research article propagating degrees of freedom in ()...

Research ArticlePropagating Degrees of Freedom in 119891(119877) Gravity

Yun Soo Myung

Institute of Basic Science and Department of Computer Simulation Inje University Gimhae 621-749 Republic of Korea

Correspondence should be addressed to Yun Soo Myung ysmyunginjeackr

Received 22 September 2016 Accepted 6 December 2016

Academic Editor George Siopsis

Copyright copy 2016 Yun Soo Myung This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We have computed the number of polarization modes of gravitational waves propagating in the Minkowski background in 119891(119877)gravity These are three of two from transverse-traceless tensor modes and one from a massive trace mode which confirms theresults found in the literature There is no massless breathing mode and the massive trace mode corresponds to the Ricci scalar Anewly defined metric tensor in 119891(119877) gravity satisfies the transverse-traceless (TT) condition as well as the TT wave equation

1 Introduction

119891(119877) gravity theory is considered as a representative theoryof modified gravities 119891(119877) gravity [1ndash4] has much attentionsas a strong candidate for explaining the current acceleratinguniverse [5 6] When choosing the Hu-Sawicki model [7]the theory could give rise to the late time cosmic accelerationwithout violating the gravity tests in the solar system andwithout affecting high redshift physics Very recently theobservational constraints on this model were reported fromweal lensing peak abundances [8] Particularly 119891(119877) = 119877 +1198772(61198722) gravity [9ndash11] has shown a strong evidence forinflation to support recent Planck data [12] An importantfeature of this model indicates that the inflationary dynamicswere driven by the purely gravitational interaction1198772 and thescale of inflation is linked to the mass parameter 1198722 Thistheory could thus provide a unified picture of both inflationin the early universe and the accelerated expansion at latertimes In addition black hole [13ndash15] and traversal wormholesolutions [16 17] have been found within 119891(119877) gravity inrecent years The recent detection of gravitational waves bythe LIGO Collaboration [18] is surely a milestone in grav-itational waves research and opens new perspectives in thestudy of Einstein gravity (general relativity) and astrophysicsHence it is meaningful to explore gravitational waves in themodified theory of gravity especially in 119891(119877) gravity Theobservation of the polarization modes of gravitational waves

will be a crucial tool to obtain valuable information about theblack holes and the physics of the early universe

It is well-known that the Einstein gravity with twopolarization degrees of freedom (DOF) is distinguished fromthe metric 119891(119877) gravity with three DOF [19] Importantly itis worth noting that the Einstein equation derived from 119891(119877)gravity contains fourth-order derivative terms A simple wayto avoid a difficulty dealing with the fourth-order equation isto transform119891(119877) gravity into a scalar-tensor theory which issurely a second-order theory

Very recently it was reported that a polynomial 119891(119877)model could provide two additional scalars of a massivelongitudinal mode (perturbed Ricci scalar 119877(1)) and a mass-less transverse mode (breathing mode ℎ119887) in addition tothe two TT tensor modes (ℎ+ ℎtimes) [20] A breathing modeseems to be overlooked in the literature because of theassumption that the application of the Lorentz gauge impliesthe TT wave equation Also it was insisted that four DOFfound in [20] is consistent with the result obtained from theNewman-Penrose (NP) formalism However the presenceof a breathing mode contradicts the well-known fact in theliterature that the 119891(119877) gravity involves three DOF of amassive longitudinal mode and two spin-2 modes Hereafterwe wish to call this the issue of DOF in 119891(119877) theories

In this work we wish to point out that 119891(119877) gravitystill involves three DOF by investigating the fourth-order

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 3901734 6 pageshttpdxdoiorg10115520163901734

2 Advances in High Energy Physics

equation composed of a second-order tensor and a fourth-order scalar

It seems that there is no breathing mode because theperturbed Ricci scalar 119877(1) is related closely to the traceldquoℎrdquo of perturbed metric tensor Hence the allocation of theRicci scalar as a newly scalar represents the trace of metrictensor Also we wish to remind the reader that the Ricciscalar equation is not an independent equation and is notseparated from the perturbed Einstein equation because itcomes out just from taking the trace of the latter equationThis implies that the Ricci scalar is an emergent mode fromℎ Furthermore it is instructive to note that in the TT gaugethere is a close connection between the metric perturbationand the linearized Riemann tensor implying that 120575119877119894119905119895119905 =minusℎTT119894119895 2 [21] This gauge is very convenient because it fixesall local gauge freedoms But it might be unclear that thereexists a close relation between the metric perturbation andthe NP formalism unless one chooses the TT gauge Onecould not naively choose the TT gauge in the perturbed 119891(119877)gravity because of ℎ = 0 whereas the Lorentz gauge iseasily implemented to eliminate the gauge DOF Howeverone might choose the TT gauge to obtain a massless spin-2in the perturbed 119891(119877) gravity when one introduces a newlymetric perturbation ℎ120583]

