# Rotating metrics and new multipole moments
from scattering amplitudes in arbitrary dimensions

Claudio Gambino<EMAIL_ADDRESS>Paolo Pani
<EMAIL_ADDRESS>Fabio Riccioni<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza Università di Roma & INFN Roma1, Piazzale
Aldo Moro 5, 00185, Roma, Italy

###### Abstract

We compute the vacuum metric generated by a generic rotating object in
arbitrary dimensions up to third post-Minkowskian order by computing the
classical contribution of scattering amplitudes describing the graviton
emission by massive spin-1 particles up to two loops. The solution depends on
the mass, angular momenta, and on up to two parameters related to generic
quadrupole moments. In $D=4$ spacetime dimensions, we recover the vacuum
Hartle-Thorne solution describing a generic spinning object to second order in
the angular momentum, of which the Kerr metric is a particular case obtained
for a specific mass quadrupole moment dictated by the uniqueness theorem. At
the level of the effective action, the case of minimal couplings corresponds
to the Kerr black hole, while any other mass quadrupole moment requires non-
minimal couplings. In $D>4$, the absence of black-hole uniqueness theorems
implies that there are multiple spinning black hole solutions with different
topology. Using scattering amplitudes, we find a generic solution depending on
the mass, angular momenta, the mass quadrupole moment, and a new stress
quadrupole moment which does not exist in $D=4$. As special cases, we recover
the Myers-Perry and the single-angular-momentum black ring solutions, to third
and first post-Minkowksian order, respectively. Interestingly, at variance
with the four dimensional case, none of these solutions corresponds to the
minimal coupling in the effective action. This shows that, from the point of
view of scattering amplitudes, black holes are the “simplest” General
Relativity vacuum solutions only in $D=4$.

###### Contents

  1. I Introduction
  2. II Metric from amplitudes
    1. II.1 Spinning vertices
    2. II.2 Loop amplitudes for any spin
    3. II.3 Spin-1/2 case
  3. III Spin-1 scattering amplitudes in arbitrary dimension
    1. III.1 Minimal vertex
    2. III.2 Non-minimal vertex
    3. III.3 Loop amplitudes
  4. IV Rotating metrics in any dimensions
    1. IV.1 Eliminating redundant parameters
    2. IV.2 Multipole moments
  5. V Particular cases
    1. V.1 Hartle-Thorne metric in $D=4$
    2. V.2 Myers-Perry black holes in $D=5$
    3. V.3 Black rings
    4. V.4 The simplest metric in any spacetime $D$ dimensions
  6. VI Conclusions
  7. A Quadrupole moments in four and five spacetime dimensions
  8. B Hartle-Thorne metric in harmonic coordinates
  9. C Myers-Perry black holes in $D=5$ in harmonic coordinates
  10. D Black rings with one angular momentum in harmonic coordinates

##  I Introduction

Despite being non-renormalizable, General Relativity can be treated as a
consistent quantum field theory if viewed as the leading-order, low-energy
effective field theory arising from a quantum theory of gravity. Specifically,
the Einstein-Hilbert action is considered as the initial term in a higher-
derivative expansion, where higher-order operators are suppressed at low
energy through inverse powers of the Planck mass. Within this view, the
gravitational interactions can be computed through the exchange of spin-2
gravitons, giving rise to graviton vertices and matter interactions order by
order in a perturbative expansion in $\hbar$, as in ordinary quantum field
theories Feynman (1963); DeWitt (1967a, b, c); ’t Hooft and Veltman (1974);
Donoghue (1994a, b).

Remarkably, loop corrections in this expansion give rise not only to Planck-
suppressed quantum terms, but also to entirely classical terms that survive in
the $\hbar\to 0$ limit Iwasaki (1971); Donoghue (1994b); Bjerrum-Bohr  _et
al._ (2003); Donoghue and Torma (1996); Holstein and Donoghue (2004); Bjerrum-
Bohr  _et al._ (2018); Kosower  _et al._ (2019); Cheung  _et al._ (2018).
Specifically, the Schwarzschild metric at first order in a post-Minkowskian
(i.e., in powers of the gravitational coupling $G$) expansion can be obtained
from the scattering amplitude of a massive scalar field emitting gravitons at
tree level, while next order post-Minkowskian corrections are obtained by the
classical contributions of each $n-$loop amplitude containing graviton
vertices111Recently, the authors of Damgaard and Lee (2024) managed to resum
the post-Minkowskian series and obtain the exact Schwarzschild solution.
Bjerrum-Bohr  _et al._ (2003). This result generalizes to the case of charged
and/or spinning geometries by computing the scattering of particles with
electric charge and/or spin, reproducing the post-Minkowskian expansion of the
Reissner-Nordstrom, Kerr, and Kerr-Newman metrics in $D=4$ spacetime
dimensions Bjerrum-Bohr  _et al._ (2003); Donoghue  _et al._ (2002); D’Onofrio
_et al._ (2022); Gambino (2022).

In a more recent development, a systematic method for extracting the classical
component of loop amplitudes involving massive scalars interacting with
gravitons in any dimension was introduced in Bjerrum-Bohr  _et al._ (2018).
This procedure not only demonstrates the agreement of these computations with
the earlier work of Duff (1973) at the second post-Minkowskian order (see also
Koemans Collado  _et al._ (2018); Jakobsen (2020)), but also reveals that the
Schwarzschild-Tangherlini metric Tangherlini (1963) in generic $D$ dimensions
at fourth order in the post-Minkowskian expansion emerges from gravitational
scattering amplitudes of massive scalars up to three loops Mougiakakos and
Vanhove (2021).

In this paper we extend this program to the case of spinning geometries both
in $D=4$ and $D>4$ dimensions. The motivation for such computation is twofold.
First, in $D=4$, unlike the case of spherically symmetric spacetimes, the
absence of Birkhoff’s theorem in axisymmetry implies that the vacuum region
outside a spinning object is not necessarily described by the Kerr geometry.
Within a post-Minkowskian expansion, the leading-order (linear) angular
momentum term is universal, but different spinning objects may have different
mass quadrupole moments Hartle (1967); Hartle and Thorne (1968); Raposo  _et
al._ (2019). Since the spin-induced quadrupole moment is quadratic in the
black-hole spin222Note that, in this paper, we occasionally refer to “spin”
both for the spin $s$ of quantum fields and for the angular momentum $J$ of
the compact object (e.g., a black hole)., in order to extend the analysis of
Bjerrum-Bohr  _et al._ (2003) it is necessary to perform computations up to
quadratic order in the angular momentum. We shall show that the computation
based on the scattering amplitudes provides the post-Minkowskian expansion of
the vacuum Hartle-Thorne solution Hartle (1967); Hartle and Thorne (1968),
which describes the spacetime of a generic spinning object up to quadratic
order in the spin. The Kerr black hole is a particular case of this family,
wherein the mass quadrupole moment is fixed by regularity at the horizon and
by the black-hole uniqueness theorem Israel (1967); Carter (1971); Hawking
(1972); Robinson (1975, 2009); Hawking and Ellis (2011); Heusler (1998);
Chrusciel  _et al._ (2012). As we shall discuss, at the level of the effective
quantum field theory, the Kerr black hole corresponds to the case of minimal
couplings, while any other choice of the quadrupole moment requires non-
minimal couplings in the action. The authors of Chung  _et al._ (2019) proved
that in $D=4$ the simplest massive $S$-matrix, defined as the term in the
3-point amplitude which behaves well in the UV, reproduces the Kerr metric in
limit in which the spin $s\rightarrow+\infty$. Furthermore, the authors of
Skvortsov and Tsulaia (2023) proved that a spin-s field minimally coupled to
gravity also reproduces the dynamics of Kerr black holes, suggesting a
correspondence between the simplest massive $S$-matrix and the minimally
coupled action in a QFT description for generic spin $s$. In this sense, one
can interpret the minimality of the quantum field theory as the scattering-
amplitude counterpart of the celebrated black-hole no-hair theorems in $D=4$
General Relativity Hawking (1972); Robinson (2009); Hawking and Ellis (2011);
Chrusciel  _et al._ (2012).

Our second motivation is that the black-hole uniqueness theorems do not hold
in $D>4$ Emparan and Reall (2008), and therefore also in this case it is
interesting to compute the metric obtained from the scattering amplitudes and
compare it with known solutions. In particular, spinning black hole solutions
in $D>4$ belong to different families and can have different topologies
Emparan and Reall (2008). Using scattering amplitudes, we find the generic
solution up to third post-Minkowskian order. This solution depends on the mass
(i.e. mass monopole moment), angular momenta (i.e. current dipole moments),
mass quadrupole moment and, interestingly, on a new quadrupole moment
parameter that we dub _stress quadrupole moment_ and is absent in $D=4$. For
specific choices of the parameters, we explicitly check that this general
solution reduces to the Myers-Perry black hole Myers and Perry (1986) in $D=5$
up to third post-Minkowskian order, and to the black ring Emparan and Reall
(2002) with single angular momentum in $D=5$ up to first post-Minkowskian
order. Remarkably, at variance with the four dimensional case, none of these
solutions corresponds to the minimal coupling in the effective action. This
provides strong evidence that black holes are not the “simplest” solutions,
from a scattering-amplitude viewpoint, to higher-dimensional General
Relativity. A more fundamental explanation for this intriguing result may
deserve further investigation. We explicitly obtain the metric corresponding
to the minimal coupling up to third post-Minkowskian order, which is then
likely sourced by some matter configuration, similarly to the $D=4$ Hartle-
Thorne solution.

In order to obtain the above results we need to compute the metric from the
scattering amplitudes of a massive spin-1 field up to the quadrupole order in
a post-Minkowskian expansion. We performed our computations up to two loops
(i.e., up to third post-Minkowskian order). The full result is provided in an
ancillary Mathematica notebook.

Recently Ref. Heynen and Mayerson (2023) worked out a generalization in $D=5$
of the Thorne formalism Thorne (1980) for the multipole description in $D=4$
General Relativity. As we shall discuss in detail, the generalization is based
on constructing a suitable coordinate system in which the multipole moments
can be read off the asymptotic behavior of the metric components. Ref. Heynen
and Mayerson (2023) identified the analog of the standard mass and current
multipole moments defined by Thorne Thorne (1980) which are related to the
fall-off of the temporal part of the metric. While we agree with their
identification of mass and current moments, in any $D>4$ we prove the
existence of a new multipole moment associated with the asymptotic behavior of
the spatial part of the metric. For this reason we call this a _stress_
multipole moment, in analogy with the mass and current moments. We expect that
in $D>4$ stress multipole moments appear at any order starting from the
quadrupolar one, and we define a generalization in arbitrary dimension of the
multipole expansion á la Thorne, including the new tower of stress moments. We
show that the new stress quadrupole moment is precisely associated to one of
the free parameters of our solution and is nonzero already for the Myers-Perry
metric.

The rest of the paper is organized as follows. In Sec. II we review the
general approach to compute the metric of a rotating object from scattering
amplitudes in arbitrary dimensions. In Sec. III we explicitly apply this
approach to the scattering of massive spin-1 particles, obtaining the post-
Minkowskian metric up to two loops, including the quadrupole moments quadratic
in the object angular momentum. This metric is given and discussed in Sec. IV
in arbitrary dimensions, with a specific focus on the multipole moments in
$D=4$ and $D=5$. In Sec. V we show that the general solution recovers the
cases of the Hartle-Thorne vacuum metric in $D=4$ and the Myers-Perry solution
as well as the single-angular-momentum black ring in $D=5$. Besides, we give
an explicit expression for the “simplest” metric, i.e. the one associated with
the minimally coupled action, in any dimension. Some details on the multipolar
expansion and on these solutions are given in various Appendices. We conclude
in Sec. VI with a discussion and future prospects.

We work in mostly negative signature with $\eta_{00}=+1$ and in natural units,
$\hbar=c=1$, keeping $G\neq 1$. The number of space-time dimensions is $D=d+1$
and the Ricci tensor is defined as $R_{\mu\nu}=R^{\alpha}_{\ \mu\alpha\nu}$.
Greek indices $\mu,\nu,...=0,1,...,d$ are meant to be contracted either by
$g_{\mu\nu}$ or $\eta_{\mu\nu}$ depending on the context, while Latin indices
$i,j,...=1,...,d$ are contracted with the Euclidean $\delta_{ij}$.

##  II Metric from amplitudes

In this section we review the general approach to recover the metric of a
rotating object in arbitrary dimension from scattering amplitudes describing
the graviton emission by massive spin-$s$ particles Mougiakakos and Vanhove
(2021); D’Onofrio  _et al._ (2022). Consider an action of a generic massive
spin-$s$ field $\Phi_{s}$ coupled to gravity333Notice that we are labelling
the massive fields as one does for an object with the same index structure in
$d=3$. In higher dimensions more representation are allowed, but this will not
be taken into account in our analysis.

$S=\int
d^{d+1}x\Bigg{(}-\frac{2}{\kappa^{2}}\sqrt{-g}R+\mathcal{L}_{m}(\Phi_{s},g_{\mu\nu})\Bigg{)}\
,$ (1)

where $\kappa^{2}=32\pi G$. Expanding the metric in a post-Minkowskian (PM)
series as

$g_{\mu\nu}=\eta_{\mu\nu}+\kappa\,h_{\mu\nu}=\eta_{\mu\nu}+\kappa\sum_{n=1}^{+\infty}h_{\mu\nu}^{(n)}\
,$ (2)

in harmonic gauge the Einstein equations can be rewritten as

$\Box
h_{\mu\nu}^{(n)}(x)=-\frac{\kappa}{2}\left(T_{\mu\nu}^{(n-1)}(x)-\frac{1}{d-1}\eta_{\mu\nu}T^{(n-1)}(x)\right)\
,$ (3)

where $\Box=\partial_{\mu}\partial_{\nu}\eta^{\mu\nu}$ is the flat
d’Alambertian operator and $T=\eta^{\mu\nu}T_{\mu\nu}$. In this expression
$T_{\mu\nu}^{(0)}(x)$ is the actual stress-energy tensor of the matter source,
while $T_{\mu\nu}^{(n)}(x)$ for $n>0$ contain graviton self-interaction terms.
Then, moving to momentum space, Eq. (3) becomes

$h_{\mu\nu}^{(n)}(x)=-\frac{\kappa}{2}\int\frac{d^{d}\vec{q}}{(2\pi)^{d}}\frac{e^{i\vec{q}\cdot\vec{x}}}{\vec{q}\,^{2}}\left(T_{\mu\nu}^{(n-1)}(q)-\frac{1}{d-1}\eta_{\mu\nu}T^{(n-1)}(q)\right)\
.$ (4)

The idea now is to compute the stress-energy tensor in momentum space by means
of 3-point off-shell scattering amplitudes describing the graviton emission
from a massive source in the classical limit. The quantization procedure
requires the introduction of a gauge-fixing term

$\mathcal{L}_{GF}=\frac{1}{\kappa^{2}}F^{\lambda}F^{\sigma}\eta_{\lambda\sigma}$
(5)

in the Lagrangian, where $F^{\lambda}$ is chosen as

$F^{\lambda}=(1-\alpha)\kappa\,\partial_{\mu}\left(h^{\mu\lambda}-\frac{1}{2}\eta^{\mu\lambda}h\right)+\alpha\,g^{\mu\nu}\Gamma^{\lambda}_{\mu\nu}\
,$ (6)

with $h=\eta^{\mu\nu}h_{\mu\nu}$ and where $\Gamma^{\lambda}_{\mu\nu}$ are the
Christoffel symbols. This gauge choice allows us to move continuously from the
de Donder gauge ($\alpha=0$) to the harmonic gauge ($\alpha=1$) keeping fixed
the expression of the graviton propagator Jakobsen (2020)

$P_{\mu\nu,\rho\sigma}=\frac{1}{2}\left(\eta_{\mu\rho}\eta_{\nu\sigma}+\eta_{\mu\sigma}\eta_{\nu\rho}-\frac{2}{d-1}\eta_{\mu\nu}\eta_{\rho\sigma}\right)\
.$ (7)

Considering a massive spin-$s$ particle as source of gravitational field, the
diagrams that contribute to the classical limit are the ones in which cutting
the internal massive lines results in tree-level diagrams involving only
gravitons Duff (1973); Bjerrum-Bohr  _et al._ (2018).

It is therefore possible to compute the classical gravitational conserved
current as

$\includegraphics[valign={c},width=195.12767pt]{Images/TreeDiagramGravExt.pdf}=-\frac{i\,\kappa}{2}(2m)^{\epsilon}\,T_{\mu\nu}^{(n-1)}(q)\,\delta_{\sigma\sigma^{\prime}}\
,$ (8)

where $\epsilon=1$ for boson sources and $\epsilon=0$ for fermions444This is
due to the fact that bosons and fermions have a different mass dimension,
hence the normalization of the stress-energy tensor is different., and where
$q=p_{1}-p_{2}$ is the transferred momentum. From Eq. (8) it is clear that
$\sum_{i=1}^{n}\ell_{i}=q$ and the order of the PM expansion ($n$) and the one
of the loop series ($\ell$) are related by $n=\ell+1$. This can be directly
seen from the form of the loop integrals. Indeed, defining the $l$-loop
“sunset” master integral

$J_{(l)}(\vec{q}\,^{2})=\int\prod_{i=1}^{l}\frac{d\vec{\ell}_{i}}{(2\pi)^{d}}\frac{\vec{q}\,^{2}}{\left(\prod_{i=1}^{l}\vec{\ell_{i}}^{2}\right)\left(\vec{q}-\vec{\ell}_{1}-\cdots-\vec{\ell}_{l}\right)^{2}}\
,$ (9)

as pointed out in Mougiakakos and Vanhove (2021); D’Onofrio  _et al._ (2022),
one can show that $T_{\mu\nu}^{(l)}(q)\propto J_{(l)}(\vec{q}\,^{2})$, and the
metric in the long range expansion is given by the Fourier transform

$\int\frac{d^{d}\vec{q}}{(2\pi)^{d}}\frac{J_{(l)}(\vec{q}\,^{2})}{\vec{q}\,^{2}}e^{i\vec{q}\cdot\vec{x}}=\left(\frac{\rho(r)}{4\pi}\right)^{l+1}\
,$ (10)

with

$\rho(r)=\frac{\Gamma(\frac{d}{2}-1)\pi^{1-d/2}}{r^{d-2}}\ .$ (11)

Notice that we are using a set of Cartesian coordinates in which
$x_{1}^{2}+\cdots+x_{d}^{2}=r^{2}$.

###  II.1 Spinning vertices

Starting from Eq.(8), we have to exploit the classical limit in order to
simplify the amplitude calculation. First of all, since we are dealing with
spinning particles, the Feynman vertex

$\includegraphics[valign={c},width=130.08731pt]{Images/SpinningVertex.pdf}=(\tau_{\Phi^{2}h})^{\mu\nu}_{a,b}(q)\
,$ (12)

will have, besides graviton indices $\mu\nu$, spin indices $a$ and $b$,
depending on the particular representation of the particles. In the amplitude
(8), while the graviton is off-shell, the massive particles are on-shell,
meaning that in order to get the stress-energy tensor we have to contract the
amplitude with the external polarizations of the particles. From the vertex,
at tree level one directly reads the amplitude as

$-\frac{i\kappa}{2}(2m)^{\epsilon}\,T_{\mu\nu}^{(0)}(q)\delta_{\sigma\sigma^{\prime}}={}^{a}\bra{p_{2};s,\sigma^{\prime}}(\tau_{\Phi^{2}h})_{\mu\nu}^{a,b}\ket{p_{1};s,\sigma}^{b}\
,$ (13)

where it is important to notice that since the source particle can be either a
fermion or a boson, the polarizations can be described either by a tensor or a
spinor, so $a$ and $b$ can be a collection of Lorentz indices or spinor
indices.

