Gravitational Waves in Massive Conformal Gravity

Eur. Phys. J. C (2020) 80:645 https://doi.org/10.1140/epjc/s10052-020-8224-z

Regular Article - Theoretical Physics

Gravitational waves in massive conformal gravity

F. F. Faria a Centro de Ciências da Natureza, Universidade Estadual do Piauí, Teresina, PI 64002-150, Brazil

Received: 20 May 2020 / Accepted: 8 July 2020 / Published online: 20 July 2020 © The Author(s) 2020

Abstract First, we obtain the plane wave solution of the lin- 2 Massive conformal gravity earized massive conformal gravity ﬁeld equations. It is shown that the theory has seven physical plane waves. In addition, Let us consider the total MCG action1 [8] we investigate the gravitational radiation from binary sys- √ 1 4 2 μ tems in massive conformal gravity. We ﬁnd that the theory S = d x −g ϕ R + 6∂ ϕ∂μϕ tot κ2 with large graviton mass can reproduce the orbit of binaries by the emission of gravitational waves. 1 αβμν 4 − C Cαβμν + d xLm, (1) 2m2

κ2 = π / ϕ 1 Introduction where 32 G 3, is a scalar ﬁeld called dilaton, m is a constant with dimension of mass,

Over the years several alternative theories of gravity have αβμν αβμν μν 2 C Cαβμν = R Rαβμν − 4R Rμν + R emerged in the attempt to solve some of the problems pre- μν 1 2 sented by the general theory of relativity, such as the dark +2 R Rμν − R (2) matter and dark energy problems. Besides solving these prob- 3 lems, for an alternative theory of gravity to be considered α α is the Weyl tensor squared, R μβν = ∂β Γμν + ··· is the consistent, it must also reproduce the successful predictions α Riemann tensor, Rμν = R μαν is the Ricci tensor, R = of general relativity. One of these recently confirmed predic- μν g Rμν is the scalar curvature, and L = L (gμν,Ψ) is tions is the existence of gravitational waves [1–4]. m m the Lagrangian density of the matter field Ψ . Among the many alternative theories of gravity that have The variation of the total action (1) with respect to gμν already studied the gravitational waves phenomenology is and ϕ gives the MCG field equations conformal gravity (CG). It was shown that the plane wave of this theory is composed of the usual plane wave of general 2 ρ ρ 2 ϕ Gμν + 6∂μϕ∂νϕ − 3gμν∂ ϕ∂ρϕ + gμν∇ ∇ρϕ relativity plus a plane wave that grows linearly in time [5], 2 −2 1 2 which causes the energy carried by the CG plane wave to −∇μ∇νϕ − m Wμν = κ Tμν, (3) 2 diverges in momentum space [6]. ρ 1 In this paper, we intend to study the behavior of gravita- ∇ ∇ρ − R ϕ = 0, (4) tional waves in another alternative theory of gravity with con- 6 formal symmetry called massive conformal gravity (MCG) where [7]. In Sect. 2, we present an introduction of the MCG theory. ρ 1 1 ρ In Sect. 3, we find the plane wave solution of the linearized Wμν =∇ ∇ρ Rμν − ∇μ∇ν R − gμν∇ ∇ρ R MCG field equations. In Sect. 4, we discuss the energy– 3 6 ρσ 1 ρσ momentum tensor of the MCG plane wave. In Sect. 5,we +2R Rμρνσ − gμν R Rρσ evaluate the radiated energy from a binary system in MCG. 2 2 1 2 Finally, in Sect. 6, we provide a brief conclusion about the − RRμν + gμν R (5) results found in the paper. 3 6 a e-mail: [email protected] (corresponding author) 1 Here we consider units in which c = h¯ = 1. 123 645 Page 2 of 7 Eur. Phys. J. C (2020) 80 :645 is the Bach tensor, as we can see by considering the variation of the modified L ϕ 1 m with respect to on the right side of (4) and comparing Gμν = Rμν − gμν R (6) the resulting field equation with the trace of (3). 