Gravitational waves in fourth order gravity

Astrophys Space Sci (2015) 358:27 DOI 10.1007/s10509-015-2425-1
ORIGINAL ARTICLE
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 invariants f(R,RαβRαβ,RαβγδRαβγδ). It is well known that when dealing with General Relativity such an approach provides massless spin-two waves as propagating degree of freedom of the gravitational field while this theory implies other ad- ditional propagating modes in the gravity spectra. We show that, in general, fourth order gravity, besides the standard massless graviton is characterized by two further massive modes with a finite-distance interaction. We find out the most general gravitational wave solutions in terms of Green functions in vacuum and in presence of matter sources. If an electromagnetic source is chosen, only the modes induced by RαβRαβ are present, otherwise, for any f(R) gravity model, we have the complete analogy with tensor modes of General Relativity. Polarizations and helicity states are clas- sified in the hypothesis of plane wave.
B S. Capozziello capozziello@na.infn.it
A. Stabile arturo.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, Palazzo Dell’Aquila Bosco Lucarelli, Corso Garibaldi, 107, 82100, Benevento, Italy
Keywords Gravitational waves·Modified theories of gravity·Weak field limit
1 Introduction
Identifying the correct theory of gravity is a crucial issue of modern physics due to the fact that General Relativity, in its standard formulation, presents shortcomings at ultraviolet and infrared limits. For the first issue, we need a theory that should deal with gravity under the same standard of the other fundamental interactions (Quantum Gravity) (Kiefer 2004; Stelle 1977). In the other case, modifications of gravity are required to deal with the vast phenomenology coming from astrophysics and cosmology, generally addressed as dark matter and dark energy issues (Peebles and Ratra 2003; Trimble 1987). This matter is rather controversial due to the fact that the ambiguity comes out from the fact that phenomenology could be explained successfully both con- sidering new material ingredients (dark matter particles ad- dressing the problem of structure formation and light scalar fields 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 the weak field limit related to General Relativity (Nojiri and Odintsov 2007, 2011; Capozziello and De Laurentis 2012; Stelle 1978; Schmidt 1986, 2008). Furthermore, there are very few investigations and experimental constraints prob- ing the gravitational field in very strong regimes. Several times, extrapolations of General Relativity are simply as- sumed without considering corrections and alternatives that could strongly affect theoretical and experimental results. With this situation in mind, it is urgent to find out some experimentum 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;StabileandScelza 2011;Stabileand Stabile 2012),that,inanycase,shouldreproducethewell- founded theoretical and experimental results of General Rel- ativity. At astrophysical level, discriminations could come from 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 quark stars could be independent signatures for modified theories of gravity considered as extensions of General Relativity in the strong field regime (Astashenok et al. 2013, 2014). Besides, discrimination could happen in the realm of gravitational 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 has been devoted to the study of gravitational radiation in the realm of General Relativity discarding the fact that modified gravity presents a huge amount of new phenomenology and features. For example, only General Relativity strictly fore- casts massless gravitons with two polarizations. In general, modified gravity and, in particular Extended Gravity, allows also massive and ghost modes and then