Theorganization of ourwork is as follows In Section 2webriefly describe 119891(119877) gravity and its scalar-tensor theory andderive two sets of perturbed equations around theMinkowskibackground in Section 3 Section 4 is focused on obtainingthe number of propagating DOF when one chooses theLorentz gauge Finally we will discuss our result which showsthat there is no breathing mode in Section 5

2 119891(119877) Gravity and Its Scalar-Tensor Theory

Instead of a polynomial model of 119891(119877) = 119877 + 1205721198772 + 1205731198773 +sdot sdot sdot [20 22] we start with a specific 119891(119877) gravity (Starobinskymodel [9])

119878119891 = 11987221198752 int1198894119909radicminus119892119891 (119877) 119891 (119877) = 119877 + 119877261198722 (1)

where the 1198772-term was originally motivated by one-loopcorrection to Einstein gravity Here the mass parameter 1198722is chosen to be a positive value which is consistent withthe stability condition of 11989110158401015840(0) gt 0 [2] This model of119891(119877) gravity is enough to find the propagating DOF aroundthe Minkowski background The Einstein equation takes theform

119877120583]1198911015840 (119877) minus 12119892120583]119891 (119877) + (119892120583]nabla2 minus nabla120583nabla]) 1198911015840 (119877) = 0 (2)

where the prime (1015840) denotes the differentiation with respectto its argument

On the other hand one might represent (1) as a scalar-tensor theory by introducing an auxiliary field 120595 [10]

119878119860 = int1198894119909radicminus119892119869 [11987221198752 119877 + 119872119875119872 119877120595 minus 31205952] (3)

where the superscript 119869 means the Jordan frame Varying 119878119860with respect to 120595 provides

120595 = 1198721198756119872119877 (4)

whichmeans that the Ricci scalar is treated as an independentscalar degree of freedom Plugging (4) into 119878119860 again leads tothe original 119891(119877) gravity (1) exactly

Making use of the conformal transformation and redefin-ing the scalar field (120595 rarr 120601)

119892119869120583] 997888rarr 11 + 2120595119872119872119875119892

119864120583] 997888rarr 119890minusradic23(120601119872119875)119892119864120583] (5)

one arrives at the Starobinskymodel in the Einstein frame [11]

119878119878 = int1198894119909radicminus119892119864 [11987221198752 119877 minus 12120597120583120601120597120583120601 minus 119881119878 (120601)] (6)

with the Starobinsky potential

119881119878 (120601) = 3119872211987511987224 [1 minus 119890minusradic23(120601119872119875)]2 (7)

At this stage we note that the conformal transformation (5) isa purely classified transformation of coordinates and resultsin one frame are classically equivalent to the ones obtained inother frame Hence it is plausible that the number of DOF inthe scalar-tensor theory (6) is three because of two TT tensormodes and one scalarmode From (6) the Einstein and scalarequations are derived as

119866120583] = 11198722119875119879120601120583]

119879120601120583] = 120597120583120601120597]120601 minus 12119892120583] [(120597120601)

2 + 119881119878] nabla2120601 minus 1198811015840119878 = 0

1198811015840119878 = radic321198721198751198722119890minusradic23(120601119872119875) [1 minus 119890minusradic23(120601119872119875)]

(8)

The above describes a process of [1198772 rarr 119877120595minus31205952 rarr minus(120597120601)2minus119881] briefly3 Two Sets of Perturbed Equations

We introduce themetric perturbation around theMinkowskibackground to find out the propagating DOF

119892120583] = 120578120583] + ℎ120583] (9)

The Taylor expansions around 119877 = 0 are employed to definethe linearized Ricci scalar 120575119877(ℎ) as [23]

119891 (119877) = 119891 (0) + 1198911015840 (0) 120575119877 (ℎ) + sdot sdot sdot 1198911015840 (119877) = 1198911015840 (0) + 11989110158401015840 (0) 120575119877 (ℎ) + sdot sdot sdot (10)

Advances in High Energy Physics 3

with 119891(0) = 0 1198911015840(0) = 1 and 11989110158401015840(0) = 131198722 We notethat 120575119877(ℎ) will be used here instead of 119877(1) in [20] Theperturbed (linearized) equation to (2) is given by the fourth-order coupled equation

120575119877120583] (ℎ) + 131198722 [120578120583] (minus

311987222 + ◻) minus 120597120583120597]] 120575119877 (ℎ)

= 0 ◻ = 1205972(11)

where the linearized Ricci tensor and scalar are given by

120575119877120583] (ℎ) = 12 [120597120588120597120583ℎ]120588 + 120597120588120597]ℎ120583120588 minus ◻ℎ120583] minus 120597120583120597]ℎ]

120575119877 (ℎ) = 120597120588120597120590ℎ120588120590 minus ◻ℎ(12)

When using (12) the linearized equation (11) becomes asecond (fourth)-order differential equation with respect toℎ120583](ℎ) Obviously it is not a tractable equation Furthermoreits trace equation leads to the linearized Ricci scalar equation

(◻ minus1198722) 120575119877 = 0 (13)

Introducing the linearized Einstein tensor 120575119866120583] = 120575119877120583] minus120578120583]1205751198772 (11) takes a compact form

