Astrophys Space Sci (2015) 358:27 DOI 10.1007/s10509-015-2425-1

O R I G I NA L A RT I C L E

Gravitational waves in fourth order gravity

S. Capozziello1,2,3 · A. Stabile4

Received: 3 June 2015 / Accepted: 18 June 2015© Springer Science+Business Media Dordrecht 2015

Abstract In the post-Minkowskian limit approximation,we study gravitational wave solutions for general fourth-order theories of gravity. Specifically, we consider a La-grangian with a generic function of curvature invariantsf (R,RαβRαβ ,Rαβγ δR

αβγ δ). It is well known that whendealing with General Relativity such an approach providesmassless spin-two waves as propagating degree of freedomof the gravitational field while this theory implies other ad-ditional propagating modes in the gravity spectra. We showthat, in general, fourth order gravity, besides the standardmassless graviton is characterized by two further massivemodes with a finite-distance interaction. We find out themost general gravitational wave solutions in terms of Greenfunctions in vacuum and in presence of matter sources. If anelectromagnetic source is chosen, only the modes inducedby RαβRαβ are present, otherwise, for any f (R) gravitymodel, we have the complete analogy with tensor modes ofGeneral Relativity. Polarizations and helicity states are clas-sified in the hypothesis of plane wave.

B S. Capozziellocapozziello@na.infn.it

A. Stabilearturo.stabile@gmail.com

1 Dipartimento di Fisica, Università di Napoli “Federico II”,Complesso Universitario di Monte Sant’Angelo, Edificio G,Via Cinthia, 80126, Napoli, Italy

2 Istituto Nazionale di Fisica Nucleare (INFN) Sezione di Napoli,Complesso Universitario di Monte Sant’Angelo, Edificio G,Via Cinthia, 80126, Napoli, Italy

3 Gran Sasso Science Institute (INFN), Viale F. Crispi, 7, 67100,L’Aquila, Italy

4 Dipartimento di Ingegneria, Università del Sannio, PalazzoDell’Aquila Bosco Lucarelli, Corso Garibaldi, 107, 82100,Benevento, Italy

Keywords Gravitational waves · Modified theories ofgravity · Weak field limit

1 Introduction

Identifying the correct theory of gravity is a crucial issue ofmodern physics due to the fact that General Relativity, in itsstandard formulation, presents shortcomings at ultravioletand infrared limits. For the first issue, we need a theory thatshould deal with gravity under the same standard of the otherfundamental interactions (Quantum Gravity) (Kiefer 2004;Stelle 1977). In the other case, modifications of gravity arerequired to deal with the vast phenomenology coming fromastrophysics and cosmology, generally addressed as darkmatter and dark energy issues (Peebles and Ratra 2003;Trimble 1987). This matter is rather controversial due tothe fact that the ambiguity comes out from the fact thatphenomenology could be explained successfully both con-sidering new material ingredients (dark matter particles ad-dressing the problem of structure formation and light scalarfields giving rise to the acceleration of the Hubble fluid)or modifying gravity that, at scales larger than Solar Sys-tem, could behave in different way with respect to theweak field limit related to General Relativity (Nojiri andOdintsov 2007, 2011; Capozziello and De Laurentis 2012;Stelle 1978; Schmidt 1986, 2008). Furthermore, there arevery few investigations and experimental constraints prob-ing the gravitational field in very strong regimes. Severaltimes, extrapolations of General Relativity are simply as-sumed without considering corrections and alternatives thatcould strongly affect theoretical and experimental results.

With this situation in mind, it is urgent to find out someexperimentum crucis or some test bed capable of discrimi-nating among concurring gravitational theories (Capozziello

27 Page 2 of 10 S. Capozziello, A. Stabile

et al. 2010a, 2015b; Stabile and Scelza 2011; Stabile andStabile 2012), that, in any case, should reproduce the well-founded theoretical and experimental results of General Rel-ativity. At astrophysical level, discriminations could comefrom anomalous stellar systems whose structures and pa-rameters do not find room in the constraints and limits im-posed by General Relativity. For example, extremely mas-sive neutron stars, magnetars or compact objects like quarkstars could be independent signatures for modified theoriesof gravity considered as extensions of General Relativity inthe strong field regime (Astashenok et al. 2013, 2014).

Besides, discrimination could happen in the realm ofgravitational wave physics. This sector of physics, practi-cally unexplored from the point of view of modified gravity,deserves a lot of attention since the large part of efforts hasbeen devoted to the study of gravitational radiation in therealm of General Relativity discarding the fact that modifiedgravity presents a huge amount of new phenomenology andfeatures. For example, only General Relativity strictly fore-casts massless gravitons with two polarizations. In general,modified gravity and, in particular Extended Gravity, allowsalso massive and ghost modes and then further polarizations(Bogdanos et al. 2010; Capozziello and De Laurentis 2011).Specifically, several authors refer this issue to the Fierz-Pauli linearized analysis of massive gravity that leads to theso called van Dam-Veltman-Zakharov (vDVZ) discontinu-ity. In such a case, we are in presence of the Boulware-Deserghosts. Such anomalies can be cured through the Vainshteinmechanism. A detailed review of these arguments can befound in de Rham (2014).

However, the possibility of these massive modes arestudied from a theoretical point of view but practicallyignored from the experimental point of view due to theenormous difficulties related to the detection of gravita-tional waves. However, the forthcoming experimental fa-cilities like VIRGO (after the Virgo supercluster of galax-ies) (http://www.ego-gw.it), LIGO (Laser InterferometerGravitational-Wave Observatory, http://www.ligo.org) col-laboration, LISA (Laser Interferometer Space Antenna,http://sci.esa.int/lisa/) etc. could be suitable, in principle, fordetecting these further modes.

