Would an alternative gravity theory developed from an improved gravitational action approach includes negative kinetic energy dynamic degrees of freedom?

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Results in Physics

journal homepage:www.elsevier.com/locate/rinp

Would an alternative gravity theory developed from an improved

gravitational action approach includes negative kinetic energy dynamic

degrees of freedom?

Veysel Binbay, F. Figen Binbay

⁎

Department of Physics, Faculty of Science, Dicle University, 21280 Diyarbakir, Turkey

A R T I C L E I N F O Keywords: General relativity Alternative models Ghost states A B S T R A C T

The observed unexpected accelerating expansion of the universe, by Riess and his collaborators in 1998, has become one of the most important problems of the contemporary physics. A considerable eﬀort has been spent by theoretical physicists to explain this observation for a while. When one looks at these attempts more closely, two of approaches attract attention: (i) Multi-dimensional alternative gravity models, (ii) Approaches which takes the more general and complex action than it is original Einstein-Hilbert form, which had been given as Ricci scalar R. The second type of these approaches must be examined carefully, because they could be gen-erically involved dynamical degrees of freedom which possess negative kinetic energy (shortly called as‘ghost states’ or simply ‘ghosts’). In this work, an alternative theory has been studied to understand if it contains ghosts or not. This alternative approach belongs to the second type of the approaches which mentioned above, and it is given as:Sgravity=∫d x4 −g f R R R( , μν μλνρRλρ)whereStotal=Sgravity+Smatter. And this model has been examined by

this way to see if this speciﬁc alternative model could be used to explain the present time acceleration of the universe or not.

Introduction

The general relativity theory is expressed by the Einstein equation given in the following[1]:

− = R Rg πG c T 1 2 8 μν μν ₄ μν _(1.1)

whereRμν,R, gμν, Tμν,c and G denotes the Ricci tensor, Ricci scalar,

metric tensor, pressure-energy-momentum tensor, speed of light in vacuum and the universal gravitational constant, respectively. How-ever, when it was understood that the equation did not allow a sta-tionary universe in this way, in 1917 the Einstein equation was added a parameter (famous cosmological constant) that would allow it to be stationary[2]. It is understandable that Einstein does this, because in those years there is no clue as to the idea that the universe is not static. Therefore, if the equations lead to dynamic universe models, it is per-ceived as a problem to be corrected. Einstein’s equations were trans-formed into the following new form

− + = R Rg g πG c T 1 2 Λ 8 . μν μν μν ₄ μν (1.2) The cosmological constant works as a parameter that balances the

recalling eﬀect of gravitational force created by matter-energy in the universe, creating a repulsive eﬀect[2]. In 1929, the famous astron-omer Edwin Hubble discovered that the universe was expanding[3]. Therefore, there is no need for correction in Eq. (1.2), i.e., cosmological constant, and Einstein re-extracted it from the Eq. (1.2). However, al-though Einstein excluded the cosmological constant from the Eq. (1.2), the discussions on the cosmological constant have grown by day-to-day until now. Cosmological constant problem is one of the biggest pro-blems of today's physics.

However, in 1998 the observation[3]that the rate of expansion of the universe increased, i.e., accelerated, led to a fundamental change in this situation[3–10]. If the big bang is considered as theﬁrst accel-eration, this observation, pointing to the second known accelaccel-eration, has become a major focus of theoretical physicists. This has been a very unexpected development. Because the theoretical expectation during the universe's enlargement process is not a rapid increase in the rate of expansion because of the recalling attraction of matter-energy in the spread of time. Attention has been drawn to the theory of general re-lativity to explain this after the observation. It is not possible to explain this observation using the original form of the General Relativity equations. Hence, it has begun to intensify the agenda of the theoretical physicists to question how it can be modiﬁed to account for this

https://doi.org/10.1016/j.rinp.2018.05.021

Received 20 March 2018; Received in revised form 9 May 2018; Accepted 10 May 2018

⁎_{Corresponding author.}

E-mail address:ﬁgenbinbay@hotmail.com(F.F. Binbay).

