Prepared for submission to JHEP

Effective actions for compact objects in an effectivefield theory of gravity

Irvin Martínez

High Energy Physics, Cosmology & Astrophysics Theory Group, Department of Mathematics &Applied Mathematics, University of Cape Town, Cape Town, 7701, South Africa

E-mail: irvinmtzrd@gmail.com

Abstract: Using the effective field theory framework for extended objects and the cosetconstruction, we build the leading order effective action for the most general compact objectallowed in an effective theory of gravity as general relativity. We derive all the covariantbuilding blocks and constraints to build up the relevant invariant terms in the action to allorders. Our derived effective action contains a relativistic point mass term, and relativisticcorrections due to spin, charge, size effects and dissipation. With our definition of therelativistic spin, we show that spin-acceleration corrections are encoded in higher orderspin-spin couplings. We derive the correction to the point particle due to charge, andbuild up the invariant operators that take into account for the internal structure. Each ofthe invariant operators in the action is accompanied by a coefficient that encapsulates theproperties of the compact object. We match the known coefficients of the effective actionfrom the literature, and point out the unknown ones that are to be derived.

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Contents

1 Introduction 1

2 Basics of the coset construction 4

3 Electrodynamics of spheres 63.1 Classical electromagnetism 63.2 Charged spheres 83.3 The effective action 12

4 Compact objects in general relativity 124.1 Effective theory of gravity 134.2 Charged spinning compact objects 154.3 Effective action for compact objects 24

5 Discussion 27

A Conventions 28

B Worldline point particle dynamics 29

C Symmetries in classical field theory 30

1 Introduction

The recent detection of gravitational waves (GWs) from various coalescing binaries [1–5] consisting of compact objects, such as black holes (BHs) and neutron stars (NSs), haveopened up the possibility to test fundamental physics in the strong regime of gravity. Withupcoming sensibility upgrades in current GW detectors, and future earth [6, 7] and spacebased detectors [8], the era of high precision gravity is arriving, and with it the potential ofgreat discoveries. One of the key potentials is to probe the internal structure of the compactobjects by matching the coefficients of the theory with GW observations [9]. Thus the needto develop theoretical models that describe the compact objects, taking into account for thedifferent effects that can play a role, such as the spin, charge, and their internal structure.

Using the coset construction [10–13] in the effective field theory (EFT) framework forextended objects [14–16], we build the most general effective action allowed in an effectivetheory of gravity such as general relativity. In this theory, BHs are constrained by the nohair theorem, which states that a BH can be described by only three parameters, its mass,spin and charge, behaving effectively as a point particle [17, 18]. In this sense, we derivethe leading order effective action for a charged spinning compact object. In the EFT for

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extended objects, we treat compact objects as point particles, with their additional effectsand internal structure encoded as higher order corrections in the action, which are madeup of the allowed invariant operators of the theory. These operators are accompanied bycoefficients which encapsulates the internal properties of the objects.

The coset construction is a very general technique from the EFT framework that canbe used whenever there is a symmetry breaking. With this technique, it is possible toderive the covariant building blocks that transform correctly under the relevant symmetries,which then can be used to form invariant operators to build up an effective action. Anystate other than vacuum will break some of the spacetime symmetries, and by correctlyidentifying the pattern of the symmetry breaking, we can derive its effective action. Withinthis approach, we can treat the coefficients that appears in front of the invariant operators asfree parameters to be fixed by experiments or observations. In this sense, we are interestedto know what was the full symmetry group, G, of the EFT, and what subgroup, H, wasrealized non-linearly, parametrized by the coset, G/H [19].

An effective theory of gravity, such as general relativity, can be derived using the cosetconstruction by weakly gauging the spacetime symmetry group of gravity, the Poincarégroup ISO(3,1), and realizing translations nonlinearly, which is parametrized by the cosetISO(3,1)/SO(3,1) [16], with SO(3,1) the Lorentz group. From this coset parametrization, itis possible to derive the widely known Einstein’s vierbein theory of curved space or tetradformalism, which is a generalization of the theory of general relativity that is independentof a coordinate frame. Once the underlying theory of gravity is developed, one can proceedto identify the symmetries that an extended object breaks to derive its effective action.For instance, a point particle breaks spatial translations and boosts, while a spinning pointparticle breaks spatial translations and the full Lorentz group [16].

Due to the Goldstone’s theorem [20] and the inverse Higgs constraint [13], the symmetrybreaking pattern for a spinning object implies the existence of a Nambu-Goldstone field,that takes the role of the angular velocity or spin of the object [16], yielding a very naturalconstruction for the action of spinning objects without introducing redundant degrees offreedom, as in the EFTs [21, 22]. Although it has been pointed out that this is a theory for"slowly" spinning compact objects, which from construction the effective action correspondsto the low energy dynamics of the theory, it has been suggested that we can safely considerthe current observed compact objects through GWs as slowly spinning [9].

Therefore, in this work we review and build on the EFT for spinning extended ob-jects introduced in [16]. We have extended it by defining the relativistic spin and the spinconstraint, and by including the electromagnetic charge and the internal structure, to de-scribe charged spinning compact objects. To incorporate charge, we add into the cosetparametrization the U(1) symmetry for electromagnetic charge as an internal symmetry,and identify the symmetry breaking pattern of a charged spinning point particle in curvedspacetime. On the symmetry breaking pattern, a point particle that is charged under aU(1) symmetry, corresponds to the eigenstate of the charge, for which does note break U(1).With this pattern, we derive the Einstein-Maxwell action and the correction to the pointparticle due to electromagnetic charge.

In this effective theory, no redundant spin degrees of freedom are needed in order to

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describe the dynamics of spinning extended objects. We derive all the covariant buildingblocks and constraints of the effective theory, which allows us to build up the tower ofinvariant operators and form the effective action to all orders. We show that our derivedeffective theory encodes the EFT for spinning objects in [22], therefore elucidating on thetheoretical foundations of spinning extended objects. With the inclusion of the electromag-netic charge, we shed light to probe whether or not charged BHs can exists, and to describecharged neutron stars, such as magnetars.

For the internal structure of the objects, we take into account for size [14], and dissi-pative effects [15, 23]. The coefficients accompanying the operators of these effects, encodethe microphysics of the object, and based on results in the literature, we can identify themwithout having to do the explicit computations. Tidal effects for the case of spinning com-pact objects have been taken into account in [9, 21, 24], as well as for noncompact objectsin [25], by considering only rotationally invariant operators and associating departures fromsphericity with higher order corrections.

We consider the size effects induced by charge as well, known as the polarizability, forwhich some of the coefficients are unknown. On the dissipative effects for spinning BHs,we consider the absorption of gravitational and electromagnetic waves [15], as well as thedissipation generated by the spin [23, 25–27]. For a NS, dissipation accounts for the energyloss due to the viscosity of the star. We match the known coefficients from the literature,and point out the unknown ones that are to be derived.

Our derived action has multiple applications in the description of the coalescence ofbinaries. The first application we have considered in [28], is on the post-Newtonian (PN)expansion in the nonrelativistic regime of gravity [14, 29–31], which is a perturbative seriesin terms of the expansion parameter v/c < 1, with v the relative velocity of the binary. ThePN expansion can be used for extracting GWs, and in particular for performing numericalsimulations in the late inspiral of the coalescence [9].

In the second application considered, we incorporate our action into the recently in-troduced EFT framework for the post-Minkowskian (PM) expansion [32–34], in which oneexpands over the gravitational constant, G, and which encodes the PN expansion. Finally,we are incorporating our derived effective action and its applications into the effective onebody (EOB) formalism [35–39], cross checking known results, and implementing new ones.The EOB, by a combination of analytical and numerical results, can take into account forthe full coalescence of a binary.

In section 2, we start with a very brief review of the basic ingredients of the cosetconstruction to derive our effective action, and refer reader to [16, 19] for a brief but morecomprehensive review. In section 3, we start with a pedagogical introduction to the cosetconstruction by deriving a classical theory of electromagnetism, as well as the effectiveaction for charged spheres in flat spacetime. This is done with particular emphasis tointroduce the tools of the coset construction with a concrete and simple known example,compared to [16], where the examples of superfluids and membranes can complicate theunderstanding for nonexperts on high energy physics. In section 4, we derive the Einstein-Maxwell action in the vierbein formalism, and then derive the covariant building blocks andconstraints to build up the effective action of a charged spinning compact object in curved

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spacetime. Then, we match the coefficients of the effective theory for compact objects fromthe literature. 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 briefbut more comprehensive review can be found in [16, 19]. We use the notation as in [16] toconsider the breaking of internal [10, 11] and spacetime symmetries [12, 40] alike.

The coset construction is a very general technique from the EFT framework that canbe used whenever there is a symmetry breaking. The breaking of some of the symmetriesimplies the existence of additional degrees of freedom, known as Nambu-Goldstone bosonsor simple as Goldstone fields.1 The coset construction is then used to derive building blocksfor the Goldstone fields that transform correctly under the relevant symmetries, blocks thatcan be used to build up invariant operators to form an effective action. Any state otherthan vacuum breaks at least some of the symmetries, and by appropriately identifying thepattern of the symmetry breaking, we can use it as a guide to derive the effective action.

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-linearlyrealized [19]. If the group is broken, G → H, due to a spontaneous symmetry breaking, thecoset recipe [16, 19] tells us that we can classify the generators into three categories:

Pa = generators of unbroken translations,

TA = generators of all other unbroken symmetries,

Xα = generators of broken symmetries,

(2.1)

where the broken generators, Xα, and the unbroken ones, TA, can be of spacetime symme-tries as well as of internal ones. We have chosen different indices to differentiate betweengenerators, but all of them run over Lorentz indices. Whenever the set of generators forbroken symmetries is nonzero, some Goldstone fields will arise. Thus, we must build aneffective action for the Goldstone fields that is invariant under the whole symmetry groupof consideration. The power of the coset construction, is that we can formulate an invariantEFT in which the broken symmetries and the unbroken translations are realized nonlinearlyon the Goldstone fields [16].