Following an approach based on Feynman diagrams Bern  _et al._ (2021), which
takes into account spin effects in gravity from the classical limit of
scattering amplitudes, we now show how to obtain an explicit expression for
the tree-level amplitude in Eq. (13). To this aim, it is convenient to define

$P^{\mu}=\frac{p_{1}^{\mu}+p_{2}^{\mu}}{2}$ (14)

so that

$p_{1}^{\mu}=P^{\mu}+\frac{1}{2}q^{\mu}\qquad\text{and}\qquad
p_{2}^{\mu}=P^{\mu}-\frac{1}{2}q^{\mu}\ .$ (15)

We are interested in the case in which the source is stationary, meaning that
$q^{0}=0$ and hence the energy is conserved. Then, since
$p_{1}^{2}=p_{2}^{2}=m^{2}$, just by stationarity one has that
$P^{\mu}=E\,u^{\mu}=m\,u^{\mu}+O(q)$, where $u^{\mu}=\delta^{\mu}_{0}$ is the
velocity of the source, and $E$ is its energy. Following Kosower  _et al._
(2019) to schematically obtain the classical limit out of scattering amplitude
calculations, for a process involving a particle with spin $s$ and transferred
momentum $q$, the algorithm is to write the amplitude in terms of the spin
tensor ${S}_{\mu\nu}$, and then make the replacement

$q\rightarrow\hbar\,q\qquad\text{and}\qquad{S}\rightarrow\frac{1}{\hbar}{S}\
,$ (16)

keeping only the terms $O(\hbar^{0})$, i.e. those that survive in the limit
$\hbar\rightarrow 0$. Therefore, terms $O(q^{n})$ with $n>0$ are quantum,
while terms $O(({S}q)^{n}$) are classical. Crucially, the presence of the spin
compensates the transferred momentum and gives rise to classical terms with
higher powers in $q$. Notice that since the two external momenta are equal up
to the transferred momentum, using the results in Bern  _et al._ (2021); Chung
_et al._ (2019) it turns out that in the stationary limit we can expand the
polarization state as

$\ket{p_{2}}=\ket{p_{1}}+O(\hbar)\ ,$ (17)

and then define the dressed vertex

$\bra{p_{2};s,\sigma^{\prime}}(\tau_{\Phi^{2}h})^{\mu\nu}\ket{p_{1};s,\sigma}=\bra{p_{1};s,\sigma^{\prime}}(\tau_{\Phi^{2}h})^{\mu\nu}\ket{p_{1};s,\sigma}+O(\hbar)=\hat{\tau}_{\Phi^{2}h}^{\mu\nu}(q,S)\delta_{\sigma\sigma^{\prime}}+O(\hbar)\
,$ (18)

where the spin-index structure is left understood.

Now, in order to extract such classical spin pieces, let us consider a field
in a particular representation of the Lorentz group, satisfying the algebra

$[M^{\mu\nu},M^{\rho\sigma}]=-i\Big{(}\eta^{\mu\rho}M^{\nu\sigma}-\eta^{\nu\rho}M^{\mu\sigma}+\eta^{\nu\sigma}M^{\mu\rho}-\eta^{\mu\sigma}M^{\nu\rho}\Big{)}\
,$ (19)

with $M$ the generators of the group. Considering then “in” and “out” states
of spin $s$ and polarizations $\sigma$ and $\sigma^{\prime}$, generalizing the
argument in Bern  _et al._ (2021) to arbitrary dimension, in the stationary
limit we have

$\displaystyle\braket{p_{2};s,\sigma^{\prime}}{p_{1};s,\sigma}$
$\displaystyle=\braket{p_{1};s,\sigma^{\prime}}{p_{1};s,\sigma}+O(\hbar)=C(s)\delta_{\sigma\sigma^{\prime}}\
,$ (20)
$\displaystyle\braket{p_{2};s,\sigma^{\prime}}{M^{\mu\nu}}{p_{1};s,\sigma}$
$\displaystyle=S^{\mu\nu}C(s)\delta_{\sigma\sigma^{\prime}}+O(\hbar^{0})\ ,$
$\displaystyle\braket{p_{2};s,\sigma^{\prime}}{\frac{1}{2}\\{M^{\mu\nu},M^{\rho\sigma}\\}}{p_{1};s,\sigma}$
$\displaystyle=S^{\mu\nu}S^{\rho\sigma}C(s)\delta_{\sigma\sigma^{\prime}}+O(\hbar^{-1})\
,$

where $\\{\cdot,\cdot\\}$ stands for the anti-commutator, the external state
can be either a fermion or a boson, $C(s)$ is the normalization of the states
which depends on the particular representation of the Lorentz group, and we
now interpret $S^{\mu\nu}$ as the spin tensor of the classical source of the
gravitational field. It is important to observe that the stationary limit
defined above corresponds to the source being in the rest frame, hence in this
frame the temporal part of the spin tensor vanishes, i.e. $S^{0i}=0$.

Finally, we observe that for a massive particle with spin $s$ the dressed
vertex contains powers up to $2s$ of the spin tensor. As we will see, this
implies that the resulting metric describes an object with only the first $2s$
multipoles turned on.

###  II.2 Loop amplitudes for any spin

We now want to discuss how, starting from the analysis above, one can
construct loop amplitudes in such a way that the spin structure arising from
the massive spin-$s$ particles is entirely given as a tensor dependence of the
dressed vertex. More specifically, in the classical limit the massive line of
Eq. (8) factorizes in the product of $n$ dressed vertices in such a way that
the loop calculation is formally the one of a scalar process. We will first
discuss explicitly the 1-loop case, and then naturally extend it to all loops.

The contribution to the amplitude arising from the massive line in Eq. (8) for
$n=2$ is

$\bra{p_{2};s,\sigma^{\prime}}(\tau_{\Phi^{2}h})^{\mu\nu}(\ell)\frac{i\,\mathcal{P}(p_{1}-\ell)}{(p_{1}-\ell)^{2}-m^{2}+i\varepsilon}(\tau_{\Phi^{2}h})^{\rho\lambda}(q-\ell)\ket{p_{1};s,\sigma}\
,$ (21)

where $\mathcal{P}$ is the numerator of the matter propagator which depends on
the spin of the massive field. For a massive spin-$s$ particle, we can always
rewrite the operator $\mathcal{P}$ in terms of the sum of the polarization
states as

$\mathcal{P}(p_{1}-\ell)=(2m)^{1-\epsilon}\sum_{\sigma^{\prime\prime}}\ket{p_{1}-\ell;s,\sigma^{\prime\prime}}\bra{p_{1}-\ell;s,\sigma^{\prime\prime}}\
.$ (22)

Plugging this back in Eq. (21) we end up with

$\frac{i\,(2m)^{1-\epsilon}}{(p_{1}-\ell)^{2}-m^{2}+i\varepsilon}\sum_{\sigma^{\prime\prime}}\bra{p_{2};s,\sigma^{\prime}}(\tau_{\Phi^{2}h})^{\mu\nu}(\ell)\ket{p_{1}-\ell;s,\sigma^{\prime\prime}}\bra{p_{1}-\ell;s,\sigma^{\prime\prime}}(\tau_{\Phi^{2}h})^{\rho\lambda}(q-\ell)\ket{p_{1};s,\sigma}\
.$ (23)

Finally, using the classical limit as in Eq. (17) we get

$\frac{i\,(2m)^{1-\epsilon}}{(p_{1}-\ell)^{2}-m^{2}+i\varepsilon}\sum_{\sigma^{\prime\prime}}\bra{p_{1};s,\sigma^{\prime}}(\tau_{\Phi^{2}h})^{\mu\nu}(\ell)\ket{p_{1};s,\sigma^{\prime\prime}}\bra{p_{1};s,\sigma^{\prime\prime}}(\tau_{\Phi^{2}h})^{\rho\lambda}(q-\ell)\ket{p_{1};s,\sigma}+O(\hbar)\
,$ (24)

from which we derive the expression written in terms of the dressed vertices
as

$\frac{i\,(2m)^{1-\epsilon}}{(p_{1}-\ell)^{2}-m^{2}+i\varepsilon}\hat{\tau}_{\Phi^{2}h}^{\mu\nu}(\ell,S)\hat{\tau}_{\Phi^{2}h}^{\rho\lambda}(q-\ell,S)\delta_{\sigma\sigma^{\prime}}\
.$ (25)

To summarize, after this dressing procedure the classical limit of a spinning
3-point 1-loop amplitude in the stationary regime completely reduces to an
amplitude in which the spin contribution arises solely from the tensor
structure of the dressed vertex, up to a normalization factor depending on
whether one considers fermions or bosons. This can be naturally generalized to
any loop, which means that we can use all the machinery of Mougiakakos and
Vanhove (2021); D’Onofrio  _et al._ (2022) in order to write down the $l$-loop
amplitude related to the stress-energy tensor, which is

$-\frac{i\,\kappa}{2}(2m)T_{\mu\nu}^{(l)}(q)=\frac{(-i)^{l+1}}{(l+1)!}\int\prod_{i=1}^{l}\frac{d^{d}\vec{\ell_{i}}}{(2\pi)^{d}}\frac{\prod_{i=1}^{l+1}\hat{\tau}_{\Phi^{2}h}^{\mu_{i}\nu_{i}}(\ell_{i},S)\prod_{i=1}^{l+1}P_{\mu_{i}\nu_{i},\alpha_{i}\beta_{i}}}{\prod_{i=1}^{l+1}\vec{\ell_{i}}^{2}}\mathcal{M}^{\alpha_{1}\beta_{1},...,\mu\nu}\
,$ (26)

where $\mathcal{M}^{\alpha_{1}\beta_{1},...,\alpha_{n}\beta_{n},\mu\nu}$
contains the sum over all the tree-level graviton diagrams as shown in Eq.
(8), and in our signature $\ell_{i}^{2}=-\vec{\ell}_{i}^{2}$, since it is
possible to show that $\ell_{i}^{0}=0$ in the classical and stationary limit.

From Eq. (26) we can systematically compute the stress-energy tensor, and
inserting it back into Eq. (4) we can read off the metric at any PM order $n$.
Since the dressed vertex contains powers up to $2s$ in the spin tensor and an
$l$-loop amplitude contains $n$ dressed vertices, as can be seen from Eq.
(26), this implies that the highest power of $S$ in the stress-energy tensor
is $2ns$. Hence, the resulting metric at PM order $n$ is expanded as

$h_{\mu\nu}^{(n)}=\sum_{j=0}^{2ns}h_{\mu\nu}^{(n,j)}\ ,$ (27)

where $j$ is the order of the expansion in powers of $S$555For instance,
$h_{\mu\nu}^{(1,1)}$ will be 1PM and linear in the spin, $h_{\mu\nu}^{(1,2)}$
will be 1PM and quadratic in the spin, and so on.. This is consistent with the
fact that all multipoles starting from $2s+1$ order vanish.

What remains to be done in this procedure is to determine the explicit
expression of the dressed vertex $\hat{\tau}_{\mu\nu}$ for a given spin-$s$
field. In the next subsection we will show as an example how the $O(S)$ terms
are derived considering spin-1/2 spinors, while in the next section we will
construct the dressed vertex for a spin-1 field.

###  II.3 Spin-1/2 case

Here we review the work of Bjerrum-Bohr  _et al._ (2003) and we extend it to
arbitrary dimensions by employing the formalism discussed in the last section.
Let us consider a massive spin-1/2 particle as source of gravitational field.
Its action reads

$S=\int d^{4}x\,e\,\overline{\psi}\left(i\,e^{\mu}_{\
a}\gamma^{a}D_{\mu}-m\right)\psi\ ,$ (28)

where $g_{\mu\nu}=e^{a}_{\ \mu}e_{a\nu}$ and

$D_{\mu}\psi=\partial_{\mu}\psi+\frac{1}{4}\omega_{\mu ab}\gamma^{ab}\psi\ ,$
(29)

with $M^{ab}=\frac{i}{2}\gamma^{ab}$ generators of the Lorentz group in the
spin-1/2 representation, $\omega_{\mu ab}$ the spin connection, and $a$ and
$b$ flat indices.

The action in (28) gives rise to the tri-linear interaction vertex

$\tau^{\mu\nu,\alpha\beta}_{\psi^{2}h}(q)=-\frac{i\,k}{2}\Bigg{(}\frac{1}{4}\gamma^{\mu}(p_{1}+p_{2})^{\nu}+\frac{1}{4}\gamma^{\nu}(p_{1}+p_{2})^{\mu}-\eta^{\mu\nu}\left(\frac{i}{2}(\not{p_{1}}+\not{p_{2}})-m\right)\Bigg{)}^{\alpha\beta}\
,$ (30)

which can be rewritten using the definition in (14) as

$\tau^{\mu\nu,\alpha\beta}_{\psi^{2}h}(q)=-\frac{i\,k}{2}\Bigg{(}\frac{1}{2}\gamma^{\mu}P^{\nu}+\frac{1}{2}\gamma^{\nu}P^{\mu}-\eta^{\mu\nu}\left(i\not{P}-m\right)\Bigg{)}^{\alpha\beta}\
,$ (31)

where $\alpha$ and $\beta$ are spinor indices. We now want to compute the
dressed vertex. Using the Dirac equations in momentum space

$\overline{u}(p_{2},\sigma^{\prime})\left(\not{P}-m\right)u(p_{1},\sigma)=0\
,$ (32)

where $u$ and $\overline{u}$ are the spinor polarizations, we obtain

$\hat{\tau}_{\psi^{2}h}^{\mu\nu}(q,S)\,\delta_{\sigma\sigma^{\prime}}=\overline{u}(p_{2},\sigma^{\prime})\tau_{\psi^{2}h}^{\mu\nu}(q)u(p_{1},\sigma)=-\frac{i\,\kappa}{2}\overline{u}(p_{2},\sigma^{\prime})\Big{(}\frac{1}{2}\gamma^{\mu}P^{\nu}+\frac{1}{2}\gamma^{\nu}P^{\mu}\Big{)}u(p_{1},\sigma)\
,$ (33)

where we keep understood the spinor indices. Now, since we are dealing with
on-shell massive states, we can simplify the vertex by means of the Gordon
identity, which reads

$\overline{u}(p_{2},\sigma^{\prime})\gamma^{\mu}u(p_{1},\sigma)=\overline{u}(p_{2},\sigma^{\prime})\Big{(}\frac{1}{m}P^{\mu}-\frac{i}{m}M^{\mu\nu}q_{\nu}\Big{)}u(p_{1},\sigma)\
,$ (34)

and finally we get

$\hat{\tau}_{\psi^{2}h}^{\mu\nu}(q,S)\,\delta_{\sigma\sigma^{\prime}}=-\frac{i\,\kappa}{2}\overline{u}(p_{2},\sigma^{\prime})\left(\frac{1}{m}P^{\mu}P^{\nu}-\frac{i}{2m}q_{\lambda}(M^{\mu\lambda}P^{\nu}+M^{\nu\lambda}P^{\mu})\right)u(p_{1},\sigma)\
.$ (35)

Then, using the relations established in (20) with the normalization
$C(1/2)=1$, we rewrite the dressed vertex in terms of the classical spin
tensor as

$\hat{\tau}_{\psi^{2}h}^{\mu\nu}(q,S)=-\frac{i\,\kappa}{2}\left(\frac{1}{m}P^{\mu}P^{\nu}-\frac{i}{2m}q_{\lambda}(S^{\mu\lambda}P^{\nu}+S^{\nu\lambda}P^{\mu})\right)\
.$ (36)

Finally, in the stationary limit in which
$P^{\mu}=m\,\delta^{\mu}_{0}+O(\hbar)$, the resulting vertex reads

$\hat{\tau}_{\psi^{2}h}^{\mu\nu}(q,S)=-\frac{i\,\kappa}{2}\left(m\,\delta^{\mu}_{0}\delta^{\nu}_{0}-\frac{i}{2}q_{\lambda}(S^{\mu\lambda}\delta^{\nu}_{0}+S^{\nu\lambda}\delta^{\mu}_{0})\right)\
.$ (37)

Notice that the dressed vertex defined above holds in arbitrary dimensions
since we have never used any explicit representation of the gamma matrices.

From the vertex, it is now straightforward to obtain the metric. At tree-
level, the stress-energy tensor reads

$T^{\mu\nu}_{(0)}(q)=\frac{2\,i}{\kappa}\hat{\tau}_{\psi^{2}h}^{\mu\nu}(q,S)=m\,\delta^{\mu}_{0}\delta^{\nu}_{0}-\frac{i}{2}q_{\lambda}(S^{\mu\lambda}\delta^{\nu}_{0}+S^{\nu\lambda}\delta^{\mu}_{0})\
,$ (38)

and using Eqs. (4) and (10) we get for the scalar part of the metric

$\displaystyle h_{00}^{(1,0)}(r)$
$\displaystyle=-\frac{4(d-2)}{d-1}Gm\,\rho(r)\ ,$ (39) $\displaystyle
h_{0i}^{(1,0)}(r)$ $\displaystyle=0\ ,$ $\displaystyle h_{ij}^{(1,0)}(r)$
$\displaystyle=-\frac{4\delta_{ij}}{d-1}Gm\,\rho(r)\ ,$

which is exactly the Schwarzschild-Tangherlini metric at 1PM, and for the
dipole part we obtain

$\displaystyle h_{00}^{(1,1)}(r)$ $\displaystyle=0\ ,$ (40) $\displaystyle
h_{0i}^{(1,1)}(r)$
$\displaystyle=-\frac{2(d-2)x^{k}S^{i}{}_{k}}{r^{2}}G\,\rho(r)\ ,$
$\displaystyle h_{ij}^{(1,1)}(r)$ $\displaystyle=0\ .$

Notice that while Greek indices are meant to be contracted by the Minkowski
metric, from now on repeated Latin indices will indicate an Euclidean
contraction. This means in particular that there is no difference between
upper and lower Latin indices.

Finally, we know from General Relativity that at dipole order, the expansion
in the far field limit is unique, and therefore we expect the metric to be
independent of any arbitrary coefficients other than the spacetime mass and
the spin tensor. From an amplitude perspective, this means that there are no
additional non-minimal couplings that can modify the classical vertex at
dipole order. In the next section we will consider a massive spin-1 field, and
we will construct the non-minimal effective action giving rise to the most
general vertex at quadrupole order. This analysis, that can be extended to
fields of arbitrary spin, will consistently show that there are no non-minimal
terms linear in $S$ that can modify the vertex.

##  III Spin-1 scattering amplitudes in arbitrary dimension

In this section we will compute the metric arising from the graviton emission
by a massive spin-1 field. This produces terms up to quadrupole order in the
multipole expansion of (27). We know from General Relativity that these are
the first terms in the expansion that discriminate between different solutions
of the Einstein equation. Therefore, from our amplitude perspective, we expect
that the dressed vertex will not be uniquely defined, corresponding to the
presence of non-minimal couplings in the action. We will first compute the
dressed vertex arising from the minimally coupled action, and then we will
include the contribution of non-minimal couplings, resulting in the most
general stress-energy tensor up to quadrupole order. Finally, we will compute
the resulting metric up to 2 loops.

