aainstitutetext: High Energy Physics, Cosmology & Astrophysics Theory Group,
Department of Mathematics & Applied Mathematics, University of Cape Town, Cape
Town, 7701, South Africabbinstitutetext: Niels Bohr International Academy &
Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej
17, DK-2100, Copenhagen, Denmarkccinstitutetext: Institute for Theoretical
Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, Netherlands,
European Union

# Compact Objects from
Spontaneous Symmetry Breaking

Irvin Martinez<EMAIL_ADDRESS>
###### Abstract

Gravitational waves from merging compact objects provides the opportunity to
explore the properties of black holes and neutron stars in the strong regime
of gravity. It is therefore of interest to explore the theoretical model that
accurately describes them. Using the coset construction, we build a worldline
Effective Field Theory that is derived from symmetry principles, does not
involve additional degrees of freedom, and describes the most general compact
object allowed in an effective Einstein-Maxwell vierbein theory. Such extended
object can be described by its mass, spin, charge and size effects: tides,
polarization and dissipation. By recognizing the symmetry breaking pattern, we
derive all the covariant building blocks and constraints to build up the
relevant invariant operators in the action to all orders. The developed theory
elucidates the description of compact objects as an Effective Field Theory.

††dedication: In memory of my father Fabián,
whose infinite curiosity and uniqueness,
remain embodied in this work.

##  1 Introduction

The recent detection of gravitational waves (GWs) from various coalescing
binaries Abbott:2016blz ; TheLIGOScientific:2017qsa ; LIGOScientific:2017ync ;
LIGOScientific:2017zic ; LIGOScientific:2021qlt  consisting of compact
objects, such as black holes (BHs) and neutron stars (NSs), have opened up the
possibility to test fundamental physics in the strong regime of gravity. With
upcoming sensibility upgrades in current GW detectors, and future earth
Punturo:2010zz ; Maggiore:2019uih  and space based detectors Barausse:2020rsu
, the era of high precision gravity is arriving, and with it the potential of
great discoveries.

One of the key potentials is to probe the internal structure of the compact
objects by matching the coefficients of the theory with GW observations
Flanagan:2007ix ; Cardoso:2017cfl . Thus the need to develop accurate
theoretical models that describe these extended objects, taking into account
for the different effects that can play a role in the waveform, such as the
spin, charge, and their internal structure due to its extended nature.

In the EFT for extended objects Goldberger:2004jt , where non-rotating objects
are introduced, they are described as worldline point particles, with higher
order operators taking into account for their properties and finite-size
structure, which are accompanied with coefficients that encapsulates their
internal structure. Since the introduction of compact objects as an EFT
Goldberger:2004jt ; Goldberger:2005cd , the framework has received a big flush
of improvements. Some of the most relevant extensions were the inclusion of
spin Porto:2005ac ; Delacretaz:2014oxa ; Levi:2015msa ; Goldberger:2020fot ;
Liu:2021zxr , charge Patil:2020dme , spin-tidal effects Porto:2005ac ;
Levi:2014gsa , dissipation Goldberger:2005cd ; Porto:2007qi ;
Goldberger:2020fot , dynamical oscillations Steinhoff:2016rfi  and
gravitational wave radiation Goldberger:2009qd . Moreover, the finding of new
methods that simplifies the computations in perturbative approximations, such
as in the Post-Newtonian (PN) Levi:2018nxp ; Kuntz:2020gan  and Post-
Minkowskian (PM) Goldberger:2009qd ; Kalin:2020mvi  expansions, have pushed
the field forward to obtain higher and higher order corrections to the
dynamics of binary compact objects, leading to an unprecedented accuracy
Levi:2008nh ; Porto:2008jj ; Porto:2008tb ; Goldberger:2009qd ; Levi:2010zu ;
Porto:2010tr ; Foffa:2011ub ; Ross:2012fc ; Levi:2015msa ; Levi:2016ofk ;
Foffa:2019yfl ; Foffa:2019hrb ; Levi:2019kgk ; Levi:2020lfn ; Levi:2020kvb ;
Levi:2020uwu ; Kalin:2020lmz ; Kalin:2020fhe ; Dlapa:2021vgp ; Cho:2022syn ;
Kalin:2022hph .

Nevertheless, certain ambiguities arises from different developed EFTs, i.e.
on spinning objects, and an effective theory that unifies the different
developed tools for the description of compact objects is still lacking.
Therefore, the purpose of this work is to develop an EFT theory using modern
theoretical tools, that can incorporate all the ingredients necessary to
describe the most general compact object allowed in a theory of gravity as GR
with classical electrodynamics, which can be charged and spinning. By
exploring the mathematical structure of such theory, we elucidate on the
description of compact objects as an EFT.

Using the coset construction Coleman:1969 ; Callan:1969sn ; Volkov:1973vd ;
Ivanov:1975zq , and following the prescription to describe spinning extended
objects in Delacretaz:2014oxa , we propose an EFT that can describe the
relativistic dynamics of compact objects without introducing additional
degrees of freedom. This theory, which is worldline re-parameterization
invariant, can naturally be used to obtain both the PN and PM expansion. This
approach allows us to disentangle multiple features of spinning objects, like
the relativistic angular velocity, and the Spin Supplementary Conditions. The
developed EFT, which is an extension of Delacretaz:2014oxa ; Endlich:2015mke ,
contains well motivated differences, which are discussed in detail. The
predictability of our theory is shown in a forthcoming work.

To describe extended objects which are charged and spinning, we first develop
an effective theory of gravity with classical electrodynamics. General
Relativity as an effective theory can be derived using the coset construction
by weakly gauging the space-time symmetry group of gravity, the Poincaré group
ISO(3,1), and realizing translations non-linearly Ivanov:1981wn . With this
procedure one can derive Einstein’s vierbein theory of curved space-time,
which is a generalization of the theory of General Relativity that is
independent of a coordinate frame, and which was introduced by Einstein
Einstein:1928  in order to unify gravity and electromagnetism. Therefore, to
the procedure carried out in Delacretaz:2014oxa , we incorporate the internal
$U(1)$ symmetry of classical electrodynamics, to derive an effective Einstein-
Maxwell theory. Once the underlying theory is developed, we proceed to
describe compact objects.

By identifying the symmetry breaking pattern of a charged spinning extended
object, we derive the covariant building blocks that are used to construct the
effective action. Due to the Goldstone’s theorem Goldstone:1961eq , the
symmetry breaking pattern of such extended object in curved space-time implies
the existence of a Nambu-Goldstone field, whose covariant derivative encodes
the angular velocity in its spatial components, and the acceleration in the
temporal-spatial ones Delacretaz:2014oxa . As such elements are derived from a
symmetry reasoning, this yields a very well motivated theory to describe
spinning extended objects. This is one of the main differences of our
effective theory with others, that all covariant quantities are derived using
symmetry principles. Although from construction the effective action
corresponds to the low energy dynamics of the theory, which implies a theory
for slowly spinning objects, we show that compact objects can be considered as
"slowly" spinning.

Therefore, with the derived building blocks, we build the leading order
invariant operators that are needed to describe a charged spinning extended
object, such as a pulsar. We discuss in detail each of the built operators and
compare them to the literature. The constructed effective action is the one of
a massive point particle, with higher order corrections made out of the
constructed operators describing the different effects: spin, charge, tides,
polarization and dissipation. The coefficients appearing in front of the
operators are free parameters to be fixed by the full theory or observations.
Beyond the discussion on the spin-orbit coupling, for which we use a specific
value of the coefficients, we leave the coefficients undetermined, to be
matched in future work.

The constructed effective action has multiple applications in the description
of the coalescence of binaries. On the PN expansion, as a perturbative series
in terms of the expansion parameter $v/c<1$, with $v$ the relative velocity of
the binary, and on the PM expansion, in which one expands over the
gravitational constant, $G$, and which encodes the PN expansion to all orders
in $v$, to a given order in $G$. The connection between the different
expansions is shown in a future work, where the dynamics to lowest order due
to all different effects are obtained. Finally, the derived building blocks
could be used in the Worldline Quantum Field Theory (WQFT) formalism
Mogull:2020sak , and to construct the effective one body (EOB) framework.

In section 2, we start with a very brief review of the basic ingredients of
the coset construction to derive our effective action. In section 3, we derive
the underlying effective theory, which is Einstein-Maxwell in the vierbein
formalism. Then, in 4, derive the covariant building blocks and constraints to
build up the worldline effective action of a charged spinning compact object
in curved space-time. We start by considering massive point-like objects, then
with charge, and then include the spin. Then we consider field-field
interactions, taking into account spin-gravity, spin-electro and gravity-
electro effects. On size effects, we first take into account static tidal
effects and polarization, which are conservative effects. Then we consider
dissipation, and generalize to consider dynamical tides with dissipation
included. After constructing all relevant invariant operators to lowest order,
we build an effective action that describes the most general compact object in
a Maxwell-Einstein effective theory. Finally in section 5, we conclude.

##  2 Basics of the Coset Construction

We start with the very basics of the coset construction to develop this paper.
A brief but more comprehensive review can be found in Ogievetsky:1974 ;
Delacretaz:2014oxa ; Penco:2020kvy . We use the notation as in
Delacretaz:2014oxa  to consider the breaking of internal Callan:1969sn ;
Coleman:1969  and space-time symmetries Volkov:1973vd ; Ivanov:1981wn  alike.

The coset construction is a very general technique from the EFT framework that
can be used whenever there is a symmetry breaking. The breaking of symmetries
implies the existence of additional degrees of freedom, known as Nambu-
Goldstone bosons or simply as Goldstone fields.111Goldstone theorem
Goldstone:1961eq  implies the existence of a Goldstone field for each broken
internal symmetry, but for the case in which space-time symmetries are broken,
there can be a mismatch on the number of degrees of freedom and broken
symmetries, for which additional constraints are needed. See the Inverse Higgs
constraint below. The coset construction is then used to derive building
blocks for the Goldstone fields that transform correctly under the relevant
symmetries, blocks that can be used to build up invariant operators to form an
effective action. Any state other than vacuum breaks at least some of the
symmetries, and by appropriately identifying the pattern of the symmetry
breaking, we can use it as a guide to derive the effective action. Within this
approach, the coefficients that appear in front of the invariant operators are
treated as free parameters to be fixed by a matching procedure to the full
theory or to observations.

We can formulate an EFT using the symmetry breaking pattern as the only input,
knowing the full symmetry group G that is broken, and the subgroup H that is
non-linearly realized Penco:2020kvy . If the group is broken, G $\rightarrow$
H, due to a spontaneous symmetry breaking, the coset recipe Delacretaz:2014oxa
; Penco:2020kvy  tells us that we can classify the generators into three
categories:

$\displaystyle\begin{split}P_{a}&=\mathrm{generators\;of\;unbroken\;translations},\\\
T_{A}&=\mathrm{generators\;of\;all\;other\;unbroken\;symmetries},\\\
X_{\alpha}&=\mathrm{generators\;of\;broken\;symmetries},\end{split}$ (1)

where the broken generators, $X_{\alpha}$, and the unbroken ones, $T_{A}$, can
be of space-time symmetries, as well as of internal ones. Whenever the set of
generators for broken symmetries is non-zero, some Goldstone fields will
arise. The broken symmetries and the unbroken translations are realized non-
linearly on the Goldstone bosons Delacretaz:2014oxa .

Following the coset recipe Delacretaz:2014oxa ; Penco:2020kvy , we do a local
parameterization of the coset, G/H0, with $H_{0}$, the subgroup of $H$
generated by the unbroken generators, $T$’s. The coset is parameterized as

$\displaystyle g(x,\pi)=e^{iy^{a}(x)P_{a}}e^{i\pi^{\alpha}(x)X_{\alpha}},$ (2)

where the factor, $e^{iy^{a}(x)P_{a}}$, describes a translation from the
origin of the coordinate system to the point, $x_{a}$, at which the Goldstone
fields, $\pi^{\alpha}(x)$, are evaluated. This factor ensures that the $\pi$’s
transform correctly under spatial translation. The group element $g$, which is
generated by the $X$’s and the $P$’s, is known as the coset parameterization.
For the case of flat space-time, the translation is simply parameterized by,
$e^{ix^{a}P_{a}}$, with $y(x)\equiv x$. To obtain building blocks that depend
on the Goldstone bosons and that have simple transformation rules, we couple
them through their derivatives Penco:2020kvy .

