Gravitational waves in conformal gravity

Physics Letters B 784 (2018) 212–216

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Physics Letters B www.elsevier.com/locate/physletb

Gravitational waves in conformal gravity Rongjia Yang a,b,∗ a b

College of Physical Science and Technology, Hebei University, Baoding 071002, China Hebei Key Lab of Optic-Electronic Information and Materials, Hebei University, Baoding 071002, China

a r t i c l e

i n f o

Article history: Received 1 May 2018 Received in revised form 25 July 2018 Accepted 3 August 2018 Available online 6 August 2018 Editor: A. Ringwald

a b s t r a c t We consider the gravitational radiation in conformal gravity theory. We perturb the metric from flat Mikowski space and obtain the wave equation after introducing the appropriate transformation for perturbation. We derive the effective energy-momentum tensor for the gravitational radiation, which can be used to determine the energy carried by gravitational waves. © 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

1. Introduction The detection of gravitational waves (GWs) by the LIGO Collaboration is a milestone in GW research and opens a new window to probe general relativity (GR) and astrophysics [1–4]. Future space-borne detectors will offer access to an unprecedented signal sensitivity [5], thus it is worthwhile to explore GWs in alternative theories of gravity. Gravitational wave were considered in f ( R ) theories [6–17], in scalar-tensor theories [18–20], in f ( T ) theories [21] and in fourth-order gravity [22]. The evolution equation for gravitational perturbation in four dimensional spacetime in presence of a spatial extra dimension has been derived in [23]. The linear perturbation of higher-order gravities has been discussed in [24]. Following the original work by Weyl [25] (for review, see [26]), conformal gravity (CG), as a possible candidate alternative to GR, attracts much attention. It can give rise to an accelerated expansion [27]. It was tested with astrophysical observations and had been confirmed that it does not suffer from an age problem [28]. It can describe the rotation curves of galaxies without dark matter [29]. Cosmological perturbations in CG were investigated in [30,31]. The particle content of linearized conformal gravity was considered in [32]. It had been shown that CG accommodates well with currently available SNIa and GRB samples [33–35]. A series of dynamical solutions in CG were found in [36]. Mass decomposition of the lens galaxies of the Sloan Lens Advanced Camera for Surveys in CG was discussed in [37]. Recently it was indicated that conformal gravity can potentially test well against all astrophysical observations to date [38]. It has been shown that CG can also give rise to an inflationary phase [39]. The holographic two-point

*

Correspondence to: College of Physical Science and Technology, Hebei University, Baoding 071002, China. E-mail address: [email protected].

functions of four dimensional conformal gravity was computed in [40]. It was shown that four dimensional conformal gravity plus a simple Neumann boundary condition can be used to get the semiclassical (or tree level) wavefunction of the universe of four dimensional asymptotically de-Sitter or Euclidean anti-de Sitter spacetimes [41]. A simple derivation of the equivalence between Einstein and Conformal Gravity (CG) with Neumann boundary conditions was provided in [42]. It was argued that Weyl action should be added to the Einstein–Hilbert action [43]. CG is also confronted with some challenges. It has been shown that CG does not agree with the predictions of general relativity in the limit of weak fields and slow motions, and it is therefore ruled out by Solar System observations [44]. It suggested that without dark matter CG can not explain the properties of X-ray galaxy clusters [45]. It is not able to describe the phenomenology of gravitational lensing [46]. The cosmological models derived from CG are not likely to reproduce the observational properties of our Universe [47]. In this paper, we will consider gravitational radiation in CG. We aim to find the equations of gravitational radiation and the energy-momentum tensor of the GWs. These results will be valuable for future observations of GWs to test gravity theories alternative to GR. This paper is organised as follows. We begin with a review of the CG theory. In Section 3, we consider GWs in CG. In Section 4, we will discuss the energy-momentum tensor of the GWs. Finally, we will briefly summarize and discuss our results. 2. Basic equations for conformal gravity Besides of the general coordinate invariance and equivalence principle structure of general relativity, CG possesses an additional local conformal symmetry in which the action is invariant under

https://doi.org/10.1016/j.physletb.2018.08.002 0370-2693/© 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

R. Yang / Physics Letters B 784 (2018) 212–216

local conformal transformations on the metric: g μν → e 2α (x) g μν . √ This symmetry forbids the presence of any  − gd4 x term in the action, so CG does not suffer from the cosmological constant problem [48]. Under such a symmetry, the action of CG in vacuum is given by