further polarizations (Bogdanos et al. 2010;CapozzielloandDeLaurentis 2011). Specifically, several authors refer this issue to the Fierz- Pauli linearized analysis of massive gravity that leads to the so called van Dam-Veltman-Zakharov (vDVZ) discontinu- ity. In such a case, we are in presence of the Boulware-Deser ghosts. Such anomalies can be cured through the Vainshtein mechanism. A detailed review of these arguments can be found in de Rham (2014). However, the possibility of these massive modes are studied from a theoretical point of view but practically ignored from the experimental point of view due to the enormous 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 Interferometer Gravitational-Wave Observatory, http://www.ligo.org)col- laboration, LISA (Laser Interferometer Space Antenna, http://sci.esa.int/lisa/)etc.couldbesuitable,inprinciple,for detecting these further modes. In this paper, we propose a systematic study of gravita- tional wave solutions in theories where generic functions of curvature invariants are considered generalizing the first par- tial outcome in the only f(R)framework (Capozziello et al. 2010b).Theseareastraightforwardgeneralizationof f(R) gravity where the degrees of freedom, related to the cur- vature invariants, is considered. However, we have to say that we are not considering R and similar terms where derivative of curvature invariants appear. We are also not considering the parity-odd Chern-Simons invariant (Alexan- der and Yunes 2009)thatentersatthesameorderincur- vatures and derivatives. Here, we are taking into account
only fourth order theories of gravity where derivatives of metric tensor gμν appear up to the fourth ones. It is inter- esting to see that relaxing the hypothesis that gravitational interaction is derived only from the Hilbert-Einstein action, linear in the Ricci curvature scalar R, further gravitational modes, polarizations and helicity states come out. This new features are directly derived from the post-Minkowskian limit 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 of these modes, implying new massive and polarization states, is the same of Wigner’s little group E(2) (Wigner 1939; Bargmann and Wigner 1948)earlierexploredinliterature by Eardley et al. (1973). Besides, f(R)theories of gravity have been very well explored in this sense. For example, it is well known that they are equivalent to scalar-tensor the- ories of gravity following the Bergmann-Wagoner formu- lation (Bergmann 1968;Wagoner 1970)whichgeneralizes the 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).Itis important to stress that being f(R)gravity equivalent to a particular class of scalar-tensor gravity theories, the gravita- tional wave content has already been characterized in both representations (Lang 2014).Alsoacomparisonbetween the two frameworks (the Einstein and Jordan frame) has been analyzed in the weak field limit (Stabile et al. 2013; Capozziello et al. 2010c).Finally,gravitationalwavekine- matics and dynamics in strong field regime and around the Minkowski background have been investigated in detail (Stein and Yunes 2011). This paper is organized as follows. In Sect. 2 we report briefly the field equations of fourth order gravity. In Sect. 3, we discuss the post-Minkowskian limit and the linearized field equations while, in Sect. 4,thegravitationalwaveso- lutions are reported. Section 5 is devoted to the discussion of all possible polarization and helicity states of the wave solutions. Conclusions are reported in Sect. 6.
2 The field equations of fourth order gravity
The most general class of gravitational theories involving curvature 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 invariants X=R, Y =RαβRαβ, and Z =RαβγδRαβγδ. The termLm 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 2
gμν −fX;μν +gμν fX +2fYRμαRαν
−2
fY Rα(μ

