On the Energy Conditions in the NonCompact 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 4dimensional Universe will be given by eq. (3).
(3)
A Kaluza–Klein metric and its relations
We assume a 5D metric as
(5)
The nonzero components of Eq.(1) are
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
A) Solution I
(12)
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
(13)
(14)
(14)
(15)
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)
where B is a function of time, F is a function of ψ and b is a constant.
The nonzero components of the energy-momentum tensor for metric (18) are
(19)
(20)
(20)
(21)
By employing (16) for relations (19)-(21) gives
(22)
where its equation of state will be described in the next section.
C) Solution III
(23)
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
(24)
where it is a spherically symmetric and called the Stephani universe.
(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.
The equation of state for the metric (23) is
(28)
which is the equation of state for vacuum energy.
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 energymomentum tensor (16) give
(29)
For a perfect fluid with the equation of state given by
(30)
where ω is, in general, a function of t and ψ, these energy conditions are reduced to
(31)
The Hubble parameter H and the deceleration parameter q in terms of the proper
time are defined, respectively, as
(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
(35)
hence, the energy conditions, relations. (31), in terms of n reduce to
(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
(37)
The Hubble parameter (32) and the deceleration parameter (33) for the metric (12) are
(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; in
Case I, we have −1 < n ≤ 0 so that only the SEC is violated and the matter is, by
definition, called abnormal. In Case II, we have n < −3 or n > 0 where all four of
the energy conditions are violated.
Case I The induced matter in our first solution (17) is dominated by abnormal matter
only 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
(43)
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
Inserting (44) and (43) into (12) we obtain
(44)
(45)
(46)
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.
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
(47)
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
(48)
Conclusions
In this paper we have, assuming that the 5D space–time is both Ricci flat, and Conformally
flat, 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