Simulations provide us with a foliation of
space-time by partial Cauchy surfaces
, each of which has a marginally trapped 2-surface
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
and if it is null
(or, equivalently, if the shear
of the outward
null normal to
is zero) it is a WIH
. 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
admits a rotational Killing
field
(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
[85
].
|
Having calculated , it is easy to
evaluate the mass
using Equation (40
):
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 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 [169] using Kerr-Schild
coordinates. Let be the length of the equator and
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
as
. The knowledge of
, together with one other quantity such as the area,
, or
, is sufficient to find the parameters
and
of the Kerr geometry. However,
difficulties arise when one wishes to use these ideas to calculate
and
for a general apparent horizon
. 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
and
! 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 in an invariant way. The north and south poles are
the points where the Killing vector vanishes, and the analog of
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
and
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 [85] 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.