In this paper, we propose a systematic study of gravita-tional wave solutions in theories where generic functions ofcurvature invariants are considered generalizing the first par-tial outcome in the only f (R) framework (Capozziello et al.2010b). These are a straightforward generalization of f (R)

gravity where the degrees of freedom, related to the cur-vature invariants, is considered. However, we have to saythat we are not considering !R and similar terms wherederivative of curvature invariants appear. We are also notconsidering the parity-odd Chern-Simons invariant (Alexan-der and Yunes 2009) that enters at the same order in cur-vatures and derivatives. Here, we are taking into account

only fourth order theories of gravity where derivatives ofmetric tensor gµν appear up to the fourth ones. It is inter-esting to see that relaxing the hypothesis that gravitationalinteraction is derived only from the Hilbert-Einstein action,linear in the Ricci curvature scalar R, further gravitationalmodes, polarizations and helicity states come out. This newfeatures are directly derived from the post-Minkowskianlimit of the theory and points out a new rich phenomenol-ogy that deserves investigation in view of possible fu-ture detection of gravitational waves. The classification ofthese modes, implying new massive and polarization states,is the same of Wigner’s little group E(2) (Wigner 1939;Bargmann and Wigner 1948) earlier explored in literatureby Eardley et al. (1973). Besides, f (R) theories of gravityhave been very well explored in this sense. For example, itis well known that they are equivalent to scalar-tensor the-ories of gravity following the Bergmann-Wagoner formu-lation (Bergmann 1968; Wagoner 1970) which generalizesthe Brans-Dicke theory. Furthermore, kinematics of gravi-tational waves in f (R) gravity have been extensively stud-ied by Berry and Gair which confronted gravitational radi-ation with Solar System tests (Berry and Gair 2011). It isimportant to stress that being f (R) gravity equivalent to aparticular class of scalar-tensor gravity theories, the gravita-tional wave content has already been characterized in bothrepresentations (Lang 2014). Also a comparison betweenthe two frameworks (the Einstein and Jordan frame) hasbeen analyzed in the weak field limit (Stabile et al. 2013;Capozziello et al. 2010c). Finally, gravitational wave kine-matics and dynamics in strong field regime and aroundthe Minkowski background have been investigated in detail(Stein and Yunes 2011).

This paper is organized as follows. In Sect. 2 we reportbriefly the field equations of fourth order gravity. In Sect. 3,we discuss the post-Minkowskian limit and the linearizedfield equations while, in Sect. 4, the gravitational wave so-lutions are reported. Section 5 is devoted to the discussionof all possible polarization and helicity states of the wavesolutions. Conclusions are reported in Sect. 6.

2 The field equations of fourth order gravity

The most general class of gravitational theories involvingcurvature invariants in four dimensions is given by the action

A =∫

d4x√−g

[f (X,Y,Z) + XLm

](1)

where f is an unspecified function of curvature invariantsX = R, Y = RαβRαβ , and Z = Rαβγ δR

αβγ δ . The term Lm

is the minimally coupled ordinary matter contribution. In the

Gravitational waves in fourth order gravity Page 3 of 10 27

metric approach, the field equations are obtained by vary-ing (1) with respect to gµν . We get the fourth-order differ-ential equations

fXRµν − f

2gµν − fX;µν + gµν!fX + 2fY Rµ

αRαν

− 2[fY Rα

(µ

];ν)α

+![fY Rµν] + [fY Rαβ ];αβgµν

+ 2fZRµαβγ Rναβγ − 4

[fZRµ

αβν

];αβ

= XTµν (2)

where Tµν = − 1√−g

δ(√−gLm)δgµν is the energy-momentum ten-

sor of matter, fX = ∂f∂X , fY = ∂f

∂Y , fZ = ∂f∂Z , ! = ;σ ;σ , and

X = 8πG.1 The conventions for Ricci’s tensor is Rµν =Rσ

µσν and for the Riemann tensor is Rαβµν = Γ α

βν,µ + · · · .The affinities are the usual Christoffel’s symbols of the met-ric: Γ

µαβ = 1

2gµσ (gασ,β + gβσ,α − gαβ,σ ). The adopted sig-nature is (+ − −−) (we follow the conventions by Landauand Lifšits 1970). The trace of field equations (2) is the fol-lowing

fXX + 2fY Y + 2fZZ − 2f +![3fX + fY X]+ 2

[(fY + 2fZ)Rαβ

];αβ

= XT (3)

where T = T σσ is the trace of energy-momentum tensor and

H = H σσ .

Some authors considered a linear Lagrangian contain-ing not only X, Y and Z but also the first power of cur-vature invariants !R and Rαβ ;αβ . Such a choice is justi-fied because all curvature invariants have the same dimen-sion (L−2) (Santos 2010). Furthermore, this dependence onthe two last invariants is only formal, since from the con-tracted Bianchi identity (2Rαβ ;αβ −!R = 0) we have onlyone independent invariant. In any linear theory of gravity(the function f is linear) the terms !R and Rαβ ;αβ give usno contribution to the field equations, because they are four-divergences. However if we consider a function of !R orRαβ ;αβ by varying the action, we still have four-divergencesbut we would have the contributions of sixth order differ-ential terms. As said in the Introduction, in this paper, weconsider only fourth order differential field equations. Thismeans that Action (1) is a fourth-order theory and no higher-order terms in derivatives are taken into account. However,as we will see below, one needs only two of the three curva-ture invariants, due to the Gauss-Bonnet topological invari-ant which fixes a constraint among the curvature terms.

3 The post Minkowskian limit

Any theory of gravity has to be discussed in the weak fieldlimit approximation. This prescription is needed to test if the

1Here we use the convention c = 1.

given theory is consistent with the well-established Newto-nian theory and with the Special Relativity as soon as thegravitational field is weak or is almost null. Both require-ments are fulfilled by General Relativity and then they canbe considered two possible paradigms to confront a giventheory, at least in the weak field limit, with the GeneralRelativity itself. The Newtonian limit of f (R)-gravity andf (R,RαβRαβ ,Rαβγ δR

αβγ δ)-gravity can be investigated al-ways remaining in the Jordan frame (Capozziello et al. 2007,2009; Capozziello and Stabile 2009; Stabile 2010a, 2010b;Stabile and Capozziello 2013) while a preliminary study ofthe post-Minkowskian limit for the f (R) gravity is providedin Capozziello et al. (2010b). Here we want to derive thepost-Minkowskian limit of f (R, RαβRαβ , Rαβγ δR

αβγ δ)

gravity, that is for generic fourth-order theory of gravity,with the aim to investigate the gravitational radiation.

