# Mimicking Kerr’s multipole moments

Béatrice Bonga  Institute for Mathematics, Astrophysics and Particle Physics,
Radboud University, 6525 AJ Nijmegen, The Netherlands  Huan Yang  Perimeter
Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
University of Guelph, Guelph, Ontario N1G 2W1, Canada

###### Abstract

Multipole moments carry a lot of information about the gravitational field.
Nonetheless, knowing all the multipole moments of an object does not determine
conclusively the nature of the object itself. In particular, the field
multipole moments of the Kerr spacetime are not unique. Here we construct
several physically motivated Newtonian objects with multipole moments
identical to those of Kerr. Moreover, we also provide a description of how to
include post-Newtonian corrections to these objects without changing their
multipole moments.

##  I  Introduction

Ground-based gravitational-wave (GW) detectors have achieved tremendous
success with the observation of merging stellar-mass black holes (BHs) and
neutron stars (NSs). These observations are not only of astrophysical
interest, but also interesting with regard to the foundations of gravity as
they allow for stringent tests of general relativity. Such tests include
searching for additional gravitational wave polarization modes Abbott et al.
(2018); Callister et al. (2017); Isi and Weinstein (2017); Chatziioannou et
al. (2021), consistency of higher harmonics with the dominant harmonics in the
signal Dhanpal et al. (2019); Kastha et al. (2018); Islam et al. (2020),
effects of dispersion during the wave propagation indicating a non-zero mass
for the graviton Finn and Sutton (2002); Mirshekari et al. (2012); Perkins and
Yunes (2019), searches for echoes Wang and Afshordi (2018); Testa and Pani
(2018); Longo Micchi et al. (2021) and parametrized tests Yunes and Pretorius
(2009); Mishra et al. (2010); Li et al. (2012a, b); Meidam et al. (2018);
Carson and Yagi (2020); Abbott et al. (2019, 2020). If the gravitational waves
are emitted by a black hole, these tests aim to determine whether these black
holes are described by the Kerr solutions of general relativity or some black
hole solution in modified theories of gravity, or some other exotic compact
objects altogether.

In this spirit, a natural question to ask is: how unique is the exterior Kerr
solution? To make this question more concrete, how unique are the multipole
moments of the Kerr black hole? In other words, are there other (stellar)
objects with the same multipole moments as those of the Kerr spacetime? Some
remarkable results in the literature answer a related question: if you know
the multipole moments of a given spacetime, how much do you know about the
spacetime itself? In the Newtonian theory, the equivalent question is
trivially answered: knowing all multipole moments, one can directly
reconstruct the gravitational potential outside sources given that the
multipole moments are simply defined as the coefficients in the $1/r$
expansion of the gravitational potential. In the general relativistic context,
the situation is significantly more challenging, nonetheless similarly rigid
results have been established. In particular, the Geroch-Hansen multipole
moments characterize a stationary, vacuum spacetime uniquely up to isometries
Beig and Simon (1980a); Herberthson (2009).111The restriction to stationary
spacetimes is necessary for the Geroch-Hansen multipole moments to be defined.
Moreover, any stationary, asymptotically flat vacuum solution approaches the
Kerr metric at infinity Beig and Simon (1980b). These results are strong and
suggest that any vacuum spacetime with _all_ field multipole moments equal to
those of the Kerr spacetime has to be the Kerr spacetime itself. This seems to
answer the question whether there are any objects with the same multipole
moments of Kerr to the negative.

However, these results all rely on a key assumption: the absence of sources.
If we relax this condition, we show by an explicit construction that there are
many stationary, axisymmetric Newtonian objects with identical multipole
moments as those of a rotating black hole in general relativity. We also
provide a constructive algorithm to go beyond the Newtonian context and
include post-Newtonian corrections, but leave a fully relativistic
generalization for future work (see McManus (1991); Bicak and Ledvinka (1993);
Pichon and Lynden-Bell (1996); Lynden-Bell (2003); Will (2009) and references
therein for earlier attempts at finding matter sources for the Kerr
metric).222These earlier works are in the context of full non-linear general
relativity and focus mostly on matching the Kerr metric, rather than just its
multipolar structure at infinity. As a result, different approximations are
made as well as different assumptions for the matter sources. The objects
constructed in this manner satisfy the dominant energy condition $\rho\geq 0$
everywhere, although their stress is not isotropic. In particular, material
elements at different locations may not satisfy the same equation of state.
While these objects may not exist in nature, this work shows explicitly that
even if one knows all the (field) multipole moments of an object, one cannot
conclusively tell the nature of the object.