We introduce a very convenient quantity that is an element of the algebra of
G, the Maurer-Cartan form, $g^{-1}\partial_{\mu}g$, which can be written as a
linear combinations of all the generators Delacretaz:2014oxa ,

$g^{-1}\partial_{\mu}g=(e_{\mu}^{\;\;a}P_{a}+\nabla_{\mu}\pi^{\alpha}X_{\alpha}+C_{\mu}^{\;\;B}T_{B}).$
(3)

The coefficients $e_{\mu}^{a}$, $\nabla_{\mu}\pi^{\alpha}$ and $C_{\mu}^{B}$,
in general are non-linear functions of the Goldstones, and are basic
ingredients of the effective theory, with
$\nabla_{\mu}\pi^{\alpha}=e_{\mu}^{\;\;a}\nabla_{a}\pi^{\alpha}$ and
$C_{\mu}^{\;\;B}=e_{\mu}^{\;\;a}C_{a}^{\;\;B}$. The explicit expression of the
aforementioned building blocks can be obtained using the algebra of the group
$G$.

We can use the coefficients of the unbroken symmetries, $C$’s, and its
operators, $T$’s, to define the covariant derivative,

$\displaystyle\nabla_{a}\equiv(e^{-1})_{a}^{\;\mu}(\partial_{\mu}+iC_{\mu}^{B}T_{B}),$
(4)

which can be used to define higher covariant derivatives on the Goldstone
fields, as well as on the building blocks that transform linearly under the
unbroken group. Then, by considering all the allowed contractions of the
building blocks including the covariant derivative, it is possible to build up
invariant operators under the full symmetry group $G$, and construct the
effective action.

In gauge symmetries, it is necessary to promote the partial derivative to a
covariant one in the Maurer-Cartan form, $\partial_{\mu}\rightarrow D_{\mu}$
Delacretaz:2014oxa . Consider the gauged generator, $E_{I}$, from a subgroup,
$G^{\prime}\subseteq G$, with corresponding gauge field, $w^{I}_{\mu}$. Thus,
by replacing the partial derivative with a covariant one, we obtain the
modified Maurer-Cartan form,

$g^{-1}\partial_{\mu}g\rightarrow
g^{-1}D_{\mu}g=g^{-1}(\partial_{\mu}+iw^{I}_{\mu}E_{I})g.$ (5)

This modification of the Maurer-Cartan form can also be written as a linear
combination of the generators as in eq. (3), with a new building block made up
of the gauge field, $w_{\mu}^{I}$, accompanying the gauged generator, $E_{I}$.
Now the building blocks can also depend on the gauge fields. The modified
Maurer-Cartan form, $g^{-1}D_{\mu}g$, is invariant under local
transformations, and its explicit components can be obtained using the
commutation relations of the generators.

### Inverse Higgs Constraint

The Goldstone’s theorem Goldstone:1961eq , which states that a Goldstone mode
exists for each broken generator, is only valid for internal symmetries. If
space-time symmetries are spontaneously broken, there can be a mismatch in the
number of broken generators and the number of bosons Low:2001bw .
Nevertheless, we can preserve all the symmetries by imposing additional local
constraints, which can be solved to write down some of the Goldstone’s modes
in terms of others Penco:2020kvy . Using the inverse Higgs constraint
Ivanov:1975zq , we can set to zero one or more of the coset covariant
derivatives, whenever $X$ and $X^{\prime}$, are two multiplets of the broken
generator, such that the commutators of the unbroken translations, $P$, and
the broken generator, $X^{\prime}$, yields a different broken generator, $X$:
$\;[P,X^{\prime}]\supset X$. If this is the case, we can set some of the
covariant derivatives of the Goldstones to zero.

##  3 Effective Theory of Gravity

There are two symmetries to consider in a theory of gravity as General
Relativity: Poincaré symmetry, and diffeomorphisms invariance or worldline
reparameterization. The former, determined by the Poincaré group,
$G=ISO(3,1)$, contains the generators for translations, $P_{a}$, and Lorentz
transformations, $J_{ab}$, with their corresponding gauge fields,
$\breve{e}_{\mu}^{a}$ and $\breve{\omega}_{\mu}^{ab}$. By considering the
principal bundle, $P(M,G)$, with base manifold, $M$, and structure group, $G$,
when weakly gauging the Poincaré group and realizing translations non-
linearly, Poincaré and diffeomorphisms are separated. The coordinates,
$x^{\mu}$, describing the position on the considered manifold are not affected
by the local Poincaré group but transformed under diffeomorphisms, while
matter fields are realized as sections of their respective fiber bundle
Delacretaz:2014oxa . The local Poincaré transformations act along the fiber,
while diffeomorphisms can be considered as relabeling the points on the
manifold.

To derive an effective theory of gravity, we proceed to gauge the Poincaré
group and realize translations non-linearly Delacretaz:2014oxa . The coset,
$ISO(3,1)/SO(3,1)$, is parameterized by

$g=e^{iy^{a}(x)P_{a}},$ (6)

which ensures the non-linear realization of translations Ivanov:1981wn . From
this coset parameterization, the covariant Maurer-Cartan form, expressed as a
linear combination of the generators of the theory, reads

$\displaystyle\begin{split}g^{-1}D_{\mu}g&=e^{-iy^{a}(x)P_{a}}\left(\partial_{\mu}+i\breve{e}_{\mu}^{\;\;a}P_{a}+\frac{i}{2}\breve{\omega}_{\mu}^{ab}J_{ab}\right)e^{iy^{a}(x)P_{a}}\\\
&=ie_{\mu}^{\;\;a}P_{a}+\frac{i}{2}\omega_{\mu}^{ab}J_{ab},\end{split}$ (7)

where we have used the commutation relation rules of the symmetries (See
appendix B).

From the linear combination of the generators in eq. (7), we can extract the
first building blocks of the theory,

$\displaystyle e_{\mu}^{\;\;a}$
$\displaystyle=\breve{e}_{\mu}^{a}+\partial_{\mu}y^{a}+\breve{\omega}_{\mu\;b}^{a}y^{b},$
(8) $\displaystyle\omega^{ab}_{\mu}$
$\displaystyle=\breve{\omega}^{ab}_{\mu}.$ (9)

The field, $e^{a}_{\mu}$, is the vierbein, which defines the metric as
$g_{\mu\nu}=\eta_{ab}e_{\mu}^{a}e_{\nu}^{b}$. It can be used to build up the
invariant element, $\mathrm{d}^{4}x\,\mathrm{det}\,e$, as well as to change
from orthogonal frame, i.e. $V_{\mu}=e_{\mu}^{\;b}V_{b}$, with $V_{a}$, a
vector field. The field, $\omega_{\mu}^{\;ab}$, is the spin connection.

We can introduce the covariant derivative for matter fields, using the
coefficients from the unbroken Lorentz generators,

$\nabla^{g}_{a}=(e^{-1})_{a}^{\;\;\mu}(\partial_{\mu}+\frac{i}{2}\omega_{\mu}^{bc}J_{bc}),$
(10)

where the upper index, $g$, denotes gravity. The only required ingredients to
describe the non-linear realizations of translations and the local
transformations of the Poincaré group, is the covariant derivatives and the
vierbein Delacretaz:2014oxa .

Moreover, we can extract building blocks from the gauge field strengths. The
curvature invariants are obtained from the covariant commutator of the
derivative in the Maurer-Cartan form,

$\displaystyle\begin{split}g^{-1}[D_{\mu},D_{\nu}]g&=iT^{a}_{\mu\nu}P_{a}+\frac{i}{2}R^{ab}_{\mu\nu}J_{ab},\\\
&=i\left(\partial_{\mu}e^{a}_{\nu}-\partial_{\nu}e^{a}_{\mu}+e_{\mu
b}\omega^{ab}_{\nu}-e_{\nu b}\omega^{ab}_{\mu}\right)P_{a}\\\
&\;\;\;+\frac{i}{2}\left(\partial_{\mu}\omega^{ab}_{\nu}-\partial_{\nu}\omega_{\mu}^{ab}+\omega^{a}_{\mu
c}\omega^{cb}_{\nu}-\omega^{a}_{\nu
c}\omega^{cb}_{\mu}\right)J_{ab},\end{split}$ (11)

with $T^{a}_{\mu\nu}=\breve{T}^{a}_{\mu\nu}+\breve{R}^{ab}_{\mu\nu}y_{b}$, and
$R^{ab}_{\mu\nu}=\breve{R}^{ab}_{\mu\nu}$, the covariant torsion and Riemann
tensor respectively. The covariant quantities have been defined in this way,
from
$[D_{\mu},D_{\nu}]=i\breve{T}^{a}_{\mu\nu}P_{a}+\frac{i}{2}\breve{R}^{ab}_{\mu\nu}J_{ab}$,
such that by construction, $T^{a}_{\mu}$ and $R^{ab}_{\mu\nu}$ transform
independently under the local transformations Delacretaz:2014oxa .

As we are interested in a gravitational theory as General Relativity, we set
the torsion tensor to be zero. From the vanishing of the torsion tensor, one
can obtain an equation for the spin connection in terms of the vierbein,

$\omega_{\mu}^{ab}(e)=\frac{1}{2}\left\\{e^{\nu
a}(\partial_{\mu}e_{\nu}^{\;\;b}-\partial_{\nu}e_{\mu}^{\;\;b})+e_{\mu
c}e^{\nu a}e^{\lambda b}\partial_{\lambda}e_{\nu}^{\;\;c}-(a\leftrightarrow
b)\right\\}.$ (12)

Then, eq. (11),

$\displaystyle\begin{split}g^{-1}[D_{\mu},D_{\nu}]g=\frac{i}{2}R^{ab}_{\mu\nu}\left(\omega_{\rho}^{cd}(e)\right)J_{ab}.\end{split}$
(13)

The Riemann tensor, $R^{ab}_{\mu\nu}$, is used to build up a Lagrangian that
describes our gravitational theory. By considering the lowest order
correction, we build

$\mathcal{S}=\int\mathrm{d}^{4}x\;\mathrm{det}\;e\;\alpha
R\;+.\;.\;.=\int\mathrm{d}^{4}x\;\mathrm{det}\;e\;\frac{1}{16\pi
G}R\;+.\;.\;.,$ (14)

with $R=R^{ab}_{\;\;\;ab}=e^{\mu}_{\;a}e^{\nu}_{\;b}R^{ab}_{\;\;\;\,\mu\nu}$,
the low energy term of the effective theory of gravity. We have matched the
coefficient $\alpha=(16\pi G)^{-1}$ in eq. (14), from the well known theory of
General Relativity, to obtain the Einstein-Hilbert action in the vierbein
formalism Ivanov:1981wn . The ellipsis stands for higher order corrections
made out of the Riemann tensor, describing a more fundamental theory of
gravity.

Given that we aim to describe not only black holes but neutron stars as well,
and that it is well know that they can possess very strong magnetic fields, we
add the internal U(1) symmetry of classical electromagnetism, with gauge
field, $\breve{A}_{\mu}$, and generator $Q$. The generator of the charge, $Q$,
is encoded as a time translation, $\bar{P}_{0}=P_{0}+Q$, which ensures the
correct transformation rules for the gauge field under the $U(1)$ symmetry.
Therefore, by considering the symmetry group, $G=U(1)\times ISO(3,1)$, we can
proceed with the coset construction as before. The coset parameterization now
reads,

$g=e^{iy^{a}(x)\bar{P}_{a}},$ (15)

with $\bar{P}_{a}=(\bar{P}_{0},P_{i})$.