###  III.1 Minimal vertex

Let us consider a massive spin-1 field minimally coupled to gravity, also
known as Proca field

$S_{\text{min}}=\int
d^{d+1}x\sqrt{-g}\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}V_{\mu}V^{\mu}\right)\
,$ (41)

where $F_{\mu\nu}=\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$ is the usual
anti-symmetric strength tensor. Our aim now is to compute the minimal dressed
vertex associated to the action (41). In order to do this, consider the tri-
linear vertex associated to this action Bjerrum-Bohr  _et al._ (2015)

$\displaystyle\varepsilon_{\beta}(p_{2}){\tau}_{V^{2}h,\text{min}}^{\mu\nu,\beta\alpha}\varepsilon_{\alpha}(p_{1})$
$\displaystyle=-\frac{i\,\kappa}{2}\Bigg{(}\varepsilon(p_{1})\cdot
p_{2}\Big{(}p_{1}^{\mu}\varepsilon^{\nu}(p_{2})+p_{1}^{\nu}\varepsilon^{\mu}(p_{2})\Big{)}+\varepsilon(p_{2})\cdot
p_{1}\Big{(}p_{2}^{\mu}\varepsilon^{\nu}(p_{1})+p_{2}^{\nu}\varepsilon^{\mu}(p_{1})\Big{)}$
(42)
$\displaystyle-\varepsilon(p_{1})\cdot\varepsilon(p_{2})\Big{(}p_{1}^{\mu}p_{2}^{\nu}+p_{2}^{\mu}p_{1}^{\nu}\Big{)}-\Big{(}p_{1}\cdot
p_{2}-m^{2}\Big{)}\Big{(}\varepsilon(p_{1})^{\mu}\varepsilon(p_{2})^{\nu}+\varepsilon(p_{2})^{\mu}\varepsilon(p_{1})^{\nu}\Big{)}$
$\displaystyle+\eta^{\mu\nu}\Big{(}\big{(}p_{1}\cdot
p_{2}-m^{2}\big{)}\varepsilon(p_{1})\cdot\varepsilon(p_{2})-p_{1}\cdot\varepsilon(p_{2})p_{2}\cdot\varepsilon(p_{1})\Big{)}\Bigg{)}\
,$

where $\varepsilon_{\alpha}(p)$ are the real polarization vectors satisfying
the transversality condition $\varepsilon(p)\cdot p=0$, due to the on-
shellness of the massive states.

We now have to manipulate the above expression in order to write it in terms
of the Lorentz generators, which in this particular representation take the
form

$M^{\mu\nu,\rho\sigma}=i\big{(}\eta^{\mu\rho}\eta^{\nu\sigma}-\eta^{\mu\sigma}\eta^{\nu\rho}\big{)}\
,$ (43)

which holds in arbitrary dimensions. The strategy is to strip off the
polarization vectors from Eq. (42) and massage the Lorentz structure in order
to rewrite everything in terms of $M^{\mu\nu,\rho\sigma},q^{\mu}$ and
$P^{\mu}$. Up to terms proportional to $p_{1}^{\alpha}$ and $p_{2}^{\beta}$,
that vanish when contracted with the polarization vectors, from Eq. (42) one
thus obtains

$\displaystyle\varepsilon_{\beta}(p_{2})\tau_{V^{2}h,\text{min}}^{\mu\nu,\beta\alpha}$
$\displaystyle\varepsilon_{\alpha}(p_{1})=\frac{i\,\kappa}{2}\varepsilon_{\beta}(p_{2})\Bigg{(}\eta^{\alpha\beta}\left(2P^{\mu}P^{\nu}+\frac{1}{2}\eta^{\mu\nu}q^{2}-\frac{1}{2}q^{\mu}q^{\nu}\right)$
(44) $\displaystyle-
iq_{\lambda}\left(P^{\mu}M^{\nu\lambda,\beta\alpha}+P^{\nu}M^{\mu\lambda,\beta\alpha}\right)-q_{\rho}q_{\sigma}\frac{1}{2}\\{M^{\mu\rho},M^{\nu\sigma}\\}^{\beta\alpha}\Bigg{)}\varepsilon_{\alpha}(p_{1})\
,$

from which, employing the relations in (20) and the definition in Eq. (18),
the minimal dressed vertex in the stationary limit reads

$\hat{\tau}_{V^{2}h,\text{min}}^{\mu\nu}(q)=-\frac{i\,\kappa}{2}\Big{(}2m^{2}\delta^{\mu}_{0}\delta^{\nu}_{0}-i\,m\,q_{\lambda}\left(S^{\mu\lambda}\delta^{\nu}_{0}+S^{\nu\lambda}\delta_{0}^{\mu}\right)-q_{\lambda}q_{\sigma}S^{\mu\lambda}S^{\nu\sigma}\Big{)}\
,$ (45)

where we have normalized the polarization vectors as $C(1)=-1$. We can see
that the scalar and dipole terms in Eq. (45), up to the normalization factor,
exactly coincide with Eq. (36), as expected from the fact that such terms are
uniquely fixed in the metric. The remaining terms are $O(S^{2})$ and
constitute the quadrupole contribution for the minimally coupled spin-1 field.

###  III.2 Non-minimal vertex

Once computed the dressed vertex associated to the Proca action, it is natural
to consider what happens for a non-minimally coupled theory. The effective
action will be constructed with several non-minimal terms with unconstrained
couplings, that we expect to enter in the expression of the general dressed
vertex as unfixed numerical parameters. In general such effective action is
more involved than the minimally-coupled one, and therefore the tri-linear
vertex associated to it will be more complicated with respect to (42). This
means that carrying out the procedure outlined in the last subsection to get
the dressed vertex is not viable.

Following the idea of Bern  _et al._ (2021), we can define a covariant
generalization of the spin tensor in order to build non-minimal operators with
an explicit spin dependency, instead of stripping it out as we did in the last
subsection. To this end, we define $\mathbb{S}^{\mu\nu}$, a well defined
covariant anti-symmetric tensor, such that

$\mathbb{S}^{\mu\nu}_{a,b}=S^{\mu\nu}\delta_{ab}+O(\kappa)\ ,$ (46)

where $a$ and $b$ are the spin indices depending on the representation of the
field over which such operator acts. Before constructing the non-minimal
effective action, we notice that since we are only interested in the long
range pieces of the metric, every local term can be neglected. In particular,
this means that at tree-level we can neglect terms that contain the squared
modulus of the transferred momentum, i.e.
$T_{\mu\nu}^{(0)}(q)=O(|\vec{q}\,|^{2})$, since they lead to local terms in
the metric. This implies that also the dressed vertex is defined up to terms
$\hat{\tau}_{\mu\nu}(q)=O(|\vec{q}\,|^{2})$.

Now we can construct the non-minimal effective action as

$\displaystyle S_{\text{non-min}}$ $\displaystyle=\int
d^{D}x\sqrt{-g}\Bigg{(}K_{0}\
R\,D^{\mu}V^{\alpha}g_{\alpha\beta}D_{\mu}V^{\beta}+K_{1}\
R\,V^{\alpha}\left(\mathbb{S}^{\mu\nu}\mathbb{S}_{\mu\nu}\right)_{\alpha\beta}V^{\beta}$
(47) $\displaystyle+K_{2}\
R_{\mu\nu}\,V^{\alpha}\left(\mathbb{S}^{\mu\lambda}\mathbb{S}_{\lambda}^{\
\nu}\right)_{\alpha\beta}V^{\beta}+K_{3}\
R_{\mu\nu\rho\sigma}\,V^{\alpha}\left(\mathbb{S}^{\mu\nu}\mathbb{S}^{\rho\sigma}\right)_{\alpha\beta}V^{\beta}$
$\displaystyle+K_{4}\
R_{\mu\nu\rho\sigma}\,D^{\nu}V^{\alpha}\left(\mathbb{S}^{\mu\lambda}\mathbb{S}_{\lambda}^{\
\sigma}\right)_{\alpha\beta}D^{\sigma}V^{\beta}+K_{5}\
D^{\nu}D^{\sigma}R_{\mu\nu\rho\sigma}\,V^{\alpha}\left(\mathbb{S}^{\mu\lambda}\mathbb{S}_{\lambda}^{\
\sigma}\right)_{\alpha\beta}V^{\beta}\Bigg{)}\ .$

It can be shown that every other terms666This includes higher curvature and
higher derivative terms. not included in Eq. (47) are either $O(\kappa^{2})$,
local contributions to the metric, or related to the terms above by symmetry
properties of the spin operator or the curvature tensor. In order to extract
from Eq. (47) a vertex that survives and does not diverge in the classical
limit, we must assign the correct dependence on $\hbar$ to the paramenters.
While $K_{1},K_{2}$ and $K_{3}$ are dimensionless parameters, $K_{0},K_{4}$
and $K_{5}$ have a mass dimension $M^{-2}$, and we observe that combining $G$
and $m$, there are only two ways to build such terms with a positive integer
power of the gravitational coupling, namely

$\left[1/m^{2}\right]=M^{-2}\qquad\text{and}\qquad\left[\frac{1}{\hbar^{2}}(Gm)^{\frac{2}{d-2}}\right]=M^{-2}\
.$ (48)

In particular, the second term leads to an integer power of $G$ only for
$d=3,4$.

By computing the vertex, one observes that for the term proportional to
$K_{0}$, the leading contribution in the transferred momentum is $O(q^{2})$,
while for the one proportional to $K_{5}$ it is $O(q^{4}S^{2})$. According to
Eq. (16), in order to get a classical contribution from both terms we have to
compensate for the extra power of $\hbar$, and this implies that both
coefficients must be proportional to $(Gm)^{\frac{2}{d-2}}$. On the other
hand, the leading term proportional to $K_{4}$ is $O(q^{2}S^{2})$, and
therefore this parameter must be proportional to $1/m^{2}$. To summarize, we
can redefine the free parameters in the action in terms of dimensionless
quantities as

$\begin{gathered}K_{0}=\frac{1}{2}\Omega_{1}\left(Gm\right)^{\frac{2}{d-2}}\
,\qquad K_{1}=-\frac{1}{4}C_{1}\ ,\qquad K_{2}=-\frac{1}{2}C_{2}\ ,\\\
K_{3}=\frac{1-H_{1}}{8}\ ,\qquad K_{4}=\frac{H_{2}}{2m^{2}}\ ,\qquad
K_{5}=\Omega_{2}\left(Gm\right)^{\frac{2}{d-2}}\ .\end{gathered}$ (49)

Explicitly, expanding (47) at $O(\kappa)$, we obtain the dressed vertex
associated to the non-minimal effective action

$\displaystyle\hat{\tau}_{V^{2}h,\text{non-
min}}^{\mu\nu}(q)=-\frac{i\,\kappa}{2}\Bigg{(}$
$\displaystyle-(H_{1}-1)\,q_{\rho}q_{\sigma}S^{\mu\rho}S^{\nu\sigma}+H_{2}\,\delta_{0}^{\mu}\delta_{0}^{\nu}q_{\rho}q_{\sigma}S^{\rho\lambda}S_{\lambda}^{\
\sigma}+C_{1}\,S^{\rho\sigma}S_{\rho\sigma}q^{\mu}q^{\nu}$ (50)
$\displaystyle+C_{2}\Big{(}\eta^{\mu\nu}q_{\rho}q_{\sigma}S^{\rho\lambda}S^{\sigma}{}_{\lambda}-q^{\lambda}\big{(}q^{\mu}S_{\lambda\sigma}S^{\nu\sigma}+q^{\nu}S_{\lambda\sigma}S^{\mu\sigma}\big{)}\Big{)}\Bigg{)}\
,$

as well as the “higher-loop” dressed vertex

$\includegraphics[valign={c},width=86.72267pt]{Images/BoxTree.pdf}=\hat{\tau}_{V^{2}h,\text{HL}}^{\mu\nu}(q)=i\,\kappa(G\,m)^{\frac{2}{d-2}}\Bigg{(}\Omega_{1}\,m^{2}\,q^{\mu}q^{\nu}+\Omega_{2}\,q^{\mu}q^{\nu}q_{\rho}q_{\sigma}S^{\rho\lambda}S_{\lambda}^{\
\sigma}\Bigg{)}\ .$ (51)

In particular, in $d=3$ a tree-level insertion of
$\hat{\tau}_{\Phi^{2}h,\text{HL}}^{\mu\nu}$ contributes to 3PM, while in $d=4$
it contributes starting from 2PM.

Considering both the minimal and non-minimal action, the dressed vertex

$\displaystyle\hat{\tau}_{V^{2}h}^{\mu\nu}(q)=\hat{\tau}_{V^{2}h,\text{min}}^{\mu\nu}(q)+\hat{\tau}_{V^{2}h,\text{non-
min}}^{\mu\nu}(q)$ (52)
$\displaystyle=-\frac{i\,\kappa}{2}\Bigg{(}2m^{2}\delta^{\mu}_{0}\delta^{\nu}_{0}-i\,m\,q_{\lambda}\left(S^{\mu\lambda}\delta^{\nu}_{0}+S^{\nu\lambda}\delta_{0}^{\mu}\right)-H_{1}q_{\lambda}q_{\sigma}S^{\mu\lambda}S^{\nu\sigma}+H_{2}\,\delta_{0}^{\mu}\delta_{0}^{\nu}q_{\rho}q_{\sigma}S^{\rho\lambda}S_{\lambda}^{\
\sigma}$
$\displaystyle+C_{1}\,S^{\rho\sigma}S_{\rho\sigma}q^{\mu}q^{\nu}+C_{2}\Big{(}\eta^{\mu\nu}q_{\rho}q_{\sigma}S^{\rho\lambda}S^{\sigma}{}_{\lambda}-q^{\lambda}\big{(}q^{\mu}S_{\lambda\sigma}S^{\nu\sigma}+q^{\nu}S_{\lambda\sigma}S^{\mu\sigma}\big{)}\Big{)}\Bigg{)}$

is associated to the stress-energy tensor of the matter source, and describes
the most general stationary rotating matter distribution at second order in
the angular momentum, which is spherically symmetric in the non-rotating
limit. Since we have no free parameters in the scalar and dipole terms (other
than the mass and angular momentum), such dressed vertex shows that at order
$O(S)$ the geometry is uniquely defined, as we commented in subsection II.3.
Moreover, the dimensionless coefficients are normalized in such a way that we
recover the minimal dressed vertex in Eq. (45) by setting

$H_{1}=1\ ,\quad H_{2}=0\ ,\quad C_{1}=0\ ,\quad C_{2}=0\ .$ (53)

Hereafter, we will refer to Eq. (53) as the minimal limit. As we will show in
detail, while the $C_{i}$ coefficients are a gauge artifact, the $H_{i}$
coefficients are physical, in the sense that they determine the multipole
structure of the source. On the other hand, the higher-loop dressed vertex in
Eq. (51) does not correspond to the stress-energy tensor $T_{\mu\nu}^{(0)}$ of
the matter distribution. Indeed, we will see how it contributes only to the
renormalization of gauge-dependent singularities in the metric that arise only
in $d=3,4$.

Finally, one can check that both dressed vertices are conserved up to local
terms, in the sense that
$q_{\mu}\hat{\tau}_{V^{2}h}^{\mu\nu}=O(|\vec{q}\,|^{2})$ and
$q_{\mu}\hat{\tau}_{V^{2}h,\text{HL}}^{\mu\nu}=O(|\vec{q}\,|^{2})$, as they
should be. This fixes the relative coefficients of the terms proportional to
$C_{2}$ in Eq. (50), and does not allow for any other terms. In fact, even
neglecting the QFT underlying the dressed vertex, we could have written the
stress-energy tensor associated to Eq. (50) just by constructing the most
generic conserved symmetric rank-2 tensor with the objects we have at our
disposal. This means that, without considering the field interpretation of the
matter source, this procedure can be easily generalized to arbitrary
multipoles to build metrics in any dimensions at any order in the PM
expansion. In this sense, the results above are universal as far as terms up
to quadrupole order are concerned. Considering a field of spin $s$, we expect
that the non-minimal couplings will produce exactly the same vertex as the
spin-1 case at quadrupole level. In other words, a spin-$s$ field will give
rise to the most general vacuum metric generated by a rotating object with
multipoles up to $2s$ order, and with all the higher multipoles vanishing.

###  III.3 Loop amplitudes

As already mentioned, the dressed vertex in Eq. (50) is directly related to
the stress-energy tensor $T_{\mu\nu}^{(0)}(q)$ of the matter source, which
through Eq. (4) leads to the metric at 1PM. In order to get higher order
terms, one has to compute self-interaction contributions to the stress-energy
tensor, which are produced by the loop amplitudes in Eq. (8). At 1-loop level
the only diagram that contributes is the one in Fig. 1,

Figure 1: 1-loop Feynman diagram that contributes to $T_{\mu\nu}^{(1)}(q)$.

while at 2-loop level the diagrams that contribute to $T_{\mu\nu}^{(2)}(q)$
are shown in Fig. 2. The computation is performed following the procedure
outlined in subsection II.1.

Figure 2: 2-loop Feynman diagrams that contribute to $T_{\mu\nu}^{(2)}(q)$.
Figure 3: 1-loop insertion of the higher-loop vertex that renormalizes the 3PM
metric in $d=4$.

The loop amplitudes exhibit infrared divergences for specific values of $d$,
and although in general the computation of $h_{\mu\nu}^{(n)}(x)$ by means of
the Fourier transform in Eq. (4) smears them, they lead to singularities in
the metric in $d=3$ and $d=4$. These divergences have to be renormalized by
the insertion of diagrams involving the higher-loop vertex in Eq. (51) used as
counter-terms, depending on the specific value of $d$ as explained in the last
subsection. At 2PM the only dimension in which a singularity appears in the
metric is $d=4$, and the singularity is cured by the tree-level insertion of
$\hat{\tau}_{V^{2}h,\text{HL}}^{\mu\nu}$. In this case, we can redefine the
coefficients in (51) as

$\displaystyle\Omega_{1}\big{|}_{d=4}$
$\displaystyle=\frac{\Omega_{1}^{\mathrm{renorm.}}}{d-4}+\Omega_{1}^{\mathrm{free}}\
,$ (54) $\displaystyle\Omega_{2}\big{|}_{d=4}$
$\displaystyle=\frac{\Omega_{2}^{\mathrm{renorm.}}}{d-4}+\Omega_{2}^{\mathrm{free}}\
,$

where $\Omega_{i}^{\mathrm{renorm.}}$ is the term that has to be fixed to
cancel the poles, and $\Omega_{i}^{\mathrm{free}}$ is the remaining free
parameter. Then, within our conventions we have to fix

$\Omega_{1}^{\mathrm{renorm.}}=\frac{1}{9\pi}\qquad\mathrm{and}\qquad\Omega_{2}^{\mathrm{renorm.}}=\frac{H_{1}+2H_{2}-1}{60\pi}\
,$ (55)

such that the metric is now finite in $d=4$ and some logarithmic terms arise
in the radial dependence of the metric.