120575119866120583] (ℎ) + 131198722 [120578120583]◻ minus 120597120583120597]] 120575119877 (ℎ) = 0 (14)

We note that the Bianchi identity is satisfied when acting 120597120583on (14)

On the other hand two linearized equations of (8)together with 120601 = 0 + 120593 take the simple forms with 120575119879120601120583] = 0and 120575119877 = 0

120575119877120583] (ℎ) = 0 (15)

(◻ minus1198722) 120593 = 0 (16)

We note that (13) and (16) are the same when replacing 120575119877by 120593 but the fourth-order coupled equation (11) is quitedifferent from the linearized Einstein equation (15) Thisindicates that (11) can be reduced to two decoupled second-order equations (15) and (16) if one employs the conformaltransformation and redefinition of scalar appropriately afterchoosing (3) leading to a canonical scalar action with theStarobinsky potential in the Einstein frame In the scalar-tensor theory approach one assigns the perturbed Ricciscalar to an independent scalar 120593 Instead it does not havethe trace of metric perturbation ℎ4 Propagating DOF with the Lorentz Gauge

In order to take into account the propagating DOF in 119891(119877)gravity it is convenient to separate the metric tensor ℎ120583] intothe traceless part ℎ119879120583] and the trace part ℎ as

ℎ120583] = ℎ119879120583] + ℎ4120578120583] (17)

with ℎ119879120583120583 equiv ℎ119879 = 0 This splitting is meaningful because theissue of DOF in 119891(119877) theories is related to the presence of ℎ

First of all let us choose the Lorentz (harmonic) gauge toeliminate the gauge DOF

120597120583ℎ120583] = 12120597]ℎ 997888rarr

120597120583ℎ119879120583] = 14120597]ℎ

(18)

Here we note that the transverse condition of 120597120583ℎ119879120583] = 0cannot be achieved in 119891(119877) gravity because of ℎ = 0 Thenthe linearized Ricci tensor and scalar are given by

120575119877 (ℎ)120583] equiv 120575119877119879120583] + 120578120583]4 120575119877 = minus12 [◻ℎ119879120583] + 120578120583]

4 ◻ℎ] 120575119877 (ℎ) = minus12◻ℎ

(19)

where the last equation indicates that the linearized Ricciscalar exists iff ℎ = 0 under the Lorentz gauge This impliesthat 120575119877 cannot be defined without ℎ That is if ℎ = 0 120575119877 = 0

Now the fourth-order equation (11) leads to

◻ℎ120583] + 131198722 [120578120583] (minus

311987222 + ◻) minus 120597120583120597]]◻ℎ = 0 (20)

The other form of (20) takes the form

◻ℎ119879120583] + 131198722 [120578120583] (minus

311987224 + ◻) minus 120597120583120597]]◻ℎ = 0 (21)

If ◻ℎ = 0 its trace equation takes the form

(◻ minus1198722) ◻ℎ = 0 (22)

which is actually the same equation as in (13) Here (22)implies

(◻ minus1198722) ℎ = 0 (23)

because ℎ could represent a massive (scalar) graviton modein 119891(119877) gravity The other case of ◻ℎ = 0 is not allowedsince if ◻ℎ = 0 one could not derive the trace equation(22) itself Importantly this issue should be carefully treatedbecause themassless mode satisfying ◻ℎ = 0may correspondto the breathing mode which is the main subject of thiswork In general it seems that the solution of (22) is givenby the sum of the massive mode and massless mode whichare independent of each other However the massless modewhich is a solution to ◻ℎ = 0 does not exist in 119891(119877) gravityWe stress that ℎ plays the role of a propagating massive modeinstead of 120575119877 Here is the additional reason to understandwhy the massless mode (breathing mode) cannot survive in119891(119877) gravity If one requires ◻ℎ = 0 [120575119877 = 0 via (19)](21) reduces to ◻ℎ119879120583] = 0 which is just the tensor equationin Einstein gravity when choosing the Lorentz gauge It isworth noting that the last fourth-order term of (21) indicatesa feature of the perturbed 119891(119877) gravity clearly If one chooses


◻ℎ = 0 this term disappears Therefore we clarify that themassless scalar mode does not exist

Acting 120597120583 on (21) leads to (18) which implies thatthe Lorentz-gauge condition is satisfied in the perturbedequation level Plugging (22) into (20) we have

◻ℎ120583] minus 131198722◻[11987222 120578120583] + 120597120583120597]] ℎ = 0 (24)

Substituting (22) into (21) implies a fourth-order equation

◻ℎ119879120583] + 131198722◻[11987224 120578120583] minus 120597120583120597]] ℎ = 0 (25)

which shows clearly that the traceless metric perturbationℎ119879120583] is closely coupled to the trace of metric perturbation ℎIt is clear that the trace mode ℎ cannot be decoupled fromthe traceless tensor mode ℎ119879120583] This is the origin of difficultymet when taking into account DOF which had arisen fromthe 119891(119877) gravity Interestingly (25) can be transformed to theRicci tensor-Ricci scalar equation

