#### Casimir force in the Gödel space-time and its possible induced cosmological inhomogeneity

Eur. Phys. J. C
Casimir force in the Gödel space-time and its possible induced cosmological inhomogeneity
Sh. Khodabakhshi 1
A. Shojai 0 1
0 Foundations of Physics Group, School of Physics, Institute for Research in Fundamental Sciences (IPM) , Tehran , Iran
1 Department of Physics, University of Tehran , Tehran , Iran
The Casimir force between two parallel plates in the Gödel universe is computed for a scalar field at finite temperature. It is observed that when the plates' separation is comparable with the scale given by the rotation of the space-time, the force becomes repulsive and then approaches zero. Since it has been shown previously that the universe may experience a Gödel phase for a small period of time, the induced inhomogeneities from the Casimir force are also studied.
1 Introduction
It was about half of the twentieth century that Gödel
proposed an exact solution of Einstein field equations which
is not in FLRW form and is compatible with an incoherent
matter distribution [
1
]. A short description of the
properties of the Gödel metric can be found in [
2–4
]. Interpreting
the dust particles as galaxies, the Gödel solution could be
a cosmological model of a rotating universe but this model
exhibits no Hubble expansion [
5
]. Thus it cannot be a
realistic model of our universe. Despite the unusual properties
of the Gödel solution, in particular allowing the existence of
closed time-like curves [
6–9
], it demonstrates a phenomenon
that we cannot easily dismiss in general relativity. It is more
rational to take it as an alternative metric which would in
principle be allowed by general relativity and use the Gödel
metric in studying the rotation of the universe.
One possible use of the Gödel solution is de Sitter–Gödel–
de Sitter phase transition, a scenario in which the Gödel’s
rotatory property induces rotation in the universe during a
phase transition from de Sitter to Gödel space-time and then
back to de Sitter space-time [
10
]. In this scenario, the
effective quantum potential of a scalar field, which may be the
inflaton field, acts like a negative cosmological constant. It
is shown that as the universe expands, it reaches a critical
temperature in which the potential shows a negative valued
minimum. Therefore it is possible to have a sequence of phase
transitions from de Sitter to Gödel and back to de Sitter
spacetime. As is shown in [
10
], using the motion of a test particle
and for a congruence of particles, it is seen that although the
local induced rotation is of order of √ c, the global rotation
is below the observational limit. This gives this scenario the
ability to describe local rotation of galaxies without
conflicting with the observed limit on the global rotation.
Assuming that the universe could make a rapid phase
transition to Gödel space-time, it is fruitful to study more the local
effects of this phenomenon. In the simulation of the induced
rotation of the mentioned scenario, a uniform distribution of a
large number (of order of 105) of particles having a Gaussian
distribution in the initial velocity has been considered and the
result shows the local induced rotation is non-zero (see Fig. 6
of [
10
]). Dividing the space into adjacent cells, due to the fact
that the quantum tunneling may happen with a probability,
each cell may or may not experience the phase transition. In
addition the symmetry axis of the Gödel space-time may be
in any direction and this can generate some anisotropy. This
produces local boundary layers, as explained later.
The universe is approximately homogeneous at large
scales. The Cosmic Microwave Background (CMB) which
is a snapshot of the oldest light in the universe, agrees well
with the predictions of the CDM model. Although it has
a thermal black body spectrum of temperature 2.72548 ±
0.00057 K [
11
] it shows very small temperature fluctuations.
Any model of the universe should explain these
observations. In the study of the thermal history of the early universe
[
12–14
], the CMB inhomogeneities are of wide interest. The
standard approach to predict inhomogeneities is to consider
the quantum fluctuation of a scalar field as the seed of
perturbations and investigate their growth during the expansion
of the universe [
15,16
].
It could be hypothesized that anisotropies generated by
the de Sitter–Gödel–de Sitter phase transition could induce an
inhomogeneity in the cosmic microwave background.
Studying the possibility of such phenomena is the aim of this paper.