2 The variation of (11) with respect to ψ and ψ gives the is the Einstein tensor, and field equations δL 2 m μ Tμν = √ γ ψ + μ ψ = , − δ μν (7) i Dμ S 0 (12) g g μ iDμψγ − μSψ = 0. (13) is the matter energy–momentum tensor. Besides being invariant under coordinate transformations, Substituting (11)into(7), and using (12) and (13), we obtain the field equations (3) and (4) are also invariant under the the matter energy–momentum tensor conformal transformations ρ Tμν = 2gμν∇ S∇ρ S − 8∇μ S∇ν S + 4S∇μ∇ν S −Δ Φ˜ = Ω(x) Φ Φ, (8) ρ 2 f 4 −4gμν S∇ ∇ρ S + 2S Gμν + Tμν − gμνλS , (14) Ω( ) where x is an arbitrary function of the spacetime coordi- where Δ Φ nates, and Φ is the scaling dimension of the field , whose − f i values are 2 for the metric field, 0 for gauge bosons, 1 Tμν = ψγμ Dνψ − Dνψγμψ + ψγν Dμψ − Dμψγνψ for scalar fields, and 3/2 for fermions. Using the conformal 4 (15) invariance of the theory, we can impose the unitary gauge ϕ = ϕ = 0 1. In this case, the field equations (3) and (4) is the fermion energy–momentum tensor. becomes Considering that, at scales below the electroweak scale, −2 1 2 the Higgs field acquires a spontaneously broken constant Gμν − m Wμν = κ Tμν, (9) 2 vacumm expectation value S0, we find that (14) reduces to R = 0. (10) = 2 + f − λ 4. Tμν 2S0 Gμν Tμν gμν S0 (16) Taking the trace of (9) and comparing with (10), we find that MCG couples only with matter whose energy– Taking the trace of (16) and substituting into the trace of (9), momentum tensor is traceless, which is a feature of con- we obtain formally invariant theories. This is the case of the standard 1 − R = −2S2 R + T f − 4λS4 , (17) model of particle physics without a Higgs mass term, for 2 0 0 instance. For simplicity, we consider the conformally invari- μν f where T f = g Tμν. The additional use of (10) then gives ant matter Lagrangian density [9] the relation √ 2 μ 4 1 Lm =− −g S R + 6∂ S∂μ S + λS λ 4 = f . S0 T (18) 4 i μ μ + ψγ Dμψ − Dμψγ ψ + μSψψ , (11) Finally, substituting this relation back into (16), we arrive at 2 2 T Tμν = 2S Gμν + Tμν, (19) where S is a scalar Higgs field, λ and μ are dimensionless 0 coupling constants, ψ = ψ†γ 0 is the adjoint fermion field, where ν ν λ μ Dμ = ∂μ +[γ ,∂μγν]/8 −[γ ,γλ]Γμν/8, and γ are the T f 1 f Tμν = Tμν − gμν T (20) general relativistic Dirac matrices, which satisfy the anticom- 4 mutation relation {γ μ,γν}=2gμν. is the traceless part of the fermion energy–momentum tensor. Before proceeding, it is worth noting that both the coordinate and the conformal symmetries of the theory allow the introduction of a quartic self-interaction term of the dilaton field in the gravitational part of the total MCG action (1). The 3 Plane gravitational waves reason why we do not consider such a term is that its inclu- sion makes the flat metric no longer a solution of the vacuum In order to find the MCG gravitational wave equations, we field equations, which invalidates the usual S-matrix formu- must perturb the metric according to lation. Additionally, we can include a coupling between the gμν = ημν + hμν, (21) dilaton and the Higgs fields in the matter Lagrangian density (11). However, we neglect this coupling because it leads to a where ημν = diag(−1, +1, +1, +1) and |hμν|1. Then, nonvanishing trace of the matter energy–momentum tensor, using (19), we find that up to first order in the perturbation 123 Eur. Phys. J. C (2020) 80 :645 Page 3 of 7 645 hμν, the field equations (9) and (10) become2 By substituting (33)into(23) and (30), and using (34), we obtain (1) − −2 (1) = 1κ2 (0), Gμν m Wμν Tμν (22) μ 1 μ 2 k aμν = kνa, q bμν = 0, b = 0, (35) ( ) R 1 = 0, (23) 2 μν μν where a = η aμν and b = η bμν. where The symmetry of the polarization tensors aμν and bμν