Then, the Maurer-Cartan form from the coset parameterization (15), reads

$\displaystyle\begin{split}g^{-1}D_{\mu}g&=e^{-iy^{a}(x)\bar{P}_{a}}\left(\partial_{\mu}+i\breve{A}_{\mu}Q+i\breve{e}_{\mu}^{\;\;a}P_{a}+\frac{i}{2}\breve{\omega}_{\mu}^{ab}J_{ab}\right)e^{iy^{a}(x)\bar{P}_{a}}\\\
&=\partial_{\mu}+iA_{\mu}Q+ie_{\mu}^{\;\;a}P_{a}+\frac{i}{2}\omega_{\mu}^{ab}J_{ab}.\end{split}$
(16)

Including the $U(1)$ symmetry, we obtain a new building block,

$\displaystyle A_{\mu}$ $\displaystyle=\breve{A}_{\mu}+\partial_{\mu}y^{0},$
(17)

which transforms as expected, $\partial_{\mu}y^{0}=\partial_{\mu}\xi(y)$. The
rest of the building blocks, eq. (8) and eq. (9), remain unchanged given that
we are considering $U(1)$ as an internal symmetry.

The coefficients from the unbroken $U(1)$ gauge field can be used as well to
define the covariant derivative for charged fields,

$\nabla^{q}_{a}=(e^{-1})_{a}^{\;\;\mu}(\partial_{\mu}+iA_{\mu}Q).$ (18)

Moreover, a new gauge field strength is obtained,

$\displaystyle\begin{split}g^{-1}[D_{\mu},D_{\nu}]g&=iF_{\mu\nu}Q+iT^{a}_{\mu\nu}P_{a}+\frac{i}{2}R^{ab}_{\mu\nu}J_{ab},\end{split}$
(19)

with $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, the
electromagnetic stress field tensor. The torsion and Riemann tensor in eq.
(19), remain unchanged, and one can proceed to construct the effective theory
as before.

After imposing the vanishing of the torsion tensor, we are now able to use
both, $R^{ab}_{\;\;\;cd}$ and $F_{ab}$, as building blocks to construct the
effective action. To lowest order,

$\mathcal{S}_{eff}=\int\mathrm{d}^{4}x\;\mathrm{det}\;e\;\left\\{-\frac{1}{4\mu_{0}}F_{ab}F^{ab}+\frac{1}{16\pi
G}R+.\;.\;.\right\\},$ (20)

where the coefficient from the first term has been matched from Maxwell’s
action in curved space-time, with $\mu_{0}$, the magnetic permeability of
vacuum. Eq. (20), is the Einstein-Maxwell action in the vierbein formalism.
Beyond lowest order, one could have the higher order correction
Camanho:2014apa ,

$\displaystyle\mathcal{S}=\int\mathrm{det}\;e\;\mathrm{d}^{4}x\;\alpha_{gq}R^{ab}_{\;\;\;cd}F_{ab}F^{cd}+.\;.\;..$
(21)

##  4 Compact Objects in Effective Field Theory

Since the introduction of the description of extended objects in the EFT
framework Goldberger:2004jt , a considerable number of extensions and
improvements to the framework have been developed. A thoroughly review is
beyond our scope, but we will refer to the relevant literature when build the
EFT. We refer the reader to Porto:2016pyg ; Levi:2018nxp ; Goldberger:2022ebt
for a more complete treatment on compact binary dynamics.

The effective theory we develop is based on Delacretaz:2014oxa , where an
effective description for spinning extended objects in curved space-time is
derived using the coset construction. In Endlich:2015mke , the model is used
in the Newtonian regime, which incorporates tidal and dissipative effects.
Therefore, this work is an extension of Delacretaz:2014oxa ; Endlich:2015mke ,
and it describes general relativistic charged and spinning massive point
particles, with higher order corrections due to the finite-size structure.

###  4.1 Charged Spinning Compact Objects

An invariant action for an extended object can be constructed by identifying
the symmetry breaking pattern that such object generates. In order to describe
a charged spinning extended object, we need to consider the full symmetry
group before being broken, $G=U(1)\times S\times ISO(3,1)$, with $S$, the
internal symmetry of a spinning extended object which characterizes the low
energy dynamics Delacretaz:2014oxa . One can choose a coset parameterization
such that the internal symmetry of the spinning extended object is unbroken,
and only the full local Poincaré group is broken.

Moreover, in the comoving frame of the object, the state of a charged point
like object is an eigenstate of the charge and does not break the U(1)
symmetry. Although in principle a neutron star can have its own charge
internal symmetries, such as the ones of a superfluid Delacretaz:2014oxa ;
Nicolis:2013lma , given that in our effective approach we consider an extended
object as a point particle with their properties encoded in higher order
operators, we leave the charge symmetry unbroken.

#### Symmetry Breaking Pattern

In the comoving frame, the group $G$ is broken into a linear combination of
internal rotations, $S_{ij}$, and spatial rotations, $J_{ij}$, such that the
symmetry breaking pattern for a charged spinning extended object reads,

$\displaystyle\begin{split}\mathrm{Unbroken\;generators}&=\begin{cases}&\bar{P}_{0}=P_{0}+Q\;\;\;\;\;\;\;\;\;\,\mathrm{time\;translations},\\\
&\bar{J}_{ij}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{internal\;and\;space-
time\;rotations.}\end{cases}\\\
\mathrm{Broken\;generators}&=\begin{cases}&P_{i}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{spatial\;translations},\\\
&J_{ab}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\mathrm{boosts\;and\;rotations},\end{cases}\end{split}$
(22)

with, $\bar{J}_{ij}$, the sum of the internal and space-time rotations
Delacretaz:2014oxa . We consider a spherical extended object at rest, for
which $S_{ij}$ are the generators of the internal $SO(3)$ group. Translations
are non-linearly realized and the local Poincaré and U(1) transformations are
considered to take place along the fiber.

A spinning extended object breaks the full local Lorentz group. Therefore, the
coset parameterization,

$g=e^{iy^{a}\bar{P}_{a}}e^{i\alpha_{ab}J^{ab}/2}=e^{iy^{a}\bar{P}_{a}}e^{i\eta^{j}J_{0j}}e^{i\xi_{ij}J_{ij}/2}=e^{iy^{a}\bar{P}_{a}}\bar{g},$
(23)

which implies a correspondence between the Goldstone fields, $\alpha_{ab}$ and
$\eta_{i}$, $\xi_{ij}$, due to the breaking of boosts and rotations,
respectively. The parameterization in eq. (23), allows us to obtain the
Maurer-Cartan form without the need to specify the explicit unbroken
generators of rotations. The residual symmetry, SO(3), requires all spatial
indices to be contracted in an SO(3) invariant manner Delacretaz:2014oxa .

#### The Building Blocks

The relevant degrees of freedom can be identified from the Maurer-Cartan form,
projected onto the worldline of the object,

$\displaystyle\begin{split}\dot{x}^{\mu}g^{-1}D_{\mu}g=&\;\dot{x}^{\mu}g^{-1}(\partial_{\mu}+i\breve{A}_{\mu}Q+i\breve{e}_{\mu}^{\;\;a}P_{a}+i\breve{\omega}_{\mu}^{\;\;ab}J_{ab})g\\\
=&\;\dot{x}^{\mu}\bar{g}^{-1}(\partial_{\mu}+iA_{\mu}Q+ie_{\mu}^{\;\;a}P_{a}+\frac{i}{2}\omega_{\mu}^{ab}J_{ab})\bar{g}\\\
=&\;ie(P_{0}+AQ+\nabla\pi^{i}P_{i}+\frac{1}{2}\nabla\alpha_{cd}J^{cd}).\end{split}$
(24)

The building blocks of the low energy dynamics are,

$\displaystyle e$
$\displaystyle=\dot{x}^{\mu}e_{\mu}^{\;\;a}\Lambda_{a}^{\;\;0},$ (25)
$\displaystyle A$
$\displaystyle=e^{-1}\dot{x}^{\mu}A_{\nu}\Lambda_{\;\;\mu}^{\nu},$ (26)
$\displaystyle\nabla\pi^{i}$
$\displaystyle=e^{-1}\dot{x}^{\mu}e_{\mu}^{\;\;a}\Lambda_{a}^{\;\;i},$ (27)
$\displaystyle\nabla\alpha^{ab}$
$\displaystyle=e^{-1}\left(\Lambda_{c}^{\;\;a}\dot{\Lambda}^{cb}+\dot{x}^{\nu}\omega_{\nu}^{cd}\Lambda_{c}^{\;\;a}\Lambda_{d}^{\;\;b}\right),$
(28)

where $\dot{x}^{\mu}=\partial_{\sigma}x^{\mu}$, is the four velocity with
$\sigma$ is the worldline parameter that traces out the trajectory of the
particle. The $\Lambda$’s, are the Lorentz transformations parameterized by
$\alpha$, or equivalently by $\eta$ and $\xi$.

In the breaking of space-time symmetries, one can impose the inverse Higgs
constraint to remove some of the Goldstones Ivanov:1975zq . Given that the
commutator between the unbroken time translations and boosts gives broken
spatial translations, $[K_{i},P_{0}]=iP_{i}$, we set to zero the covariant
derivative of the Goldstone,

$\nabla\pi^{i}=e^{-1}(\dot{x}^{\nu}e_{\nu}^{a}\Lambda_{a}^{\;\;i})=e^{-1}(\dot{x}^{\nu}e_{\nu}^{0}\Lambda_{0}^{\;\;i}+\dot{x}^{\nu}e_{\nu}^{j}\Lambda_{j}^{\;\;i})=0.$
(29)

From this constraint, one can define the velocity Delacretaz:2014oxa

$\displaystyle\beta^{i}=\frac{\dot{x}^{\mu}e_{\mu}^{i}}{\dot{x}^{\nu}e_{\nu}^{0}}=\frac{\dot{x}^{i}}{e},$
(30)

which imply that $\Lambda^{a}_{\;\;0}=\dot{x}^{a}$. Therefore, $\beta^{i}$
parameterize the the boost necessary to get into the particle’s rest frame.

The physical interpretation of the inverse Higgs constraint can be seen as
well if we rewrite Delacretaz:2014oxa ,

$\displaystyle\begin{split}e=&\sqrt{\dot{x}^{\mu}e_{\mu}^{\;\;a}\Lambda_{a}^{\;\;0}\dot{x}^{\nu}e_{\nu}^{\;\;b}\Lambda_{b0}^{\;\;}}=\sqrt{(e\nabla\pi^{i})^{2}-(\dot{x}^{\nu}e_{\nu}^{\;\;a}\Lambda_{a}^{\;\;c}\dot{x}^{\mu}e_{\mu}^{b}\Lambda_{bc})}\\\
=&\sqrt{-\eta_{ab}e_{\nu}^{\;\;a}e_{\mu}^{\;\;b}\dot{x}^{\nu}\dot{x}^{\mu}}=\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}=\frac{\mathrm{d}\tau}{\mathrm{d}\sigma},\end{split}$
(31)

with $\sigma$ the worldline parameter, and $\tau$ the proper time measured in
the particle’s rest frame. To obtain last equation, we have imposed the
inverse Higgs constraint, and the property of the Lorentz matrices,
$\Lambda_{a}^{\;\;b}\Lambda_{c}^{\;\;a}=\delta^{b}_{\;\;c}$. We identify the
building block $e$, as the einbein.

