The situation in exact, non-linear general
relativity is not so simple. Using the geometric structure of the
gravitational field near spatial infinity, the field multipoles for
stationary space-times were studied by Geroch, Hansen, Beig, Simon,
and others [97, 107, 43, 41, 40
, 42
]. They found that,
just as in electrodynamics, the gravitational field has two sets of
multipoles: The mass multipoles
and the angular
momentum multipoles
. The knowledge of these multipole
moments suffices to determine the space-time geometry in a
neighborhood of spatial infinity [41, 40, 42]. Thus, at least in the context of
stationary space-times, the field multipole moments are well
understood. However, in problems involving equations of motion, it
is the source multipoles that are of more direct interest.
It is natural to ask if these can be defined for black holes.
The answer is affirmative for black holes in
equilibrium, which can be represented by isolated horizons. For
simplicity, we will consider only type II (i.e., axisymmetric),
non-extremal isolated horizons in vacuum. The source multipoles are
two sets and
of numbers which
provide a diffeomorphism invariant characterization of the horizon
geometry.
As before, let be a cross-section
of
. We denote the intrinsic Riemannian metric on it by
, the corresponding area 2-form by
, and the derivative operator by
. Since the horizon is of type II, there exists a
vector field
on
such that
. The two points where
vanishes are called
the poles of
. The integral curves of
are natural
candidates for the ‘lines of latitude’ on
, and the lines of longitude are the curves which
connect the two poles and are orthogonal to
. This leads to an invariantly defined coordinate
- the analog of the function
in usual spherical coordinates - defined by
Recall from Section 2.1.3 that the invariant content in the
geometry of an isolated horizon is coded in (the value of its area
and) . The real part of
is proportional to
the scalar curvature
of
and captures distortions [98, 88], while the imaginary
part of
yields the curl of
and captures the angular momentum information. (The
free function
in Equation (66
) determines and is
completely determined by the scalar curvature
.) Multipoles are constructed directly from
. The angular momentum multipoles are defined as
These multipoles have a number of physically desired properties:
There is a one-one correspondence between the
multipole moments and the geometry of the
horizon: Given the horizon area
and multipoles
, assuming the multipoles satisfy a convergence
condition for large
, we can reconstruct a non-extremal
isolated horizon geometry
, uniquely
up to diffeomorphisms, such that the area of
is
and its multipole moments are the given
. In vacuum, stationary space-times, the multipole
moments also suffice to determine the space-time geometry in the
vicinity of the horizon. Thus, we see that the horizon multipole
moments have the expected properties. In the extremal case, because
of a surprising uniqueness result [143], the
are universal - the same as those on the extremal
Kerr IH and the ‘true multipoles’ which can distinguish one
extremal IH from another are constructed using different fields in
place of
[24
]. Finally, note that
there is no a-priori reason for these source multipoles to agree
with the field multipoles at infinity; there could be matter fields
or radiation outside the horizon which contribute to the field
multipoles at infinity. The two sets of quantities need not agree
even for stationary, vacuum space-times because of contributions
from the gravitational field in the exterior region. For the Kerr
space-time, the source and field moments are indeed different for
. However, the difference is small for low
[24
].
See [24] for further
discussion and for the inclusion of electromagnetic fields,
and [48
] for the numerical
implementation of these results.