Following the coset recipe [16, 19], we do a local parametrization of the coset, G/H0,with H0, the subgroup of H generated by the unbroken generators, T ’s. The coset isparametrized as

g(x, π) = eiya(x)Paeiπ

α(x)Xα , (2.2)

1Goldstone theorem [20] implies the existence of a Goldstone field for each broken internal symmetry,but for the case in which spacetime symmetries are broken, there can be a mismatch on the number ofdegrees of freedom and broken symmetries, for which additional constraints are needed. See the InverseHiggs constraint below.

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where the factor, eiya(x)Pa , describes a translation from the origin of the coordinate systemto the point, xa, at which the Goldstone fields, πα(x), are evaluated. This factor ensuresthat the π’s transform correctly under spatial translation. The group element g, which isgenerated by the X’s and the P ’s, is known as the coset parametrization. For the case offlat spacetime, the translation is simply parametrized by, eixaPa , with y(x) ≡ x.

To derive the building blocks that depend on the Goldstone bosons and that havesimple transformation rules, we first note that the Goldstone fields, when appearing inthe Lagrangian, they are coupled through its derivatives. Then, we introduce the Maurer-Cartan form, g−1∂µg, a very convenient quantity that is an element of the algebra of G,and that can be written as a linear combinations of all the generators [16, 19],

g−1∂µg = (e aµ Pa +∇µπαXα + C B

µ TB). (2.3)

The coefficients eaµ, ∇µπα and CBµ , in general are nonlinear functions of the Goldstones,and form the building blocks to build the EFT, with ∇ α

µ = e aµ ∇ α

a and C Bµ = e a

µ CBa .

As we shall see, we can compute the explicit expression of the building blocks with thealgebra of the group G.

From the coset recipe, we can use the coefficients, C’s, and its operators, T ’s, to definethe covariant derivative that acts on matter fields [16, 19],

∇a ≡ (e−1) µa (∂µ + iCBµ TB). (2.4)

This covariant derivative can be used on the building blocks, as well as to consider highercovariant derivatives of the Goldstone fields.

Furthermore, in gauge symmetries, it is necessary to promote the partial derivative toa covariant one in the Maurer-Cartan form, ∂µ → Dµ [16]. Consider the gauged generator,EI , from a subgroup, G′ ⊆ G, with corresponding gauge field, wIµ. Thus, by replacing thepartial derivative with a covariant one, we obtain the modified Maurer-Cartan form,

g−1∂µg → g−1Dµg = g−1(∂µ + iwIµEI)g. (2.5)

This modification of the Maurer-Cartan form can also be written as a linear combinationof the generators as in eq. (2.3), with a new building block made up of the gauge field,wIµ, accompanying the gauged generator, EI . Now the building blocks can also depend onthe included gauge fields. The modified Maurer-Cartan form, g−1Dµg, is invariant underlocal transformations, and its explicit components can be obtained using the commutationrelations of the generators.

Inverse Higgs constraint

The Goldstone’s theorem [20], which states that a Goldstone mode exists for eachbroken generator, is only valid for internal symmetries. If spacetime symmetries are spon-taneously broken, there can be a mismatch in the number of broken generators and thenumber of bosons [41]. Nevertheless, we can preserve all the symmetries by imposing addi-tional local constraints, which can be solved to write down some of the Goldstone’s modes

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in terms of others. Using the inverse Higgs constraint [13], we can set to zero one or moreof the coset covariant derivatives, whenever X and X ′, are two multiplets of the brokengenerator, such that the commutators of the unbroken translations, P , and the broken gen-erator, X ′, yields a different broken generator, X: [P ,X ′] ⊃ X. If this is the case, we canset some of the covariant derivatives of the Goldstones to zero. By imposing all possibleinverse Higgs constraints, one obtains the only relevant building blocks.

3 Electrodynamics of spheres

3.1 Classical electromagnetism

It is well known that a classical theory of electromagnetism obeys the symmetries ofspecial relativity determined by the Poincaré group, ISO(3,1), as well as the symmetriesof the U(1) charge symmetry group. Thus, the full group is, G = U(1)×ISO(3,1), whichcontains the generators for Lorentz transformations, Jab, and gauge field, ωabµ , the generatorsof translations, Pa, and gauge field, eaµ, and the generator of charge, Q, and a gauge fieldAµ.2 Charge corresponds to a time invariant generator of the internal symmetry group,U(1), while the symmetry of the Poincaré group is a spacetime symmetry (See appendix Cfor more details).

Therefore, we parametrize the coset, U(1)×ISO(3,1)/U(1)×SO(3,1), separating trans-lations from the rest of the group. The coset parametrization in flat spacetime is given bythe group element

g = eixaPa = eix

aPaeix0Q, (3.1)

which contains all the unbroken translation, Pa = P0 + Pi = Pa + Q, with P0 = P0 + Q.Then, the Maurer-Cartan form reads

g−1Dµg = e−ixaPa

(∂µ + iAµQ+ ie a

µ Pa +i

2ωabµ Jab

)eix

aPa

= ∂µ + iAµQ+ ie aµ Pa +

i

2ωabµ Jab,

(3.2)

where we have used the commutation relation rules of the symmetries (See appendix C) toobtain

e aµ = e a

µ + ∂µxa + ωabµ xb, (3.3)

Aµ = Aµ + ∂µξ(x), (3.4)

ωabµ = ωabµ . (3.5)

2Note that we have defined the gauge fields, o’s, compared to [16], in which the gauge fields are definedas o’s. We will reserve the tilde to refer to quantities in the comoving frame.

– 6 –

Before going any further, we consider the case of flat spacetime. From a geometricalperspective, this limit implies that the curvature and the torsion tensor are equal to zero,which can be measured from the commutator of the two covariant derivatives, [Dµ, Dν ]. Byconsidering the covariant derivative appearing in the Maurer-Cartan form, (3.2), we cancast the commutator as,

[Dµ, Dν ] =iFµνQ+ iT aµνPa +i

2RabµνJab,

=i(∂µAν − ∂νAµ

)Q+ i

(∂µe

aν − ∂ν eaµ + eµbω

abν − eνbωabµ

)Pa

+i

2

(∂µω

abν − ∂ν ωabµ + ωaµcω

cbν − ωaνcωcbµ

)Jab.

(3.6)

Given that in flat spacetime, Tµνa = 0 and Rµνab = 0, it implies that

ωabµ = 0 and e aµ = 0, (3.7)

for which eq. (3.3), takes the simple form

e aµ = ∂µx

a = δ aµ . (3.8)

The field, e aµ , is known as the vierbein, which is used for deriving an invariant volumeelement d4x det e, and which for the case of flat spacetime is simply d4x. Given the cosetrecipe, we can use the coefficients of the unbroken U(1) generator in eq. (3.2), to define thegauge covariant derivative,

∇a ≡ ∂a + iQAa, (3.9)

which is the usual covariant derivative in a classical field theory of electromagnetism. Havingdefined the vierbein and the gauge covariant derivative, we can proceed to build invariantactions [16].

The first building block can be extracted from the commutator of the covariant deriva-tives, eq. (3.6), which in the flat spacetime limit,

g−1[Dµ, Dν ]g = i(∂µAν − ∂νAµ)Q = iFµνQ, (3.10)

with Fµν = ∂µAν − ∂νAµ, the electromagnetic tensor. Thus, the first invariant term thatcan be build with our covariant building block, Fµν , is,

S = −α∫

d4xFµνFµν . (3.11)

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As pointed out earlier, the coefficients of the theory can be treated as free parameters thatare to be fixed by experiments or from the literature. Thus, to reproduce Maxwell’s theory,we find the coefficient, α = (4µ0)−1, with µ0, the magnetic permeability of vacuum, suchthat we have the well known Maxwell action,

S = − 1

4µ0

∫d4xFabF

ab, (3.12)

where we have used, F ab = e aµ e

bν F

µν . We separate the space into the bulk and theworldline, such that eq. (3.12) lives in the bulk, while the action that describes the sphere,will live in the worldline.

3.2 Charged spheres

With the underlying theory of classical electromagnetism, we identify the symmetrypattern for a charged point particle, and derive its building blocks to construct an effectiveaction. Then, by considering spherical objects at rest, and departures from sphericity, weadd size effects to describe charged spheres.

Charged point particles

In general, a point particle breaks spatial translations, and boosts by choosing a pre-ferred reference frame. Under a U(1) symmetry, the state of a charged point particle isan eigenstate of the charge, and does not break U(1) the symmetry. Thus, the symmetrybreaking pattern reads

Unbroken generators =

P0 ≡ P0 +Q time translations,

Jij spatial rotations.