The post-Minkowskian limit of any theory of gravityarises when the regime of small field is considered with-out any prescription on the propagation of the field. Thiscase has to be clearly distinguished with respect to the New-tonian limit which, differently, requires both the small ve-locity and the weak field approximations. Often, in litera-ture, such a distinction is not clearly remarked and severalcases of pathological analysis can be accounted. The post-Minkowskian limit of General Relativity gives rise to mass-less gravitational waves and reproduces the Special Relativ-ity. An analogous study can be pursued considering, insteadof the Hilbert-Einstein Lagrangian, linear in the Ricci scalarR, a most general function f of curvature invariants.

However, working with post-Minkowskian limit, one hasto be extremely careful with the issue whether or not thereare ghost modes that could render the theory meaningless.A standard way to deal with ghost modes is to consider anyalternative theory as an effective field theory close to Gen-eral Relativity, since it is well known that General Relativityhas no ghosts. In fact, effective field theory has a restrictedregime of validity, and the ghost modes only appear onceone leaves such a regime. See, for examples Dyda et al.(2012), Capozziello et al. (2015a) where such a problemis discussed for the Chern-Simons and f (R) theories re-spectively. Essentially, the approach consists in decouplingthe modes in the weak limit. In Stein and Yunes (2011), itwas shown that in the decoupling limit, and approachingthe asymptotically flat region far from any matter sources,a very generic class of theories with higher curvature invari-ants leads exactly to the same gravitational wave modes asin General Relativity (see for example Bamba et al. 2013 forthe case of f (T ) teleparallel gravity).

Another question is related the Cauchy problem of thetheory, and in particular with its well-formulation and well-posed initial value formulation. For f (R) gravity, thisproblem has been highly debated and discussed (Lanahan-Tremblay and Faraoni 2007; Capozziello and Vignolo

27 Page 4 of 10 S. Capozziello, A. Stabile

2009a, 2009b; Salgado 2006). In any case, consideringtheories with higher derivatives, the Cauchy problem hasto be treated with care because the related Hamiltonianscould suffer, in general, with the Ostrogradski instability(see Querella 1998 for a detailed study). In particular, theexistence of a well-posed initial value problem for higher-order theories of gravity involves the introduction of aux-iliary degrees of freedom to reduce the derivative order. Insome sense, the mechanism is the same acting in confor-mal transformations where the further degrees of freedomin the Jordan frame are disentangled in scalar fields mini-mally coupled to gravity in the Einstein frame (Capozzielloand De Laurentis 2011). This fact allows to evaluate howmany propagating dynamical degrees of freedom are presentin the theory. Alternatively, treating the theory as an effec-tive field theory can contribute to address the question of theexistence of a well-posed initial value formulation (Delsateet al. 2015).

In order to perform the post-Minkowskian limit of fieldequations, one has to perturb Eqs. (2) on the Minkowskibackground ηµν . In such a case, we obtain

gµν = ηµν + hµν (4)

with hµν small (O(hµν2) ≪ 1). Then the curvature invari-

ants X, Y , Z become

X ∼ X(1) + O(hµν

2)

Y ∼ Y (2) + O(hµν

3)

Z ∼ Z(2) + O(hµν

3)(5)

and the function f can be developed as

f (X,Y,Z) ∼ f (0) + fX(0)X(1) + 12fXX(0)X(1)2

+ fY (0)Y (2) + fZ(0)Z(2) + O(hµν

3) (6)

Analogous relations for partial derivatives of f are obtained.From lowest order of field equations (2) and (3) we have thecondition f (0) = 0, while at O(1)-order, we have2

fX(0)R(1)µν +

[fY (0) + 4fZ(0)

]!ηR

(1)µν − fX(0)

2X(1)ηµν

+[fXX(0) + fY (0)

2

]ηµν!ηX

(1) − fXX(0)X(1),µν

−[fY (0) + 4fZ(0)

]R(1)α

µ,να − fY (0)R(1)αν,µα

= X T (0)µν − fX(0)X(1)

+[3fXX(0) + 2fY (0) + 2fZ(0)

]!ηX

(1) = XT (0) (7)

2We are using the properties: 2Rαβ;αβ − !R = 0 and Rµ

αβν;αβ =

Rµα

;να −!Rµν .

where !η is the d’Alembert operator in the flat space. Tµν isfixed at zero-order in (7) since, in this perturbation scheme,the first order on Minkowski space has to be connected withthe zero order, with respect to hµν , of the standard matter en-ergy momentum tensor. This means that Tµν is independentof hµν , and satisfies the standard conservation conditionsT µν

,µ = 0. By introducing the quantities

m12 .= − fX(0)

3fXX(0) + 2fY (0) + 2fZ(0)

m22 .= fX(0)

fY (0) + 4fZ(0)

(8)

we get two differential equations for curvature invariant X(1)

and Ricci tensor R(1)µν

3

(!η + m2

2)R(1)µν −

[m1

2 − m22

3m12 ∂2µν + ηµν

(m2

2

2

+ m12 + 2m2

2

6m12 !η

)]X(1) = m2

2XT (0)µν

(!η + m1

2)X(1) = −m12XT (0)

(9)

We note that in the case of f (X)-theory we obtain onlya massive mode (with mass m1) of Ricci scalar (X). Infact if fY = fZ = 0 from the mass definition (8) m1

2 →(−3fXX(0))−1 and m2

2 → ∞ we recover the equations off (X)-gravity (Capozziello et al. 2010b)

R(1)µν − X(1)

2ηµν +

∂2µν − ηµν!η

3m12 X(1) = XT (0)µν

(!η + m1

2)X(1) = −m12XT (0)

(10)

while f (X,Y,Z)-theory we have an additional massivepropagation (with mass m2) of Ricci tensor. Finally in thecase of f → X also m1

2 → ∞ and we recover the GeneralRelativity.