Available online 17 May 2018

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surprising development, at the same time, without disturbing the in-tegrity and coherence of the equations, by continuing to account for events which can be well explained with at least the same degree of truth. The goal is to obtain an alternative theory of gravity that can explain Einstein's equation and explain how the universe accelerated in time. As far as we can understand[10], only about 4.9% of our universe consists of ordinary matter (baryonic matter) we know. The remaining part consists of 27.8% dark matter and 67.3% dark energy, which we do not know much about. As it is understood, dark matter-energy problem and cosmological constant problem are closely related problems. Thence, the earliest theoretical candidate of the dark energy is the fa-mous cosmological constant. Although, the cosmological constant with cold dark matter (ΛCDM) scenario yields excellent conclusions for the accelerated expansion phenomenon suffers from some issues such as the fine-tuning and cosmological coincidence. From this point of view, many physicists have tried to obtain different proposals in order to explain the speedy expansion phase theoretically: scalarfield minimally coupled with gravity[11–14], unified dark matter-energy expressions [15–20], assuming the existence of extra dimensions [21–26] and modified gravity models[27–38].

While these eﬀorts have brought about various approaches, two approaches have come to the forefront:

1) Approaches with extra dimensions,

2) Approaches that use more complicated expressions instead of cur-vature scalar R contained in the 4-dimensional action expression. One of the main diﬀerences between the two approaches is the in-ﬁnite number of degrees of freedom of the extra dimensions in the ef-fective 4-dimensional theorem [39]. Kaluza-Klein theory and DGP model are examples of this approach[40]. The second group approach, on the other hand, is the total action expression given in the original form of the general theory of relativity, as below.

= +

Stotal Sgravity Smatter (1.4)

where

∫

= −

Sgravity d x4 g R. _(1.5)

As given and in the act of gravity, also known as the Einstein-Hilbert action, instead of the Ricci scalar chosen asRas the simplest possible form, are approaches based on more complex functions of R that can be expressed by f R( ). In this case, the gravitational action is expressed in the following manner:

∫

= −

Sgravity d x4 g f R( ) _(1.6)

As is known, it is possible to obtain Einsteinﬁeld equations given by (1.1) byﬁnding the extremum points of the obtained expression by taking the variation of the action according to the metric tensor with the Palatini approach, starting from the original action expression given by Eq.(1.5) [2]. Moreover, when a more complex function ofR, such as

f R( ), is taken instead of the Ricci scaleRin the action expression, the field equations obtained by variation management in general will be different from the Einstein motion equations given by the Eq.(1.1). Already the goal of the approach is to solve the problem of accelerating expansion using this difference. However, it has recently been shown that such a modification corresponds to or is reduced to Einstein's ori-ginal theory by adding an extra scalar[41–43]. In this case, instead of the Ricci scalar R in the original action expression, a more general variation like f R R R( , μν μν,RμναβRμναβ, )… involving combinations of Ricci and Riemann tensors has been introduced [43,44]. Studies on the subject showed that; making such modifications in the expression of action may lead to situations that have negative normality and there-fore negative probability, negative energy in the term kinetic energy, and move forward in time. The existence of these conditions, which correspond to physically unacceptable incoherent and unstable

solutions, removes the coherence of the theory that includes them. Therefore, in alternative gravitational theory, which is a candidate for solving the problem of accelerating expansion of the universe, which will be obtained by modifying the original action expression, a con-sistency control mustﬁrst be made whether such situations are included or not. These conditions are also referred to brieﬂy as ghosts in the literature.

Two scientists from New York University, A. Nunez and S. Solganik [43], prove that an alternative gravitation theory, which can be ob-tained by choosing a function f R R R( , μν μν,RμναβRμναβ)instead ofR, is the most common case and therefore not useful. Moreover, even if this can be removed with specialﬁne parameter adjustments, they show that the theory to be obtained by this adjustment is reduced to scalar-tensor gravitational theories, and so it is again insuﬃcient[45].

The theme of this work is to check whether an original alternative gravitational theory which is a candidate for solving the problem of the accelerating expansion of the universe involves primarily negative ki-netic energetic conditions that lead to inconsistency. If the theory seems consistent in this respect, then it is possible to discuss how the para-meters of the alternative action function can be adjusted so that the universe can account for the acceleration problem in terms of ob-servation values[46].

