Prepared for submission to JHEP Spinning gravitating objects in the effective field theory in the post-Newtonian scheme Michele Levia,b and Jan Steinhoffc,d a 5 Universit´e Pierre et Marie Curie-Paris VI, CNRS-UMR 7095, Institut d’Astrophysique de Paris, 1 98 bis Boulevard Arago, 75014 Paris, France 0 b Sorbonne Universit´es, Institut Lagrange de Paris, 2 98 bis Boulevard Arago, 75014 Paris, France t c c Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute), O Am Mu¨hlenberg 1, 14476 Potsdam-Golm, Germany 5 d Centro Multidisciplinar de Astrofisica, Instituto Superior Tecnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal ] c q E-mail: [email protected], [email protected] - r g [ Abstract: We introduce a formulation for spinning gravitating objects in the effective field theory in the post-Newtonian scheme in the context of the binary inspiral problem. 3 v We aim at an effective action, where all field modes below the orbital scale are integrated 6 out. Wespellouttherelevant degrees of freedom, inparticular therotational ones, andthe 5 9 associated symmetries. Building on these symmetries, we introduce the minimal coupling 4 part of the point particle action in terms of gauge rotational variables, and construct the 0 . spin-induced nonminimal couplings, where we obtain the leading order couplings to all 1 0 orders in spin. We specify the gauge for the rotational variables, where the unphysical 5 degrees of freedom are eliminated already from the Feynman rules, and all the orbital field 1 : modes are integrated out. The equations of motion of the spin can be directly obtained v i via a proper variation of the action, and Hamiltonians may be straightforwardly derived. X We implement this effective field theory for spin to derive all spin dependent potentials up r a to next-to-leading order to quadratic level in spin, namely up to the third post-Newtonian order for rapidly rotating compact objects. In particular, the proper next-to-leading order spin-squared potential and Hamiltonian for generic compact objects are also derived. For the implementations we use the nonrelativistic gravitational field decomposition, which is found here to eliminate higher-loop Feynman diagrams also in spin dependent sectors, and facilitates derivations. This formulation for spin is thus ideal for treatment of higher order spin dependent sectors. Contents 1 Introduction 1 2 Setup of EFT for gravitating spinning objects 4 2.1 Setup and goal 4 2.2 Degrees of freedom 6 2.3 Symmetries 7 3 Formulation of EFT for spin 8 3.1 EFT with the worldline spin as a further DOF 8 3.2 Unfixing the gauge of the rotational variables 9 3.3 Extra term from minimal coupling 14 4 EFT for spin: Nonminimal couplings 15 4.1 Construction of spin nonminimal couplings 16 4.2 Nonminimal couplings to all orders in spin 19 5 Integrating out the orbital scale 21 5.1 Tetrad field gauge 22 5.2 Fixing the gauge of the rotational variables 23 5.3 Feynman rules 27 5.4 EOM of the spin 29 6 Spin potentials to NLO 30 6.1 LO spin sectors 30 6.1.1 LO spin-orbit sector 30 6.1.2 LO quadratic in spin sectors 32 6.2 NLO spin-orbit sector 32 6.3 NLO spin1-spin2 sector 35 6.4 NLO spin-squared sector 38 7 Conclusions 41 1 Introduction The anticipated direct detection of gravitational waves (GWs) may be realized soon with the upcoming operation of second-generation ground-based interferometers, such as the twin Advanced LIGO [1] detectors in the US, Advanced Virgo [2] in Europe, and in a few years also KAGRA [3] in Japan. This will open a new era of observational gravitational wave astronomy, where also space-based detectors, such as eLISA [4, 5] are planned to – 1 – extendtheobservedfrequencyrangetothelowfrequencyband. Binariesofcompactobjects are the most promising sources in the accessible frequency band of such experiments. The post-Newtonian (PN) approximation of General Relativity stands out among the various and complementary approaches to model these systems, as it enables to treat analytically the inspiral phase of their evolution [6]. The search for GW signals from such sources employs the matched-filtering technique, and thus accurate theoretical template waveforms are crucial to obtain a successful detec- tion. Even relative high order PN corrections, such as the fourth PN (4PN) order, have an impact on the waveform templates for the binary inspiral, and further they are required to gain information about the inner structure of the components of the binary [7]. Moreover, astrophysical observations indicate that such black hole components have near extreme spin [8]. Hence, PN spin effects for rapidly rotating compact