At 2-loop level, the metric is divergent in both $d=3$ and $d=4$. As we
already discussed, in $d=4$ the 3PM metric is renormalized by a 1-loop
insertion of the higher-loop vertex, as in Fig. 3. Consistently, the
parameters defined in Eq. (55) renormalize the 3PM metric in $d=4$, and the
final result is finite. In the $d=3$ case, the metric is renormalized by a
tree-level insertion of the higher-loop vertex. However, even if the same
renormalization procedure that we carried out for $d=4$ can be pursued,
obtaining a result for every value of the gauge parameter $\alpha$, we notice
that in $d=3$ the harmonic gauge ($\alpha=1$) naturally eliminates every
poles, and the metric becomes finite777In this case the tree-level insertion
of the higher-loop vertex only adds two gauge redundancies to the metric.. We
checked this explicitly up to 2-loops, and at the best of our knowledge there
is no principle for which such gauge guaranties finiteness at every PM and
multipole orders. In fact in $d=4$ at 1-loop, one can define a particular
gauge choice (choosing a specific value of $\alpha$) in which
$h_{\mu\nu}^{(2,0)}(r)$ is finite, but $h_{\mu\nu}^{(2,2)}(r)$ is not.

From the point of view of the Einstein equations, the divergences discussed
above are interpreted as a consequence of the fact that, for a specific gauge
choice, imposing that the metric is a series expansion in powers of $\rho(r)$
is not consistent. Similarly, the presence of the free parameters
$\Omega_{i}^{\mathrm{free}}$ is interpreted as a redundancy of the gauge
defined in Eq. (6). In the next section we will discuss in detail the
properties of the metric and give its full expression at 1PM. Although we have
also managed to perform such calculation up to 2-loops and for an arbitrary
value of $\alpha$, higher orders in the PM expansion do not give any more
physical information than the leading order, and since the full expressions
are quite involved, we only provide the analytic form of $h_{\mu\nu}^{(2)}(r)$
and $h_{\mu\nu}^{(3)}(r)$ in an auxiliary Mathematica file.

##  IV Rotating metrics in any dimensions

From Eq. (52), by considering the Fourier transform in Eq. (4), we can compute
the metric induced by the most generic rotating matter distribution (which is
spherically symmetric in the non-rotating case) at quadrupole order at 1PM. At
tree-level we have already computed the metric up to $O(S)$ in Sec. II.3, so
we just need to determine the quadrupole part, which reads

$\displaystyle h_{00}^{(1,2)}(r)$
$\displaystyle=\frac{2(d-2)\Big{(}H_{2}(d-2)+H_{1}\Big{)}}{d-1}\frac{r^{2}S_{k_{1}k_{2}}S^{k_{1}k_{2}}-d\,x^{k_{1}}x^{k_{2}}S_{k_{1}}{}^{k_{3}}S_{k_{2}k_{3}}}{mr^{4}}G\rho(r)\
,$ (56) $\displaystyle h_{0i}^{(1,2)}(r)$ $\displaystyle=0\ ,$ $\displaystyle
h_{ij}^{(1,2)}(r)$
$\displaystyle=-\frac{2(d-2)}{(d-1)mr^{4}}\Bigg{(}-C_{1}(d-1)d\,x_{i}x_{j}S_{k_{1}k_{2}}S^{k_{1}k_{2}}-r^{2}(d-1)\Big{(}2C_{2}+H_{1}\Big{)}S_{ik}S_{j}{}^{k}$
$\displaystyle+r^{2}\Big{(}C_{1}(d-1)+H_{1}-H_{2}\Big{)}S_{k_{1}k_{2}}S^{k_{1}k_{2}}\delta_{ij}+d\,C_{2}(d-1)x^{k_{1}}S_{k_{1}k_{2}}\Big{(}x_{j}S_{i}{}^{k_{2}}+x_{i}S_{j}{}^{k_{2}}\Big{)}$
$\displaystyle+d\,x^{k_{1}}x^{k_{2}}\Big{(}(d-1)H_{1}S_{ik_{1}}S_{jk_{2}}+(H_{2}-H_{1})S_{k_{1}}{}^{k_{3}}S_{k_{2}k_{3}}\delta_{ij}\Big{)}\Bigg{)}G\rho(r)\
.$

We notice that, at this level, the metric does not depend on the gauge
parameter $\alpha$ introduced in Eq. (6) since the differences between the de
Donder and the harmonic gauge start to appear at 2PM in the expression of the
self-interacting graviton vertices.

###  IV.1 Eliminating redundant parameters

In the last subsection we recovered the metric induced by a rotating matter
source at quadrupole order, which depends on four free parameters. However, it
can be shown that some of these are gauge artifacts, which can be eliminated
by an infinitesimal coordinate transformation.

Let us restrict to the case of $\alpha=1$ (harmonic gauge), even though the
same argument can be generalized. Consider a generic infinitesimal coordinate
transformation $x^{\prime}_{\mu}=x_{\mu}+\xi_{\mu}(x)$, such that the metric
perturbation in the new frame reads

$h^{\prime}_{\mu\nu}=h_{\mu\nu}-(\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu})\
.$ (57)

By definition, in the harmonic gauge

$\Box x^{\mu}=0\ ,$ (58)

and if we want to make a coordinate transformation preserving this gauge, we
just have to impose

$\Box x^{\prime\mu}=0\quad\rightarrow\quad\Box\xi^{\mu}=0\ .$ (59)

Moreover, in order to preserve stationarity the infinitesimal shift
$\xi^{\mu}$ must be time-independent and in addition we choose it such that
$\xi^{0}=0$, so that only the spatial components of the metric are
transformed. With these choices, since in arbitrary dimension

$\Box\rho(r)=0\ ,$ (60)

the most generic harmonic shift is

$\xi^{i}=\sum_{\ell=0}^{+\infty}\mathcal{T}^{i,A_{\ell}}\partial_{A_{\ell}}\rho(r)\
,$ (61)

where $\mathcal{T}$ is a generic constant tensor, and we will use hereafter
the shorthand notation $A_{\ell}=a_{1}\cdots a_{\ell}$, with
$\partial_{A_{\ell}}=\partial_{a_{1}}\cdots\partial_{a_{\ell}}$. Finally,
since $\xi^{i}$ must have the dimensions of a length, at a certain order in
the derivatives of the harmonic function we have to define a dimensionful
quantity that compensates for the extra length dimensions.

The most generic harmonic function linear in $G$ that parameterizes a
coordinate transformation at quadrupole order inside the gauge reads

$\xi^{i}=\frac{G}{m}\Big{(}A\,S^{ik}S_{k}{}^{j}+B\,S^{lm}S_{lm}\delta^{ij}\Big{)}\partial_{j}\rho(r)\
.$ (62)

Choosing the dimensionless coefficients as

$A=2C_{2}\qquad\text{and}\qquad B=C_{1}\ $ (63)

and computing Eq. (57) we will end up with a metric independent of $C_{1}$ and
$C_{2}$. This means that such coefficients in Eq. (56) are only a gauge
artifact, while $H_{1}$ and $H_{2}$ are physical because there is no
transformation inside the gauge that can cancel them.

There are two additional shifts that can be defined at higher PM order up to
quadrupole terms. In particular, at order $O(S^{0})$ one can consider

$\xi^{i}_{1}=(Gm)^{\frac{d}{d-2}}\,\tilde{\Omega}_{1}\,\partial^{i}\rho(r)\ ,$
(64)

while at order $O(S^{2})$ one gets

$\xi^{i}_{2}=\frac{1}{m^{2}}(Gm)^{\frac{d}{d-2}}\,\tilde{\Omega}_{2}\,S_{l}{}^{k}S_{km}\partial^{i}\partial^{l}\partial^{m}\rho(r)\
.$ (65)

From Eqs. (64) and (65) one can repeat the previous argument and show that the
coefficients $\Omega_{1}$ and $\Omega_{2}$ in Eq. (51) are a gauge artifact.

In summary, besides the spacetime mass $m$ and spin tensor $S^{\mu\nu}$, the
solution depends on _two_ further physical parameters, namely $H_{1}$ and
$H_{2}$. However, in $d=3$ this is not the end of the story. Indeed, such case
is special since we can write $S^{ij}=\epsilon^{ijk}S_{k}$, where $S^{k}$ is
the angular momentum vector and $\epsilon^{ijk}$ is the Levi-Civita symbol.
Replacing this relation inside the metric in $d=3$ and fixing

$A=H_{2}-H_{1}\qquad\text{and}\qquad B=-\frac{1}{2}H_{1}\ ,$ (66)

we find that the entire metric depends on $H_{1}$ and $H_{2}$ only through the
combination $H_{1}+H_{2}$, showing that the $D=4$ metric effectively depends
only on one extra parameter. The physical interpretation of $H_{1}$ and
$H_{2}$, as well as their degeneracy in four dimensions, will be discussed in
the next subsection.

###  IV.2 Multipole moments

Having identified the physical parameters of the solution, we now turn our
attention to their physical interpretation. The metric in Eq. (56), in which
the coefficients $H_{1}$ and $H_{2}$ enter, is truncated at the leading
quadrupole order in a multipole expansion, and so it is natural to think that
such coefficients are associated with the quadrupole moment(s) of the source.

The concept of multipole expansion in General Relativity has a long history,
pioneered by Geroch and Hansen Geroch (1970a, b); Hansen (1974), who defined a
multipole expansion in a gauge invariant framework. Later, Thorne introduced
the so-called Asymptotically Cartesian Mass Centered (ACMC) coordinates, a
particular reference frame in which it is possible to unambiguously extract
the multipole moments directly from the asymptotic behavior of the metric
Thorne (1980). Albeit not gauge invariant, Thorne’s formalism is more
intuitive, and it can be shown that the two definitions of multipole moments
coincide Gürsel (1983). While most of the knowledge on this topic is limited
to the $D=4$ case (see Refs. Cardoso and Gualtieri (2016); Mayerson (2023) for
a review), there have been attempts to generalize the multipole expansion
formalism to $D=5$ in the Geroch-Hansen framework Tanabe  _et al._ (2010) and
using the ACMC coordinates à la Thorne Heynen and Mayerson (2023).

Let us discuss the original formalism of Thorne Thorne (1980), which is also
used in Heynen and Mayerson (2023). Considering the linearized metric

$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h^{(1)}_{\mu\nu}+...\ ,$ (67)

which is nothing but Eq. (2) truncated at first order, and defining the trace-
reversed perturbation as

$\gamma_{\mu\nu}=\kappa h^{(1)}_{\mu\nu}-\frac{\kappa}{2}\eta_{\mu\nu}h^{(1)}\
,$ (68)

the harmonic gauge condition and the Einstein equations respectively are

$\partial^{\mu}\gamma_{\mu\nu}=0\qquad\text{and}\qquad\Box\gamma_{\mu\nu}=0\
.$ (69)

As we already noticed in subsection IV.1, we can build harmonic functions in
arbitrary dimensions at any order by means of derivatives of the harmonic
function $\rho(r)$. In this framework, the most generic solution of the
linearized vacuum Einstein equations can be written as

$\displaystyle\gamma_{00}$
$\displaystyle=\sum_{\ell=0}^{+\infty}\mathcal{M}_{A_{\ell}}\partial_{A_{\ell}}\rho(r)\
,$ (70) $\displaystyle\gamma_{0i}$
$\displaystyle=\sum_{\ell=0}^{+\infty}\mathcal{J}_{i,A_{\ell}}\partial_{A_{\ell}}\rho(r)\
,$ $\displaystyle\gamma_{ij}$
$\displaystyle=\sum_{\ell=0}^{+\infty}\mathcal{G}_{ij,A_{\ell}}\partial_{A_{\ell}}\rho(r)\
,$

where $\mathcal{M}_{A_{\ell}}$, $\mathcal{J}_{i,A_{\ell}}$ and
$\mathcal{G}_{ij,A_{\ell}}$ are generic constant tensors which are completely
symmetric and traceless in the indices $A_{\ell}$, and following Thorne
(1980); Heynen and Mayerson (2023) we will denote these indices as
$\\{A_{\ell}\\}_{\text{STF}}$ (symmetric and trace-free); moreover,
$\mathcal{G}_{ij,A_{\ell}}$ is also symmetric with respect to $ij$ but not
traceless, so in a shorthand notation we express all these symmetries by
writing $\mathcal{G}_{(ij),\\{A_{\ell}\\}_{\text{STF}}}$. In particular, in
$d$ spatial dimensions, these multipole tensors are $SO(d)$ tensors, and they
can be decomposed in irreducible representations of such rotation group. For
the moment, let us restrict to the quadrupole case ($\ell=2$) and $d=3$
spatial dimensions. We can rewrite the trace-reversed perturbation in terms of
irreducible representations as (see Appendix A)

$\displaystyle\gamma_{00}\big{|}_{\ell=2}^{d=3}=\mathcal{M}_{\\{a_{1}a_{2}\\}_{\text{STF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{r}\right)\
,$ (71)
$\displaystyle\gamma_{0i}\big{|}_{\ell=2}^{d=3}=\mathcal{J}^{(1)}_{a_{1}}\partial_{a_{1}}\partial_{i}\left(\frac{1}{r}\right)+\mathcal{J}^{(2)}_{\\{ia_{1}a_{2}\\}_{\text{STF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{r}\right)+\epsilon_{ia_{1}a_{2}}\mathcal{J}^{(3)}_{\\{a_{1}a_{3}\\}_{\text{STF}}}\partial_{a_{2}}\partial_{a_{3}}\left(\frac{1}{r}\right)\
,$
$\displaystyle\gamma_{ij}\big{|}_{\ell=2}^{d=3}=\delta_{ij}\mathcal{G}^{(1)}_{\\{a_{1}a_{2}\\}_{\text{STF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{r}\right)+\mathcal{G}^{(2)}\partial_{i}\partial_{j}\left(\frac{1}{r}\right)+\mathcal{G}^{(3)}_{\\{(i|a_{1}\\}_{\text{STF}}}\partial_{|j)}\partial_{a_{1}}\left(\frac{1}{r}\right)$
$\displaystyle+\mathcal{G}^{(4)}_{\\{ija_{1}a_{2}\\}_{\text{STF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{r}\right)+\epsilon_{(i|a_{1}a_{2}}\mathcal{G}^{(5)}_{a_{1}}\partial_{a_{2}}\partial_{|j)}\left(\frac{1}{r}\right)+\epsilon_{(i|a_{1}a_{2}}\mathcal{G}^{(6)}_{\\{a_{1}a_{3}|j)\\}_{\text{STF}}}\partial_{a_{2}}\partial_{a_{3}}\left(\frac{1}{r}\right)\
,$

where all the tensors are constant and the superscripts are just labelling the
different tensors. Now imposing the harmonic gauge condition in Eq. (69) one
has to fix

$\mathcal{J}^{(2)}=0\ ,\qquad\mathcal{G}^{(1)}=-\frac{1}{2}\mathcal{G}^{(3)}\
,\qquad\mathcal{G}^{(4)}=0\ ,\qquad\mathcal{G}^{(6)}=0\ .$ (72)

Moreover, as we did in the last subsection, we can define a coordinate
transformation inside the harmonic gauge where

$\gamma^{\prime}_{\mu\nu}=\gamma_{\mu\nu}-\partial_{\mu}\xi_{\nu}-\partial_{\nu}\xi_{\mu}+\eta_{\mu\nu}\partial^{\alpha}\xi_{\alpha}\
.$ (73)

We can exploit this gauge freedom to choose

$\begin{gathered}\xi_{0}=J^{(1)}_{a_{1}}\partial_{a_{1}}\left(\frac{1}{r}\right)\
,\\\
\xi_{i}=-\mathcal{G}^{(1)}_{ja_{1}}\partial_{a_{1}}\left(\frac{1}{r}\right)+\frac{1}{2}\epsilon_{ja_{1}a_{2}}\mathcal{G}^{(5)}_{a_{1}}\partial_{a_{2}}\left(\frac{1}{r}\right)+\frac{1}{2}\mathcal{G}^{(2)}\partial_{j}\left(\frac{1}{r}\right)\
,\end{gathered}$ (74)

obtaining a trace-reversed perturbation that reads

$\displaystyle\gamma_{00}\big{|}_{\ell=2}^{d=3}$
$\displaystyle=\mathcal{M}_{\\{a_{1}a_{2}\\}_{\text{STF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{r}\right)\
,$ (75) $\displaystyle\gamma_{0i}\big{|}_{\ell=2}^{d=3}$
$\displaystyle=\epsilon_{ia_{1}a_{2}}\mathcal{J}^{(3)}_{\\{a_{1}a_{3}\\}_{\text{STF}}}\partial_{a_{2}}\partial_{a_{3}}\left(\frac{1}{r}\right)\
,$ $\displaystyle\gamma_{ij}\big{|}_{\ell=2}^{d=3}$ $\displaystyle=0\ .$

This argument can be generalized to arbitrary multipole orders, exactly as in
Thorne (1980), and the final outcome is that in $d=3$ there are two
independent towers of multipole tensors, namely the mass multipoles
$\mathcal{M}_{\\{A_{\ell}\\}_{\text{STF}}}$ and the current multipoles
$\mathcal{J}^{(3)}_{\\{A_{\ell}\\}_{\text{STF}}}$, which are gauge invariant
within ACMC transformations.

Let us now discuss the quadrupolar case ($\ell=2$) but in $d=4$. In this case
we have to decompose the perturbation in terms of $SO(4)$ irreducible
representations and, besides STF tensors, there will appear two more
structures (see Appendix A for details). Following Heynen and Mayerson (2023),
we will indicate as ASTF (anti-symmetric and trace-free) a tensor
$\mathcal{T}_{\\{b_{1}b_{2},A_{\ell}\\}_{\text{ASTF}}}$, which is STF with
respect to $A_{\ell}$, anti-symmetric on the $b$’s, and trace-free with
respect to all indices. Furthermore, irreducible representations of $SO(4)$
lead to another kind of tensors (not recognized in Heynen and Mayerson (2023))
that we call RSTF (Riemann-symmetric and trace-free), and we indicate them as
$\mathcal{T}_{\\{ib_{1},jb_{2},A_{\ell}\\}_{\text{RSTF}}}$. They have the same
symmetries of the Riemann tensor for the first four indices, are STF with
respect to $A_{\ell}$’s, and are trace-free with respect to all indices. We
can now express the trace-reversed metric perturbation in terms of irreducible
representations as

$\displaystyle\gamma_{00}\big{|}_{\ell=2}^{d=4}$
$\displaystyle=\mathcal{M}_{\\{a_{1}a_{2}\\}_{\text{STF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{\pi
r^{2}}\right)\ ,$ (76) $\displaystyle\gamma_{0i}\big{|}_{\ell=2}^{d=4}$
$\displaystyle=\mathcal{J}^{(1)}_{a_{1}}\partial_{a_{1}}\partial_{i}\left(\frac{1}{\pi
r^{2}}\right)+\mathcal{J}^{(2)}_{\\{ia_{1}a_{2}\\}_{\text{STF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{\pi
r^{2}}\right)+\epsilon_{ib_{1}b_{2}a_{1}}\mathcal{J}^{(3)}_{\\{b_{1}b_{2},a_{2}\\}_{\text{ASTF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{\pi
r^{2}}\right)\ ,$ $\displaystyle\gamma_{ij}\big{|}_{\ell=2}^{d=4}$
$\displaystyle=\delta_{ij}\mathcal{G}^{(1)}_{\\{a_{1}a_{2}\\}_{\text{STF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{\pi
r^{2}}\right)+\mathcal{G}^{(2)}\partial_{i}\partial_{j}\left(\frac{1}{\pi
r^{2}}\right)+\mathcal{G}^{(3)}_{\\{(i|a_{1}\\}_{\text{STF}}}\partial_{|j)}\partial_{a_{1}}\left(\frac{1}{\pi
r^{2}}\right)$
$\displaystyle+\mathcal{G}^{(4)}_{\\{ija_{1}a_{2}\\}_{\text{STF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{\pi
r^{2}}\right)+\epsilon_{(i|b_{1}b_{2}a_{1}}\mathcal{G}^{(5)}_{\\{b_{1}b_{2}\\}_{\text{ASTF}}}\partial_{a_{1}}\partial_{|j)}\left(\frac{1}{\pi
r^{2}}\right)$
$\displaystyle+\epsilon_{(i|b_{1}b_{2}a_{1}}\mathcal{G}^{(6)}_{\\{b_{1}b_{2},a_{2}|j)\\}_{\text{ASTF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{\pi
r^{2}}\right)+\mathcal{G}^{(7)}_{\\{ib_{1},jb_{2}\\}_{\text{RSTF}}}\partial_{b_{1}}\partial_{b_{2}}\left(\frac{1}{\pi
r^{2}}\right)\ .$

Notice that $\mathcal{G}^{(7)}$ is not manifestly symmetric in $ij$ since it
has a Riemann-like symmetry; however, once symmetrized in $b_{1}b_{2}$ (due to
the contraction with the derivatives), it becomes symmetric also with respect
to $ij$.