Therefore, with eq. (31), we can rewrite the inverse Higgs constraint in a way
that makes manifest its physical interpretation of Delacretaz:2014oxa . For
rotations, $\Lambda^{0}_{\;\;a}(\xi)=\delta^{0}_{a}$ and
$\Lambda^{i}_{\;j}(\xi)=\mathcal{R}^{i}_{\;\;j}(\xi)$, with $\mathcal{R}(\xi)$
an $SO(3)$ matrix, such that the constraint (29), now reads

$u^{a}\Lambda_{a}^{\;\;i}(\eta)\mathcal{R}_{i}^{\;\;j}(\xi)=0,$ (32)

with, $u^{a}=e_{\mu}^{\;\;a}\partial_{\tau}x^{\mu}$, the velocity measured in
the local co-moving frame defined by the vierbein. Given that the matrix
$\mathcal{R}^{\;\;i}_{j}(\xi)$ is invertible, we obtain

$u^{a}\Lambda_{a}^{\;\;i}(\eta)=0.$ (33)

These quantities now have a clear geometrical interpretation: the set of local
Lorentz vectors,

${\hat{n}^{a}_{\;\;(0)}\equiv
u^{a}=\Lambda^{a}_{\;\;0}(\eta)\;,\;\;\;\;\hat{n}^{a}_{(i)}\equiv\Lambda^{a}_{\;\;i}(\eta)},$
(34)

define an orthonormal local basis with respect to the local flat metric,
$\eta_{ab}$, in the frame that is moving with the particle’s trajectory
Delacretaz:2014oxa . One can also define the orthonormal basis in terms of the
space-time vectors, $\hat{n}^{\mu}_{\;(b)}\equiv
e^{\mu}_{a}\hat{n}^{a}_{(b)}$, with respect to the full metric, $g_{\mu\nu}$.
Moreover, an additional set of orthonormal vectors is obtained
Delacretaz:2014oxa ,

$\hat{m}^{b}_{\;\;(a)}\equiv\Lambda^{b}_{\;\;a}(\alpha)=\Lambda^{b}_{\;\;c}(\eta)\mathcal{R}^{c}_{\;\;a}(\xi),$
(35)

with the zeroth vector, $\hat{m}^{b}_{\;\;(0)}=\hat{n}^{b}_{\;\;(0)}=u^{b}$,
coinciding. The rest of the vectors differs by a rotation, $\mathcal{R}(\xi)$.
Therefore, the set of vectors in eq. (35), contain the information about the
rotation, parameterized by the degrees of freedom of $\xi$.

With these identifications, in the co-moving and co-rotating frame of the
particle, the covariant derivatives of the Goldstone, $\nabla\alpha^{0i}$, can
be expressed as Delacretaz:2014oxa

$\nabla\alpha^{0i}=\mathcal{R}_{j}^{\;\;i}(\xi)\Lambda_{a}^{\;\;j}(\eta)(\partial_{\tau}u^{a}+u^{\mu}\omega_{\mu\;\;c}^{\;\;a}u^{c})=\Lambda_{a}^{\;\;i}(\alpha)u^{\mu}\nabla_{\mu}u^{a}=\Lambda_{a}^{\;\;i}(\alpha)e_{\mu}^{a}a^{\mu}$
(36)

which is the acceleration projected into the orthonormal basis defined by the
$\hat{m}$’s. In the co-moving frame, $\nabla\alpha^{0i}=0$, by definition. The
rest of the covariant derivatives of the Goldstone, reads Delacretaz:2014oxa

$\nabla\alpha^{ij}=\Lambda^{\;\;i}_{k}(\alpha)(\eta^{kl}\partial_{\tau}+\omega_{\mu}^{kl}u^{\mu})\Lambda_{l}^{\;\;j}(\alpha)=\Omega^{ij}(\tau),$
(37)

which is the angular velocity of the object in the co-moving frame.

The gauge field strengths can be used as well as building blocks in the
worldline of the particle. To use these operators, we define the
transformation Delacretaz:2014oxa ,

$R_{abcd}=g^{-1}_{L}\;\tilde{R}=(\Lambda^{-1})_{a}^{\;\;e}(\Lambda^{-1})_{b}^{\;\;f}(\Lambda^{-1})_{c}^{\;\;g}(\Lambda^{-1})_{d}^{\;\;h}\tilde{R}_{efgh},$
(38)

with, $g_{L}$, the Lorentz part of the parameterization in eq. (23), and
$R_{abcd}$, the Riemann tensor in the local rest frame of the object. To
describe induced moments on the worldline, one works instead with the Weyl
tensor, $W_{abcd}$ which has the physical content Goldberger:2004jt . The Weyl
tensor is obtained by subtracting out various traces from the Riemann tensor.
The electromagnetic stress tensor transformation,

$F_{ab}=(\Lambda^{-1})_{a}^{\;\;c}(\Lambda^{-1})_{b}^{\;\;d}\tilde{F}_{cd}.$
(39)

#### Massive Objects

We start by considering the einbein, $e$. The action that can be built with
this building block is simply Delacretaz:2014oxa

$\displaystyle\begin{split}\mathcal{S}=&-n_{e}\int\mathrm{d}\sigma\,e=-m\int\mathrm{d}\sigma\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}},\end{split}$
(40)

where the coefficient, $n_{e}$, has been identified as the mass of the object,
$m$. Eq. (40), which is invariant under re-parameterizations of the particle’s
trajectory, is the usual point particle action that can be expanded around a
small gravitational perturbation $h_{\mu\nu}=g_{\mu\nu}-\eta_{\mu\nu}$, and in
powers of the velocity parameter, $v$. Nevertheless, this equation for the
point particle has a disadvantage, that when expanding around $h_{\mu\nu}$, it
will generate an infinite number of powers of $h$.

This can be fixed by recasting the point particle action in a Polyakov form
Green:1987sp ,

$\displaystyle\mathcal{S}=-\frac{1}{2}\int\mathrm{d}\sigma
e\left(m^{2}-e^{-2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\right).$ (41)

which no longer has a square root on the metric. This action now linear in the
metric, implies a one point function only. Non-linearities to the point
particle are encoded in the einbeins Kuntz:2020gan .

The physical interpretation of the inverse einbein become manifest when
integrating it out from the effective action, $\delta\mathcal{S}/\delta e=0$.
It leads to the mass-shell constraint

$\displaystyle g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}+e^{2}m^{2}=0.$ (42)

In the proper frame of the particle, $\dot{x}^{2}=-1$, and the einbein,
$1/e=m$, is the mass. Sitting outside the rest frame, we can parameterize the
velocity $\dot{x}^{\mu}=(1,v^{i})$, and the inverse einbein is now the
relativistic mass,

$\displaystyle\frac{1}{e}=\frac{m}{\dot{x}^{2}}=\frac{m}{\sqrt{1-v^{2}}}=m\gamma,$
(43)

where we have considered the special relativistic limit. Therefore, in
$\beta^{i}$, the einbein parameterizes the boost to the particle’s rest frame.

The definition of the Polyakov action used in (41), gives a transparent
construction of the action in terms of invariant quantities using the
relativistic momentum, $p^{\mu}=\dot{x}^{\mu}/e=m\gamma\dot{x}^{\mu}$. The
inverse einbein (43), can be seen as the energy of the particle. Substituting
the explicit value of the einbein (43) in the effective action (41), one
recovers the point particle action in eq. (40). Nontheless, with such
definition of the Polyakov action, each einbein is mass dependent and we would
need to construct invariant operators that are supressed by mass factors of
the order of the number of einbeins that the operator posses.

This is remediated by using the definition of the Polyakov action currently
used in the EFT for compact objects Kuntz:2020gan ; Kalin:2020mvi ;
Mogull:2020sak ,

$\displaystyle\mathcal{S}=-\frac{m}{2}\int\mathrm{d}\sigma
e\left(1-\frac{1}{e^{2}}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\right).$ (44)

Integrating out the einbein, the constraint now leads to the einbein derived
in (31),

$\displaystyle e^{2}=-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu},$ (45)

from which one recovers the point particle action in (40) as well. Now the on-
shell condition implies that $e^{2}=1$.

Therefore, the only difference between using eq. (41) and (44), is how we
parameterize our invariant quantities. We choose to use eq. (44), such that
the einbein has no dependence on the mass, and the relativistic momentum is
defined as

$\displaystyle p^{\mu}=m\frac{\dot{x}^{\mu}}{e}.$ (46)

With such parameterization, in the special relativistic limit, $1/e=\gamma$,
and $\beta^{i}$, is the relativistic velocity which is worldline re-
parameterization covariant. By choosing eq. (44), the Polyakov action can be
exploited only through the covariant quantity, $\dot{x}^{\mu}/e$.

#### Charged Point Particles

Now we consider the building block, $A$. The effective action of a massive
charged point particle reads

$\displaystyle\begin{split}\mathcal{S}=&\int\mathrm{d}\sigma
e(-m+n_{A}A)=\int\mathrm{d}\sigma
e(-m+q\frac{\dot{x}^{\mu}}{e}A_{\mu})=-m\int\mathrm{d}\sigma
e+q\int\mathrm{d}\sigma\dot{x}^{\mu}A_{\mu},\end{split}$ (47)

where we have matched the coefficient, $n_{A}=q$, from the action of a charged
point particle Goldberger:2004jt ; Patil:2020dme  with net charge, $q$. The
new correction due to charge is invariant under worldline re-parameterizations
as well, and it is independent of the einbein. Therefore, in eq. (47), we
could re-writte the point particle term in a Polyakov form, and the on-shell
constraint, (45), would remain the same.

#### Spinning Objects

Now we include the building block, $\nabla\alpha^{ab}$, neglecting charge for
simplicity. To lowest order, the effective action for a spinning object reads,

$\mathcal{S}=\int\mathrm{d}\sigma
e\left(-m+n_{\alpha}\nabla\alpha_{ab}\nabla\alpha^{ab}+.\;.\;.\right),$ (48)

where a term linear in $\nabla\alpha$ has been discarded by time reversal
symmetry, and we have considered spherical objects at rest Delacretaz:2014oxa
. The ellipses denotes higher order corrections made out of the Goldstone
field.

In contrast to Delacretaz:2014oxa , where boosts and rotations are treated as
independent building blocks, we propose that the natural covariant object to
work with when describing relativistic spinning objects, is the covariant
derivative of the Goldstone boson. Therefore, we define the relativistic
angular velocity through,

$\displaystyle\nabla\alpha^{ab}=e^{-1}\Omega^{ab}(\sigma)=e^{-1}\Lambda^{\;\;a}_{c}(\eta^{cd}\partial_{\sigma}+\omega_{\mu}^{cd}\dot{x}^{\mu})\Lambda_{d}^{\;\;b}.$
(49)

The implications of this definition for the relativistic angular velocity, is
that $\Omega^{0b}=a^{b}$. This is in agreement with the fact that
$\Omega^{0b}$ is a gauge choice Steinhoff:2015ksa , and that in the rest frame
of the particle, $\Omega^{0b}=a^{b}=0$.

Comparing our action to the one of a relativistic spinning point particle
HANSON1974498 , we can match the coefficient, $n_{\Omega}=I/4$
Delacretaz:2014oxa , with $I$ the moment of inertia. The relativistic action
for a spinning point particle in curved space-time,

$\mathcal{S}=\int\mathrm{d}\sigma
e\left(-m+\frac{I}{4}\nabla\alpha^{ab}\nabla\alpha_{ab}+.\;.\;.\right)=\int\mathrm{d}\sigma
e\left(-m+\frac{I}{4}e^{-2}\Omega_{ab}\Omega^{ab}+.\;.\;.\right).$ (50)

The correction to the point particle due to rotation is worldline re-
parameterization invariant. To lowest order on the second term, the einbein
can be set to one, and one recovers the usual correction of the angular
velocity HANSON1974498 ; Porto:2005ac . But in contrast to any other effective
action for spinning objects, beyond lowest order we have einbeins which are in
general are different from one.