This also has theoretical implications on the conjecture put forward in
Bianchi et al. (2020), in which it is suggested that the multipole moments of
a Kerr black hole are minimal in some sense. In particular, they provide
numerical evidence that all multipole moments of a large family of horizonless
microstate geometries known as fuzzballs are larger (in absolute value) than
those of a Kerr black hole with the same mass and spin. The simple Newtonian
objects we construct show that the Kerr values are not minimal, in the sense
that Newtonian objects with the same mass and spin as a Kerr black hole may
have smaller multipole moments than the Kerr values. While their conjecture
may still hold for the class of specific modified black hole solutions they
considered, it cannot be more generically true for all possible compact
objects within general relativity.333In fact, the minimalness conjecture rests
on very specific families of microstate geometries Bah et al. (2021).
Generically, there is no such bound on the multipolar structure of microstate
geometries. The counterexamples in Bah et al. (2021) are provided in the
context of almost-BPS microstate geometries, which are slightly different from
the supersymmetric ones in Bianchi et al. (2020), but a construction of
counterexamples in the SUSY setting is entirely analogous. Moreover, the
results in Bena and Mayerson (2020, 2021) also indirectly support the idea
that the minimalness conjecture does not hold for generic perturbations off
the Kerr spacetime, at least within the framework of those papers, i.e. string
theory black holes in 4 dimensions.

In recent years there is a revolutionary development of the Post-Newtonian and
Post-Minkowskian theory based on effective field theory calculations of
scattering amplitudes Damour (2020); Bini et al. (2020a, b). In particular, it
was shown that the scattering experiment of “minimally” coupled spin fields
gives rise to the Kerr multipoles up to high orders in black hole spin Vaidya
(2015); Guevara (2019); Cachazo and Guevara (2020); Arkani-Hamed et al.
(2017); Guevara et al. (2019); Aoude et al. (2020); Chung et al. (2019). This
is a rather surprising result as “minimally coupled” spin fields and Kerr
black holes are rather different objects in nature. One possible way to build
a connection between these two systems is to require that the Kerr moments are
special in certain sense, e.g. minimal for any object with the same mass and
angular momentum. The analysis presented in this work provides counterexamples
to such an intuitive explanation, as one can construct objects with smaller
multipole moments than the Kerr values within general relativity. The
coincidence observed in Vaidya (2015); Guevara (2019); Cachazo and Guevara
(2020); Arkani-Hamed et al. (2017); Guevara et al. (2019) ought to have a
deeper origin.

This paper is organized as follows. In Sec. II, we discuss the various notions
of multipole moments and make sharp the comparison we make in this paper
regarding the multipole moments of the Kerr spacetime and our Newtonian
mimicker. Thorne’s field multipole moments for the Kerr spacetime are also
presented in that section. In Sec. III, we construct a Newtonian object whose
field multipole moments have the same value as those of the Kerr exterior
region. A generalization to include post-Newtonian corrections is described in
Sec. IV. We conclude in Sec. V. Our conventions are: We set Newton’s constant
$G$ and the speed of light $c$ both equal to one, the spacetime metric has
signature -+++ and for the normalization of the various harmonics, we use the
conventions in Thorne (1980).

##  II Many multipole moments

There are two distinct notions of multipole moments: source multipole moments,
defined as integrals over the source, and field multipole moments, defined
from the gravitational field near infinity. In Newtonian gravity, the source
multipole moments describe the way in which mass is distributed, while the
field multipole moments are important to determine the motion of extended or
nearby objects. While the definitions are distinct, the source and field
multipole moments of a single object are the same in the Newtonian context
(see e.g. Eqs. (1.139) and (1.140) in Poisson and Will (2014)). In general
relativity, the story is more complicated and generically one cannot even
define the source multipole moments rigorously in the full non-linear context
Dixon (1974). The exception are spacetimes with black holes described by
axisymmetric isolated horizons (for which the symmetry restriction only
applies to the horizon geometry and not the entire spacetime) Ashtekar et al.
(2004). Another exception is the scenario in which a post-Newtonian
description of the sources applies Thorne (1980). Field multipole moments are
well-defined for a large class of spacetimes. The Geroch-Hansen multipole
moments are an example of such field multipoles Geroch (1970); Hansen (1974).
Here we will not use the geometric definition by Geroch and Hansen, but we use
the definition due to Thorne instead Thorne (1980). We make this choice
because the Newtonian limit is more transparent in Thorne’s approach. In
addition, despite the clear difference in their definitions, Thorne’s
multipole moments of a stationary, asymptotically flat spacetime are identical
to the field multipole moments of Geroch-Hansen, up to a normalization factor
Gürsel (1983).

The post-Newtonian field and source multipole moments are equivalent when no
gravitational radiation is present. In this paper, we will use this
equivalence to relate the (post-)Newtonian _source_ multipole moments of the
objects we construct to the relativistic _field_ multipole moments of the Kerr
spacetime.