The alternative theory to be investigated deals with a function chosen as f R R R( , μν μναβRαβ) instead ofR in the original action expres-sion.

The model

The action for the theory of gravity is generally expressed as below:

= +

S Sgravity Smatter (3.1)

Here Sgravitystands for the gravitational part of the total action while

Smatter indicates the matter part including radiation. In the general

theory of relativity, the action of gravity is given by the following re-lation

∫

= −

Sgravity d x4 g R _(3.2)

which is also known as the Einstein-Hilbert action. The main task of this study is to make a consistency check by investigating whether the al-ternative action approach, originally to be considered as follows, has led to dynamic freedom degrees with negative kinetic energy; this al-ternative approach is to show if there is a candidate approach to explain the problem of the accelerating expansion of the universe. Hence, we focus on the following extended form of the gravitational action

∫

= −

Sgravity d x4 g f R R R( , μν μλνρRλρ). _(3.4)

In this signiﬁcant approach, the total action transforms the fol-lowing new version:

∫

= − +

Stotal d x4 g f R R R( , μν μλνρRλρ) Smatter. _(3.5)

To control the presence of negative kinetic energetic states, equa-tions of motion willﬁrst be obtained using the Palatini approach. For this purpose, we need to take the variation according to the metric tensor of the action expression and equal to zero. If the variation of action is taken, then it is found that

∫

= − +

δS δ d x4 g f R R R( , μν μλνρRλρ) δSmatter _(3.6)

Theﬁrst term on the right-hand side of the equation is calculated as =

R Rμν μλνρRλρ P. Consequently, one can write

∫

− =

∫

− +

∫

−

δ d x4 g f R R R( , μν μλνρRλρ) d xf R P δ4 ( , ) g d x4 g δf R P( , ) (3.7) The following expression can be written for theﬁrst term on the right side of the equation

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∫

d x δ( −g f) = −1

∫

d x −g g δg f

2 μν

μν

4 4

(3.8) wheref R P( , )=f. On the other hand, the following expression can be written for the second term on the right side of Eq.(3.7):

∫

d x4 −g δf R P( , )=

∫

d x4 −g f δR_R +

∫

d x4 −g f δP(_P ), (3.9) where =∂ ∂ f_R f R P R ( , ) and =∂ ∂ f_P f R P P ( , )

. If theﬁrst term is calculated by re-membering that it is given as R=g Rμν

μνat this point, the following

expression is obtained:

∫

d x4 −g f δR(_r )=

∫

d x4 −g f R δg_R μν μν+

∫

d x4 −g f g δRR μν μν

(3.10) Next, after the second term in the above result is computed, the following relation is obtained:

∫

d x −g f g δR_R μν =

∫

d x −g f [V ] μν R α α 4 4 ; _(3.11) where − = gμα( Γ ) (δ g δΓ ) V μτ τ μν μν α α (3.12) .

While performing the above calculation, the following Palatini equation is used[2]:

= −

δRμν ( Γ )δ τμτ;ν ( Γ )δ τμν;τ (3.13) .

Now, if partial integration is applied to equation(3.10), one can obtain the following result

∫

d x −g f_R[Vα] = −

∫

d x −g f[ ] V

α R α α

4

; 4 ; _(3.14)

Note that here we use

= + − δΓ 1g δg δg δg 2 ( ) μν α αλ μλ ν; νλ μ; μν λ; _(3.15)

If Christoﬀel’s symbol is used, we reach the following expression

= − = − Vα gμα( Γ ) (δ g δΓ ) δg g δg μτ τ μν μν α β αβ μν μν α ; ; (3.16) It can be seen that in this interesting case the Eq. (3.11) still contains derivatives of δgμν_{. If we proceed from Eq.}_(3.14)_{to get rid of this term,}

the following expression is found:

∫

d x −g f[ ]_{R α}Vα= −

∫

d x −g g δg [ ]f +

∫

d x −g δg [ ]f

μν μν R αα μν R μν

4

; 4 ;; 4 ;