Comparing this result with the work in Heynen and Mayerson (2023), one can
notice that in Eq. (76) there is an extra structure, namely
$\mathcal{G}^{(7)}$. The existence of such new tensor is very clear from a
group theory point of view (see Appendix A), in which by counting the number
of degrees of freedom one gets

$\\#\Bigg{[}\mathcal{G}^{(7)}_{\\{ib_{1},jb_{2}\\}_{\text{RSTF}}}+\mathcal{G}^{(6)}_{\\{b_{1}b_{2},a_{1}a_{2}\\}_{\text{ASTF}}}\Bigg{]}=40\
,$ (77)

conversely to what is discussed in Heynen and Mayerson (2023), in which 40
components are assigned to $\mathcal{G}^{(6)}$ only.

As done previously for the $d=3$ case, after imposing the harmonic gauge
condition, we can find a suitable coordinate transformation inside the gauge
such that the perturbation finally reads

$\displaystyle\gamma_{00}\big{|}_{\ell=2}^{d=4}$
$\displaystyle=\mathcal{M}_{\\{a_{1}a_{2}\\}_{\text{STF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{\pi
r^{2}}\right)\ ,$ (78) $\displaystyle\gamma_{0i}\big{|}_{\ell=2}^{d=4}$
$\displaystyle=\epsilon_{ib_{1}b_{2}a_{1}}\mathcal{J}^{(3)}_{\\{b_{1}b_{2},a_{2}\\}_{\text{ASTF}}}\partial_{a_{1}}\partial_{a_{2}}\left(\frac{1}{\pi
r^{2}}\right)\ ,$ $\displaystyle\gamma_{ij}\big{|}_{\ell=2}^{d=4}$
$\displaystyle=\mathcal{G}^{(7)}_{\\{ib_{1},jb_{2}\\}_{\text{RSTF}}}\partial_{b_{1}}\partial_{b_{2}}\left(\frac{1}{\pi
r^{2}}\right)\ .$

Since now $\gamma_{ij}\neq 0$, we have an extra degree of freedom at
quadrupole order with respect to the $d=3$ case. This argument can be
generalized to arbitrary dimensions and to any multipole order, showing the
existence of a new tower of independent multipole tensors
$\mathcal{G}^{(7)}_{\\{ib_{1},jb_{2},A_{\ell}\\}_{\text{RSTF}}}$ that we call
stress multipoles888The name is inspired by the fact that $g_{ij}$ is induced
by $T_{ij}$, which is the stress-part of the matter source.. To summarize, Eq.
(78) shows that at each multipole order the spacetime is characterized by
_three_ independent multipole tensors, which however in the case of $D=4$
reduce to two only, due to group theory properties of $SO(3)$ and the specific
form of the Einstein equations.

Inspired by the above argument and by the metric in Eq. (56), we conjecture
the possibility of defining an ACMC coordinate system in arbitrary dimensions
and the existence of a new independent multipole tensor associated with the
spatial part of the metric. To this end we give the generic expression of a
stationary metric in arbitrary dimensions expanded in a multipole series as

$\displaystyle g_{00}$
$\displaystyle=1-\frac{4(d-2)}{d-1}Gm\rho(r)+\sum_{\ell=1}^{+\infty}\frac{2(d-2)}{d-1}\frac{Gm\rho(r)}{r^{\ell}}\mathbb{M}^{(\ell)}_{A_{\ell}}N_{A_{\ell}}+\cdots\
,$ (79) $\displaystyle g_{0i}$
$\displaystyle=-2(d-2)\frac{Gm\rho(r)}{r}\left(\frac{1}{m}S^{ik}\frac{x_{k}}{r}\right)+2(d-2)\sum_{\ell=2}^{+\infty}\frac{Gm\rho(r)}{r^{\ell}}\mathbb{J}^{(\ell)}_{i,A_{\ell}}N_{A_{\ell}}+\cdots\
,$ $\displaystyle g_{ij}$
$\displaystyle=-\delta_{ij}-\frac{4}{d-1}Gm\rho(r)\delta_{ij}+\sum_{\ell=1}^{+\infty}\frac{2(d-2)}{d-1}\frac{Gm\rho(r)}{r^{\ell}}\tilde{\mathbb{G}}^{(\ell)}_{ij,A_{\ell}}N_{A_{\ell}}+\cdots\
,$

with $N_{A_{\ell}}=\frac{x_{a_{1}}\cdots\,x_{a_{\ell}}}{r^{\ell}}$ and where
the ellipses stand for non-gauge invariant contributions. Indeed, in order to
assure gauge invariance, it is important to define the quadrupole tensors up
to terms $\delta_{a_{m}a_{n}},\delta_{ia_{m}}$ or $\delta_{ja_{m}}$. Loosely
speaking, in Eq. (79) we are neglecting terms in which $N_{A_{\ell}}$’s
contract some $\delta$’s. Therefore, within a “generalized” ACMC coordinate
transformation in arbitrary dimensions, the expression written in Eq. (79) is
conjectured to be invariant. $\mathbb{M}^{(\ell)}_{A_{\ell}}$ are the standard
mass multipoles, $\mathbb{J}^{(\ell)}_{A_{\ell}}$ correspond to the so-called
current multipoles, and $\tilde{\mathbb{G}}^{(\ell)}_{ij,A_{\ell}}$ contain
the new multipole tensors that we have just discovered, whose expression is

$\mathbb{G}^{(\ell)}_{ij,A_{\ell}}=\tilde{\mathbb{G}}^{(\ell)}_{ij,A_{\ell}}+\frac{1}{2}\delta_{ij}\left(\mathbb{M}^{(\ell)}_{A_{\ell}}-\tilde{\mathbb{G}}^{(\ell)}_{kk,A_{\ell}}\right)\
,$ (80)

as can be easily derived from the definition (68). Since the latter multipoles
are associated with the spatial part of the metric, it is natural to dub them
_stress_ multipole moments.

It is important to notice that in Eq. (79) the explicit dependence on $G$ is
included for dimensional reasons, and that in general the multipole tensors
themselves can have an explicit dependence on the Newton constant. The
formalism described above is indeed completely general, and takes into account
also the possibility of having _intrinsic_ , i.e. not necessarily spin-
induced, multipoles999This is the case, for example, of a static body deformed
away from spherical symmetry Raposo  _et al._ (2019); Raposo and Pani (2020)..
In that case, in addition to the mass and the angular momenta, a new length
scale is present, and it is always possible to rewrite it in units of the
fundamental scale $Gm$, thus introducing extra powers of $G$ in the multipolar
expansion (an example of this case is the black ring discussed in Sec. V.3).
In order to consider such intrinsic multipoles in a bottom-up approach, from
the amplitude perspective one can consider a non-minimal action that, instead
of terms quadratic in the spin tensor like in Eq. (47), contains new
independent tensors for each term, defined similarly to Eq. (46).

Let us now restrict to spin-induced multipole moments. In the specific case in
which the matter source is axis-symmetric101010In $D=4$ this corresponds to
the usual symmetry along the rotational axis, in $D>4$ one can define a
similar symmetry in which every rotational axis can be exchanged., the mass
and the new stress multipoles contain only even orders of the spherical
harmonics (even $\ell$), while the current multipoles are formed only by odd
orders of them (odd $\ell$). The metric in Eq. (56) describes the most generic
stationary axis-symmetric matter configuration at quadrupole order, and
therefore, according to our definitions, we can write the spin-induced mass
and stress quadrupoles of a generic rotating source in arbitrary dimensions as

$\displaystyle\mathbb{M}^{(2)}_{a_{1}a_{2}}$
$\displaystyle=-\frac{1}{m^{2}}d\Big{(}H_{1}+(d-2)H_{2}\Big{)}S_{a_{1}k}S_{a_{2}}{}^{k}\
,$ (81) $\displaystyle\mathbb{G}^{(2)}_{ij,a_{1}a_{2}}$
$\displaystyle=-\frac{1}{m^{2}}d(d-1)H_{1}S_{(i|a_{1}}S_{|j)a_{2}}\ .$ (82)

It is now evident that the parameters $H_{1}$ and $H_{2}$ of our generic
solution are actually related to the mass quadrupole moment
$\mathbb{M}^{(2)}_{a_{1}a_{2}}$ and to the new stress quadrupole moment
$\mathbb{G}^{(2)}_{ij,a_{1}a_{2}}$. In the next section we will show that this
is indeed the case for the Myers-Perry and black ring solutions, which have a
non-vanishing stress quadrupole moment.

In order to see the degeneracy that occurs in $d=3$ in a “gauge invariant”
way, we just need to rewrite the spin tensor as a spin vector, and the result
reads

$\displaystyle\mathbb{M}^{(2)}_{a_{1}a_{2}}N_{a_{1}a_{2}}\Big{|}_{d=3}$
$\displaystyle=\frac{3}{m^{2}}(H_{1}+H_{2})(S\cdot x)^{2}+\cdots\ ,$ (83)
$\displaystyle\tilde{\mathbb{G}}^{(2)}_{ij,a_{1}a_{2}}N_{a_{1}a_{2}}\Big{|}_{d=3}$
$\displaystyle=\delta_{ij}\mathbb{M}^{(2)}_{a_{1}a_{2}}N_{a_{1}a_{2}}\Big{|}_{d=3}+\cdots\
,$ (84)

where again the ellipses indicate that the expressions are equal up to terms
that contain contraction of the coordinates and some $\delta$’s. Note that,
only in this $d=3$ case, $\tilde{\mathbb{G}}^{(2)}_{ij,a_{1}a_{2}}$ is fixed
in terms of $\mathbb{M}^{(2)}_{a_{1}a_{2}}$ and therefore
$\mathbb{G}^{(2)}_{ij,a_{1}a_{2}}=0$. This result generalizes to all orders in
the multipole expansion.

##  V Particular cases

In this section we compare the metric obtained with our amplitude-based
approach to the PM expansion of known vacuum solutions in $D\geq 4$, and
discuss their multipolar structure. We will see that, in $D=4$, in order to
reproduce a specific solution at quadrupole order we only need to fix a
combination of $H_{1}$ and $H_{2}$, while in $D=5$ we have to fix both of them
independently, proving explicitly the existence of the stress multipole
moment. Furthermore, we define the “simplest” metric as the metric associated
with a minimally-coupled theory. We put it in relation with specific cases,
proving that Kerr black holes are the simplest solution in $D=4$, while Myers-
Perry black holes and black rings in $D=5$ are not.

###  V.1 Hartle-Thorne metric in $D=4$

The Hartle-Thorne metric Hartle (1967); Hartle and Thorne (1968) is the metric
induced by the most generic rotating matter distribution in $D=4$ up to spin-
induced quadrupole order. As shown in Appendix B, we can rewrite the Hartle-
Thorne metric in harmonic coordinates in order to compare it with the
amplitude result. This transformation introduces some free numerical
coefficients due to gauge redundancies that can be mapped to the free
parameters of our solution. With such an identification, up to 3PM, we have
proved that the Hartle-Thorne metric and the amplitude-based one match
exactly. Such comparison has been done in the special frame in which the
Hartle-Thorne metric is defined, namely with the angular momentum aligned with
the $z$-axis. In order to move to such frame, in terms of the spin tensor we
have to choose

$S_{ij}=\pNiceMatrix[columns-width=auto]0&J0\\\ -J00\\\ 000\ ,$ (85)

where $J\equiv a/m$ can be identified with the physical angular momentum of
the spacetime through an expansion in the far-field limit.

The temporal component of the Hartle-Thorne metric in harmonic coordinates
reads

$g_{00}^{\rm
HT}=1-\frac{2Gm}{r}+\frac{a^{2}Gm\zeta}{r^{3}}\left(3\frac{z^{2}}{r^{2}}-1\right)+O(G^{2},a^{3})\
,$ (86)

where $\zeta$ parametrizes the mass quadrupole111111Due to the axisymmetry,
the quadrupole moment tensor is defined by a single parameter.. By setting

$H_{1}+H_{2}=\zeta\ ,$ (87)

the amplitude-based metric fully reproduces the Hartle-Thorne solution.
Therefore, in agreement with Eq. (83), only the combination $H_{1}+H_{2}$
enters in the definition of the mass quadrupole, and since the stress
quadrupole is not independent of it, as shown in Eq. (84), there is only one
degree of freedom. Indeed, as previously discussed, one can find a coordinate
transformation such that the entire metric depends on $H_{1}$ and $H_{2}$ only
through the combination $H_{1}+H_{2}$.

However, since the Hartle-Thorne metric describe the most generic rotating
object in $D=4$, it contains the case in which such an object is a black hole.
In this special scenario, no-hair theorems in $D=4$ state that the only vacuum
solution with a regular horizon is the Kerr metric, which has a well defined
tower of multipole moments only determined by the mass and spin Geroch
(1970b); Hansen (1974). In the Hartle-Thorne formalism, the Kerr metric
corresponds to the solution with $\zeta=1$ and, in terms of Eq. (56), this
means that a Kerr black hole is reproduced by fixing

$H_{1}+H_{2}=1\ .$ (88)

From the amplitude perspective, this condition is satisfied by an infinite
number of non-minimally coupled theories, since $H_{1}$ and $H_{2}$ can be
chosen freely as long as the Kerr condition in Eq. (88) is satisfied. This can
be associated to the degeneracy occurring in $D=4$, that reduces the number of
physical degrees of freedom to one, still having two independent free
parameters. In addition to non-minimal QFTs, the minimally-coupled theory,
that gives rise to what we define the simplest metric121212This name is
inspired by the fact that a minimally-coupled theory is the simplest possible
theory from a QFT perspective., also satisfies the Kerr condition. Thus, from
the point of view of the effective QFT, the scattering amplitude computation
unveils that the simplest vacuum solution in $D=4$ is the Kerr black hole.

###  V.2 Myers-Perry black holes in $D=5$

Let us now consider higher dimensional solutions in General Relativity, and in
particular Myers-Perry black holes in $D=5$ Myers and Perry (1986). First of
all notice that in $D=5$ the number of Casimir associated to the rotation
group $SO(4)$ is two, namely an object can rotate independently with respect
to two independent axes. This means that the Myers-Perry metric in $D=5$ has
two independent angular momenta, and since the classical metric is written in
terms of $(x_{1},y_{1},x_{2},y_{2})$ coordinates, defined in such a way that
the plane $(x_{i},y_{i})$ is orthogonal to the angular momentum $J_{i}$, we
have to block diagonalize the spin tensor as

$S_{ij}=\pNiceMatrix[columns-width=auto]0&J_{1}00\\\ -J_{1}000\\\ 000J_{2}\\\
00-J_{2}0\ ,$ (89)

where the match between the physical angular momenta $J_{i}$ and the spin
parameter of the Myers-Perry metric $a_{i}$ (written in Eq. (117)) is given
by131313Notice that with respect to the appendix C $a_{1}=a$ and $a_{2}=b$.

$m\,a_{i}=\frac{D-2}{2}J_{i}\ .$ (90)

Hence, rewriting the metric in harmonic coordinates as shown in Appendix C,
and fixing the physical parameters as

$H_{1}=\frac{3}{8}\qquad\text{and}\qquad H_{2}=\frac{15}{16}\ ,$ (91)

this metric matches exactly with our solution obtained from amplitudes up to
the order of our expansion (3PM; we believe this is true at all PM orders).

Furthermore, we can see that in this case the degrees of freedom manifestly
match, and $H_{1}$ and $H_{2}$ are independently fixed to a specific numerical
value. From the discussion of the previous sections, this corresponds to
having independent mass and stress multipole moments

$\mathbb{M}^{(2)}_{a_{1}a_{2}}\Big{|}_{d=4}^{\rm
MP}=-\frac{9}{m^{2}}S_{a_{1}k}S_{a_{2}}{}^{k}\ ,$ (92)
$\mathbb{G}^{(2)}_{ij,a_{1}a_{2}}\Big{|}_{d=4}^{\rm
MP}=-\frac{9}{2\,m^{2}}S_{(i|a_{1}}S_{|j)a_{2}}\ .$ (93)

This also shows that, from a QFT point of view, the Myers-Perry solution does
not correspond to a minimally coupled theory. From this perspective, unlike
the Kerr black hole, the Myers-Perry black hole is not the simplest vacuum
solution in General Relativity for $D=5$, indeed Eq. (91) does not coincide
with the minimal limit in (53). Since the Myers-Perry solution is the natural
generalization of the Kerr solution in arbitrary dimensions Emparan and Reall
(2008), one can conclude that, generally speaking, black holes are not the
simplest vacuum solutions in arbitrary dimensions from a scattering amplitude
perspective.

Finally, since when $J_{1}=J_{2}$ the full Myers-Perry metric features an
enhanced symmetry becoming cohomogeneity-1 Myers and Perry (1986), one can
observe that, in terms of multipole expansion, in the limit of equal angular
momenta the mass quadruple vanishes Heynen and Mayerson (2023). Explicitly, it
is possible to see that

$\lim_{J_{1}\rightarrow J_{2}}\mathbb{M}^{(2)}_{a_{1}a_{2}}\Big{|}_{d=4}^{\rm
MP}N_{a_{1}a_{2}}=0\ ,$ (94)

while the stress quadrupole does not vanish. This is not only true for the
Myers-Perry metric Myers and Perry (1986), but actually for the general metric
obtained from the amplitudes for any odd spacetime dimension $D$ when all the
angular momenta are the same. This means that, in general,

$\lim_{J_{i}\rightarrow
J}\mathbb{M}^{(2)}_{a_{1}a_{2}}\Big{|}_{d=\text{even}}N_{a_{1}a_{2}}=0\ .$
(95)

We interpret this as a generic property of the gravitational field sourced by
a spinning point-like mass.

###  V.3 Black rings

As we already mentioned, in $D>4$ Myers-Perry black holes are not the only
vacuum solutions with horizons. One solution beyond the Myers-Perry black
holes is the black ring solution Emparan and Reall (2006), which has ring
topology at the horizon (at variance with the Kerr and Myers-Perry solutions
that instead have spherical topology).