√

C μνκ λ C μνκ λ − gd4 x

I = −αg

  = −αg

(1)



  = −2αg



where |hμν | ∼  which is a small parameter. We will consider terms up to O ( ). Thus the inverse metric is g μν = ημν − hμν where the indices are raised by used the Minkowski metric. To the first-order, the covariant derivative of any perturbed quantity will be the same as the partial derivative, so the connection and the Riemann tensor are, respectively, given by

1 ρλ η (∂μ hν λ + ∂ν hμλ − ∂λ hμν ), 2 1 (1)λ R μνρ = (∂μ ∂ρ hλν + ∂ λ ∂ν hμρ − ∂μ ∂ν hλρ − ∂ λ ∂ρ hμν ). 2 (1 )ρ

μν =



√ 1 R μνα β R μνα β − 2R μν R μν + R 2 − gd4 x 3

√ 1 R μν R μν − R 2 − gd4 x,

213

(5) (6)

Contracting the Riemann tensor gives the Ricci tensor

3

(1 )

μν =

1

(2hμν + ∂μ ∂ν h − ∂μ ∂λ hλν − ∂ν ∂λ hλμ ),

where C μνκ λ the Weyl tensor and αg is a dimensionless coupling constant, unlike general relativity. To obtain the last equation, we have taken into account the fact that the Gauss–Bonnet term, R μνα β R μνα β − 4R μν R μν + R 2 , is a total derivative in fourdimension spacetime. Here we take the signature of the metric as (−, +, +, +) and the Rieman tensor is defined as R λμνκ =

where the d’Alembertian operator is 2 = ημν ∂μ ∂ν . Contracting the Ricci tensor gives the first-order Ricci scalar

∂κ μν − ∂ν μκ + αμν λακ − αμκ λαν from which the Ricci tensor is obtained R μν = g α β R αμβ ν , and the Ricci scalar is defined as R = g μν R μν . If to include the matter in the action, the

Inserting Eqs. (7) and (8) into (3) and retaining terms to the firstorder, we obtain

λ

λ

energy-momentum tensor of matter should also be Weyl invariant. Due to the conformal invariance one can choose a gauge in which the scalar field is constant. Then it is obvious that the Ricci scalar term belongs to the gravity part of the theory and cannot be discarded in the vacuum case. The only way this can be achieved is by choosing the vacuum expectation value of the scalar as zero. But this means that the fermion mass term also vanishes, which means that one can consider only massless matter with this theory. Hence, one should either say that one only considers massless matter or one has to take the Ricci term into account (for details see [49]) in CG. The most general local matter action for a generic scalar and spinor field coupled conformally to gravity has been proposed in [49]. Here we focus on the vacuum case corresponding to conformal gravity with massless matter. Variation with respect to the metric generates the field equations

4αg W μν = 0,

(2)

where

2 ;λ ;λ ;λ ;λ g μν R ;λ + R ;μ;ν + R μν ;λ − R λν ;μ − R λμ;ν 6 3 2 1 1 + R R μν − 2R μλ R νλ + gμν R λκ R λκ − gμν R 2 . (3) 3 2 6

W μν = −

1

Since W μν is obtained from an action that is both conformal invariant and general coordinate invariant, it is traceless and μ kinematically covariantly conserved: W μ ≡ g μν W μν = 0 and μν W ;ν = 0.

2

R (1) = 2h − ∂μ ∂ν hμν .

W

(1 )

μν = −

1 6

(7)

(8)

2

1) (1 ) (1) ,λ (1) ,λ ημν 2 R (1) + R ,(μν + 2 R μν − R λν ,μ − R λμ,ν .

3

(9) In general relativity, if we define the trace-reversed perturbation h¯ μν = hμν − 12 ημν h and impose the Lorenz gauge ∂ μ h¯ μν = 0, the linearized vacuum Einstein field equations reduce to the wave equation

2h¯ μν = 0.