objects, which first appear at 1.5PN order, should be obtained at least to 4PN order, which was recently completed in the non-spinning case [9]. Several efforts have been made in recent years to push ahead the formulation for gravitating spinning objects in the context of the binary inspiral problem. An action formalism plays a central role in the various approaches, building in particular on the seminal works in [10] and [11] for flat and curved spacetime, respectively, and see also section 11 of [6] for a review of spinning compact binaries for gravitational radiation. The self-contained Effective Field Theory (EFT) approach for the binary inspiral as introduced in [12, 13] for non-spinning objects seems then to provide a solid path to obtain such a formulation. In the EFT formulation manifest power counting in the small expansion parameter (here v c = 1) is achieved by performinga decomposition of the gravitational ≪ field at the level of the action into modes with definite scaling properties, followed by integrating out the off-shell modes [12]. TheEFT approach provides a systematic methodology to construct the action to arbi- trarily high accuracy, in terms of operators with Wilson coefficients ordered by relevance, which is indispensable beyond the point-mass approximation. In that respect it should be pointed out, that even just the point-mass approximation, which past work was using, is naturally incorporated already in the EFT framework. It should also be noted that some effective action with derivative expansion to model finite size effects was already discussed in [14] in the context of alternative theories of gravity. The EFT approach also provides a natural framework to handle the regularization required for higher order corrections in the PN approximation within the standard renormalization scheme. It formulates the pertur- bative calculation efficiently by applying the standard tools from Quantum Field Theory, such as Feynman diagrams (a related basic diagrammatic expansion was used already in [15]). Consequently, the EFT approach then benefits from existent developed Feynman integral calculus at its disposal. For spinning objects such EFT techniques were first used in [16], and revised in [17], where eventually a Routhian approach from [18] was adopted. Our goal in this work is indeed to obtain an EFT formulation for gravitating spinning objects for the binary inspiral problem, building on [12, 13], and on several observations made in a series of works, mainly [19,20], [21] and [22]. Theessential obstacle that one has todealwithinextendingtheformulationfromagravitatingpoint-masstoaspinningobject – 2 – in terms of an EFT point particle approach is just the intrinsic conflict between the actual spinning object, which must be extended for its rotational velocity not to be superluminal, and its view as a point particle. The elusive notion of a ‘center’, which would serve as a reference point within the object, in relativistic physics, similar to the center of mass in Newtonian physics, is the origin of ambiguities in the description of relativistic spinning objects. Ever since the first treatment in 1959 of the leading order (LO) PN correction, which involves spin, in the spin-orbit sector, the essential choice of such a center has been a puzzling issue, see Tulczyjew’s paper and errata in [23]. In this work we aim at an effective action, that incorporates the essential requirement, that all field modes below the orbital scale are integrated out. We aim to attain accuracy at the 4PN order for rapidly rotating compact objects, and indeed the formulation in this paper holds as it stands to this high PN order, and it may hold until dissipative effects start to play a role as of the 5PN order [24]. Here, we spell out the relevant degrees of freedom (DOFs), in particular the rotational ones, and most importantly the associated symmetries. Building on these symmetries, we start with the minimal coupling part of the point particle action, stressing the role of the worldline spin as a further worldline rotational DOF. We proceed to construct the spin-induced nonminimal couplings, where we obtain the LO couplings to all orders in spin. We then introduce the gauge freedom of the rotational variables into the action, and express it in terms of gauge rotational variables. Again, this spin gauge invariance was not addressed previously in the action. From introducing this spin gauge freedom we get that the minimal coupling part of the spinin theaction, wouldcontribute tothefinitesizeeffects, whichis justthemanifestation of the aforementioned conflict between