A first consideration regarding the masses (8) inducedby f (X,Y,Z)-gravity is necessary at this point. The secondmass m2 is originated by the presence, in the Lagrangian, ofRicci and Riemann tensor square, but also a theory contain-ing only Ricci tensor square gives rise to the same outcome.Obviously the same is valid also with the Riemann tensorsquare alone. Then such a modification of theory enables amassive propagation of Ricci Tensor and, as it is well knownin the literature, a substitution of the Ricci scalar with anyfunction of the Ricci scalar enables a massive propagation ofRicci scalar. We can conclude that a Lagrangian containingany function of only Ricci scalar and Ricci tensor square isnot restrictive. This result is coming from the Gauss-Bonnettopological invariant GGB defined as GGB = X2 − 4Y + Z

3We set fX = 1 i.e. G → fX(0)G.

Gravitational waves in fourth order gravity Page 5 of 10 27

(de Witt 1965). In fact, by applying the variation with re-spect to the metric tensor gµν in 4 dimensions to the quantity∫

d4x√−g GGB = 0, we have

δ

∫d4x

√−gGGB

=∫

d4x√−g

(HX2

µν − 4HYµν + HZ

µν

)δgµν

=∫

d4x√−gHGB

µν δgµν (11)

where HX2

µν = 1√−g

δ(√−gX2)δgµν , HY

µν = 1√−g

δ(√−gY )δgµν and

HZµν = 1√−g

δ(√−gZ)δgµν . In four dimensions we have that HGB

µν

is identically zero. For this reason the variation of the Gauss-Bonnet invariant generates a dimension-dependent identity,that is

HX2

µν − 4HYµν + HZ

µν = 0 (12)

By substituting condition (12) in Eqs. (2), at post-Minkows-kian level, we find the same equations (9) with a redefinitionof masses (8). In the weak field limit approximation, we canconsider as Lagrangian in the action (1), the quantity (Sta-bile 2010b)

f (X,Y,Z) = aX + bX2 + cY (13)

Then the masses (8) become

m12 = − a

2(3b + c)

m22 = a

c

(14)

These masses have real values if the conditions a > 0, b < 0and 0 < c < −3b hold. This fact is extremely important inorder to establishing a no-ghost constraint on such a the-ory. For gravity theories explicitly containing the Gauss-Bonnet term in the post-Newtonian limit see De Laurentisand Lopez-Revelles (2014).

4 Gravitational wave solutions

Once developed the post-Minkowskian limit, one can searchfor gravitational wave solutions. The general solution offield equations (9) is given by

hµν = hµν + hµν (15)

where hµν is the (homogeneous) solution in the vacuum andhµν is the (particular) one in the matter. First we try to find

the solution hµν . From the second line of (9), by introduc-ing the Green function GKG,i(x, x′) of Klein-Gordon fielddefined as follows

(!η + mi

2)GKG,i

(x, x′) = δ(4)

(x − x′) (16)

with i = 1,2 and δ(4)(x − x′) is the Dirac delta function infour dimensions, we find

X(1) = −m12X

∫d4x′GKG,1

(x, x′)T (0)

(x′) (17)

where x = xµ = (t,x) = (t, x1, x2, x3). The first line of (9)can be recast as follows

(!η + m2

2)R(1)µν

= X[m2

2 T (0)µν − m1

2 + 2m22

6ηµνT

(0)

]

− (m12 − m2

2)X3

∫d4x′

[∂2µν − m1

2

2ηµν

]

× GKG,1(x, x′)T (0)

(x′) (18)

so the solution for the Ricci tensor is obtained

R(1)µν = X

∫d4x′GKG,2

(x, x′)

×[m2

2T (0)µν

(x′) − m1

2 + 2m22

6ηµνT

(0)(x′)

]

− (m12 − m2

2)X3

∫d4x′ d4x′′GKG,2

(x, x′)

×[∂2µ′ν′ − m1

2

2ηµν

]GKG,1

(x′, x′′)T (0)

(x′′) (19)

The Ricci tensor, in terms of the metric (4), is given by

R(1)µν = hσ

(µ,ν)σ − 12!ηhµν − 1

2h,µν (20)

where h = hσσ . Since we can use the harmonic gauge con-

dition gρσ Γ αρσ = 0, we set hµσ

,σ − 1/2h,µ = 0, then theRicci tensor becomes R

(1)µν = − 1

2!ηhµν . The solution ofEq. (19) is

hµν = 2(m12 − m2

2)X3

×∫

d4x′ d4x′′ d4x′′′GGR

(x, x′)GKG,2

(x′, x′′)

×[∂2µ′′ν′′ − m1

2

2ηµν

]GKG,1

(x′′, x′′′)T (0)

(x′′′)

− 2X∫

d4x′ d4x′′GGR

(x, x′)GKG,2

(x′, x′′)

27 Page 6 of 10 S. Capozziello, A. Stabile

×[m2

2T (0)µν

(x′′) − m1

2 + 2m22

6ηµνT

(0)(x′′)

](21)

where GGR(x, x′) is the Green function defined as

!ηGGR

(x, x′) = δ(4)

(x − x′) (22)

Considering the expressions of the Green functionsGGR(x, x′) and GKG,1,2(x, x′) in terms of plane waves itis possible to rewrite the solution (21) as follows

hµν(x) = X∫

d4x′[Z2(x, x′)T (0)

µν

(x′)

+(Z

(x, x′)ηµν + Zµν

(x, x′))T (0)

(x′)]

h(x) = X∫

d4x′Z1(x, x′)T (0)

(x′)

(23)

where

Zs

(x, x′) = 2ms

2(−1)1+s

∫d4k

(2π)4

ejk(x−x′)

k2(k2 − ms2)

Z(x, x′) =

∫d4k

(2π)4

(m12 + 2m2

2)k2 − 3m12m2

2

3k2(k2 − m12)(k2 − m22)

× ejk(x−x′)

Zµν

(x, x′) =

∫d4k

(2π)4

2(m12 − m2

2)kµkν

3k2(k2 − m12)(k2 − m22)

× ejk(x−x′)