Broken generators =

Pi spatial translations,

J0i ≡ Ki boosts,

(3.13)

where we have included the charge generator as a time invariant generator of translations[16]. Given this pattern, we can parametrize the coset as

g = eixa(λ)Paeiη

i(λ)Ki = eixa(λ)Pa g. (3.14)

with λ, the worldline parameter that traces out the trajectory of the particle, ηi, theGoldstone mode, and g = eiη

i(λ)Ki .The important building blocks are contained in the Maurer-Cartan form projected

into the particle’s trajectory, xµg−1Dµg. It can be casted as a linear combination of thegenerators,

– 8 –

xµg−1Dµg = xµg−1(∂µ + iAµQ+ ie aµ Pa)g

= xµg−1(∂µ + iAµQ+ ie aµ Pa)g

= ixµ(AνΛνµQ+ e bµΛ 0b P0 + e bµΛ i

b Pi) + i(Λ−1)0cΛ

ciKi,

= iE(P0 +AQ+∇πiPi +∇ηiKi

),

(3.15)

where the dot means derivative with respect to the worldline parameter, λ, and Λab =

Λab(η) ≡ (eiηjKj )ab, being the boost matrix of the Lorentz transformations, which is a

function of the Goldstone field, ηi. Thus, the covariant quantities are

E = xµe bµ Λ 0

b ,

A = E−1xµAνΛνµ,

∇πi = E−1xµe bν Λ i

b ,

∇ηi = E−1(Λ−1)0cΛ

ci.

(3.16)

These are some of the building blocks that we will use to build up an invariant action.Nevertheless, as pointed out before, given that we are working with spacetime symmetries,there are some subtleties with the counting of Goldstone fields, for which the Inverse Higgsconstraint can be placed.

In this case, the commutator between boosts and unbroken time translations givesbroken spatial translations [16]. Therefore, by imposing the inverse Higgs constraint, weset to zero the covariant derivative, ∇πi = 0, such that we obtain the relation

∇πi = E−1xµe bµ Λ i

b = E−1(xµe 0µ Λ i

0 + xµeµΛ ij ) = 0. (3.17)

Solving for this constraint, we obtain the velocity (See eq. C.18)

βi ≡ ηi

ηtanh η =

xνe jν

xνe 0ν

= ui, (3.18)

where we have used the fact that, in flat spacetime, e aν = δ a

ν , such that ui = ∂τxi, the

four velocity measured in the proper frame. Using the inverse Higgs constraint we haveexpressed the η’s in terms of the π’s. This result can be interpreted as the Goldstones, η’s,or in terms of the, β’s, as parametrizing the boost necessary to get into the moving particlerest frame [16].

The building block, E = |E| =√E2, can be rewritten

|E| =√

(E∇πi)2 + (E2 − (E∇πi)2)

=√

(E∇πi)2 − (uν e aν u

µe bµ ),

=√−ηabe a

ν ebµ u

ν uµ =√−ηabuaub =

dτ

dλ,

(3.19)

– 9 –

where we have imposed the inverse Higgs constraint, and used the orthogonality propertyof the boost matrices, Λ b

a Λac = δbc, and e aµ = δ a

µ . Therefore, we can rewrite the inverseHiggs constraint in a way that the physical interpretation is transparent [16]. This is doneby expressing eq. (3.17), in terms of the velocity in the proper frame, ua, as

uaΛ ia = 0, (3.20)

and defining the set of Lorentz vectors, na(b),

na(0) ≡ ua = Λa0(η) , na(i) ≡ Λai(η). (3.21)

This set of Lorentz vectors define an orthonormal basis in the particle’s comoving frame.Then, we analyse the covariant derivative of the boost Goldstone, ηi, in eq. (3.16),

which can be rewritten as

∇ηi = n(i)a ∂τ u

a = n(i)b e

bµ a

µ = ai, (3.22)

with, ai, the acceleration of the particle in the comoving frame. Thus, the covariant deriva-tives, ∇ηi, corresponds to the component of the acceleration projected on the i-th vectordefined by the set of Lorentz vectors, na(b), measured in the proper frame of the particle [16].In the absence of external forces, ∇ηi = 0. Nevertheless, this building block is needed totake into account for a full description of the system, i.e. when the charge from an externalbody is relevant, such that the acceleration is nonzero.

Thus, we are left with the building blocks,

E = uµe 0µ ,

A = uµAµ,

∇ηi = ai.

(3.23)

with e aν = e b

ν Λ ab , and Aν = AνΛνµ, which are to be used in order to build an invariant

action. We can build an invariant term made up of the building block E,

S = −cE∫

dλE = −cE∫

dλdτ

dλ= −m

∫dτ, (3.24)

and match the coefficient, cE = m, to the action of a free relativistic point particle withmass, m. The last expression shows the action for a free relativistic point particle to allorders in the absence of external forces [16].

For the next covariant block, A, we add the point particle correction due to charge as,

S =

∫dλE(−mc2 + cAA) =

∫dτ(−mc2 + quaAa), (3.25)

with, q, the net charge of the charged particle, where the coefficient, cA, has been matchedfrom the classical theory of electrodynamics. Therefore, equation (3.25), is the action inthe proper frame that describes charged point particles in classical electromagnetism. Wenow move into the description of size effects, to describe a charged sphere.

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Size effects

In the context of the EFT for extended objects, the addition of size effects in the actionof a point particle was introduced in [14], which can be done in a systematic fashion. Sizecorrections must respect the symmetries of the theory, Lorentz and gauge U(1) symmetry,and thus contained in an infinite series expansion of all possible invariant operators madeup of our covariant building blocks. To build up invariant terms made out of Fab, we firstdefine its transformation properties under local Lorentz transformations. By consideringthe Lorentz parametrization of eq. (3.14), gL, the electromagnetic tensor transforms as

F ≡ g−1L F, (3.26)

such that F is transformed in a linear representation under a Lorentz transformation asexpected. The explicit transformation reads

Fab = (Λ−1) ca (Λ−1) d

b Fcd. (3.27)

Having the correct transformation rules, we can form rotationally invariant objectsmade out of Fab, ua, and ab, to build up the invariant action. These corrections in theaction reads,

S =∑n

∫dτcnOn(Fab, uc, ad), (3.28)

with, On, the invariant operators, cn, their corresponding coefficients, and n, being chosento the desired accuracy. The leading order, electric parity terms are

S =

∫dτ(nqF

baFcaubuc + nqauaabF

ab + . . .), (3.29)

where the ellipsis denotes higher order corrections. The first term correspond to the in-duced, electric parity, dipolar moment, while the second term can be seen as a higher ordercorrection to the latter due to the acceleration of the body.

The electric parity building block is identified, Ea = Fbaub [15], such that the first term

in the last equation, ∝ EaEa. For the size effects in the gravitational case, which are takeninto account with invariant combinations of the Riemann tensor defined in next section,the magnetic parity is subleading with respect to the electric one, therefore restrictingour discussion to the electric parity terms. Nevertheless, a term with magnetic parity,Ba = 1

2εabcdFbcud, can be build as well, ∝ BaBa. The leading order corrections in (3.29),

with their corresponding coefficients, cq, and cqa, encode the short distance information, orthe size structure, which are responsible for the polarization. The polarizability accountsfor the deformation of the sphere in the presence of an external electromagnetic field. Ingeneral, the coefficients, cq and cqa, are dependent on the radius of the sphere, thereforetaking into account for the fact that a sphere is an extended object. The specific form ofthese coefficient for a charged sphere is beyond the scope of this work.

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Dissipative effects

Dissipation in the EFT of extended objects was introduced in [15]. Dissipative effectsof the sphere takes into account for its absorption of electromagnetic waves. These largenumber of degrees of freedom can be encoded in operators allowed by the symmetries ofthe object. For instance, for a rigid sphere, the allowed operators for rotations, SO(3), aswell as parity eigenvalue, allows us to build the action [15]

S =

∫dτ Pa(τ)ubFab, (3.30)

with P(τ), a composite dynamical operator corresponding to the electric parity, electro-magnetic dipole moment. The specific form of the operator, P(τ), and its coefficient, arediscussed in detail in the next section.

3.3 The effective action

Finally, by considering, the electric parity only, we write down the leading order effectiveaction of a charged sphere in a classical theory of electromagnetism,

Seff =

∫dτ(−mc2 + quaAa + cqE

aEa + cqaEaaa + PaFabub + . . .

)+ S0, (3.31)

with

S0 = −∫

d4x1

4µ0FabF

ab. (3.32)

We have developed a theory of classical electromagnetism and described a charged sphereas a charged point particle with the finite size structure encoded in higher order operatorsin the action. In this approach, the description of the sphere lives in the worldline, whilethe interaction, eq. (3.32), lives in the bulk. In a similar manner, we proceed into thederivation of the leading order effective action for the most general compact object in atheory of gravity.

4 Compact objects in general relativity

Following the same methodology as in the previous section, before constructing ourtheory of compact objects, we review how a theory of gravity can be derived from thecoset construction as in [16], where a frame independent generalization of general relativity,known as Einstein’s vierbein field theory, is derived. This is a theory that can naturallyincorporate spinning objects. The difference of our construction compared to the one in[16], is the inclusion of the gauge symmetry group, U(1), which allows us to derive theEinstein-Maxwell action in the vierbein formalism, as well as the correction to the pointparticle due to charge. Once the underlying theory of gravity has been developed, then wederive the leading order action for charged spinning compact objects in the effective theoryof general relativity (vierbein formalism).

– 12 –

4.1 Effective theory of gravity

There are two symmetries in gravity to consider: Diffeomorphisms invariance, andPoincaré symmetry, determined by the Poincaré group, ISO(3,1), which contains the gener-ators for translations, Pa, and Lorentz transformations, Jab, with their corresponding gaugefields, eaµ and ωabµ . Both of the aforementioned symmetries of the system can be separatedby considering the principal bundle, P(M,G), with base manifold, M, and structure group,G. In this way, we realize the matter fields as sections of their respective fiber bundle [16].The coordinates, xµ, describing the position on the considered manifold, M, are not af-fected by the local Poincaré group, but it is transformed under diffeomorphisms. The localPoincaré transformations act along the fiber, while diffeomorphisms can be considered asrelabeling the points on the manifold.