###  II.1 Multipole moments of Kerr

The mass and current multipole moments of the Kerr spacetime are only non-zero
for $m=0$ due to its axisymmetry and are given by:

$\displaystyle I^{\ell 0}_{\rm Kerr}\\!\\!$
$\displaystyle=\\!\begin{cases}\\!M(ia)^{\ell}\sqrt{4\pi}\sqrt{\frac{(2\ell+1)(\ell+1)(\ell+2)}{2\ell(\ell-1)}}\frac{2^{\ell+1}(\ell-1)!}{(2\ell-1)!(2\ell+1)}\\\
\\!0\end{cases}$ (1) $\displaystyle S^{\ell 0}_{\rm Kerr}\\!\\!$
$\displaystyle=\\!\begin{cases}\\!0\\\
\\!iM(ia)^{\ell}\sqrt{4\pi}\sqrt{\frac{(2\ell+1)(\ell+1)(\ell+2)}{2\ell(\ell-1)}}\frac{2^{\ell+1}(\ell-1)!}{(2\ell-1)!(2\ell+1)}\end{cases}$
(2)

where the first line indicates the result for $\ell$ being even and the second
for $\ell$ being odd. Here, $M$ is the mass parameter of the Kerr spacetime
while $\ell$ and $m$ indicate the degree and order of the spherical harmonic
decomposition. The above formulas are only valid for $\ell\geq 2$. For
$\ell=0$ the mass multipole moment of Kerr is simply $M$ and the current
multipole is not defined, and for $\ell=1$ the mass multipole moment vanishes
and the current multiple moment is $aM$. The vanishing of the odd mass
multipole moments and even current multipole moments is due to the reflection-
symmetry of Kerr in its equatorial plane. The dependence on $\ell$ has been
derived using the relation between the Geroch-Hansen multipole moments and
Thorne’s symmetric trace-free multipole moments in Gürsel (1983) and
translating those results to the multipole moments in the spherical harmonic
decomposition. We have explicitly checked these numerical factors up to
$\ell=6$ using the ACMC-6 coordinate system in Sopuerta and Yunes (2011).

##  III A Newtonian mimicker

In this section, we will first discuss the mass multipole moments and next
consider the spin multipole moments. Since the Kerr spacetime is stationary,
the configuration of the Newtonian object should be time-independent, i.e., no
explicit dependence on the time coordinate $t$.

###  III.1 Mass multipole moments

We start with a general Newtonian star with a density profile
$\rho(r,\theta,\phi)$. Decomposing the mass density as

$\rho(r,\theta,\phi)=\sum_{\ell,m}\rho_{\ell m}(r)Y_{\ell m}(\theta,\phi),$
(3)

the Newtonian mass multipoles are given by

$I^{\ell m}_{\rm
Newtonian}=\frac{16\pi}{(2\ell+1)!}\sqrt{\frac{(\ell+2)(\ell+1)}{2\ell(\ell-1)}}\int\\!dr\;r^{\ell+2}\rho_{\ell
m}(r)\;.$ (4)

One immediate observation of Eq. (4) is that there are many possible
$\rho_{\ell m}$ giving rise to the same set of $I^{\ell m}_{\rm Newtonian}$,
because of the nature of the integration equation. In this study we shall
focus on the axis-symmetric scenarios, with no explicit dependence on $\phi$.
As a first example, we assume a thin shell of matter with mass density
$\rho_{\ell
m}=\alpha_{\ell}\delta_{m,0}\frac{M}{R^{2}}\left(\frac{a}{R}\right)^{\ell}\delta(r-R)$
with $R$ being the radius of the thin shell, $M$ its mass and $a$ its spin
parameter. The constant $\alpha_{\ell}$ is determined by matching the
Newtonian mass multipole moments with those of the Kerr black hole

$\alpha_{\ell}=\begin{cases}i^{\ell}\sqrt{\frac{2\ell+1}{4\pi}}\\\
0\end{cases}.$ (5)

For this choice, whenever $\rho_{\ell m}(R)$ is not zero, it is positive in
roughly half of the cases ($\ell=0,4,8,12,\ldots$) and negative in the other
half ($\ell=2,6,10,14,\ldots$). Similarly, the combination $\rho_{\ell
m}(r)Y_{\ell m}(\theta,\phi)$ for given $\ell$ and $m=0$ can be negative at
specific polar angles even if $\rho_{\ell m}$ is positive. With the monopole
piece ($\ell=0$) included, the mass density itself $\rho(r,\theta,\phi)$ is
nowhere negative as long as $a/R\leq 0.57735$. This is illustrated in Fig. 1,
which shows that the mass density is concentrated near the equator and
minimized near its pole. As the spin to radius ratio increases, this feature
becomes more pronounced. The $\ell$ multipoles of the mass density of the thin
shell at the poles (where $\phi$ is undefined) is given by

$\displaystyle\rho_{\ell 0}(r=R)Y_{\ell 0}(\theta=0,\phi)$
$\displaystyle=i^{\ell}\frac{2\ell+1}{4\pi}\left(\frac{a}{R}\right)^{\ell}\;.$
(6)

When the ratio $a/R$ is increased beyond the numerical value 0.57735, the mass
density becomes negative near the poles. The restriction on the ratio $a/R$
certainly seems reasonable to assume, because if we were to take the spin
parameter to be equal to a maximally spinning Kerr black hole (i.e., $a=M$),
the compactness of this object as measured by the ratio $M/R$ would be close
to a half. In this regime, one certainly would need relativistic corrections
as this is close to the compactness of a Schwarzschild black hole.

