Black hole multipole moments

5.3 Black hole multipole moments

Let us begin by considering the notion of source and field multipole moments in Newtonian gravity and in flat space electrodynamics. Field multipoles appear in the asymptotic expansions of the fields at infinity while the source multipoles are defined in terms of the mass or charge distribution of the source. These two sets of multipole moments are related to each other via field equations. The same is true in linearized general relativity [165 ]. Also, the well known quadrupole formula relates the rate of change of the quadrupole moment to the energy flux at infinity due to gravitational waves.

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 $Mn$ and the angular momentum multipoles $Jn$ . 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 $Mn$ and $Jn$ of numbers which provide a diffeomorphism invariant characterization of the horizon geometry.

As before, let $S$ be a cross-section of $Δ$ . We denote the intrinsic Riemannian metric on it by $&tidle;q ab$ , the corresponding area 2-form by $&tidle;ε ab$ , and the derivative operator by $D&tidle; a$ . Since the horizon is of type II, there exists a vector field $a ϕ$ on $S$ such that $ℒ ϕ&tidle;qab = 0$ . The two points where $ϕa$ vanishes are called the poles of $S$ . The integral curves of $ϕa$ are natural candidates for the ‘lines of latitude’ on $S$ , and the lines of longitude are the curves which connect the two poles and are orthogonal to $ϕa$ . This leads to an invariantly defined coordinate $ζ ∈ [− 1,1]$ – the analog of the function $cosθ$ in usual spherical coordinates – defined by

where $R$ is the area radius of $S$ . The freedom of adding a constant to $ζ$ is removed by requiring $∮ S ζ&tidle;ε = 0$ . With $φ ∈ [0,2π )$ being an affine parameter along $ϕa$ , the 2-metric $&tidle;qab$ takes the canonical form

where $a 2 f = ϕaϕ ∕R$ . The only remaining freedom in the choice of coordinates is a rigid shift in $φ$ . Thus, in any axi-symmetric geometry, there is an invariantly defined coordinate $ζ$ , and multipole moments are defined using the Legendre polynomials in $ζ$ as weight functions.

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) $Ψ2$ . The real part of $Ψ2$ is proportional to the scalar curvature $R&tidle;$ of $&tidle;qab$ and captures distortions [98 , 87 ], while the imaginary part of $Ψ2$ yields the curl of $ωa$ and captures the angular momentum information. (The free function $f$ in Equation (66 ) determines and is completely determined by the scalar curvature $&tidle;R$ .) Multipoles are constructed directly from $Ψ 2$ . The angular momentum multipoles are defined as

where $0 Yn(ζ)$ are the spherical harmonics. (Axi-symmetry ensures that $m Yn$ vanish if $m ⁄= 0$ .) The normalization factors have been chosen to ensure that the dimensions of multipoles are the physically expected ones and $J 1$ is the angular momentum of the isolated horizon. The mass multipoles are defined similarly as the moments of $&tidle; R$ :

where the normalization factor has been chosen in order to ensure the correct physical dimensions and $M0 = M Δ$ .

These multipoles have a number of physically desired properties:

The angular momentum monopole vanishes, $J0 = 0$ . (This is because we are considering only smooth fields. When $ωa$ has a wire singularity similar to the Dirac monopole in electrodynamics, i.e., when the horizon has NUT charge, then $J0$ would be non-zero.)
The mass dipole vanishes, $M = 0 1$ . This tells us that we are in the rest frame of the horizon.
For a type I (i.e., spherically symmetric) isolated horizon, $Im [Ψ2] = 0$ and $Re [Ψ2]$ is constant. This implies that $M0$ is the only non-zero multipole moment.
If the horizon is symmetric under reflections as in the Kerr solution (i.e., $Ψ2 ↦→ Ψ ⋆2$ when $ζ ↦→ − ζ$ ), then $M n$ vanishes for odd $n$ while $J n$ vanishes for even $n$ .

There is a one-one correspondence between the multipole moments ${Jn,Mn }$ and the geometry of the horizon: Given the horizon area $aΔ$ and multipoles ${Jn,Mn }$ , assuming the multipoles satisfy a convergence condition for large $n$ , we can reconstruct a non-extremal isolated horizon geometry $(Δ, qab,𝒟a)$ , uniquely up to diffeomorphisms, such that the area of $Δ$ is $a Δ$ and its multipole moments are the given ${Jn, Mn }$ . 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 $M ,J n n$ 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 $Ψ2$ [23 ]. 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 $n ≥ 2$ . However, the difference is small for low $n$ [23 ].

See [23 ] for further discussion and for the inclusion of electromagnetic fields, and [46 ] for the numerical implementation of these results.