To incorporate electrodynamics in the theory of gravity, we add the U(1) symmetryof electromagnetism with its gauge field, Aµ, and generator Q, and proceed with the cosetconstruction by gauging the Poincaré group and realizing translations nonlinearly [40]. Thecoset then reads, U(1) × ISO(3,1)/U(1) × SO(3,1), with the coset parametrization

g = eiya(x)Pa , (4.1)

where Pa = P0 + Pi = Pa + Q. We now compute the Maurer-Cartan form from the cosetparametrization (4.1), which is expressed as a linear combination of the generators of thetheory,

g−1Dµg = e−iy(x)aPa

(∂µ + iAµQ+ ie a

µ Pa +i

2ωabµ Jab

)eiy(x)aPa

= ∂µ + iAµQ+ ie aµ Pa +

i

2ωabµ Jab,

(4.2)

where we have used the commutation relation rules of the symmetries (See appendix C) toobtain

e aµ = eaµ + ∂µy

a + ωaµ byb, (4.3)

Aµ = Aµ + ∂µξ(y), (4.4)

ωabµ = ωabµ . (4.5)

The field, eaµ, is the vierbein, that appears in the tetrad formalism, which defines the metricas gµν = ηabe

aµebν . It can be used to build up the invariant element, d4x det e, as well as

to change from orthogonal frame, i.e. Aµ = e bµAb. The field, ω abµ , is known as the spin

connection, and it is named given its transformation properties, which transforms as agauge field [19] (See appendix C).

Following the coset recipe, we can introduce the covariant derivative for matter fieldsby using the coefficients from the unbroken Lorentz generators, which transform in a linearrepresentation under Lorentz transformations. The covariant derivative for matter fieldsreads

– 13 –

∇ga = (e−1) µa (∂µ +

i

2ωbcµ Jbc), (4.6)

where the upper index, g, denotes gravity. It transforms as, ∇ga = (Λ−1) ba ∇

gb , under Lorentz

transformations. In a similar manner we can obtain the covariant derivative for chargedfields, which transform in a linear representation under U(1) transformations, by using thecoefficients from the unbroken U(1) generators. The covariant derivative for charged fieldsreads

∇qa = (e−1) µa (∂µ + iAµQ), (4.7)

which transforms in the same way under Lorentz transformations as eq. (4.6). The onlyrequired ingredients to describe the nonlinear realizations of translations and the localtransformations of the Poincaré and U(1) group, are the covariant derivatives and thevierbein [16]. Having defined these elements, we proceed to identify the building blocks ofthe bulk.

The curvature invariants can be obtained from the covariant version of the commutatorof the covariant derivative that appear in the Maurer-Cartan form,

g−1[Dµ, Dν ]g = iFµνQ+ iT aµνPa +i

2RabµνJab,

= i(∂µAν − ∂νAµ

)Q+ i

(∂µe

aν − ∂νeaµ + eµbω

abν − eνbωabµ

)Pa

+i

2

(∂µω

abν − ∂νωabµ + ωaµcω

cbν − ωaνcωcbµ

)Jab,

(4.8)

with T aµν = T aµν + Rabµνyb, and Rabµν = Rabµν , the covariant torsion and Riemann tensorrespectively. The covariant quantities have been defined in this way, from, [Dµ, Dν ] =

iFµνQ+ iT aµνPa + i2R

abµνJab, in eq. (3.6), such that by construction, T aµ and Rabµν transforms

independently under the local transformations [16].We are interested in a gravitational theory as general relativity, where the torsion

tensor is zero. Solving for the vanishing torsion tensor, we obtain an equation for the spinconnection in terms of the vierbein [16],

ωabµ (e) =1

2

eνa(∂µe

bν − ∂νe b

µ ) + eµceνaeλb∂λe

cν − (a↔ b)

. (4.9)

From the expression for the torsion in eq. (4.8), one can read the Christoffel symbols,T aµν = Γaµν − Γaνµ. Therefore, the Christoffel symbol reads

Γaµν = ∂µeaν − eνbωabµ , (4.10)

which can be used to express the Riemann tensor in terms of the Christoffel symbols asusual,

Rcdab = ∂aΓcbd − ∂bΓcad + ΓcfaΓ

fdb − ΓcfbΓ

fda. (4.11)

– 14 –

After imposing the condition that the torsion tensor vanishes, eq. (4.8), reads

g−1[Dµ, Dν ]g = iFµνQ+i

2RabµνJab, (4.12)

Therefore, we can use the tensors, Fµν and Rabµν , to build up a Lagrangian that matchesto the one of the theory of general relativity, as well as the theory of electrodynamics incurved spacetime. Such action reads:

S =

∫det e d4x

− 1

4µ0FabF

ab +1

16πGR+ . . .

, (4.13)

with, R = Rabab = eµaeνbRabµν , the Ricci scalar, which is the low energy term of the

theory of gravity. The second term is the well known Einstein-Hilbert action, with G, thegravitational constant, while the first term is the Maxwell’s action in curved spacetime,with µ0, the magnetic permeability of vacuum. The coefficients of eq. (4.13), have beenmatched from the known theory, which allows us to obtain the Einstein-Maxwell action inthe vierbein formalism. The ellipsis stands for higher order terms from a more fundamentaltheory of gravity.

One can easily obtain the well known gravitational action by changing to the spacetimeindices:

S =

∫ √−g d4x

1

16πGR, (4.14)

with R = Rµνµν = eµaeνbRabµν , and where we have used gµν = eaµe

bνηab. Furthermore, one

can also recover the Christoffel symbol, eq. (4.10), in terms of the metric via, Γαµν = eαaΓaµν .

4.2 Charged spinning compact objects

Extended objects in the EFT framework was first introduced in [14, 15] for non-spinningobjects. Then, spin was introduce in [21], and later in [42], theories whose constructionsdiffer. Finally the effective theory for spinning objects derived using the coset constructionwas introduced in [16]. Although in [15], BHs electrodynamics is introduced, it was until[43] that charge was considered in the EFT for compact objects to obtain the dynamics. Inthe following we will use the coset construction and extend the work on spinning objectsin [16], to include the U(1) symmetry and describe charged spinning compact objects.

Charged spinning particles

In the previous section we constructed an invariant action for a charged extended objectwithout spin in flat spacetime, by identifying the symmetry breaking pattern of a chargedpoint particle, which breaks spatial translations and boosts. The state of a charged pointparticle is an eigenstate of the charge and does not break the U(1) symmetry. Now, in thecurved spacetime case, we consider spinning extended objects, which break the full Poincarégroup, such that now the system is described by the additional breaking of rotations.

BHs and NSs have their own internal symmetry, which characterizes the low energydynamics of the effective theory. The additional symmetry group for a compact object, is

– 15 –

the internal symmetry group, S ⊆ SO(3), so that G = U(1) × ISO(3,1) × S, with G thefull symmetry before being broken [16]. In principle, S, can be a discrete group, but withthe appropriate choice of the coset parametrization, we can treat in the same way bothcontinuous and discrete symmetries.

In the comoving frame, the group G is broken into a linear combination of internalrotations, Sij , and spatial rotations, Jij , such that the symmetry breaking pattern for thecharged spinning point particle reads,

Unbroken generators =

P0 = P0 +Q time translations,

Jij internal and spacetime rotations.

Broken generators =

Pi spatial translations,

Jab boosts and rotations,

(4.15)

with, Jij , the sum of the internal and spacetime rotations [16], and where we have considereda spherical star at rest, such that, Sij , are the generators of the internal SO(3) group, andJij = Sij + Jij . All translations are nonlinearly realized. Therefore, the local Poincaré andU(1) transformations are consider to take place along the fiber.

Given that Lorentz transformations can be parametrized as the matrix product of aboost and a rotation, the coset parametrization reads

g = eiyaPaeiαabJ

ab/2 = eiyaPaeiη

jJ0jeiξijJij/2 = eiyaPa g , (4.16)

which implies a one to one correspondence between the Goldstone fields, αab and ηi, ξij[16]. By parametrizing the coset with the generators of the broken spatial rotations, theMaurer-Cartan form can be computed without the need to specify the explicit unbrokengenerators of rotations, Jij .

We proceed to identify the relevant degrees of freedom, from the projected Maurer-Cartan form to the worldline of the object,

xµg−1Dµg =xµg−1(∂µ + iAµQ+ ie aµ Pa + iω ab

µ Jab)g

=xµg−1(∂µ + iAµQ+ ie aµ Pa +

i

2ωabµ Jab)g

=iE(P0 +AQ+∇πiPi +1

2∇αcdJcd).

(4.17)

The building blocks of the low energy dynamics are,

E = xµe aµ Λ 0

a (4.18)

A = E−1xµAνΛνµ (4.19)

∇πi = E−1xµe aµ Λ i

a (4.20)

∇αab = E−1(

Λ ac Λcb + xνωcdν Λ a

c Λ bd

)(4.21)

– 16 –

with the Λ’s, the Lorentz transformations parametrized by α, or equivalently by η and ξ.As one can observe, one of the advantages of using the parametrization (4.16), is that noconnection proportional to J appears, which make the building blocks independent of theresidual symmetry group [16]. The residual symmetry, SO(3), requires all spatial indices tobe contracted in an SO(3) invariant manner.