(24)

k = kµ = (ω,k) = (ω, k1, k2, k3), kx = kσ xσ = ωt − k ·x, k2 = kσ kσ = ω2 − |k|2 and s = 1,2. We note thatin the case f → X (from the mass definitions (8) wehave m1,m2 → ∞) we find Zs(k) → 2(−1)sk−2, Z(k) →−k−2, Zµν(k) → 0 and the solutions (23) become those ofGeneral Relativity:

hµν(x) = −2X∫

d4x′GGR

(x, x′)S(0)

µν

(x′), (25)

where S(0)µν = T

(0)µν − ηµνT

(0)/2.Finally we can recast the propagators (24) in terms of the

Green functions GGR(x, x′) and GKG,s(x, x′) and we cansee immediately the propagation of the massless interactionand of two massive interactions. We get

Zs

(x, x′) = 2(−1)1+s

[GGR

(x, x′) − GKG,s

(x, x′)]

Z(x, x′) = GGR

(x, x′) − 1

3GKG,1

(x, x′) − 2

3GKG,2

(x, x′)

Zµν

(x, x′) = 2

3∂2µν

[m1

2 − m22

m12m22 GGR

(x, x′)

+ GKG,1(x, x′)m12 − GKG,2(x, x′)

m22

](26)

It is important to stress that the limit m → 0 could becomeproblematic for the above propagators since the theory ap-pears strongly coupled in this limit. However the source termcan be assumed decoupled as soon as m → 0. In such a casethe problem is solved. See de Rham (2014) for a discussion.

A very interesting feature is achieved when we considera traceless source with T (0) = 0. In this case, solutions(23) are unaffected by the functions of the Ricci scalar butonly by the presence of the Ricci tensor square terms intothe interaction Lagrangian. Moreover, from the second lineof (23), we have also that the gravitational waves must betraceless. A prototype of traceless source is the electromag-netic field where we have

hemµν (x) = X

∫d4x′Z2

(x, x′)T (0),em

µν

(x′) (27)

with hem(x) = 0. Finally, the solution of the first field equa-tions (9) in the vacuum (Tµν = 0) can be derived. By usingagain the hypothesis of harmonic gauge and the principle ofplane wave superposition, we get

hµν = m12 − m2

2

3m22

∫d4x′Z2

(x, x′)

[∂2µ′ν′

m12 − ηµν

2

]

× X(1)(hs)

(x′) − 2

∫d4x′GGR

(x, x′)R(1)

µν(hs)

(x′)

+ hµν(hs) (28)

where X(1)(hs), R

(1)µν(hs), hµν(hs) are respectively the homoge-

neous solutions of the Klein-Gordon equation for the Ricciscalar and tensor and the solution of wave equation for themetric. The homogeneous solutions are chosen in such away that they satisfy the boundary conditions and the gaugeharmonic condition hµρ

,ρ −1/2h ,µ = 0. Also here, the limitm → 0 becomes problematic since it is strongly coupled. Asabove, the remedy is that the sources decouples in this limit(de Rham 2014).

An important remark is necessary at this point. In theabove calculations, we are using the harmonic gauge con-dition that allows to transform admissible field configura-tions if one considers sufficiently small regions of space-time. However, such a condition has to work over more thanone gravitational wavelengths and the effects of backgroundcurvature on wave propagation have to be considered. A de-tailed discussion on harmonic gauge condition, shortwaveapproximation and propagation on curved background canbe found in Misner et al. (1971). Here we adopted the sameapproach (see § 35.13, § 35.14 therein).

Below we will consider polarizations and helicity statesin vacuum starting from this result.

Gravitational waves in fourth order gravity Page 7 of 10 27

Table 1 Classification of solutions in the vacuum of field equa-tions (29). Specifically, the case A is one of General Relativity; thecase B is massive and pure trace; the case C is massive, trace-free andtransverse and the case D is massless where the trace-reverse of h istransverse

Case Choices Gauge condition

A k2 = 0any hµν with h = 0

hµσ kσ = 0

B k2 = m12

hµν =[ kµkν

m12 + ηµν

2

] h3

with h = 0

Verified

C k2 = m22

any hµν with h = 0hµσ kσ = 0

D k2 = 0any hµν with h = 0

hµσ kσ − 1/2hkµ = 0

5 Polarizations and helicity states in vacuum

We can analyze the propagation in vacuum by performingthe Fourier analysis of field equations (9). In the Fourierspace, we have

k2{(

m22 − k2)hµν −

[m2

2 − m12

3m12 kµkν

+ ηµν

(m2

2

2− m1

2 + 2m22

6m12 k2)]

h

}= 0

k2(m12 − k2)h = 0

(29)

where hµν , h are the Fourier representation of hµν , h. Thegauge condition now becomes hµσ kσ − 1/2hkµ = 0. Thesolutions are shown in Table 1.

The case A corresponds to the standard massless gravi-tational waves of General Relativity. The solution B is thesame of the pure f (X)-gravity, i.e. the choice k2 = m1

2

reduces automatically the field equations of f (X,Y,Z)-gravity to those of f (X)-gravity. Cases B and C are verydifferent. In fact the solution C is traceless while the solu-tion B satisfies directly the gauge condition.

The solution of case B for hµν is given by the followingexpression

hBµν = 1

3

∫d4k

(2π)4

[kµkν

m12 + ηµν

2

]hB(k)ejkx

= 13

[ηµν

2−

∂2µν

m12

]∫d4k

(2π)4 hB(k)ejkx

= 13

[ηµν

2−

∂2µν

m12

]hB(x), (30)

where the trace of metric is a generic Klein-Gordon functionwith k = (ω1,k), where ω1 =

√|k|2 + m12. We can write

the general solution in terms of its Fourier modes which areplane waves, that is

hB(t,x) =∫

d3k(2π)3 C(k)ej(ω1t−k·x) (31)

where C(k) is the Fourier representation of the trace hB .If we consider a propagating trace in the z-direction (weset x3 = z) hB(t, z) = h0e

j(ω1t−kzz), where k = (ω1,0,0, kz)

with ω12 − kz

2 = m12, the solution (30) is given by

hBµν(t, z) = ϵB

µν ej(ω1t−kzz)