There exist a tower of operators made out of the Goldstone boson in eq. (50).
Some of the corrections that can play a role in the evolution of the angular
velocity of the object are Delacretaz:2014oxa ; Endlich:2015mke ;
HANSON1974498 ,

$\displaystyle\begin{split}\mathcal{S}=&\int\mathrm{d}\sigma
e\left(-m+\frac{I}{4}e^{-2}\Omega_{ab}\Omega^{ab}+n_{\Omega^{2}}e^{-4}(\Omega_{ab}\Omega^{ab})^{2}+e^{-4}n_{\Omega,u}\Omega_{ac}\Omega^{cb}\dot{x}^{a}\dot{x}_{b}+.\;.\;.\right).\end{split}$
(51)

For a compact object, the frequency of the normal modes, $\omega_{N}\sim
c/\ell$, with $\ell$ the radius of the extended object. The tower of higher
order operators made out of the angular velocity is valid in perturbation
theory as long as the rotational frequency is much less than the speed of
sound of the material $c_{s}$ Delacretaz:2014oxa , which for relativistic
objects, $c_{s}\sim c$.

For a rotating extended object in the classical limit, the angular velocity is
constrained by the dimensionless spin,

$\chi=\frac{J}{GM^{2}}=\frac{I\Omega}{GM^{2}},$ (52)

which $\chi\leq 1$. Using the well constrained properties of compact objects
LIGOScientific:2018mvr ; Abbott:2020gyp , one can realize that the expansion
over the parameter, $\Omega/c$, is under perturbative control as long as we
are considering similar properties to those so far detected by the LIGO-Virgo
observatories.

#### Spin Supplementary Condition

Since the inverse Higgs constraint implies that
$e^{-1}(\dot{x}^{a}\Lambda_{a}^{\;\;i})=0$, then the Goldstone boson is
orthogonal to the four velocity as well, $\dot{x}_{a}\nabla\alpha^{ab}=0$.
This has a direct implication on what is known as the Spin Supplementary
Condition. Expanding the components of the Goldstone field, we obtain

$\displaystyle\begin{split}e^{-1}\dot{x}_{a}\nabla\alpha^{ab}=&-e^{-1}\nabla\alpha^{0b}+e^{-1}\dot{x}_{i}\nabla\alpha^{ib}=e^{-2}\dot{x}_{\mu}\nabla^{\mu}\dot{x}^{b}+e^{-2}\dot{x}_{i}\Omega^{ib}=0.\end{split}$
(53)

Removing the einbeins, we are left with the constraint,

$\displaystyle\begin{split}\dot{x}_{a}\Omega^{ab}=&\dot{x}_{0}\Omega^{0b}+\dot{x}_{i}\Omega^{ib}=\dot{x}_{\mu}\nabla^{\mu}\dot{x}^{b}+\dot{x}_{i}\Omega^{ib}=0,\end{split}$
(54)

which is dependent on the chosen worldline, $x^{\mu}$. This is the analog of
the spin supplementary condition (SSC), written in a generic fashion and in
terms of the angular velocity. The condition is fixed when choosing a
worldline to describe the dynamics, but it is needed as well to choose a frame
to describe the rotation.

In the rest frame of the particle, $\Omega^{0b}=0$, and $\dot{x}^{i}=0$.
Therefore, we end up with the constraint, $\dot{x}_{a}\Omega^{ab}=0$, known as
the Covariant SSC. In this frame, $\sigma=\tau$, we have that
$\partial_{\tau}x_{0}=u_{0}=\sqrt{-u^{2}}$, which is an invariant quantity.
Thus, we can replace, $\dot{x}_{0}=\sqrt{-\dot{x}^{2}}$. Then, to describe the
dynamics on another worldline with a different set of local orthonormal basis,
one transforms the local indices with Lorentz matrices,
$\tilde{\Omega}^{ab}=\Lambda^{a}_{\;\;c}\Lambda^{b}_{\;\;d}\Omega^{cd}$ and
choose a different einbein or worldline parameter. A useful parameterization
for the velocity is, $\dot{x}^{a}=\partial_{t}x^{a}=(1,v^{i})$, with
$\sigma=t$, the coordinate time. In this frame, the SSC condition,

$\displaystyle\begin{split}\dot{x}_{a}\tilde{\Omega}^{ab}=&\;e\tilde{\Omega}^{0b}+\dot{x}_{i}\tilde{\Omega}^{ib}=0\\\
\end{split}$ (55)

is the analog to the Canonical SSC Steinhoff:2015ksa ; Levi:2015msa , with
$e=\sqrt{-\dot{x}^{2}}$, and $\tilde{\Omega}^{ab}$, analog to the canonical
spin Levi:2015msa . Although to lowest order, in eq. (LABEL:eq:ssccoordinate),
$\tilde{\Omega}^{0b}=0$, when working outside the co-rotating frame of the
particle, one should consider the SSC in eq. (LABEL:eq:ssccoordinate), to
remove the temporal components of the Goldstone boson accordingly.

#### Field-Field Interactions

From an EFT point of view, we should consider all possible combinations
between fields that are allowed by the symmetries. For a charged spinning
object, a first possible operator to consider, even in the absence of gravity,
is the angular velocity or the spin coupled with the electromagnetic tensor.
Such Spin-Field coupling,

$\displaystyle\mathcal{S}=n_{\Omega,q}\int\mathrm{d}\sigma
e\;\nabla\alpha_{ab}F^{ab}=n_{\Omega,q}\int\mathrm{d}\sigma\;\Omega_{ab}F^{ab},$
(56)

corresponds to a Pauli interaction term Pauli:1941zz , which needs to be taken
into account Skagerstam:1981xp ; vanHolten:1990we . The gauge field of the
unbroken U(1) generator, $A_{a}$, enters into the dynamics, first with the
invariant combination, $e^{-1}A_{\mu}\dot{x}^{\mu}$, and then, through the
covariant quantity, $F_{ab}$.

Therefore, on the gravitational side, we can use the coefficients from the
unbroken Lorentz generators (from the bulk theory) with the four velocity. The
covariant building block made out of the spin connection is,
$e^{-1}\omega_{\mu}^{ab}\dot{x}^{\mu}$. Therefore, we can build up the
correction,

$\displaystyle\mathcal{S}=n_{\Omega,g}\int\mathrm{d}\sigma
e\;\left(\nabla\alpha_{ab}e^{-1}\omega^{ab}_{\mu}\dot{x}^{\mu}\right)=\int\mathrm{d}\sigma
e^{-1}\;n_{\Omega,g}\Omega_{ab}\omega^{ab}_{\mu}\dot{x}^{\mu}.$ (57)

Comparison to the full theory shows that the coefficient $n_{\Omega,g}=I/2$,
with $I$ the moment of inertia, and therefore the spin is defined as
$S^{ab}=I\Omega^{ab}$. To lowest order with $e=1$, one recovers the well known
results in both the PN and PM expansions for the Spin-Orbit coupling in curved
space-time. This is a crucial difference of our EFT with respect to others, to
allow the covariant quantity, $e^{-1}\omega_{\mu}^{ab}\dot{x}^{\mu}$ to be a
building block. Therefore, in eq. (57), there is an inverse einbein in the
spin-orbit correction, for which beyond lowest order, the equations of motion
will differ from to the ones obtained using conjugate variables, i.e.
Porto:2005ac ; Levi:2015msa ; Liu:2021zxr .

Another implication is that the complete Spin-Orbit correction is described by
two terms,

$\displaystyle\mathcal{S}=\int\mathrm{d}\sigma
e\;\left(n_{\Omega,g}e^{-2}\Omega_{ab}\omega^{ab}_{\mu}\dot{x}^{\mu}+n_{\Omega,g,u}e^{-4}\Omega_{ab}\dot{x}^{a}\omega_{\mu}^{cb}\dot{x}^{\mu}\dot{x}_{c}\right).$
(58)

This extra term comes from the possible combinations of the angular velocity
with the spin connection and the four velocity. The extra term does not spoil
the effective theory. To lowest order, in the PM expansion the extra term is
removed by the corresponding SSC, recovering the results in Liu:2021zxr
following their procedure. In the PN expansion, this term can be seen as the
contribution from the relativistic acceleration in Levi:2015msa , which plays
a role on the Spin-Orbit dynamics of the PN expansion.

The next set of corrections that can be built with the spin connection and the
angular velocity that satisfies parity and time reversal symmetry,

$\displaystyle\begin{split}\mathcal{S}=\int\mathrm{d}\sigma
e\left(n_{\omega^{2}\Omega^{2}}e^{-2}\omega_{\nu}^{ab}\omega^{\nu
cd}\Omega_{ab}\Omega_{cd}+n_{R\Omega^{2}}e^{-4}R_{abcd}\Omega^{ab}\Omega^{cd}\right),\end{split}$
(59)

with the Riemann tensor, $R_{abcd}=R_{abcd}(\omega_{\mu}^{ab})$. Both of these
corrections are considered in Vines:2016unv  in the Hamiltonian formalism, and
the second term is considered as well in Bern:2020buy  and as the lowest order
correction in Bern:2022kto  from a tower of correction made out of the the
covariant derivative and the projected spin tensor, tensor which is defined
below. Another possible operator that can be built is

$\displaystyle\mathcal{S}=\int\mathrm{d}\sigma
e\;n_{\Omega^{2}R}e^{-4}\Omega^{ab}\Omega^{cd}\dot{x}_{a}R_{bfcd}\dot{x}^{f},$
(60)

correction that appears in the Routhian of Liu:2021zxr , and without going
into details into the Routhian formalism, we suggest that this is a term to
consider when building the EFT, given that is built out of the building blocks
and is allowed by the symmetries.

We can couple the electromagnetic field tensor as well with the Lorentz gauge
field, and build the Electro-Gravity couplings,

$\displaystyle\mathcal{S}=\int\mathrm{d}\sigma
e\;\left(n_{F,g}e^{-1}F_{ab}\omega^{ab}_{\mu}\dot{x}^{\mu}+n_{F,g,u}e^{-3}F_{ab}\dot{x}^{b}\omega^{ca}_{\mu}\dot{x}^{\mu}\dot{x}_{c}+.\;.\;.\right).$
(61)

In general we should couple all of the covariant operators in all the possible
ways that are allowed by the symmetries. By including all possible operators
there may be redundancies via the equations of motion, but even if all of them
are included, one will get the same answer for anything physical.

#### Extended Objects

We now describe the properties of an extended object, beyond the point
particle approximation. Invariant operators can be built to account for the
tidal deformation of the compact object using the electric and magnetic
components of the electromagnetic and Weyl tensor in the worldline
Goldberger:2004jt . It is useful to work with multipole expansions at the
level of the action Ross:2012fc , where induced multipole moments are
organized in irreducible representations of the rotation SO(3) group. This
makes the task of making invariant operators more transparent and ensures that
multipole moments are not mixed.

The projected electromagnetic and Weyl tensor are used to define their
electric, $E_{a.\,.\,.}$ and magnetic, $B_{a.\,.\,.}$, components. Size
effects are made out of the curvature,

$\displaystyle\begin{split}E_{ab}=&e^{-2}W_{acbd}\dot{x}^{c}\dot{x}^{d},\\\
B_{ab}=&\epsilon_{cdea}e^{-2}W^{cd}_{\;\;\;\;fb}\dot{x}^{e}\dot{x}^{f}.\end{split}$
(62)

The polarization is accounted by the electromagnetic tensor,

$\displaystyle\begin{split}E_{a}=&e^{-1}F_{ab}\dot{x}^{b},\\\
B_{a}=&\epsilon_{abcd}e^{-1}F^{bc}\dot{x}^{d}.\end{split}$ (63)

These operators, which are traceless and transverse to $\dot{x}^{a}$, are
covariant under worldline re-parameterizations. They couple to all other
derived building blocks, in the possible ways allowed by the symmetries, to
construct higher order operators.