Figure 1: This figure shows the angular dependence of the mass density $\rho$
for the thin shell model at radius $R$ for different ratios of $a/R$. It is
clear that the mass density is positive for all $a/R\leq 0.57735$. In all
cases, the mass density is peaked near the equator and minimal at the poles.
For plotting purposes, the sum over $\ell$ in Eq. (3) is truncated at
$\ell=200$. This truncation does not alter the plot as higher order terms are
suppressed (in fact, for $a/R=0.1$ truncating at $\ell=4$ does not alter the
plot, but as $a/R$ increases more terms are needed).

Other straight-forward examples with mass multiple moments equal to those of
the Kerr spacetime include a stellar object with a constant radial profile and
an elementary decaying profile:

$\displaystyle\rho_{\ell
m}=\begin{cases}\beta_{\ell}\delta_{m,0}\frac{M}{R^{3}}\left(\frac{a}{R}\right)^{\ell}&\text{constant
radial profile}\\\
\gamma_{\ell}\delta_{m,0}\frac{M}{R^{3}}\left(\frac{a}{R}\right)^{\ell}\cos\left(\frac{\pi
r}{4R}\right)&\text{decaying radial profile}\end{cases}$

with the constant $\beta_{\ell}$ and $\gamma_{\ell}$ given by

$\displaystyle\beta_{\ell}$ $\displaystyle=\left(\ell+3\right)\alpha_{\ell}$
$\displaystyle\gamma_{\ell}$ $\displaystyle=\frac{\pi^{3+\ell}}{64\cdot
2^{2\ell}}\frac{1}{\text{Re}\left[i^{\ell+1}\Gamma(\ell+3,-\frac{i\pi}{4})\right]+\Gamma(3+\ell)\sin\left(\frac{\pi\ell}{2}\right)}\alpha_{\ell}\;.$

For simplicity, we will only consider the thin shell model and the constant
radial profile from hereon. The mass density for the constant radial density
profile is non-negative for all polar angles as long as $a/R\leq 0.42583$. A
comparison with the thin shell model is shown in Fig. 2.

Figure 2: This figure shows the angular dependence of the mass density $\rho$
evaluated at $r=R$ for the thin shell model versus the constant density
profile for different ratios of $a/R$. The value of $n$ on the vertical axis
is 2 for the thin shell model and 3 for the constant density profile.

###  III.2 Current multipole moments

As a second step, we match the values of the current multipole moments. For a
generic Newtonian body, they are given by

$\displaystyle S^{\ell m}_{\rm Newtonian}$
$\displaystyle=\frac{32\pi}{(2\ell+1)!!}\sqrt{\frac{(\ell+2)(2\ell+1)}{2(\ell+1)(\ell-1)}}\times$
$\displaystyle\quad\int d^{2}\Omega\int
dr\;r^{\ell+2}\;\epsilon_{ijk}\bar{Y}^{i}_{\ell-1,\ell m}v^{j}\rho\,n^{k}$ (7)

where $n^{i}$ is the radial unit vector and $\bar{Y}^{i}_{\ell-1,\ell
m}=(-1)^{m}Y^{i}_{\ell-1,\ell-m}$ are the pure-orbital harmonics. The latter
are related to the spin-weighted spherical harmonics ${}_{-1}Y_{\ell-m}$
through

$Y^{i}_{\ell-1,\ell-m}=\sqrt{\frac{\ell+1}{2(2\ell+1)}}\left({}_{-1}Y_{\ell-m}m^{i}-_{1}Y_{\ell-m}\bar{m}^{i}\right)$
(8)

with $m^{i}$ and its complex conjugate $\bar{m}^{i}$ being the Newman-Penrose
complex null vectors on the sphere (for more details, see (Thorne, 1980, Sec.
2)). The velocity field is most intuitively decomposed into the pure-spin
vector harmonics:

$v^{i}=\sum_{\ell,m}\left[R_{\ell m}{}^{R}Y_{\ell m}^{i}+E_{\ell
m}{}^{E}Y_{\ell m}^{i}+B_{\ell m}{}^{B}Y_{\ell m}^{i}\right]$ (9)

where ${}^{R}Y_{\ell m}^{i},{}^{E}Y_{\ell m}^{i}$ and ${}^{B}Y_{\ell m}^{i}$
are normalized as in (Thorne, 1980, Eq. (2.18)). Due to the cross product of
$v^{i}$ with the radial unit vector $n^{i}$ in the definition of the current
multipole moments, $R_{\ell m}$ does not contribute to $S^{\ell m}_{\rm
Newtonian}$ and thus is not constrained by the requirement that the current
multipole moments are equal to those of Kerr. The $E_{\ell m}$ contribute an
imaginary part to the current multipole moments when $m$ is odd, and are thus
required to vanish in order to match the current multipole moments of Kerr:
$E_{\ell m}=0$. Therefore, the only relevant coefficients are $B_{\ell m}$.
Nevertheless, the coefficients $R_{\ell m}$ are not entirely free as the
continuity equation constrains its behavior. For both the thin shell model and
the constant density profile, the continuity equation implies that the
divergence of the velocity vector field has to vanish. Since the part
proportional to $B_{\ell m}$ is by construction divergence-free, this imposes
the following constraint on $R_{\ell m}$

