Gravitational Waves in Non-Local Gravity

Home , Petrov classification

arXiv:2107.06972v1 [gr-qc] 14 Jul 2021 ewe h fields dispersions, the temporal or between f possible spatial is when Lagrangians it media, point-like level, continuous of classical by at governed already However, are and motion. point, same the Introduction 1 gravitatio gravity; modified gravity; Non-local 98.80.-k Keywords: 04.30.-w, 04.50.Kd, numbers: PACS 1 4 ∗ iatmnod iia".Pnii,Uiest iNpl “ Napoli di Università Pancini", "E. Fisica di Dipartimento † iatmnod aeaiaUiest iSlro i Giov via Salerno, di Università Matematica di Dipartimento hoiso hsc eciigeeetr neatosar interactions elementary describing physics of Theories [email protected] [email protected] 2 NNSzoed aoi op.Ui.d ot .Agl,Edifi Angelo, S. Monte di Univ. Compl. Napoli, di Sezione INFN nv iMneS neo dfii ,VaCnha -02,Na I-80126, Cinthia, Via G, Edificio Angelo, S. Monte di Univ. 3 sabetigmd,wih ttelws re fa effective an of order lowest the at which, mode, breathing a is httepeec o bec)o rvttoa aemdsi modes wave gravitational of absence) (or presence the that ieec ewe erynl n ulpaewvs Finally waves. plane null and null nearly between difference asv oewt eiiy0adaglrfrequency angular and 0 helicity with mode massive ls o h o-oa rvt stype is gravity non-local the for class lsadcosplrztos eiiy2adaglrfreque angular and t 2 standard helicity the polarizations, to cross addition and In plus Lagrangian. Hilbert-Einstein a.Ter omlg,TmkSaeUiest fCnrlSy Control of University State Tomsk Cosmology, Theor. Lab. edrv rvttoa ae nater ihnnlclcur non-local with theory a in waves gravitational derive We rvttoa ae nnnlclgravity non-local in waves Gravitational avtr Capozziello Salvatore ( aieetois(UU) 300Tmk Russia. Tomsk, 634050 (TUSUR), Radioelectronics D , H ) and ( iti,I816 aoi Italy, Napoli, I-80126, Cinthia, D E icao AI804 Italy. I-84084, SA Fisciano, , i B ( x ) = ) cu 1 ,3 ]as 4] 3, 2, [1, occur , Z uy1,2021 16, July N 3 d ∗1,2,3 codn otePto lsicto.Ti means This classification. Petrov the to according , Abstract 4 x ′ ǫ 1 ij n arzoCapriolo Maurizio and ( x ′ ) E ω oosrennlclt nElectrodynamics in non-locality observe to j 2 a waves. nal hs oaiaini rnvre It transverse. is polarization whose , ( u otennlclcntttv relation constitutive non-local the to due x ncy oa,ta s ed r vlae at evaluated are fields is, that local, e − h quasi-Lorentz the , observer-independent. s omsls esrmds with modes, tensor massless wo o hc n eie qain of equations derives one which rom x ′ ω parameter ) 1 . eoti ute scalar further a obtain we , auecretost the to corrections vature eeioI” Compl. II”, Federico niPooI,132, II, Paolo anni γ rsnsaspeed a presents , 4 † E oi Italy, poli, (2) tm and stems i ,Via G, cio -invariant (1.1) It is a sort of memory-dependent phenomenon taking into account both the past history of the fields and their values taken in other points of the medium. Also at quantum level, some effective actions show non-local terms and therefore, the associated field equations are integro-differential ones. Recently, non-locality has been considered in cosmology taking into account non-local models to explain early and late-time cosmic acceleration as well as structure formation, without introducing dark energy and dark matter. Specifically, non-locality can play interesting roles to address problems like cosmological constant, Big Bang and black hole singularities, and, in general, coincidence and fine-tuning problems, which affect the ΛCDM model [5, 6, 7, 8, 9, 10]. Some non-local field theories are of infinite order because they have an infinite number of derivatives. This feature is due to the presence of operators like f () which can be expanded in series, assuming that f is an analytic function, as

∞ n f()= an . (1.2) n=0 X Here is the d’Alembert operator. This procedure is aimed to make the theory ghost free [11, 12, 13, 14]. In particular, we know that General Relativity describes gravity as a local interaction while Quantum Mechanics shows non-local aspects. Several approaches have been proposed to achieve a self-consistent Quantum Gravity as discussed, for example, in [15, 16]. A possibility towards Quantum Gravity is considering non-local corrections to the Hilbert-Einstein action [17, 18]. It is a natural way to cure ultraviolet and infrared behaviors of General Relativity. Introducing non-local terms can work also in alternative theories like teleparallel gravity [19]. In all these approaches, it is important to study the linearized versions of the theories and to derive gravitational waves (GWs). In fact, gravitational radiation allows to detect possible effects of non-local gravity [20] as well as to classify the degrees of freedom of a given theory [21, 22, 23]. In this paper, we want to investigate the effect of non-locality in a theory of gravity where the Ricci scalar R of General Relativity is corrected by R−1R, which is the first interesting non-local curvature term. Specifically, we want to investigate how polarization, helicity, and mass of GWs are affected by this kind of terms and how the E(2) Petrov classification changes. Sec. 2 is devoted to a procedure for the localization of the non-local gravitational action by the method of Lagrangian multipliers. Starting from this localized action, it is possible to derive the field equations. Subsequently, in Sec. 3, we linearize the field equations and solve them in harmonic gauge and in a further gauge in view of eliminating ghost modes. Finally, we get the GWs. In Sec.4 and 5, polarizations are analyzed by using both geodesic deviation and the Newman-Penrose formalism. The method consists in expanding the nearly null massive plane waves in term of the exactly null plane waves. The expansion is achieved in terms of a parameter γk. In Sec. 6, results are summarized and possible future developments are discussed.