;ν)α+ [fY Rμν]+[fYRαβ];αβgμν +2fZRμαβγ Rναβγ −4
fZRμαβν
;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 αβν,μ+···. The affinities are the usual Christoffel’s symbols of the met- ric: Γ μ αβ = 1 2gμσ(gασ,β +gβσ,α −gαβ,σ ). The adopted sig- nature is (+−−− ) (we follow the conventions by Landau and Lifšits 1970).Thetraceoffieldequations(2)isthefol- lowing fXX+2fYY +2fZZ−2f + [3fX +fY X] +2
(fY +2fZ)Rαβ
;αβ =XT (3) where T =T σ σ is the trace of energy-momentum tensor and H =Hσ σ . 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,thisdependenceon the two last invariants is only formal, since from the con- tracted Bianchi identity (2Rαβ;αβ − R = 0) we have only one independent invariant. In any linear theory of gravity (the function f is linear) the terms R and Rαβ;αβ give us no contribution to the field equations, because they are four- divergences. However if we consider a function of R or Rαβ;αβ by varying the action, we still have four-divergences but we would have the contributions of sixth order differ- ential terms. As said in the Introduction, in this paper, we consider only fourth order differential field equations. This means 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 field limit approximation. Thisprescription 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 the gravitational field is weak or is almost null. Both require- ments are fulfilled by General Relativity and then they can be considered two possible paradigms to confront a given theory, at least in the weak field limit, with the General Relativity itself. The Newtonian limit of f(R)-gravity and f(R,RαβRαβ,RαβγδRαβγδ)-gravity can be investigated al- ways remaining in the Jordan frame (Capozziello et al. 2007, 2009;CapozzielloandStabile 2009;Stabile 2010a, 2010b; Stabile and Capozziello 2013)whileapreliminarystudyof the post-Minkowskian limit for the f(R)gravity is provided in Capozziello et al. (2010b). Here we want to derive the post-Minkowskian limit of f(R,RαβRαβ,RαβγδRαβγδ) 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 gravity arises when the regime of small field is considered with- out any prescription on the propagation of the field. This case 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 several cases 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, instead of the Hilbert-Einstein Lagrangian, linear in the Ricci scalar R, a most general function f of curvature invariants. However, working with post-Minkowskian limit, one has to be extremely careful with the issue whether or not there are ghost modes that could render the theory meaningless. A standard way to deal with ghost modes is to consider any alternative theory as an effective field theory close to Gen- eral Relativity, since it is well known that General Relativity has no ghosts. In fact, effective field theory has a restricted regime of validity, and the ghost modes only appear once one leaves such a regime. See, for examples Dyda et al. (2012),Capozzielloetal.(2015a)wheresuchaproblem is discussed for the Chern-Simons and f(R) theories re- spectively. Essentially, the approach consists in decoupling the modes in the weak limit. In Stein and Yunes (2011), it was shown that in the decoupling limit, and approaching the 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 as in General Relativity (see for example Bamba et al. 2013 for the case of f(T)teleparallel gravity). Another question is related the Cauchy problem of the theory, and in particular with its well-formulation and well- posed initial value formulation. For f(R) gravity, this problem has been highly debated and discussed (Lanahan- Tremblay and Faraoni 2007;CapozzielloandVignolo
 27 Page 4 of 10 S. Capozziello, A. Stabile
2009a, 2009b;Salgado 2006).Inanycase,considering theories with higher derivatives, the Cauchy problem has to be treated with care because the related Hamiltonians could suffer, in general, with the Ostrogradski instability (see Querella 1998 for a detailed study). In particular, the existence 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. In some sense, the mechanism is the same acting in confor- mal transformations where the further degrees of freedom in the Jordan frame are disentangled in scalar fields mini- mally coupled to gravity in the Einstein frame (Capozziello and De Laurentis 2011).Thisfactallowstoevaluatehow many propagating dynamical degrees of freedom are present in the theory. Alternatively, treating the theory as an effec- tive field theory can contribute to address the question of the existence of a well-posed initial value formulation (Delsate et al. 2015). In order to perform the post-Minkowskian limit of field equations, one has to perturb Eqs. (2) on the Minkowski background 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
hμν2

