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, 169, 40, 39, 41]. 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 [40, 39, 41]. 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, 87], 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 , 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 . 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 .
See  for further discussion and for the inclusion of electromagnetic fields, and  for the numerical implementation of these results.
© Max Planck Society and the author(s)