2 Localization of non-local gravity action via Lagrange multipliers approach

Let us study gravitational interaction governed by the following non-local action 1 S[g]= d4x√ g R + a R−1R + d4x√ g [g] , (2.1) 2κ2 − 1 − Lm Z Z 2 where k2 = 8πG/c4. Its field equations are non-linear integro-differential equations due to the non-local term. We introduce the auxiliary field φ(x) defined as φ(x)= −1R, (2.2) and then the Ricci scalar is R = φ(x) . (2.3) According to this definition, a Lagrange multiplier can be considered so that the gravitational action becomes [5] 1 S [g, φ, λ]= d4x√ g [R (1 + a φ)+ λ (φ R)] , (2.4) g 2κ2 − 1 − Z where λ(x) is a further scalar field. Using integration by parts and imposing that fields and their derivatives vanish onto the boundary of integration domain, we obtain 1 S [g, φ, λ]= d4x√ g [R (1+ a φ λ) ν λ φ] . (2.5) g 2κ2 − 1 − − ∇ ∇ν Z Varying with respect to φ, we get 1 δS 1 = [a R + λ]=0 λ = a R, (2.6) √ g δφ 2κ2 1 ⇒ − 1 − while varying with respect to λ, the functional derivative takes the form 1 δS 1 = [ R + φ]=0 φ = R. (2.7) √ g δλ 2κ2 − ⇒ − Finally, the variation with respect to the metric gµν gives 2κ2 δS 1 g = (G + g )(1+ a φ λ) φ λ + g σφ λ , (2.8) √ g δgµν µν µν − ∇µ∇ν 1 − − ∇(µ ∇ν) 2 µν ∇ ∇σ − and δ (√ g ) √ g − Lm = − T . (2.9) δgµν − 2 µν The final field equations are

1 (G + g )(1+ a φ λ) φ λ + g σφ λ = κ2T , (2.10) µν µν − ∇µ∇ν 1 − − ∇(µ ∇ν) 2 µν ∇ ∇σ µν

φ = R , (2.11)

λ = a R (2.12) − 1 where Gµν is the Einstein tensor 1 G = R η R. (2.13) µν µν − 2 µν According to Eqs. (2.11) and (2.12), the trace of Eq. (2.10) is

[1 + a (φ 6) λ] R ρφ λ = κ2T , (2.14) 1 − − − ∇ ∇ρ − and now we can start our considerations on the weak ﬁeld behavior of this theory.

3 3 The weak ﬁled limit and gravitational waves

In order to analyze gravitational radiation, let us perturb the metric tensor gµν around the ﬂat metric ηµν and the two scalar ﬁelds φ and λ around their values in Minkowskian spacetime φ0 and λ0. It is

g η + h , (3.1) µν ∼ µν µν φ φ + δφ , (3.2) ∼ 0 λ λ + δλ . (3.3) ∼ 0

At ﬁrst order in hµν , the Ricci tensor Rµν and the Ricci scalar R become 1 R(1) = ∂ ∂ hσ + ∂ ∂ hσ ∂ ∂ h h , (3.4) µν 2 σ µ ν σ ν µ − µ ν − µν R(1) = ∂ ∂ hµν h , (3.5) µ ν − µν where h is the trace of perturbation hµν . In vacuum and under Lorentz gauge, Eqs. (3.4), (3.5) and the Einstein tensor Gµν become 1 R(1) = h , (3.6) µν −2 µν 1 R(1) = h , (3.7) −2 1 1 G(1) = h + η h . (3.8) µν −2 µν 4 µν Later, according to Eqs. (2.10), (2.11) and (2.12), the linearized ﬁeld equations in vacuum are

(1 + a φ λ )(2h η h)+4∂ ∂ (a δφ δλ)+4a η h =0 , (3.9) 1 0 − 0 µν − µν µ ν 1 − 1 µν (2δφ + h)=0 , (3.10) (2δλ a h)=0 , (3.11) − 1 and, from Eqs. (3.10) and (3.11), the linearized trace equation is

[1 + a (φ 6) λ ] h =0 . (3.12) 1 0 − − 0 The trace equation (3.12) admits solutions if

1+ a φ λ =6a h =0 , (3.13) 1 0 − 0 6 1 → or 1+ a φ λ =6a h =0 and h =0 . (3.14) 1 0 − 0 1 → 6 In the case (3.13), Eqs. (3.9), (3.10) and (3.11) become

(1+ a φ λ ) h +2∂ ∂ (a δφ δλ)=0 , (3.15) 1 0 − 0 µν µ ν 1 − δφ =0 , (3.16)

4 δλ =0 . (3.17) In k-space, considering the Fourier transform, Eq. (3.15) takes the following form

(1 + a φ λ ) k2h˜ (k)+2k k a δφ˜ (k) δλ˜ (k) =0 . (3.18) 1 0 − 0 µν µ ν 1 − It implies h =0 k2h˜ (k)=0 k2 =0 , (3.19) ⇒ ⇒ and, putting (3.19) into (3.18), we get a solution if

a1δφ˜ (k)= δλ˜ (k) , (3.20) which, in x-space, becomes a1δφ (x)= δλ (x) . (3.21) Inserting (3.21) into (3.15), for 1+ a φ λ =0, we obtain 1 0 − 0 6

hµν =0 , (3.22) which, together with Eqs. (3.16) and (3.17), yield massless, 2-helicity transverse waves solutions for 2 k1 =0, namely the standard gravitational waves of General Relativity. In the case (3.14), excluding h =0 because the previous massless case returns, Eqs. (3.9), (3.10) and (3.11) become a coupled partial diﬀerential equations system, i.e.