120575119877119879120583] + 131198722 [

11987224 120578120583] minus 120597120583120597]] 120575119877 = 0 (26)

which indicates that the traceless Ricci tensor is coupled tothe Ricci scalar

At this stage we observe that (25) may become amasslesspropagating tensor equation for ℎ120583] as

ℎ120583] equiv ℎ119879120583] + 13 (120578120583]

4 minus 120597120583120597]1198722 )ℎ ◻ℎ120583] = 0 (27)

which suggests a way of defining a massless spin-2 in 119891(119877)gravity That is plugging ℎ119879120583] in (27) into (25) leads to ◻ℎ120583] =0 We find from (17) that ℎ119879120583] differs from ℎ120583] by ℎ Inparticular the splitting of ℎ in ℎ120583] is nontrivial which reflectsa feature of 119891(119877) gravity This is compared to that of ℎ120583] inEinstein gravity For this purpose we may express ℎ120583] as

ℎ120583] = ℎ120583] minus 13 (120578120583]

2 + 120597120583120597]1198722 )ℎ (28)

= ℎ120583] minus 13 (120578120583] minus 120597120583120597]

1198722 )ℎ (29)

where ℎ120583] = ℎ120583] minus 120578120583]ℎ2 is the trace-reversed metricperturbation (ℎ = minusℎ) [21] Here the Lorentz gauge is givenby 120597120583ℎ120583] = 0 We may rewrite (20) in terms of ℎ120583]

◻ℎ120583] minus 131198722 [120578120583]◻ minus 120597120583120597]] ◻ℎ = 0 (30)

which is surely the same equation found in [19] Substitutingℎ120583] defined in (29) into that in (30) arrives at ◻ℎ120583] = 0 whichis the same equation as in (27)

Using the trace equation (23) and the Lorentz gauge (18)we may impose the TT condition for ℎ120583]

120597120583ℎ120583] = 0ℎ120583120583 = 0 (31)

which indicates that ℎ120583] is a newly tensor mode defined in119891(119877) gravityThismay imply that119891(119877) gravity accommodatesthree DOF of two from ℎ120583] and one from ℎ We note that ifℎ = 0 ℎ120583] reduces to ℎ119879120583] and to ℎTT120583] finally leading to Einsteingravity

Considering a gravitational wave that propagates in 119911direction (27) together with (31) exhibits the fact that gravi-tational waves have two polarization components Explicitly(31) implies

ℎ119905119905 = ℎ119905119894 = 0ℎ119894119894 = 0

120597119894ℎ119894119895 = 0(32)

where the last two expressions correspond to the TT gaugeℎTT119894119895 = ℎTT119894119895 (119905 minus 119911) is a valid solution to the TT wave equation

◻ℎTT119894119895 = 0 The TT gauge condition of 120597119911ℎTT119911119895 = 0 implies

ℎTT119911119895 (119905minus119911) is constant and however this component should bezero to satisfy a condition of the asymptotic flatness ℎ120583] rarr 0as 119911 rarr infin The remaining nonzero components of ℎTT119894119895 are

given by ℎTT119909119909 ℎTT119909119910 ℎTT119910119909 and ℎTT119910119910 Requiring the symmetry andtraceless condition leads to the two independent components

ℎTT119909119909 = minusℎTT119910119910 equiv ℎ+ (119905 minus 119911) ℎTT119909119910 = ℎTT119910119909 equiv ℎtimes (119905 minus 119911)

(33)

Now considering (23) one finds the trace solution [24]

ℎ = ℎ0119890119894119896120583119909120583 997888rarr ℎ (119905 minus V119866119911) (34)

where V119866 = 119896120596 = radic1205962 minus1198722120596 lt 1 (1205962 = 1198722 + 1198962) is thegroup velocity of a massive scalar graviton

Finally we obtain ℎ120583] as the solution to (24) with (28)

ℎ120583] (119905 119911) = ℎ+ (119905 minus 119911) 119890(+)120583] + ℎtimes (119905 minus 119911) 119890(times)120583]+ 13 (120578120583]

2 minus 119896120583119896]1198722 )ℎ (119905 minus V119866119911)

(35)

The other solution ℎ120583] as the solution to (30) with (29) takesthe form

ℎ120583] (119905 119911) = ℎ+ (119905 minus 119911) 119890(+)120583] + ℎtimes (119905 minus 119911) 119890(times)120583]+ 13 (120578120583] + 119896120583119896]

1198722 )ℎ (119905 minus V119866119911) (36)


which is the same solution found in [19] This encodes thatthree DOF of (ℎ+ ℎtimes ℎ) are found from 119891(119877) gravity

In order to compare (35) and (36) with the Ricci scalar-like solution [20 24 25] we write down its solution byreplacing 120575119877 = minus◻ℎ2 with ℎ119891

ℎ119877120583] (119905 119911) = 119860+ (119905 minus 119911) 119890(+)120583] + 119860times (119905 minus 119911) 119890(times)120583]+ ℎ119891 (119905 minus V119866119911) 120578120583]

(37)