Y ∼Y(2)+O
hμν3

Z∼Z(2)+O
hμν3

(5)
and the function f can be developed as f(X,Y,Z)∼f(0)+fX(0)X(1)+ 1 2 fXX(0)X(1)2 +fY(0)Y(2)+fZ(0)Z(2)+O
hμν3

(6)
Analogous relations for partial derivatives of f are obtained. From lowest order of field equations (2) and (3) we have the condition f(0) = 0, while atO(1)-order, we have2 fX(0)R(1) μν +
fY (0)+4fZ(0)

ηR(1) μν −
fX(0) 2
X(1)ημν
+
fXX(0)+
fY (0) 2
ημν ηX(1)−fXX(0)X(1),μν
−
fY(0)+4fZ(0)
R(1)αμ,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 andRμαβ ν;αβ = Rμα;να− Rμν.
where η is the d’Alembert operator in the flat space. Tμν is fixed at zero-order in (7) since, in this perturbation scheme, the first order on Minkowski space has to be connected with the zero order, with respect to hμν, of the standard matter en- ergy momentum tensor. This means that Tμν is independent of hμν, and satisfies the standard conservation conditions T 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 +m22
R(1) μν −

m12−m22 3m12
∂2 μν +ημν
m22 2
+
m12+2m22 6m12 η


X(1) =m22XT (0) μν
η +m12
X(1) =−m12XT (0)
(9)
We note that in the case of f(X)-theory we obtain only a massive mode (with mass m1) of Ricci scalar (X). In fact if fY = fZ = 0 from the mass definition (8) m12 → (−3fXX(0))−1 and m22 →∞we recover the equations of f(X)-gravity (Capozziello et al. 2010b)
R(1) μν −
X(1) 2
ημν +
∂2 μν −ημν η 3m12
X(1) =XT (0) μν
η +m12
X(1) =−m12XT (0)
(10)
while f(X,Y,Z)-theory we have an additional massive propagation (with mass m2) of Ricci tensor. Finally in the case of f →X also m12 →∞and we recover the General Relativity. A first consideration regarding the masses (8) induced by f(X,Y,Z)-gravity is necessary at this point. The second mass m2 is originated by the presence, in the Lagrangian, of Ricci 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 tensor square alone. Then such a modification of theory enables a massive propagation of Ricci Tensor and, as it is well known in the literature, a substitution of the Ricci scalar with any function of the Ricci scalar enables a massive propagation of Ricci scalar. We can conclude that a Lagrangian containing any function of only Ricci scalar and Ricci tensor square is not restrictive. This result is coming from the Gauss-Bonnet topological 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).Infact,byapplyingthevariationwithre- spect to the metric tensor gμν in 4 dimensions to the quantity d4x√−gGGB =0, we have
δ

d4x√−gGGB =

d4x√−g
HX2 μν −4HY μν +HZ μν
δgμν
=

d4x√−gHGB μν δgμν (11)
where HX2 μν = 1 √−g δ(√−gX2) δgμν , HY μν = 1 √−g δ(√−gY) δgμν and HZ μν = 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 −4HY μν +HZ μν =0 (12)
By substituting condition (12) in Eqs. (2), at post-Minkows- kian level, we find the same equations (9) with a redefinition of masses (8). In the weak field limit approximation, we can consider 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<0 and 0 <c<−3b hold. This fact is extremely important in order 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 Laurentis and Lopez-Revelles (2014).
4 Gravitational wave solutions
Once developed the post-Minkowskian limit, one can search for gravitational wave solutions. The general solution of field equations (9) is given by
hμν =hμν +hμν (15)
where hμν is the (homogeneous) solution in the vacuum and hμν 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 field defined as follows
η +mi2
GKG,i
x,x

=δ(4)
x−x

(16) with i =1,2 and δ(4)(x −x ) is the Dirac delta function in four 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 +m22
R(1) μν
=X
m22 T (0) μν −
m12+2m22 6
ημνT (0)

−
(m12−m22)X 3

d4x
∂2 μν −
m12 2
ημν

×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

×

m22T (0) μν
x

−
m12+2m22 6
ημνT (0)
x


−
(m12−m22)X 3

d4x d4x GKG,2
x,x

×

∂2 μ ν −
m12 2
ημν

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 −
1 2
ηhμν − 1 2
h,μν (20)
where h=hσ σ. Since we can use the harmonic gauge con- dition gρσΓ αρσ =0, we set hμσ ,σ −1/2h,μ =0, then the Ricci tensor becomes R(1) μν =− 1 2 ηhμν. The solution of Eq. (19) is
hμν =
2(m12−m22)X 3 ×

d4x d4x d4x GGR
x,x
GKG,2
x ,x

×

∂2 μ ν −
m12 2
ημν

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
×

m22T (0) μν
x

−
m12+2m22 6
ημνT (0)
x


(21)
whereGGR(x,x ) is the Green function defined as ηGGR
x,x

=δ(4)
x−x

(22)
Considering the expressions of the Green functions GGR(x,x ) and GKG,1,2(x,x ) in terms of plane waves it is 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

=2ms2(−1)1+s

d4k (2π)4
ejk(x−x ) k2(k2−ms2)
Z
x,x

=

d4k (2π)4
(m12+2m22)k2−3m12m22 3k2(k2−m12)(k2−m22) ×ejk(x−x )
Zμν
x,x

=

d4k (2π)4
2(m12−m22)kμkν 3k2(k2−m12)(k2−m22) ×ejk(x−x )
(24)
k = kμ = (ω,k) = (ω,k1,k2,k3), kx = kσxσ = ωt − k · x, k2 = kσkσ = ω2 −| k|2 and s = 1,2. We note that in the case f → X (from the mass definitions (8) we have m1,m2 →∞) we find Zs(k)→2(−1)sk−2, Z(k)→ −k−2, Zμν(k) →0 and the solutions (23) become those of General 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 can see immediately the propagation of the massless interaction and 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
∂2 μν