= h0

3

⎛

⎜⎜⎜⎜⎝

12 + ω1

2

m12 0 0 −ω1kz

m12

0 − 12 0 0

0 0 − 12 0

−ω1kz

m12 0 0 − 12 + kz

2

m12

⎞

⎟⎟⎟⎟⎠ej(ω1t−kzz)

(32)

where ϵBµν is the polarization tensor. By a change of coor-

dinates xµ → x′µ = xµ + ζµ(x) with O(ζ 2) ≪ 1, we cantransform the metric hµν into a new metric h′

µν = hµν −ζµ,ν − ζν,µ. Let us suppose that we choose ζµ(x) = jθµejkx ,the metric hµν(x) becomes h′

µν(x) = ϵ′µνe

jkx where ϵ′µν =

ϵµν + kµθν + kνθµ. By performing a change of coordinatesand choosing θµ = θB

µ = (− 14ω1

− ω12m12 ,0,0,

kz

2m12 − kz

4ω12 )

the polarization tensor ϵBµν in Eq. (32) becomes

ϵ′Bµν = h0

3

⎛

⎜⎜⎜⎜⎜⎜⎝

0 0 0 0

0 − 12 0 0

0 0 − 12 0

0 0 0 − 12 + kz

2

2ω12

⎞

⎟⎟⎟⎟⎟⎟⎠

= −h0

6

⎛

⎜⎜⎝

0 0 0 00 1 0 00 0 1 00 0 0 1

⎞

⎟⎟⎠ + kz2h0

6ω12

⎛

⎜⎜⎝

0 0 0 00 0 0 00 0 0 00 0 0 1

⎞

⎟⎟⎠ . (33)

In the case C, we have

hCµν(t,x) =

∫d3k

(2π)3 Cµν(k)ej(ω2t−k·x) (34)

where Cµν(k) is the Fourier representation of the grav-

itational wave hCµν and ω2 =

√|k|2 + m22. Also in this

case, by considering a propagating wave in the z-directionhC

µν(t, z) = ϵCµνe

j(ω2t−kzz), where the polarization tensor ϵCµν

satisfies the traceless condition ηρσ ϵCρσ = 0 and after also

the harmonic gauge ϵCµσ kσ = 0 (in the Fourier space), we

27 Page 8 of 10 S. Capozziello, A. Stabile

find the solution

hCµν(t, z) =

⎛

⎜⎜⎜⎜⎝

ϵ00 ϵ01 ϵ02 − ω2kz

ϵ00

ϵ01 ϵ11 ϵ12 − ω2kz

ϵ01

ϵ02 ϵ12 − m22

kz2 ϵ00 − ϵ11 − ω2

kzϵ02

− ω2kz

ϵ00 − ω2kz

ϵ01 − ω2kz

ϵ02ω2

2

kz2 ϵ00

⎞

⎟⎟⎟⎟⎠ej(ω2t−kzz)

(35)

where ϵ00, ϵ01, ϵ02, ϵ11, ϵ12 are unspecified values. By per-forming also in this case a change of coordinates (Bogdanoset al. 2010; Capozziello and De Laurentis 2011) and choos-ing θµ = θC

µ = (− ϵ002ω2

,− ϵ01ω2

,− ϵ02ω2

, ϵ00kz

− kzϵ002ω22 ) the polariza-

tion tensor ϵCµν in Eq. (35) becomes

ϵ′Cµν =

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 0

0 ϵ11 ϵ12 − m22

kzω2ϵ01

0 ϵ12 −m22

kz2 ϵ00 − ϵ11 − m2

2

kzω2ϵ02

0 − m22

kzω2ϵ01 − m2

2

kzω2ϵ02

m24

kz2ω22 ϵ00

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

= ϵ11

⎛

⎜⎜⎝

0 0 0 00 1 0 00 0 −1 00 0 0 0

⎞

⎟⎟⎠ + ϵ12

⎛

⎜⎜⎝

0 0 0 00 0 1 00 1 0 00 0 0 0

⎞

⎟⎟⎠

− m22ϵ00

kz2

⎛

⎜⎜⎝

0 0 0 00 0 0 00 0 1 00 0 0 0

⎞

⎟⎟⎠ + m24ϵ00

kz2ω22

⎛

⎜⎜⎝

0 0 0 00 0 0 00 0 0 00 0 0 1

⎞

⎟⎟⎠

− m22ϵ01

kzω2

⎛

⎜⎜⎝

0 0 0 00 0 0 10 0 0 00 1 0 0

⎞

⎟⎟⎠ − m22ϵ02

kzω2

⎛

⎜⎜⎝

0 0 0 00 0 0 00 0 0 10 0 1 0

⎞

⎟⎟⎠ . (36)

Finally for the case D we have

hDµν(t, z) =

⎛

⎜⎜⎜⎝

ϵ2 − ϵ03 ϵ01 ϵ02 ϵ03

ϵ01 ϵ11 ϵ12 −ϵ01ϵ02 ϵ12 −ϵ11 −ϵ02ϵ03 −ϵ01 −ϵ02 − ϵ

2 − ϵ03

⎞

⎟⎟⎟⎠ejω(t−z) (37)

where ϵ01, ϵ02, ϵ03, ϵ11, ϵ12 are unspecified values and ϵ isthe trace of polarization tensor ϵD

µν . Here if we choose θµ =θDµ = ( 2ϵ03−ϵ

4ω ,− ϵ01ω ,− ϵ02

ω ,− 2ϵ03+ϵ4ω ) the polarization tensor

ϵDµν in Eq. (37) becomes

ϵ′Dµν =

⎛

⎜⎜⎝

0 0 0 00 ϵ11 ϵ12 ϵ01 + ϵ130 ϵ12 −ϵ11 ϵ02 + ϵ230 ϵ01 + ϵ13 ϵ02 + ϵ23 0

⎞

⎟⎟⎠

= ϵ11

⎛

⎜⎜⎝

0 0 0 00 1 0 00 0 −1 00 0 0 0

⎞

⎟⎟⎠ + ϵ12

⎛

⎜⎜⎝

0 0 0 00 0 1 00 1 0 00 0 0 0

⎞

⎟⎟⎠

+ (ϵ01 + ϵ13)

⎛

⎜⎜⎝

0 0 0 00 0 0 10 0 0 00 1 0 0

⎞

⎟⎟⎠ + (ϵ02 + ϵ23)

⎛

⎜⎜⎝

0 0 0 00 0 0 00 0 0 10 0 1 0

⎞

⎟⎟⎠

(38)

However the last two polarizations in (38) are not physicaland practically we obtained the same outcome of GeneralRelativity (case A).