No one doubts the importance of the Casimir effect in
modern physics. Since the Casimir prediction of a force
between neutral conducting plates, which is an effect due to
the vacuum fluctuations of quantum fields, a lot of research
has been done both theoretically and experimentally on the
Casimir effect [
17–19
]. Although the Casimir effect was
originally predicted for electromagnetic fields it has also been
calculated for scalar and other fields using different
boundary conditions [
20
]. Besides studying the Casimir effect in
flat space-time, investigating the Casimir effect in the curved
space-time is increasingly making progress [
21
]. For
cosmological models, to get closer results than the actual ones,
one has to calculate the finite temperature Casimir energy
[
24,25
].
An important property of the Gödel space-time is that it
has an axis of symmetry. This leads to different values of
Casimir force for different directions. Thus, in principle the
Casimir force could induce an inhomogeneity in the cosmic
microwave background, because it is direction dependent in
Gödel space. We shall investigate such an effect in the
following section.
In Sect. 2 we consider a scalar field living in the Gödel
space-time subjected to the Dirichlet boundary conditions on
two parallel plates and compute the Casimir force at finite
temperature. Then in Sect. 3, using the obtained results and
[
10
], the possibility of contributing to the inhomogeneities
via the Casimir force is investigated.
2 Finite temperature Casimir force in the Gödel
space-time
As is explained in the Introduction, recently we have shown
[
10
] that the universe may experience a Gödel phase in a small
period of time. Since this phase transition occurs locally,
the Casimir force between the particles could produce some
inhomogeneity in the early universe. Here we are interested
in investigating the production of such inhomogeneities. In
order to do this we first calculate the Casimir energy and force
in Gödel background, using the method of effective action
and zeta function regularization [
22,23
].
We consider the action functional of a massive scalar field
coupled to a curved background, thus1:
1 In general we can use the action S[φ] =
d4x √−g 21 gμν ∂μφ∂ν φ − m22 φ2 − ξ2 Rφ2 − 4λ! φ4 . In the mean
field approximation, the only difference is that in what follows, one
should replace m2 by M2 = m2 + ξ R + 21 λφ2 mean field.
where
= gμν ∇μ∇ν
and μ is an arbitrary parameter with dimension of a mass
appearing in the regularization procedure. One of the best
methods for the evaluation of the effective action is the zeta
function method. Since the trace can be evaluated as the sum
of the eigenvalues of the ( + m2) operator, it is useful to
define the zeta function as
ζ
+m2 (s) =
μ2
n
η−s
n
1
eff = − 2 ζ
+m2 (0).
μ2
where ηn’s are the eigenvalues. Therefore the effective action
is given by
S[φ] =
d4x √−g
1 gμν ∂μφ∂ν φ −
2
In order to consider the quantum effects such as the Casimir
effect, we can use the effective action given by
eff = − ln
To obtain the finite temperature effective action, we have
to use the Euclidean action. The Casimir energy at finite
temperature then can be obtained from the effective action
[
17
]:
1
ECasimir = − 2β ζ
+m2 (β, 0).
μ2
Choosing the background space-time to be the Gödel
space-time, whose line element is given by
ds2 = (dt + eαx dy)2 − dx 2 − 21 e2αx dy2 − dz2,
where α is related to the intrinsic angular four-velocity by
β = (0, 0, 0, √2α),
and following the calculations of [
26
], the eigenvalues of the
operator ( + m2) can be derived:
1 2 1 2 1
η = kz2 +m2 +α2 n + 2 + 4 α − ω + √2αε n + 2
2
(9)
where ε = ky /|ky |. Note that the wave function is of the
form exp(i ky y + i kz z − i ωt )ψ (x ) and the quantum number
corresponding to the frequency is n = 0, 1, 2, . . ..
Computations at finite temperature can be performed via
the Matsubara replacement, in which the time dependence
would be replaced with a periodic function. For Gödel
spacetime, this leads to [
26
]
ω +
√
1
2εα n + 2
2π il
→ β
,
in which l = 0, ±1, ±2, . . ..