Let us consider for simplicity the case with only one angular momentum, where
$S_{21}=-S_{12}=J$ are the only non-vanishing components of the spin tensor.
The original solution is written in terms of three parameters
$(\mathcal{R},\nu,\lambda)$ with $0<\nu\leq 1$, which encode mass, spin, and
shape of the ring. As shown in Appendix D, we can replace $\mathcal{R}$ and
$\nu$ in favor of $m$ and $J$, which are respectively the mass and the spin of
the solution. In terms of $(m,J,\lambda)$, we can write the metric in harmonic
coordinates in a PM expansion. For a generic value of $\lambda$, the black
ring solution has a naked conical singularity unless

$\lambda=\frac{2\nu(m,J)}{1+\nu^{2}(m,J)}\ ,$ (96)

which in turn implies that mass and spin are not independent. This condition
also corresponds to the existence of equilibrium solutions in the absence of
external forces, and sets a lower bound on the angular momentum. However, we
keep $0<\lambda\leq 1$ as a free parameter, thus describing a family of
solutions, of which only one is free of naked conical singularities for a
given $m$ and $J$. The Myers-Perry solution is then recovered from the black
ring if $\lambda\rightarrow 1$, keeping fixed $m$ and $J$ (this is outside the
equilibrium curve of the ring).

With the aim of comparing the black ring metric with Eq. (56), consider the
quadrupole structure described in (79). Both mass and stress quadrupole
moments have a dimension of a length squared (in $\hbar=c=1$ units), meaning
that in $D=5$ they can be written in terms of dimensionful quantities
$S^{2}/m^{2}$ or $Gm$141414We recall that in higher dimension $[Gm]=L^{D-3}$..
If a new fundamental length scale $\Lambda$ is present in the gravitational
source, schematically the intrinsic quadrupole reads

$\mathbb{M}^{(2)}\sim\Lambda=\sigma\,Gm\ ,$ (97)

where $\sigma$ is a dimensionless parameter. This means that, if one writes
$\Lambda$ in units of $Gm$, such non-spin-induced moment would enter at 2PM in
the expansion of the metric, even though it is associated with a tree-level
scattering amplitude. This is the case for the black ring solution, that has a
ring topology and hence a non-vanishing intrinsic quadrupole moment even in
the non-spinning limit. However, since the amplitude-based approach presented
in this paper focuses on spin-induced multipole moments, we cannot match the
metric at quadrupole level including intrinsic moments. We restrict in this
case to 1PM order, in which only spin-induced moments are present and we
expect to be able to fully reconstruct the metric and its spin-induced
multipoles.

Then, from the results of Appendix D, fixing the physical parameters as

$H_{1}=\frac{3}{4(1+\lambda)}\ ,\qquad
H_{2}=\frac{3(6\lambda-1)}{8(1+\lambda)}\ ,$ (98)

we have verified the agreement of the black ring metric with the amplitude-
based one, up to 1PM. Indeed, we can easily check that for $\lambda=1$ we
recover exactly the expected coefficients $H_{1}$ and $H_{2}$ of the Myers-
Perry solution in Eq. (91). As in the Myers-Perry black holes, in Eq. (98) for
a given value of $\lambda$, the parameters $H_{1}$ and $H_{2}$ are
independently fixed, and so also black rings in $D=5$ have independent mass
and stress quadrupole moments. For completeness, their explicit expression is

$\displaystyle\mathbb{M}^{(2)}_{a_{1}a_{2}}\Big{|}_{d=4}^{\text{BR}}$
$\displaystyle=-\frac{1}{m^{2}}\frac{18\lambda}{1+\lambda}S_{a_{1}k}S_{a_{2}}{}^{k}+O(Gm)\
,$ (99)
$\displaystyle\mathbb{G}^{(2)}_{ij,a_{1}a_{2}}\Big{|}_{d=4}^{\text{BR}}$
$\displaystyle=-\frac{1}{m^{2}}\frac{9}{1+\lambda}S_{(i|a_{1}}S_{|j)a_{2}}+O(Gm)\
,$ (100)

where we are neglecting the intrinsic (i.e. non-spin-induced) quadrupole
moments. Given the extra parameter $\lambda$ (in addition to the mass and
angular momentum), one could check whether there exists a particular black
ring solution corresponding to the simplest metric generated by the minimal
vertex of the QFT. However, its easy to verify that there is no value of
$\lambda$ such that $H_{1}=1$ and $H_{2}=0$ at the same time. This shows that
all black ring solutions with single angular momentum are generated by non-
minimally coupled theories from the scattering amplitude perspective.

###  V.4 The simplest metric in any spacetime $D$ dimensions

For completeness, we give here the explicit form of the simplest solution in
generic spacetime $D$ dimensions, obtained from the minimal vertex in (45) and
corresponding to $H_{1}=1$ and $H_{2}=0$:

$\displaystyle h_{00}^{(1,2)}(r)$
$\displaystyle=\frac{2(D-3)}{D-2}\frac{r^{2}S_{k_{1}k_{2}}S^{k_{1}k_{2}}-(D-1)\,x^{k_{1}}x^{k_{2}}S_{k_{1}}{}^{k_{3}}S_{k_{2}k_{3}}}{mr^{4}}G\rho(r)\
,$ (101) $\displaystyle h_{0i}^{(1,2)}(r)$ $\displaystyle=0\ ,$ $\displaystyle
h_{ij}^{(1,2)}(r)$
$\displaystyle=-\frac{2(D-3)}{(D-2)mr^{4}}\Bigg{(}-r^{2}(D-2)S_{ik}S_{j}{}^{k}+r^{2}S_{k_{1}k_{2}}S^{k_{1}k_{2}}\delta_{ij}$
$\displaystyle+(D-1)\,x^{k_{1}}x^{k_{2}}\Big{(}(D-2)S_{ik_{1}}S_{jk_{2}}-S_{k_{1}}{}^{k_{3}}S_{k_{2}k_{3}}\delta_{ij}\Big{)}\Bigg{)}G\rho(r)\
.$

We stress again that the 1PM order contains all the physics that we want to
capture, and we consider only the quadrupole term because monopole and dipole
moments are uniquely fixed. In this regard the quadrupole moments of this
solution are

$\displaystyle\left.\mathbb{M}^{(2)}_{a_{1}a_{2}}\right|^{\rm simplest}$
$\displaystyle=-\frac{D-1}{m^{2}}S_{a_{1}k}S_{a_{2}}{}^{k}\ ,$ (102)
$\displaystyle\left.\mathbb{G}^{(2)}_{ij,a_{1}a_{2}}\right|^{\rm simplest}$
$\displaystyle=-\frac{(D-1)(D-2)}{m^{2}}S_{(i|a_{1}}S_{|j)a_{2}}\ .$ (103)

As previously discussed, in $D=4$ this solution corresponds to the Kerr
metric. However, when considering $D=5$, it does not match either the Myers-
Perry metric or the black ring with a single angular momentum. Since our
analysis shows that the minimal solution is not a black hole, we leave it as
an interesting open problem to identify whether an exact solution exists that
corresponds to this metric, and possibly what is the matter content sourcing
such solution, similarly to the Hartle-Thorne metric in $D=4$. Moreover, one
could in principle still construct a black hole solution whose spin-induced
part of each multipole matches the one of the simplest metric. At tree-level
the theory is linear and this could be done by superimposing various solutions
and their corresponding moments. However, in general such superposition is
invalid beyond the linear level, so constructing such solution (if it exists)
is highly nontrivial.

Another interesting possibility is the existence of different simplest
solutions in $D>4$. As we briefly mentioned in Sec. II, when we refer to
spin-$s$ fields we are assuming a completely symmetric and trace-less field
representation of the Lorentz group, but in higher dimensions there are more
possible representations, and for example in $D=5$ we can build an action that
couples an anti-symmetric field with gravity. The minimal vertex associated to
this theory could be either equal to (45), thus suggesting the fundamental
nature of the simplest metric, or different, leading to an another simplest
solution that can be compared to Myers-Perry or other black hole solutions.
Likewise, one could consider a theory for a minimally coupled field with spin
$s>1$ and check whether the simplest metric is universal, at least up to the
quadrupole order.

Finally, it is important to stress that Eq. (52) (and therefore the most
generic metric in $D$ dimensions) could also be derived without the use of
scattering amplitudes, but simply by constructing the most generic stress-
energy tensor compatible to its symmetries in momentum space. However, having
an underlying QFT allows us to identify the simplest solution among the
general family, and gives to us the possibility to interpret black holes from
a different point of view. The fact that in $D=4$ the simplest solution
corresponds to the Kerr metric seems to resemble the no-hair theorem in a QFT
language, in which the absence of hair corresponds to the absence of extra
coupling in the effective action. In $D>4$ the story is more complicated, and
the absence of black hole uniqueness theorems seems to spoil the connection
between simplest metrics and black holes solutions, although our work suggests
further investigations along this line.

##  VI Conclusions

In this work we computed the vacuum solution describing the metric of a
generic rotating object in arbitrary dimensions, up to spin-induced quadrupole
order and within a PM expansion. Our computation is based on the the classical
contributions of scattering amplitudes describing the emission of gravitons
out of massive spin-1 particles and provides valuable insights into the nature
of spinning compact objects in arbitrary dimensions.

In the context of $D=4$ spacetime dimensions, our computations have
successfully recovered the PM expansion of the well-known vacuum Hartle-Thorne
solution, describing the metric of a spinning compact object in General
Relativity up to second order in the angular momentum. The Kerr metric
describing a spinning black hole is a special case arising for a specific
choice of the mass quadrupole moment. At the level of the effective quantum
field theory, this choice corresponds to the case of a Proca field minimally
coupled to gravity. This reaffirms black holes as the simplest solutions to
General Relativity in four dimensions also from a scattering-amplitude
perspective.

However, our investigation reveals that the situation is different and richer
in higher ($D>4$) dimensions. Here, through our scattering amplitude analysis
up to 2-loop calculations, we have obtained the generic solution including
quadrupole-moments terms quadratic in the object’s angular momentum.
Interestingly, at variance with the $D=4$ case, we have found that the
solution depends on _two_ quadrupole moment parameters, namely the standard
mass quadrupole and a new stress quadrupole; the latter does not exist in
$D=4$ and was previously missed in the analysis of Heynen and Mayerson (2023).
We have successfully identified and characterized different $D>4$ black-hole
solutions, such as the Myers-Perry and the black ring metrics, as particular
cases obtained for specific choices of the parameters of our general PM
solution. Notably, unlike the four-dimensional case, none of these solutions
corresponds to the choice of minimal couplings in the effective action,
highlighting that the nature of black holes in higher dimensions is more
intricate.

It would be interesting to assess whether this property is somehow linked to
the absence of a black-hole uniqueness and no-hair theorem in higher-
dimensional General Relativity. Nevertheless, it is known that the Myers-Perry
solution is the unique stationary, non-extremal, asymptotically flat, vacuum
black hole solution with spherical topology Emparan and Reall (2008); Morisawa
and Ida (2004), which are exactly the hypotheses behind our simplest metric.
This means that if the Myers-Perry solution is not the simplest one, strictly
speaking no other solution can be within our framework. However, an
interesting follow-up is to understand whether the concept of simplest metric
is universal in $D>4$, for example by constructing the metric arising from a
minimally coupled theory other than Proca, in this case it could happen that
another simplest solution matches the Myers-Perry one.

Even though we focused on the case of solutions with spin-induced multipole
moments, it should be possible to extend our framework to describe generically
deformed compact objects (an example being the black ring solution that has
intrinsic moments in addition to the spin-induced ones), also featuring
higher-order multipole moments or moments that break the Kerr symmetries
(e.g., current quadrupoles, mass and stress octupoles that break the
equatorial symmetry, or generically moment tensors that break the axisymmetry)
Raposo  _et al._ (2019); Bena and Mayerson (2020); Bianchi  _et al._ (2020);
Bena and Mayerson (2021); Bianchi  _et al._ (2021); Fransen and Mayerson
(2022).

Another interesting avenue of exploration would be to identify the solution
corresponding to the minimal coupling in $D>4$ at the full nonlinear level.
Our results suggest that such a solution (if it exists) should be stellar-
like, i.e. it requires matter fields coupled to gravity, or perhaps it
corresponds to black holes with complex topology as those obtained by
“superimposing” different black hole solutions using the inverse scattering
method, as in the case of the black saturn Elvang and Figueras (2007), multi-
ring, or other Emparan and Figueras (2010); Elvang  _et al._ (2007), possibly
yet unknown, solutions Emparan and Reall (2008). The study of compact objects
other than black holes and their multipolar structure in higher dimensions is
seldom explored and worth investigating to address this question.

We conclude by highlighting that the discovery of stress multipoles opens
several interesting avenues for future investigation, both at the
phenomenological level and for a more fundamental understanding of gravity.
Although we have shown that the stress moments do not exist in $D=4$ General
Relativity, it would be very interesting to assess whether this happens also
for other gravity theories. We expect that other gravitational theories with
more degrees of freedom (e.g. massive gravity or generic metric theories
propagating up to six polarizations at the linearized level) should indeed
have non-vanishing stress multipoles. If this is the case, it would be
interesting to understand the phenomenology associated to these new moments.
This might also change the way in which we build stationary solutions beyond
General Relativity or perform tests of exotic compact objects (see, e.g,
Loutrel  _et al._ (2022)). Furthermore, in higher dimensions these
considerations have to be taken into account already within General
Relativity, potentially impacting on the way in which we look for new
solutions and study their linearized dynamics. For example, even for a
spherical object the new multipole moments can be induced by an external tidal
field, giving rise to _stress Love numbers_ , associated with tensor
perturbations of compact objects in $D>4$ Kodama and Ishibashi (2003), which
are indeed absent in $D=4$ (see Kol and Smolkin (2012); Cardoso  _et al._
(2019); Hui  _et al._ (2021) for a discussion of the tidal Love numbers in
higher dimensions).

Finally, even though the stress multipoles could have been found within a
classical General Relativity framework, considering scattering amplitudes was
really key to identify their existence. In particular, we realized that
working in momentum space it is relatively easy to write down the most generic
stress-energy tensor at arbitrary high-order in the multipole expansion, at
variance with working in position space (the natural setting of General
Relativity), in which deriving the most generic expression of $T_{\mu\nu}$
would be more and more tedious as the order of multipoles increases. This is a
perfect example of how recovering classical gravity from scattering amplitudes
not only can give us insights on the phenomenology, but can also open new
perspectives to deepen our understanding of gravity at a more fundamental
level.

###### Acknowledgements.

We thank George Pappas for interesting correspondence, and Roberto Emparan,
Leonardo Gualtieri, Jef Heynen and Daniel Mayerson for useful comments on the
draft. This work is partially supported by the MIUR PRIN Grant 2020KR4KN2
“String Theory as a bridge between Gauge Theories and Quantum Gravity”, by the
FARE programme (GW-NEXT, CUP: B84I20000100001), by the EU Horizon 2020
Research and Innovation Programme under the Marie Sklodowska-Curie Grant
Agreement No. 101007855, and by the INFN TEONGRAV initiative.

##  Appendix A Quadrupole moments in four and five spacetime dimensions

In this appendix we perform in detail the decomposition in irreducible $SO(d)$
representations of the constant tensor $\mathcal{G}_{ij,ab}$ which enters the
expansion in Eq. (70) and it is relevant for the analysis of the quadrupole
moments. As observed in subsection IV.2, the tensor is symmetric in the
indices $ij$ and $ab$, and it is also traceless in $ab$. We will consider
explicitly the five-dimensional case, but the same results apply in any
dimension $D>4$. To highlight the difference with the four-dimensional case,
we will first quickly review how the latter works, and then move to $D=5$.

In four dimensions the tensor $\mathcal{G}_{ij,ab}$ belongs to the $({\bf
5}\oplus{\bf 1})\otimes{\bf 5}$ of $SO(3)$. This decomposes in irreducible
representations as

$({\bf 5}\oplus{\bf 1})\otimes{\bf 5}={\bf 9}\oplus{\bf 7}\oplus 2\times{\bf
5}\oplus{\bf 3}\oplus{\bf 1}$ (104)

which correspond to symmetric traceless tensors with 4, 3, 2, 1 and 0 indices
of $SO(3)$, respectively. It is straightforward to recognize these
representations as the ones of the tensors $\mathcal{G}^{(4)}$,
$\mathcal{G}^{(6)}$, $\mathcal{G}^{(1)}$, $\mathcal{G}^{(3)}$,
$\mathcal{G}^{(5)}$ and $\mathcal{G}^{(2)}$, respectively, that appear in Eq.
(71).

We can now move to five dimensions. In this case the indices belong to
$SO(4)$, which is isomorphic to $SU(2)\times SU(2)$. We can then label the
representations in terms of the ones of each of the two $SU(2)$’s, and in
particular the tensor $\mathcal{G}_{ij,ab}$ belongs to the $(({\bf 3},{\bf
3})\oplus({\bf 1},{\bf 1}))\otimes({\bf 3},{\bf 3})$, which decomposes in
irreducible representations as

$\displaystyle(({\bf 3},{\bf 3})\oplus({\bf 1},{\bf 1}))\otimes({\bf 3},{\bf
3})=({\bf 5},{\bf 5})\oplus({\bf 5},{\bf 3})\oplus({\bf 3},{\bf 5})$ (105)
$\displaystyle\oplus({\bf 5},{\bf 1})\oplus({\bf 1},{\bf 5})\oplus
2\times({\bf 3},{\bf 3})\oplus({\bf 3},{\bf 1})\oplus({\bf 1},{\bf
3})\oplus({\bf 1},{\bf 1})\ .$

We want to identify such representations with the index structure and
constraints of the tensors in Eq. (76). The $({\bf 5},{\bf 5})$ representation
corresponds to the four-index symmetric traceless tensor $\mathcal{G}^{(4)}$,
while the $({\bf 3},{\bf 3})$’s correspond to the two-index symmetric
traceless tensors $\mathcal{G}^{(1)}$ and $\mathcal{G}^{(3)}$, and the $({\bf
1},{\bf 1})$ is the singlet $\mathcal{G}^{(2)}$. The $({\bf 3},{\bf
1})\oplus({\bf 1},{\bf 3})$ identifies the antisymmetric tensor
$\mathcal{G}^{(5)}$. The remaining representations, namely $({\bf 5},{\bf
3})\oplus({\bf 3},{\bf 5})$ and $({\bf 5},{\bf 1})\oplus({\bf 1},{\bf 5})$,
correspond to tensors with mixed symmetry.

The tensor $\mathcal{G}^{(6)}$ in the $({\bf 5},{\bf 3})\oplus({\bf 3},{\bf
5})$ is the one identified in Heynen and Mayerson (2023) as the ASTF tensor
$\mathcal{H}_{ab,cd}$ with four indices, antisymmetric in the first two and
symmetric traceless in the last two. Such tensor, being irreducible, satisfies
the additional constraints151515To avoid overburderning with notation, in this
appendix we leave the STF, ASTF, and RSTF notation implicit, for instance
$\mathcal{G}^{(6)}_{ab,cd}\equiv\mathcal{G}^{(6)}_{\\{ab,cd\\}_{\rm ASTF}}$
and $\mathcal{G}^{(7)}_{ab,cd}\equiv\mathcal{G}^{(7)}_{\\{ab,cd\\}_{\rm
RSTF}}$.