From the coset perspective, we can remove some of the Goldstones using the inverseHiggs constraint [13], given that the commutator between the unbroken time translationsand boosts, gives spatial translations, [Ki, P 0] = iP i. Thus, we set to zero the covariantderivative of the Goldstone,

∇πi = E−1(xνe0νΛ i

0 + xνejνΛ ij ) = 0, (4.22)

and solve this constraint, to express the boost as a function of velocities [16],

βi ≡ηiη

tanh η =xνe i

ν

xνe 0ν

(4.23)

where, η, is identified as the rapidity (See appendix C). To obtain the physical interpretationof last equation, we rewrite the building block, E, such that

E =√

(E∇πi)2 − (xνe aν Λ c

a xµebµΛbc)

=√−ηabe a

ν ebµ x

ν xµ =√−gµν xµxµ =

dτ

dλ,

(4.24)

with, λ, the worldline parameter that traces out the trajectory of the particle. To obtainlast equation, we have imposed the inverse Higgs constraint, and the property of the boostmatrices, Λ b

a Λac = δbc.Therefore, with eq. (4.24), we rewrite the constraint, eq. (4.22), in a way that makes

manifest its physical interpretation [16]. For rotations, we can write, Λ0a(ξ) = δ0

a andΛij(ξ) = Ri j(ξ), with R(ξ) an SO(3) matrix, such that the constraint (4.22) now reads

uaΛ ia (η)R j

i (ξ) = 0, (4.25)

with, ua ≡ e aµ ∂τx

µ, the Lorentz velocity measured in the local inertial frame defined bythe vierbein. Given that the matrix R i

j (ξ) is invertible, we obtain

uaΛ ia (η) = 0. (4.26)

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

na(0) ≡ ua = Λa0(η) , na(i) ≡ Λai(η), (4.27)

define an orthonormal local basis with respect to the local flat metric, ηab, in the framethat is moving with the particle [16]. Orthogonality is obtained from the boost matrices,Λ ba Λac = δbc. Therefore, the na(b), is a set of vierbeins that defines the local inertial frame on

– 17 –

the particle worldline. One can also define the orthonormal basis in terms of the spacetimevectors, nµ(b) ≡ e

µa na(b), with respect to the full metric, gµν .

Moreover, an additional set of orthonormal vectors is obtained [16],

mb(a) ≡ Λba(α) = Λbc(η)Rca(ξ), (4.28)

with the zeroth vector, mb(0) = nb (0) = ub, the velocity of the compact object in the proper

frame. The rest of the vectors differs by a rotation, R(ξ), compared to (4.27), from whichone can observe that the set of vectors, mb

(a), contain information about the rotation,parametrized by the three degrees of freedom of ξ.

With these definitions, the covariant derivatives, ∇α0i, can be expressed as

∇α0i = R ij (ξ)Λ (j)

a (η)(∂τ ua + uµω a

µ cuc) = R i

j (ξ)Λ (j)a (η)aa = ai, (4.29)

which is a rotated version of the acceleration, projected into the orthonormal basis definedby the n’s basis in the proper frame of the particle. As in the case of charged spheres, in theabsence of external forces, this building block is zero. Nevertheless, it must be consideredto build up invariant operators when an external force, such as the one from an externalgravitating object, or form the charge of another object, is strong enough to make thisbuilding block relevant.

Thus, we are left with the building blocks

E = uµe aµ Λ 0

a , (4.30)

A = uµAµ, (4.31)

∇αab =(

Λ ac Λcb + xνωcdν Λ a

c Λ bd

), (4.32)

(4.33)

with Aµ = AνΛνµ. We proceed to consider the blocks A and E, to derive the effectiveaction of a charged point particle in curved spacetime. The action takes the form

S =

∫dλE(−cE + cAA) =

∫dτ(−mc2 + quaAa), (4.34)

where we have matched the coefficient, cE = mc2, from the action of a point particle [14]with mass, m, and the coefficient, cA = q, to the one of a charged point particle [14, 43]with net charge, q.

Now we include the building block, ∇αab, by neglecting the charge for simplicity. Wecan build the invariant action [16]

S =

∫dλE

(−m+ cΩ∇αab∇αab + . . .

), (4.35)

where a term linear in ∇α has been discarded by time reversal symmetry. We have consid-ered spherical objects at rest. The ellipses denotes higher order corrections. In the usualmechanics of rotational dynamics, to characterize a rigid sphere only two parameters are

– 18 –

needed, which is the mass m and moment of inertia I. By comparing our action to the oneof a relativistic spinning point particle [44], and to the currently used EFTs for spinningobjects [21, 22], we match the coefficient, cΩ = I/2, to obtain the relativistic action forspinning particles in curved spacetime,

S =

∫dτ

−m+

I

2ΩabΩ

ab + . . .

, (4.36)

where we have identified, Ωab = ∇αab, the angular velocity of the star in the rest frame.Therefore, from (4.32), the explicit form of the angular velocity in the proper frame, Ωij ,reads

Ωab = Λ ac (ηcd∂τ + ∂τx

µωcdµ )Λ bd . (4.37)

It is also possible to define the spin vectors with the epsilon tensor, for which theangular velocity vector reads

Ωa = −1

2εabcΛ

bd (ηde∂τ + ∂τx

µωdeµ )Λ ce . (4.38)

Nevertheless, eq. (4.38), does not transform ideally under rotations and it is necessary toconsider the transformation, Ωij = ΛkiΛ

ljΩ

kl, which transforms correctly under rotations.Thus, the Lorentz invariant object in the lab frame to consider is, Ωab

Ωab = ΛacΛbdΩ

cd = ΛacΛbd[Λ

ce (ηef∂τ + ∂τx

µω efµ )Λ d

f ] (4.39)

= −Λac∂τΛbc + ∂τxµω ab

µ . (4.40)

Eq. (4.37) and eq. (4.40) are the relativistic angular velocity or spin in the proper and labframe respectively. The spatial part of eq. (4.37) is the usual relativistic spin tensor

Ωij = Λ ik (ηkl∂τ + ∂τx

µωklµ )Λ jl , (4.41)

while the temporal part, from eq. (4.29), implies that

Ω0i =(

Λ 0j Λji + uν ωkiν uk

)= ai. (4.42)

The relativistic spin encoding the acceleration have implications in the known post-Newtonian expansion for spinning objects. For instance, it has been shown in [22], that aterm with a spin-acceleration coupling is needed to obtain the correct results. Our definitionfor the spin in eq. (4.37), implies that the spin correction considered in [22], is encoded inthe higher order correction

S =

∫dτ

. . . + cΩ,uΩacΩ

bc uaubu2

+ . . .

. (4.43)

– 19 –

By changing to the lab frame, and setting ua = va with v0 = 1, and λ = t, the expectedcorrection, ∝ cΩ,uΩijviaj , is obtained. Nevertheless, the latter acceleration dependentcorrection, is just a low order correction, from the complete higher order, correction in eq.(4.43). This analysis allows us to set cΩ,u = I, for which one obtain the expected Feynmanrules [22] for the PN expansion [28].

More on the implications of our defined relativistic spin, can be seen in what is knownas the spin supplementary condition. The covariant quantity, Ωab, is already gauge fixedby the Goldstones, η and ξ. We can fix, ηi → βi, and ξij → θij = εijkθ

k, with the θ’s, beingthe Euler angles that describe the orientation of the compact object, and the β’s defined byeq. (4.23). The gauge fixing condition can be obtained from the inverse Higgs constrainton the Goldstone ∇αab. Therefore, in the proper frame, the gauge fixing condition reads

uaΩab = u0Ω0b + uiΩ

ib =√u2Ω0b + uiΩ

ib = 0, (4.44)

which is equivalent to the relativistic Price-Newton-Wigner spin supplementary condition[22]. This constraint allows us to rewrite the temporal part in terms of the spatial com-ponents of the angular spin, and therefore by applying the spin supplementary condition,eq. (4.44), altogether with eq. (4.42), we can express the dynamics in terms of the spatialrelativistic spin, with no corrections proportional to the acceleration. This fact is shownwhen obtaining the PN expansion for the derived action in [28], in which we recover thewell known results for the PN expansion for spinning objects in [22], but with the lack ofterms proportional to the acceleration.

With all of our derived invariant operators, we write down the leading order effectiveaction of a charged spinning point particle in the particle’s rest frame,

S =

∫dτ

−mc2 + quaAb +

I

2ΩabΩ

ab + IΩacΩbc u

aubu2

+ . . .

. (4.45)

Size effects

Size effects in the EFT for extended objects was introduced in [14], which can besystematically taken into account by building invariant operators made up of the Weylcurvature tensor, Wabcd. The Weyl tensor is obtained from the Riemann tensor, Rabcd, bysubtracting out various traces. Defining the transformation [16]

R ≡ g−1L R, (4.46)

with, gL, the Lorentz part of the parametrization in eq. (4.16), the Riemann tensor trans-forms linearly under Lorentz transformations as expected. The explicit transformationreads

Rabcd = (Λ−1) ea (Λ−1) f

b (Λ−1) gc (Λ−1) h

d Refgh, (4.47)

with, Rabcd, the Riemann tensor in the local rest frame of the object. Having the correcttransformations, we define the Weyl tensor as usual,

– 20 –

Wabcd = Rabcd +1

2(Radgbc −Racgbd +Rbcgad −Rbdgac) +

1

6R(gacgbd − gadgbc), (4.48)

which have the physical content [14]. The Weyl tensor measures the curvature of thespacetime and contains the tidal force exerted on an extended particle that is moving alongthe worldline, taking into account for how the shape of the body is distorted. It transformsin the same way as eq. (4.47).

Furthermore, we can also use the electromagnetic tensor to build invariant operatorsthat take into account for the polarizability of the object. Following the above discussion,we define its transformation rule,

F ≡ g−1L F. (4.49)

Having defined the correct transformation rules for the Riemman and the electromagnetictensor, we can now proceed to form rotationally invariant objects.