To obtain the Casimir energy, we assume we have two
plates located in the z-direction, separated by a distance d.
This yields the quantization of kz as follows:
kz =
nz π
d
where nz = 1, 2, . . .
It is not necessary to be worried about the existence of ky
in the eigenfunction; the eigenvalues are independent of ky
and it can be integrated out like what have been done in [
26
].
Therefore the finite temperature ζ -function would be
ζ( +m2)(β, s) =
∞
∞
∞
l=−∞ n=0 nz=1
πd22 n2z + α2 n + 2
.
2 For all ci = 0, one should exclude ni = 0 from the summation.
(10)
(11)
(12)
(13)
(14)
(15)
×
≡
n1,..,nk ∈Z
In order to evaluate the zeta function, we shall write it in the
form of a generalized Epstein (or Epstein–Hurwitz) multiple
series, defined as2:
Ek (s; a1, . . . , ak ; c1, . . . , ck ; c)
[a1(n1 + c1)2 + · · · + ak (nk + ck )2 + c]−s .
In doing this we notice that for small α the summand can be
replaced by A defined by
A(n, nz , l) ≡ α−2s π 2 2 2
d2 nz + n +
¯
= α−2s A¯(n, nz , l)
where the dimensionless quantities are
m
β¯ = βα, d¯ = αd, m¯ = α ,
4π 2l2
β¯2
1 −s
+ m¯ 2 + 2
and we can rewrite the zeta function
ζ( +m2)(β, s) = α−2s
∞
∞
∞
l=−∞ n=0 nz=1
α−2s
A prime over the summation means that the zero is excluded.
In terms of Epstein series one obtains
Before proceeding, one should note that as usual [
17
] one
should investigate the zero temperature (β → ∞) separately.
The normalized Casimir energy (E¯ = E /α) will then be
E¯Casimir(d¯, β¯) = E¯0(d¯) + ¯ F.T.(d¯, β¯).
¯ F.T.(d¯, β¯) is the finite temperature part and E¯0(d¯) is the
zero temperature contribution defined by
E¯0(d¯) = lim
β¯→∞
−ζ (β¯, 0)
2β¯
.
Introducing ξ 2 = 4βπ¯22 l2, the zero temperature limit of the
sum over l could be replaced by an integral in the following
way:
1
lim
β¯→∞ β¯ l=−∞
∞
f (ξ ) =
∞ dξ
−∞ 2π
f (ξ ).
All the relations can be simplified using the Epstein
recursion formula [
27
],
(16)
(17)
(18)
(19)
(20)
(21)
Ek−1(s; a1, . . . , ak−1; c1, . . . , ck−1; c)
a j (n j + c j )2 + c⎦
and the relations
K−1/2(z) =
and
1
(s) = s − γ + O(s).
π
exp(−z)
2z
For the zero temperature contribution after some lengthy
calculations we have
1
E¯0(d¯) = divergent terms − 4π
1
− 8π
∞ 1
k
n∈Z k=1
−∞
k=1
∞ e−2πkc/a2
k
∞ dξ e−2πkξ2/a2
−∞
∞ dξ e−2πk√(a1n2+c+ξ2)/a2 .
(25)
The divergent parts have to be renormalized by counter terms
and the parameter μ, leaving the finite part:
E¯ 0(d¯) = − 4√√a22π Li3/2(e−2π c/a2 )
1 ∞ 1
− 8π k
∞ dξ e−2π k√(a1n2+c+ξ 2)/a2
n∈Z k=1
π 1
− 24d¯ − 4√2d¯
−∞
exp
−2
1
2 + m¯ 2
d2
¯
π
where Lin (z) is the polygamma function.