$\delta^{bc}\mathcal{G}^{(6)}_{ab,cd}=0\qquad\epsilon^{abce}\mathcal{G}^{(6)}_{ab,cd}=0\
.$ (106)

To show that these constraints lead to 30 surviving components161616Note that
Heynen and Mayerson (2023) ascribes 40 components to this tensor. We believe
this is a mistake. Indeed, the 40 components corresponds to the 30 components
of $\mathcal{G}^{(6)}$ plus the 10 components of the tensor
$\mathcal{G}^{(7)}$ described below. we can start by observing that the number
of components of the tensor before imposing the constraints is $6\times 9=54$.
The first constraint in Eq. (106) removes 15 components, because both the
indices $a$ and $d$ can take any value but the trace in $ad$ of the constraint
is identically zero. The second constraint in Eq. (106) removes 9 additional
components. Indeed, given the previous constraint,
$\epsilon^{abce}\mathcal{G}^{(6)}_{ab,cd}$ is symmetric traceless in $ed$. The
fact that it is traceless is due to the symmetry in $cd$, while the fact that
it is symmetric can be seen using the Fierz identity

$\epsilon^{[abce}\mathcal{G}^{(6)}_{ab,c}{}^{d]}=0\ ,$ (107)

which implies that the antisymmetric part of
$\epsilon^{abce}\mathcal{G}^{(6)}_{ab,cd}$ vanishes identically. We are thus
left with $54-15-9=30$ components.

The tensor $\mathcal{G}^{(7)}$ in the $({\bf 5},{\bf 1})\oplus({\bf 1},{\bf
5})$, which is missing in Heynen and Mayerson (2023), is antisymmetric in both
the pairs $ab$ and $cd$, which makes in total 36 components. Besides, it
satisfies the constraints

$\delta^{bc}\mathcal{G}^{(7)}_{ab,cd}=0\qquad\epsilon^{abce}\mathcal{G}^{(7)}_{ab,cd}=0\
.$ (108)

Using arguments similar to the ones above, one can show that the first
constraint removes 16 components and the second removes 10 components, leaving
in total $36-16-10=10$ components.

The analysis performed here in five dimensions can be generalized to any
$D>4$, implying that only in four dimensions one can choose a gauge such that
$\gamma_{ij}=0$ as in Eq. (75).

##  Appendix B Hartle-Thorne metric in harmonic coordinates

The most generic rotating object in $D=4$ is described up to spin-induced
quadrupole order by the Hartle-Thorne metric, which is exact in $G$ Hartle
(1967); Hartle and Thorne (1968). Given a spherical set of coordinates
$(t,r,\theta,\phi)$, the explicit expression of the metric up to 3PM order is

$\displaystyle g_{tt}$
$\displaystyle=1-\frac{2Gm}{r}+\frac{a^{2}Gm\zeta(3\cos(2\theta)+1)}{2r^{3}}$
(109)
$\displaystyle+\frac{a^{2}G^{2}m^{2}((3\zeta-2)\cos(2\theta)+\zeta-2)}{2r^{4}}+\frac{a^{2}G^{3}m^{3}(4\zeta-11)(3\cos(2\theta)+1)}{7r^{5}}+O(G^{4},a^{3})\
,$ $\displaystyle g_{t\phi}$
$\displaystyle=+\frac{2aGm\sin^{2}(\theta)}{r}+O(G^{4},a^{3})\ ,$
$\displaystyle g_{rr}$
$\displaystyle=-1-\frac{2Gm}{r}+\frac{(a^{2}Gm\zeta(3\cos(2\theta)+1))}{2r^{3}}-\frac{4G^{2}m^{2}}{r^{2}}-\frac{a^{2}G^{2}m^{2}(-5\zeta-3(5\zeta-8)\cos(2\theta)+4)}{2r^{4}}$
$\displaystyle-\frac{8G^{3}m^{3}}{r^{3}}-\frac{a^{2}G^{3}m^{3}(-60\zeta-9(20\zeta-27)\cos(2\theta)+25)}{7r^{5}}+O(G^{4},a^{3})\
,$ $\displaystyle g_{\theta\theta}$
$\displaystyle=-r^{2}+\frac{a^{2}Gm\zeta(3\cos(2\theta)+1)}{2r}+\frac{a^{2}G^{2}m^{2}(5\zeta-1)(3\cos(2\theta)+1)}{4r^{2}}$
$\displaystyle+\frac{18a^{2}G^{3}m^{3}(\zeta-1)(3\cos(2\theta)+1)}{7r^{3}}+O(G^{4},a^{3})\
,$ $\displaystyle g_{\phi\phi}$
$\displaystyle=-r^{2}\sin^{2}(\theta)+\frac{a^{2}Gm\zeta(3\cos(2\theta)+1)\sin^{2}(\theta)}{2r}+\frac{\left(a^{2}G^{2}m^{2}(5\zeta-1)(3\cos(2\theta)+1)\sin^{2}(\theta)\right)a^{2}}{4r^{2}}$
$\displaystyle+\frac{18\left(a^{2}G^{3}m^{3}(\zeta-1)(3\cos(2\theta)+1)\sin^{2}(\theta)\right)}{7r^{3}}+O(G^{4},a^{3})\
,$

where $a=J/m$ is the angular momentum per unit mass and $\zeta$ is the
quadrupole mass moment normalized such that for $\zeta=1$ we have the Kerr
limit.

In order to compare such metric with the one derived by scattering amplitudes,
we have to impose the harmonic gauge. The reason why this reference frame is
particularly suitable is twofold: on one hand this is a “universal gauge”, in
the sense that there exists a gauge fixing term at the level of the action
such that every metric can be put in this frame, and on the other hand it is
classically defined by the covariant d’Alambertian of the associated Cartesian
coordinates, that makes it possible to define the transformation on both the
original and the transformed metric. Defining the spherical harmonic
coordinates as $(T,R,\Theta,\Phi)$, and the associated Cartesian coordinates
as

$\begin{cases}x=R\sin(\Theta)\cos(\Phi)\ ,\\\ y=R\sin(\Theta)\sin(\Phi)\ ,\\\
z=R\cos(\Theta)\ ,\end{cases}$ (110)

the equation that the metric in harmonic spherical coordinates has to satisfy
is

$g^{\mu\nu}D_{\mu}\partial_{\nu}(T,x,y,z)=0\ ,$ (111)

where each coordinate is treated as a scalar. Noticing that the metric in
(109) does not depend on the azimuthal angle $\phi$, we can define a
coordinate transformation as

$T=t\ ,\qquad R=r(R,\Theta)\ ,\qquad\Theta=\theta(R,\Theta)\ ,\qquad\Phi=\phi\
.$ (112)

We can now define an ansatz for the relation between the two set of
coordinates as

$r(R,\Theta)=R\sum_{i=0}^{n\text{PM}}\left(\frac{Gm}{R}\right)^{i}\sum_{j=0}^{\lfloor
n\text{Pole}/2\rfloor}\left(\frac{a}{R}\right)^{2j}\sum_{k=0}^{j}\mathcal{C}_{i,2j,k}^{(R)}P_{2k}(\cos\Theta)\
,$ (113)
$\cos\theta(R,\Theta)=\cos(\Theta)\Bigg{(}1+\sum_{i=0}^{n\text{PM}}\left(\frac{Gm}{R}\right)^{i}\sum_{j=1}^{\lfloor
n\text{Pole}/2\rfloor}\left(\frac{a}{R}\right)^{2j}\sum_{k=0}^{j}\mathcal{C}_{i,2j,k}^{(\Theta)}P_{2k}(\cos\Theta)\Bigg{)}\
,$ (114)

where $P_{n}$ are the Legendre polynomials and $\lfloor\cdot\rfloor$ stands
for the integer part. The ansatz is motivated as follows:

  * •

In the limit in which $R\rightarrow+\infty$ the two reference frames must
coincide, which implies $\mathcal{C}^{(R)}_{0,0,0}=1$.

  * •

In the non-spinning limit ($a\rightarrow 0$), the Schwarzschild metric must be
recovered, and so in (114) for $a=0$ we have to impose $\theta=\Theta$.

  * •

In the original metric at a given spin power, the order of the Legendre
polynomials is always equal or lower than the spin power itself, hence we set
$k<j$.

  * •

Since we have to respect the time-reversal symmetry in which $t\rightarrow-t$
and $\Phi\rightarrow-\Phi$, only even powers of the angular momentum are
allowed.

Finally, performing the coordinate transformation and imposing equation (111),
we end up with

$\begin{gathered}\mathcal{C}^{(R)}_{0,0,0}=1\
,\quad\mathcal{C}^{(R)}_{0,2,2}=0\ ,\quad\mathcal{C}^{(R)}_{0,2,0}=0\
,\quad\mathcal{C}^{(\Theta)}_{0,2,2}=0\
,\quad\mathcal{C}^{(\Theta)}_{0,2,0}=0\ ,\\\ \mathcal{C}^{(R)}_{1,0,0}=1\
,\quad\mathcal{C}^{(\Theta)}_{1,2,2}=-\mathcal{C}^{(R)}_{1,2,2}\
,\quad\mathcal{C}^{(\Theta)}_{1,2,0}=\mathcal{C}^{(R)}_{1,2,0}\ ,\\\
\mathcal{C}^{(R)}_{2,0,0}=0\
,\quad\mathcal{C}^{(\Theta)}_{2,2,2}=\frac{\zeta-4}{3}\
,\quad\mathcal{C}^{\Theta}_{2,2,0}=\frac{4-\zeta}{3}\
,\quad\mathcal{C}^{(R)}_{2,2,2}=\frac{7}{3}\
,\quad\mathcal{C}^{(R)}_{2,2,0}=-\frac{1}{3}\ ,\\\
\mathcal{C}^{(\Theta)}_{3,2,2}=\frac{64}{9}-\mathcal{C}^{(R)}_{1,2,2}+\frac{2}{3}\mathcal{C}^{(R)}_{3,2,2}\
,\quad\mathcal{C}^{(\Theta)}_{3,2,0}=-\frac{64}{9}+\mathcal{C}^{(R)}_{1,2,2}-\frac{2}{3}\mathcal{C}^{(R)}_{3,2,2}\
,\quad\mathcal{C}_{3,2,0}^{(R)}=\frac{2}{3}+\frac{3}{5}\mathcal{C}^{(R)}_{1,2,0}\
.\end{gathered}$ (115)

Once the coordinate transformation is fixed, we can find the metric in
harmonic gauge at the chosen perturbative order by means of (112). We notice
that the coordinate transformation has some gauge redundancies since there are
coefficients which are unfixed. At 1PM and 2PM the redundancy is parameterized
by two coefficients, namely $\mathcal{C}^{(R)}_{1,2,0}$ and
$\mathcal{C}^{(R)}_{1,2,2}$. At 3PM a new redundancy arises by means of the
unfixed coefficients $\mathcal{C}^{(R)}_{3,0,0}$ and
$\mathcal{C}^{(R)}_{3,2,2}$. While the full 3PM result is given in the
attached Mathematica notebook, at 1PM the explicit expression of the Hartle-
Thorne metric in harmonic gauge reads

$\displaystyle g_{tt}$
$\displaystyle=1-\frac{2Gm}{R}+\frac{a^{2}Gm\zeta(3\cos(2\Theta)+1)}{2R^{3}}+O(G^{2},a^{3})\
,$ (116) $\displaystyle g_{t\Phi}$
$\displaystyle=\frac{2aGm\sin^{2}(\Theta)}{R}+O(G^{2},a^{3})\ ,$
$\displaystyle g_{RR}$
$\displaystyle=-1-\frac{2Gm}{R}+a^{2}Gm\frac{8\mathcal{C}^{(R)}_{1,2,0}+(3\cos(2\Theta)+1)(\zeta+2\mathcal{C}^{(R)}_{1,2,2})}{2R^{3}}+O(G^{2},a^{3})\
,$ $\displaystyle g_{\Theta\Theta}$
$\displaystyle=-R^{2}-2GmR+a^{2}Gm\frac{\zeta(3\cos(2\Theta)+1)+\mathcal{C}^{(R)}_{1,2,2}(3\cos(2\Theta)-1)-4\mathcal{C}^{(R)}_{1,2,0}}{2R}+O(G^{2},a^{3})\
,$ $\displaystyle g_{R\Theta}$
$\displaystyle=-\frac{3Gma^{2}\sin^{2}(\Theta)\mathcal{C}^{(R)}_{1,2,2}}{4R^{2}}+O(G^{2},a^{3})\
,$ $\displaystyle g_{\Phi\Phi}$
$\displaystyle=-R^{2}\sin^{2}(\Theta)-2GmR\sin^{2}(\Theta)$
$\displaystyle+a^{2}Gm\sin^{2}(\Theta)\frac{\zeta(3\cos(2\Theta)+1)-4\mathcal{C}^{(R)}_{1,2,0}+2\mathcal{C}^{(R)}_{1,2,2}}{2R}+O(G^{2},a^{3})\
,$

and moving to Cartesian coordinates such metric can be directly compared with
the amplitude-based one.

This perturbative approach to the harmonic coordinate transformation can be
compared directly with Aguirregabiria  _et al._ (2001), in which a similar
procedure is described. Considering only terms up to 3PM and spin-square
orders, we find disagreement with the number of gauge redundancies present in
the metric after the transformation. In particular at 1PM we find two free
parameters, while in the mentioned work there is only one present. Our result
is also confirmed by the amplitude approach, in which as we showed in
subsection V.1 there are two gauge degrees of freedom coming from the dressed
vertex in $D=4$.

##  Appendix C Myers-Perry black holes in $D=5$ in harmonic coordinates

Myers-Perry black holes are a class of stationary vacuum solutions of General
Relativity built to be the generalization of the Kerr metric in arbitrary
dimensions Myers and Perry (1986). In $D=5$ the metric reads

$\displaystyle ds^{2}=dt^{2}$
$\displaystyle-\frac{\mu}{\Sigma}\left(dt+a\,\sin^{2}\theta\,d\phi_{1}+b\,\cos^{2}\theta\,d\phi_{2}\right)^{2}-\frac{r^{2}\Sigma}{\Pi-\mu
r^{2}}dr^{2}$ (117) $\displaystyle-\Sigma
d\theta^{2}-(r^{2}+a^{2})\sin^{2}\theta\,d\phi_{1}^{2}-(r^{2}+b^{2})\cos^{2}\theta\,d\phi_{2}^{2}\
,$

where

$\Sigma=r^{2}+a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta\
,\qquad\Pi=(r^{2}+a^{2})(r^{2}+b^{2})\ ,$ (118)

and $a$ and $b$ are two independent angular momenta and

$\mu=\frac{16\pi Gm}{(D-2)\Omega_{D-2}}$ (119)

with $\Omega_{D}$ the surface of a $D$-sphere. As we did in the $D=4$ case, we
can define a set of Cartesian harmonic coordinates as

$\begin{cases}x_{1}=R\sin(\Theta)\cos(\Phi_{1})\ ,\\\
y_{1}=R\sin(\Theta)\sin(\Phi_{1})\ ,\\\ x_{2}=R\cos(\Theta)\cos(\Phi_{1})\
,\\\ y_{2}=R\cos(\Theta)\sin(\Phi_{1})\ ,\end{cases}$ (120)

related to the original coordinates through

$T=t\ ,\qquad R=r(R,\Theta)\ ,\qquad\Theta=\theta(R,\Theta)\
,\qquad\Phi_{1}=\phi_{1}\ ,\qquad\Phi_{2}=\phi_{2}\ ,$ (121)

and such that

$g^{\mu\nu}D_{\mu}\partial_{\nu}(T,x_{1},y_{1},x_{2},y_{2})=0\ .$ (122)

Motivated by the same reasons of appendix B, the ansatz for the perturbative
coordinate transformation is the following

$\displaystyle
r(R,\Theta)=R\sum_{i=0}^{n\text{PM}}\left(Gm\rho(R)\right)^{i}\sum_{j=0}^{\lfloor
n\text{Pole}/2\rfloor}\sum_{k=0}^{j}\mathcal{C}_{i,2j,2k}^{(R)}\Bigg{(}\left(\frac{a}{R}\right)^{2j}P_{2k}(\cos\Theta)+\left(\frac{b}{R}\right)^{2j}P_{2k}(\sin\Theta)\Bigg{)}$
(123)
$\displaystyle+R\sum_{i=2}^{nPM}\left(Gm\rho(R)\right)^{i}\log(mG\rho(R))\sum_{j=0}^{\lfloor
n\text{Pole}/2\rfloor}\sum_{k=0}^{j}\mathcal{C}_{i,2j,2k}^{(\text{log},R)}\Bigg{(}\left(\frac{a}{R}\right)^{2j}P_{2k}(\cos\Theta)+\left(\frac{b}{R}\right)^{2j}P_{2k}(\sin\Theta)\Bigg{)}\
,$
$\displaystyle\cos\theta(R,\Theta)=\cos\theta\sum_{i=0}^{n\text{PM}}\left(Gm\rho(R)\right)^{i}\sum_{j=0}^{\lfloor
n\text{Pole}/2\rfloor}\sum_{k=0}^{j}\Bigg{(}\mathcal{C}_{i,2j,2k}^{(\Theta,a)}\left(\frac{a}{R}\right)^{2j}P_{2k}(\cos\Theta)+\mathcal{C}_{i,2j,2k}^{(\Theta,b)}\left(\frac{b}{R}\right)^{2j}P_{2k}(\sin\Theta)\Bigg{)}$
$\displaystyle+R\sum_{i=2}^{nPM}\left(Gm\rho(R)\right)^{i}\log(mG\rho(R))\\!\\!\\!\\!\sum_{j=0}^{\lfloor
n\text{Pole}/2\rfloor}\\!\\!\\!\sum_{k=0}^{j}\Bigg{(}\mathcal{C}_{i,2j,2k}^{(\text{log},\Theta,a)}\left(\frac{a}{R}\right)^{2j}P_{2k}(\cos\Theta)+\mathcal{C}_{i,2j,2k}^{(\text{log},\Theta,b)}\left(\frac{b}{R}\right)^{2j}P_{2k}(\sin\Theta)\Bigg{)}\
.$

Notice that differently from the $D=4$ case, here we have to consider
logarithmic pieces starting from 2PM. This is consistent with the amplitude
based approach in which we observe the presence of singularities in $D=5$ even
in the harmonic gauge, renormalized by the insertion of higher-loop vertices
giving rise to logarithmic terms. Moreover the coordinate transformation in
Eqs. (123) takes into account the symmetries of the original metric, such that
the symmetry $a\rightarrow b$ and $\Theta\rightarrow\Theta+\pi/2$.

As we did in the Hartle-Thorne case, replacing Eq. (123) in the harmonic
equation, one is able to fix the coefficients of the coordinate
transformation. In this case the equivalent of Eq. (115) for the Myers-Perry
solution is way more involved, and since there is no physical information in
it, we provide it in the attached Mathematica notebook. However, it is worth
mentioning that coherently to what we know from the amplitude calculation, the
coordinate transformation develops two independent gauge redundancies at
orders $O(Ga^{2})$, and two more at orders $O(G^{2}a^{0})$ and
$O(G^{2}a^{2})$. In particular the redundancies at 2PM are related to the
logarithmic terms and are the classical counterpart of the higher-loop vertex
in Eq. (51).