$\frac{d}{dr}R_{\ell m}+\frac{2}{r}R_{\ell m}=0\;.$ (10)

A solution to this equation is $R_{\ell m}(r)\sim 1/r^{2}$, but this is not
well-defined at the origin and therefore we are required to set $R_{\ell m}=0$
for the model with a constant radial profile. We will also set $R_{\ell m}=0$
for the thin shell model.

The $B_{\ell m}$ part in the velocity field can also be written in terms of
the spin-weighted harmonics as:

$v^{i}_{B}=-\frac{i}{\sqrt{2}}\sum_{\ell,m}B_{\ell m}(r)\left[{}_{-1}Y_{\ell
m}(\theta,\phi)m^{i}+{}_{1}Y_{\ell m}(\theta,\phi)\bar{m}^{i}\right]\;.$ (11)

Substituting this decomposition as well as the decomposition for the mass
density in Eq. (3) into the current multipole moments in Eq. (7), and using
that $\epsilon_{ijk}n^{k}=2i\;m_{[i}\bar{m}_{j]}$, we obtain

$\displaystyle S^{\ell m}_{\rm Newtonian}$
$\displaystyle=\frac{32\pi}{(2\ell+1)!!}\sqrt{\frac{(\ell+2)(2\ell+1)}{2(\ell+1)(\ell-1)}}\;\times$
$\displaystyle\int
dr\;r^{\ell+2}\frac{(-1)^{m}}{2}\sqrt{\frac{\ell+1}{2\ell+1}}\sum_{\ell^{\prime},m^{\prime}}\sum_{\ell^{\prime\prime},m^{\prime\prime}}\rho_{\ell^{\prime}m^{\prime}}B_{\ell^{\prime\prime}m^{\prime\prime}}$
$\displaystyle\int
d^{2}\Omega\;\left({}_{0}Y_{\ell^{\prime}m^{\prime}}\;{}_{-1}Y_{\ell^{\prime\prime}m^{\prime\prime}}\;{}_{1}Y_{\ell-m}\right.$
$\displaystyle\left.\qquad\qquad+\;{}_{0}Y_{\ell^{\prime}m^{\prime}}\;{}_{1}Y_{\ell^{\prime\prime}m^{\prime\prime}}\;{}_{-1}Y_{\ell-m}\right)\;.$
(12)

The angular integrals over the two sets of three spin-weighted spherical
harmonics are given by the product of two $3j$-symbols(DLMF, , Eq. (34.3.22)).
Since $\rho_{\ell m}$ is only non-zero for $m=0$ and $\ell$ even, the
$3j$-symbols simplify significantly. Moreover, as we assume these current
multipole moments to match those of a rotating black hole, this expression can
be further simplified by setting $B_{\ell m}$ equal to zero for all $m\neq 0$
and $\ell$ even. After these simplifications, the infinite sums over
$\ell^{\prime}$ and $\ell^{\prime\prime}$ still remain:

$\displaystyle S^{\ell m}_{\rm Newtonian}$
$\displaystyle=\frac{32\pi}{(2\ell+1)!!}\sqrt{\frac{(\ell+2)(2\ell+1)}{2(\ell+1)(\ell-1)}}\times$
$\displaystyle\quad\int
dr\;r^{\ell+2}\frac{(-1)^{m}}{2}\sqrt{\frac{\ell+1}{2\ell+1}}\sum_{\ell^{\prime},\ell^{\prime\prime}}\rho_{\ell^{\prime}0}B_{\ell^{\prime\prime}0}$
$\displaystyle\quad\sqrt{\frac{(2\ell+1)(2\ell^{\prime}+1)(2\ell^{\prime\prime}+1)}{4\pi}}\begin{pmatrix}\ell^{\prime}&\ell^{\prime\prime}&\ell\\\
0&0&0\end{pmatrix}$
$\displaystyle\quad\left[\begin{pmatrix}\ell^{\prime}&\ell^{\prime\prime}&\ell\\\
0&1&-1\end{pmatrix}+\begin{pmatrix}\ell^{\prime}&\ell^{\prime\prime}&\ell\\\
0&-1&1\end{pmatrix}\right].$ (13)

Setting these current multipole moments equal to those of Kerr yields a very
large but invertible matrix equation for $B_{\ell m}$, which is in principle
solvable. Here we will only provide a perturbative solution in $a/R$ up to
$\mathcal{O}\left(\frac{a}{R}\right)^{6}$. For both the thin-shell model and
the model with a constant radial profile, we find