m12−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 become problematic for the above propagators since the theory ap- pears strongly coupled in this limit. However the source term can be assumed decoupled as soon as m→0. In such a case the problem is solved. See de Rham (2014) for a discussion. A very interesting feature is achieved when we consider a traceless source with T (0) = 0. In this case, solutions (23)areunaffectedbythefunctionsoftheRicciscalarbut only by the presence of the Ricci tensor square terms into the interaction Lagrangian. Moreover, from the second line of (23), we have also that the gravitational waves must be traceless. A prototype of traceless source is the electromag- netic field where we have
hem μν(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 using again the hypothesis of harmonic gauge and the principle of plane wave superposition, we get
hμν =
m12−m22 3m22

d4x Z2
x,x


∂2 μ 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 Ricci scalar and tensor and the solution of wave equation for the metric. The homogeneous solutions are chosen in such a way that they satisfy the boundary conditions and the gauge harmonic condition hμρ ,ρ−1/2h,μ =0. Also here, the limit m→0 becomes problematic since it is strongly coupled. As above, the remedy is that the sources decouples in this limit (de Rham 2014). An important remark is necessary at this point. In the above 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 than one gravitational wavelengths and the effects of background curvature on wave propagation have to be considered. A de- tailed discussion on harmonic gauge condition, shortwave approximation and propagation on curved background can be found in Misner et al. (1971). Here we adopted the same approach (see § 35.13, § 35.14 therein). Below we will consider polarizations and helicity states in 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; the case B is massive and pure trace; the case C is massive, trace-free and transverse and the case D is massless where the trace-reverse of h is transverse
Case Choices Gauge condition
A k2 =0 any ˜ hμν with ˜ h=0
˜ hμσ kσ =0
B k2 =m12 ˜ hμν =

kμkν m12 + ημν 2

˜ h 3
with ˜ h =0
Verified
C k2 =m22 any ˜ hμν with ˜ h=0
˜ hμσ kσ =0
D k2 =0 any ˜ hμν with ˜ h =0
˜ hμσ kσ −1/2˜ hkμ =0
5 Polarizations and helicity states in vacuum
We can analyze the propagation in vacuum by performing the Fourier analysis of field equations (9). In the Fourier space, we have
k2
m22−k2

˜ hμν −

m22−m12 3m12
kμkν
+ημν
m22 2 −
m12+2m22 6m12
k2


˜ h

=0
k2
m12−k2

˜ h=0
(29)
where ˜ hμν, ˜ h are the Fourier representation of hμν, h. The gauge condition now becomes ˜ hμσkσ −1/2˜ hkμ = 0. The solutions are shown in Table 1. The case A corresponds to the standard massless gravi- tational waves of General Relativity. The solution B is the same of the pure f(X)-gravity, i.e. the choice k2 = m12 reduces automatically the field equations of f(X,Y,Z)- gravity to those of f(X)-gravity. Cases B and C are very different. 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 following expression
hB μν =
1 3

d4k (2π)4

kμkν m12 +
ημν 2

hB(k)ejkx
=
1 3

ημν 2 −
∂2 μν m12


d4k (2π)4
hB(k)ejkx
=
1 3

ημν 2 −
∂2 μν m12
hB(x), (30)
where the trace of metric is a generic Klein-Gordon function with k =(ω1,k), where ω1 =

|k|2+m12. We can write
the general solution in terms of its Fourier modes which are plane 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 (we set x3 =z) hB(t,z)=h0ej(ω1t−kzz), where k=(ω1,0,0,kz) with ω12−kz2 =m12, the solution (30) is given by hB μν(t,z)= B μν ej(ω1t−kzz)
=
h0 3 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
1 2 + ω12 m12
00 −ω1kz m12 0 −1 2 00 00 −1 2 0 −ω1kz m12 00 −1 2 + kz2 m12 ⎞ ⎟ ⎟ ⎟ ⎟
⎠
ej(ω1t−kzz)
(32)
where B μν is the polarization tensor. By a change of coor- dinates xμ →x μ =xμ +ζμ(x) with O(ζ2) 1, we can transform 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)= μνejkx where μν = μν +kμθν +kνθμ. By performing a change of coordinates and choosing θμ =θB μ =(− 1 4ω1 − ω1 2m12 ,0,0, kz 2m12 − kz 4ω12 ) the polarization tensor B μν in Eq. (32) becomes
B μν =
h0 3 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0 0 0 0 0−1 2 00 00−1 2 0 0 0 0−1 2 + kz2 2ω12 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
⎠
=− h0 6 ⎛ ⎜ ⎜ ⎝
0000 0100 0010 0001 ⎞ ⎟ ⎟
⎠
+
kz2h0 6ω12 ⎛ ⎜ ⎜ ⎝
0000 0000 0000 0001 ⎞ ⎟ ⎟
⎠
. (33)
In the case C, we have
hC μν(t,x)=