(10)

We can apply this similar standard reasoning within the CG framework and find a quantity h¯ μν that satisfies a wave equation when linearizing the field equations (3). The rank-two tensors in lin(1) earized conformal gravity are: hμν , ημν , R μν , and ∂μ ∂ν . In order (1)

to eliminate R μν , we will try the simper combination ημν R (1) . The linearized field equations (3) is forth-order, we hope to get secondorder wave equations which can be easily solved, so we look for a solution with the following form

h¯ μν = 2hμν + αημν 2h + β ημν R (1) , where

(11)

α and β are constants. Taking the trace of Eq. (11) yields

h¯ = (4α + 1)2h + 4β R (1) .

(12)

So we can eliminate hμν in terms of h¯ μν to give

2hμν = h¯ μν −

α 4α + 1

ημν h¯ −

β 4α + 1

ημν R (1) ,

(13)

and

3. Gravitational waves in conformal gravity Here we are interested in vacuum GWs of CG. Recently the GWs of CG with matter are discussed in [50], however, the results in vacuum case cannot be derived simply from the non-vacuum case by letting the energy-momentum tensor as zero. The linearized framework provides a natural way to study gravitational waves, which is a weak-field approximation that assumes small deviations from a flat background

g μν = ημν + hμν ,

R

(4)

2h =

1 4α + 1

h¯ −

4β 4α + 1

R (1 ) .

(14) (1)

Inserting Eqs. (13) and (14) into 2 R μν yields (1 )

2 R μν =

1



2

+

2h¯ μν −

2α + 1 4α + 1

αημν 4α + 1

∂μ ∂ν h¯ −

2h¯ − ∂μ ∂ λ h¯ λν − ∂ν ∂ λ h¯ λμ 2β 4α + 1

∂μ ∂ν R

(1 )

 β ημν (1 ) . − 2R 4α + 1 (15)

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R. Yang / Physics Letters B 784 (2018) 212–216

h¯ = −22∂σ ξ σ , we then have h¯  = 0. We may impose further constrains upon qμν : q0ν = 0, and can define

(1)

Using Eq. (15), W μν can be rewritten as (1 )

W μν =

1



2h¯ μν −

αημν

2h¯ − ∂μ ∂ λ h¯ λν 4α + 1 2α + 1 − ∂ν ∂ λ h¯ λμ + ∂μ ∂ν h¯ 4α + 1  3β + 4α + 1  − 6∂μ ∂ν R (1) + ημν 2 R (1) , 3(4α + 1)

2

(1) ,λ

⎡

0 0   ⎢ 0 q+ ⎢ qμν = ⎣ 0 q× 0 0 (16)

(1) ,λ

where we have used R λν ,μ = R λμ,ν = 12 ∂μ ∂ν R (1) . The third and fourth term can be eliminated if Lorentz gauge conditions are taking. This can be done via coordinate transformation. Considering an infinitesimal coordinate transformation, xμ → x μ = xμ + ξ μ , we get

hμν = hμν − ∂μ ξν − ∂ν ξμ ,

0 q× −q+ 0

⎤

0 0⎥ ⎥, 0⎦ 0

(27)

where q+ and q× are constants representing the amplitudes of the two transverse polarizations of gravitational radiation. Here we note that equations (26) and (27) may mislead one to conclude that there are only two independent components. The degree of freedom of the linearized conformal gravity was discussed in detail in [32], which was consistent with the results obtained by using the canonical analysis [51]. In [32], Riegert introduced h¯ μν = hμν − 14 ημν h and obtain fourth-order wave equations 22 h¯ μν = 0

gauge conditions, the third and fourth term in (16) are zero. To obtain the wave equation, the fifth and the last term should also be set to zero. These two terms can not be eliminated by choosing gauge conditions. We can, however, choose freely the parameters α and β in the equation (11) to eliminate these two terms, and this can be done by setting the coefficients of these two terms to zero. It is easily to find when taking α = − 12 and β = 13 , the fifth and the last term can be eliminated. Then we obtain the first-order linear vacuum equations (2) as