(0) 2 (0) T (0) f (0) 1 f (0) means that each of them has ten independent components. Tμν = 2S Gμν + Tμν = Tμν − ημν T (24) 0 4 The conditions (35) reduce the independent components of aμν to six and of bμν to five. In addition, we can choose a is the zero-order matter energy–momentum tensor, solution of (31) to impose four more conditions on aμν.For (1) (1) 1 (1) 1 (1) instance, choosing Wμν = Rμν − ∂μ∂ν R − ημνR (25) 3 6 ρ ξμ = μ sin(kρ x ), (36) is the first-order Bach tensor, and substituting into (29) together with (33), we arrive at (1) (1) 1 (1) Gμν = Rμν − ημν R (26) 2 aμν = aμν + kμν + kνμ. (37) is the first-order Einstein tensor, Since μ is arbitrary, we can select it to impose four more (1) 1 ρ ρ conditions on aμν. In particular, we can choose μ such that Rμν = ∂μ∂ hρν + ∂ν∂ hρμ − hμν − ∂μ∂ν h (27) 2 ij a0i = 0,δaij = 0, (38) is the first-order Ricci tensor, and which reduce the independent components of aμν to just two. (1) μ ν R = ∂ ∂ hμν − h (28) Thus, we conclude that the MCG plane gravitational wave μ (33) has seven propagating degrees of freedom, represented is the first-order scalar curvature, with = ∂ ∂μ and h = μν by the two independent components of aμν and the five inde- η hμν. pendent components of bμν, which is consistent with recent It is not difficult to see that both (22) and (23) are invariant results obtained in the literature [8,10]. under the coordinate gauge transformation hμν = hμν + ∂μξν + ∂νξμ, (29) 4 Gravitational energy–momentum tensor where ξμ is an arbitrary spacetime dependent vector field. By imposing the gauge condition We can see from (32) that the total linearized MCG μ 1 Lagrangian density is dynamically equivalent to ∂ hμν − ∂ν h = 0, (30) 2 1 −2 μν ρ μν Ltot =− m h hμν + ∂ h ∂ρ hμν which fix the coordinate gauge freedom up to residual gauge 4κ2 parameter satisfying the subsidiary condition 1 μν (0) + h Tμν . (39) 2 ξμ = 0, (31) Inserting the gravitational part of (39) into the canonical and using (23), we find that (22) reduces to energy–momentum tensor [11] ∂L ∂L −2 2 (0) μν ν m − 1 hμν = κ Tμν . (32) t = ∂ρ − ∂ hαβ ∂(∂μ∂ρ hαβ ) ∂(∂μhαβ )

