Approximate Killing Vectors on S^2

Abstract

We present a new method for computing the best approximation to a Killing vector on closed 2-surfaces that are topologically S^2. When solutions of Killing's equation do not exist, this method is shown to yield results superior to those produced by existing methods. In addition, this method appears to provide a new tool for studying the horizon geometry of distorted black holes.

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arXiv:0706.0199v1 [gr-qc] 1 Jun 2007
Approximate Killing Vectors on S2
Gregory B. Cook1, ∗and Bernard F. Whiting2, †
1Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27006
2Department of Physics, University of Florida, Gainesville, FL
(Dated: February 1, 2008)
32611
We present a new method for computing the best approximation to a Killing vector on closed
2-surfaces that are topologically S2. When solutions of Killing’s equation do not exist, this method
is shown to yield results superior to those produced by existing methods. In addition, this method
appears to provide a new tool for studying the horizon geometry of distorted black holes.
PACS numbers: 02.40.-k, 04.70.Bw, 04.25.Dm, 04.70.-s
INTRODUCTION
An exact geometric sphere possesses a two parameter
family of rotational Killing vectors while even a slightly
distorted sphere may possess no Killing vectors what-
soever.Nevertheless, one could imagine defining per-
turbations of these initial vectors which would, in some
well-defined “best” sense, represent the closest available
approximation to vectors which almost satisfy Killing’s
equation on such a slightly distorted sphere. In this paper
we introduce a definition which is best in a least squared
sense and discuss some of its attributes.
In general relativity, rotational Killing vectors play an
important role in providing a quasi-local definition for
the spin of a rotating body. A system of astrophysical
interest, such as a pair of orbiting black holes, possesses
no global rotational Killing vectors. In this case, angular
momentum can only be rigorously defined for the sys-
tem as a whole in terms of asymptotic rotational Killing
vectors.
A quantity of great importance in the evolution of
black-hole binaries is the spin of the individual black
holes. The spin of such black holes can only be deter-
mined by some approximate quasi-local definition (see
Ref.[1] for a review). There exist many different quasi-
local definitions for angular momentum, but they all take
the form of an integral over a 2-surface with topology S2,
and all require a rotational Killing vector on this surface.
In numerical relativity, the quasi-local spin of a black
hole is most often expressed as
S(ξ)=
1
8πG
?
S
Kijξjd2Si, (1)
where Kij is the extrinsic curvature of a spatial hyper-
surface with metric γij, d2Siis the area element of an
S2surface of integration S taken to be a black hole’s
apparent horizon, and ξiis a Killing vector of the met-
ric hij induced on S by γij. This form was derived by
Brown and York[2] and later within the Isolated Horizons
framework (see Ref.[3] for a review). It gives the angular
momentum of the rotation associated with the rotational
Killing vector ξi. Unfortunately, the induced metric hij
will not admit a solution of Killing’s equation for the case
of orbiting black hole binaries. In this situation, one has
no recourse but to find some reasonable approximation
for the Killing vector required in Eq. (1).
In some cases, conformal Killing vectors have been
used, and have yielded physically reasonable results[4].
Better still, a “Killing Transport” (KT) technique[5],
which finds exact Killing vectors when they are present,
has been recently adopted, and appears to give physi-
cally reasonable results (cf Refs. [4, 6]). Our definition is
shown to be even better in a well-defined least squared
sense and, for coalescing binary black holes, yields other
interesting results worthy of further investigation.
EQUATIONS FOR THE BEST APPROXIMATE
KILLING VECTOR
An arbitrary vector field ξion S2can be decomposed
into two scalars d and v
ξi≡ Did + ǫijDjυ, (2)
where Diis the covariant derivative compatible with the
metric hij induced on the surface S and ǫij is the Levi-
Civita tensor. Similarly, the general gradient of a 1-form
can be expressed as
Diξj≡ Lǫij+ hijΛ + Sij, (3)
where L and Λ are scalars, and Sij is symmetric and
trace-free. Eqs. (2) and (3) immediately imply that
Λ =
L = −1
Sij = DiDjd −1
+1
1
2DiDid,
2DiDiv,
(4)
(5)
2hijDkDkd
?ǫikDjDkv + ǫjkDiDkv?.
(6)
2
For ξito be Killing, it must satisfy Killing’s equation
D(iξj)= 0 where the parentheses denote symmetrization.
This implies that Λ and Sij must vanish if ξiis Killing.