We can also build spin-induced multipoles Porto:2005ac ; Levi:2014gsa , using
the angular velocity as the Pauli-Lubanski spin tensor,

$\displaystyle
e^{-2}\Omega_{a}=\frac{1}{2}\epsilon_{abcd}e^{-2}\Omega^{cd}\dot{x}^{b}.$ (64)

In the proper frame of the object,

$\Omega_{i}=-\frac{1}{2}\epsilon_{ijk}\Lambda_{a}^{\;\;j}(\eta^{ab}\partial_{\tau}+\omega_{\mu}^{ab}u^{\mu})\Lambda_{b}^{\;\;k}.$
(65)

Considering the covariant operators and their corresponding covariant
derivatives, to lowest order we build

$\displaystyle\begin{split}\mathcal{O}(\Omega,E,B)&=\begin{cases}&e^{-4}E^{ab}E_{ab},\,e^{-4}B^{ab}B_{ab}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{Gravity},\\\
&e^{-6}\Omega^{a}\Omega^{b}E_{ab},\,e^{-9}\Omega^{a}\Omega^{b}\Omega^{c}\nabla_{c}B_{ab}\,\;\;\;\;\;\;\;\;\;\;\;\mathrm{Spin-
Gravity},\\\
&e^{-2}E^{a}E_{a},\,e^{-2}B^{a}B_{a}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;\;\;\;\;\;\;\;\;\;\mathrm{Electromagnetic,}\\\
&e^{-3}\Omega^{a}B_{a},\;e^{-6}\Omega^{a}\Omega^{b}\nabla_{b}E_{a}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;\;\;\mathrm{Spin-
Electro,}\\\ \end{cases}\end{split}$ (66)

which are parity and time reversal invariant. The operators from the first
Goldberger:2005cd  and second line Porto:2005ac ; Levi:2015msa  in (66),
encode the tidal deformation due to an induced multipole. The rest of the
operators takes into account for the polarization of the extended object. The
ones from the third line were first considered in Goldberger:2004jt , and from
the fourth line we have not found a reference yet. The latter could play a
crucial role in the description of pulsars: neutron stars with considerable
spin and strong magnetic field. In the effective action, each of the operators
are accompanied by a Wilson coefficient that encodes the internal structure of
the compact object. The operators in eq. (66), contains linear corrections due
to the tidal deformation and polarization. Non-linear effects Bern:2020uwk
can be considered using the blocks from eq. (62) and eq. (63).

The covariant derivative can be used in the Goldstone’s building blocks, and
therefore we should build terms made out of $\nabla_{b}\Omega_{a}$ as well.
Moreover, in principle one can couple two different multipole expansions,
respecting parity invariance and time reversal symmetry. From such
considerations, we build the following operators,

$\displaystyle\begin{split}\mathcal{O}(\Omega,E,B)&=\begin{cases}&e^{-4}\nabla_{a}E_{b}E^{ab},\,e^{-4}\nabla_{a}B_{b}B^{ab},\,e^{-5}\nabla_{b}\Omega_{a}B^{ab},\,e^{-5}\Omega_{a}B_{b}E^{ab}\;.\;.\;..\\\
\end{cases}\end{split}$ (67)

Although to lowest order, $\nabla_{b}\Omega_{a}=0$, the spin evolves through
time due to spin-size effects even in the Newtonian limit Endlich:2015mke ,
and therefore one should carefully treat such term.

It should be noted that size effects can be seen as encoded in a composite
operator $Q^{ab}$ Goldberger:2005cd , which we discuss in detail below. By
considering the tidal deformation as a dynamical operator, we can consider
considering dissipative effects and dynamical tides.

#### Dynamical Operators

One can introduce composite dynamical operators, $Q_{ab}(\sigma)$ and
$Q_{a}(\sigma)$, to account for the tidal and polarization response function
respectively. The simplest operators that can be built are Goldberger:2005cd

$\displaystyle\begin{split}\mathcal{O}(E)&=\begin{cases}&e^{-4}Q^{ab}(\sigma)E_{ab}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\textrm{Gravity},\\\
&e^{-2}Q^{a}(\sigma)E_{a}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\,\textrm{Electro},\end{cases}\end{split}$
(68)

with $Q^{ab}$ and $Q^{a}$, composed of the worldline degrees of freedom,
$E^{ab}$ and $E^{a}$ respectively. The covariant building blocks with magnetic
like parity, $B^{ab}$ and $B^{a}$, are used as well to construct invariant
dynamical operators Goldberger:2005cd .

The operators, $Q^{ab}$ and $Q^{a}$, are dependent on the internal degrees of
freedom of the compact object in an unspecified way, but which explicit form
is not necessary to obtain the dynamics Goldberger:2005cd ; Goldberger:2020fot
. The dynamics of the system can be obtained using the in-in closed time path
Jordan:1986ug , a formalism that allows us to treat the tidal response
function with dissipative effects in a time asymmetric fashion
Goldberger:2005cd . The internal structure is encoded in the Wilson
coefficients coming from the response function.

The interactions of the mass quadrupole $Q^{ab}$, up to linear order, can be
described in terms of an invertible Hermitian linear operator
$\mathcal{O}^{ab}_{\;\;\;cd}$,

$\displaystyle\mathcal{O}$
$\displaystyle=\begin{cases}&e^{-4}Q_{ab}\mathcal{O}^{ab}_{\;\;\;cd}Q^{cd},\end{cases}$
(69)

The effective action that describes the dynamics of the dynamical variable
$Q^{ab}$,

$\displaystyle\mathcal{S}=\int d\sigma
e\left(e^{-4}E_{ab}Q^{ab}+e^{-4}Q_{ab}\mathcal{O}^{ab}_{\;\;\;cd}Q^{cd}\right),$
(70)

leads to the equations of motion for $Q^{ab}$,

$\displaystyle\mathcal{O}^{ab}_{\;\;\;cd}Q^{cd}=-\frac{1}{2}E^{ab}.$ (71)

The Green’s function for $\mathcal{O}^{ab}_{\;\;\;cd}$, is the solution to the
above equation with the delta function as a source,
$\mathcal{O}^{ab}_{\;\;\;cd}G^{\;\;\;cd}_{ab}=\delta(\sigma-\sigma^{\prime})$,
such that the Green’s function can be given in terms of the inverse operator,
$\mathcal{O}^{-1\;ab}_{\;\;\;\;\;\,\;\;\;\;cd}$.

The expectation value, $\braket{Q^{ab}(\sigma)}$, is obtained by integrating
the Green’s function over the source $E^{ab}$, which is the expectation value
in the initial state of the internal degrees of freedom. The requirement that
the external field is zero at the initial state of the interaction requires to
choose the retarded Green’s function. Considering the linear response in a
weak external field, the in-in formalism implies the expectation value
Goldberger:2020fot

$\displaystyle\braket{Q^{ab}(\sigma)}=\int\mathrm{d}\sigma^{\prime}e^{\prime}G^{ab,cd}_{R}(\sigma-\sigma^{\prime})e^{\prime-2}E_{cd}(\sigma^{\prime})+O\left(e^{-4}E^{2}\right),$
(72)

with retarded Green’s function,

$\displaystyle
G^{ab,cd}_{R}(\sigma-\sigma^{\prime})=-i\theta(\sigma-\sigma^{\prime})\braket{[Q^{ab}(\sigma),Q^{cd}(\sigma^{\prime})]},$
(73)

The Green’s function provides the response function of the quadrupole under
external gravitational forces.

By considering low frequencies, from which we assume that the degrees of
freedom from the operator, $Q^{ab}$, are near equilibrium, the time ordered
two point correlation function imply that the Fourier transform, $G_{R}$, must
be an odd, analytic function of the frequency, $\omega>0$. Given that the
operator $Q_{ab}$ is a symmetric trace-free tensor, the Green’s function is
projected with the symmetric trace-free projection operator. In terms of the
worldline parameter, $\omega\sim\partial_{\sigma}$, the retarded correlation
function Goldberger:2005cd

$G^{ab,cd}_{R}(\sigma)\simeq
F(\sigma)\left(\delta^{ac}\delta^{bd}+\delta^{ad}\delta^{cb}-\frac{2}{3}\delta^{ab}\delta^{cd}\right),$
(74)

with $F(\sigma)$,

$\displaystyle
F(\sigma)=n_{g}+ic_{g}\frac{\mathrm{d}}{\mathrm{d}\sigma}+n_{g}^{\prime}\frac{\mathrm{d}^{2}}{\mathrm{d}\sigma^{2}}+.\;.\;..$
(75)

Considering the response of the interaction to be nearly instantaneous, the
expectation value of the dynamical operator encoding the tidal response
function in the adiabatic approximation reads Goldberger:2005cd ;
Goldberger:2020fot

$\displaystyle\braket{Q_{ab}(\sigma)}\simeq
n_{g}E_{ab}+ic_{g}\frac{\mathrm{d}}{\mathrm{d}\sigma}E_{ab}+n_{g}^{{}^{\prime}}\frac{\mathrm{d}^{2}}{\mathrm{d}\sigma^{2}}E_{ab}+\,.\;\;.\;\;.\,.$
(76)

From the first term in the expansion, it yields the lowest order operator
built for the tidal deformation in eq. (66). The second one accounts for
dissipative effects to lowest order, and the third for dynamical oscillations
in the quasi-static limit. The Wilson coefficients appearing in eq. (76),
encode the internal structure.

On the electromagnetic side, an analog procedure can be taken, for which
retarded correlation function, Goldberger:2005cd

$G^{ab}_{R}(\sigma)\simeq\left(n_{q}+ic_{q}\frac{\mathrm{d}}{\mathrm{d}\sigma}+.\;.\;.\right)\delta^{ab},$
(77)

Thus, the dynamical operator to account for the polarizability in the quasi-
static limit reads

$\displaystyle\braket{Q_{a}(\sigma)}\simeq
n_{q}e^{-1}E_{a}+ic_{q}\frac{\mathrm{d}}{\mathrm{d}\sigma}e^{-1}E_{a}+\,.\;\;.\;\;.\,.$
(78)

#### Dissipative Effects

In the EFT description of extended objects, dissipative effects arise due to
the existence of gapless modes that are localized on the worldline of the
particle, taking into account for the energy and momentum loss from the
interaction with external sources Goldberger:2005cd . These effects occur due
to a relative time-dependence between the object and its tidal environment,
even if it is not rotating. If the extended object is spinning, in the co-
rotating frame the external environment rotates at the frequency of the
angular velocity, generating spin dependent dissipative effects.

After dissipation in a worldline EFT description for a static compact object
was introduced Goldberger:2004jt , dissipative effects for spinning extended
objects in EFT were incorporated in Porto:2007qi  for the PN expansion in
phase space. Using the model from the coset Delacretaz:2014oxa  in the
Newtonian limit Endlich:2015mke , spin dissipative effects were obtained from
the effective action when considering a modified variation that takes into
account for non-conservative effects Galley:2012hx . Dissipation for maximally
spinning extended objects were introduced in Goldberger:2020fot , which upon
taking the Newtonian limit, recovers the results from Endlich:2015mke . We
generalize Endlich:2015mke  for the relativistic case.

These large number of degrees of freedom can be encoded in operators allowed
by the symmetries of the object as shown above. Given the imaginary dependence
of the dissipative effects, and in order to extract the dynamics using a
modified variation, it is useful to separate the conservative from the non-
conservative part from the tidal response function. Therefore, from the
operator in eq. (78), we extract the imaginary part, which to lowest order
Goldberger:2005cd ; Endlich:2015mke :

$\displaystyle\begin{split}\mathcal{O}(E)&=\begin{cases}&e^{-2}\mathcal{D}^{a}(\sigma)E_{a}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{Electro},\\\
&e^{-4}\mathcal{D}^{ab}(\sigma)E_{ab}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\mathrm{Gravity},\end{cases}\end{split}$
(79)

with the covariant quantities, $\mathcal{D}^{a}(\sigma)$ and
$\mathcal{D}^{ab}(\sigma)$, composite dynamical operators made out of
$e^{-1}E^{a}$ and $e^{-2}E^{ab}$ respectively, encoding the dissipative
degrees of freedom. The covariant building blocks with magnetic like parity,
$B^{a}$ and $B^{ab}$, are used as well to construct invariant operators
Goldberger:2005cd ; Endlich:2015mke ; Goldberger:2020fot .

