International Journal of Astronomy and Astrophysics , 2011, 1, 183-199
doi:10.4236/ijaa.2011.14024 Published Online December 2011 (http://www.SciRP.org/journal/ijaa)
Copyright © 2011 SciRes. IJAA
An Astrophysical Peek into Einstein’s Static Universe:
No Dark Energy
Abhas Mitra
Theoretical Astrophysics Section, Bhabha Atomic Research Centre, Mumbai, India
Homi Bhabha National Institute, Mumbai, India
E-mail: amitra@barc.gov.in
Received June 26, 2011; revised August 28, 2011; accepted September 13, 2011
Abstract
It is shown that in order that the fluid pressure and acceleration are uniform and finite in Einstein’s Static
Universe (ESU), , the cosmological constant, is zero.
being a fundamental constant, should be the
same everywhere including the Friedman model. Independent proofs show that it must be so. Accordingly,
the supposed acceleration of the universe and the attendant concept of a “Dark Energy” (DE) could be an
illusion; an artifact of explaining cosmological observations in terms of an oversimplified model which is
fundamentally inappropriate. Indeed observations show that the actual universe is lumpy and inhomogeneous
at the largest scales. Further in order that there is no preferred centre, such an inhomogeneity might be ex-
pressed in terms of infinite hierarchial fractals. Also, the recent finding that the Friedman model intrinsically
corresponds to zero pressure (and hence zero temperature) in accordance with the fact that an ideal Hubble
flow implies no collision, no randomness (Mitra, Astrophys. Sp. Sc., 333,351, 2011) too shows that the
Friedman model cannot represent the real universe having pressure, temperature and radiation. Dark Energy
might also be an artifact of the neglect of dust absorption of distant Type 1a supernovae coupled with likely
evolution of supernovae luminosities or imprecise calibration of cosmic distance ladders or other systemetic
errors (White, Rep. Prog. Phys., 70, 883, 2007). In reality, observations may not rule out an inhomogeneous
static universe (Ellis, Gen. Rel. Grav. 9, 87, 1978), if the fundamental “constant”s are indeed constant.
Keywords: General Relativity, Cosmological Constant, Cosmology, Dark Energy, Fractal Cosmology, Static
Universe, Big Bang Theory
1. Introduction
The “cosmological principle” demands that not only does
the center of the universe lie everywhere, but also any
fundamental observer must see the universe as isotropic
and homogeneous. This means that, the space must be
“maximally symmetric” [1]. By this consideration, one
can directly obtain not only the metric for Einstein’s sta-
tic universe (ESU) but the non-static Friedmann-Rober-
tson-Walker (FRW) metric as well [1,2] without using Eins-
tein’s equation at all (). Let us in-
vert it in the form
=8π
ik ik
GT ==1Gc
1
=8π
ik ik
TG
(1)
The Einstein tensor ik , on the right hand side (RHS)
of this equation, comprises only geometric quantities, the
structure of space time
G
ik
g
. On the other hand, the left
hand side (LHS) of the same equation contains much
more tangible quantity, ik , the energy momentum ten-
sor of the matter generating the space time structure ik
T
g
or ik . Thus any constraint imposed on ik
G
g
from
symmetry or other considerations must take its toll on the
admissible form of as well. This means that con-
straints imposed on ik must be reflected on the ik of
the fluid generating the ik . This latter physics aspect
however may remain hidden in a purely symmetry driven
mathematical derivation of the cosmological metric
which does not invoke Einstein equation at all. And in
this paper, we would like to explore the tacit constraints
imposed on ik by the requirements of the cosmological
principle, “there is no preferred center”, or “center lies
everywhere”. All the results obtained during this process
will be also ratified by physical considerations as well.
ik
T
G T
G
T
In effect we shall be dealing with spherically symmet-
ric static solutions of Einstein equation with the inclusion
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184
of a cosmological constant
, [3,4]. In recent times
several authors have shown keen interest in this impor-
tant problem [5-8]. However such studies mostly focus
attention on isolated objects or stars where there (1) must
be a pressure gradient, (2) in general, a density gradient,
(3) a natural boundary where fluid pressure
must vanish, and (4) where a discontinuity
in the fluid density
=b
RR

=0
b
pR
might also occur. Further in such
generalized studies, has not been considered as a
fixed fundamental constant contrary to what was in Ein-
stein’s mind when it was first introduced.
In contrast, in the present study
E
instein is a basic
constant and there is no question of modifying its value
according to the needs of a general fluid solution. And of
course, there must not be any natural “boundary” at all
for the universe so that Copernican principle remains
valid. It may be recalled that the ancient Ionian philoso-
pher “Aristarchos of Samos” had proposed the heliocen-
tric theory much before Copernicus and hence we may
also call this as “Aristarchian Principle”. It will be found
that both the metric for a star and ESU have a “coordi-
nate singularity at