= 0. Riegert under conformal gauge conditions V μ = 13 h¯ α β,μ has shown that the plane wave solutions of linearized conformal gravity propagate six physical degrees of freedom, corresponding to massless spin-2 and spin-1 ordinary particles and a massless spin-2 ghost particle. Here we introduce the variable of field (11) and the Lorentz gauge conditions ∂ μ h¯ μν = 0 and obtain 2h¯ μν = 0 which is second-order equations differentiating from the fourthorder wave equations obtained in [32], this is because the variable of field (11) introduced here is proportional to the second derivative of the metric perturbation. Though if we insert h¯ μν (as functions of hμν ) into the wave equations, both method lead to the same equations 22 hμν − 14 ημν 22 h = 0 but with gauge conditions ∂μ ∂ α ∂ β hα β = 32∂ α hαμ in [32] and ∂μ ∂ α ∂ β hα β = 2∂ α hαμ here, it is hμν , not the second derivative of the metric, that is the actual perturbation. To obtain the solution for the original metric perturbation from solution (26) for the new metric perturbation, one should assume the form of the solution for the original metric perturbation as done in [32], then insert it into the equation (11) for h¯ μν , comparing with the solution (26), one can obtain the solution for the original metric perturbation. The particle content of linearized conformal gravity should be investigated in the fourth-order wave equations, see [32,50] for details.

1

4. Energy-momentum tensor of the gravitational waves

(17)

h  = h − 2∂ σ ξ σ , h¯ 



(18) 

μν = 2hμν + αημν 2h + β ημν R

 (1)

= h¯ μν − 2(∂μ ξν + ∂ν ξμ ) + ημν 2∂σ ξ σ , h¯  = h¯ + 22∂σ ξ σ ,

(19) (20)

where the index was raised or lowered by using the Minkowski metric ημν . If choosing ξμ so that it satisfies the equation

22 ξν = ∂ μ h¯ μν ,

(21)

then we have the Lorentz gauge condition ∂ μ h¯ μν = 0. Using this

2

1 2h¯ μν − ημν 2h¯ = 0. 4

(22)

Taking trace and yields

1 2

2h¯ − 2h¯ = 0.

(23)

This equation leads to

2h¯ = 0.

(24)

Combining Eqs. (22) and (24), we finally obtain

2h¯ μν = 0,

(25)

which seemingly has the same form of the GWs has in general relativity, they are different from each other in fact: with different h¯ μν and with different energy-momentum tensor as shown below. The solution takes the form

h¯ μν = qμν exp (ikλ xλ ),

(26)

where kλ is a four-wavevector and satisfies ημν kμ kν = 0 and kμ qμν = 0. For a wave travelling along the z-axis, kμ = ω(1, 0, 0, 1) with ω the angular frequency. The Lorentz gauge condition can not fix the gauge freedom completely, it leaves a residual coordinate transformation with 22 ξν = 0. If ξμ also satisfies the equation

,α ,β

Physically, we could expect the gravitational field to carry energy-momentum, but as it is well known that it is difficult to define an energy-momentum tensor for a gravitational field. Nevertheless, one can regard the linearised theory as describing a simple rank-2 tensor field hμν in Cartesian inertial coordinates propagating in a fixed Minkowski spacetime background, and then can assign an energy-momentum tensor to this field in Minkowski spacetime. As was discussed above, the linearised gravitational theory ignores the energy-momentum associated with the gravitational field itself. To include this contribution, and thereby go beyond the linearised theory, we must modify the linearised field equations to read (1 )

W μν =

1 4αg

t μν ,

(28)

where t μν is the energy-momentum tensor of the gravitational field itself. Rearranging this equation gives (1 )

W μν −

1 4αg

t μν = 0.

(29)

Returning to the equations (3), we can expand beyond first order to obtain

R. Yang / Physics Letters B 784 (2018) 212–216

(1 )

(2 )

W μν ≡ W μν + W μν + ... = 0,

(30)

where indexes (i ) indicate the order of the expansion in hμν . To a good approximation (up to O ( 3 )), we should make the identification (2 )

t μν = −4αg W μν .

(31)

Since the energy-momentum of a gravitational field at a point in spacetime has no real meaning, this suggests that in order to probe (2) the physical curvature of the spacetime one should average W μν over a small region at each point in spacetime, which gives a gauge-invariant measure of the gravitational field strength of the GWs themselves. The average is carried out over a region of length scale d with λ  d  L, where λ is the GW wavelength and L is the background radius. The average of a tensor is defined as [52, 53]





     4  jα μ (x, x ) j ν (x, x ) A α β (x ) f (x, x ) − g¯ (x )d x , β

A μν =

(32)

where · · · denotes the averaging process, g¯ μν is the metric of the background, j α μ is the bivector of geodesic parallel displacement, and f (x, x ) is a weight function that  falls quicklyand smoothly to zero when | x − x |> d, such that allspace f (x, x ) − g¯ (x )d4 x = 1. Therefore we should replace Eq. (31) by



(2 ) 

t μν = −4αg W μν .