the actual spinning extended object and its view as a point particle. We then fix a canonical gauge for the rotational variables, where the unphysical DOFs are eliminated already from the Feynman rules, and all the orbital field modes are conveniently integrated out. The equations of motion (EOM) of the spin are then directly obtained via a proper variation of theaction, wherethey take ona simpleform. ThecorrespondingHamiltonians arealsostraightforwardly obtainedfromthepotentials, derivedviathisformulation, dueto the canonical gauge fixing. We implement this EFT formulation for spin to derive all spin dependent potentials up to next-to-leading order (NLO) to quadratic level in spin, i.e. up to the 3PN order for rapidly rotating compact objects. In particular, the proper next- to-leading order spin-squared potential and Hamiltonian for generic compact objects are also derived. For the implementations we use the nonrelativistic gravitational (NRG) field decomposition[25,26],whichisfoundheretoeliminatehigher-loopFeynmandiagramsalso for spin dependent sectors, and facilitates derivations. Hence, with the simple EOM of the spin,andtheadditionaladvantageous usefulnessoftheHamiltonian forthestraightforward obtainment of gauge-invariant quantities, andforimplementations withintheeffective one- body formulation [27], the EFT formulation for spin here is ideal for treatment of higher order spin dependent sectors. Indeed, the application of the EFT formulation for spin presented here has led to the completion of the spin dependent conservative sector up to the 4PN order in the recent works [28], [29], and [30], which obtained the LO cubic and quartic in spin, NNLO spin-orbit, and NNLO spin-squared sectors, respectively. – 3 – Theoutline of the paperis as follows. In section 2we present thesetup and goal of our EFT formulation for gravitating spinning objects, and detail the relevant DOFs, and the associated symmetries. Insection 3westartbypresentingtheminimalcouplingpartofthe action, andthenexpressitintermsofrotationalgaugevariables,whichyieldsanextraterm from minimal coupling. In section 4 we construct the spin-induced nonminimal coupling part of the action, where we obtain the LO couplings to all orders in spin. In section 5 we fix all ingredients in order to integrate out the orbital field modes: we disentangle the tetrad field from the worldline tetrad, we fix the gauge of the tetrad field and of the rotational variables, and present the resulting Feynman rules. We also discuss how the EOM of the spin are then directly obtained after the orbital modes have been integrated out. In section 6 we implement this EFT for spin to derive all spin dependent potentials and Hamiltonians upto NLO toquadratic level inspin, i.e. upto the3PNorder for rapidly rotating compact objects. In section 7 we summarize our main conclusions. Throughoutthis paper we use c 1, η Diag[1, 1, 1, 1], and the convention for µν ≡ ≡ − − − the Riemann tensor is Rµ ∂ Γµ ∂ Γµ +Γµ Γλ Γµ Γλ . Greek letters denote ναβ ≡ α νβ − β να λα νβ − λβ να indices in the global coordinate frame, lowercase Latin letters from the beginning of the alphabet denote indices in the local Lorentz frame, and upper case Latin letters from the beginning of the alphabet denote the worldline tetrad frame. All indices run from 0 to 3, while spatial tensor indices from 1 to 3, are denoted with lowercase Latin letters from the middle of the alphabet. Square brackets on indices denote that they are in the worldline tetrad frame. Uppercase Latin letters from the middle of the alphabet denote particle labels. The scalar triple product appears here with no brackets, i.e.~a ~b ~c (~a ~b) ~c. × · ≡ × · 2 Setup of EFT for gravitating spinning objects 2.1 Setup and goal We begin by recalling the general setup of an EFT for the binary inspiral problem in terms of a tower of EFTs, building on [12, 13]. The binary inspiral problem involves two intermediate scales below the radiation wavelength scale, λ, which are the scale of internal structure of each of the compact components of the binary, r m, where m is the mass s ∼ of the compact object, and the orbital separation scale, r. It holds that r r /v2 λv, s ∼ ∼ where v is the typical nonrelativistic orbital velocity at the inspiral phase, that is v 1. ≪ Hence, there is a hierarchy of scales in the binary inspiral problem, which makes it ideal for an EFT treatment. We note that we consider here gravitating objects, which are in general spinning. Therefore, to obtain an EFT describing the radiation from the binary, one should proceed in two stages: 1. First, we should have an EFT that removes the