arXiv:1505.07225v1 [gr-qc] 27 May 2015 h Hsi scluae ypromn h following the performing by calculated approach is this spin In BH gen- [10]. on the the horizons based and dynamical is [9] to method formalism eralization This horizon isolated [8]. so-called in the the described of momentum is angular horizon the BH of magnitude paper. the this measure in present we space. work in the direction the inspired its both also study and measure a spin sim- Such to BH of the necessary kind of obviously such magnitude of is analysis it quantitative order ulations, In a out [7]. torus accretes carry the it to from as momentum angular undergoes and BH mass the nutation the and investigating precession BHs, around disks of accretion gravitating simulations have relativity we Recently, numerical simulations. performed bi- merger in of star encountered form neutron commonly the nary situation in a e.g. disks, as, in- accretion matter spacetimes by non-vacuum surrounded in BHs volving role non-negligible a play its after plane orbital the from [6]. displaced where formation is super-kicks hang- so-called BH the orbital final of the the presence the of and occurrence [5] signif- up the to as The led discoveries, has vectors. ratio icant parameters spin initial initial mass these their BH of of investigation the components initial six are The the recent simulations and simulations). a BBH BBH for of of therein parameters status references the and space of [4] overview parameter e.g. initial vast (see have the possible systems of the these exploration simulate in the ever- made to advances [1–3], used and methods ago numerical resources decade a computational about growing of simulations through n ftesadr ehd nnmrclrltvt to relativity numerical in methods standard the of One also may orientation its particular in and spin BH The break- merger (BBH) hole black binary the Since esrn h lc oesi ieto n3 atsa nume Cartesian 3D in direction spin hole black the Measuring 3 a-lnkIsiuefu srpyi,Karl-Schwarzsch f¨ur Astrophysik, Max-Planck-Institute xdt asinnra oriae,wielaigtecoor the curvature leaving extrinsic while the coordinates, and normal Gaussian to fixed ASnmes 42.m 53.f 97.60.Lf 95.30.Sf, 04.25.Dm, coordinates. numbers: Cartesian PACS of use explicit expressio the integral on in rely present not normally is which invariance, nate nerli oito dpe oteaiymtyo h spa lapse the evolved of the axisymmetry on the depend gauge to explicitly this adapted not in foliation no a pseudotensor Gaussian surfac in the using the integral of when from integration surface derived the horizon be Moreover, apparent can the spin over hole sor black a of components eso httes-aldfltsaerttoa iln vec Killing rotational flat-space so-called the that show We .INTRODUCTION I. 1 eatmnod srnmı srfıia nvria d Astrof´ısica, Universitat Astronom´ıa y de Departamento /Ctd´tc o´ et´n2 68,Ptra(Val`enci Paterna 46980, Catedr´atico Jos´e Beltr´an 2, C/ aslo Mewes, Vassilios r oie 0 60,Brast(Val`encia), Spain Burjassot 46100, 50, Moliner Dr. 2 bevtr srnoi,Uiesttd Val`encia, de Astron`omic, Universitat Observatori K ij nhne.Sc ag xn nostemto ihcoordi- with method the endows fixing gauge Such unchanged. 1 tilted o´ .Font, Jos´e A. simulations self- α n shift and l-t.1 54,Grhn e M¨unchen, Germany bei Garching 85748, 1, ild-Str. n ntvco omlt h horizon, the to normal vector unit ing h oio hthst edtrie ueial (see numerically finding of determined method be a to for has [8] that horizon the ehdcnb eie ypromn ufc nerlof integral surface a performing by derived be can to method have not timeslice. each does on it numerically therefore found constant, be is and alytically field vector Killing approximate The ishes). where uvtr ntehrznsraeand surface horizon the on curvature w onsweeteailysmercvector the symmetric (i.e. horizon axially coor- de- the the the on simply where poles to points two is tangent two the vector direction joining unit line spin Euclidean dinate BH the the as [11]. fined approach by suggested this approach