6a h a η h +2∂ ∂ (a δφ δλ)=0 , (3.23) 1 µν − 1 µν µ ν 1 − 1 δφ = h , (3.24) −2 a δλ = 1 h , (3.25) 2 with the additional condition h =0 . (3.26) 6 From Eq. (3.26), in the momentum space, we have

k2h˜ (k) =0 k2 =0 and h˜ (k) =0 , (3.27) 2 6 ⇒ 2 6 6 µ k 2 2 where (k2) = (ω2, ) is the wave four-vector and k2 = M . We assume that Eq. (3.26) is

h (x)= g (x) , (3.28) choosing g(x) with any k2 =0 such as 6 1 k x g(x)= d3k q˜(k) eiϕeiω2te−i · + c.c. (3.29) (2π)3/2 Z 1 = d3k 2˜q (k)cos(ω t k x + ϕ) , (3.30) (2π)3/2 2 − · Z

5 where q˜(k) L2 R3 is a square integrable function in R3 and ϕ is the phase. By performing the Fourier transform∈ with respect to the spatial coordinates, the non-homogeneous wave equation (3.28) becomes, ink-space,

∂2 + k 2 h˜ (t, k)=˜q (k) eiϕeiω2t , (3.31) 0 | | which has a particular solution

q˜(k) eiϕ ˜ k iω2t ˜ k iω2t h (t, )= 2 e = A ( ) e , (3.32) − k2

2 2 2 where ω2 = M + k . The value M corresponds to the squared mass of a new effective scalar field Ψ which can be| defined| as the combination p λ (x) Ψ (x)= φ (x) , (3.33) a1 − whose linear perturbation δΨ satisfies the Klein-Gordon equation

+ M 2 δΨ=0, (3.34) under the assumptions that g(x) behaves as a wave packet with k2 exactly equal to M 2. It is worth noticing that the linear perturbation of Ψ in k-space, δΨ(˜ k), corresponds to h˜(k) = A˜(k) see Eq. (3.32). It is the trace of metric perturbation in k-space. From Eqs. (3.24), (3.25) and (3.28), it is

λ (x) Ψ (x)= φ (x) = h, (3.35) a − 1 which, in k-space with non-null k2, becomes

δλ˜ (k) δΨ˜ (k)= δφ˜(k)= A˜ (k) . (3.36) a1 − With these considerations in mind, a solution of Eq. (3.28) is 1 h(x)= d3k A˜ (k) eik2·x + c.c. , (3.37) (2π)3/2 Z and, according to the two wave Eqs. (3.24), (3.25) in k-space, it is

∂2 + k 2 2 δφ˜ (t, k)+ h˜ (t, k) =0 , (3.38) 0 | | h i ∂2 + k 2 2 δλ˜ (t, k) a h˜ (t, k) =0 . (3.39) 0 | | − 1 From Eq. (3.32), we obtain two particular h solutions for ﬁxed ki 1 δφ˜ (t, k)= A˜ (k) eiω2t , (3.40) −2 a δλ˜ (t, k)= 1 A˜ (k) eiω2t , (3.41) 2

6 2 2 2 that allows us to derive the waves k1 = 0 and k2 = M for the two scalar linear perturbations δφ and δλ, i.e. we have

1 1 1 δφ(x)= d3k D˜ (k) eik1·x + d3k A˜ (k) eik2·x + c.c. , (3.42) 3/2 (2π)3/2 −2 (2π) Z Z and

1 3 ik1·x 1 3 a1 ik2·x δλ(x)= d k a1D˜ (k) e + d k A˜ (k) e + c.c. . (3.43) (2π)3/2 − (2π)3/2 2 Z Z Now, carrying out the spatial Fourier transform of Eq. (3.23) and by means of Eqs. (3.32), (3.40) and (3.41), we get

6a ∂2 + k 2 h˜ (t, k)+ a η k2A˜ (k) eiω2t +2a (k ) (k ) A˜ (k) eiω2t =0 , (3.44) 1 0 | | µν 1 µν 2 1 2 µ 2 ν that simplifying gives

2 ηµν k + 2 (k2) (k2) 2 k 2 ˜ k 2 µ ν ˜ k iω2t ∂0 + hµν (t, )= A ( ) e . (3.45) | | − 6 ! To solve the non-homogeneous Eq. (3.45), ﬁrst we get the related homogeneous solution linked to 2 the massless wave k1 =0,

˜(o) k (1) k iω1t (2) k −iω1t hµν (t, )= Cµν ( ) e + Cµν ( ) e , (3.46)

2 2 and then a particular solution, linked to the massive wave k2 = M , is

2 ηµν k + 2 (k2) (k2) ˜(p) k 2 µ ν ˜ k iω2t hµν (t, )= 2 A ( ) e . (3.47) 6k2 Remembering that the solutions must be real, from the following decomposition of spatial coordinates, we have 1 3k ˜ k −ik·x hµν (x)= 3/2 d hµν (t, ) e , (3.48) (2π) Z and then we can reconstruct hµν (x) as

1 3 ik1·x hµν (x)= d k Cµν (k) e (2π)3/2 Z , (3.49) 1 1 η (k2) (k2) + d3k µν + µ ν A˜ (k) eik2·x + c.c. 3/2 3 2 k2 (2π) Z 2 that is, the gravitational waves in non-local linear gravity. For a similar approach in higher order gravity theories see [24]. It is worth noticing that the part of solution due to non-locality, related to k2 =0, appears only if the constraint (3.14) is veriﬁed. 2 6

7 Eq.(3.49) has the disadvantage of presenting ghost modes, so it is more useful to choose a suitable gauge in order to suppress waves without physical meaning. In this perspective, perturbing ﬁeld Eqs. (2.10), (2.11) and (2.12) to ﬁrst order in hµν , δφ and δλ, we get