Even though (119860+ 119860times ℎ119891) are similar to (ℎ+ ℎtimes ℎ) the lastterm of solution ℎ119877120583] (37) differs from that of ℎ120583] (35)

On the other side of the scalar-tensor theory (15) togetherwith 120575119877 = 0 reduces to

120575119877119879120583] = 0 997888rarr◻ℎ119879120583] = 0 (38)

Also as was shown in (16) the scalar mode 120593 is decoupledcompletely from ℎ119879120583] Requiring the transverse condition of120597120583ℎ119879120583] = 0 (Lorentz condition with ℎ = 0) leads to twoDOF of (ℎ+ ℎtimes) which describe the general relativity Henceit is obvious that the scalar-tensor theory has three DOF (onescalar DOF + two tensor DOF)

5 Discussions

First of all we note that three DOF of (ℎ+ ℎtimes ℎ) (35) arefound from analyzing the perturbed 119891(119877) gravity Here wedid not introduce theRicci scalarmode (120575119877 = 119877(1)) separatelybecause it is closely related to the trace of metric tensor ℎ Wehave solved the fourth-order coupled equation (24) togetherwith the trace equation (23) directly

We have found that there is no breathing mode in 119891(119877)gravity The four DOF including breathing mode have beenobtained in [20] by assuming that the traceless condition ofℎ = 0 cannot be imposed on the perturbed 119891(119877) gravityafter counting the Ricci scalar modeThe authors in [20] havediscovered the breathing mode (ℎ119887) from the condition ofℎ = 0 when the background space-time is not MinkowskiThe approach used in [20] was based on the observation that119877(1) is considered as a different mode from ℎ initially [25]This might lead to overcounting of DOF However noting anexpression of 120575119877 = minus◻ℎ2 (19) in the Lorentz gauge impliesthat 120575119877 is closely connected to ℎ

One might attempt to argue from (22) that the masslessmode satisfying ◻ℎ = 0 may correspond to the breathingmode In general it seems that the solution to the fourth-order equation (22) is given by the sum of the massive modeand massless mode which are independent of each otherHowever the massless mode which is a solution to ◻ℎ =0 (ℎ = 0) does not exist in 119891(119877) gravity since ◻ℎ = 0 [via(19)] means 120575119877 = 0 in the Lorentz gauge

Consequently we have clarified the issue of DOF in 119891(119877)theories The number of polarization modes of gravitationalwaves in 119891(119877) gravity is still ldquothreerdquo in Minkowski space-time which is consistent with the results in the literature

(especially for [19]) Also we would like to mention that theDOF counting of 119891(119877) theories should be independent ofpropagating space-time

Competing Interests

The author declares no competing interests

Acknowledgments

This work was supported by the 2016 Inje University researchgrant

References

[1] S Nojiri and S D Odintsov ldquoIntroduction to modified gravityand gravitational alternative for dark energyrdquo InternationalJournal of Geometric Methods in Modern Physics vol 4 no 1article 115 2007

[2] T P Sotiriou and V Faraoni ldquof (R) theories of gravityrdquo Reviewsof Modern Physics vol 82 pp 451ndash497 2010

[3] A De Felice and S Tsujikawa ldquof(R) theoriesrdquo Living Reviews inRelativity vol 13 2010

[4] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[5] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 no 2 1999

[6] A G Riess A V Filippenko P Challis et al ldquoObservationalevidence from supernovae for an accelerating universe and acosmological constantrdquo The Astronomical Journal vol 116 no3 p 1009 1998

[7] W Hu and I Sawicki ldquoModels of f (R) cosmic acceleration thatevade solar system testsrdquo Physical Review D vol 76 no 6Article ID 064004 13 pages 2007

[8] X Liu B Li G-B Zhao et al ldquoConstraining f (R) gravity theoryusing weak lensing peak statistics from the Canada-France-Hawaii-Telescope Lensing Surveyrdquo Physical Review Letters vol117 no 5 Article ID 051101 2016

[9] A Starobinsky ldquoA new type of isotropic cosmological modelswithout singularityrdquo Physics Letters B vol 91 no 1 pp 99ndash1021980

[10] BWhitt ldquoFourth-order gravity as general relativity plusmatterrdquoPhysics Letters B vol 145 no 3-4 pp 176ndash178 1984

[11] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[12] P A R Ade N AghanimM Arnaud et al ldquoPlanck 2015 resultsXX Constraints on inflationrdquo Astronomy amp Astrophysics vol594 article A20 65 pages 2016

[13] T Multamaki and I Vilja ldquoSpherically symmetric solutions ofmodified field equations in 119891(119877) theories of gravityrdquo PhysicalReview D vol 74 no 6 Article ID 064022 2006

[14] A de la Cruz-Dombriz A Dobado and A L Maroto ldquoBlackholes in f(119877) theoriesrdquo Physical Review D vol 80 no 12 ArticleID 124011 2009 Erratum to Physical Review D vol 83 no 2Article ID 029903 2011

[15] T Moon Y S Myung and E J Son ldquo119891(119877) black holesrdquoGeneralRelativity and Gravitation vol 43 no 11 pp 3079ndash3098 2011