( ) 0 = ∂L ν μν The simplest physical solution to (32) in vacuum (Tμν − ∂ ∂ρ hαβ + η L , (40) 0) is the plane wave ∂(∂μ∂ρ hαβ ) ρ ρ integrating by parts, and using (32) in vacuum, we obtain hμν = aμν cos(kρ x ) + bμν cos(qρ x ), (33) 1 −2 αβ αβ where aμν and bμν are symmetric wave polarization tensors, tμν = 2m ∂μ∂ν h hαβ + ∂μh ∂ν hαβ , (41) 2κ2 and kμ and qμ are wave vectors, which satisfies where the angle brackets denote the average over a macro- ρ ρ 2 k kρ = 0, q qρ =−m . (34) scopic region. The substitution of the plane wave (33)into(41)gives 2 From now on the indexes (i) indicate the order of the expansion in 1 αβ αβ tμν = a aαβ kμkν − b bαβ qμqν , (42) hμν . 2κ2 123 645 Page 4 of 7 Eur. Phys. J. C (2020) 80 :645 where we used (34). We can see from (42) that the ener- for m2 <ω2, and gies of the waves with the two physical polarizations aμν are −k |r| ˜ e m Θ(m −|ω|) positive, while those of the ones with the five physical polar- G B(ω, r) =− (54) 4π|r| izations bμν are negative. Since these waves do not interact with each other, energy cannot flow between them so that for m2 >ω2, where c.c. is the complex conjugate of the there is no violation of energy conservation. exponential√ function, Θ√is the Heaviside step function, kω = 2 2 2 2 ω − m , and km = m − ω . Substituting (52)into(48), transforming back to real 5 Gravitational waves from a binary system space, and using the far zone approximation (|x|≈|x − x |), we can write the spatial components of Aμν in the form In order to analyze the gravitational waves created from κ2 +∞ ω d −iω(t−r) 2 ˜ binary systems, we need to solve the fourth-order differen- Aij(t, r) = e ω Qij(ω), (55) (0) 8πr −∞ 2π tial equation (32) in the presence of matter (Tμν = 0). For simplicity, we can split hμν according to where r =|x| is the distance between the observer and the source, and Q˜ (ω) is the Fourier transform of the reduced = + , ij hμν Aμν Bμν (43) quadrupole moment where Aμν and Bμν obey the second-order differential equa- 3 1 2 (0) Qij(t) = d r xi x j − δijr T (t, r). (56) tions 3 00 2 (0) Aμν =−κ T , (44) Following the same steps for Bμν, but now with the substi- μν 2 2 (0) tution of (53) and (54)into(49) , we obtain − m Bμν = κ Tμν . (45) κ2 ∞ ω d −iωt ikωr Bij(t, r) =− e e In the frequency domain, the field equations (44) and (45) 8πr 2π m become −m dω − ω − + e i t e ikωr ω2 Q˜ (ω) (57) 2 2 ˜ 2 ˜ (0) π ij ∇ + ω Aμν(ω, x) =−κ Tμν (ω, x), (46) −∞ 2

2 2 2 ˜ 2 ˜ (0) for m2 <ω2, and ∇ + ω − m Bμν(ω, x) = κ T (ω, x), (47) μν κ2 m dω −iωt −kmr 2 ˜ ∇2 ω Bij(t, r) =− e e ω Qij(ω) (58) where is the Laplacian, is the frequency of the wave, and 8πr −∞ 2π the tilde denotes the Fourier transform. The general solutions to (46) and (47)aregivenby for m2 >ω2. In the case of a circular binary system formed by a pair of ˜ 2 3 ˜ ˜ (0) Aμν(ω, x) =−κ d x G A(ω, r)Tμν (ω, x ), (48) masses m1 and m2, separated by a distance d, orbiting each other in the xy-plane with frequency ωs = ω/2, we have ( ) ˜ 2 3 ˜ ˜ (0) f 0 ( , ) = μδ0 δ0δ3( ) Bμν(ω, x) = κ d x G B(ω, r)Tμν (ω, x ), (49) Tμν t x μ ν x . The substitution of this value into (24)gives ˜ (ω, ) where the frequency domain Green functions G A r and (0) 3 ˜ T (t, r) = μδ3(r), (59) G B(ω, r) are deﬁned by 00 4 √ 2 2 ˜ (3) where μ = m1m2/(m1 +m2) is the reduced mass. By insert- ∇ + ω G A(ω, r) = 8πδ (r), (50) √ ing (59) and the relative coordinates 2 2 2 ˜ (3) ∇ + ω − m G B(ω, r) = 8πδ (r), (51) x1 =−d sin(ωst), x2 = d cos(ωst), x3 = 0, (60) with r = x − x being the difference between the positions into (56), and taking the Fourier transform, we ﬁnd of the observer (x) and the source (x ). 2 ˜ 3πμd It follows from (50) that Q (ω) = [δ(ω) − δ(ω + 2ωs) − δ(ω − 2ωs)] , 11 8 iω|r| (61) ˜ (ω, ) =−e , G A r (52) 2 4π|r| ˜ 3πμd Q22(ω) = [δ(ω) + δ(ω + 2ωs) + δ(ω − 2ωs)] , and from (51) that 8 (62) ikω|r|Θ(ω − ) + Θ(−ω − ) πμ 2 ˜ (ω, ) =−e m c.c. m ˜ 3 d G B r (53) Q (ω) = [δ(ω − 2ωs) − δ(ω + 2ωs)] , (63) 4π|r| 12 8i 123 Eur. Phys. J. C (2020) 80 :645 Page 5 of 7 645 where we omitted the term proportional to δij in (56) because from (48), (49), (55), (57) and (58) that Aμν and Bμν obey it does not contribute to the radiated energy. the traceless-transverse conditions

