Numerical computation of black hole mass and angular momentum

5.1 Numerical computation of black hole mass and angular momentum

As we saw in Section 4, the mechanics of IHs and DHs provides expressions of angular momentum and mass of the horizon. These expressions involve geometric quantities defined intrinsically on the IH $Δ$ and DH $H$ . Numerical simulations, on the other hand, deal with the 3-metric $q¯ab$ and extrinsic curvature $K¯ab$ on (partial) Cauchy surfaces $M$ . Therefore, the first task is to recast the formulas in terms of this Cauchy data.

Simulations provide us with a foliation of space-time $ℳ$ by partial Cauchy surfaces $M$ , each of which has a marginally trapped 2-surface $S$ as (a connected component of) its inner boundary. The world tube of these 2-surfaces is a candidate for a DH or an IH. If it is space-like, it is a DH $H$ and if it is null (or, equivalently, if the shear $σab$ of the outward null normal to $S$ is zero) it is a WIH $Δ$ . The situation is depicted in Figure 7 . It is rather simple to numerically verify if these restrictions are met. To calculate mass and angular momentum, one assumes that the intrinsic 2-metric on the cross-sections $S$ admits a rotational Killing field $a ϕ$ (see, however, Section 8 for weakening of this assumption). A rather general and convenient method, based on the notion of Killing transport, has been introduced and numerically implemented to explicitly find this vector field $ϕa$ [84 ].

Figure 7:

The world tube of apparent horizons and a Cauchy surface $M$ intersect in a 2-sphere $S$ . $Ta$ is the unit time-like normal to $M$ and $Ra$ is the unit space-like normal to $S$ within $M$ .

Let us first suppose that, in a neighborhood of the cross-section $S$ of interest, the world tube of marginally trapped surfaces constitutes an IH. Then the task is to recast Equation (32 ) in terms of the Cauchy data $(¯qab, ¯Kab)$ on $M$ . This task is also straightforward [84 ] and one arrives at⁵ :

where $Ra$ is the unit radial normal to $S$ in $M$ . This formula is particularly convenient numerically since it involves the integral of a single component of the extrinsic curvature.

Now consider the dynamical regime, i.e., assume that, in a small neighborhood of $S$ , the world tube of marginally trapped surfaces is a DH $H$ . The angular momentum formula (46 ) on DHs involves the Cauchy data on $H$ . However, it is easy to show [30 ] that it equals the expression (58 ) involving Cauchy data on $M$ . Thus, Equation (58 ) is in fact applicable in both the isolated and dynamical regimes. The availability of a single formula is extremely convenient in numerical simulations. From now on, we will drop the subscript $Δ$ and denote the angular momentum of $S$ simply by $(ϕ) JS$ :

$(ϕ) JS$ is the intrinsic angular momentum (i.e., spin) of the black hole at the instant of ‘time’ represented by $S$ . Note that Equation (59

) differs from the standard formula for the ADM angular momentum $JADM$ only in that the integral is over the apparent horizon instead of the sphere at infinity. Finally, we emphasize that one only assumes that $a ϕ$ is a rotational Killing field of the intrinsic 2-metric on $S$ ; it does not have to extend to a Killing field even to a neighborhood of $S$ . In fact, the expressions (32

) and (46

) on IHs and DHs are meaningful if $ϕa$ is replaced by any vector field $φ&tidle;a$ tangential to $S$ ; on $Δ$ the expression is conserved and on $H$ it satisfies a balance law. Furthermore, they both equal Equation (59

) if $a &tidle;ϕ$ is divergence-free; it has to Lie-drag only the area 2-form (rather than the intrinsic metric) on $S$ . However, since $S$ admits an infinity of divergence-free vector fields $&tidle;ϕa$ , there is no a priori reason to interpret $J(S&tidle;φ)$ as the ‘angular momentum’ of the black hole. For this interpretation to be meaningful, $&tidle;ϕa$ must be chosen ‘canonically’. The most unambiguous way to achieve this is to require that it be a Killing field of the intrinsic geometry of $S$ . However, in Section 8 we will introduce a candidate vector field which could be used in more general situations which are only ‘near axi-symmetry’.

Having calculated $J S$ , it is easy to evaluate the mass $M S$ using Equation (40 ):

This method for calculating $JS$ and $MS$ has already been used in numerical simulations, particularly in the analysis of the gravitational collapse of a neutron star to a black hole [33 , 46

]. It is found that this method is numerically more accurate than the other commonly used methods.

Comparison with other methods

The approach summarized above has the key advantage that it is rooted in Hamiltonian methods which can be universally used to define conserved quantities. In particular, it is a natural extension of the approach used to arrive at the ADM and Bondi–Sachs quantities at spatial and null infinity [7 , 17 , 185

]. From more ‘practical’ considerations, it has three important features:

The procedure does not presuppose that the horizon geometry is precisely that of the Kerr horizon.
It is coordinate independent.
It only requires data that is intrinsic to the apparent horizon.

None of the commonly used alternatives share all three of these features.