In the by measured commonly simulation. the relativity of numerical frame reference a Cartesian of 3D grid the direction computational in the spin give BH however, the not, of does method This ment. iln etrmto a h rcia datg that advantage flat-space practical The the vector has the the vector. method from resulting vector magnitude the Killing its of and norm spin Euclidean BH Cartesian coordi- the the of calculate flat-space to components the vectors using Killing finding direction, rotational distorted. for nate and too method become magnitude another as not spin present results does satisfactory horizon [11] gives BH Moreover, and the slice as of initial param- long definition spin the Bowen-York This on the eter zones. reproduces typi- grid vector is angular spin direction few the spin a the about in cally accuracy co- the spherical-polar and using ordinates numerically obtained is horizon ufc nerlo h paethrzn(H fteBH the of (AH) horizon apparent the on integral surface ,2 1, nti ae eso o h a-pc iln vector Killing flat-space the how show we paper this In h ieto fteB pni ueia eaiiyis relativity numerical in spin BH the of direction The β n er .Montero J. Pedro and suigWibr’ suoesr sthey as pseudotensor, Weinberg’s using ns i ψ ntersetv ielc,a hyare they as timeslice, respective the on o ehdfrmauigteCartesian the measuring for method tor a ilsteKmraglrmomentum angular Komar the yields iaelbl ftesailmetric spatial the of labels dinate eie sarsl,temto does method the result, a As cetime. sa prxmt oainlKligvco on vector Killing rotational approximate an is mlcodntsi h integration. the in coordinates rmal ψ nerlo enegspseudoten- Weinberg’s of integral e a J AH sdi h ufc nerl( integral surface the in used = 8 Val`encia,e ) Spain a), 1 π Z S ψ ψ 3 a a ), R ia relativity rical b R K b ab steotadpoint- outward the is dS
K dS, ab stesraeele- surface the is γ steextrinsic the is ij 1 sgvnan- given is ) ψ a ψ nthe on a van- (1) 2

Weinberg’s energy-momentum pseudotensor [12]. By us- and indices of linearized quantities are raised and lowered ing the 3+1 split of spacetime and Gaussian coordinates, with ηµν . it is possible to express the angular momentum of a given Using the pseudotensor, the volume integrals giving volume using Weinberg’s energy-momentum pseudoten- the total four-momentum of the volume are given by sor in a simple form that allows for a straightforward i0µ calculation of the spin vector of the BH horizon. Wein- µ 0µ 3 − 1 ∂Q 3 P = τ d x = i d x. (5) berg’s energy-momentum pseudotensor is a symmetric ZV 8π ZV  ∂x  pseudotensor derived by writing Einstein’s equations µν using a coordinate system that is quasi-Minkowskian, Furthermore, the pseudotensor τ defined by Eq. (3) i.e. with the four-dimensional metric gµν approaching the is symmetric, which allows one to use it to calculate the Minkowski metric ηµν at infinity. Although it is not gen- total angular momentum in a volume V using the follow- erally covariant, the pseudotensor is Lorentz covariant, ing volume integral: and with the appropriate choice of coordinates it pro- vides a measure of the total angular momentum of the J µν = xµτ 0ν − xν τ 0µ d3x Z system. In the following Greek indices run from 0 to 3 V
i0ν i0µ while Latin indices run from 1 to 3. We use geometrized − 1 µ ∂Q − ν ∂Q 3 = x i x i d x. (6) units (G = c = 1) throughout. 8π ZV  ∂x ∂x  As Weinberg remarks, the physically interesting Carte- II. 3+1 SURFACE INTEGRAL OF sian components of the angular momentum contained in WEINBERGS’S PSEUDOTENSOR IN GAUSSIAN the volume are COORDINATES 23 31 12 Jx ≡ J , Jy ≡ J , Jz ≡ J . (7) In this section, we first briefly review the calculation Using Gauss’ law the volume integral can be trans- of the angular momentum contained in a volume using formed to the following surface integral over the bound- Weinberg’s pseudotensor. Next, we express the resulting ing surface: surface integral in terms of the 3+1 spacetime variables on a given timeslice. Finally, we show that by choosing ij − 1 − ∂h0j ∂h0i ∂hjk Gaussian coordinates the integral reduces in complexity J = xi + xj + xi 16π ZZ  ∂xk ∂xk ∂t and is analytically equivalent to the flat-space rotational S ∂hik k Killing vector method. − xj + h0j δki − h0iδkj n dS, (8) ∂t 