=R
, where is the luminosity
distance and
R
is an appropriate integration constant.
This singularity gets propagated in the hydrostatic bal-
ance equation as well. And in order to ensure that iso-
tropic pressure gradient does not blow up at this
singularity, relativistic stars are constructed so that
p
>
b
R
. Here a prime denotes differentiation by R. Ac-
cordingly one can almost forget this singularity while
studying static stars. But for the ESU one has to live with
this singularity. And in this paper, we would study the
behavior of physical quantities like pressure and accel-
eration despite the presence of this singularity. And the
conclusion is that ESU has to be vacuous with =0
in
order to avoid such singularities anywhere.
At the very beginning, let us remind the difference
between a fundamantal constant and a model parameter.
Suppose we are studying a room temperature homoge-
neous static gas and we have obtained some expression
for pressure , temperature T and particle number
density . But since , in principle, we may
obtain a numerical value of the Boltzmann constant .
Once we assume that is a fundamental constant and
not just a model related parameter, the value of must
be independent of time or position or any other
variable. Thus, if we would be studying a gas in a situa-
tion where the gas may be inhomogeneous or may have
bulk motion, we must be able to use the same numerical
value of obtained in an ideal static homogeneous
case. Similarly, here we would study the case of the
supposed fundamental constant though it will be in the
context of ESU. Therefore, we would expect
p
n
k
=pn
k
t
kT k
k
=0
r
for
the dynamic FRW model too. It would be found that in
order that the timelike worldlines of test particles always
remain timelike and no trapped surface is formed, FRW
model too has to be vacuous with . Then the real
universe must be something entirely different from the
isotropic, homogeneous and continuous FRW/ESU mod-
els. Indeed galaxies and structures are found to be dis-
tributed in discrete, lumpy and inhomogeneous manner
even at the largest scales. Nonetheless such matter dis-
tribution can still satisfy the “Copernican Principle” of
no unique centre if it would form infinite hierarchal
fractal pattern. At the beginning, however, we focus at-
tention on the ESU.
=0
22
dR
2. Formulation
We start with the assumption of spherical symmetry and
consider a general form a static metric [2]
22
d=ddseteR

2
 (2)
where 22
2
d=d d
sin 2

 and is the area coor-
dinate. In particular, in a static universe, the luminosity
distance turns out to be exactly [2]
R
R
d=
LR (3)
We also assume the cosmic fluid to be perfect with
=
iki k
Tp
ik
uupg (4)
where,
is the fluid density, is the isotropic pres-
sure, and is fluid 4-velocity. As is clear from Equa-
tion (1), at the beginning, we do not consider any cos-
mological constant
p
i
u
; and the 0
0 component of the
Einstein equation reads
0
02
1
8π=8π=Te
RR
2
1
R


(5)
This can be integrated to yield

=1 R
eR
(6)
where

2
0
=8πd
R
RR

R (7)
Here the condition (0) = 0
has been used to ensure
that e
is regular at . Thus whether, it is the in-
terior solution of a star or the static universe, general
form of the metric is
=0R
2
22 2
d
d=d d
1
R
set R
R
2

(8)
Note, as of now, we need not necessarily interpret
()R
as related to the observed mass though what we
have done looks like finding the interior of a relativistic
star by working out the Schwarzschild interior solution.
The important difference, however, is that while for a
Copyright © 2011 SciRes. IJAA
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star there is a unique center and a boundary where the
density may be discontinuous, for the universe, center is
everywhere and there is no boundary, no exterior solu-
tion, and no density discontinuity. Further even for a
constant density star, there must be a pressure gradient
because there is a unique center. On the other hand, for
an isotropic and homogeneous continuous universe, not
only and
p
, but all physically meaningful quanti-
ties, all geometrical scalars must the same everywhere.
Such requirements may demand severe restriction on the
admissible equation of state (EOS) of the cosmic fluid.
In contrast for a star, at best only
can be uniform in
which case there would be a discontinuity at its boundary
.
=Rb
R
Now we introduce the condition that the fluid must be
homogeneous: =uniform
and thus both for the uni-
verse as well as a constant density star, one has
3
8π
=3R
(9)
and the metric assumes the form

2
22 22
d
d d
18π3
R
t R

2
R
d=se (10)
If we introduce a parameter
3
=8π
S
(11)
Equation (10) would acquire the form
2
22 22
d
d=d d
1
R
set R
RS

22 (12)
It may be noted that Tolman [4] too obtained the static
cosmological metric in a similar way by using the scalar
as the radial variable. In the static case, is also a
comoving coordinate for the interior solution because a
given shell encloses a fixed number of bar-
yons/particles.
R R
=fixR ed
Further, if we introduce a new coordinate
=rRS (13)
We can rewrite the above metric as
2
222 2
2
d2
d d
1
r
setS r
r




d= (14)
Note, the time dependent Robertson-Walker metric
uses this as the (comoving) radial coordinate.
r
Hence, the spatial section of both the universe and the
interior of a constant density star is that of a 3-sphere, a
space of constant curvature, a fact noted by Weyl [3].
This becomes clearer if we express
==sinrRS
(15)
to rewrite Equation (14) as
2222
2
d=dd d
sin
setS