(1 + a φ λ)(0) G(1) + (g )(0) (1 + a φ λ)(1) = κ2T (0) , (3.50) 1 − µν µν − ∇µ∇ν 1 − µν that is (1 + a φ λ ) G(1) + (η ∂ ∂ ) (a δφ δλ)= κ2T (0) , (3.51) 1 0 − 0 µν µν − µ ν 1 − µν µν where = η ∂µ∂ν , and δφ = R(1) , (3.52) δλ = a R(1) . (3.53) − 1 In a particular coordinates frame xµ , being our equations gauge invariant, we deﬁne a new gauge as { } 1 η h¯ = h η h µν (a δφ δλ) , (3.54) µν µν − 2 µν − 1+ a φ λ 1 − 1 0 − 0 such that, in our reference frame, it is µν ∂µh¯ =0 , (3.55) with 1+ a φ λ =0. The trace of Eq. (3.54) yields 1 0 − 0 6 4 h¯ = h (a δφ δλ) . (3.56) − − 1+ a φ λ 1 − 1 0 − 0 In this gauge, we get the following expressions to ﬁrst order for the Ricci tensor Rµν , the Ricci scalar R and the Einstein tensor Gµν 1 1 R(1) = h + ∂ ∂ (a δφ δλ) , (3.57) µν −2 µν 1+ a φ λ µ ν 1 − 1 0 − 0 1 1 R(1) = h + (a δφ δλ) , (3.58) −2 1+ a φ λ 1 − 1 0 − 0 1 1 1 1 G(1) = h + η h + ∂ ∂ η (a δφ δλ) . (3.59) µν −2 µν 4 µν 1+ a φ λ µ ν − 2 µν 1 − 1 0 − 0 In terms of barred quantities h¯µν and h¯, we obtain 1 3 R(1) = h¯ + (a δφ δλ) , (3.60) 2 1+ a φ λ 1 − 1 0 − 0 1 1 G(1) = h¯ (η ∂ ∂ ) (a δφ δλ) , (3.61) µν −2 µν − 1+ a φ λ µν − µ ν 1 − 1 0 − 0 which replaced in Eqs. (3.51), (3.52) and (3.53) give, in vacuum,

h¯µν =0 , (3.62) 3 δφ = (a δφ δλ) , (3.63) 1+ a φ λ 1 − 1 0 − 0

8 and 3a δλ = 1 (a δφ δλ) , (3.64) −1+ a φ λ 1 − 1 0 − 0 because Eq. (3.62) implies h¯ =0. Eqs. (3.63) and (3.64) can be rewritten as

1+ a φ λ 1 0 − 0 a δφ + δλ =0 , (3.65) 3 − 1 and 1+ a φ λ a δφ + 1 0 − 0 1 δλ =0 . (3.66) 1 3a − 1 The homogeneous linear system (3.65) and (3.66), in δφ and δλ variables, admits trivial and non-trivial solutions depending on the determinant of the following matrix

1+a1φ0−λ0 a 1 1 det 3 − 1 = (1 + a φ λ 6a )(1+ a φ λ ) , (3.67) a 1+a1φ0−λ0 1 a 1 0 − 0 − 1 1 0 − 0 1 3a1 − ! 1 which can be equal to zero or non-equal to zero, depending on 1+ a1φ0 λ0 = 0. So, if the determinant does not vanish, under the constraint − 6

1+ a φ λ =6a , (3.68) 1 0 − 0 6 1 we get the trivial solution δφ = δλ =0 , (3.69) from which, it follows (a δφ δλ)=0 . (3.70) 1 − A particular solution of Eq. (3.70) is

a1δφ (x)= δλ (x) . (3.71)

Putting this solution together with Eq. (3.62) and Eq. (3.54), we get massless transverse gravita- 2 tional waves with helicity 2, associated to k1 =0 under the constraint (3.68)

1 (k1) 3k ˜ k ik1·x hµν (x)= 3/2 d Cµν ( ) e + c.c. , (3.72) (2π) Z

˜ ˜σ ¯µν where Cµν is the traceless polarization tensor, i.e. C σ =0, in a suitable gauge that leaves ∂µh =0 invariant, as in the case (3.13). The non-trivial solution is obtained by imposing that the determinant of the above matrix is equal to zero, which means 1+ a φ λ =6a , (3.73) 1 0 − 0 1 that gives again a δφ δλ =0 , (3.74) 1 − 6 or equivalently h =0 , (3.75) 6

9 as in the case (3.14). Then, from the inverse gauge under the constraint (3.73)

1 ηµν hµν = h¯µν ηµν h¯ (a1δφ δλ) , (3.76) − 2 − 6a1 − we choose the non-homogeneous wave equation 2 (a δφ δλ)= a h = a g (x) , (3.77) 3 1 − − 1 − 1 with g (x) L(R4), as already deﬁned in (3.29). The particular solution of Eq. (3.77) is ∈ 1 3 a δφ(x) δλ(x)= d3k a A˜ (k) eik2·x + c.c. , (3.78) 1 − (2π)3/2 −2 1 Z 2 2 with any k2 = M . Inserting Eq. (3.78) into Eq. (3.76), we obtain a massive wave 1 η h(k2) (x)= d3k µν A˜ (k) eik2·x + c.c. . (3.79) µν (2π)3/2 4 Z In general under the constraint (3.73), the gravitational waves for non-local f(R, −1R) gravity are both massless and massive waves, that is

1 3 ik1·x hµν (x)= d k C˜µν (k) e (2π)3/2 Z . (3.80) 1 η + d3k µν A˜ (k) eik2·x + c.c. (2π)3/2 4 Z 2 Also here the solution component, related to k2 = 0, appears if the constraint (3.73) is satisﬁed. With these results in mind, let us investigate gravitationa6 l wave polarizations in non-local gravity.