i A. Angular momentum with Weinberg’s where n is the unit normal to the surface of integration pseudotensor and dS the surface element. The convergence of the four-momentum volume inte- µν Weinberg’s energy-momentum pseudotensor is ob- grals (5) involving the pseudotensor τ critically de- tained by writing the Einstein equations in a coordinate pends on the rate at which the metric gµν approaches system that is quasi-Minkowskian in Cartesian coordi- the Minkowski reference metric at large distances. Given the following behaviour of h as r → ∞, nates, so that the metric gµν approaches the Cartesian µν − Minkowski metric ηµν = diag( 1, 1, 1, 1) at infinity as O −1 follows hµν = (r ), ∂hµν − g = η + h , (2) = O(r 2), µν µν µν ∂xσ (9) 2 where hµν does not necessarily have to be small every- ∂ h − µν = O(r 3), where. Then, by writing the Einstein equations in parts ∂xσ∂xρ linear in hµν , one arrives at an energy-momentum pseu- µν 1 dotensor τ , which is the total energy-momentum “ten- where r = (x2 + y2 + z2) 2 , it can be shown that the sor” of the matter fields, Tλκ, and of the gravitational energy-momentum “tensor” of the gravitational field, field, tλκ, tµν , behaves at large distances as

1 ∂ − τ µν = ηµληνκ [T + t ]= Qσµν , (3) t = O(r 4), (10) λκ λκ 8π ∂xσ µν where Qσµν is the superpotential given by which in turn shows that the four-momentum volume λ λ λµ integral (5) converges. The convergence of the total an- σµν 1 ∂hλ σν − ∂hλ µν − ∂h σν Q = η η λ η gular momentum volume integral (6) and of the corre- 2 ∂xµ ∂xσ ∂x sponding surface integral (8) is more problematic, due to λσ µν σν µ ∂h µν ∂h − ∂h the appearance of x in the volume integral. This is also + λ η + , (4) ∂x ∂xσ ∂xµ  observed in the convergence properties of the integrals 3 of the ADM quantities [13], where the surface integrals where Lβ is the Lie derivative with respect to the shift for the ADM mass and linear momentum converge when vector βi, to see that the time derivative of the spatial imposing fall-off conditions like those of Eq. (9), while metric ∂γij /∂t in Gaussian coordinates is simply the calculation of the ADM angular momentum gener- ∂γ ally requires stronger asymptotic fall-off conditions [14]. ij = −2K . (17) We shall return to the issue of the convergence of Eq. (8) ∂t ij after we have expressed it in terms of the 3+1 variables and in Gaussian normal coordinates in the next section. Substituting Eq. (17) in Eq. (15), we find that 1 J ij = (x K − x K ) nkdS. (18) 8π i jk j ik B. The angular momentum pseudotensor integral ZZS in Gaussian coordinates Finally, using Eq. (7), the three components of the Cartesian angular momentum vector of a volume are We can express the total angular momentum given by given by Eq. (8) in Gaussian normal coordinates (also called syn- chronous coordinates), which represent free-falling ob- 23 1 k Jx = J = (yK3k − zK2k) n dS , servers. We start by doing a 3+1 decomposition of the 8π ZZS four-dimensional metric g , µν 31 1 k Jy = J = (zK1k − xK3k) n dS , (19) 2 i j 8π ZZS −α + βiβ γij β gµν = j , (11) 12 1 k  γij β γij  Jz = J = (xK2k − yK1k) n dS . 8π ZZS i where α is the lapse function, β the shift vector, and γij Introducing the components of the three Cartesian the spatial metric induced on the hypersurface. From the Killing vectors of the rotational symmetry of Minkowski requirement that the metric gµν approaches Cartesian space Minkowski space at infinity (2), we see that ξ = (0, −z,y) − 2 i j x α + βiβ +1 γij β − hµν = j . (12) ξy = (z, 0, x) (20)  γij β γij − δij  ξz = (−y, x, 0) If we now express the angular momentum surface inte- gral, Eq. (8), in terms of the 3+1 variables we find that we can rewrite the surface integrals of the three Cartesian J ij can be written as components of the angular momentum