2

(16)
There is clearly a singularity in one of the metric coef-
ficients in Equations (12) and (14) at or
=RS
=1R
or . On the other hand, metric (16) has a
similar singularity at
=1r
=0
. But one cannot make any a
priori comment on the nature of such singularities with-
out studying the behavior of relevant scalar quantities.
Accordingly, we would try to study the behavior of sca-
lars at appropriate regions instead of having any a priori
debate/discussion on these singularities.
Let us suppose that somehow the density of the star
would be reduced to zero everywhere. In such a case, the
space time must become flat with a metric
22 22
d=d ddstRR
2
 (17)
From the viewpoint of metric(8) it would immediately
be apparent that one should then have and
=1e
=0R
. But from the viewpoint of metric (14), it would
appear that in such a case one would have =S
,
and . Why would we have in
the latter case? This is so because the general form of a
space time with constant spatial curvature is
=0r=RSr0=0r
2
222 2
2
d
d=d d
1
r
setS r
Kr

2



(18)
where =1,0,1K
corresponds to closed, flat and
open space time respectively. And once we presume
=1
K
and adopt metric (14), we presume >0
and
then it becomes difficult to have a smooth transition to a
0
case. Hence, if one would try to arrive at a flat
space
=0K
=0r
by imagining that it is due to a closed
space of infinite radius, one would land up with the con-
dition . In fact, if in Equation (4), had we taken,
0
0=TK
(19)
instead of
, we would indeed have obtained Equation
(18) directly in lieu of Equation (14). Then we would
have found that the condition for flatness is . But
when we do take
=0K
0
0=T
and implicitly assume >0
,
we presume and exclude the possibility that one
might have too. In such a case, the condition for
flatness would appear as as mentioned above.
Note, the fact that the comoving coordinate sig-
nifies that there is no baryon/particle at all; i.e., it is all
vacuum.
=1
=0
K
K
=0r
=0r
Cosmological Constant
It is well known that in order to obtain a static universe
which would be closed and finite radius, Einstein modi-
fied Equation (1) into
=8π
ik ikik
Gg T
(20)
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186
apparently implying that either is a fundamental
constant like
4
8πGc or a basic scalar, like the Ricci
scalar appearing within ik
G. It is also well known
that, from a purely mathematical view point, one can
incorporate the effect of , by replacing [7]
R
=8π
e


(21)
and
=8π
e
ppp
 (22)
It should be borne in mind that such a mathematical
clubbing however does not really make a new form
of matter. This is so because while real matter repre-
sented by
generates global negative self-gravita-
tional energy, pure vacuum represented by
is not
associated with any negative self-gravitational energy.
So, as far as mathematics is concerned, instead of
Equation (7), one now obtains
()
=1 eR
eR
(23)
where
2
0
=8πd
R
ee
RR

(24)
One also finds
2
22 2
d
d=d d
1e
R
set R
R

2
(25)
And for a constant density case, one has

2
22 2
2
d
d=d d
18π3e
R
set R
R

2
(26)
and
3
=8πe
S
(27)
Equations (12)-(16) remain unaltered in the presence
of . Thus now, if the space time has to be flat, in ad-
dition to , one must have
=1e
0
e
(28)
This shows that if the original definition of “vacuum”
is =0
, as if, in the presence of a , it gets modified
to
=0
e
.
3. Acceleration Scalar and Singularity
The fact that for a spherically symmetric static system,
one should have <1R
as has already been investi-
gated [8-10]. The basic reason for this is not difficult to
see. In general, static or non-static, for spherically sym-
metric space time occurrence of (,) >rt R
corre-
sponds to a formation of a “trapped surface” and the
condition (,)=1rt R
marks the formation of an “ap-
parent horizon”. Though for a non-static system, it is
conjectured to be possible to have trapped surfaces or
horizons, for a static system they are not allowed. This is
so because once a trapped surface would be there, stellar
matter would be inexorably pulled towards the central
point of symmetry and thus matter would soon end up in
a point singularity rather than as an extended static object.
While this is definitely not allowed for a static star, this
problem would be much more severe for cosmology be-
cause “center of symmetry lies everywhere”.
Following the case of a Schwarzschild black hole
space time, generally, it is believed that this =R
singularity is a mere coordinate one even in the presence
of matter. But as we would see below, the coordinate
independent scalar acceleration of the fluid would blow
up unless severe constraints are imposed on the fluid
EOS.
It may be recalled that Einstein was very much con-
cerned about this =R
singularity and constructed a
static model of a fluid where test particles are moving in
randomly oriented circular orbits under their own gravi-
tational field. While the radial stress of the fluid is zero,
tangential stresses are finite. This configuaration is
known as “Einstein Cluster” and Einstein showed that,
the speed of the orbiting particles would be equal to the
speed of light at c=R
[11]. Thus, he pleaded that
there cannot be any =R
or <R
situation. Later
it turned out that Einstein’s cluster indeed corresponded
to some well defined interior Schwarzschild solution
studied by Florides [12,13]. And now Bohmer & Lobo
[14] have shown that Einstein’s intuition was correct,
and the =R
singularity is indeed a curvature singu-
larity. This however does not at all mean that all =R
singularities are curvature singularities. In fact, in the pre-
sent static case of isotropic pressure, it would be found
that they should be regions with zero curvature singularity
to avoid an acceleration or pressure gradient singularities.
For any static spherically symmetric fluid, one can
easily find the acceleration [8]
=
ik
k
au ui
(29)
In spherical symmetry, only one component of accele-
ration survives
=2
Re
a
(30)
How to evaluate this
R
a? Again, for the sake of easy
understanding, we first do not consider any
and
write down the
R
R
component of Einstein Equation (1),
Copyright © 2011 SciRes. IJAA
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2
11
8π=8π=
R
R
Tpe
RRR