We form all leading order invariant operators that contribute to the dynamics, bycombining all our covariant quantities: Wabcd, Fab, ua and Ωa, in all possible ways allowedby the symmetries. In particular, for the electromagnetic and Weyl tensor, the buildingblocks are the electric like parity tensors, Ea = Fabu

b, and Eab = Wacbducud, respectively

[15]. By considering the electromagnetic dipolar and gravitational quadrupolar moments,we build the following leading order relevant operators for finite-size effects:

O(ua, Ωa, Eab, Ea) =

EabEab Gravity,

ΩaΩbEab Spin− gravity,

EaEa Electromagnetic,

ΩaΩbEaub Spin− electro,

EaEbEab Gravity − electro.

(4.50)

One can consider the magnetic parity operators as well, Bab = (1/2)εcdeaWcdfbu

euf

and Ba = εabcdFbcud [15], for the gravitational and electromagnetic case respectively, which

are subleading with respect to the electric parity terms (at least for the gravitational case),therefore restricting our discussion to the electric parity action. The leading order magneticparity operators can be build in analog to (4.50) i.e. BabBab and BaBa.

Moreover, as shown in [22], terms proportional to the acceleration are needed to re-produce the correct PN expansion, ∝ Ωabv

aab. Therefore, we shall consider as well thefollowing corrections that are relevant in the lab frame,

O(va, aa,Ωa, Eab, Ea) =

Ωabvaab Spin,

Ebab Electromagnetic,

ΩaΩbEaab Spin− electro,

ΩaΩbEacvbac Spin− gravity,

Eabaaab Gravity.

(4.51)

– 21 –

In the proper frame, these higher order corrections are encoded in the following invariantoperators,

O(ua, Ωa, Eab, Ea) =

EaΩ

abub Electromagnetic,

ΩabEaΩbcuc Spin− electro,

ΩaΩbEacubΩcdud Spin− gravity,

EabΩacΩbducud Gravity.

(4.52)

There are no corrections that are acceleration dependent from the invariant operatorsin eq. (4.50), given that the relevant dynamics are encoded in the spatial components, Eiand Eij , for which any term that is acceleration dependent will vanish. Higher order termsto the ones shown in eqs. (4.50) and (4.51), can be built from the derived building blocksand the covariant derivatives, eqs. (4.6) and (4.7), i.e. ∇gcEab and ∇qbEa. Furthermore, itis worth noting that size effects can be seen as encoded in a composite operator Qab, whichwe comment below.

Dissipative effects

Dissipation, due to the internal structure of an extended object, was introduced in EFTdescription in [15], where the existence of gapless modes that are localized on the worldlineof the particle take into account for the energy and momentum loss from the interactionwith external sources. Dissipative effects for slowly spinning objects were considered in[25, 26], and for maximally spinning in [23].

These large number of degrees of freedom can be encoded in operators allowed by thesymmetries of the object. For a compact object, the allowed operators due to its symmetriesgives rise to the invariant operators [15, 23]:

Dissipative operators =

Pa(τ)Fabu

b Electro,

Dab(τ)Wacbducud Gravity,

(4.53)

with P(τ) and D(τ), composite operators corresponding to the electric parity of the elec-tromagnetic dipole and the gravitational quadrupole moment respectively, encoding thedissipative degrees of freedom.

For a non-spinning BH, dissipation takes into account for the absorption of electro-magnetic and gravitational waves, while for a non-spinning NS, dissipative effects take intoaccount for the energy loss during the interaction with an external source given the internalviscosity of its equation of state of matter. On spinning objects, the spin has a time de-pendence between the object and its environment which generates dissipative effects. Theoperators in eq. (4.53), take into account for the spin dissipative degrees of freedom as well.

The coefficients encoding the internal structure are encoded in the dynamical moments,P and D, which are dependent on the internal degrees of freedom of the compact object inan unspecified way, but which explicit form is not necessary to obtain the dynamics [15, 23].

– 22 –

The dynamics of the system containing the dissipative degrees of freedom can be obtainedusing the in-in closed time path [45], a formalism that allows us to treat dissipative effectsin a time asymmetric approach [15].

The expectation values of these operators, 〈Pa(τ)〉 , 〈Dab(τ)〉, are defined through thein-in path integral, which is the expectation value in the initial state of the internal degrees offreedom, and which in general is a functional of the building blocks, Ea and Eab, respectively[23]. In the gravitational case, by considering the linear response in a weak external field,the in-in formalism implies the form of the expectation value [23]

〈Dab(τ)〉 =

∫dτ ′Gab,cdR (τ − τ ′)Ecd(τ ′) +O

(E2), (4.54)

where the expectation values of the retarded Green’s function,

Gab,cdR (τ − τ ′) = iθ(τ − τ ′) 〈[Dab(τ), Dcd(τ ′)]〉 , (4.55)

are obtained at the initial state of the interaction where the external field is zero.By considering low frequencies, from which we assume that the degrees of freedom from

the operator, Dab, are near equilibrium, the time ordered two point correlation functionimply that the Fourier transform, GR, must be an odd, analytic function of the frequency,ω > 0 [15]. Therefore, the retarded correlation function, GR, reads

Gab,cdR (ω) ' icgω(δacδbd + δadδcb − 2

3δabδcd

), (4.56)

with the coefficient for dissipative effects, cg ≥ 0. Note that, in contrast to [15], we haveabsorbed the 1/2 factor appearing in front of (4.56) into the dissipative coefficient.

By considering the response of the interaction to be nearly instantaneous, the operatordue to gravitational dissipative effects takes the form [15, 23]

〈Dab(τ)〉 ' 2icgd

dτEab + . . . . (4.57)

In general, one can use the dynamical composite operator, Qab(τ), to account for the tidalresponse function, for which the above reasoning leads to [15, 23]

〈Qab(τ)〉 ' ngEab + 2icgd

dτEab + n

′g

d2

dτ2Eab + . . . . (4.58)

The first term in last equation is the static quadrupolar tidal effect considered above, whilethe third term is a dynamical tidal effect in the quasi-static limit. The second term, which isthe imaginary part of the response function, takes into account for the dissipative degrees offreedom. In our work we consider the static tidal response only, and use the operator, Dab,to contain only the dissipative degrees of freedom. For NSs, dynamical oscillations in the

– 23 –

quasi-static limit have been taken into account in [9]. Nevertheless, dynamical oscillationsshould be considered in a different fashion, as in [38], which is beyond our scope.

On the electromagnetic side, an analog procedure can be taken, for which retardedcorrelation function, GR, reads [15]

GabR (ω) ' icqδabω, (4.59)

with the coefficient, cq ≥ 0. Therefore, the operator for electromagnetic dissipative effectsreads

〈Pa(τ)〉 ' icqd

dτEa + . . . . (4.60)

4.3 Effective action for compact objects

Gathering all our results in the proper frame, we construct the most general, leadingorder and electric like parity, effective action for a compact object in the theory of generalrelativity,

Seff =

∫dτ

−mc2 + quaAa +

I

2ΩabΩ

ab + IΩacΩbc u

aubu2

+ nq,ΩΩaΩbEaub + ng,ΩΩaΩbEab + nqEaEa

+ngEabEab + nq,gE

aEbEab + PaEa + DabEab + . . .

+ S0,

(4.61)

with the electric parity tensor, Eab = Wacbducud, and Ea = Fabu

b, that corresponds to thegravitational quadrupole and electromagnetic dipole moment respectively. The interactionaction,

S0 =

∫det e d4x

− 1

4µ0FabF

ab +1

16πGR+ . . .

, (4.62)

is the Einstein-Maxwell action. The action describing the compact object lives in theworldline, while S0 lives in the bulk. Eq. (4.61), describes charged spinning compactobjects in our effective theory.

Coefficients of the effective theory

The coefficients of the effective theory encode the microphysics of the compact objects,which are determined through a matching procedure to the full known theory, and ulti-mately from GW observations. We identify them from the results in literature, withoutthe need to do the explicit calculations here. We have already pointed out the coefficientsappearing in the action describing a charged spinning point particle. The coefficient of thepoint particle term [14], cE = mc2, the coefficients in the spin corrections, cΩ = I/2 andcΩ,u = I [21, 22], and the coefficient from the correction due to electromagnetic charge,

– 24 –

cA = q [14, 43]. Now, we proceed to point out the rest of the coefficients due to the internalstructure of the compact object.

We start with the coefficient due to static tidal effects, ng, which is a coefficient thatdepends on the internal structure of the star through a parameter known as the Lovenumber. Any stellar object that can be described by an equation of state of matter, canbe described approximately in terms of its Love numbers. In the case of BHs, it has beenfound that their Love numbers vanishes [46], which also occurs for the case of spinningBHs [27, 47], therefore setting ng = 0 and ng,Ω = 0. Furthermore, it has been shown that,for both non-spinning [48] and spinning BHs [47], the coefficients due to the polarizabilityvanishes as well, nq = 0 and nq,Ω = 0.

For a NS, the coefficient is different from zero. The leading order quadrupolar Lovecoefficient reads [49]

ng =2`5k2

3G, (4.63)

with k2, the quadrupolar dimensionless Love number, and ` the radius of the star.3 The Lovenumbers are dimensionless parameters that measure the rigidity and tidal deformability ofthe compact object, and varies given different equations of state of matter [50]. This numberis related to other parameters of the star from the relativistic I-Love-Q relations [50, 51],which relates the moment of inertia, I, the tidal deformability parameter, or Love number,k2, and the quadrupole moment. These relations, which are empirically found to hold fora wide range of equations of state of matter, are only approximate relations and are notexact first principles relations.