Before the availability of the IH and DH frameworks, standard procedures of calculating angular momentum were based on properties of the Kerr geometry. The motivation comes from the common belief, based on black-hole uniqueness theorems, that a black hole created in a violent event radiates away all its higher multipole moments and as it settles down, its near horizon geometry can be approximated by that of the Kerr solution. The strategy then is to identify the geometry of $S$ with that of a suitable member of the Kerr family and read off the corresponding angular momentum and mass parameters.

The very considerations that lead one to this strategy also show that it is not suitable in the dynamical regime where the horizon may be distorted and not well-approximated by any Kerr horizon. For horizons which have very nearly reached equilibrium, the strategy is physically well motivated. However, even in this case, one has to find a way to match the horizon of the numerical simulation with that of a specific member in the Kerr family. This is non-trivial because the coordinate system used in the given simulation will, generically, not bear any relation to any of the standard coordinate systems used to describe the Kerr solution. Thus, one cannot just look at, say, a metric component to extract mass and angular momentum.

A semi-heuristic but most-commonly used procedure is the great circle method. It is based on an observation of the properties of the Kerr horizon made by Smarr [170 ] using Kerr–Schild coordinates. Let $Le$ be the length of the equator and $Lp$ the length of a polar meridian on the Kerr horizon, where the equator is the coordinate great circle of maximum proper length and a polar meridian is a great circle of minimum proper length. Define a distortion parameter $δ$ as $δ = (Le − Lp)∕Le$ . The knowledge of $δ$ , together with one other quantity such as the area, $Le$ , or $Lp$ , is sufficient to find the parameters $m$ and $a$ of the Kerr geometry. However, difficulties arise when one wishes to use these ideas to calculate $M$ and $a$ for a general apparent horizon $S$ . For, notions such as great circles, equator or polar meridian are all highly coordinate dependent. Indeed, if we represent the standard two-metric on the Kerr horizon in different coordinates, the great circles in one coordinate system will not agree with great circles in the other system. Therefore, already for the Kerr horizon, two coordinate systems will lead to different answers for $M$ and $a$ ! In certain specific situations where one has a good intuition about the coordinate system being used and the physical situation being modelled, this method can be useful as a quick way of estimating angular momentum. However, it has the conceptual drawback that it is not derived from a well-founded, general principle and the practical drawback that it suffers from too many ambiguities. Therefore it is inadequate as a general method.

Problems associated with coordinate dependence can be satisfactorily resolved on axi-symmetric horizons, even when the coordinate system used in the numerical code is not adapted to the axial symmetry. The idea is to use the orbits of the Killing vector as analogs of the lines of latitude on a metric two-sphere. The analog of the equator is then the orbit of the Killing vector which has maximum proper length. This defines $Le$ in an invariant way. The north and south poles are the points where the Killing vector vanishes, and the analog of $Lp$ is the length of a geodesic joining these two points. (Because of axial symmetry, all geodesics joining the poles will have the same length). This geodesic is necessarily perpendicular to the Killing vector. Hence one just needs to find the length of a curve joining the north and south poles which is everywhere perpendicular to the Killing orbits. With $Le$ and $Lp$ defined in this coordinate invariant way, one can follow the same procedure as in the great circle method to calculate the mass and angular momentum. This procedure has been named [84 ] the generalized great circle method.

How does the generalized great circle method compare to that based on IHs and DHs? Since one must find the Killing vector, the first step is the same in the two cases. In the IH and DH method, one is then left simply with an integration of a component of the extrinsic curvature on the horizon. In the generalized great circle method, by contrast, one has to determine the orbit of the Killing vector with maximum length and also to calculate the length of a curve joining the poles which is everywhere orthogonal to the Killing orbits. Numerically, this requires more work and the numerical errors are at least as large as those in the IH-DH method. Thus, even if one ignores conceptual considerations involving the fundamental meaning of conserved quantities, and furthermore restricts oneself to the non-dynamical regime, the practical simplicity of the great circle method is lost when it is made coordinate invariant. To summarize, conceptually, Equations (59 ) and (60 ) provide the fundamental definitions of angular momentum and mass, while the great circle method provides a quick way of estimating these quantities in suitable situations. By comparing with Equations (59 ) and (60 ), one can calculate errors and sharpen intuition on the reliability of the great circle method.

A completely different approach to finding the mass and angular momentum of a black hole in a numerical solution is to use the concept of a Killing horizon. Assume the existence of Killing vectors in the neighborhood of the horizon so that mass and angular momentum are defined as the appropriate Komar integrals. This method is coordinate independent and does not assume, at least for angular momentum, that the near horizon geometry is isometric with the Kerr geometry. But it has two disadvantages. First, since the Komar integral can involve derivatives of the Killing field away from the horizon, one has to find the Killing fields in a neighborhood of the horizon. Second, existence of such a stationary Killing vector is a strong assumption; it will not be satisfied in the dynamical regime. Even when the Killing field exists, computationally it is much more expensive to find it in a neighborhood of the horizon rather than the horizon itself. Finally, it is not a priori clear how the stationary Killing vector is to be normalized if it is only known in a neighborhood of the horizon. In a precise sense, the isolated horizon framework extracts just the minimum amount of information from a Killing horizon in order to carry out the Hamiltonian analysis and define conserved quantities by by-passing these obstacles.