The finite temperature part can also be simplified using
the Epstein recursion relation:
¯ F.T.(d¯, β¯) = β1¯ ln ⎣
⎡
1 − exp −
2 21 + m¯ 2 d¯2
π
4 5 6
Normalized distance
7
8
9
10
The Casimir energy and force are plotted in Figs. 1, 2, 3
and 4 as a function of normalized temperature and separation.
An interesting feature of the Casimir force in the Gödel
space-time is that, in contrast to the flat space-time, the force
somewhere near d¯ = 1 goes repulsive and for larger
normalized distance approaches zero. This effect can be
understood using a semiclassical argument. The Casimir force is
4 5 6
Normalized distance
7
8
9
10
in fact a result of the polarization of plates by trapping the
virtual pairs created in vacuum (see Fig. 5a). The effect of
Gödel space-time on the motion of particles is to introduce
a helical motion with radius of order of α. Considering the
fact that the virtual particles live for a time given by the
uncertainty relation, one observes that for a large value of
the rotation factor (i.e. d¯ ∼ 1) the virtual particles spend
much of their time in spirals and cannot arrive at the plates.
This is some depolarization effect that neutralizes the plates
near d¯ ∼ 1 and a small decaying repulsive force for larger
distances.
Finally, it should be noted that putting the boundary
conditions in the x or y direction, there would be no noticeable
change in the Casimir force with respect to the flat
spacetime case. This can be seen from the form of the eigenvalues.
Therefore it is expected that some inhomogeneities would
be induced into the universe. We shall investigate this in the
next section.
As is stated previously, since the de Sitter–Gödel–de Sitter
phase transition can happen locally [
10
], some
inhomogeneity would be induced in the universe. This is because the
Casimir force is sensitive to the direction of the rotation of
the (local) Gödel space-time.
To investigate the Casimir force we have to clarify what
the local boundary layers acting as boundary conditions are.
It may be local layers of matter which could be produced
in many ways. One way is to assume that these layers were
produced by gravitational waves after the inflation era which
can produce more dense regions.
But what is of our interest here is that the de Sitter–Gödel–
de Sitter phase transition scenario creates a natural boundary
condition. As explained in the Introduction, because of the
fact that the quantum tunneling to Gödel space-time happens
with a certain probability, it may or may not happen. As a
result we can have local regions where such a transition is
made or not in the neighborhood. The boundary between
these neighboring regions would have different densities and
act like the boundary layers for computation of the Casimir
force. Since these neighboring regions are completely
distinct, we can adopt Dirichlet boundary conditions.
As a result, local layers of matter would be redistributed in
an inhomogeneous way. Since this phase transition may
happen in the inflation era [
10
], the Casimir force could amend
the inhomogeneities of the cosmic microwave background
radiation.
In the de Sitter–Gödel–de Sitter phase transition scenario
[
10
] the one-loop Euclidean effective potential of a scalar
field plays the role of the cosmological constant. In the first
phase (i.e. de Sitter phase), the effective potential acts like
a positive cosmological constant and it is much larger than
the dust density. After reaching the critical temperature, the
phase transition happens and the effective potential plays the
role of a negative cosmological constant which must be equal
to −ρdust/2 to have the Gödel space-time. Finally the scalar
field slowly rolls so that the universe goes to the de Sitter
phase again. This process is shown in Fig. 6 [
10
].
As is shown in [
10
], the cooling of the universe that is
needed to go to the Gödel phase is achieved via expansion and
the critical temperature is given by the relation dd2φVe2ff |φ=0=
0. Then in the Gödel phase, slow rolling takes t˜ seconds,
given by
t˜ =
% hG2ρdust
¯
c7
= tPlanck
Godel
de Sitter
where tPlanck is the Planck time, Godel and de Sitter are
the cosmological constant in the Gödel and de Sitter phase,
respectively.
(29)
We have shown in [
10
] that being in the Gödel phase for
t˜ seconds induces some amount of rotation in the motion of
test particles which is of order √ . The induced rotation on
a congruence of 1.5 × 105 particles is simulated in (Fig. 7) of
[
10
]. The result shows that with such a mechanism a global
rotation within the observational acceptable limit, while
having local rotation, can be obtained.
