On the Energy Conditions in the Non-Compact Kaluza-Klein Gravity

S. M. M. Rasouli and S. Jalalzadeh

Department of Physics, Shahid Beheshti University, Evin,Tehran-Iran

Grasscosmofun'09 ,Szczecin-Poland

Abstract

We investigate a few cosmological solutions in non-compact five-dimensional space-time which is both Ricci-flat and Conformally flat. We then study the reduced solutions on the brane in detail by employing the four energy conditions, which categorize our model in some classes. These classes of solutions can describe an universe including accelerating expansion, a phantom model and a Stephani universe.

• Equations In Space-Time-Matter Theory

• A Kaluza–Klein metric and its relations

• Energy conditions and deceleration parameter

[General Relativity and Gravitation, Vol. P8, No. 3, 1996]

Postulate III. The energy-momentum tensor which describes the matter content of the 4-dimensional Universe will be given by eq. (3).

)3(

A Kaluza–Klein metric and its relations

We assume a 5D metric as

The nonzero components of Eq.(1) are

(5)

In the following we have derived some cosmological solutions of IMT by assuming that the bulk space is conformally flat.In the other words the set of nontrivial equations of the 5D Ricci flat and conformally flat space time (5) are obtained. We have obtained three classes of solutions as

(12)

(13)

(14)

where B is a function of time, F is a function of ψ and n is a constant.

For the above metric the nonzero components of the induced energy momentum tensor are

(15)

(14)

A) Solution I

We consider a perfect fluid as

(16)(17)

From (13)-(16) we find that

(17)

Which can describe different kinds of matter that will discussed in the following.

B) Solution II

(18)

(19)

(20)

where B is a function of time, F is a function of ψ and b is a constant.

(21)

(20)

The nonzero components of the energy-momentum tensor for metric (18) are

By employing (16) for relations (19)-(21) gives

(22)

(23)

where its equation of state will be described in the next section.

C) Solution III

Where F is a function of ψ and Where we have

One can write this metric on each hypersurface with constant ψ in a familiar form as

where it is a spherically symmetric and called the Stephani universe.

(24)

(25)

(26)

Stephani universe is the most general class of non-static, perfect fluid solutions of the Einstein’s equations that are conformally flat. The Stephani model is not expansion free and has a non-vanishing density.

which is the equation of state for vacuum energy.

The equation of state for the metric (23) is

(28)

Energy conditions and deceleration parameter

In this section, we employ the energy conditions to study properties of the induced matter and also obtain the deceleration parameter, q, for each solution.

The standard classical energy conditions are the null energy condition (NEC), weak energy condition (WEC), strong energy condition (SEC) and dominant energy condition (DEC). As usual, when all these four energy conditions are satisfied, matter is called “normal”, when matter specifically violates the SEC, it is called “abnormal”, and it is called “non-normal” otherwise.

Basic definitions of these energy conditions for the diagonalised energy-momentum tensor (16) give

(29)

For a perfect fluid with the equation of state given by

where ω is, in general, a function of t and ψ, these energy conditions are reduced to

The Hubble parameter H and the deceleration parameter q in terms of the proper time are defined, respectively, as

(30)

(31)

(32)

(33)

(34)

We will now study the energy conditions (31) for the solutions derived in previous section and calculate H and q for each solution.

Solution I

From relations (17) and (30) one has

hence, the energy conditions, relations. (31), in terms of n reduce to

(35)

(36)

4.1.1 Models dominated by normal matter

As was derived above for a positive energy density, in the metric (12), when n is restricted to −3 ≤ n ≤ −1, all the energy conditions are satisfied and the matter is therefore normal, that is

The Hubble parameter (32) and the deceleration parameter (33) for the metric (12) are

(37)

(38)

From (37) and (38) one has

(39)

The results (37) and (39), are the same as those in 4D standard FRW cosmology with normal matter and zero spatial curvature. This implies that, as in standard FRW cosmology, the universe dominated by normal matter has either accelerating contraction or decelerating expansion i.e. the gravitational force is attracting for normal matter.

4.1.2 Models dominated by non-normal matter

The matter is called non-normal whenever at least one of the four energy conditions)29 (is violated. Since the special case where only the SEC is violated is of special

importance, we follow our discussion of non-normal matter in two separate cases; inCase I, we have −1 < n ≤ 0 so that only the SEC is violated and the matter is, bydefinition, called abnormal. In Case II, we have n < −3 or n > 0 where all four ofthe energy conditions are violated.

Case I The induced matter in our first solution (17) is dominated by abnormal matteronly if

(40)

With (40) satisfied, the deceleration parameter (38) is restricted to be

(41)

Case II Whenever n is less than −3 or greater than 0, the induced matter in our first solution (17) is dominated by non-normal matter, that is

(42)

For n < −3, only the DEC is violated and the deceleration parameter satisfies q > 2,which cannot describe a universe with accelerating expansion, while for For n > 0,

all four of the energy conditions (36) are violated and the deceleration parameter satisfies q < −1. Since B is a function of time only, we may define a time parameter T as

where the positive sign corresponds to the case where T and t grow in the same direction, and the minus sign corresponds to the case where T and t possess opposite directions (we may always choose B such that d(B−n)/dt is positive for all t). Considering the fact that choosing the positive sign in (43) leads to eλ ∝ T −2/n ,which, since n is positive, decreases as T increases, we are left with the choice of the minus sign to describe an expanding universe. Carrying out the integration in (43), and choosing the minus sign we have

(43)

Inserting (44) and (43) into (12) we obtain

(44) (45)

As T approaches the scale factor in (46)

approaches infinity, so that the universe will undergo a big rip. This metric is the same as the 5D late-time attractor solution in which the induced matter is described by a phantom model of the dark energy.

(46)

Solution II

Comparing (30) and (22) we find ω = −1/3, which, based on the energy conditions)31 ,(describes Normal matter. The Hubble parameter and the deceleration parameter for

the metric (18) are

which describe a uniformly expanding universe.

Solution III

Inserting (28) into (30) we find ω = −1, for which, based on the energy conditions)31 ,(the matter is abnormal. The Hubble parameter and the deceleration parameter for the

metric (23) are

(47)

(48)

ConclusionsIn this paper we have, assuming that the 5D space–time is both Ricci flat, and Conformallyflat, derived three solutions, Solution I, II and III, each with its corresponding 4D induced energy–momentum tensor as a perfect fluid.

We have then used the four energy conditions, NEC, WEC, SEC and DEC to study the properties of the solutions further. The Solution I , can describe non-normal matter, as well as normal matter, depending on the value of n; the latter is quite similar to the standard FRW cosmology with zero spatial curvature, as is evident in equation, while the former can describe a universe with accelerating expansion, with either abnormal matter, Case I, or a phantom model of dark energy, Case II. The Solution II, is dominated by normal matter resulting in a uniformly expanding universe. The Solution III describes a Stephani universe with a non-constant deceleration parameter, and is dominated by abnormal matter describing vacuum energy.

Thanks for you attention