Therefore, the non-conservative expectation value of the gravitational
dissipation in the quasi-static limit,

$\displaystyle\braket{\mathcal{D}_{ab}(\sigma)}\simeq
ic_{g}\frac{\mathrm{d}}{\mathrm{d}\sigma}e^{-2}E_{ab}+\,.\;\;.\;\;.\,.$ (80)

Once we have the effective action, the non-conservative dynamics can be
obtained from the modified variation Galley:2012hx

$\delta\mathcal{S}+i\int\mathrm{d}\sigma\mathrm{d}\sigma^{\prime}\delta
E_{ab}(\sigma)G^{ab,cd}_{R}(\sigma-\sigma^{\prime})E_{cd}(\sigma^{\prime})=0.$
(81)

On the electromagnetic dissipation an analog procedure is taken.

For a non-spinning black hole, dissipation takes into account for the
absorption of electromagnetic and gravitational waves. For a non-spinning
neutron star, dissipative effects can be generated due to the internal
viscosity of the star Goldberger:2005cd , or due to a time asymmetry in the
formation of a tidal bulge Hut1981 . When one is dealing with spinning
extended objects, the spin has a time dependence between the object and its
environment, which contributes to dissipation Endlich:2015mke .

#### Dynamical Oscillations

One can use the composite operator, $Q_{ab}(\sigma)$, in eq. (76), to account
for the dynamical tidal response in the adiabatic approximation, where the
last term is a dynamical correction in the quasi-static limit. For black
holes, eq. (76) might describe them accurately given that the coefficients
$n_{g}$ and $n_{g}^{{}^{\prime}}$ vanishes Poisson:2004cw ; Hui:2021vcv ;
Chia:2020yla . Deviations from a more fundamental theory of gravity might
imply the existence of a non-zero coefficients Cardoso:2018ptl , for which
then eq. (76) can be used to test gravity.

For neutron stars, the adiabatic approximation is not enough. It is possible
to treat $Q^{ab}$ as an independent degree of freedom Steinhoff:2016rfi , and
build the effective action

$\displaystyle\begin{split}\mathcal{S}=&\int d\sigma
e\left(.\;.\;.+e^{-4}E_{ab}Q^{ab}+e^{-4}Q_{ab}F^{-1}(\sigma)Q^{ab}\right)\\\
=&\int d\sigma
e\left(.\;.\;.+e^{-4}E_{ab}Q^{ab}+n_{o}e^{-4}Q_{ab}Q^{ab}+n_{o}^{\prime}e^{-4}\dot{Q}_{ab}\dot{Q}^{ab}+c_{o}e^{-4}\dot{Q}_{ab}Q^{ab}\right).\end{split}$
(82)

The third and fourth term in the last line correspond to the dynamical
internal oscillations of the compact object, induced by an external tidal
force. The last term is not time-reversal invariant and therefore a non-
conservative effect. The perturbative dynamics of the binary in terms of
$Q^{ab}$ can be obtained from the effective action Steinhoff:2016rfi ;
Steinhoff:2021dsn .

###  4.2 The Effective Action

Collecting the constructed invariant operators, we can build an effective
action that describes a compact object. There are some differences in
describing the two currently detected compact objects, black holes and neutron
stars, which is reflected in the values for the Wilson coefficients. Despite
their differences, we can build a generic effective action. To lowest order, a
compact object that is charged and spinning is described by the effective
action in the rest frame,

$\displaystyle\begin{split}\mathcal{S}_{eff}=&\int\mathrm{d}\tau\big{(}-\frac{m}{2}\left(1-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\right)+q\dot{x}^{\mu}A_{\mu}+\frac{I}{4}\Omega_{ab}\Omega^{ab}\\\
&\;\;\;\;\;\;\;\;\;\;\;\;+n_{\Omega,q}\Omega_{ab}F^{ab}+n_{\Omega,g}\Omega_{ab}\omega^{ab}_{\mu}\dot{x}^{\mu}+n_{q,g}F_{ab}\omega^{ab}_{\mu}\dot{x}^{\mu}\\\
&\;\;\;\;\;\;\;\;\;\;\;\;+n_{\Omega,B_{q}}\Omega^{a}B_{a}+n_{\Omega,E_{g}}\Omega^{a}\Omega^{b}E_{ab}+n_{E_{q}}E^{a}E_{a}\\\
&\;\;\;\;\;\;\;\;\;\;\;\;+n_{E_{g}}E^{ab}E_{ab}+ic_{E_{g}}E^{ab}\dot{E}_{ab}+n^{{}^{\prime}}_{E_{g}}\dot{E}^{ab}\dot{E}_{ab}+.\;.\;.\;\big{)}\\\
&\;\;\;\;\;+\mathcal{S}_{0},\\\ \end{split}$ (83)

in the quasi-static limit, with the ellipsis representing all other operators
including the ones built in this work, and all others that can be constructed
and are allowed by the symmetries. The interaction action, $\mathcal{S}_{0}$,
is the Einstein-Maxwell action, eq. (20).

The first line of eq. (83) describes a massive charged spinning point
particle, with their coefficients matched to the well known theories. The
second line describes the interaction between the different degrees of
freedom, which includes the spin-orbit correction to lowest order. The third
and fourth line are multipole moments taking into account for the tidal
deformation and the polarization. In the fourth line we have included the
leading order dynamical tidal effects in the quasi-static limit, including
dissipation. The rest of the coefficients are left unmatched, which will be
done in a future work. The action describing the charged spinning compact
object lives in the worldline, while $\mathcal{S}_{0}$ lives in the bulk.

##  5 Conclusions

We have constructed an EFT that describes compact objects as point particles
with higher order corrections made out of the allowed couplings between the
derived covariant building blocks. It is based in the EFT for spinning
extended objects introduced in Delacretaz:2014oxa , which is derived using the
powerful method of the coset construction, that allows us to build effective
theories from the symmetry breaking pattern as the only input. The developed
worldline effective theory, describes compact objects that are characterized
by their mass, spin and charge, as well as their finite-size structure, with
an underlying effective Einstein-Maxwell bulk theory. The EFT is described in
the vierbein formalism.

The development of the framework brings various advantages. It allows to
obtain relativistic dynamics without going to phase space, thus being a
Lagrangian framework. It incorporates multiple different developed tools in
the EFT for extended objects into a single framework, without the need of
introducing additional degrees of freedom other than the ones derived from the
breaking of symmetries, and the one needed to take into account for dynamical
oscillations beyond the quasi-static limit. It provides a connection between
the different developed theories for spinning objects, shedding light onto a
common framework to describe the spin dynamics. It allows to build invariant
operators with explicit dependence on the einbeins, which then have
implications on the obtention of the dynamics using the Polyakov action,
providing new perturbative approaches to obtain the dynamics.

Moreover, it provides a transparent connection between the different inertial
frames from which the dynamics are described, as naturally expected from the
vierbein formalism. Therefore, the effective action in eq. (83) is valid to
obtain both the PN and PM expansion, and one can simply change from one frame
to another by choosing the appropriate worldline parameter $\sigma$ and
worldline parameterization $\dot{x}^{\mu}$, or in other words, by choosing an
einbein, $e$. The dynamics of binary systems using the derived effective
action are obtained in an upcoming work, where the vierbein formalism with the
use of einbeins is exploited.

###### Acknowledgements.

I.M. is very thankful to A. Weltman, R. Penco, I. Rothstein, J. Steinhoff, A.
Luna, T. Hinderer, S. Mougiakakos, G. Mogull, G. Creci, I. Meijer for the many
enlightening conversations and discussions. I.M. gratefully acknowledge
support from the University of Cape Town Vice Chancellor’s Future Leaders 2030
Awards programme which has generously funded this research, support from the
South African Research Chairs Initiative of the Department of Science and
Technology and the NRF. This research was supported in part by the National
Science Foundation under Grant No. NSF PHY-1748958, and from the Educafin-
JuventudEsGto Talentos de Exportacion programme.

##  Appendix A Conventions

We differentiate between space-time and local Lorentz indices as in
Delacretaz:2014oxa :

  * •

$\mu,\nu,\sigma,\rho...$ denote space-time indices.

  * •

$a,b,c,d...$ denote Lorentz indices.

  * •

$i,j,k,l...$ denote spatial components of the Lorentz indices.

We denote the time of occurrence and the location in space of an event with
the four component vector, $x^{a}=(x^{0},x^{1},x^{2},x^{3})=(t,\vec{x})$, and
define the flat space-time interval, $\mathrm{d}s$, between two events,
$x^{a}$ and $x^{a}+\mathrm{d}x^{a}$, by the relation

$\mathrm{d}s^{2}=-c^{2}\mathrm{d}t^{2}+\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2},$
(84)

which we write using the notation

$\mathrm{d}s^{2}=\eta_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b}\,\mathrm{;}\,\,\,\,\,\;\;\;\;\eta_{ab}=\mathrm{diag}(-1,+1,+1,+1).$
(85)

##  Appendix B Symmetries in Classical Field Theory

We consider symmetries that can be labelled by a continuous parameter,
$\theta$. Working with the Lie algebras of a group $G$, we write a group
element as a matrix exponential

$U=e^{i\theta^{u}T_{u}},$ (86)

where the generators $T_{u}$, with $u=1,...,n$, form a basis of the Lie
algebra of G. The $T$’s generators are hermitian if $U$ is unitary. For each
group generator, a corresponding field arises, which in this case is the
field, $\theta$.

The properties of a group G, are encoded in its group multiplication law

$[T_{u},T_{v}]=T_{u}T_{v}-T_{v}T_{u}=ic_{uvw}T_{w},$ (87)

where, $c_{uvw}$, are the structure constant coefficients. The last expression
defines the Lie algebra of the group G. The Lie bracket is a measure of the
non-commutativity between two generators.

In the local framework of field theory, it is also possible to consider
continuous symmetries that have a position dependent parameter,
$\theta=\theta(x)$. The space-time dependent symmetry transformation rules are
called local or gauge symmetries. For global symmetries, $\theta$ do not
depends on the position. There is also a distinction between internal
symmetries and space-time symmetries, on whether they act or not on space-time
position. An example of an internal symmetry, where $x$ is unchanged, is

$\phi^{u}(x)\rightarrow U^{-1}\phi^{u}(x)U=\mathcal{U}_{v}^{\;u}\phi^{v}(x),$
(88)

while an example for a space-time symmetry is the transformation

$\phi^{u}(x)\rightarrow
V^{-1}\phi^{u}(x)V=\mathcal{V}_{v}^{\;u}\phi^{v}(x^{\prime}),$ (89)

with $x^{{}^{\prime}u}=\mathcal{V}_{v}^{\;u}x^{v}$. Both internal and space-
time symmetries can arise in global or gauged varieties.

### Symmetries of Special Relativity

The full symmetry of special relativity is determined by the Poincaré
symmetry. Its Lie group, known as the Poincaré group, G = ISO(3,1), is the
group of Minkowski space-time isometries that includes all translations and
Lorentz transformations.

#### The Lorentz Group

The Lorentz group, SO(3,1), is the group of linear coordinate transformations

$x^{a}\rightarrow x^{\prime a}=\Lambda^{a}_{\;\;b}x^{b},$ (90)

that leave invariant the quantity

$\eta_{ab}x^{a}x^{b}=-(ct)^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2},$ (91)

with $\mathrm{det}\Lambda=1$. In order for eq. (91) to be invariant, $\Lambda$
must satisfy

$\eta_{ab}x^{\prime a}x^{\prime
b}=\eta_{ab}(\Lambda^{a}_{\;\;c}x^{c})(\Lambda^{b}_{\;\;d}x^{d})=\eta_{cd}x^{c}x^{d},$
(92)

which implies the transformation of the metric as

$\eta_{cd}=\eta_{ab}\Lambda^{a}_{\;\;c}\Lambda^{b}_{\;\;d}.$ (93)

Consider an infinitesimal Lorentz transformation, with the Lorentz generators
$J_{ab}$, and its corresponding field, $\alpha_{ab}$. We can expand

$\Lambda^{a}_{\;\;b}=(e^{\frac{i}{2}\alpha^{cd}J_{cd}})^{a}_{\;\;b}=(e^{\alpha})^{a}_{\;\;b}\approx\delta^{a}_{\;\;b}+\alpha^{a}_{\;\;b}.$
(94)

From equation (93) we find

$\alpha_{ab}=-\alpha_{ba},$ (95)

which is an antisymmetric 4x4 matrix with six components that are independent.
Thus, the six independent parameters of the Lorentz group from the
antisymmetric matrix, $\alpha_{ab}$, corresponds to six generators which are
also antisymmetric $J^{ab}=-J^{ba}$.