On the coefficient due to the coupling of spin-gravity size effects for spinning NSs, inthe slow rotation limit, the relativistic coefficient, ng,Ω, is obtained through the Love-Q partof the I-Love-Q relations [51]. The latter coefficient in the Newtonian limit is the same asthe one for static tides, ng,Ω = ng [51]. On the charge-gravity, and charge-spin size effects,given that charge in compact objects has been mostly neglected, the rest of the coefficients,nq,Ω, nq and nq,g, are unknown, and are to be derived by analytical and numerical means.

Moving on into dissipative effects, although their coefficients are not explicitly shownin the action, they are encoded in the operators, P and D, as shown above. We point outthe known coefficients for dissipative effects in BH interactions from the existing literature.The operator Dab, for non-spinning BHs, contains the coefficient, cg, which encodes thecapacity of the BH to absorb GWs, and which can be read off from the response functionderived in [15],

cg =16

90

G5M6

c13=

`6s360Gc

, (4.64)

with M the mass of the BH, and `s its radius. The coefficient in eq. (4.64), includes thefactor of 1/2 commented in eq. (4.56). In the same way, we can obtain the coefficient forelectromagnetic dissipative effects from the composite operator, P(τ) [15],

3We have chosen to denote the radius of the object with ` as in [9, 25], rather than with R as commonlyused, given that we use R for the Ricci scalar.

– 25 –

cq =2π`4s3µ0c

, (4.65)

which encodes the capacity of the BH to absorb electromagnetic waves.On the dissipative effects of rotating BHs, the coefficients can be obtained from the

response function derived in [27] using the Teukolsky equation, for both non-spinning andspinning case. We start by reading off the coefficient for a non-spinning BH from theresponse function in [27], which in their notation, reads

1

2N2`

5sFSch2ma =i

`5sM

90c3ω +O(ω3) ' i `6s

360Gcω, (4.66)

with ma, the azimuthal number, `s, the Schwarzschild radius, and where we have used theleading order mode, l = 2, of the angular momentum number, such that N2 = 1/3. Toobtain eq. (4.66) as eq. (4.64), we have substituted the mass of the BH in terms of itsradius, M = `sc

2/2G, and considered the extra factor of 1/2, as for eq. (4.64).In the spinning case, the response function reads [27]

FI,Kerr2ma=− i

30Gc

iama

(`+ − `−)+

i

15c3

`+M

(`+ − `−)ω +O(ω3)

'− i

30Gc

Ima

Mc(`+ − `−)Ω +

i

15c3

`+M

(`+ − `−)ω,

(4.67)

where a = J/Mc = IΩ/Mc, with J = IΩ, the scalar value of the angular momentum, and`+ and `−, the outer and inner radius of the Kerr BH.4 The moment of inertia of a BH isI = 4GM2/c4. Therefore, the response function with its normalization constant, reads

1

2N2`

5+F

I,Kerr2ma

'− i

180

Ima`5+

MGc(`+ − `−)Ω +

i

90c3

`6+M

(`+ − `−)ω. (4.68)

We have obtained a response function for the dissipative effects of the form, 〈Dab(τ,Ω)〉 ∝i(cg,ΩΩ+cgd/dτ)Eab. We can identify that, for a rotating extended object, tidal dissipationarises due to two separate contributions. The first term on the right hand side of eq. (4.68),for which is nonzero even in the case of ω = 0, arises given that the spin of the body hasa time dependence between the object and its tidal environment. This can be seen, fromthe object perspective in its proper frame, as the external environment rotating with thefrequency of the spin of the object.

Furthermore, now we have an expression which is explicit on the azimuthal numbers,ma. For the dominant perturbation mode, l = 2, then, ma = [−2,−1, 0, 1, 2]. Nev-ertheless, we can not identify a specific value of ma to be dominant. Therefore, it is

4To make the reading of the coefficients accessible, we have written the response function as in [27], andthen converted it to our notation.

– 26 –

necessary to sum over all possible values of ma. This can be done by considering theelectric tidal moment, Elma , for which leading order is E2ma . Then, we one can pro-ceed to identify the different elements of E2ma , given the possible values of ma, to thencouple each response function with its electric tidal moment. We would couple schemati-cally, Q2ma ∝ D2−2E2−2 + D2−1E2−1 + D20E20 + D21E21 + D22E22. Similar procedure formaximally spinning BHs has been performed in [23]. The same reasoning applies for themagnetic like parity tidal moments. On the dissipative effects for charged spinning BHs,the coefficient, cq,Ω, is still unknown.

The coefficients for NSs from dissipative effects are still unknown, and must be deter-mined from hydrodynamical simulations, and ultimately from observations. Nevertheless,it is worth commenting that for a stellar object that is described by an equation of stateof matter, there exists the weak friction model [52]. In the latter model, the coefficient,cg = Θng [52], with Θ, being the time lag, which accounts for the tidal bulge formed in thestellar object during the interaction with another compact object. It is unknown whether ornot this model applies for NSs, but perhaps such coefficient can allows us to get an insightinto the dissipative description for NSs.

5 Discussion

In this work we have reviewed and extended the model for spinning extended objectsintroduced in [16], which is derived using the coset construction [11, 40], a very powerfulmethod that allows us to derive an effective theory from the symmetry breaking patternas the only input. In this approach, a spinning extended object whose ground state breaksspacetime symmetries, is coupled to a gravitational theory formulated as a gauge theorywith local Poincaré symmetry and translations being nonlinearly realized. We have includedthe internal structure [14, 15, 25] and electromagnetic charge [15, 43], such that we describecharged spinning extended objects, the most general extended object allowed in a theoryof gravity such as general relativity.

We have derived all the covariant building blocks of the effective theory, which are usedto build up invariant operators to form an action. The first action that we have derived,where we have matched the coefficients from the literature, is the well known Einstein-Maxwell action in the vierbein formalism. Then, by recognizing the symmetry breakingpattern of a charged spinning extended object, we have described it as a worldline pointparticle, with its properties and internal structure encoded in higher order corrections inthe action. By considering the leading order invariant operators that are allowed by thesymmetries, and matching the coefficients from the literature, we have described chargedspinning compact objects, such as BHs and NSs.

This theory for spinning extended objects [16], which is a low energy description ofthe dynamics, is well suited to describe the so far detected compact objects [53], which fitinto the description of slowly rotating [9]. It has the advantage that from construction,redundant degrees of freedom have been gauge fixed, and that all the covariant buildingblocks to construct the tower of invariant operators have been derived. In the theories forspinning extended objects [21, 22], these objects are coupled to gravity by promoting the

– 27 –

flat spacetime theory for spinning point like objects to the curved spacetime case, whichintroduces redundancies in order to derive the rotational covariant degrees of freedom. Inour case, using the coset construction, these spin degrees of freedom naturally arise fromthe symmetry breaking pattern that such object generates.

In contrast to [16] and [25], we have considered a different definition for the relativisticspin, in which the acceleration is encoded in the temporal components. With this definition,we have shown that the spin-acceleration correction considered in [22], is in fact encoded ina higher order spin-spin coupling, from which we have matched the coefficient. The derivedspin supplementary condition, which is equivalent to the covariant condition used in [22],implies that acceleration dependent terms in the action can be rewritten in terms of thespatial components of the spin.

The most direct application of our derived action is on the PN expansion [28], wherewe have shown that our theory reproduces the well known results for spinning [22] andcharged [43] objects. Moreover, novel results in the PN expansion have been derived onthe internal structure of charged spinning compact objects [28]. Therefore, our theorylays on the foundations to obtain new state of the art results for spinning and chargedextended objects, and to build up the effective theory of compact objects to all orders.More applications of the derived effective action are coming in the next series of papers, inwhich we approach the post-Minkowskian expansion and the effective one body.

Acknowledgments

I.M. is very thankful to R. Penco and H. S. Chia for very valuable discussions, andto T. Hinderer for very valuable comments. I.M. gratefully acknowledge support fromthe University of Cape Town Vice Chancellor’s Future Leaders 2030 Awards programmewhich has generously funded this research, support from the South African Research ChairsInitiative of the Department of Science and Technology and the NRF and support from theEducafin-JuventudEsGto Talentos de Exportacion programme.

A Conventions

We differentiate between spacetime and local Lorentz indices as in [16]:

• µ, ν, σ, ρ... 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 fourcomponent vector, xa = (x0, x1, x2, x3) = (t, ~x), and define the flat spacetime interval ds

between two events, xa and xa + dxa, by the relation

ds2 = −c2dt2 + dx2 + dy2 + dz2, (A.1)

– 28 –

which we write using the notation

ds2 = ηabdxadxb ; ηab = diag(−1,+1,+1,+1). (A.2)

B Worldline point particle dynamics

Our most important consideration when modelling compact objects with EFTs is thatthey can be treated as point particles with corrections. To describe the dynamics of pointparticles, it is necessary to specify a continuous sequence of events in the space-time bygiving the coordinates, xa(λ), of the events along a parametrized curve, defined in termsof a suitable parameter, λ. There is one curve among all possible curves in the spacetime,which describes the trajectory of a material particle moving along some specified path,known as the worldline. We can consider it as a curve in the spacetime, with λ = cτ , actingas a parameter so that xa = (cτ, ~x(τ)). The existence of a maximum velocity |~u| < c, whichrequires the curve to be time-like everywhere, ds < 0, allows us to have a direct physicalinterpretation for the arc length along a curve.