4 Polarizations via geodesic deviation

A useful tool to study the polarization of gravitational radiation is the use of geodetic deviation produced by the wave when it invests a small region of spacetime, as the relative acceleration measured between nearby geodesics. Hence, we start from a wave h (t v z) propagating in +ˆz µν − g direction in a local proper reference frame, where vg is the group velocity in units where c = 1 deﬁned as 2 dω c kz vg = = , (4.1) dkz ω and let us consider the equation for geodesic deviation x¨i = Ri xk , (4.2) − 0k0 i where the Latin index range over the set 1, 2, 3 and R 0k0 are the only measurable components called the electric ones [25]. After replacing{ the} linearized electric components of the Riemann (1) tensor R i0j0, expressed in terms of the metric perturbation hµν 1 R(1) = (h + h h h ) , (4.3) i0j0 2 i0,j0 j0,i0 − ij,00 − 00,ij

10 in Eq.(4.2), we get a linear non-homogeneus system of diﬀerential equations

1 1 1 x¨(t)= 2 h11,00 x 2 h12,00 y + 2 (h10,03 h13,00) z −1 − 1 1 − y¨(t)= 2 h12,00 x 2 h22,00 y + 2 (h02,03 h23,00) z (4.4) −1 − 1 − 1 z¨(t)= (h01,03 h13,00) x + (h02,03 h23,00) y + (2h03,03 h33,00 h00,33) z . 2 − 2 − 2 − −  2 For a massless plane wave travelling in +ˆz direction, that is k1 = 0, which propagates at speed c, k µ if we keep ﬁxed and k1 = (ω1, 0, 0, kz), Eq.(3.72) yields

(k1) (+) (+) (×) (×) iω1(t−z) hµν (t,z)= √2 ǫ˜ (ω1) ǫµν +˜ǫ (ω1) ǫµν e + c.c. , (4.5) h i where 00 0 0 1 01 0 0 ǫ(+) =   , (4.6) µν √2 0 0 1 0 − 00 0 0     0 0 0 0 1 0 0 1 0 ǫ(×) =   , (4.7) µν √2 0 1 0 0 0 0 0 0     2 and ω1 = kz. Furthermore, for a massive plane wave propagating in +ˆz direction, that is k2 = 0, k µ 6 always keeping ﬁxed and instead with k2 = (ω2, 0, 0, kz), Eq.(3.79) becomes

A˜ (k ) h(k2) (t,z)= z η ei(ω2t−kz z) + c.c. , (4.8) µν 4 µν where here the propagation speed is less than c. In a more compact form, the metric linear perturbation hµν , travelling in the +ˆz direction, assuming k ﬁxed, is

(+) (+) (×) (×) iω1(t−z) (s) i(ω2t−kzz) hµν (t,z)= √2 ǫ˜ (ω1) ǫµν +˜ǫ (ω1) ǫµν e +˜ǫµν (kz) e + c.c. , (4.9) h i (s) where ǫ˜µν is the polarization tensor associated to the mixed scalar mode

A˜ (k ) ǫ˜(s) (k )= ǫ(T T ) √2ǫ(b) ǫ(l) z . (4.10) µν z µν − µν − µν 4 and the polarization tensors are explicitly given by

10 0 0 0 0 00 0 00 0 00 0 0 1 0 1 00 0 00 0 ǫ(T T ) =   , ǫ(b) =   , ǫ(l) =   . (4.11) µν 00 0 0 µν √2 0 0 10 µν 0 00 0 00 0 0 0 0 00 0 00 1             (+) (×) (T T ) (b) (l) where the set of polarization tensors ǫµν ,ǫµν ,ǫµν ,ǫµν ,ǫµν satisfy the orthonormality relations n o Tr ǫ(a)ǫ(b) = ǫ(a)ǫ(b)µν = δa,b with a,b +, ,TT,b,l . (4.12) µν ∈{ × } n o 11 (s) The scalar mode ǫ˜µν is a mixed state obtained from a combination of longitudinal and transverse scalar modes which is produced by the single degree of freedom A˜ as for f(R) gravity which has three d.o.f.: ǫ˜(+), ˜ǫ(×) and A˜ [26, 27]. However, as we will see, the transverse component weighs more (s) than the longitudinal one. In fact, the polarization tensor ǫ˜µν , restricted to spatial components (s) ǫ˜i,j , is provided by A˜ (k ) ˜ǫ(s) = √2ǫ(b) + ǫ(l) z , (4.13) i,j i,j i,j 4 (k1) where (i, j) range over (1, 2, 3). Hence, from Eqs.(4.4) in the case of massless plane waves hµν , we have 1 2 (+) (×) iω1(t−z) x¨(t)= 2 ω1 ˜ǫ (ω1) x +˜ǫ (ω1) y e + c.c.  y¨(t)= 1 ω2 ǫ˜(×) (ω ) x ˜ǫ(+) (ω ) y eiω1(t−z) + c.c. , (4.14)  2 1 1 − 1   z¨(t)=0   that give us the two standard transverse tensor polarizations predicted by General Relativity, con- ventionally called plus and cross modes.

(k2) 2 2 2 Otherwise, in the case of a massive plane wave hµν with M = ω2 kz , the linearized geodesic deviation Eq.(4.4) becomes −

x¨(t)= 1 ω2A˜ (k ) xei(ω2t−kzz) + c.c. − 8 2 z   1 2 ˜ i(ω2t−kz z) y¨(t)= 8 ω2A (kz) ye + c.c. , (4.15)  −  1 2 i(ω2t−kz z) z¨(t)= M A˜ (kz) ze + c.c.  − 8   that can be integrated, assuming that hµν (t,z) is small, as

1 ˜ i(ω2t−kz z) x(t)= x(0) + 8 A (kz) x(0)e + c.c.   1 ˜ i(ω2t−kz z) y(t)= y(0) + 8 A (kz) y(0)e + c.c. . (4.16)   1 2 i(ω2t−kz z) z(t)= z(0) + 2 M A˜ (kz) z(0)e + c.c.  8ω2   If we suppose that M 2is very small, which happens when we are suﬃciently far from the radiation source, and that becomes zero for exactly null plane waves, k2 =0, then we can expand our results with respect to a parameter γ deﬁned as [28]

c 2 γ = 1 , (4.17) v − g which takes into account the diﬀerence in speed between nearly null waves with speed vg and null ones with speed c. Thus, keeping k ﬁxed and by using Landau symbols, namely little-o and big- z O