[16] F S Lobo and M A Oliveira ldquoWormhole geometries in f (R)modified theories of gravityrdquo Physical Review D vol 80 no 102009

[17] K A Bronnikov M V Skvortsova and A A StarobinskyldquoNotes on wormhole existence in scalar-tensor and F(R) grav-ityrdquoGravitation and Cosmology vol 16 no 3 pp 216ndash222 2010

[18] B P Abbott R Abbott T D Abbott et al ldquoObservation ofgravitational waves from a binary black hole mergerrdquo PhysicalReview Letters vol 116 no 6 Article ID 061102 16 pages 2016

[19] E Barausse and T P Sotiriou ldquoPerturbed kerr black holescan probe deviations from general relativityrdquo Physical ReviewLetters vol 101 no 9 Article ID 099001 2008

[20] H R Kausar L Philippoz and P Jetzer ldquoGravitational wavepolarization modes in f (R) theoriesrdquo Physical Review D vol 93no 12 Article ID 124071 2016

[21] E E Flanagan and S A Hughes ldquoThe basics of gravitationalwave theoryrdquo New Journal of Physics vol 7 article 204 2005

[22] Y S Myung ldquoInstability of a rotating black hole in a limitedform of 119891(119877) gravityrdquo Physical Review D vol 84 no 2 ArticleID 024048 2011

[23] Y S Myung T Moon and E J Son ldquoStability of f (R) blackholesrdquo Physical Review D vol 83 no 12 Article ID 124009 7pages 2011

[24] S Capozziello C Corda and M F De Laurentis ldquoMassivegravitational waves from f (R) theories of gravity potentialdetection with LISArdquo Physics Letters B vol 669 no 5 pp 255ndash259 2008

[25] C P L Berry and J R Gair ldquoLinearized 119891(119877) gravity gravita-tional radiation and Solar System testsrdquo Physical Review D vol83 no 10 Article ID 104022 2011 Erratum to Physical ReviewD vol 85 no 8 Article ID 089906 2012

Submit your manuscripts athttpwwwhindawicom

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equation composed of a second-order tensor and a fourth-order scalar

It seems that there is no breathing mode because theperturbed Ricci scalar 119877(1) is related closely to the traceldquoℎrdquo of perturbed metric tensor Hence the allocation of theRicci scalar as a newly scalar represents the trace of metrictensor Also we wish to remind the reader that the Ricciscalar equation is not an independent equation and is notseparated from the perturbed Einstein equation because itcomes out just from taking the trace of the latter equationThis implies that the Ricci scalar is an emergent mode fromℎ Furthermore it is instructive to note that in the TT gaugethere is a close connection between the metric perturbationand the linearized Riemann tensor implying that 120575119877119894119905119895119905 =minusℎTT119894119895 2 [21] This gauge is very convenient because it fixesall local gauge freedoms But it might be unclear that thereexists a close relation between the metric perturbation andthe NP formalism unless one chooses the TT gauge Onecould not naively choose the TT gauge in the perturbed 119891(119877)gravity because of ℎ = 0 whereas the Lorentz gauge iseasily implemented to eliminate the gauge DOF Howeverone might choose the TT gauge to obtain a massless spin-2in the perturbed 119891(119877) gravity when one introduces a newlymetric perturbation ℎ120583]

Theorganization of ourwork is as follows In Section 2webriefly describe 119891(119877) gravity and its scalar-tensor theory andderive two sets of perturbed equations around theMinkowskibackground in Section 3 Section 4 is focused on obtainingthe number of propagating DOF when one chooses theLorentz gauge Finally we will discuss our result which showsthat there is no breathing mode in Section 5

2 119891(119877) Gravity and Its Scalar-Tensor Theory

Instead of a polynomial model of 119891(119877) = 119877 + 1205721198772 + 1205731198773 +sdot sdot sdot [20 22] we start with a specific 119891(119877) gravity (Starobinskymodel [9])

119878119891 = 11987221198752 int1198894119909radicminus119892119891 (119877) 119891 (119877) = 119877 + 119877261198722 (1)

where the 1198772-term was originally motivated by one-loopcorrection to Einstein gravity Here the mass parameter 1198722is chosen to be a positive value which is consistent withthe stability condition of 11989110158401015840(0) gt 0 [2] This model of119891(119877) gravity is enough to find the propagating DOF aroundthe Minkowski background The Einstein equation takes theform

119877120583]1198911015840 (119877) minus 12119892120583]119891 (119877) + (119892120583]nabla2 minus nabla120583nabla]) 1198911015840 (119877) = 0 (2)

where the prime (1015840) denotes the differentiation with respectto its argument

On the other hand one might represent (1) as a scalar-tensor theory by introducing an auxiliary field 120595 [10]

119878119860 = int1198894119909radicminus119892119869 [11987221198752 119877 + 119872119875119872 119877120595 minus 31205952] (3)

where the superscript 119869 means the Jordan frame Varying 119878119860with respect to 120595 provides

120595 = 1198721198756119872119877 (4)

whichmeans that the Ricci scalar is treated as an independentscalar degree of freedom Plugging (4) into 119878119860 again leads tothe original 119891(119877) gravity (1) exactly