scale of the compact objects, r , s from the purely gravitational action of the isolated compact object, which is just the Einstein-Hilbert action 1 S[g ] = d4x√gR. (2.1) µν −16πG Z – 4 – We integrate out the strong field modes, gs , where g gs +g¯ , by writing down µν µν µν µν ≡ an effective action containing the most general set of worldline operators consistent with the symmetries of the theory. According to the decoupling theorem [31] the ef- fective action canbeexpressedbyintroducinganinfinitetower ofworldlineoperators O (σ), such that i 1 S yµ,eµ,g¯ = d4x√g¯R[g¯ ]+ C dσO (σ) , (2.2) eff A µν −16πG µν i i Z i Z (cid:2) (cid:3) X Spp≡pointparticleaction where yµ and eµ are the particle worldline coordinate| and w{ozrldline }tetrad degrees A of freedom (DOFs), discussed in the following section. All UV dependence shows up only in the Wilson coefficients C (r ) in the point particle action, S , and the i s pp worldline operators O (σ) must respect the symmetries of the relevant DOFs at this i scale. In sections 2.2 and 2.3 below we elaborate on the degrees of freedom and the symmetries, considering gravitating spinning objects. We note that a spinning point particle is characterized by two parameters, its mass, m, and spin length, S2, to bedefined in sections 3.1 and 4.1. Yet, since S . m2 r2, s ∼ then indeed r is the only scale in the full theory. In addition, dissipative effects s from the absorption of gravitational waves by the compact objects, as considered in e.g. [24], which modify the mass and spin of the objects, enter only as of the 5PN order. Hence, the mass and spin length can be considered as constant for all relevant implementations. 2. The following EFT in the tower should have the orbital scale of the binary removed. The metric field is again decomposed into the modes g¯ η + H + h , (2.3) µν µν µν µν ≡ orbital radiation e and we note that |{z} |{z} v v ∂ H H , ∂ h h , (2.4) t µν µν ρ µν µν ∼ r ∼ r whereas e e 1 ∂ H H . (2.5) i µν µν ∼ r This EFT of the binary, which is regarded now as a single composite object, is obtained by explicitly integrating out the field modes below the orbital scale, H . µν Starting from an effective action of a binary, given by 1 S yµ,yµ,e µ,e µ,g¯ = d4x√g¯R[g¯ ]+S +S , (2.6) eff 1 2 (1)A (2)A µν −16πG µν (1)pp (2)pp Z (cid:2) (cid:3) the effective action of the composite object is defined by the functional integral eiSeff(composite)[yµ,eµA,ehµν] Hµν eiSeff[y1µ,y2µ,e(1)µA,e(2)µA,g¯µν], (2.7) ≡ D Z – 5 – considering the classical limit, i.e. evaluating the relevant Feynman diagrams in the tree level approximation. Here yµ and eµ are the worldline coordinate and tetrad, A i.e. ηABe µ(σ)e ν(σ) = ηµν +hµν, of the composite particle, respectively. A B To obtain the final EFT in the tower, an EFT of radiation, the field DOFs should all e be integrated out. Therefore, in general one has to proceed to a third stage, where also the radiation modes, h , are integrated out. Yet, in the conservative sector, where no µν radiation modes are present, and from which the conservative dynamics is inferred, the e EFT construction process ends after the aforementioned two stages, that is after having integrated out the field modes below the orbital scale. Indeed, in this paper we focus on the imperative two stage process for the conservative sector, where the end goal of this process should be an effective action, i.e. an action without any remaining orbital scale field DOFs. Naturally, this also involves eliminating all unphysical DOFs from the action, in particular those associated with the rotational DOFs, see section 3 in [22]. By definition, e.g. in eqs. (2.3) and (2.7), an effective action should not contain any remaining field DOFs of modes of the scale, which it removes. These should all be integrated out. It should also be noted that this construction of the EFT, starting from the scale of the internal structure of the compact objects, r , should be supplemented below this scale s for compact stars, rather than just black holes. This becomes relevant, when nonminimal couplings should be taken into account, and we comment on that in section 4. 