12 notation, we can expand in terms of our parameter γ the following quantities

M 2 1 ω2 ω 2 = 2 k2 = 2 1= γ , (4.18) k2 k2 c2 − z ck − z z z and in units where c =1 ω 1 2 = 1+ γ = 1+ γ + o (γ) , (4.19) k 2 z p ei(ω2t−kzz) = eikz (t−z) + (γ) , (4.20) O that implies M 2 2 = γ + o (γ) . (4.21) ω2

Within the ﬁrst order in γ, the solution (4.16) for the mode ω2 give us

x(t)= x(0) + 1 A˜ (k ) x(0)eikz (t−z) + (γ)+ c.c. 8 z O   1 ˜ ikz(t−z) y(t)= y(0) + 8 A (kz) y(0)e + (γ)+ c.c. , (4.22)  O  z(t)= z(0) + 1 γA˜ (k ) z(0)eikz(t−z) + γ2 + c.c.  8 z  O  that is  ∆z(t)= o (∆x(t)) for γ 0 , (4.23) → and then only the breathing tensor polarization ǫ(b) survives because longitudinal modes are in- ﬁnitesimal in higher order than transverse modes when γ tends to zero. Thus when a GW strikes a sphere of particles of radius r = x2(0) + y2(0) + z2(0), this will be distorted into an ellipsoid described by p x 2 y 2 z 2 + + = r2 , (4.24) ρ (t) ρ (t) ρ (t) 1 1 2 where ρ (t)=1+ 1 A˜ (k ) cos [k (t z)+ φ]+ (γ) and ρ (t)=1+ (γ), that, at zero order in γ 1 4 z z − O 2 O only ρ1, is varying between their maximum and minimum values. This ellipsoid swings only on xy- plane between two circumferences of minimum and maximum radius and represents an additional transverse scalar polarization which has zero helicity within the lowest order in γ [29]. According to these considerations, the d.o.f. of non-local f(R, −1R) gravity are three: two of these, ǫ˜(+) and ˜ǫ(×), give rise to the standard tensor modes of General Relativity while the degree of freedom A˜ generates a further breathing scalar mode. In summary, f(R, −1R) gravity has three polarizations, namely two massless 2-helicity tensor modes and one massive 0-helicity scalar mode, all purely transverse within the lowest order in γ, exactly like f(R) gravity (see for a discussion [27, 30, 31, 32, 33, 34, 35]).

5 Polarizations via Newman-Penrose formalism

A further approach to study polarizations can be obtained by adopting the Newman-Penrose (NP) formalism for low-mass gravitational waves [28]. Even if it is not directly applicable to massive

13 waves because it was, in origin, worked out for massless waves, it is possible to generalize it to low- mass waves propagating along nearly null geodesics [36]. It is worth noticing that the little group E (2) classification fails for massive waves but can be recovered within the first order in γ. Let us introduce a new basis, namely a local quasi-orthonormal null tetrad basis (k, l, m, m¯ ) defined as [29, 37, 38] 1 k = ∂ + ∂ , l = (∂ ∂ ) , (5.1) t z 2 t − z 1 1 m = (∂x + i∂y) , m¯ = (∂x i∂y) , (5.2) √2 √2 − which, adopting the Minkowski metric tensor η of signature 2, satisfies the relations µν − k l = m m¯ =1 , · − · k k = l l = m m =m ¯ m¯ =0 , (5.3) · · · · k m = k m¯ = l m = l m¯ =0 , · · · · that is ηµν =2k(µlν) 2m(µm¯ ν) . (5.4) − Therefore we can raise and lower the tetrad indices by the metric of the tetrad ηab 01 0 0 10 0 0 η = ηab = , (5.5) ab 00 0 1 − 0 0 1 0   −    where (a,b) run over (k, l, m, m¯ ). We now split the Riemann tensor into three irreducible parts, namely: Weyl tensor, traceless Ricci tensor and Ricci scalar, known as Newmann-Penrose quantities. The four-dimensional Weyl tensor Cµνρσ is defined as 1 C = R 2g R + g g R, (5.6) µνρσ µνρσ − [µ|[ρ σ]|ν] 3 µ[ρ σ]ν and in tetrad form becomes 1 C = R 2η R + η η R, (5.7) abcd abcd − [a|[c d]|b] 3 a[c d]b taking into account that the tetrad components of the generic tensor Pabcd... express in terms of the local coordinate basis are P = P aαbβcγ dδ (5.8) abcd... α,βγδ... ··· where (a,b,c,d,... ) run over (k, l, m, m¯ ). The fifteen NP-amplitudes are specifically the five complex Weyl-NP Ψ scalars, expressed in tetrad components of the Weyl tensor as

Ψ C , 0 ≡− kmkm Ψ C , 1 ≡− klkm Ψ C , (5.9) 2 ≡− kmml¯ Ψ C , 3 ≡− klml¯ Ψ C , 4 ≡− ml¯ ml¯

14 and the ten Ricci-NP scalars Φ, Λ, expressed in tetrad components of Ricci tensor as 1 Φ R 02 ≡−2 mm ,

Φ 1 R 01 ≡− 2 km , Φ 1 R ( 12 ≡− 2 lm 1 Φ00 2 Rkk ≡− 1 Φ11 4 (Rkl + Rmm¯ ) , ≡− 1 Φ22 Rll ≡− 2  Φ 1 R =Φ∗ 10 ≡− 2 km¯ 01 , Φ 1 R =Φ∗ ( 21 ≡− 2 lm¯ 21