$\displaystyle B_{\ell 0}$
$\displaystyle=\begin{cases}\sqrt{6\pi}\frac{a}{R}\left[1-\left(\frac{a}{R}\right)^{2}+\frac{11}{7}\left(\frac{a}{R}\right)^{4}\right]+\mathcal{O}\left(\frac{a}{R}\right)^{6}\\\
\sqrt{\frac{16\pi}{21}}\left(\frac{a}{R}\right)^{3}\left[1+\left(\frac{a}{R}\right)^{2}\right]+\mathcal{O}\left(\frac{a}{R}\right)^{6}\\\
\frac{16}{7}\sqrt{\frac{2\pi}{165}}\left(\frac{a}{R}\right)^{5}+\mathcal{O}\left(\frac{a}{R}\right)^{6}\\\
\ldots\end{cases}$ (14)

with the different lines indicating the results for $\ell=1,3,5,\ldots$. (Both
models have the same velocity vector field as the extra factor of $\ell+3$ in
$\beta_{\ell}$ compared to $\alpha_{\ell}$ is canceled after performing the
integration over $r$.) Using Mathematica, these expressions are easily
obtained for much larger $\ell$ (we do not show these here as the expressions
are not particularly informative). It is clear from this particular solution
that the coefficients $B_{\ell 0}$ are all of the form

$B_{\ell
0}=\sum_{\ell^{\prime}=\ell}^{\infty}c_{\ell^{\prime}}^{(\ell)}\left(\frac{a}{R}\right)^{\ell^{\prime}}$
(15)

with $c_{\ell^{\prime}}^{\ell}$ numerical coefficients that are non-zero only
for $\ell,\ell^{\prime}$ odd. For instance, we already know from Eq. (14) that
$c_{1}^{(1)}=\sqrt{6\pi},c_{3}^{(1)}=-\sqrt{6\pi},c_{5}^{(1)}=\frac{11}{7}\sqrt{6\pi}$
and $c_{1}^{(3)}=\sqrt{16\pi/21}$.

Substituting the solution for $B_{\ell 0}$ into the expression for the
velocity field, we find that the only non-zero component of the velocity
vector field is given by

$\displaystyle v^{i}_{B}\hat{e}^{(\phi)}_{i}$
$\displaystyle=\frac{1}{2}\left[-3\frac{a}{R}+\left(\frac{a}{R}\right)^{3}\left(4-5\cos^{2}\theta\right)\right.$
(16)
$\displaystyle\quad\left.-\left(\frac{a}{R}\right)^{5}\left(6\cos^{4}\theta+\cos^{2}\theta+4\right)\right]\sin\theta+\mathcal{O}\left(\frac{a}{R}\right)^{6}$

where $\hat{e}_{i}^{(\phi)}$ is the orthonormal vector in the
$\phi$-direction. Fig. 3 shows the angular dependence of the velocity above
for different values of $a/R$ (for plotting purposes, we used expressions
accurate to $\mathcal{O}\left(\frac{a}{R}\right)^{20}$). This also shows that
the magnitude of the velocity $v^{i}$ is less than the speed of light for all
polar angles, and it vanishes at the poles.

Figure 3: This plot shows the angular dependence of $\phi$-component of the
velocity $v_{B}^{i}$ in an orthonormal basis for different ratios of $a/R$.
The maximal velocity increases as the ratio $a/R$ increases, but is always
less than the speed of light. To make this plot, we truncated the sum over
$\ell$ in Eq. (11) at $\ell=20$ and verified that the higher order terms do
not change the result. (For the model with a constant radial profile, the
ratio $a/R$ needs to be restricted to $a/R\leq 0.42583$.)

##  IV Going the extra mile: including post-Newtonian corrections

In the previous sections, we have demonstrated how to construct Newtonian
objects with identical moments to those of Kerr. The construction is not
unique and there are in fact many possible objects which share the same values
for the multipole moments as those of Kerr. In this section, we give a
constructive argument how to extend those results to include post-Newtonian
corrections.

First note that the expression for the mass multipole moments with leading
order post-Newtonian corrections involves significantly more terms than at
Newtonian order (Thorne, 1980, Eq. (5.31))

$\displaystyle I^{\ell m}$
$\displaystyle=\frac{16\pi}{(2\ell+1)!}\sqrt{\frac{(\ell+2)(\ell+1)}{2\ell(\ell-1)}}\int\\!d^{2}\Omega\int\\!dr\;r^{\ell+2}$
$\displaystyle\quad\left[\tau_{00}\bar{Y}^{\ell
m}+\frac{1}{2}\sqrt{\frac{(2\ell+1)\ell(\ell-1)}{2\ell-1}}\bar{T}_{ij}^{2\;\;\ell-2,\ell
m}\tau^{ij}\right.$
$\displaystyle\quad\left.\sqrt{\frac{6\ell(\ell-1)^{2}}{(2\ell+3)(\ell+1)(2\ell-1)}}\bar{T}_{ij}^{2\;\;\ell,\ell
m}\tau^{ij}\right]$ (17)

where $\bar{T}_{ij}^{2\;\;\ell(-2),\ell m}$ are tensor harmonics (Thorne,
1980, Eq. (2.27)-(2.28)) and $\tau_{\mu\nu}$ is the effective stress-energy
tensor evaluated at the first post-Newtonian order in the post-Newtonian de
Donger gauge so that