2.2 Degrees of freedom We should specify and keep track of our degrees of freedom in the process of constructing the EFTs. We should consider here three kinds of DOFs: 1. The gravitational field. For the effective action in eq. (2.2) we have the field DOFs in thepurelygravitational action, andinthenon-spinningpointparticleactions, simply represented by the metric g (x) (the overbar notation of the metric is dropped here µν and henceforth). For the point particle actions beyond the mass monopole, which also involve the spins, the tetrad field, ηabe˜ µ(x)e˜ ν(x) = gµν(x), which couples to a b the multipoles of the objects, also represents the field DOFs. After gauge fixing the purelygravitational action, and the tetrad field, both the metric and the tetrad fields are left with 6 DOFs. 2. The particle worldline coordinate. yµ(σ)isafunctionofanarbitraryaffineparameter σ. The time coordinate is used to fix the gauge of the affine parameter, and we have the 3 DOFs, giving the position of the particle. The particle worldline position does not in general coincide with the ‘center’ of the object, that is the reference point within the actual extended object. The ‘center’ is uniquely defined in Newtonian physics, but not in relativity theory. 3. The particle worldline rotating DOFs. Initially, we consider the worldline tetrad, an orthonormal frame ηABe µ(σ)e ν(σ) =gµν, localized on the particle worldline, con- A B necting the body-fixedand general coordinate frames. From this tetrad we definethe – 6 – worldline angular velocity Ωµν(σ), and then we add its conjugate, the worldline spin, S (σ), as a further DOF. We then have 6+6 DOFs. The field DOFs, represented µν by the tetrad field, which satisfies e˜ µ(y(σ)) = Λ A(σ)e µ(σ), are then disentangled a a A from the worldline tetrad DOFs, such that we are left with the worldline Lorentz matrices DOFs, ηABΛ a(σ)Λ b(σ) = ηab, and the conjugate worldline spin, S (σ), A B ab projected to the local frame. After gauge fixingthe rotational DOFs, we are left only with the 3+3 physical DOFs. 2.3 Symmetries The aforementioned degrees of freedom should be coupled in all possible ways allowed by the symmetries of the problem in order to construct the effective action. The following symmetries should then be considered: 1. General coordinate invariance, and in particular parity invariance is also included, which holds for macroscopic objects in General Relativity, and is relevant for non- minimal couplings in the point particle action, see section 4. 2. Worldline reparametrization invariance. This is used to construct the minimal cou- pling as well as the non minimal coupling parts of the point particle action, see sections 3.1 and 4, respectively. 3. Internal Lorentz invariance of the local frame field. We use the 3+3 DOFs of local Lorentz transformations to fix the gauge of the tetrad field, which in general has 16 DOFs, such that it is represented by the 10 DOFs of the metric (beforegauge fixing), see section 5.1. µ 4. SO(3) invariance of the body-fixed spatial triad, e , consisting of the 3 spacelike [i] vectors. This follows from the 3 rotational DOFs to orient the massive particle in space in the body-fixed frame. In consequence, the worldline spin DOFs are SO(3) tensors in the body-fixed frame, which is also relevant for nonminimal couplings in thepointparticle action, seesection 4. Thisis also discussedin section 3.2in relation with the spin gauge invariance. 5. Spin gauge invariance, that is an invariance under the choice of a completion of the body-fixed spatial triad through a timelike vector. This is a gauge of the rotational variables, i.e. of the worldline tetrad and of the worldline spin. It is considered in section 3.2, and further discussed in section 5.2. 6. We assume that the isolated object has no intrinsic permanent multipole moments beyond the mass monopole and the spin dipole. This is used in sections 3.1 and 4. PermanentmultipolemomentsmaybeincludedthroughconstantSO(3)tensors. Yet, recall that mass and spin are conserved for isolated objects, but higher multipoles are not. Westressthattime-reversalsymmetryisnotassumedhere,butinsteadtermswhichviolate it are shown not to contribute at the considered order, see also section 4. – 7 – 3 Formulation of EFT for spin 3.1 EFT with the worldline spin as a further DOF First, we briefly review the essential basic definitions as in, e.g. section III of [20]. We µ start by considering the worldline tetrad, an orthonormal frame e (σ), localized on the A particle worldline, connecting the body-fixed and general coordinate frames, such that ηABeµeν = gµν with ηAB diag[1,-1,-1,-1] the flat spacetime Minkowski metric. We recall A B ≡ that the reciprocal tetrad is defined by eµA ηABeµ. The projections of any tensor onto ≡ B the tetrad frame, and the converse projection onto the coordinate frame are then defined as, e.g. for a vector, V eµV , and V eAV , respectively. A ≡ A µ µ ≡ µ A We proceed to define the antisymmetric angular velocity tensor by DeAν Ωµν eµ , (3.1) ≡ A Dσ where D/Dσ is the covariant derivative with respect to the worldline parameter σ, and this