(3.17) After substituting our results into the Eq. (3.6), the following equation is obtained collectively:

∫

− − + − − − + − + − = − d x g g δg f d x g g δg f d x g δg f d x g f R δg d x g f δP d x g δg T 1 2 [ ] [ ] ( ) μν μν μν μν R αα μν R μν R μν μν R μν μν 4 4 ; ; 4 ; 4 4 4 (3.18) Here, we assume

∫

= − δSmatter d x4 g δg Tμν μν

In this case, only the last term on the left side of the equation re-mains to be calculated so that Eq.(3.18)can be modiﬁed. Subsequently, we get

∫

− = − + − ∇ − − ∇ ∇ − − ∇ ∇ d x g f δP d x g δg R R R f d x g δg g f R R d x g δg g f R R d x g δg f R R { ( ) ( ) ( ) 2 ( ) P μν ζη μζηρ νρ P μν μν R λρ λρ μν μν ζ η P ζληρ λρ μν μ ν P λρ λρ 4 4 4 2 4 4 (3.19) If these calculations are substituted in the Eq.(3.18), the following

conclusion is obtained[46] + ∇ ∇ − ∇ − + + ∇ − ∇ ∇ − ∇ ∇ = fg g R f R R R f g f R R g f R R f R R T 1 2 ( ) [ ] ( ) 2 ( ) μν μ ν μν μν R ζη μζηρ νρ P μν P λρ λρ μν ζ η P ζληρ λρ μ ν P λρ λρ μν 2 2 (3.20) which denotes the original equation of motion obtained for the pro-posed alternative gravitational action. Whether this equation of motion leads to negative kinetic energy states can be understood by obtaining propagator expression[46]. For this aim, the equation of motion needs to be linearized over the maximum symmetric space-time with constant curvature.R=R0in the case of a maximum symmetrical background

with constant curvature, Riemann and Ricci tensors are given as follows [2,43]: = − R R g g g g 12( ) λμνσ λν μσ λσ μν 0 (3.21) = R R g 4 μν μν 0 (3.22) In order to linearize the equation of motion (3.20), we can choose the metric as follows:

= +

g_μν g_μν(0) hμν. _(3.23)

Here, g_μν(0)_{shows the solution corresponding to the maximum}

sym-metric space-time with constant curvature of the equation of motion =

R R0, hμν is considered as a small perturbation[43]. To reach the

propagator expression we need to open the Eq. (3.20) to include only the linear components of hμν. When we do this, we get to an

arrange-ment like the one below from the Eq. (3.20): =

Oμναβhαβ Tμν (3.24)

where Oμναβ is known as the reverse propagator expression. The

pro-pagator expression can be reached by taking the inverse of this op-erator. However, the presence or absence of negative kinetic energy states can be understood from the reverse propagators instead of the propagator itself. Terms with a higher derivative than the second order in the reverse propagator lead to the existence of negative kinetic en-ergy states[43].

It is clear that theﬁrst term in the equation of motion (3.20) will not produce components containing higher order derivatives of hμν.

Therefore, if the second term is considered, the following expression is obtained: ∇ ∇ = ∂ ∇ − ∇ = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ − ∂ + ∂ f f f f R f R f P f P f R f P ( ) Γ ( ) ( )( ) ( ) ( )( ) ( ) Γ ( ) μ ν R μ ν R μνλ ν R μ RR ν RR μ ν μ RP ν RP μ ν μνλ RR ν RP ν (3.25) where we define = ∂ ∂ = ∂ ∂ ∂ = ∂ ∂ −∇ f f R f f R PR h h RR RP α β αβ 2 2 2 2 (3.26) Only nonlinear components from thefirst, third and fifth terms on the right side of Eq. (3.25) will be neglected. Therefore, the second and fourth terms that produce the remaining and linear terms must be calculated to linearize the equation. In this case, the Eq. (3.25) can be written as: ∇ ∇ = ∂ ∂ ∂ ∂ − ∂ ∂ ∇ + ∂ ∂ ∂ ∂ − ∂ ∂ ∇ f f h g h R f h g h ( ) 3 16 ( ) μ ν R RR μ ν α β αβ αβ μ ν αβ RP μ ν α β αβ αβ μ ν αβ 2 2 2 (3.27) where ∇ = ∇ ∇2 α αand ∇ = ∇ ∇ ∇ ∇2 α α β β. The resulting form appears to

consist of terms containing linear and fourth order derivatives. As noted earlier, higher order terms in the second order can lead to the existence of negative kinetic energy states; all terms with higher order derivatives that are obtained by linearizing the equation of motion together with these terms should be considered together. If we take the third term of the motion Eq. (3.20), we get