The insertion of (61)–(63)into(55)gives i ij ∂ Aij = 0, A0i = 0,δAij = 0, (75) 2μd2ω2 ∂i = , = ,δij = . A (t, r) =−A (t, r) = s cos (2ω t ), (64) Bij 0 B0i 0 Bij 0 (76) 11 22 r s ret i = 2μd2ω2 By using the combination of (72)–(76), and n ni 1, we A (t, r) = A (t, r) = s sin (2ω t ), (65) 12 21 r s ret can write (71)as where tret = t − r is the retarded time. In the same way, 0i 1 ij kl ij kl t ni ≈ Λijkl ∂0 A ∂0 A − vm∂0 B ∂0 B (77) substituting (61)–(63)into(57) and (58), we arrive at 2κ2 2 < ω2 2μd2ω2 for m 4 s , and B (t, r) =−B (t, r) =− s cos (2ω t ), (66) 11 22 r s m 0i ≈ 1 Λ ∂ ij∂ kl − ij∂ kl μ 2ω2 t ni ijkl 0 A 0 A km B 0 B (78) 2 d s 2κ2 B12(t, r) = B21(t, r) =− sin (2ωstm), (67) r 2 > ω2 for m 4 s , where for m2 < 4ω2, and s 1 Λ = P P − P P (79) 2μd2ω2 ijkl ik jl ij kl s −kmr 2 B11(t, r) =−B22(t, r) =− e cos (2ωst), r is the Lambda tensor, with (68) = δ − 2μd2ω2 Pij ij ni n j (80) s −kmr B12(t, r) = B21(t, r) =− e sin (2ωst), (69) r being the traceless-transverse projection operator. 2 > ω2 = − v for m 4 s , where tm t mr is the travel time, with Inserting (77) and (78)into(70), and using the surface 2 2 vm = 1 − m /(4ω ) being the speed of the massive grav- integral s itational wave. 2π The rate of energy loss from a source, in the far ﬁeld limit, dΩΛijkl = 11δikδ jl − 4δijδkl + δilδ jk , (81) ∂ 15 is given by V we ﬁnd ˙ =− 2 Ω 0i , E r d t ni (70) 3r 2 ∂ ˙ ij ij V E ≈− ∂ A ∂ Aij − vm∂ B ∂ Bij (82) 40 0 0 0 0 where the dot is the derivative with respect to time, dΩ = 2 < ω2 sin θdθdφ is the differential solid angle, ∂V is the surface of for m 4 s , and a spherical shell with volume V centered around the source, 2 ˙ 3r ij ij and n are the components of the spatial unit vector pointing E ≈− ∂ A ∂ Aij − km B ∂ Bij (83) i 40 0 0 0 from the source to the observer. 2 > ω2 By substituting (43)into(41), integrating by parts, and for m 4 s . using (44) and (45) in vacuum, we obtain Finally, substituting (64)–(69)into(82) and (83), we arrive at 0i 1 0 αβ i 0 αβ i t ni = ni ∂ A ∂ Aαβ − ∂ B ∂ Bαβ . (71) 2κ2 3 m2 E˙ ≈ 1 − 1 − E˙ (84) ω2 GR We can see from (64) and (65) that 8 4 s