in the following way: m m ij − 1 − ∂(γjmβ ) ∂(γimβ ) 1 J = xi k + xj k j k 16π ZZS  ∂x ∂x Ji = Kjk(ξi) n dS . (21) 8π ZZS ∂(γ − δ ) ∂(γ − δ ) + x jk jk − x ik ik i ∂t j ∂t Thus, Weinberg’s identification of the (2,3), (3,1) and (1,2) components as being the physically interesting ones m m k + γjmβ δki − γimβ δkj n dS. (13)  is now clearly seen from Eq. (21), as it is the rotational Killing vectors of Minkowski space that enter in the cal- Moreover, in terms of the 3+1 variables, Gaussian co- culation of the Cartesian components of the total angular ordinates are defined by the following choice of the lapse momentum of the volume. and shift vector: Note that this form of the angular momentum is re- markably similar to that of the ADM angular momentum α =1, βi =0, (14) [14]: 1 so that h00 = h0i = hi0 = 0. In this gauge, Eq. (13) J = lim (K − Kγ ) (ξ )j nkdS . (22) i →∞ jk jk i considerably simplifies to 8π r ZZS If the integration is done over a sphere, the components ij 1 ∂γjk ∂γik k J = − xi − xj n dS. (15) of the surface normal nk are given by 16π ZZS  ∂t ∂t  i x y z We can now use the definition of the extrinsic curvature n = , , , (23)  r r r  Kij , j k so that (ξi) and n are orthogonal vectors, 1 ∂γij Kij = − − Lβγij , (16) j k k 2α  ∂t  γjk(ξi) n = (ξi) nk =0 ∀ i. (24) 4

Therefore, the part of the integral containing the trace of expressions in a numerical relativity 3D Cartesian code Kij in Eq. (22) vanishes for spherical surfaces and there- based on the 3+1 decomposition. For instance, if using fore equations (21) and (22) are identical. We have thus the widely adopted BSSN formulation [18–20], the extrin- shown that by using Weinberg’s pseudotensor in Gaus- sic curvature Kij of the spatial slices is closely related to sian coordinates we obtain the total ADM angular mo- one of the evolved variables, namely the traceless part mentum evaluated at spatial infinity, when the integra- of the conformally related extrinsic curvature. We note tion surface is a sphere. We might still need to impose that in present-day numerical relativity simulations one a stricter asymptotic behaviour than the asymptotic Eu- does not typically use Gaussian coordinates for the ac- clidean flatness of [14] (for instance the quasi-isotropic tual numerical evolutions. This has to do with the fact or asymptotic maximal gauge), but as [15] noted, the that Gaussian coordinates can only be used in the close j k Kjk(ξi) n part of Eq. (22) converges in practice. We vicinity of a spatial hypersurface, as the geodesics ema- are, however, interested in evaluating Eq. (21) quasi- nating from the hypersurfaces will eventually cross and locally, that is, associated with finite 2-surfaces (in our form caustics in a finite time [15]. Furthermore the foli- actual applications, these will be apparent horizons of ation is not singularity-avoiding, which means Gaussian black holes [7]). coordinates are unsuitable for the numerical evolution of For an axisymmetric spacetime, the angular momen- spacetimes containing curvature singularities. Instead, tum can be calculated via the so-called Komar angular the gauge conditions most commonly employed today in momentum [16], which is defined as (following again the numerical relativity belong to the family of the so-called notation of [14, 15]): moving puncture gauges, which consist of the “1+log” condition for the lapse function [21] and the Gamma 1 µ ν driver condition for the shift vector [22]. However, one JK = ∇ φ dSµν , (25) 16π ZZS can use the numerical solution for the extrinsic curvature K in Eq. (21) due to the freedom to choose any gauge ν ij where φ is the axial Killing vector. Note the extra factor for calculations done on each timeslice. of 2 in the denominator, known as Komar’s anomalous In addition, Eq. (21) is actually equivalent to the factor [17]. The Komar angular momentum integral does method proposed by [11] for the calculation of the an- not have to be evaluated at spatial infinity, but is valid gular momentum of a volume