 


2
(31)
In view of Equation (30), let us rewrite this equation
as
3
3
8π
=
e
RR
pR
(32)
or
3
2
8π
=2
RRp
aR
(33)
Since there is only one component of , the scalar
acceleration becomes
i
a
/2
==||=
2
iR
iRR
e
aaaag
(34)
Using Equation (32), we obtain
3
2
8π
=21
Rp
aRR
(35)
If would be included, then the acceleration scalar
would be given by
3
2
8π
=21
e
e
Rp
aRR
e
(36)
It is now clear that if a static fluid would tread upon
the =e
R
singularity, its acceleration could be infinite
and it cannot be at rest. And this is the physical reason
that for a static fluid one must have >e
R
everywhere.
And in case, the manifold would cover =e
R
, one
must satisfy
3
8π=0; =
ee
RpRe
(37)
in an attempt to keep regular. Using a=e
R
in the
foregoing equation, we have

2
18π=0
eee
p

(38)
i.e., one must have either
=0
e
(39)
or,

2
1
==
8π
ee e
pR
(40)
or both of the above two conditions. If we assume that
the minimum value of is zero and it cannot be nega-
tive, we will have
p
=8π
e
p . Then Equation (40)
would yield
1
==RS
(41)
But one is still not sure whether the additional condi-
tion such as Equation (39) is needed to really ensure that
is indeed finite at
a=e
R
. For a fluid with >0
e
,
the safest way to avoid the =e
R
singularity will sim-
ply be to ensure that >e
R
. The fluid can do so by
choosing an appropriate density profile and by ensuring
that its outer boundary
>
b
Re
(42)
For a constant density star or the ESU, we have
2
3
=
12
ee
e
Rp
aR

(43)
where =4π3
. In terms of , we obtain r

2
3
=
1
ee
rp
aS r
(44)
It is clear that the sufficient condition for avoiding the
=e
R
singularity in this case is
>; <
b
RRS
(45)
In terms of density, this means
2
3
>8π
eb
R
(46)
Thus, if one would imagine a region with =0
, i.e.,
=8π
e
, one should restrict the radius of this vacuum
as
3
<
b
R; (47)
And in case, one would have =4π
, then the
above restriction would become
1
<
b
R (48)
and which cannot be satisfied. Thus for a constant den-
sity star with =1
b
R
, one must have >4π
.
And in case this condition would be violated, one must
critically analyze the additional constraint on the EOS
which would prevent e
p
from blowing up at =e
R
.
It may be seen that the =4π
EOS corresponds to
3=0
ee
p
(49)
if .
=0p
4. TOV Equation
Recall that local energy momentum conservation equa-
tion ;0
k
ik
T
immediately leads to [2]
2
=p
p
(50)
Further if we again consider the
R
R
component of
Copyright © 2011 SciRes. IJAA
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188
Einstein Equation and combine it with Equation (50), we
will obtain the TOV equation for hydrostatic balance for
any self-gravitating static fluid:



3
2
8π
=21
pR
pRR


p
(51)
Since the effect of gets included if one replaces
and
p
by their “effective” values[7]



3
2
8π
=21
eee e
e
pR
pRR


p
(52)
Further since , in a complete symmetric man-
ner, we rewrite the above equation as
=e
pp



3
2
8π
=21
eee e
ee
pR
pRR


p
(53)
This equation strongly suggests that if the definition of
a “dust” in the absence of
is , as if, in the
presence of , dust EOS should be . Similarly,
if the original EOS of “vacuum” is
=0p
e
p
=0
=0
, in the pres-
ence of , the vacuum EOS is
=0
e
, a hint we have
already found.
Note, even now, there is no need to interpret
in
terms of any exterior boundary condition, and thus TOV
equation is valid in any spherically symmetric static GR
problem including ESU. By using Equation (36), it is
interesting to rewrite the TOV Equation in terms of the
acceleration scalar