Here we want to notice that during the Gödel phase the
adjacent matter layers experience a Casimir force which,
according to the results of the previous section, is sensible to
the direction and this produces inhomogeneity.
Consider a cube cell with length, width and height d. It
would shrink by an amount δ in the (random) z-direction
because of the Casimir force during the Gödel phase. A
computer simulation for a 50 × 50 × 50 lattice of such cubes has
been done. For each cell, it is assumed that the transition is
such that the axis of the local Gödel phase is in a uniformly
distributed random direction. Then using the Casimir force
obtained the new sizes of the sides of each cell are
calculated. The resulting matter distribution is shown in Fig. 7.
As is clear from this figure, small perturbations to the global
homogeneity would be produced.
The density of inhomogeneities can be estimated as
follows. The maximum difference in the density that will occur
after the Gödel phase is
In order to compute the magnitude of δ, we should solve the
geodesic equation of a test particle under the Casimir force
in the Gödel universe. We have
d2z
dτ 2 +
z dx μ dx ν
μν dτ
dτ =
FCasimir(z)
m
.
But because of the fact that τ = t˜ is small (see [
10
]), the
acceleration is nearly constant and the Newtonian limit will
work well; δ is simply
1
δ = 2m FCasimir(d)t˜2.
Recovering all c, h¯ and G’s and noting that the force behaves
like 1/β¯d¯2 for small β¯ and d¯, we get
1 √
βd4
Godel
1
3
de Sitter lPlanckmPlanckc2
.
Using the fact that during the inflation, a number of e-foldings
of order 65 is needed and that inflation happens between the
time ti ∼ 10−36s and t f ∼ 10−32s, we have
δρ
ρ
c2
3
de Sitter(t f − ti ) ∼ 65;
and noting that the temperature is between 1026 K and 1030 K
and the density is between 10−23 mkg3 and 10−26 mkg3 , one
observes that this gives density perturbations between 10−6
and 10−5. This is quite acceptable. Numerical simulation
gives the same result.
It should be noted here that an important feature of the
observed CMB temperature perturbations is that they are
predominantly Gaussian. The quantum fluctuations are random
and this suggests that one has a Gaussian distribution [
28
].
This is usually expressed in terms of the power spectrum
index n S . Gaussianity leads one to expect n S 1.
Observations show that n S = 0.960 ± 0.014 [
29
]. Our simple model
clearly does not have the ability to explain the exact
properties of the cosmological perturbations like Gaussianity.
In order to investigate the possibility of explanation of all
the properties of the cosmic perturbations in this way, we
have to simulate the de Sitter–Gödel transition considering
the quantum tunneling probability. Then the resulting
curvature perturbation and its growth during the expansion of
δρ
ρ =
1 1
d3 − d2(d−δ) .
1
d3
Taking δ small, we have
δρ δ
ρ = d .
(30)
(31)
(32)
(33)
(34)
(35)
the universe should be considered. In this way the predicted
power spectrum of perturbations can be obtained. Here we
have only shown that it is possible to get inhomogeneities
from the de Sitter–Gödel–de Sitter transition scenario. Exact
properties of the induced inhomogeneities will be
investigated in a forthcoming work.
4 Conclusion
In the framework of our previously presented scenario[
10
],
the universe may experience a (local) Gödel phase during
the inflation era. Since the symmetry axis of this local Gödel
space-time is randomly chosen via a spontaneous
symmetry breaking mechanism, one expects that the Casimir force
between layers of matter introduces some inhomogeneity.
This is because of the fact that the Casimir force in Gödel
space-time depends on the direction of the rotation axis of
the local Gödel phase.
Here we calculated the Casimir force for a scalar field in
the Gödel space-time using the techniques of finite
temperature quantum field theory. It is observed that there is some
induced inhomogeneity during the ending period of inflation
and that its value is not far from the expected inhomogeneity.
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