Measuring the black hole spin direction in 3D Cartesian numerical relativity simulations Vassilios Mewes,1 Jos´eA. Font,1, 2 and Pedro J. Montero3 1Departamento de Astronom´ıa y Astrof´ısica, Universitat de Val`encia, Dr. Moliner 50, 46100, Burjassot (Val`encia), Spain 2Observatori Astron`omic, Universitat de Val`encia, C/ Catedr´atico Jos´eBeltr´an 2, 46980, Paterna (Val`encia), Spain 3Max-Planck-Institute f¨ur Astrophysik, Karl-Schwarzschild-Str. 1, 85748, Garching bei M¨unchen, Germany We show that the so-called flat-space rotational Killing vector method for measuring the Cartesian components of a black hole spin can be derived from the surface integral of Weinberg’s pseudoten- sor over the apparent horizon surface when using Gaussian normal coordinates in the integration. Moreover, the integration of the pseudotensor in this gauge yields the Komar angular momentum integral in a foliation adapted to the axisymmetry of the spacetime. As a result, the method does not explicitly depend on the evolved lapse α and shift βi on the respective timeslice, as they are fixed to Gaussian normal coordinates, while leaving the coordinate labels of the spatial metric γij and the extrinsic curvature Kij unchanged. Such gauge fixing endows the method with coordi- nate invariance, which is not present in integral expressions using Weinberg’s pseudotensor, as they normally rely on the explicit use of Cartesian coordinates. PACS numbers: 04.25.Dm, 95.30.Sf, 97.60.Lf I. INTRODUCTION surface integral on the apparent horizon (AH) of the BH 1 a b JAH = ψ R Kab dS, (1) Since the binary black hole (BBH) merger break- 8π ZS through simulations of about a decade ago [1–3], ever- a growing computational resources and advances in the where ψ is an approximate rotational Killing vector on numerical methods used to simulate these systems have the horizon that has to be determined numerically (see a b made the exploration of the vast initial parameter space [8] for a method of finding ψ ), R is the outward point- possible (see e.g. [4] and references therein for a recent ing unit vector normal to the horizon, Kab is the extrinsic overview of the status of BBH simulations). The initial curvature on the horizon surface and dS is the surface ele- parameters of BBH simulations are the BH mass ratio ment. This method does not, however, give the direction and the six components of their initial spin vectors. The of the BH spin in the 3D Cartesian reference frame of the investigation of these initial parameters has led to signif- computational grid of a numerical relativity simulation. icant discoveries, as the occurrence of the orbital hang- The direction of the BH spin in numerical relativity is up [5] and the presence of the so-called super-kicks where commonly measured by the approach suggested by [11]. the final BH is displaced from the orbital plane after its In this approach the BH spin direction is simply de- formation [6]. fined as the Euclidean unit vector tangent to the coor- dinate line joining the two poles on the horizon (i.e. the The BH spin and in particular its orientation may also two points where the axially symmetric vector ψa van- play a non-negligible role in non-vacuum spacetimes in- ishes). The approximate Killing vector field ψa on the volving BHs surrounded by matter as, e.g. in the form of arXiv:1505.07225v1 [gr-qc] 27 May 2015 horizon is obtained numerically using spherical-polar co- accretion disks, a situation commonly encountered in bi- ordinates and the accuracy in the spin direction is typi- nary neutron star merger simulations. Recently, we have tilted cally about a few angular grid zones. This definition of performed numerical relativity simulations of self- the spin vector reproduces the Bowen-York spin param- gravitating accretion disks around BHs, investigating the eter on the initial slice and gives satisfactory results as precession and nutation the BH undergoes as it accretes long as the BH horizon does not become too distorted. mass and angular momentum from the torus [7]. In order Moreover, [11] present another method for finding the to carry out a quantitative analysis of such kind of sim- spin magnitude and direction, using flat-space coordi- ulations, it is obviously necessary to measure both the nate rotational Killing vectors to calculate the Cartesian magnitude of the BH spin and also its direction in space. components of the BH spin and its magnitude from the Such a study inspired the work we present in this paper. Euclidean norm of the resulting vector. The flat-space One of the standard methods in numerical relativity to Killing vector method has the practical advantage that measure the magnitude of the angular momentum of the the vector ψa used in the surface integral (1) is given an- BH horizon is described in [8]. This method is based on alytically and is constant, therefore it does not have to the so-called isolated horizon formalism [9] and the gen- be found numerically on each timeslice. eralization to dynamical horizons [10]. In this approach In this paper we show how the flat-space Killing vector the BH spin is calculated by performing the following method can be derived by performing a surface integral of 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.