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− ∇ = − ∇ ∂ ∂ + ∇ − ∇ ∂ ∂ + ∇ g f g f h g g f h g R f h R g g f h 3 16 3 16 μν R μν RR α β αβ μν αβ RR αβ μν RP α β αβ μν αβ RP αβ 2 2 4 2 2 2 4 (3.28) If the sixth term in the equation of motion (3.20) is taken into ac-count, the following expression is reached:

⎜ ⎟ ∇ = ⎛ ⎝ + ⎞ ⎠ ∇ ∂ ∂ −∇ + ∇ g f R R g r f R f h h g g f R R ( ) 4 3 16 ( ( )) ( ). μν P λρ λρ μν PR PP α β αβ μν αβR αβ 2 2 2 2 2 2 (3.29) At this point, for the equation we are trying to linearize, we have

= ∂ ∂ + ∂ ∂ −∂ ∂ −∇ R 1 h h h h 2( ) αβ σ β ασ σ α βσ α β 2 αβ _(3.30) And = ∂ ∂ + ∂ ∂ −∂ ∂ −∂ ∂ R 1 h h h h 2( ). μνρσ ρ ν μσ σ μ νρ σ ν μρ ρ μ νσ _(3.31)

So, one can reach the following result

∇ = ∇ ∂ ∂ + ∇ ∂ ∂ − ∇ − ∇ + ∇ ∂ ∂ − ∇ g f R R R f g h R f g h R f g g h R f g g h f Rg h f Rg g h ( ) 4 3 64 4 3 64 . μν P λρ λρ PR μν α β αβ PP μν α β αβ PR μν αβ αβ PP μν αβ αβ P μν α β αβ P μν αβ αβ 2 2 2 4 2 2 4 4 4 2 4 (3.32) Making use of the 7th term in the equation of motion (3.20), one can get

−g_{μν ε}∇ ∇η(f RP εληρRλρ)= −gμν( (∂ ∇ε η(f RP εληρRλρ)))+Γ (εεγ∇η(f RR γληρRλρ)). (3.33) It is important to mention here that the second term here is ne-glected because it produces non-linear contributions. Therefore, it can be found that − ∇ ∇ = − ∂ ∂ + + g f R R g f R R f R R f R R ( ) ( ( ( ))) Γ ( ) Γ ( ). μλ ε η P εληρ λρ μν ε η P εληρ λρ ηγε P γληρ λρ ηθ η P ελθρ λρ _(3.34)

Again, the second and third terms are neglected because they do not produce linear components. Hence, the following expression is ob-tained: ∇ = − ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ∇ g η f R R g f R R f R R f R R f R R ( ) (( ) ( )( ( ))) ( )( ( )) ( ). μν P εληρ λρ μν ε η P εληρ λρ η P ε εληρ λρ ε P η εληρ λρ P ε η εληρ λρ (3.35)

Here, the linear component contributions from the second and third terms can be neglected. From this point of view, we get the following expression ⎜ ⎟ − ∇ ∇ = − ⎛ ⎝ ∂ ∂ + ∂ ∂ ⎞ ⎠ − ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ g f R R g R R f R R f R g f R R R R R R R R ( ) 3 16 (( ) ( )( ) ( )( ) ( )). μν ε η P εληρ λρ μν εληρ λρ PR ε η PP ε η μν P ε η εληρ λρ ρ εληρ ε λρ ε εληρ η λρ εληρ ε η λρ 2 (3.36) Note that, in the above result, the fourth andﬁfth terms were ne-glected because they give non-linear components. As a result, weﬁnd