$\displaystyle\tau_{00}$
$\displaystyle=\rho\left(1+4U+v^{2}+\Pi\right)-\frac{3}{8\pi}\nabla_{k}U\nabla^{k}U$
(18a) $\displaystyle\tau_{ij}$
$\displaystyle=\rho\;v_{i}v_{j}-\frac{1}{4\pi}\nabla_{i}U\nabla_{j}U-\frac{1}{2\pi}U\nabla_{i}\nabla_{j}U$
$\displaystyle\quad+\left(P+\frac{3}{8\pi}\nabla_{k}U\nabla^{k}U-2\rho
U\right)g_{ij}-2\Sigma_{ij}\;.$ (18b)

Here the mass density $\rho$ is the Newtonian mass density and $v^{i}$ the
Newtonian velocity vector field. The Newtonian potential $U$ and pressure $P$
as well as the specific internal energy density $\Pi$ and stress tensor
$\Sigma_{ij}$ are determined by the Poisson, Euler and conservation of energy
equation

$\displaystyle\nabla^{2}U=-4\pi\;\rho$ (19)
$\displaystyle\nabla_{i}P-2\nabla_{j}\Sigma_{ij}=\rho\nabla_{i}U+\rho\,\Omega^{2}\left(n_{i}-r\cos\theta\;\hat{e}^{(z)}_{i}\right)$
(20) $\displaystyle\rho\;v^{i}\nabla_{i}\Pi=-P\nabla_{i}v^{i}\;,$ (21)

where $\hat{e}^{(z)}_{i}$ is the unit-vector in the $z$-direction. The second
term on the right hand side of the Euler equation is the centrifugal force for
which the angular velocity $\Omega$ is determined by
$\Omega^{2}=v_{B}^{i}v^{B}_{i}/r^{2}$ (and thus, $\Omega$ also depends on the
polar angle $\theta$). In the above equations, we restricted ourselves to the
case in which all fields are time-independent.

Since the leading order terms themselves already match the multipole moments
of Kerr, the post-Newtonian corrections have to vanish. Therefore, we need to
establish whether there is enough functional freedom to ensure this. The
strategy is to take $\rho$ as the energy density previously obtained plus some
perturbation at post-Newtonian level, say $\delta\rho$, chosen such that
exactly all the additional terms introduced at the first post-Newtonian order
vanish. Concretely, one first needs to solve for $U$ given $\rho$ in Eq. (19)
(a solution is easily constructed by decomposing $U$ into spherical harmonics
and using (Poisson and Will, 2014, Eq. (1.128)))

$\displaystyle U_{\ell m}=\frac{4\pi}{2\ell+1}$
$\displaystyle\left[r^{\ell}\int_{r}^{\infty}\rho_{\ell
m}(r^{\prime})r^{\prime-\ell+1}dr^{\prime}\right.$
$\displaystyle\left.\quad+\frac{1}{r^{\ell+1}}\int_{0}^{r}\rho_{\ell
m}(r^{\prime})r^{\prime\ell+2}dr^{\prime}\right]\;.$ (22)

This Newtonian potential will then serve as a source for the pressure $P$ and
stress tensor $\Sigma_{ij}$ in Eq. (20). Since the stress tensor is symmetric
and traceless by definition and we are interested in the axially symmetric
case, we can decompose the stress tensor simply as a linear combination of
four tensor harmonics

$\displaystyle\Sigma_{ij}$ $\displaystyle=\sum_{\ell}\left[\sigma^{(1)}_{\ell
m}\;Y_{\ell m}\;\left(g_{ij}-3n_{i}n_{j}\right)\right.$
$\displaystyle\left.\quad+\sigma_{\ell m}^{(2)}\left({}_{-1}Y_{\ell
m}\left(m_{i}n_{j}+n_{i}m_{j}\right)-\,{}_{1}Y_{\ell
m}\left(\bar{m}_{i}n_{j}+n_{i}\bar{m}_{j}\right)\right)\right.$
$\displaystyle\left.\quad+\sigma_{\ell m}^{(3)}\left({}_{-2}Y_{\ell
m}m_{i}m_{j}+\,{}_{2}Y_{\ell m}\bar{m}_{i}\bar{m}_{j}\right)\right]$ (23)

with $\sigma^{(i)}_{\ell m}=0$ whenever $m\neq 0$. In the case of the thin
shell model, the stress tensor satisfies $\Sigma_{ij}n^{j}=0$ and
$\sigma_{lm}^{(1)}=0$. The $\phi$-component of the Euler equation is trivially
satisfied, but the $\theta$\- and $r$-component yield two differential
equations. Since the source on the right-hand side of these equations is the
product of $\rho$ and $\nabla_{i}U$, we obtain $3j$-symbols again which result
into large matrix equations (similar to the situation for $B_{\ell 0}$). These
equations can also be solved to any desired order using, for instance,
Mathematica. The solutions are not unique given that there are two equations
and four free functions $P_{\ell m}$ and $\sigma_{\ell m}^{(i)}$ (three in the
case of the thin shell model). One can use this freedom to simplify the
solutions. A simple solution for $\Pi$ in Eq. (21) would be to take $\Pi=0$
(as $v^{i}$ is divergence free). Knowing all the relevant components of
$\tau_{\mu\nu}$, one can finally calculate the leading order post-Newtonian
correction to the mass multipole moments. The last step is to determine
$\delta\rho$ such that