We may choose Λ to vanish, in which case Eq. (4) implies
that d is harmonic. But, assuming a non-singular metric,

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the only harmonic function on S2is a constant, which
makes no contribution to ξi. Thus, we are left with
ξi≡ ǫijDjυ,
Diξj ≡ Lǫij+ Sij.
(7)
(8)
Our goal is to find an approximate Killing vector that
minimizes the non-Killing aspects of ξi. Clearly, having
already set Λ to zero, a solution with Sijas close to zero
as possible is what we desire, so we proceed by finding a
vector ξithat minimizes SijSij.
From Eqs. (7) and (8) it follows that
L =1
2ǫijDiξj,(9)
and
SijSij= (DiDjv)(DiDjv) −1
2(DkDkv)2.(10)
Now, we wish to extremize SijSijwith respect to v. But,
we must do this in a way that is independent of the nor-
malization of ξi. Using |ξ|2= (Div)(Div), we choose the
following scalar function on S
L ≡ SijSij+1
2
2RΘ(Dkv)(Dkv), (11)
where2R is the Ricci scalar associated with hij and Θ
is a dimensionless constant. Varying with respect to v,
δL/δv = 0 yields a fourth-order scalar elliptic equation
for v that can be rewritten as a pair of second-order scalar
elliptic equations for v and L
DiDiL − (1 − Θ)?1
2(Di2R)Div −2RL?
DiDiv + 2L = 0. (13)
= 0, (12)
Eqs. (12) and (13) can be solved for L and v, with
the Lagrange multiplier Θ fixed by the requirement that
Eq. (12) be integrable on S2. Given a solution for v, the
approximate rotational Killing vector is given by Eq. (7)
once it is normalized to have affine length 2π. We note
that Eq. (12) is satisfied by a Killing vector ξiwhen Θ = 0
(with L defined by Eq. (9) and Div = (2/2R)DiL).
It has been noted[7, 8] that an approximate Killing
vector should be divergence free Diξi= 0, in part be-
cause the angular momenta computed using such an ap-
proximate Killing vector possess a certain gauge invari-
ance. Approximate Killing vectors constructed using the
KT method are not guaranteed to be divergenceless, al-
though at least in certain cases[4] this can be enforced a
posteriori. These approximate Killing vectors also inherit
an additional problem. A defining equation of the KT
method[5], which can be written in the form of Eq. (9),
is not satisfied by solutions of the KT equations unless
there is a Killing vector. This is due to the path depen-
dence of the solution scheme. Moreover, Eq. (9) cannot
be enforced a posteriori.
Because our approximate Killing vector is defined from
the solution of Eqs. (12) and (13) via Eq. (7), it is guar-
anteed to be divergenceless. Furthermore, because our
solution is obtained via a global solution of elliptic equa-
tions, Eq. (9) is also guaranteed to be satisfied.
TESTS
We have implemented a code to solve Eqs. (12) and
(13) for situations where the metric hijis conformal to a
unit 2-sphere:
ds2= ψ4r2(dθ2+ sin2θdφ2).(14)
Details of the solution scheme will be presented in a fu-
ture paper[9]. While this form for the metric may seem to
be a strong simplification, a scheme based on this form is
suitably general since any sufficiently smooth metric on
S2is conformally equivalent to a unit 2-sphere.
To explore our method, we first consider the case where
the conformal factor is given by an ℓ = 2, m = 0 scalar
spherical harmonic with its axis of symmetry rotated to
a direction given by (θ′,φ′)
ψ(θ,φ) = A + B
2
?
m=−2
Y2m(θ,φ)Y∗
2m(θ′,φ′), (15)
and A and B are real constants chosen to guarantee that
ψ > 0 everywhere. The form of Eq. (15) guarantees that
the metric possesses a rotational Killing vector and in
all the cases we attempted, solving Eqs. (12) and (13)
returned a solution where the axis of symmetry was cor-
rectly rotated by (θ′,φ′) and for which Θ = 0. As men-
tioned above, the solutions are divergenceless and satisfy
Eq. (9) to the level of roundoff error. Furthermore, we
find SijSij= 0 also to the level of roundoff error, as
expected for a solution that yields a true Killing vector.
Interestingly, Eqs. (12) and (13) in general have multi-
ple solutions. In fact for the example given by Eq. (15),
there exist an infinitely degenerate set of solutions where
the axis of approximate symmetry lies anywhere in the
rotated equatorial plane. For these solutions, Θ ?= 0 and
SijSij?= 0 since these solutions are not true Killing vec-
tors. Again, the solutions are divergenceless and satisfy
Eq. (9) to the level of roundoff error.