d3k (2π)3
Cμν(k)ej(ω2t−k·x) (34)
where Cμν(k) is the Fourier representation of the grav- itational wave hC μν and ω2 =

|k|2+m22. Also in this case, by considering a propagating wave in the z-direction hC μν(t,z)= C μνej(ω2t−kzz), where the polarization tensor C μν satisfies the traceless condition ηρσ C ρσ =0 and after also the harmonic gauge C μσkσ =0 (in the Fourier space), we
 27 Page 8 of 10 S. Capozziello, A. Stabile
find the solution
hC μν(t,z)= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
00 01 02 −ω2 kz
00
01 11 12 −ω2 kz
01
02 12 −m22 kz2
00 − 11 −ω2 kz
02
−ω2 kz
00 −ω2 kz
01 −ω2 kz
02 ω22 kz2
00 ⎞ ⎟ ⎟ ⎟ ⎟
⎠
ej(ω2t−kzz)
(35)
where 00, 01, 02, 11, 12 are unspecified values. By per- forming also in this case a change of coordinates (Bogdanos et al. 2010; Capozziello and De Laurentis 2011) and choos- ing θμ =θC μ =(− 00 2ω2 ,− 01 ω2 ,− 02 ω2 , 00 kz − kz 00 2ω22 ) the polariza- tion tensor C μν in Eq. (35) becomes
C μν = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
0 0 0 0 0 11 12 −m22 kzω2 01 0 12 −m22 kz2 00− 11 −m22 kzω2 02 0−m22 kzω2 01 −m22 kzω2 02 m24 kz2ω22 00 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
⎠
= 11 ⎛ ⎜ ⎜ ⎝
0 0 0 0 0 1 0 0 00−10 0 0 0 0 ⎞ ⎟ ⎟
⎠
+ 12 ⎛ ⎜ ⎜ ⎝
0000 0010 0100 0000 ⎞ ⎟ ⎟
⎠
−
m22 00 kz2 ⎛ ⎜ ⎜ ⎝
0000 0000 0010 0000 ⎞ ⎟ ⎟
⎠
+
m24 00 kz2ω22 ⎛ ⎜ ⎜ ⎝
0000 0000 0000 0001 ⎞ ⎟ ⎟
⎠
−
m22 01 kzω2 ⎛ ⎜ ⎜ ⎝
0000 0001 0000 0100 ⎞ ⎟ ⎟
⎠
−
m22 02 kzω2 ⎛ ⎜ ⎜ ⎝
0000 0000 0001 0010 ⎞ ⎟ ⎟
⎠
. (36)
Finally for the case D we have
hD μν(t,z)= ⎛ ⎜ ⎜ ⎜ ⎝
2 − 03 01 02 03 01 11 12 − 01 02 12 − 11 − 02 03 − 01 − 02 − 2 − 03 ⎞ ⎟ ⎟ ⎟
⎠
ejω(t−z) (37)
where 01, 02, 03, 11, 12 are unspecified values and is the trace of polarization tensor D μν. Here if we choose θμ = θD μ =(2 03− 4ω ,− 01 ω ,− 02 ω ,−2 03+ 4ω ) the polarization tensor D μν in Eq. (37) becomes
D μν = ⎛ ⎜ ⎜ ⎝
0 0 0 0 0 11 12 01+ 13 0 12 − 11 02+ 23 0 01+ 13 02+ 23 0 ⎞ ⎟ ⎟
⎠
= 11 ⎛ ⎜ ⎜ ⎝
0 0 0 0 0 1 0 0 00−10 0 0 0 0 ⎞ ⎟ ⎟
⎠
+ 12 ⎛ ⎜ ⎜ ⎝
0000 0010 0100 0000 ⎞ ⎟ ⎟
⎠
+( 01+ 13) ⎛ ⎜ ⎜ ⎝
0000 0001 0000 0100 ⎞ ⎟ ⎟
⎠
+( 02+ 23) ⎛ ⎜ ⎜ ⎝
0000 0000 0001 0010 ⎞ ⎟ ⎟
⎠
(38)
However the last two polarizations in (38) are not physical and practically we obtained the same outcome of General Relativity (case A). If we introduce an independent basis (see Table 2) for the polarizations (33) and (36), the general solution of field equations (9) in vacuum is
hμν(t,z)=
H1 (+) μν +H2 (×) μν
ejkz(t−z)
+H3

