ) (cid:19) ∂∂ ¯ ζ , (3)2here the expansions of the Weyl curvature areΨ = Ψ r + Ψ r + O ( r − ) , Ψ = Ψ r + Ψ r + Ψ r + O ( r − ) , Ψ = Ψ r + Ψ r + Ψ r + O ( r − ) , Ψ = Ψ r + Ψ r + O ( r − ) , (4)Ψ = Ψ r + Ψ r + Ψ r + Ψ r + O ( r − ) . We find that for stationary spacetimes, the timelike Killing vector field, canbe expressed as t a = T l a + n a + ¯ Am a + A ¯ m a , where T and A were computedin [15] from the Killing equations, and found to be T = 12 + Ψ r − ¯ ð Ψ + ð ¯Ψ r + ¯ ð Ψ + ð ¯Ψ r + ¯ ð Ψ + ð ¯Ψ r − | Ψ | r + O ( r − ) ,A = − Ψ r + ¯ ð Ψ r + ¯ ð Ψ r + O ( r − ) . (5) Expressed in terms of the coordinate basis, the Killing vector is t a = ∂∂u + O ( r − ) ∂∂r + O ( r − ) ∂∂ζ + O ( r − ) ∂∂ ¯ ζ . (6)For further calculations, we need expansions of the metric components. Thecontravariant metric is given by g ab = 2 l ( a n b ) − m ( a ¯ m b ) . Matrix inversion thengives the covariant metric g ab = (cid:0) r − − (¯ ð Ψ + ð ¯Ψ ) r − + (¯ ð Ψ + ð ¯Ψ ) r − + O ( r − ) (cid:1) du + (cid:18) √ ζ ¯ ζ ) r − − ð ¯Ψ √ ζ ¯ ζ ) r − + O ( r − ) (cid:19) dudζ + (cid:18) √ ζ ¯ ζ ) r − − ¯ ð Ψ √ ζ ¯ ζ ) r − + O ( r − ) (cid:19) dud ¯ ζ + 2 dudr (7)+ (cid:18) ζ ¯ ζ ) r − + O ( r − ) (cid:19) dζ + (cid:18) ζ ¯ ζ ) r − + O ( r − ) (cid:19) d ¯ ζ + (cid:18) − ζ ¯ ζ ) r + O ( r − ) (cid:19) dζd ¯ ζ. The norm λ = t a t a = 2 T − A ¯ A is λ = 1 + 2Ψ r − − (¯ ð Ψ + ð ¯Ψ ) r − + (¯ ð Ψ + ð ¯Ψ ) r − + O ( r − ) (8)Furthermore, the twist ω a = − ε abcd t b ∇ c t d has a potential ω , which is definedvia ∇ a ω = ω a and ω → r → ∞ . Observe that the sign convention alter-nates throughout the literature. A change of the sign corresponds to complexconjugation of the multipole moments. From the metric we compute( ∂∂r ) a ω a = i ( ð ¯Ψ − ¯ ð Ψ ) r − − i ( ð ¯Ψ − ¯ ð Ψ ) r − + O ( r − ) . (9)3n integration then yields ω = − i ( ð ¯Ψ − ¯ ð Ψ ) r − + i ( ð ¯Ψ − ¯ ð Ψ ) r − + O ( r − ) . (10)The equations for the other components are then satisfied due to the vacuumfield equations.Now consider a conformal compactification V of the 3-manifold of trajecto-ries of t a with metric h ab = Ω ( − λg ab + t a t b ). We want to choose Ω such thatwe can add a point Λ (the infinity point) such that h ab extends smoothly to Λ.We also demand Ω = 0 , D a Ω = 0 , D a D b Ω = 2 h ab at Λ , (11)where D a is the covariant derivative on h ab . The following choice of conformalfactor turns out to be adequate:Ω = ( r − − Ψ r − + (Ψ ) r − ) . (12)The coefficients are chosen so as to make the limit of the Ricci tensor of h ab tovanish.The coordinates r, ζ, ¯ ζ will naturally induce coordinates on V . With a slightabuse of notation we will use the same name for the induced coordinates. Notethat r will be a radial coordinate on V for large r . Unfortunately, the compo-nents of the metric h ab will not extend smoothly to Λ in the Cartesian coor-dinates corresponding to the coordinates ( R = r − , ζ, ¯ ζ ). Therefore, we needbetter coordinates to verify that our choice of conformal factor is good. Oneway to find good coordinates is to compute harmonic coordinates. Hence, we willuse asymptotically Euclidian harmonic coordinates ( x, y, z ). For computationalpurposes, we also use the corresponding spherical coordinates with complexstereographic angles. Thus, x = ρ η + ¯ η η ¯ η , y = − iρ η − ¯ η η ¯ η , z = ρ η ¯ η −
11 + η ¯ η . (13)A fairly straightforward computation gives us the new coordinates expressed interms of the old ones: ρ = r − − Ψ r − + (Ψ ) r − + O ( r − ) ,η = ζ − √ (1 + ζ ¯ ζ )Ψ r − + O ( r − ) . (14)The conformal metric and the conformal factor are then found to be h ab = dx + dy + dz + O ( ρ ) , Ω = ρ + (Ψ ) ρ + O ( ρ ) . (15)Now we easily see that Ω → ρ →
0; thus, ρ = 0 will now representthe infinity Λ on our 3-manifold. The smoothness of h ab and the conditions (11)can now be easily verified. The Ricci tensor R ab of h ab is R ab = O ( ρ ). For the computation of the multipole moments, we actually do not need better coordinates,but to verify smoothness, we do. Geroch–Hansen multipole moments