Finally, we report the 3PM Myers-Perry metric in harmonic coordinates in the
attached Mathematica file, giving here for the sake of clarity only the
expression at 1PM in the case in which $\mathcal{C}_{1,2,0}^{(R)}=0$ and
$\mathcal{C}_{1,2,2}^{(R)}=-2/9$ such that $g_{R\Theta}=0$, resulting in the
simplified expression

$\displaystyle g_{tt}$ $\displaystyle=1-\frac{8Gm}{3\pi
R^{2}}+\frac{8Gm(a^{2}-b^{2})\cos(2\Theta)}{3\pi R^{4}}+O(G^{2},a^{3})\ ,$
(124) $\displaystyle g_{t\Phi_{1}}$
$\displaystyle=-\frac{8aGm\sin^{2}(\Theta)}{3\pi R^{2}}+O(G^{2},a^{3})\ ,$
$\displaystyle g_{t\Phi_{2}}$ $\displaystyle=-\frac{8bGm\cos^{2}(\Theta)}{3\pi
R^{2}}+O(G^{2},a^{3})\ ,$ $\displaystyle g_{RR}$
$\displaystyle=-1-\frac{4Gm}{3\pi
R^{2}}+\frac{4Gm(a^{2}-b^{2})\cos(2\Theta)}{3\pi R^{4}}+O(G^{2},a^{3})\ ,$
$\displaystyle g_{\Theta\Theta}$
$\displaystyle=-\frac{4Gm}{3\pi}-R^{2}+\frac{4Gm}{9\pi
R^{2}}\Big{(}a^{2}+b^{2}+3(a^{2}-b^{2})\cos(2\Theta)\Big{)}+O(G^{2},a^{3})\ ,$
$\displaystyle g_{\Phi_{1}\Phi_{1}}$
$\displaystyle=-\frac{4Gm\sin^{2}(\Theta)}{3\pi}-R^{2}\sin^{2}(\Theta)-\frac{2Gm\sin^{2}(\Theta)}{9\pi
R^{2}}\Big{(}a^{2}+b^{2}-3(3a^{2}-b^{2})\cos(2\Theta)\Big{)}+O(G^{2},a^{3})\
,$ $\displaystyle g_{\Phi_{2}\Phi_{2}}$
$\displaystyle=-\frac{4Gm\cos^{2}(\Theta)}{3\pi}-R^{2}\cos^{2}(\Theta)-\frac{2Gm\cos^{2}(\Theta)}{9\pi
R^{2}}\Big{(}a^{2}+b^{2}-3(a^{2}-3b^{2})\cos(2\Theta)\Big{)}+O(G^{2},a^{3})\
,$ $\displaystyle g_{\Phi_{1}\Phi_{2}}$
$\displaystyle=-\frac{2abGm\sin^{2}(2\Theta)}{3\pi R^{2}}+O(G^{2},a^{3})\ .$

To the best of our knowledge, there is no other derivation of the Myers-Perry
metric in such coordinates in the literature to be compared with Eq. (124).
However, moving to Cartesian coordinates, we can directly match it with the
amplitude-based metric, finding a perfect agreement up to the order of our
calculation, i.e. 3PM. Even though all the physical information is contained
in the tree-level part of the metric, higher-order calculations are an
important and non-trivial consistency check that this approach passes
perfectly.

##  Appendix D Black rings with one angular momentum in harmonic coordinates

It is known that in $D>4$ there is no black hole uniqueness theorem, meaning
that if one considers non-spherical topologies, the Myers-Perry black hole is
not the only solution with a horizon. Black rings are indeed another black
hole solution in $D=5$ Emparan and Reall (2002). Limiting to the case of one
angular momentum along the $\varphi_{1}$-direction, in the so-called black
ring coordinates $(x,y,\varphi_{1},\varphi_{2})$ the explicit metric reads
Emparan and Reall (2002)

$\displaystyle ds^{2}=$
$\displaystyle-\frac{A(y)}{A(x)}\left(dt-C\,\mathcal{R}\frac{1+y}{A(y)}d\varphi_{1}\right)^{2}$
(125)
$\displaystyle+\frac{\mathcal{R}^{2}}{(x-y)^{2}}A(x)\left(-\frac{B(y)}{A(y)}d\varphi_{1}^{2}-\frac{dy^{2}}{B(y)}+\frac{dx^{2}}{B(x)}+\frac{B(x)}{A(x)}d\varphi_{2}^{2}\right)\
,$

where

$A(z)=1+\lambda z\ ,\qquad B(z)=(1-z^{2})(1+\nu z)\ ,\qquad
C=\sqrt{\lambda(\lambda-\nu)\frac{1+\lambda}{1-\lambda}}\ ,$ (126)

and $0<\nu\leq\lambda<1$ are dimensionless parameters, while $\mathcal{R}$ is
a dimensionful radius. To avoid naked conical singularities in Eq. (125), one
needs to impose

$\lambda=\frac{2\nu}{1+\nu^{2}}\ ,$ (127)

and since $\lambda$ and $\nu$ are parameters related to the shape and angular
momentum of the ring, the physical interpretation of such condition is to
require that the centrifugal force exactly compensate the gravitational self-
attraction. Therefore, one can refer to Eq. (127) as the equilibrium
condition. As explained in the main text, we will not enforce such condition,
so the black ring solution is described by three parameters.

Our goal now is to move from ring coordinates to harmonic coordinates. In
order to do so, we need an intermediate step in which we simply consider
asymptotically flat coordinates through the transformation Heynen and Mayerson
(2023)

$\begin{gathered}x=-\left(\frac{1-\lambda}{1-\nu}\right)\frac{r^{2}-2\left(\frac{1-\lambda}{1-\nu}\right)\mathcal{R}^{2}\cos^{2}(\theta)}{r^{2}}\
,\qquad
y=-\left(\frac{1-\lambda}{1-\nu}\right)\frac{r^{2}+2\left(\frac{1-\lambda}{1-\nu}\right)\mathcal{R}^{2}\sin^{2}(\theta)}{r^{2}}\
,\\\
(\varphi_{1},\varphi_{2})=\frac{\sqrt{1-\lambda}}{1-\nu}(\phi_{1},\phi_{2})\
,\end{gathered}$ (128)

in which one may now express the metric in terms of the coordinates
$(t,r,\theta,\phi_{1},\phi_{2})$. Furthermore, it is convenient to make the
dependence on the mass $m$ and the angular momentum $J$ manifest in the black
ring metric. Considering the asymptotic long-range behavior of the solution,
the mass monopole and the spin dipole, i.e. the mass and the spin itself,
respectively read

$m=\frac{3\pi\mathcal{R}^{2}}{4G}\ ,\qquad
J=\frac{\pi\mathcal{R}^{3}}{2G}\frac{\sqrt{\lambda(\lambda-\nu)(1+\lambda)}}{(1-\nu)^{2}}\
.$ (129)

Keeping $\lambda$ a free parameter, in the metric written in asymptotically
flat coordinates we can now replace $(\mathcal{R},\nu)$ in favor of $(m,J)$,
having finally a metric described by three independent parameters
$(m,J,\lambda)$.

We can finally move to harmonic coordinates by considering the transformation
(121), that through the definition of harmonic Cartesian coordinates in Eq.
(120) have to satisfy the relation (122). The ansatz for the harmonic
coordinate transformation is different from the Myers-Perry one since the
topology is more complicated. Moreover, for the reasons explained in Sec. V.3,
we will consider only terms up to 1PM and second order in the spin, so that
the coordinate transformation reads171717Notice that at higher PM orders one
should include logarithmic pieces as in the $D=5$ Myers-Perry case.

$\displaystyle
r(R,\Theta)=R\sum_{i=0}^{1}\left(Gm\rho(R)\right)^{i}\sum_{j=0}^{1}\sum_{k=0}^{2}\mathcal{C}_{i,2j,2k}^{(R)}(\lambda)\left(\frac{a}{R}\right)^{2j}P_{2k}(\cos\Theta)+O(G^{2},a^{3})\
,$ (130)
$\displaystyle\cos\theta(R,\Theta)=\cos\theta\sum_{i=0}^{1}\left(Gm\rho(R)\right)^{i}\sum_{j=0}^{1}\sum_{k=0}^{2}\mathcal{C}_{i,2j,2k}^{(\Theta)}(\lambda)\left(\frac{a}{R}\right)^{2j}P_{2k}(\cos\Theta)+O(G^{2},a^{3})\
,$

where now the coefficients that have to be fixed will be functions of the free
parameter $\lambda$. Notice that among all the constraints to impose to this
ansatz, we have to ensure that the monopole term in the space-part of the
metric is independent of $\lambda$, since it must carry only the information
of the physical mass. This extra constraint, which is not a priori satisfied
from Eq. (130), fixes to two the total number of gauge redundant parameters at
1PM, exactly as the previous cases.

Finally, considering the explicit expression of Eq. (130) (reported in the
Mathematica file), the black ring metric in harmonic coordinates (in a
suitable gauge), in which we are neglecting quadrupoles that are not spin-
induced, reads

$\displaystyle g_{tt}$ $\displaystyle=1-\frac{8Gm}{3\pi
R^{2}}+\frac{12Gma^{2}\lambda\cos(2\Theta)}{\pi
R^{4}(1+\lambda)}+O(G^{2},a^{3})\ ,$ (131) $\displaystyle g_{t\Phi_{1}}$
$\displaystyle=\frac{4aGm\sin^{2}(\Theta)}{\pi R^{2}}+O(G^{2},a^{3})\ ,$
$\displaystyle g_{RR}$ $\displaystyle=-1-\frac{4Gm}{3\pi
R^{2}}+\frac{6Gma^{2}\lambda\cos(2\Theta)}{\pi
R^{4}(1+\lambda)}+O(G^{2},a^{3})\ ,$ $\displaystyle g_{\Theta\Theta}$
$\displaystyle=-\frac{4Gm}{3\pi}-R^{2}+\frac{2Gma^{2}(1+3\lambda\cos(2\Theta))}{\pi
R^{2}(1+\lambda)}+O(G^{2},a^{3})\ ,$ $\displaystyle g_{\Phi_{1}\Phi_{1}}$
$\displaystyle=-\frac{4Gm\sin^{2}(\Theta)}{3\pi}-R^{2}\sin^{2}(\Theta)+\frac{Gma^{2}\sin^{2}(\Theta)}{\pi
R^{2}}\Big{(}-1+3(1+3\lambda)\cos(2\Theta)\Big{)}+O(G^{2},a^{3})\ ,$
$\displaystyle g_{\Phi_{2}\Phi_{2}}$
$\displaystyle=-\frac{4Gm\cos^{2}(\Theta)}{3\pi}-R^{2}\cos^{2}(\Theta)+\frac{Gma^{2}\cos^{2}(\Theta)}{\pi
R^{2}}\Big{(}-1+3(-1+3\lambda)\cos(2\Theta)\Big{)}+O(G^{2},a^{3})\ .$

This metric can be now compared directly with the amplitude-based one,
obtaining a perfect match up to the order we are considering.

## References

  * Feynman (1963) R. P. Feynman, Acta Phys. Polon. 24, 697 (1963).
  * DeWitt (1967a) B. S. DeWitt, Phys. Rev. 160, 1113 (1967a).
  * DeWitt (1967b) B. S. DeWitt, Phys. Rev. 162, 1195 (1967b).
  * DeWitt (1967c) B. S. DeWitt, Phys. Rev. 162, 1239 (1967c).
  * ’t Hooft and Veltman (1974) G. ’t Hooft and M. J. G. Veltman, Ann. Inst. H. Poincare Phys. Theor. A 20, 69 (1974).
  * Donoghue (1994a) J. F. Donoghue, Phys. Rev. Lett. 72, 2996 (1994a), arXiv:gr-qc/9310024 .
  * Donoghue (1994b) J. F. Donoghue, Phys. Rev. D 50, 3874 (1994b), arXiv:gr-qc/9405057 .
  * Iwasaki (1971) Y. Iwasaki, Prog. Theor. Phys. 46, 1587 (1971).
  * Bjerrum-Bohr  _et al._ (2003) N. E. J. Bjerrum-Bohr, J. F. Donoghue, and B. R. Holstein, Phys. Rev. D 68, 084005 (2003), [Erratum: Phys.Rev.D 71, 069904 (2005)], arXiv:hep-th/0211071 .
  * Donoghue and Torma (1996) J. F. Donoghue and T. Torma, Phys. Rev. D 54, 4963 (1996), arXiv:hep-th/9602121 .
  * Holstein and Donoghue (2004) B. R. Holstein and J. F. Donoghue, Phys. Rev. Lett. 93, 201602 (2004), arXiv:hep-th/0405239 .
  * Bjerrum-Bohr  _et al._ (2018) N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Planté, and P. Vanhove, Phys. Rev. Lett. 121, 171601 (2018), arXiv:1806.04920 [hep-th] .
  * Kosower  _et al._ (2019) D. A. Kosower, B. Maybee, and D. O’Connell, JHEP 02, 137 (2019), arXiv:1811.10950 [hep-th] .
  * Cheung  _et al._ (2018) C. Cheung, I. Z. Rothstein, and M. P. Solon, Phys. Rev. Lett. 121, 251101 (2018), arXiv:1808.02489 [hep-th] .
  * Damgaard and Lee (2024) P. H. Damgaard and K. Lee, (2024), arXiv:2403.13216 [hep-th] .
  * Donoghue  _et al._ (2002) J. F. Donoghue, B. R. Holstein, B. Garbrecht, and T. Konstandin, Phys. Lett. B 529, 132 (2002), [Erratum: Phys.Lett.B 612, 311–312 (2005)], arXiv:hep-th/0112237 .
  * D’Onofrio  _et al._ (2022) S. D’Onofrio, F. Fragomeno, C. Gambino, and F. Riccioni, JHEP 09, 013 (2022), arXiv:2207.05841 [hep-th] .
  * Gambino (2022) C. Gambino,  _The Reissner-Nordström-Tangherlini solution from graviton and photon emission processes_ , Master’s thesis, Rome U. (2022), arXiv:2210.13190 [hep-th] .
  * Duff (1973) M. J. Duff, Phys. Rev. D 7, 2317 (1973).
  * Koemans Collado  _et al._ (2018) A. Koemans Collado, P. Di Vecchia, R. Russo, and S. Thomas, JHEP 10, 038 (2018), arXiv:1807.04588 [hep-th] .
  * Jakobsen (2020) G. U. Jakobsen, Phys. Rev. D 102, 104065 (2020), arXiv:2006.01734 [hep-th] .
  * Tangherlini (1963) F. R. Tangherlini, Nuovo Cim. 27, 636 (1963).
  * Mougiakakos and Vanhove (2021) S. Mougiakakos and P. Vanhove, Phys. Rev. D 103, 026001 (2021), arXiv:2010.08882 [hep-th] .
  * Hartle (1967) J. B. Hartle, Astrophys. J. 150, 1005 (1967).
  * Hartle and Thorne (1968) J. B. Hartle and K. S. Thorne, Astrophys. J. 153, 807 (1968).
  * Raposo  _et al._ (2019) G. Raposo, P. Pani, and R. Emparan, Phys. Rev. D 99, 104050 (2019), arXiv:1812.07615 [gr-qc] .
  * Israel (1967) W. Israel, Phys. Rev. 164, 1776 (1967).
  * Carter (1971) B. Carter, Phys. Rev. Lett. 26, 331 (1971).
  * Hawking (1972) S. W. Hawking, Commun. Math. Phys. 25, 152 (1972).
  * Robinson (1975) D. C. Robinson, Phys. Rev. Lett. 34, 905 (1975).
  * Robinson (2009) D. Robinson,  _Four decades of black holes uniqueness theorems_ (Cambridge University Press, 2009).
  * Hawking and Ellis (2011) S. W. Hawking and G. F. R. Ellis,  _The Large Scale Structure of Space-Time_ , Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2011).
  * Heusler (1998) M. Heusler, Living Rev. Relativity 1 (1998).
  * Chrusciel  _et al._ (2012) P. T. Chrusciel, J. L. Costa, and M. Heusler, Living Rev.Rel. 15, 7 (2012), arXiv:1205.6112 [gr-qc] .
  * Chung  _et al._ (2019) M.-Z. Chung, Y.-T. Huang, J.-W. Kim, and S. Lee, JHEP 04, 156 (2019), arXiv:1812.08752 [hep-th] .
  * Skvortsov and Tsulaia (2023) E. Skvortsov and M. Tsulaia, (2023), arXiv:2312.08184 [hep-th] .
  * Emparan and Reall (2008) R. Emparan and H. S. Reall, Living Rev. Rel. 11, 6 (2008), arXiv:0801.3471 [hep-th] .
  * Myers and Perry (1986) R. C. Myers and M. J. Perry, Annals Phys. 172, 304 (1986).
  * Emparan and Reall (2002) R. Emparan and H. S. Reall, Phys. Rev. Lett. 88, 101101 (2002), arXiv:hep-th/0110260 .
  * Heynen and Mayerson (2023) J. Heynen and D. R. Mayerson, (2023), arXiv:2312.04352 [gr-qc] .
  * Thorne (1980) K. S. Thorne, Rev. Mod. Phys. 52, 299 (1980).
  * Bern  _et al._ (2021) Z. Bern, A. Luna, R. Roiban, C.-H. Shen, and M. Zeng, Phys. Rev. D 104, 065014 (2021), arXiv:2005.03071 [hep-th] .
  * Bjerrum-Bohr  _et al._ (2015) N. E. J. Bjerrum-Bohr, B. R. Holstein, L. Planté, and P. Vanhove, Phys. Rev. D 91, 064008 (2015), arXiv:1410.4148 [gr-qc] .
  * Geroch (1970a) R. P. Geroch, J. Math. Phys. 11, 1955 (1970a).
  * Geroch (1970b) R. P. Geroch, J. Math. Phys. 11, 2580 (1970b).
  * Hansen (1974) R. O. Hansen, J. Math. Phys. 15, 46 (1974).
  * Gürsel (1983) Y. Gürsel, General Relativity and Gravitation 15, 737 (1983).
  * Cardoso and Gualtieri (2016) V. Cardoso and L. Gualtieri, Class. Quant. Grav. 33, 174001 (2016), arXiv:1607.03133 [gr-qc] .
  * Mayerson (2023) D. R. Mayerson, SciPost Phys. 15, 154 (2023), arXiv:2210.05687 [gr-qc] .
  * Tanabe  _et al._ (2010) K. Tanabe, S. Ohashi, and T. Shiromizu, Phys. Rev. D 82, 104042 (2010), arXiv:1009.1486 [gr-qc] .
  * Raposo and Pani (2020) G. Raposo and P. Pani, Phys. Rev. D 102, 044045 (2020), arXiv:2002.02555 [gr-qc] .
  * Emparan and Reall (2006) R. Emparan and H. S. Reall, Class. Quant. Grav. 23, R169 (2006), arXiv:hep-th/0608012 .
  * Morisawa and Ida (2004) Y. Morisawa and D. Ida, Phys. Rev. D 69, 124005 (2004), arXiv:gr-qc/0401100 .
  * Bena and Mayerson (2020) I. Bena and D. R. Mayerson, Phys. Rev. Lett. 125, 221602 (2020), arXiv:2006.10750 [hep-th] .
  * Bianchi  _et al._ (2020) M. Bianchi, D. Consoli, A. Grillo, J. F. Morales, P. Pani, and G. Raposo, Phys. Rev. Lett. 125, 221601 (2020), arXiv:2007.01743 [hep-th] .