$\displaystyle\frac{16\pi}{(2\ell+1)!}\sqrt{\frac{(\ell+2)(\ell+1)}{2\ell(\ell-1)}}\int\\!dr\;r^{\ell+2}\delta\rho_{\ell
m}(r)=-I^{\ell m}_{\rm 1PN}\;.$ (24)

The same argument applies to the current multipole moments. This algorithm
shows that the Newtonian construction in Sec. III can be easily extended to
the leading post-Newtonian order. The expectation is that this will also hold
for higher-order post-Newtonian corrections.

##  V Discussion

The field multipole moments of the Kerr spacetime are not unique: we
constructed several examples in the Newtonian theory with identical multipole
moments as those of a Kerr black hole. Therefore, knowing all the (field)
multipole moments of an object does not conclusively tell us the nature of the
object. That we were able to construct such examples is not surprising in
light of the fact that the uniqueness results mentioned in the Introduction
rests on ellipticity of the field equations. The differential equation for the
Newtonian potential is clearly elliptic. Einstein’s equations are elliptic
provided that they describe stationary vacuum spacetimes and are formulated on
the manifold of trajectories of the time-like Killing vector field (and
written in suitable coordinates)Friedrich (2007); Hagen (1970).444The
ellipticity also applies to the conformally completed spacetime constructed
for the formulation of the Geroch-Hansen multipole moments. Accordingly,
uniqueness typically fails when ellipticity of the equations is lost. The
presence of matter is one such way in which ellipticity and consequently
uniqueness no longer applies.

The construction of the explicit examples in this paper also shows that the
Kerr multipole moments are not minimal in the sense that their absolute value
is minimized for objects with the same mass and angular momentum, as the
construction in this paper can also be used to find objects with multipole
moments smaller than those of the Kerr spacetime. This point can be further
elaborated by considering an example of a solid star (so that non-isotropic
stress is allowed) with a mountain in the north pole. If the star is rotating,
the spin-induced quadrupole may perfectly cancel the mountain-associated
quadrupole, so that there is no net quadrupole moment for such star. It is
straightforward to see that such construction leads to a quadrupole moment
smaller than the Kerr value, assuming the mass and angular momentum are the
same. Of course, the “minimalness” conjecture may still hold if we restrict
the matter sources to be fluid stars.

There are several important drawbacks to this analysis. First, to mimic the
multipole moments of very compact objects with $R$ not much greater than $M$,
the Newtonian analysis is not enough and one is required to include post-
Newtonian corrections (possibly many). This complicates the analysis, but is
in principle calculable.

Second, the star profiles that mimic Kerr moments suggest materials with non-
isotropic stress and likely position-dependent equation of state, or different
types of materials at different locations. This “naturalness” problem may be
used to argue against the likelihood of finding such object(s) in nature, as
the required compositions are difficult to be naturally fabricated. It is
however worth noting that the naturalness problem can be split into two sub-
problems. The first one is whether the object is allowed by the laws of nature
(e.g., general relativity), and the second one is whether its formation is
natural, without the intervention of intelligence. Our study can only address
the first question. Despite this possible objection, it is still of
fundamental interest whether one can also construct fully relativistic objects
with the same multipole moment structure as those of the Kerr spacetime.

Third, while the objects we constructed have a non-negative mass density
everywhere and velocity smaller than the speed of light, we did not
investigate their stability under generic linear perturbations. Such a task
should be more technically complicated than analyzing the perturbation of
barotropic stars because of the non-isotropic stress and non-homogeneous
equation of state. We shall leave this for future work.

## Acknowledgments

We thank Eric Poisson for reading a draft of this paper. We also would like to
thank him and Vojtěch Witzany for pointing us to McManus (1991); Bicak and
Ledvinka (1993); Pichon and Lynden-Bell (1996); Lynden-Bell (2003); Will
(2009) and the general program of finding matter sources for the Kerr
solution. B.B. also thanks Daniel Mayerson for enlightening discussions on
microstate geometries and the minimalness conjecture. H. Y. is supported by
the Natural Sciences and Engineering Research Council of Canada and in part by
Perimeter Institute for Theoretical Physics. Research at Perimeter Institute
is supported in part by the Government of Canada through the Department of
Innovation, Science and Economic Development Canada and by the Province of
Ontario through the Ministry of Colleges and Universities.

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