Under Lorentz transformations, a scalar field is invariant,

$\phi^{\prime}(x^{\prime})=\phi(x).$ (96)

A covariant vector field, $V^{a}$, transforms in a representation of the
Lorentz group,

$V^{a}\rightarrow(e^{\frac{i}{2}\alpha_{cd}J^{cd}})^{a}_{\;\;b}V^{b}.$ (97)

If we consider an infinitesimal transformation, the variation of $V^{a}$
reads,

$\delta V^{a}=\frac{i}{2}\alpha_{cd}(J^{cd})^{a}_{\;\;b}V^{b},$ (98)

which is an irreducible representation.

The explicit form of the matrix $(J^{ab})^{c}_{\;\;d}$, reads

$\displaystyle(J_{ab})^{c}_{\;\;d}=-i(\delta^{c}_{\;\;a}\eta_{bd}-\delta^{c}_{\;\;b}\eta_{ad}).$
(99)

Using the form of the generator in eq. (99), we can compute the commutator

$[J_{ab},J_{cd}]=i(\eta_{ac}J_{bd}-\eta_{bc}J_{ad}+\eta_{bd}J_{ac}-\eta_{ad}J_{bc}),$
(100)

to find the Lie algebra. The components of $J^{ab}$ can be rearranged into two
spatial vectors

$J_{i}=\frac{1}{2}\epsilon_{ijk}J^{jk},\;\;\;K^{i}=J^{i0},$ (101)

with, $J^{ij}$ and $K^{i}$, the generators of rotations and boosts,
respectively.

The Lorentz group has six parameters: Three rotations in three $2D$ planes
that can be formed with the $(x,y,z)$ coordinates that leave $ct$ invariant,
which is the SO(3) rotation group, and three boost transformations in the
$(ct,x)$, $(ct,y)$ and $(ct,z)$ planes that leave invariant $-(ct)^{2}+x^{2}$,
$-(ct)^{2}+y^{2}$ and $-(ct)^{2}+z^{2}$, respectively. We parameterize the
Lorentz matrix as

$\Lambda^{0}_{\;\;0}=\gamma,\;\;\Lambda^{0}_{\;\;i}=\gamma\beta_{i},\;\;\Lambda^{i}_{\;\;0}=\gamma\beta^{i},\;\;\Lambda^{i}_{\;\;j}=\delta^{i}_{\;\;j}+(\gamma-1)\frac{\beta^{i}\beta_{j}}{\beta^{2}},\;\;$
(102)

with $\gamma=(1-v^{2}/c^{2})^{-1/2}$, the Lorentz factor, and $\beta^{i}$, the
velocity

$\beta^{i}\equiv\frac{\eta^{i}}{\eta}\tanh\eta,$ (103)

where $\eta$ is the rapidity, defined as the hyperbolic angle that
differentiates two inertial frames of reference that are moving relative to
each other.

Therefore, the four vectors, $V^{a}$ and $V_{a}$, transforms under the Lorentz
group as

$\displaystyle V^{a}(x)\rightarrow
V^{{}^{\prime}a}(x^{\prime})=\Lambda^{a}_{\;\;b}V^{b}(x),\;\;\;\;\;\;V_{a}(x)\rightarrow
V_{a}^{{}^{\prime}}(x^{\prime})=\Lambda_{a}^{\;\;b}V_{b}(x),$ (104)

with $\Lambda_{a}^{\;\;b}=\eta_{ac}\eta^{bd}\Lambda^{c}_{\;\;d}$. The vectors
are related via $V_{a}=\eta_{ab}V^{b}$. A tensor, $T^{ab}$, transforms as

$T^{ab}(x)\rightarrow T^{\prime
ab}(x^{\prime})=\Lambda^{a}_{\;\;c}\Lambda^{b}_{\;\;d}T^{cd}(x).$ (105)

In general, any tensor with arbitrary upper and lower indices transforms with
a $\Lambda^{a}_{\;\;b}$ matrix for each upper index, and with
$\Lambda_{a}^{\;\;b}$ for each lower one. We denote Lorentz transformations
simply as, $V^{{}^{\prime}a}=\Lambda^{a}_{\;\;b}V^{b}$ and
$T^{{}^{\prime}ab}=\Lambda^{a}_{\;\;c}\Lambda^{b}_{\;\;d}T^{cd}$.

#### The Poincaré group

To complete the Poincaré group, in addition to Lorentz invariance, we also
require invariance under space-time translations. We can write a general
element of the group of translations in the following form,

$U=e^{iz^{a}P_{a}},$ (106)

where $z^{a}$ are the components of the translation,

$x^{a}\rightarrow x^{a}+z^{a},$ (107)

and $P^{a}$ their generators. Lorentz transformations plus translations form
the Poincaré group, ISO(3,1). The Poincaré group algebra reads

$\displaystyle[P_{a},P_{b}]$ $\displaystyle=0$ (108)
$\displaystyle[P_{a},J_{bc}]$ $\displaystyle=i(\eta_{ac}P_{b}-\eta_{ab}P_{c})$
(109) $\displaystyle[J_{ab},J_{cd}]$
$\displaystyle=i(\eta_{ac}J_{bd}-\eta_{bc}J_{ad}+\eta_{bd}J_{ac}-\eta_{ad}J_{bc}).$
(110)

### Gauge Symmetry of Classical Electromagnetism

The gauge symmetry of classical electromagnetism is invariance under the U(1)
gauge transformation. This is an internal symmetry for which the charge
generator, Q, correspond to a time invariant generator, with its corresponding
gauge field, $A_{\mu}(x)$, which is the electromagnetic or photon gauge field.
The local gauge symmetry is parameterized by the parameter,
$\theta=\theta(x)$, with the group element

$\displaystyle U(x)=e^{\theta(x)}.$ (111)

The gauge field, $A_{\mu}$, transforms under the U(1) symmetry as

$\displaystyle A_{\mu}(x)\rightarrow A_{\mu}(x)+\partial_{\mu}\theta(x).$
(112)

Therefore, under Lorentz and U(1) transformations, the gauge field transforms
as

$\displaystyle
A_{\mu}(x)\rightarrow\Lambda_{\mu}^{\;\;\nu}A_{\nu}(x)+\partial_{\mu}\theta(x,\Lambda).$
(113)

The commutations relations of the charge generator, $Q$, with the generators
of the Poincaré group, are constrained by the Coleman-Mandula theorem
Coleman:1967ad . This theorem constraints the kinds of continuous space-time
symmetries that can be present in an interacting relativistic field theory,
and states that the most general possible transformation can be parameterized
by

$U=\exp\left\\{i\left(z^{a}P_{a}+\sigma^{a}\mathcal{O}_{a}+\frac{1}{2}\omega^{ab}J_{ab}\right)\right\\},$
(114)

with $P_{a}$, the generators of translations, $J_{ab}$, of Lorentz
transformations, and $\mathcal{O}_{a}$, the rest of the generators. The
generators, $\mathcal{O}_{a}$, must be from internal symmetries, and although
they can fail to commute with themselves,
$[\mathcal{O}_{a},\mathcal{O}_{b}]\neq 0$, they must always commute with the
space-time symmetry generators, $[P_{a},\mathcal{O}_{b}]=0$ and
$[J_{ab},\mathcal{O}_{c}]=0$. The charge operator for the U(1) symmetry of
electromagnetism commutes with itself, thus obtaining the commutation
relations: $[P_{a},Q]=0$, $[J_{ab},Q]=0$ and $[Q,Q]=0$.

### Transformation Properties of Gauge Fields

The transformation properties of the gauge fields $\breve{e}_{\mu}^{a}$,
$\breve{A}_{\mu}$ and $\breve{\omega}_{\mu}^{ab}$, introduced in eq. (7),
under local translations, $e^{iz^{a}P_{a}}$, local Lorentz transformations,
$e^{\frac{i}{2}\alpha_{cd}J^{cd}}$ and the local U(1) transformation,
$e^{i\theta}$, read

$\displaystyle\begin{split}U&=\;\,e^{i\theta}\;:\begin{cases}\;\breve{A}_{\mu}\;\rightarrow\;\breve{A}_{\mu}-\partial_{\mu}\theta,\\\
\;\breve{e}_{\mu}^{\;a}\;\rightarrow\;\breve{e}_{\mu}^{\;a},\\\
\;\breve{\omega}_{\mu}^{ab}\rightarrow\;\breve{\omega}_{\mu}^{ab}.\end{cases}\\\
U&=\,e^{icP}:\begin{cases}\,\breve{A}_{\mu}\;\rightarrow\;\breve{A}_{\mu},\\\
\;\breve{e}_{\mu}^{\;a}\;\rightarrow\;\breve{e}^{\;a}_{\mu}-\breve{\omega}^{a}_{\mu
b}z^{b}-\partial_{\mu}z^{a},\\\
\;\breve{\omega}_{\mu}^{ab}\rightarrow\;\breve{\omega}_{\mu}^{\;ab}.\end{cases}\\\
U&=e^{i\alpha
J}:\begin{cases}\;\breve{A}_{\mu}\;\rightarrow\;\Lambda^{\;\;\nu}_{\mu}\breve{A}_{\nu},\\\
\;\breve{e}_{\mu}^{\;a}\;\rightarrow\;\,\Lambda^{a}_{\;b}\breve{e}^{b}_{\mu}=\;\breve{e}^{a}_{\mu}+\alpha^{a}_{\;b}\breve{e}^{b}_{\mu},\\\
\,\breve{\omega}_{\mu}^{ab}\,\rightarrow\;\,\Lambda^{a}_{\;c}\Lambda^{b}_{\;d}\breve{\omega}^{cd}_{\mu}+\Lambda^{a}_{\;c}\partial_{\mu}(\Lambda^{-1})^{cb}=\breve{\omega}^{ab}_{\mu}+\breve{\omega}_{\mu}^{ac}\alpha^{b}_{\;c}+\breve{\omega}^{cb}_{\mu}\alpha^{a}_{\;c}-\partial_{\mu}\alpha^{ab}.\end{cases}\\\
\end{split},$ (115)

The gauge field, $\breve{e}$, transforms inhomogeneously under local
translations. Under Lorentz transformations, $\breve{e}$ and $\breve{A}$,
transforms linearly, while $\breve{\omega}^{ab}_{\mu}$ transforms as a
connection. Under the U(1) transformation, only the gauge field, $\breve{A}$,
is transformed.

Finally, the photon field, the vierbein and spin connection, under
diffeomorphisms transforms as

$\displaystyle\begin{split}A_{\mu}(x)\xrightarrow{\text{diffeo}}&A_{\mu}(x)-A_{\nu}(x)\partial_{\mu}\xi^{\nu}-\xi^{\nu}(x)\partial_{\nu}A_{\mu}(x),\\\
e_{\mu}^{a}(x)\xrightarrow{\text{diffeo}}&\;e_{\mu}^{a}(x)-e_{\nu}^{a}(x)\partial_{\mu}\xi-\xi^{\nu}(x)\partial_{\nu}e_{\mu}^{a}(x),\\\
\omega_{\mu}^{ab}(x)\xrightarrow{\text{diffeo}}&\;\omega_{\mu}^{ab}(x)-\omega_{\nu}^{ab}(x)\partial_{\mu}\xi-\xi^{\nu}(x)\partial_{\nu}\omega_{\mu}^{ab},\end{split}$
(116)

which all of them transforms in the same way.

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