⎜ ⎟ − ⎛ ⎝ ∂ ∂ + ∂ ∂ ⎞ ⎠ − ∂ ∂ + ∂ ∂ = − ∇ ∂ ∂ + ∇ − ∇ ∂ ∂ + ∇ − ∇ ∂ ∂ + ∇ g R R f R R f R g f R R R R Rf g h R f g g h R f g h R f g g h R f g h R f g g h 3 16 (( ) ( )) 3 3 16 16 3 256 3 256 . μν εληρ λρ PR ε η PP ε η μν P ε η εληρ λρ εληρ ε η λρ P μν α β αβ P μν αβ αβ RP μν α β αβ RP μν αβ αβ PP μν α β αβ PP μν αβ αβ 2 2 4 2 2 2 4 4 2 4 4 (3.37) If we focus on the 8th term in the equation of motion, we can easily get

− ∇ ∇2 μ ν(f R RP λρ λρ)= − ∂ ∇2 (μ ν(f R RP λρ λρ)) 2Γ (− μνγ ∇γ(f R RP λρ λρ)) (3.38) In is seen that the 2nd term introduces non-linear components, thus omitting this term yields

− ∇ ∇ = − ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ f R R f R R f R R R f R R f R R 2 ( ) 2(( ) ( ) ( ) ( ) ( ) ( )). μ ν P λρ λρ μ ν P λρ λρ ν P μ λρ λρ λρ μ P ν λρ λρ P μ ν λρ λρ (3.39)

Now, only non-linear contributions from the second and third terms have been neglected. In this case, we reach the following result

− ∇ ∇ = − ∂ ∂ ∂ ∂ + ∂ ∂ ∇ − ∂ ∂ ∂ ∂ + ∂ ∂ ∇ − ∂ ∂ ∂ ∂ + ∂ ∂ ∇ f R R Rf h Rf g h R f h R f g h R f h R f g h 2 ( ) 2 2 1 2 1 2 3 32 3 32 . μ ν P λρ λρ P μ ν α β αβ P αβ μ ν αβ PR μ ν α β αβ PR αβ μ ν αβ PP μ ν α β αβ PP αβ μ ν αβ 2 2 2 2 4 4 2 (3.40) Not that, in this way, all the terms that are derived from the line-arization of the equation of motion (3.20) including the derivatives of the higher order (fourth order) are calculated carefully.

Discussions

In this study, we focused on an interesting idea which can be a candidate to solve the problem of the accelerating expansion phase of our universe. The possibility of solving the problem in a consistent manner is investigated by using an alternative gravitational theory using an improved action expression instead of the original Einstein-Hilbert description. For this purpose, it has beenﬁrstly checked whe-ther the proposed theory leads to the existence of negative kinetic en-ergy dynamic degrees of freedom corresponding to physical incon-sistencies, which can be caused by such alternative action approaches. However, it has been understood that the analysis leads to the existence of these undesirable conditions, which are considered as inconsistencies in the alternative theory. Therefore, it has been concluded that the proposed theory cannot produce a consistent solution to the problem of the accelerating expansion of the universe.

It has been observed that negative kinetic energetic states can only be removed if R itself or f_Pis zero. However, the fact that these two alternatives are unacceptable is also the result of the fact that the si-tuation is inevitable for this approach. Although theR=0 state seems to oﬀer a consistent alternative for at least space, it is not possible for space-time to remain stable and consistent in any curvature case. Moreover, given the quantum vacuumﬂuctuations, it is understandable that the local curvature cannot remain steadily constant even in space. When assessed in conjunction with the previous work on the sub-ject, it has generally been understood that approaches with such action generally lead to the existence of such inconsistent situations. Therefore, it is suggested to use super-symmetric models with extra dimensions approaches or a much more diverse alternative gravity approaches to solve the speedy expansion puzzle.

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Acknowledgments

We would like to thank Prof. Dr. Durmuş Ali DEMİR for oﬀering this current and pleasurable problem and Dr. Nurettin PİRİNÇÇİOĞLU for his valuable and indispensable contributions.

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