=1
ee
e
e
ap
pR
(54)
Clearly, there is a singularity in the denominator of
TOV Equation at =e
R
. If we write
=1 e
x
R
(55)
It is seen that while the singularity in acceleration
, for the pressure gradient it is much stronger
. And if the fluid would cover the
1
ax
2
px
=e
R
sin-
gularity, regularity of may not be sufficient to ensure
regularity of . In fact Equation (54) suggests that one
might require the additional constraint
a
p
=0;=
ee
pRS
(56)
to tame the singularity at .
p=RS
=
Now let us use the condition euniform
which is
equally valid for the interior solution of a constant den-
sity star as well as the ESU:

2
3
=12
eee e
e
p
pR R



p
(57)
In terms of the normalized coordinate , one finds
r
 
2
3
=1
eee e
rpp
pS r


(58)
And in term of , we have a
2
=
12
ee
e
ap
pR

(59)
and

2
=
1
ee
ap
pr
(60)
5. Constant Density Star
For the interior solution of a constant density relativistic
star one, one finds [7]
2
=
c
p
ep


e
(61)
where c is the central pressure. This relation is inter-
esting because it does not involve any exterior bounday
condition. Also note that since
p
=e
pp
(62)
we can rewrite Equation (61) as
2
=
c
ee
ee
p
ep


(63)
Since for a star, there must be a pressure gradient,
c
ppR, and one really cannot reduce
eR
to a
constant value independent of . But if, from the
mathematical viewpoint, one would still demand that it
should be possible to set up a time orthogonal Gaussian
coordinate system where
R
2
d2
=dt
, one would un-
knowingly kill the pressure gradient and set in
Equation (61). In the absence of , for a static con-
figuration,
=
c
pp
=0
if , and the star would vanish
under the assumption of !
=0pe
=1
When
is present, the expression for pressure for a
constant density star is [7]
4π1cos
()= 3cos
C
pR

(64)
where
3
34π
==
c
ce
cc
ee
p
p
Cpp
e



(65)
Now it transpires that even when to honor
, there would still be a finite density
=0p
=1e
=4π
.
But one attains this finite density at a huge cost because
it now turns out that the boundary of the star merges with
the coordinate singularity where the metric
=
b
RS
Copyright © 2011 SciRes. IJAA
A. MITRA189
behaves badly. For a real star, this singularity can always
be avoided by allowing which in turn means by
having
>1e
>4π
. Thus a “world time” cannot be in-
troduced for a general relativistic star having a finite
density.
Verification from TOV Equation
Although the case of constant density stars in the pres-
ence of cosmological constants has been studied in great
detail, to our knowledge, nobody actually verified the
validity of the solutions in case one would tread on the
coordinate singularity at . Since the denominator
of Equation (53) becomes zero at , in order that
does not blow up there, one must have either
=RS
=RS
p
=0; =
ee
pRS
(66)
or
3=0; =
ee
pRS
(67)
or both
=3=0;=
ee ee
ppRS
 (68)
If the last constraint would be satisfied, one would
immediately obtain ==0
ee
p
. However, it is possible
that Equation (68) is not needed and only one of the less
rigourous conditions (66) or (67) is satisfied. In particu-
lar, to avoid occurrence of negative pressure, let us as-
sume that only Equation (67) is satisfied at .
Since attains a
=RS
p00 form at , let us study
the nature of at this singularity by using L’ Hospital
rule. For this, let us first write
=RS
p
()
=()
e
f
R
p
g
R
(69)
where
()= 3
eee e
f
RR pp
 

(70)
and

2
=2 1
e
gR R

(71)
Since =0
, we find that
 

=3
33
eee e
eeeee
fpp
Rpp p




(72)
Also, since, we have already considered 3=0
ee
p
,
we obtain a reduced expression for
=3 ee e
f
Rp p

(73)
On the other hand,
=4 e
g
R

(74)
So that
=3 4
ee
ee
p
fg p
 (75)
And at the singularity, , by l’ Hospital rule, we
obtain
=RS
=lim
eRS
f
p
g
(76)
From Equations (75) and (76) we obtain the required
condition:
3=
ee
p4
e
(77)
i.e.,
(=)=3(=)
ee
RS pRS
(78)
which looks like the EOS of incoherent radiation! By
combining Equations (67) and (78), it becomes clear that
in order that p
does not blow up at the “coordinate
singularity” at , the constant density star must
have
=RS
(= )=(= )=0
ee
RS pRS
(79)
Thus instead of we would obtain at
. This means that if indeed , the solution
must avoid singularity. More importantly, since
=0p=0
e
p
=RS
=
eco
>0
=RS
ntnsta
, we find that, if the solution would indeed
extend upto , we must have
=RS =0
e
. Thus all
finite density stars must avoid the =1
eR
or
singularity.
=1r
6. Static Universe
In cosmological case, there must be a universal time
which would be the proper time of all fundamental ob-
servers, i.e., 2
=dtd 2
. This demands that one must be
able to set so that, the Equation (14) would be-
come the ESU metric
=1e
2
222 22
2
d
d=d d
1
r
stS r
r