1 Φ R =Φ∗ , 20 ≡−2 m¯ m¯ 02 R Λ= . (5.10) 24 In order to expand the low-mass gravitational radiation in terms of null plane waves, we ﬁrst ′ µ deﬁne the "wave" four-vector (k2) in units where c = 1 associated to the nearly null plane wave propagating in positive z direction as

µ ′µ k2 = (ω2, 0, 0, kz)= ω2k2 , (5.11) with ′µ k2 = (1, 0, 0, vg) , (5.12) and we set the time retarded u˜ as u˜ = t v z , (5.13) − g that gives us u˜ = (k′ ) . (5.14) ∇µ 2 µ ′ Now, we expand k2 with respect to our null tetrads basis

′µ µ µ µ µ k2 = (1+ γk) k + γll + γmm +¯γmm¯ , (5.15) where the expansion coeﬃcients γk,γl,γm are of same order of the γ of previous section. Given the arbitrariness of the observer to orient its reference system, it is possible to perform orientation 0 0 3 3 0 3 in such a way that k2 = k and k2 k , where k and k are the angular frequency and the third component of vector wave of its null∝ wave, respectively. Therefore we obtain, from Eq. (5.15) ,γ = 2γ and γ =0 that yields l − k m k′µ = kµ + γ (kµ 2lµ) , (5.16) 2 k − or ′µ k2 = (1, 0, 0, 1+2γk) , (5.17)

15 and, as already observed according to 1 γk = − γ + o (γ) , (5.18) 4 the parameters γ and γk are of the same order. The derivatives of Riemann tensor can be expressed as ∂R R = αβγδ u˜ = (k′ ) R˙ , (5.19) αβγδ,µ ∂u˜ ∇µ 2 µ αβγδ where the superscripted dot means the derative with respect to u˜. The following identities

R = 2γ R˙ , (5.20) αβγδ,k − k αβγδ

Rαβγδ,l = (1+ γk) R˙ αβγδ , (5.21)

Rαβγδ,m =0 , (5.22) combined with diﬀerential Bianchi identity

Rab[cd,e] =0 , (5.23) involve that the only non-zero tetrad components of Riemann tensor Rαβγδ to zero order in γk are ¯ terms of form Rlplq with (p, q) range over (k, m, m). In the nearly null plane waves framework, only four complex NP tetrad components are independent and non-vanishing within the ﬁrst order in γ, that is, from Eq. (5.7), they are 1 Ψ (˜u)= R + (γ ) , (5.24) 2 −6 lklk O k 1 Ψ (˜u)= R + (γ ) , (5.25) 3 −2 lklm¯ O k Ψ (˜u)= R , (5.26) 4 − lml¯ m¯ Φ (˜u)= R . (5.27) 22 − lmlm¯ (5.28)

In terms of tetrad components of metric perturbation hab, they become 1 Ψ (˜u)= h¨ + (γ ) , (5.29) 2 12 kk O k 1 Ψ (˜u)= h¨ + (γ ) , (5.30) 3 4 km¯ O k 1 Ψ (˜u)= h¨ + (γ ) , (5.31) 4 2 m¯ m¯ O k 1 Φ (˜u)= h¨ + (γ ) . (5.32) 22 2 mm¯ O k (5.33)

These real amplitudes, under the subgroup of Lorentz transformations which leaves k2 unchanged, namely the little group E(2), show the following helicity values s:

Ψ2 s =0 , Ψ4 s =2 , (5.34)

Ψ3 s =1 , Φ22 s =0 . (5.35)

16 They allow the Petrov classiﬁcation for a set of quasi-Lorentz invariant GWs. Remembering that our gravitational radiation travels along the positive +ˆz axis, we obtain the following four NP amplitudes expressed both in terms of the electric components of the Riemann tensor Ri0j0 and its linearized components [39, 40, 41, 42]. That is, taking into account the identities

R =0 R + R = R + R (5.36) kmlm¯ → 1313 2323 0101 0202 R =0 2R 2R = 2 (R + R ) (5.37) kmkm¯ → − 0131 − 0232 0101 0202 R =0 R = R and R = R (5.38) klkm → 0301 − 0331 0302 − 0332 R =0 R = R and R R = R R (5.39) kmlm → 3132 0102 3131 − 3232 0101 − 0202 2R3101 2R3202 = R3131 + R3232 R0101 + R0202 Rkmkm =0 − − − (5.40) → R = R R R , ( 0102 − 0132 − 0231 − 3132 we get, in terms of Ri0j0, 1 Ψ (˜u)= R + (γ ) , (5.41) 2 − 6 0303 O k 1 1 Ψ3 (˜u)= R0301 + iR0302 + (γk) , (5.42) − 2√2 2√2 O 1 1 Ψ (˜u)= R + R + iR + (γ ) , (5.43) 4 − 2 0101 2 0202 0102 O k 1 1 Φ (˜u)= R R + (γ ) , (5.44) 22 − 2 0101 − 2 0202 O k while, in term of metric perturbation, the NP scalars take the form 1 Ψ (˜u)= (2h h h )+ (γ ) , (5.45) 2 − 12 03,03 − 00,33 − 33,00 O k 1 1 Ψ3 (˜u)= (h01,03 h13,00)+ i (h02,03 h23,00)+ (γk) , (5.46) − 4√2 − 4√2 − O 1 Ψ (˜u)= (h h ) 2ih + (γ ) , (5.47) 4 4 11,00 − 22,00 − 12,00 O k 1 Φ (˜u)= (h + h )+ (γ ) . (5.48) 22 4 11,00 22,00 O k From Eqs.(4.5) and (4.8) for non-null and null geodesic congruences of GWs, we get, for a massless mode ω1 and a massive mode ω2, at k ﬁxed at the lowest order in γk, the expressions