using flat-space coordinate for every surface. In [14, 15] it is shown that using a rotational Killing vectors (cf. Eq. (20)). To see this, con- slicing adapted to the axisymmetry of the spacetime, and sider the definition of the Killing vectors in Cartesian expressing Eq. (25) in terms of the 3+1 variables, the coordinates given by [11]: Komar angular momentum becomes a ψ = [0, −(z − zc), (y − yc)], 1 i k x JK = Kij φ n dS. (26) a − − − 8π ZZS ψy = [(z zc), 0, (x xc)], (27) a ψ = [−(y − yc), (x − xc), 0], In [15] the above integral is evaluated for a Kerr BH z in spherical Boyer-Lindquist coordinates, and the angu- where (x ,y ,z ) is the coordinate centroid of the appar- lar momentum is found to be J = Ma, as expected, c c c K ent horizon, which has to be subtracted to avoid including where M and a are the black hole mass and spin pa- contributions from a possible orbital angular momentum rameter, respectively. As the two integrals (21) and (26) of the BH about the center of the computational grid in have exactly the same structure, and the latter is co- the calculation of its spin. Upon substituting their flat- ordinate (but not foliation) invariant, we arrive at the space coordinate rotational Killing vectors into Eq. (1), conclusion that the introduction of Gaussian coordinates we find that has led to a coordinate invariant expression for the an- gular momentum derived from Weinberg’s pseudotensor, 1 b namely the Komar angular momentum. Note the ab- Jx = (yK3b − zK2b) n dS , 8π ZZ sence of the anomalous factor of 2 in our final expression S 1 b (21). It therefore seems that it is possible to relax the Jy = (zK1b − xK3b) n dS , (28) restriction of using Cartesian coordinates in calculations 8π ZZS involving Weinberg’s pseudotensor. 1 b Jz = (xK2b − yK1b) n dS , 8π ZZS

C. Measuring the angular momentum in numerical where we have set xc = yc = zc = 0 for simplicity. We see relativity simulations that the two sets of expressions for the Cartesian compo- nents of the angular momentum vector of the AH, those It is easy to check that not only the choice of Gaussian from Weinberg’s pseudotensor evaluated in Gaussian co- coordinates simplifies the calculation of the total angular ordinates and those from the flat space rotational Killing momentum via Weinberg’s pseudotensor, but also that it vector method, are equivalent and equal to the Komar makes straightforward the implementation of the above angular momentum in an axisymmetric spacetime. 5

III. DISCUSSION results for all times because the method is not gauge in- variant. However, as we have seen, such method can be As we have shown, the flat-space rotational Killing vec- derived from the integration of Weinberg’s total angu- tor method of [11] can be derived from Weinberg’s pseu- lar momentum pseudotensor over the apparent horizon dotensor when using Gaussian coordinates. These coor- surface when using Gaussian normal coordinates in the dinates have two interesting properties that make them integration. As a result, the method does not depend on particularly useful for the evaluation of the angular mo- the evolving lapse and shift, as the gauge is fixed to the mentum pseudotensor integral, Eq. (8). First, as we have Gaussian normal coordinates on the respective timeslice. shown, the complicated integral (8) reduces to the much We stress that the evolution of the lapse and shift during simpler expressions given by Eq. (21) and this final ex- the free evolution of the spacetime does not enter the cal- pression is equal to the Komar angular momentum in- culation, given the coordinates evolve in such a way that tegral in a foliation adapted to the axisymmetry of the an AH is found at all times during the evolution, which system. As a result, one does not need the knowledge of is usually the case in puncture evolutions with the BSSN the shift vector and of its spatial derivatives on the sur- system. There is a dependence on the gauge evolution face of integration, which in practice would involve more