(80)
Note that the value of for ESU is still given by
Equation (27) and thus one is justified in deducing ESU
metric indeed by setting . Since the Einstein equa-
tion tells that the metric is essentially determined by ik,
such an important change of setting on the LHS
of this equation must be endorsed by the RHS, i.e., by
the admissible forms of fluid EOS. What are those con-
ditions? To explore them, we note from Equations (50)
and (51) that the most general form of static, spherically
symmetric Einstein Equation yields
S
=1e
T
=1e

3
2
8π
=1
e
e
Rp
RR
e
(81)
In constant density case, this condition becomes
2
3
=2 12
ee
e
Rp
R
 
(82)
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190
And in term of
r
, this condition becomes

2
3
=2 1
ee
rp
Sr

(83)
In order that , one must have
=1e
=0
, and from
the foregoing equation, it is immediately clear that then
we must have atleast
3=0
ee
p
(84)
In any case, we must have everywhere in-
cluding or . Then if we follow the L’ Hos-
pital treatment of the previous section, we would find,
we must have
=0
e
p
=RS=1r
=0
e
(85)
everywhere because =
econstant
. Although, we have
already obtained this important result, it would be inter-
esting to obtain this result from somewhat different
routes.
First consider Equation (59) and write
()= ee
f
Ra p
(86)
and

2
=12
e
g
R

 R
(87)
so that
=e
f
pa
(88)
and

12
2
=21 2
ee
g
 
R
(89)
Now applying L’Hospital rule at , we find
=RS
2
12
=2
e
e
aR
pp R


 (90)
Since the lim= 1xx irrespective of whether
or
0x
x
 or =
x
finite , we can cancel from both
sides of the foregoing equation. Then, using Equation
(43) in the above Equation , we find
p
3=1
2
ee
e
p
(91)
This again implies
=3
e
pe
(92)
In conjunction with (68), this will lead to ==
ee
p0
.
It may be also of some interest to extend this study by
directly considering the radial variable as rather than
. Note that
r
R
1
=d d
ee
pp
S
r (93)
Now if we denote differentiation by with a prime,
we rewrite Equation (60) as
r

2
=
1
ee
Sa p
pr
(94)
so that
()=ee
f
rSap
(95)
and
2
()= 1
g
rr (96)
Then we have
=
f
Sap
(97)
and
2
=
1
r
gr
(98)
Again applying L’Hospital rule for the limit of p
at
, we find
=RS
2
1=1
Sa r
r
(99)
Inserting the expression for from Equation (44),
we obtain the interesting relation
a
23=
ee
Sp

1
(100)
or
3
=4π3
ee
Sp
(101)
Comparison of Equations (27) and (101) would again
convey one of the hidden messages for the ESU as well
as for a constant density star which is attemting to sup-
press its pressure gradient, namely e
3=
e
p
. The sum
and substance of the entire exercise is that for the ESU,
in oder that e
p
is indeed zero everywhere, one must
have
=3 =0
ee
p
(102)
But unlike the constant density stellar case now we
cannot escape confronting this singularity by demanding
that >4π
. And hence we must accept the fact that
for the ESU fluid ==0
ee
p
everywhere. This means
that
8π=0; =8π

 (103)
atleast for the ESU. The weak energy condition demands
that 0
and thus we find that ! =0
7. Einstein’s Solution
If one would directly use Equation (80) into Einstein
Equation (20), one would be led to[4]
Copyright © 2011 SciRes. IJAA
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
4π3=p
(104)
and

2
1
4π=
pS
(105)
Einstein chose, EOS which led to
=0p
21
1=; =
SS (106)
Note that Equations (104) and (105) may be rewritten
as

4π3=0
ee
p
(107)
and

2
1
4π=
ee
pS
(108)
But, if we recall, Equation (92), actually, =0
ee
p
so that . Further Equations (100)-(101) showed that
=S

2
3
4π3=
ee
pS
(109)
And a comparison of Equations (108) and (109) again
shows that, and
=S=0
e
. As mentioned before,
for weak energy condition, this would mean ==0
.
Note that it is indeed possible to have a situation where
1
reste
, where is the proper internal energy den-
sity. In such a case, one might approximately write
. But this does not mean that pressure is strictly
zero. A strict EOS is possible only when
e
0p
=0
p=0
.
In such a case, Equation (104) would again yield =0
.
However, if we would ignore such physical and
mathematical regularity or self-consistency considera-
tions, Einstein’s solution (106) would apparently suggest
2
=4π>0Gc
.
Further, if one would assume that (1) ESU correctly
describes the real universe and (2) the true mean density
of the real universe in its totality (about which we may
never have an absolute knowledge) is equal to the mean
density of the patch of the observed universe, i.e.,
31 3
==10 g cm
patch true