Ψ (˜u)= (γ ) , 2 O k Ψ (˜u)= (γ ) 3 O k Ψ (˜u)= ω2 ǫ(+) (ω ) eiω1(t−z) + c.c. i ǫ(×) (ω ) eiω1(t−z) + c.c. + (γ ) 4 − 1 1 − 1 O k k2 h i Φ (˜u)= z A (k ) eikz(t−z) + (γ )+ c.c. , 22 8 z O k considering ω = k + (γ ) , (5.49) 2 z O k

17 in units c = 1. Hence, being Ψ2 = Ψ3 = 0 and Φ22 = 0 at zero order in γk, the quasi-Lorentz 6 −1 invariant E(2) class of non-local gravitational theory R + a1R R is N3, according to the Petrov classiﬁcation. Here the presence or absence of all modes is observer-independent. The driving-force matrix S(t) can be expressed in terms of the six new basis polarization matrices WA(ez) along the wave direction kˆ = ez as S (t)= pA (ez,t) WA (ez) , (5.50) XA where the index A ranges over 1, 2, 3, 4, 5, 6 [37, 38] and explicitly { } 0 0 0 0 0 1 W1 (ez)= 6 0 0 0 , W2 (ez)= 2√2 0 0 0 , − 0 0 1 − 1 0 0 0 0 0 1 0 0  W3 (ez) =2√2 0 0 1 , W4 (ez)= 0 1 0 , 0 1 0 − 0− 0 0 0 1 0  1 0 0  W5 (ez)= 1 0 0 , W6 (ez)= 0 1 0 . (5.51) 0 0 0 − 0 0 0     (l) (x) Consequently, there are six polarizations modes: the longitudinal mode p1 , the vector-x mode p2 , (y) (+) (×) (b) the vector-y mode p3 , the plus mode p4 , the cross mode p5 , and the breathing mode p6 . Here pA (ez,t) are the amplitudes of the wave at the detector in the frame origin [30, 31, 32, 43, 44]. Taking into account that the spatial components of matrix S(t) are the electric components of Riemann tensor, that is Sij (t)= Ri0j0 , (5.52) we can get the six polarization amplitudes pA (ez,t) in terms of the NP scalars for our non-local waves

p(l) (e ,t)=Ψ = (γ ) , 1 z 2 O k p(x) (e ,t)= Re Ψ = p(y) (e ,t)= Im Ψ = (γ ) , 2 z 3 3 z 3 O k p(+) (e ,t)= Re Ψ = ω2ǫ˜(+) (ω ) eiω1t + c.c. + (γ ) , 4 z 4 − 1 1 O k p(×) (e ,t)= Im Ψ = ω2˜ǫ(×) (ω ) eiω1t + c.c. + (γ ) , 5 z 4 1 1 O k k2 p(b) (e ,t)=Φ = z A (k ) eikz t + (γ )+ c.c. . (5.53) 6 z 22 8 z O k

(x) (y) It is clear, from Eqs.(5.53), that the two vector modes p2 , and p3 , together with the longitudinal (l) scalar mode p1 , are suppressed at the ﬁrst order in γk, while the two standard plus and cross (+) (×) transverse tensor polarization modes, p4 and p5 of frequency ω1, survive together with the (b) transverse breathing scalar mode p6 of frequency ω2 at zero order in γk. −1 In summary, the gravitational radiation for non-local gravity R + a1R R shows three polarizations: (+) and ( ) massless 2-spin transverse tensor modes of frequency ω1, governed by two ×(+) (+) degrees of freedom ǫ (ω1) and ǫ (ω2) and a massive 0-spin transverse scalar mode of frequency

18 Constraints Order Frequency Polarization Type d.o.f. Modes Helicity Mass Petrov Class

1+ a1φ0 − λ0 =6 6a1 2th ω1 2, transverse tensor 2 (+), (×) 2 0

1+ a1φ0 − λ0 = 6a1 2th ω1 3, transverse tensor 3 (+), (×), (b) 2 0 ω2 scalar N3 0 M

Table 1: Polarizations and modes for gravitational waves in a theory of gravity with non-local corrections.

ω2, namely the breathing mode, governed by one d.o.f. A(kz). The main results of this paper are summarized in the table below It is worth noticing that the same approach can be used also for higher-order theories of gravity both in metric and in teleparallel formalisms. For details see [45, 46].

6 Conclusions

The main result of this paper is the presence of a massive scalar gravitational mode in addition −1 to the standard massless tensor ones in a non-local gravity theory of the form R + a1R R. This model can be considered as a straightforward extension of General Relativity where a non-local correction is taken into account. In this sense, it can be considered as an Extended Theory of Gravity where the Einstein theory is a particular case [47, 48, 49]. The further scalar mode is achieved in the limit of plane waves, that is, the observer is assumed far from the wave source. It exhibits only a transverse polarization and not a mixed one because the longitudinal polarization is suppressed if we retain only the lowest order terms in the small parameters γ and γk. Such parameters take into account deviations of the waves from exactly massless ones propagating at the light speed. In addition, this further massive transverse scalar mode has helicity 2 and it is governed by one degree of freedom A(kz). Via the NP formalism, we have found that the gravitational waves belong to the N3, E(2) classes of Petrov classification. These results have been obtained using both the geodesic deviation and the NP formalism. The approach can be easily extended to models containing more general terms like R−kR which appear as effective non-local quantum corrections. Beside the renormalization and regularization of gravitational field at ultraviolet scales [17, 18], these models can be ghost-free [11] and their infrared counterparts can be interesting at astrophysical [50] and cosmological scales [19] to address the dark side issues. Finally, detecting further modes as the scalar massive one derived here is a major signature to break the degeneracy of modified theories of gravity which could be discriminated at fundamental level [51]. In a forthcoming paper, we will match these non-local theories with GW observations.

Acknowledgements

SC is supported by the INFN sezione di Napoli, iniziative speciﬁche MOONLIGHT2 and QGSKY.

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