Making use of the conformal transformation and redefin-ing the scalar field (120595 rarr 120601)

119892119869120583] 997888rarr 11 + 2120595119872119872119875119892

119864120583] 997888rarr 119890minusradic23(120601119872119875)119892119864120583] (5)

one arrives at the Starobinskymodel in the Einstein frame [11]

119878119878 = int1198894119909radicminus119892119864 [11987221198752 119877 minus 12120597120583120601120597120583120601 minus 119881119878 (120601)] (6)

with the Starobinsky potential

119881119878 (120601) = 3119872211987511987224 [1 minus 119890minusradic23(120601119872119875)]2 (7)

At this stage we note that the conformal transformation (5) isa purely classified transformation of coordinates and resultsin one frame are classically equivalent to the ones obtained inother frame Hence it is plausible that the number of DOF inthe scalar-tensor theory (6) is three because of two TT tensormodes and one scalarmode From (6) the Einstein and scalarequations are derived as

119866120583] = 11198722119875119879120601120583]

119879120601120583] = 120597120583120601120597]120601 minus 12119892120583] [(120597120601)

2 + 119881119878] nabla2120601 minus 1198811015840119878 = 0

1198811015840119878 = radic321198721198751198722119890minusradic23(120601119872119875) [1 minus 119890minusradic23(120601119872119875)]

(8)

The above describes a process of [1198772 rarr 119877120595minus31205952 rarr minus(120597120601)2minus119881] briefly3 Two Sets of Perturbed Equations

We introduce themetric perturbation around theMinkowskibackground to find out the propagating DOF

119892120583] = 120578120583] + ℎ120583] (9)

The Taylor expansions around 119877 = 0 are employed to definethe linearized Ricci scalar 120575119877(ℎ) as [23]

119891 (119877) = 119891 (0) + 1198911015840 (0) 120575119877 (ℎ) + sdot sdot sdot 1198911015840 (119877) = 1198911015840 (0) + 11989110158401015840 (0) 120575119877 (ℎ) + sdot sdot sdot (10)



120575119877120583] (ℎ) + 131198722 [120578120583] (minus

311987222 + ◻) minus 120597120583120597]] 120575119877 (ℎ)

= 0 ◻ = 1205972(11)



120575119877 (ℎ) = 120597120588120597120590ℎ120588120590 minus ◻ℎ(12)


(◻ minus1198722) 120575119877 = 0 (13)


120575119866120583] (ℎ) + 131198722 [120578120583]◻ minus 120597120583120597]] 120575119877 (ℎ) = 0 (14)



120575119877120583] (ℎ) = 0 (15)

(◻ minus1198722) 120593 = 0 (16)



ℎ120583] = ℎ119879120583] + ℎ4120578120583] (17)



120597120583ℎ120583] = 12120597]ℎ 997888rarr

120597120583ℎ119879120583] = 14120597]ℎ

(18)


120575119877 (ℎ)120583] equiv 120575119877119879120583] + 120578120583]4 120575119877 = minus12 [◻ℎ119879120583] + 120578120583]

4 ◻ℎ] 120575119877 (ℎ) = minus12◻ℎ

(19)



◻ℎ120583] + 131198722 [120578120583] (minus

311987222 + ◻) minus 120597120583120597]]◻ℎ = 0 (20)


◻ℎ119879120583] + 131198722 [120578120583] (minus

311987224 + ◻) minus 120597120583120597]]◻ℎ = 0 (21)


(◻ minus1198722) ◻ℎ = 0 (22)


(◻ minus1198722) ℎ = 0 (23)





◻ℎ120583] minus 131198722◻[11987222 120578120583] + 120597120583120597]] ℎ = 0 (24)


◻ℎ119879120583] + 131198722◻[11987224 120578120583] minus 120597120583120597]] ℎ = 0 (25)


120575119877119879120583] + 131198722 [

11987224 120578120583] minus 120597120583120597]] 120575119877 = 0 (26)



ℎ120583] equiv ℎ119879120583] + 13 (120578120583]

4 minus 120597120583120597]1198722 )ℎ ◻ℎ120583] = 0 (27)


ℎ120583] = ℎ120583] minus 13 (120578120583]

2 + 120597120583120597]1198722 )ℎ (28)

= ℎ120583] minus 13 (120578120583] minus 120597120583120597]

1198722 )ℎ (29)


◻ℎ120583] minus 131198722 [120578120583]◻ minus 120597120583120597]] ◻ℎ = 0 (30)



120597120583ℎ120583] = 0ℎ120583120583 = 0 (31)



ℎ119905119905 = ℎ119905119894 = 0ℎ119894119894 = 0

120597119894ℎ119894119895 = 0(32)






(33)


ℎ = ℎ0119890119894119896120583119909120583 997888rarr ℎ (119905 minus V119866119911) (34)




2 minus 119896120583119896]1198722 )ℎ (119905 minus V119866119911)

(35)



1198722 )ℎ (119905 minus V119866119911) (36)