(110)
one would obtain[2]
58 2
10 cm
(111)
Now ponder over the fact that until the advent of big
optical telescopes in 1920 or so, for hundreds of years,
most of the astronomers thought that total universe was
nothing but our Milkyway galaxy. And if would have
been possible to measure the mean density of the Mil-
kyway, one might have concluded that
13 3
==10 g cm
patch true

(112)
and accordingly
40 2
10 cm before 1920
(113)
As late as 1970, we hardly had any idea that the gal-
axy clusters and superclusters are actually distributed as
“filaments” and “walls” around huge voids whose di-
mensions could be as large as Mpc [15]. The re-
cent Sloan Digital Sky Survey has revealed structures of
dimension Mpc [16]. Thus, we cannot rule out
the possibility that eventually, it might be found that the
entire patch of presently observed universe also lies on
the wall of a larger void. If the mean density of observed
universe would be revised by future observations, would
we again revise the value of the fundamental constant
280
500
?
At the cost of sounding repetitive, let us again raise the
old question: if
would indeed be due to some quan-
tum mechanical effect, why the value of obtained
under the assumption of
=
p
atch true

falls short of the
theoretical value by an approximate factor of ?
And if the various field modes would cancel one another
to generate a small
120
10
, why do they not cancel exactly
to result =0
? In fact there are some theoretical esti-
mates by which =0
[17].
8. Summary So Far
Both a spherically symmetric static star and the ESU
result from the same spherically symmetrical form of
Einstein equations. If
R
R
component of the Einstein
equation would be studied, one would obtain, the expres-
sion for acceleration scalar and condition for hydrostatic
balance in both the cases. In fact, Tolman [4] did obtain
one of the constraints to be followed by the ESU fluid by
demanding that the pressure gradient must vanish
for the ESU: e
p
3
p=0
e
. We found that the singularity
appearing in the
,RR component of the metric tensor
in all spherically symmetric static configurations do
propagate in the expressions for acceleration scalar
and pressure gradient
a
p
. While, for the former, the
nature of singularity is , for the latter it is much
stronger
1
x
a
2
xp
. In fact, the requirement 3
e
p=0
e
is the basic condition for ensuring that the does not
blow up at or . This basic condition how-
ever need not be the sufficient condition for ensuring that
is finite at . Further, since tends to blow
up much faster at , one must require an additional
condition to ensure the regularity of . We found that
the latter requirement is e
a
=0x
=0x
=0x
=1r
ap
p
=3
e
p
. This means that if the
constant density static star would have , one
must have
=
b
RS
=0
e
. But one can always avoid this singu-
larity by ensuring that >4π
. However, previous
detail studies of perfect fluid spheres were primarily in-
Copyright © 2011 SciRes. IJAA
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192
tended for isolated systems and not for the universe, and
to our knowledge, nobody tried to study the remedy for
either the 1
x
or 2
x
singularities at . =0x
Incidentally, it has been found that even for a sup-
posed constant density star with <
b
R
, the “constant
density” is nothing but =0
[18].
9. Buchdahl Inequality
It is well known that any spherically symmetric static
configuration, homogeneous or inhomogeneous, sup-
ported by isotropic pressure, satisfies a constraint much
stronger than the >
b
R
or <b
R
constraint. In the
absence of a , this is known as Buchdahl inequality
[19]:
8
<9
b
R
(114)
If one would have
=89b
R
, central pressure
would blow up (assuming >0
). For a uniform density
star this borderline would correspond to
=8 9r (115)
But suppose there is a solution which appears to vio-
late this constraint, i.e., 89r. Then how to ensure
that central pressure is still finite? The only solution out
of this dilemma would be to set =0
or =0
. From
this more stringent condition, it should be clear why Ein-
stein’s original static universe must have =0
.
Does the fate of the ESU improve after the incorpora-
tion of a positive ? From Equation (3.35) of ref.[7], it
turns out that now the modified Buchdhal constraint is
2
<4
94π
e
R


(116)
Thus with a positive , this constraint gets even
tighter. With
=4π
, one obtains
2
<3
e
R
(117)
In view of Equations (9), (11) and (33), this means
that for a constant density star, one must have
<23r (118)
to ensure that pressure does not blow up at the center
! For the ESU, this would mean blowing up of
pressure everywhere because “center would lie every-
where”. And the only solution to get rid of this problem
is is accept the fact that
=0r
=0
e
for ESU.
9.1. Mass Function
Since for a fluid in an asympotictially flat space time,
one can define a “mass function” in terms of which ac-
celeration scalar
3
2
4π
=12
M
Rp
aRM
R
(119)
we can now identify ()R
in Equation (36) as twice
the quasilocal mass-energy of the fluid
()=2 ()RMR
(120)
In the presence of a
, the quasilocal mass will be
=2
ee
M
.
In general relativity, the gravitational mass of a sta-
tionary system is[3]
003
00
= d
M
Tt gx
(121)
where is energy momentum pseudo tensor and
k
i
t
g
is the determinant of the metric tensor. One can work out
from the metric ik
0
0
t
g
of the universe. Probably start-
ing from Rosen [20], many authors have worked out the
mass energy of a closed universe and all of them have
concluded that for a closed universe [21-25].
This too would suggest that, in the absence of
=0M
,
=0
, and, in general =0
e
. For the ESU, it is possi-
ble to confirm this result irrespective of the value of .
In all coordinate systems, for a static fluid, one obtains
[2-4]
0
0
t