Define the complex potential P = 1 − λ − iω (1 + λ + iω ) √ Ω . (16)This potential as well as the choice of sign in the definition of the twist is takenfrom [5]. There are many different possible choices of potential, but large classesof potentials do produce the same moments [14]. The Geroch-Hansen multipolemoments [8, 9] are given by the limits of P a ...a n = C (cid:20) D a P a ...a n − ( n − n − R a a P a ...a n (cid:21) , (17)as one approaches Λ. Here C [ · ] represents the totally symmetric and trace-freepart.Hence, with monopole (mass) M , dipole C a , and quadrupole Q ab expressedin Cartesian coordinates, we by definition havelim ρ → P = M lim ρ → P a = C x dx + C y dy + C z dz lim ρ → P ab = Q xx dx + Q yy dy − ( Q xx + Q yy ) dz + 2 Q xy dxdy + 2 Q xz dxdz + 2 Q yz dydz. (18)Under a translation Ω ′ = Ω(1 + xT x + yT y + zT z ) the dipole will transformlike C ′ j = C j − M T j , while the quadrupole will transform like Q ′ xx = Q xx − T x C x + T y C y + T z C z − M (cid:0) − T x + T y + T z (cid:1) ,Q ′ yy = Q yy + T x C x − T y C y + T z C z − M (cid:0) T x − T y + T z (cid:1) ,Q ′ xy = Q xy − T x C y − T y C x + M T x T y ,Q ′ xz = Q xz − T x C z − T z C x + M T x T z ,Q ′ yz = Q yz − T y C z − T z C y + M T y T z . (19)We expand Ψ , Ψ and Ψ in terms of spin-weighted spherical harmonics:Ψ = X m = − A m Y ,m = √ A − + 2 ζA − + √ ζ A + 2 ζ A + ζ A √ π (1 + ζ ¯ ζ ) , Ψ = X m = − B m Y ,m = −√ B − + √ ζB + ζ B √ π (1 + ζ ¯ ζ ) , Ψ = C. (20)Here C is real, B m and A m are complex.A series expansion of the potential yiels P = − C + 2¯ ηB − + √ η ¯ η − B − ηB √ π (1 + η ¯ η ) ρ − √ η ¯ η − η ¯ η + 1) A √ π (1 + η ¯ η ) ρ − √ η A − + ¯ η ( η ¯ η − A − − η ( η ¯ η − A + η A )4 √ π (1 + η ¯ η ) ρ + 3 C ρ + O ( ρ )(21)5ne then easily obtains the multipole moments by changing to Cartesiancoordinates and taking limits:lim ρ → P = − C, lim ρ → P a = lim ρ → D a P = √ √ π ( B − − B ) dx − i √ √ π ( B − + B ) dy + √ √ π B dz, lim ρ → P ab = lim ρ → ( D a D b P − h ab D c D c P ) = √ √ π ( √ A − A − A − ) dx + √ √ π ( √ A + 3 A + 3 A − ) dy + i √ √ π ( − A + A − ) dxdy − √ √ π A dz + √ √ π ( A − A − ) dxdz + i √ √ π ( A + A − ) dydz. (22) By comparing the limits (18) and (22), one can conclude that A − = − q π ( Q xx − Q yy + 2 iQ xy ) , B − = √ π ( C x + iC y ) ,A − = − q π ( Q xz + iQ yz ) , B = 2 √ πC z ,A = 2 q π ( Q xx + Q yy ) , B = √ π ( − C x + iC y ) , (23) A = 4 q π ( Q xz − iQ yz ) , C = − M,A = 2 q π ( − Q xx + Q yy + 2 iQ xy ) . The NP constants { G m } can then be computed from G m = Z π Z π Ψ Y ,m sin θdθdφ = Z π Z π ( Ψ − Ψ ) Y ,m sin θdθdφ. (24)Here the spin-weighted spherical harmonics are as in the definition (2). Forthe integration, the variables are changed to ( θ, φ ) via ζ = e iφ cot θ . Observethat we do not change the spin frame to be adapted to the new coordinates.Expansions of the integrands can, in principle, be taken from [15] eq (51), butthey do use a different spin-frame in that section the paper, hence it is easier toredo the calculations than translating the result.The integration gives G − = − √ π (3 C y − C x + M Q xx − M Q yy + 2 iM Q xy − iC x C y ) ,G − = − √ π ( iM Q yz − C x C z − iC y C z + M Q xz ) ,G = 2 √ π ( − C x − C y + 2 C z + M Q xx + M Q yy ) , (25) G = − √ π ( iM Q yz + 3 C x C z − iC y C z − M Q xz ) ,G = − √ π (3 C y − C x + M Q xx − M Q yy − iM Q xy + 6 iC x C y ) . As expected, this is the same form as in the original paper by Newman andPenrose [12], i.e., linear combinations of dipole squared and monopole times6uadrupole. From the translation rules (19), it is easy to see that the NPconstants are invariant under translations. Hence, they are independent of thechoice of conformal factor. As the NP constants are expansion coefficients forspin-weighted spherical harmonics, they will depend on the spin-frame though.For the axisymmetric case, we see that G − = G − = G = G = 0 and G = 2 √ π (2 C z − M Q zz ), where Q zz = − Q xx = − Q yy is the zz -componentof the quadrupole.We can conclude that the NP constants are, in general, not zero, but for someimportant solutions they are. For instance, the Kerr solution has C z = iM a , Q zz = − Q xx = − Q yy = − M a , and all other components of C a and Q ab arezero. This yields the well-known fact that all NP constants are zero for the Kerrsolution. In fact, they are zero for all stationary, algebraically special solutions[15]. This work was supported by the Wenner-Gren foundations. Thanks to Juan A.Valiente Kroon, for helpful discussions. I would also like to thank Lars Anders-son for asking about the relation between multipole moments and Newman–Penrose constants.
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