(33)

To calculate the energy-momentum of GWs, we must expand each team in W μν up to the second order in hμν . The second-order Ricci tensor is (2 )

(2 )σ

R μν = ∂ν

(1 )ρ

(2 )σ

μσ − ∂σ

μν +

(1 )ρ

(1 )σ

μσ

ρν −

(1 )σ

μν

ρσ ,

(34) where

(1)σ

μν is presented in equation (6) and

215

( g μν R 2 )(2) = ημν R (1)2 . (1)

Inserting the R (1) , R μν , and

(43) (1)λ

μν into these equations, and using

the wave equations (25), the vanishing trace h¯ = 0, and the Lorentz gauge ∂ μ h¯ μν = 0, we obtain after a rather cumbersome calculation



(2 ) 

t μν = −4αg W μν  1 3 = −4αg ∂μ ∂ν hα β ∂α ∂ λ hλβ + 2h∂μ ∂ν h 2 32  1 − ∂μ ∂ν hα β 2hα β 2

 ∂μ ∂ν ¯ α β  = 2αg h¯ α β h , 2

(44)

(45) (46)

where 1/2 = 1/k2 in momentum space. It is obvious that this energy-momentum tensor is symmetric. Using the vanishing trace, (1) (1) the Lorentz gauge, and W μν = 0 (implying that 2 R μν = 1 6

ημν 2 R (1) + 13 ∂μ ∂ν R (1) ), it can also been shown that this energy-

momentum tensor is traceless. Because of the operator 1/2, the energy carried by GWs in vacuum case diverges in momentum space, this due to the massless ghost contained in CG. 5. Conclusions and discussions We have discussed the gravitational radiation in conformal gravity theory. We have linearized the field equations and obtain the second-order wave equation after introducing the suitable transformation for hμν . We have also derived the effective energymomentum tensor for the gravitational radiation. Unfortunately, however, the energy carried by GWs in vacuum case diverge in momentum space. The methods presented here can be applied to investigate other alternative gravity theories. Acknowledgements

(2)σ

μν is given by

1 (2)σμν = − hσ λ (∂μ hν λ + ∂ν hμλ − ∂λ hμν ) 2

(35)

Since we average over all directions at each point, first derivatives average to zero: ∂μ f = 0 for any function of position f . This has the important consequence that ∂μ f g = − ∂μ g f [54,

We thank Doctors Yungui Gong, Liu Zhao, and Bin Hu for useful discussions. This study is supported in part by National Natural Science Foundation of China (Grant Nos. 11273010 and 11147028), Hebei Provincial Natural Science Foundation of China (Grant No. A2014201068), the Outstanding Youth Fund of Hebei University (No. 2012JQ02), and the Midwest universities comprehensive strength promotion project.

(2)

55]. Repeated application of this, we obtain the terms in W μν as follows, respectively (1)λ

( g μν ∇ α ∇α R )(2) = ημν ηα β ∂λ

αβ R

+ 2ημν ∂ α (1)λαρ R

(1 )

(1 )ρ λ

− ημν hα β ∂α β R (1)

(1 )

− ημν hρσ 2 R ρσ + hμν 2 R (1) ,

(36)

(1)λ (1 ) (1)λ (1 ) (∇ α ∇α R μν )(2) = ηα β ∂λ α β R μν + ∂ α αμ R λν 1) + ∂ α (1)λαν R λ(1μ) − hα β ∂α β R (μν

,

(37)

(∇ α ∇μ R να )(2) = −hα β ∂α ∂μ R (ν1β) + ∂λ (1)λαμ R (1)να (1 )α (1)λ (1 ) αβ αν R λ + η ∂μ α β R ν λ , (38)

(1)λ

+ ∂μ

(∇ν ∇μ R )(2) = −hα β ∂μ ∂ν R (α1β) + ∂λ (1)λμν R (1) (1 )σ νσ R λ ,

(1)λ

+ 2∂ μ

(1 )

(39)

(1 ) ( R μα R να )(2) = ηα β R μα R ν β ,

(40)

1) ( g μν R α β R α β )(2) = ημν ηαρ ηβ σ R (α1β) R (ρσ

,

(41)

( R R μν )(2) = R

(1 ) (1 )

R μν ,

(42)

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