∂ Aαβ ≈−n ∂ Aαβ , (72) 2 < ω2 i i 0 for m 4 s , and and from (66)–(69) that 3 E˙ ≈ E˙ (85) 8 GR ∂i Bαβ ≈−ni vm∂0 Bαβ (73) for m2 > 4ω2, where 2 < ω2 s for m 4 s , and 32Gμ2d4ω6 ∂ ≈− E˙ =− s (86) i Bαβ ni km Bαβ (74) GR 5 2 > ω2 / 2 for m 4 s , where we neglected terms of order 1 r .In is the standard energy loss of general relativity. addition, considering the conservation and the traceless con- Experiments on the inverse square law of the MCG grav- (0) dition of Tμν , and the traceless condition of Qij,itfollows itational potential of a point particle with a mass M, which 123 645 Page 6 of 7 Eur. Phys. J. C (2020) 80 :645 is given by [8] Taking this into account, the theory with a large graviton − mass (m > 10 2 eV) can explain the decrease of the orbital GM −mr φ(r) =− 1 − e , (87) period of binary systems. r constrain the graviton mass to the ranges m < 10−22 eV for m2 < 4ω2 and m > 10−2 eV for m2 > 4ω2 [12]. In s s 6 Final remarks particular, considering the orbital frequency ωs ≈ 1.3 × 10−20 eV of the binary system PSR J1012 + 5307 formed Here we have shown that the MCG plane wave has seven by a neutron star and a white dwarf in quasi-circular motion − propagating degrees of freedom, two of which are massless [13–15], we have that m2/4ω2 < 10 5 for m2 < 4ω2.By s s and carry positive energies and the other five are massive and substituting this value into (84), we find carry negative energies. Despite the presence of the waves ˙ −6 ˙ E 10 EGR (88) with negative energies, the lack of interaction between them and the waves with positive energies means that the energies for m2 < 4ω2. s of all the seven MCG plane waves do not diverge. The energy loss of the binary system results in a decay of The study on the radiated energy from a binary system its orbital period P, which can be written as restricts the MCG to large graviton mass, which give rise P˙ d˙ φ˙ to a modification of general relativity only at high energies = − , (89) P 2d 2φ and small distances. Although this modification is the cause of MCG being renormalizable [18,19], it makes the theory where φ(d) = μ−1∂ U(d) is the derivative of the gravita- d unable to explain galaxy rotation curves without dark matter. tional potential φ with respect to d, with U being the gravi- However, the conformal symmetry of the matter part of MCG tational potential energy. It follows from (87) that allows us to consider that the Higgs mass is generated by the Gm1m2 − symmetry breaking of an extra scalar field, which may be a U(d) =− 1 − e md . (90) d good candidate for dark matter. Further studies are needed to Substituting this result into (89), we find figure this out. P˙ 3 |E˙| Data Availability Statement This manuscript has no associated data or ≈− (91) P 2 |E | the data will not be deposited. [Authors’ comment: This is a theoretical GR study and no experimental data has been listed.] 2 < ω2 2 > ω2 | |= for both m 4 s and m 4 s , where EGR 2 2 2 2 Open Access This article is licensed under a Creative Commons Attri- Gm1m2/2d and we considered m d 1form < 4ω . s bution 4.0 International License, which permits use, sharing, adaptation, The insertion of (88) and (85)into(91) then gives distribution and reproduction in any medium or format, as long as you ˙ ˙ give appropriate credit to the original author(s) and the source, pro- P − P 10 6 GR (92) vide a link to the Creative Commons licence, and indicate if changes P PGR were made. The images or other third party material in this article 2 < ω2 are included in the article’s Creative Commons licence, unless indi- for m 4 s , and cated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended P˙ 3 P˙ ≈ GR (93) use is not permitted by statutory regulation or exceeds the permit- P 8 PGR ted use, you will need to obtain permission directly from the copy- right holder. To view a copy of this licence, visit http://creativecomm 2 > ω2 ˙ / =−| ˙ |/ | | for m 4 s , where PGR PGR 3 EGR 2 EGR .We ons.org/licenses/by/4.0/. can see from (92) that the MCG decay of the orbital period Funded by SCOAP3. 2 < ω2 for m 4 s is several orders of magnitude smaller than in general relativity, which rules out the theory with a small < −22 graviton mass (m 10 eV). On the other hand, the dis- References crepancy seen in (93) between the MCG decay of the orbital 2 > ω2 period for m 4 s and the general relativity result disap- 1. B.P. Abbott et al., Phys. Rev. Lett. 116, 061102 (2016) 2 (1) 2. B.P. Abbott et al., Phys. Rev. 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