is a generalization of the flat spacetime definition given by Ωab Λa dΛAb [10, 11]. ≡ A dσ Considering the degrees of freedom and symmetries of the problem, noted in the previous section, the point particle Lagrangian should be a function of the coordinate velocity, uµ dyµ/dσ, the angular velocity from eq. (3.1), and the metric, that is L [uµ,Ωµν,g ], pp µν ≡ where the dependence in the metric is extended beyond minimal coupling to include the Riemanntensorandfurthercovariantderivatives. Thespinisthendefinedastheconjugate to the angular velocity, i.e. ∂L S 2 . (3.2) µν ≡ − ∂Ωµν The minus sign in this definition is chosen to give the correct form in the nonrelativistic limit. It is then beneficial to construct the Lagrangian with the spin as a further worldline DOF since it makes sense to utilize the spin dipole moment, sourcingthe gravitons, similar to the mass monopole, as a classical source on the worldline. Another advantage is then that the spin becomes an independent variational variable, and the equations of motion (EOM)of thespinarethendirectly andconveniently obtained viaanappropriatevariation of the effective action [22]. Therefore, the point particle action from eq. (2.2) can be written as 1 S = dσ m√u2 S Ωµν +L [uµ,S ,g (yµ)] , (3.3) pp µν SI µν µν − − 2 Z (cid:20) (cid:21) wherethefirsttwotermsarejustthepoint-mass androtational minimalcouplings retained from flat spacetime [10, 11], which are inferred from reparametrization invariance. L SI stands for the nonminimal coupling part of the action, which according to the symmetries spelled out in section 2.3 contains only spin-induced multipoles, and as will be further illustrated in section 4, only depends on the worldline DOFs uµ and S . The conjugate µν to the 4-velocity uµ is the linear momentum, given by ∂L p . (3.4) µ ≡ −∂uµ – 8 – Clearly, it is Lagrangian dependent and is modified as higher multipoles, i.e. nonminimal couplings, are introduced, and we have uµ pµ = m + (S2). (3.5) √u2 O For an isolated compact object the linear momentum pµ can be obtained from surface integrals at spatial infinity. In this case finite size effects are not taken into account, and the mass is matched as m2 = p pµ. µ We note that we can also express the rotational minimal coupling term, using the spin projected to the body-fixed frame, where the spin is a permanent multipole moment. Indeed, the components of the spin in this frame are constant, which can be seen most directly using the EOM following from the action in eq. (3.3). Using the Ricci rotation coefficients, defined by ω ab eb D eaν, (3.6) µ ν µ ≡ it holds that 1 1 S Ωµν = S ωABuµ. (3.7) 2 µν 2 AB µ We shall see that only the spatial SO(3) components in the body-fixed frame are non- vanishing here. Considering the scalar mass monopole from eq. (3.3), and this form, where the spin dipole is also represented as a constant antisymmetric SO(3) tensor, we shall be able to construct the nonminimal coupling part of the action in a rather straightforward manner, as will be detailed in section 4. As we are working in an action approach, there is no impediment to the implementa- tion of gauge constraints on the rotational DOFs. Moreover, as we shall see these gauge constraints should be implemented at the level of the point particle action in order to ul- timately arrive at an effective action without any remaining orbital scale field degrees of freedom. We shall also see in the following section 3.2, that in order to arrive at a generic point particle action, where the gauge of the rotational variables is not fixed, we should initially implement the covariant gauge. Yet it is crucial to point out that in the point particle Lagrangian in eq. (3.3), we have both DOFs of the angular velocity and of the spin, and therefore it is necessary to implement gauge fixing both on the worldline tetrad DOFs and on the spin DOFs, rather than only on the latter ones. We shall explicitly see in section 5.2, that we cannot obtain an effective action formulated with the worldline spin, if the gauge of the spin is fixed without gauge fixing its conjugate DOFs. These are principal statements in this paper. 3.2 Unfixing the gauge of the rotational variables Aswenotedinsection2.3thereisaspingaugefreedominthechoiceofatimelike vectorfor the worldline tetrad. This is a choice of a ‘center’ point within the spinning object, which must have a finite size due to its spin. This gauge is fixed using some spin supplementary conditions (SSC), corresponding to a gauge choice of the timelike basis vector for the worldline tetrad. The covariant SSC by Tulczyjew [32], given by S pν = 0, (3.8) µν – 9 –