Consider a clock attached to the particle frame, or the proper frame F, which is movingrelative to some other inertial frame F on an arbitrary trajectory. As measured in the frameF, during a time interval between t and t + dt, the clock moves through a distance |d~x|.The proper frame F, which is moving with same velocity as the clock, will have d~x = 0. Ifthe clock indicates a lapse of time, dt ≡ dτ , the invariance of the space-time interval, eq.(A.1), implies that

ds2 = −c2dt2 + dx2 + dy2 + dz2 = ds2 = −c2dτ2. (B.1)

Thus, we obtain the lapse of time in a moving clock

dτ = dt

√1− v2

c2, (B.2)

along the trajectory of the clock. The total time that has elapsed in a moving clock betweentwo events, known as the proper time τ , is denoted as

τ =

∫dτ =

∫ t2

t1

dt

√1− v2

c2. (B.3)

We now proceed to build the action of a free point particle. The Lagrangian mustbe constructed from the trajectory, xa(τ), of the particle, and should be invariant underLorentz transformations. The only possible term should be proportional to the integral ofdτ , yielding the action

S = −α∫τ

dτ = −α∫ √

1− v2

c2dt, (B.4)

where α is a dimensionful constant. To recover the action of a free point particle fromnon-relativistic mechanics, we take the limit c → ∞, for which the Lagrangian yieldsL = αv2/2c2. By comparing our point particle Lagrangian with the non-relativistic one,

– 29 –

L = (1/2)mv2, with m the mass of the particle, we find that, α = mc2. Thus, the actionfor a relativistic point particle is

S = −mc2

∫dτ, (B.5)

which corresponds to the arc length of two connecting points in the spacetime.

C Symmetries in classical field theory

We consider symmetries that can be labelled by a continuous parameter, θ. Workingwith the Lie algebras of a group G, we write a group element as a matrix exponential

U = eiθuTu , (C.1)

where the generators Tu, with u = 1, ..., n, form a basis of the Lie algebra of G. The T ’sgenerators are hermitian if g is unitary. For each group generator, a corresponding gaugefield arises, which in this case is the field θ.

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

[Tu, Tv] = TuTv − TvTu = icuvwTw, (C.2)

where, cuvw, are the structure constant coefficients. The last expression defines the Liealgebra of the group G. The Lie bracket is a measure of the non-commutativity betweentwo generators.

It is also possible to consider in the local framework of field theory, continuous sym-metries that have a position dependent symmetry parameters, θ = θ(x). The spacetimedependent symmetry transformation rules are called local or gauge symmetries. For globalsymmetries, θ and g do not depend on spacetime position. There is also a distinctionbetween internal symmetries and spacetime symmetries, on whether they act or not onspacetime position. An example of an internal symmetry, where x is unchanged, is

φu(x)→ Uφu(x)U−1 = U uv φ

v(x), (C.3)

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

φu(x)→ V φu(x)V −1 = V uv φ

v(x′), (C.4)

with x′u = V uv x

v. Both internal and space-time symmetries can arise in global or gaugedvarieties.

Symmetries of special relativity

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

– 30 –

The Lorentz group

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

xa → x′a = Λabxb, (C.5)

that leave invariant the quantity

ηabxaxb = −(ct)2 + x2

1 + x22 + x2

3, (C.6)

with detΛ = 1. In order for eq. (C.6) to be invariant, Λ must satisfy

ηabx′ax′b = ηab(Λ

acxc)(Λbdx

d) = ηcdxcxd, (C.7)

which implies the transformation of the metric as

ηcd = ηabΛacΛ

bd. (C.8)

Consider an infinitesimal Lorentz transformation, with the Lorentz generators Jab, andits corresponding field αab. We can expand

Λab = (ei2αcdJcd)ab = (eα)ab ≈ δab + αab, (C.9)

where the factor of, 12 , is taking into account the sum over all a and b. From equation (C.8)

we find

αab = −αba, (C.10)

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, αab,corresponds to six generators which are also antisymmetric Jab = −Jba.

Under Lorentz transformations, a scalar field is invariant,

φ′(x′) = φ(x). (C.11)

A covariant vector field, V a, transforms in a representation of the Lorentz group as

V a → (ei2αcdJ

cd)abV

b, (C.12)

where the exponential is the matrix representation of the Lorentz group. If we consider aninfinitesimal transformation, the variation of V a reads,

δV a =i

2αcd(J

cd)abVb, (C.13)

which is a irreducible representation.The explicit form of the matrix (Jab)cd, reads

(Jab)cd = −i(δcaηbd − δcbηad). (C.14)

– 31 –

Using the form of the generator in eq. (C.14), we can compute the commutator

[Jab, Jcd] = i(ηacJbd − ηbcJad + ηbdJac − ηadJbc), (C.15)

to find the Lie algebra, SO(3,1). The components of Jab can be rearranged into two spatialvectors

Ji =1

2εijkJ

jk, Ki = J i0, (C.16)

with, J ij and Ki, 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) rotationgroup, and three boost transformations in the (ct, x), (ct, y) and (ct, z) planes that leaveinvariant −(ct)2 +x2, −(ct)2 +y2 and −(ct)2 +z2, respectively. We parametrize the Lorentzmatrix as

Λ00 = γ, Λ0

i = γβi, Λi 0 = γβi, Λi j = δi j + (γ − 1)βiβjβ2

, (C.17)

with γ = (1− v2/c2)−1/2, the Lorentz factor, and βi, the velocity

βi ≡ ηi

ηtanh η, (C.18)

where η is the rapidity, defined as the hyperbolic angle that differentiates two inertial framesof reference that are moving relative to each other.

Therefore, the four vectors, V a and Va, transforms under the Lorentz group as

V a(x)→ V′a(x′) = ΛabV

b(x), Va(x)→ V′a(x′) = Λ b

a Vb(x), (C.19)

with Λ ba = ηacηbdΛ

cd. The vectors are related via Va = ηabV

b. A tensor, T ab, transformsas

T ab(x)→ T ′ab(x′) = ΛacΛbdT

cd(x). (C.20)

In general, any tensor with arbitrary upper and lower indices transforms with a Λab matrixfor each upper index, and with Λ b

a for each lower one. We will simple denote Lorentztransformations as, V ′a = ΛabV

b and T ′ab = ΛacΛbdT

cd.

The Poincaré group

To complete the Poincaré group, in addition to Lorentz invariance, we also requireinvariance under spacetime translations. We can write a general element of the group oftranslations in the following form,

U = eizaPa , (C.21)

where za are the components of the translation,

– 32 –

xa → xa + za, (C.22)

and P a its generators. Lorentz transformations plus translations form the Poincaré group,ISO(3,1). The Poincaré group algebra reads

[Pa, Pb] = 0 (C.23)

[Pa, Jbc] = i(ηacPb − ηabPc) (C.24)

[Jab, Jcd] = i(ηacJbd − ηbcJad + ηbdJac − ηadJbc). (C.25)

Gauge symmetry of classical electromagnetism

The gauge symmetry of classical electromagnetism is invariance under the U(1) gaugetransformation. This is an internal symmetry for which the charge generator, Q, corre-spond to a time invariant generator, with its corresponding gauge field, Aµ(x), which isthe electromagnetic gauge field. The local gauge symmetry is parametrized by a parameterθ = θ(x), and the group element is

U(x) = eθ(x). (C.26)

The gauge field, Aµ, transforms under the U(1) symmetry as

Aµ(x)→ Aµ(x) + ∂µθ(x). (C.27)

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

Aµ(x)→ Λ νµ Aν(x) + ∂µθ(x,Λ). (C.28)

The commutations relations of the charge generator, Q, with the generators of thePoincaré group, are constrained by the Coleman-Mandula theorem [54]. This theorem con-straints the kinds of continuous spacetime symmetries that can be present in an interactingrelativistic field theory and states that the most general possible transformations are

U = exp

i

(zaPa + iσaOa +

i

2ωabJab

)(C.29)

with Pa, the generators of translations, Jab, of Lorentz transformations, and Oa, the rest ofthe generators. The generators, Oa, must be from internal symmetries, and although theycan fail to commute with themselves, [Oa,Ob] 6= 0, they must always commute with thespacetime symmetry generators [Pa,Ob] = 0 and [Jab,Oc] = 0. Nevertheless, the chargeoperator for the U(1) symmetry of electromagnetism, commutes with itself, thus obtainingthe commutation relations: [Pa, Q] = 0, [Jab, Q] = 0 and [Q,Q] = 0.

– 33 –

Transformation properties of gauge fields

The transformation properties of the gauge fields eaµ, Aµ and ωabµ , introduced in eq.(4.2), under local translations, eizaPa , local Lorentz transformations, e

i2αcdJ

cdand the local

U(1) transformation, eiθ, read

U = eiθ :

Aµ → Aµ − ∂µθ,e aµ → e aµ ,

ωabµ → ωabµ .

U = eicP :

Aµ → Aµ,

e aµ → e aµ − ωaµbzb − ∂µza,ωabµ → ω ab

µ .

U = eiαJ :

Aµ → Λ ν

µ Aν ,

e aµ → Λabebµ = eaµ + αabe

bµ,

ωabµ → ΛacΛbdω

cdµ + Λac∂µ(Λ−1)cb = ωabµ + ωacµ α

bc + ωcbµ α

ac − ∂µαab.

,

(C.30)

where the indices are lowered and raised using the metric ηab. The gauge field, e, trans-forms inhomogeneously under local translations. Under Lorentz transformations, e and A,transforms linearly, while ωabµ transforms as a connection. Under the U(1) transformation,only the gauge field, A, transforms.

Finally, the field for electromagnetism, the vierbein and spin connection, under diffeo-morphisms transforms as

Aµ(x)diffeo−−−→Aµ(x)−Aν(x)∂µξ

ν − ξν(x)∂νAµ(x),

eaµ(x)diffeo−−−→ eaµ(x)− eaν(x)∂µξ − ξν(x)∂νe

aµ(x),

ωabµ (x)diffeo−−−→ ωabµ (x)− ωabν (x)∂µξ − ξν(x)∂νω

abµ ,

(C.31)

which all of them transforms in the same way.

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