(37)



120575119877119879120583] = 0 997888rarr◻ℎ119879120583] = 0 (38)


5 Discussions






Competing Interests


Acknowledgments


References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120575119877120583] (ℎ) + 131198722 [120578120583] (minus

311987222 + ◻) minus 120597120583120597]] 120575119877 (ℎ)

= 0 ◻ = 1205972(11)



120575119877 (ℎ) = 120597120588120597120590ℎ120588120590 minus ◻ℎ(12)


(◻ minus1198722) 120575119877 = 0 (13)


120575119866120583] (ℎ) + 131198722 [120578120583]◻ minus 120597120583120597]] 120575119877 (ℎ) = 0 (14)



120575119877120583] (ℎ) = 0 (15)

(◻ minus1198722) 120593 = 0 (16)



ℎ120583] = ℎ119879120583] + ℎ4120578120583] (17)



120597120583ℎ120583] = 12120597]ℎ 997888rarr

120597120583ℎ119879120583] = 14120597]ℎ

(18)


120575119877 (ℎ)120583] equiv 120575119877119879120583] + 120578120583]4 120575119877 = minus12 [◻ℎ119879120583] + 120578120583]

4 ◻ℎ] 120575119877 (ℎ) = minus12◻ℎ

(19)



◻ℎ120583] + 131198722 [120578120583] (minus

311987222 + ◻) minus 120597120583120597]]◻ℎ = 0 (20)


◻ℎ119879120583] + 131198722 [120578120583] (minus

311987224 + ◻) minus 120597120583120597]]◻ℎ = 0 (21)


(◻ minus1198722) ◻ℎ = 0 (22)


(◻ minus1198722) ℎ = 0 (23)





◻ℎ120583] minus 131198722◻[11987222 120578120583] + 120597120583120597]] ℎ = 0 (24)


◻ℎ119879120583] + 131198722◻[11987224 120578120583] minus 120597120583120597]] ℎ = 0 (25)


120575119877119879120583] + 131198722 [

11987224 120578120583] minus 120597120583120597]] 120575119877 = 0 (26)



ℎ120583] equiv ℎ119879120583] + 13 (120578120583]

4 minus 120597120583120597]1198722 )ℎ ◻ℎ120583] = 0 (27)


ℎ120583] = ℎ120583] minus 13 (120578120583]

2 + 120597120583120597]1198722 )ℎ (28)

= ℎ120583] minus 13 (120578120583] minus 120597120583120597]

1198722 )ℎ (29)


◻ℎ120583] minus 131198722 [120578120583]◻ minus 120597120583120597]] ◻ℎ = 0 (30)



120597120583ℎ120583] = 0ℎ120583120583 = 0 (31)



ℎ119905119905 = ℎ119905119894 = 0ℎ119894119894 = 0

120597119894ℎ119894119895 = 0(32)






(33)


ℎ = ℎ0119890119894119896120583119909120583 997888rarr ℎ (119905 minus V119866119911) (34)




2 minus 119896120583119896]1198722 )ℎ (119905 minus V119866119911)

(35)



1198722 )ℎ (119905 minus V119866119911) (36)





(37)



120575119877119879120583] = 0 997888rarr◻ℎ119879120583] = 0 (38)


5 Discussions






Competing Interests


Acknowledgments


References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






◻ℎ120583] minus 131198722◻[11987222 120578120583] + 120597120583120597]] ℎ = 0 (24)


◻ℎ119879120583] + 131198722◻[11987224 120578120583] minus 120597120583120597]] ℎ = 0 (25)


120575119877119879120583] + 131198722 [

11987224 120578120583] minus 120597120583120597]] 120575119877 = 0 (26)



ℎ120583] equiv ℎ119879120583] + 13 (120578120583]

4 minus 120597120583120597]1198722 )ℎ ◻ℎ120583] = 0 (27)


ℎ120583] = ℎ120583] minus 13 (120578120583]

2 + 120597120583120597]1198722 )ℎ (28)

= ℎ120583] minus 13 (120578120583] minus 120597120583120597]

1198722 )ℎ (29)


◻ℎ120583] minus 131198722 [120578120583]◻ minus 120597120583120597]] ◻ℎ = 0 (30)



120597120583ℎ120583] = 0ℎ120583120583 = 0 (31)



ℎ119905119905 = ℎ119905119894 = 0ℎ119894119894 = 0

120597119894ℎ119894119895 = 0(32)






(33)


ℎ = ℎ0119890119894119896120583119909120583 997888rarr ℎ (119905 minus V119866119911) (34)




2 minus 119896120583119896]1198722 )ℎ (119905 minus V119866119911)

(35)



1198722 )ℎ (119905 minus V119866119911) (36)





(37)



120575119877119879120583] = 0 997888rarr◻ℎ119879120583] = 0 (38)


5 Discussions






Competing Interests


Acknowledgments


References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







(37)



120575119877119879120583] = 0 997888rarr◻ℎ119879120583] = 0 (38)


5 Discussions






Competing Interests


Acknowledgments


References
































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article propagating degrees of freedom in ()...

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