We have also tested the system of equations against
the numerically generated initial data for corotating and
non-spinning equal-mass black-hole binaries as described
in Ref.[4]. We use data for different orbital separations,
parameterized by the dimensionless orbital angular ve-
locity MΩ0, where M is the total irreducible mass of
the binary. For the case of corotating black holes, the
spin of each black hole is aligned with the direction of
the orbital angular momentum. In all cases tested, we
have found that the measured spins based on an approx-
imate Killing vector obtained using Eqs. (12) and (13) are
nearly identical to those based on the KT method, with
differences growing only to a few parts in 107. While
the resulting spins are nearly identical, our solutions are
measurably different from the results of the KT method,
and we find that our system of equations produces so-
lutions for which ?SijSij? ≡ (4π)−1?SijSijdΩ is always

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0
0.025 0.050.075
0.1
MΩ0
0
5⋅10-6
10-5
1.5⋅10-5
<SijSij>
CKV
KT
AKV
Θ
-5⋅10-6
-4⋅10-6
-3⋅10-6
Θ
-2⋅10-6
-10-6
0
0
0.025 0.05 0.075
0.1
10-7
10-6
10-5
10-4
FIG. 1: The value of ?SijSij? for the new method is dis-
played as a solid(black) line and its value when using the KT
method is shown as a dashed(red) line. The analogous quan-
tity ?2Λ2? when using conformal Killing vectors is shown as
a dot-dot-dashed(black) line visible in the inset where a loga-
rithmic scaling is used. The value of the Lagrange multiplier
Θ obtained using the new method is also displayed as a dot-
dashed(blue) line and its scale is shown on the right. All data
is from one corotating black hole in an equal-mass binary.
0
0.5
1
1.5
radians
0
20
40
60
80
degrees
φ1
φ2- π/2
0
0.0250.05 0.075
0.1
MΩ0
-5⋅10-6
0
5⋅10-6
10-5
∆φ(radians)
0.0172 0.01722
0
0.5
1
1.5
FIG. 2: The top portion of the figure displays the directions
of the additional approximate symmetry axes with φ = 0 in
the positive x direction. The axis direction of the Solution 1
is displayed as a solid(black) curve and the direction for So-
lution 2, with π/2 subtracted, is displayed as a dashed(red)
line. The lines are visually coincident. The lower portion of
the figure displays ∆φ ≡ φ1−φ2+π/2 showing the degree to
which the two axes are approximately orthogonal. All data is
from one corotating black hole in an equal-mass binary.
smaller than that produced by the KT method. Note
that in both cases, the approximate Killing vectors have
been properly normalized to have affine length 2π. Fig-
ure 1 shows ?SijSij? as computed by our method and the
KT method, and the analogous quantity ?2Λ2? obtained
by using conformal Killing vectors. On a separate scale,
Fig. 1 also shows the value of Θ obtained by our method
for the same data.
As with the first example given by Eq. (15), our equa-
tions yield multiple solutions for the numerically gen-
erated black-hole data. The solutions shown in Fig. 1
have their axis of approximate symmetry aligned with
the orbital angular momentum (z-axis) as expected. In
all cases, we find two additional solutions. Both have
their axis of approximate symmetry in the orbital plane
and we refer to them as Solutions 1 and 2.
that for all orbital separations, the directions of these
two symmetry axes are orthogonal to the level of nu-
merical error in the code (truncation error). For large
separation (small MΩ0), Solution 1 has its axis of ap-
proximate symmetry pointed roughly in the direction of
motion of the black hole (y-axis) at an azimuthal angle
of φ1, and Solution 2 has its axis pointed roughly toward
the companion black hole (x-axis) at an angle of φ2. The
directions φ1and φ2(mod π) for these two solutions are
displayed in Fig. 2. Here, zero azimuthal angle is in the
direction of the positive x-axis. For small separations,
the roles are reversed and we find that Solution 1 points
roughly toward the x-axis and Solution 2 toward the y
axis.
The regime between large and small orbital separa-
tions, where the two solutions swap orientations, is quite
interesting. Over most of the range of MΩ0considered,
φ1 and φ2 change gradually. However, over a narrow
range of MΩ0, the angles change rapidly but smoothly,
rotating by an angle of approximately π/2. Interestingly,
the value of ?SijSij? for Solution 1 is smaller than that
for Solution 2 for all separations. At the point where
φ1= φ2−π/2 = π/4, curves of ?SijSij? for the two solu-
tions “appear” to cross. However, a careful examination
shows this to be an “avoided” crossing. We shall examine
this behavior in more detail in a future paper[9].