(1) μν −
kz2 √3ω12 (S) μν
ej(ω1t−kzz)
+
H4 (+) μν +H5 (×) μν
+H6
√3 (1) μν −√2 (+) μν −
ω22+2m22 ω22
(S) μν

+H7 (2) μν +H8 (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 =− √3h0 6 , H4 = 11, H5 = 12, H6 =−m22 00 2kz2 , H7 =− √2m22 01 kzω2 , H8 =− √2m22 02 kzω2 .
Table 2 The basis of the polarizations. Each polarization satisfies the condition ρσ ρσ =1

(+) μν = 1 √2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
0 0 0 0 0 1 0 0 00−10 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟
⎠

(×) μν = 1 √2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
0000 0010 0100 0000 ⎞ ⎟ ⎟ ⎟ ⎟
⎠

(S) μν = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
0000 0000 0000 0001 ⎞ ⎟ ⎟ ⎟ ⎟
⎠

(1) μν = 1 √3 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
0000 0100 0010 0001 ⎞ ⎟ ⎟ ⎟ ⎟
⎠

(2) μν = 1 √2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
0000 0001 0000 0100 ⎞ ⎟ ⎟ ⎟ ⎟
⎠

(3) μν = 1 √2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
0000 0000 0001 0010 ⎞ ⎟ ⎟ ⎟ ⎟
⎠
Gravitational waves in fourth order gravity Page 9 of 10 27
The different components of polarization tensor can be distinguished if we ask how μν changes when the coordi- nate system undergoes a rotation of a given angle ϕ about the z-axis (Weinberg 1972). This is a Lorentz transforma- tion of the form
Rμν = ⎛ ⎜ ⎜ ⎝
1 0 0 0 0 cosϕ sinϕ 0 0−sinϕ cosϕ 0 0 0 0 1 ⎞ ⎟ ⎟
⎠
(40)
and since it leaves kμ invariant (Rμσkσ =kμ), the only ef- fect is to transform μν into ˜ μν =RμρRνσ ρσ. In the case B, the polarization tensor B μν is unchanged, then we can state that the helicity is null.4 In the case C, we have
˜ C ± =e±j2ϕ C ± +
m22 00 kz2
sinϕe±j(ϕ+π/2)
˜ l C ± =e±jϕl C ±
(41)
where ±
. = 11 ∓j 12 and l± . = 13 ∓j 23. The helicity is ±1 for the states l±, while for ± we cannot identify the condition ˜ C ± =e±jξϕ C ± . As such condition holds only if 00 =0 (and H6 =0 in Eq. (39)). In other words, we have the possible helicity values 0,±1,±2. Clearly, all the polar- ization tensors discussed here can be referred to the Wigner little 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 of fourth-order gravity theories of gravity in four dimensional space-time. We considered a generic action constructed with curvature invariants derived from the Riemann tensor, that are the Ricci curvature scalar, the squared Ricci tensor and the squared Riemann tensor. Thanks to the Gauss-Bonnet topological 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 the budget 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 all two further massive scalar modes emerge in relation to the non linearity in the Ricci scalar and tensor terms. This means that massive gravitons are a characteristic of these theories and their effective masses are strictly related to the form of the action f . As discussed in Buchbinder et al. (1992), these terms play an important role in effective action in quantum
4Any plane wave ψ transforming under a rotation of an angle ϕ about the 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 and helicity 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 interest for the gravitational wave detection since running and forth- coming experiments could take advantage from this theoret- ical result and investigate new scenarios. In a forthcoming paper, 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.
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