If we introduce an independent basis (see Table 2) forthe polarizations (33) and (36), the general solution of fieldequations (9) in vacuum is

hµν(t, z) =[H1ϵ

(+)µν + H2ϵ

(×)µν

]ejkz(t−z)

+ H3

[ϵ(1)µν − kz

2√

3ω12ϵ(S)µν

]ej(ω1t−kzz)

+[H4ϵ

(+)µν + H5ϵ

(×)µν

+ H6

(√3ϵ(1)

µν −√

2ϵ(+)µν − ω2

2 + 2m22

ω22 ϵ(S)µν

)

+ H7ϵ(2)µν + H8ϵ

(3)µν

]ej(ω2t−kzz) (39)

where H1, H2 are arbitrary constants related to the prop-agation modes of gravitational waves in General Relativ-

ity and the other constants are defined as H3 = −√

3h06 ,

H4 = ϵ11, H5 = ϵ12, H6 = −m22ϵ00

2kz2 , H7 = −

√2m2

2ϵ01kzω2

,

H8 = −√

2m22ϵ02

kzω2.

Table 2 The basis of the polarizations. Each polarization satisfies thecondition ϵρσ ϵρσ = 1

ϵ(+)µν = 1√

2

⎛

⎜⎜⎜⎜⎝

0 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 0

⎞

⎟⎟⎟⎟⎠ϵ(×)µν = 1√

2

⎛

⎜⎜⎜⎜⎝

0 0 0 0

0 0 1 0

0 1 0 0

0 0 0 0

⎞

⎟⎟⎟⎟⎠

ϵ(S)µν =

⎛

⎜⎜⎜⎜⎝

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 1

⎞

⎟⎟⎟⎟⎠ϵ(1)µν = 1√

3

⎛

⎜⎜⎜⎜⎝

0 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎞

⎟⎟⎟⎟⎠

ϵ(2)µν = 1√

2

⎛

⎜⎜⎜⎜⎝

0 0 0 0

0 0 0 1

0 0 0 0

0 1 0 0

⎞

⎟⎟⎟⎟⎠ϵ(3)µν = 1√

2

⎛

⎜⎜⎜⎜⎝

0 0 0 0

0 0 0 0

0 0 0 1

0 0 1 0

⎞

⎟⎟⎟⎟⎠

Gravitational waves in fourth order gravity Page 9 of 10 27

The different components of polarization tensor can bedistinguished if we ask how ϵµν changes when the coordi-nate system undergoes a rotation of a given angle ϕ aboutthe z-axis (Weinberg 1972). This is a Lorentz transforma-tion of the form

Rµν =

⎛

⎜⎜⎝

1 0 0 00 cosϕ sinϕ 00 − sinϕ cosϕ 00 0 0 1

⎞

⎟⎟⎠ (40)

and since it leaves kµ invariant (Rµσ kσ = kµ), the only ef-

fect is to transform ϵµν into ϵµν = RµρRν

σ ϵρσ . In the caseB, the polarization tensor ϵ′B

µν is unchanged, then we canstate that the helicity is null.4 In the case C, we have

ϵ′C± = e±j2ϕϵ′C

± + m22ϵ00

kz2 sinϕe±j(ϕ+π/2)

l′C± = e±jϕ l′C±

(41)

where ϵ±.= ϵ11 ∓ jϵ12 and l±

.= ϵ13 ∓ jϵ23. The helicity is±1 for the states l±, while for ϵ± we cannot identify thecondition ϵ′C

± = e±jξϕϵ′C± . As such condition holds only if

ϵ00 = 0 (and H6 = 0 in Eq. (39)). In other words, we havethe possible helicity values 0, ±1, ±2. Clearly, all the polar-ization tensors discussed here can be referred to the Wignerlittle group E(2) as discussed in Eardley et al. (1973).

6 Conclusions

In this paper, we presented a complete study of post-Minkowskian limit and gravitational wave solutions offourth-order gravity theories of gravity in four dimensionalspace-time. We considered a generic action constructed withcurvature invariants derived from the Riemann tensor, thatare the Ricci curvature scalar, the squared Ricci tensor andthe squared Riemann tensor. Thanks to the Gauss-Bonnettopological invariant, it is possible to consider only two cur-vature invariants since the third is always related to the oth-ers by the Gauss-Bonnet constraint. In this sense, all thebudget of degrees of freedom can be related to R and Rαβ .

With respect to the standard General Relativity, new fea-tures come out in the post-Minkowskian limit. First of alltwo further massive scalar modes emerge in relation to thenon linearity in the Ricci scalar and tensor terms. This meansthat massive gravitons are a characteristic of these theoriesand their effective masses are strictly related to the form ofthe action f . As discussed in Buchbinder et al. (1992), theseterms play an important role in effective action in quantum

4Any plane wave ψ transforming under a rotation of an angle ϕ aboutthe direction of propagation into ψ = ejξϕψ has helicity ξ .

gravity. Finally, as supposed by several authors, these par-ticles could play a fundamental role for the dark matter is-sue that, in this case, could directly come from the gravita-tional part of cosmic dynamics (van Dam and Veltman 1970;Zakharov 1970; Meszaros 1985).

Furthermore, the total number of polarizations is six andhelicity can come into three distinct states. It is worth notic-ing that we have not to choose arbitrarily × and + polar-izations as in General Relativity but all possible polariza-tions naturally come out. This fact could be of great interestfor the gravitational wave detection since running and forth-coming experiments could take advantage from this theoret-ical result and investigate new scenarios. In a forthcomingpaper, a detailed study of sources compatible with these re-sults will be pursued.