Finally, if we measure the spin of the black holes us-
ing the approximate Killing vectors associate with So-
lutions 1 and 2, all cases yield zero to truncation error.
So, for the case of corotating equal-mass black hole bina-
ries, we find three “orthogonal” solutions and only one
of them yields a non-vanishing spin.
We find similar results for the case of non-spinning
black-hole-binary initial data. There is a solution with
an axis of approximate symmetry aligned with the or-
bital angular momentum and two additional solutions
with their axes in the orbital plane and similarly orthog-
onal to each other. As discussed in Ref. [4], the non-
spinning black hole data we use are defined by setting
the spin measured by the KT method to zero. The cor-
responding spins measured by our method again differ at
most by a few parts in 107, and in all cases we find the
value of ?SijSij? for the new method to be smaller than
the value from the KT solution. The behavior of the
additional solutions is qualitatively the same as seen for
the case of corotation and is displayed in Fig. 3. While
the behaviors are generally similar, the directions of the
approximate symmetry axes are somewhat different and
the rapid change in the direction of the solutions occurs
We find

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0
0.025 0.050.075
0.1
MΩ0
0
0.5
1
1.5
radians
0
20
40
60
80
degrees
φ1
φ2- π/2
FIG. 3: The directions of the additional approximate symme-
try axes from one non-spinning black hole in an equal-mass
binary. See description for Fig. 2.
at much smaller separation.
DISCUSSION
We have mentioned two previous methods defined in
the literature for use in computing the spin of rotat-
ing black holes that lack axial symmetry.
been considered useful in the past, and yet both have
shortcomings. When a Killing vector does not exist, the
conformal Killing approach returns a vector for which
Diξi?= 0. By contrast, the Killing Transport method
constructs a ξithat can often be made divergenceless. It
also constructs the scalar L (see Eq. (3)), but generally
this does not satisfy Eq. (9). Our new method not only
ensures both that ξiis divergenceless and that Eq. (9)
is satisfied, but it also is best in the sense that ?SijSij?
is minimal. Since a Killing vector cannot be produced
where one does not exist, the usefulness of our results for
an approximate Killing vector will depend on the extent
to which physical questions (such as concern black hole
spins) can be given meaningful answers. In particular, it
may have immediate application in giving a more refined
definition for binaries containing black holes without in-
dividual spin.
One sense in which our results already appear mean-
ingful relates to the vectors found to reside in our xy-
plane, for which the corresponding spins are computed to
be zero to the level of truncation error. Any other out-
come for these would have been somewhat unpalatable.
Their actual orientation for binary black holes at large
separation can be easily interpreted in terms of boosted
frames. Closer in, their combined dramatic rotation at
some critical separation warrants further investigation,
as does the apparent occurrence of avoided crossings for
?SijSij? and for Θ at the critical separation.
Another sense in which our results appear meaningful
Both have
relates to the solution associated with our z-direction.
For sufficiently large black hole separation, our result
leads to spins which are in good agreement with those of
the KT method. This is not unreasonable, even though
we find small differences between the two solutions for
ξiover the surface of the apparent horizon. However,
for highly distorted black holes — such as near merger
or with near maximal rotation, or for unaligned spins —
we can imagine that our results could prove to be more
robust. More extensive comparisons than we have been
able to carry out here will be necessary before this ex-
pectation might be practically substantiated.
To a high degree of precision, it appears that the axes
of the approximate symmetries we find form an orthog-
onal basis in all the cases we have examined. This is
of considerable interest, because this basis is directly re-
lated to intrinsic properties of the apparent horizons we
have studied. Thus, in addition to the reasonableness of
our results associated with both the rotation axis and the
orbital plane, our method appears to give a new tool for
studying the horizon geometry of distorted black holes
in general, and the volatile dynamics of black holes dur-
ing collision in particular. Further investigation of the
usefulness of this new tool will be forthcoming.
We wish to thank for their hospitality the Yukawa
Institute and the organizers of the Post-YKIS2005
mini-workshop on the ‘Frontiers of Gravitational Wave
Physics’ where this work was started. G.B.C. acknowl-
edges support by NSF grant PHY-0555617 and the Z.
Smith Reynolds Foundation. B.F.W. acknowledges sup-
port by NSF grant PHY-0555484. Computations were
performed on the Wake Forest University DEAC Clus-
ter.
∗Electronic address: cookgb@wfu.edu
†Electronic address: bernard@phys.ufl.edu
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