via the extrinsic curvature Kij that is interpolated onto quantities that one would need to interpolate onto the the AH for the calculation of the spin direction, but the horizon surface for the calculation of the spin, thus also same is true for the expression of the spin magnitude in avoiding the numerical error associated with the com- Eq. (1). putation of the finite difference approximation to those In [24] the authors have shown that Eq. (1) will give the spatial derivatives. Second, Gaussian coordinates triv- spin magnitude provided an approximate Killing vector ially satisfy the necessary falloff conditions for the lapse can be found on the horizon and is gauge independent on and shift. Moreover, by using Gaussian coordinates we the respective time-slice if the approximate Killing vec- a recover the ADM angular momentum evaluated at spa- tor field ψ is divergence-free. Here we have shown that tial infinity, provided we use a spherical surface of inte- both methods (i.e. either via Weinberg’s pseudotensor in gration. Gaussian coordinates or via flat-space rotational Killing It is generally known that the various energy- vectors) yield the Komar angular momentum when the momentum pseudotensors proposed in the literature are latter is expressed in a foliation adapted to the axisym- not covariant and care has to be taken when evaluat- metry. We note that the restriction to axisymmetry turns ing them in different coordinate systems and gauges. out in practice not to be a major weakness, as numerical (See [23] for a review on quasi-local mass and angular relativity simulations repeatedly show that the remnants momentum in General Relativity, where the problems of binary black hole mergers and perturbed Kerr black arising when using pseudotensors for the calculation of holes typically settle down to the axisymmetric Kerr so- mass and angular momentum are also discussed.) The lution quickly [24, 25]. Moreover, both methods provide derivation of Weinberg’s pseudotensor relies crucially on a measure of the BH spin magnitude and direction that the reference space being Cartesian Minkowski. In his is not explicitly dependent on the lapse and the shift on textbook [12] Weinberg states that a spherical polar co- the respective time-slice. As both methods use the fixed ordinate system would lead to a gravitational energy den- rotational Killing vectors of Minkowski space, they mea- sity concentrated at infinity. While being non-covariant sure the spin contribution from the axisymmetry of the is generally not desirable, Weinberg’s method is em- AH. ployed in a Cartesian grid, and Gaussian coordinates guarantee the correct asymptotic behaviour of the lapse and shift, irrespective of the asymptotic behaviour the Acknowledgments evolved lapse and shift may posses, which are, as pre- viously stated, not explicitly used in the calculation of It is a pleasure to thank I. Cordero-Carri´on, the AH spin on the respective timeslice. Furthermore, D. Hilditch, J. L. Jaramillo, R. Lapiedra and E. Schnet- we have shown that Gaussian coordinates transform the ter for useful discussions. V. Mewes would like to pseudotensor angular momentum surface integral (13) to thank Ewald M¨uller and the MPA for the support dur- the Komar angular momentum integral (26) which is co- ing his visit. This work was supported by the Spanish ordinate independent. The use of Gaussian coordinates Ministry of Economy and Competitiveness (MINECO) (as an explicit gauge-fixing) seems to therefore remove through grants AYA2010-21097-C03-01 and AYA2013- the coordinate restrictions of the pseudotensor. 40979-P, by the Generalitat Valenciana (PROMETEOII- When [11] introduced the flat-space rotational Killing 2014-069) and by the Deutsche Forschungsgemeinschaft vectors for the calculation of the BH spin direction, the (DFG) through its Transregional Center SFB/TR7 authors stated that they could not guarantee the correct “Gravitational Wave Astronomy”.

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