Acknowledgements SC acknowledge INFN Sez. di Napoli (Inizia-tive Specifiche QGSKY, and TEONGRAV) for financial support.

References

Alexander, S., Yunes, N.: Phys. Rep. 480, 1 (2009)Astashenok, A., Capozziello, S., Odintsov, S.D.: J. Cosmol. Astropart.

Phys. 12, 040 (2013)Astashenok, A., Capozziello, S., Odintsov, S.D.: Phys. Rev. D 89,

103509 (2014)Bamba, K., Capozziello, S., De Laurentis, M., Nojiri, S., Saez-Gómez,

D.: Phys. Lett. B 727, 194 (2013)Bargmann, V., Wigner, E.P.: Proc. Natl. Acad. Sci. USA 34, 211

(1948)Bergmann, P.G.: Int. J. Theor. Phys. 1, 25 (1968)Berry, C.P.L., Gair, J.R.: Phys. Rev. D 83, 104022 (2011)Bogdanos, C., Capozziello, S., De Laurentis, M., Nesseris, S.: As-

tropart. Phys. 34, 236 (2010)Buchbinder, I.L., Odintsov, S.D., Shapiro, I.L.: Effective Action in

Quantum Gravity. IOP Publ., Bristol (1992)Capozziello, S., De Laurentis, M.: Phys. Rep. 509, 167 (2011)Capozziello, S., De Laurentis, M.: Ann. Phys. 524, 545 (2012)Capozziello, S., Stabile, A.: Class. Quantum Gravity 26, 085019

(2009)Capozziello, S., Vignolo, S.: Class. Quantum Gravity 26, 175013

(2009a)Capozziello, S., Vignolo, S.: Class. Quantum Gravity 26, 168001

(2009b)Capozziello, S., Stabile, A., Troisi, A.: Phys. Rev. D 76, 104019 (2007)Capozziello, S., Stabile, A., Troisi, A.: Mod. Phys. Lett. A 24(9), 659

(2009)Capozziello, S., De Laurentis, M., Stabile, A.: Class. Quantum Gravity

27, 165008 (2010a)Capozziello, S., Stabile, A., Troisi, A.: Int. J. Theor. Phys. 49, 1251

(2010b)Capozziello, S., Stabile, A., Troisi, A.: Phys. Lett. B 686, 79 (2010c)Capozziello, S., De Laurentis, M., Paolella, M., Ricciardi, G.: Int. J.

Geom. Methods Mod. Phys. 12, 1550004 (2015a)Capozziello, S., Lambiase, G., Sakellariadou, M., Stabile, A., Stabile,

An.: Phys. Rev. D 85, 044012 (2015b)De Laurentis, M., Lopez-Revelles, A.J.: Int. J. Geom. Methods Mod.

Phys. 11, 1450082 (2014)de Rham, C.: Living Rev. Relativ. 17, 7 (2014)de Witt, B.S.: Dynamical Theory of Groups and Fields. Gordon and

Breach, New York (1965)

27 Page 10 of 10 S. Capozziello, A. Stabile

Delsate, T., Hilditch, D., Witek, H.: Phys. Rev. D 91, 024027 (2015)Dyda, S., Flanagan, E.E., Kamionkowski, M.: Phys. Rev. D 86, 124031

(2012). Johns Hopkins U.Eardley, D.M., Lee, D.L., Lightman, A.P., Wagoner, R.V., Will, C.M.:

Phys. Rev. Lett. 30, 884 (1973)Kiefer, C.: Quantum Gravity. Oxford Univ. Press, Oxford (2004)Lanahan-Tremblay, N., Faraoni, V.: Class. Quantum Gravity 24, 5667

(2007)Landau, L.D., Lifšits, E.M.: Theorie des Champs, vol. II. Mir, Moscow

(1970)Lang, R.N.: Phys. Rev. D 89, 084014 (2014)Meszaros, A.: Astrophys. Space Sci. 111, 399 (1985)Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman and

Co., New York (1971)Nojiri, S., Odintsov, S.D.: Int. J. Geom. Methods Mod. Phys. 4, 115

(2007)Nojiri, S., Odintsov, S.D.: Phys. Rep. 505, 59 (2011)Peebles, P.J.E., Ratra, B.: Rev. Mod. Phys. 75(2), 559 (2003)Querella, L.: Variational principles and cosmological models in higher

order gravity. Ph.D. Thesis (1998). arXiv:gr-qc/9902044Salgado, M.: Class. Quantum Gravity 23, 4719 (2006)

Santos, E.: Phys. Rev. D 81, 064030 (2010)Schmidt, H.J.: Astron. Nachr. 307, 339 (1986)Schmidt, H.J.: Phys. Rev. D 78, 023512 (2008)Stabile, A.: Phys. Rev. D 82, 064021 (2010a)Stabile, A.: Phys. Rev. D 82, 124026 (2010b)Stabile, A., Capozziello, S.: Phys. Rev. D 87, 064002 (2013)Stabile, A., Scelza, G.: Phys. Rev. D 84, 124023 (2011)Stabile, A., Stabile, An.: Phys. Rev. D 84, 044014 (2012)Stabile, A., Stabile, An., Capozziello, S.: Phys. Rev. D 88, 124011

(2013)Stein, L.C., Yunes, N.: Phys. Rev. D 83, 064038 (2011)Stelle, K.S.: Phys. Rev. D 16, 953 (1977)Stelle, K.S.: Gen. Relativ. Gravit. 9, 353 (1978)Trimble, V.: Annu. Rev. Astron. Astrophys. 25, 425 (1987)van Dam, H., Veltman, M.G.: Nucl. Phys. B 22, 397 (1970)Wagoner, R.V.: Phys. Rev. D 1, 3209 (1970)Weinberg, S.: Gravitation and Cosmology. Wiley, New York (1972)Wigner, E.P.: Ann. Math. 40, 149 (1939)Zakharov, V.I.: JETP Lett. 12, 312 (1970)