3
3
=34π d
=3 d
e
ee
M
pg
pgx
 

x
(122)
In view of Equation (84) we thus directly obtain
=0
e
M (123)
Further, from Equation (24), we obtain =0
e
9.2. Poisson Equation
It is known that the RHS of Poission equation indicates
the source of gravity. For a spherically symmetric static
fluid the Poission’s equation is [26,27]


2
00 00
00
=4π34π
=4π3
ee
ggp
gp


(124)
This shows that the source density of gravity is
00
=3
g
ee
g
p

(125)
And when , one will obtain
00 =1g
=3
g
ee
p

(126)
In either case, the RHS of Equation (124) is zero when
3=0
ee
p
, which is the case for both a constant den-
sity star (if it would be extended up to ) or the
=RS
Copyright © 2011 SciRes. IJAA
A. MITRA193
ESU. In all such cases, one must necessarily have
00 ==
g
e constant
=0
e
. And since source of gravity is zero,
the space time must be flat. And a space time is flat when
, as obtained by us. Alternatively, if one would
first focus attention on the LHS of Equation (124), it
seems that if one would set , the source of
gravity will vanish, again in which case, one should have
=e constant
=0
e
.
10. A Simple Reason
Suppose one chooses to ignore most of the previous
proofs that =0
e
for ESU in order to avoid any
physically singular behavior. Even then, one can arrive
at the same conclusion by simply demanding that all
physical quantities must be position independent in ac-
cordance with the inherent assumption in the following
manner:
For a constant density case, we may rearrange the
TOV equation(58) as
2
=
3)1()(
e e
pSr
ppr

 
ee
(127)
We may also rewrite the acceleration Equation (44) as
2
=
31
ee
aS
pr
r
(128)
The RHS of Equations (127) and (128) obviously de-
pends on r if r would indeed be a free parameter.
For a constant density star, the LHS of the same equa-
tions too must depend on r. And they do depend n r
becaus )
ee
even though
o
e =(ppr =
econstant
. But
suppose we would like to freeze the dependence of
to make . Then the value of on the RHS of
Equations (127) and (128) too must be frozen. And since
during the freezing process, the only solu-
tion here would be to adopt
r
rp =1e
=0RSr
=0
K
=
ik
TK
=; =0;0SrR (129)
We have already discussed that occurrence of Equa-
tion (129) actually signifies that the spatial section is flat:
. This would have been more transparent had we
taken
123
123
===TTT Kp (130)
alongwith Equation (19), i.e., had we written

i kik
puupg



(131)
instead of Equation (4). Further had we also taken the
cosmological constant as
K
instead of , Equations
(127) and (128) would have appeared as

2
=
31
eee e
pKSr
pp
K
r

 
(132)
and
2
=
31
ee
aKS
p
r
K
r
(133)
respectively. In such a case, it would have been seen
immediately that the requirement that the RHS of Equa-
tions (132) and (133) are independent of , one must
have is . And since we have already set , it
means, the space time must be flat to ensure that the
RHS of Equations (127) and (128) or (132) and (133) are
indeed independent of . Hence one must have
r
=0K=1e
r
=0
e
K
in such a case. So will be the case for ESU.
This means that in order that all physically meaningful
quantities are position independent, the mean density of
the ESU must be zero.
11. Non-Static FRW Universe: Big Bang
Model
In physics, we have various fundamental constants like
, , , e etc. But in no case there is any evidence
that the values of such constants directly depend of am-
bient factors like mean density of universe. In fact, all
truly fundamental factors allow themselves to be deter-
mined with high precision by judicious combination of
theory and experiments. The only exception here is the
supposed
Gch m
! This anomaly seems to be resolved with
the result that there is no ! On the other hand, cos-
mologists have found that distant Type 1a supernovae
appear to be fainter than expected if one would assume 1)
them to be standard candles and there is no evolutionary
effect in the intrinsic luminosities of the distant superno-
vae, 2) Assume that cosmological luminosity distance
measurements to be perfect; i.e., assume that standard
Big Bang Model (BBM) is correct, 3) Assume that there
is no dust absorption of lights from distant supernovae.
And by ignoring such questions, in the paradigm of the
BBM, such extra faintness was interpreted as the proof
of existence of
or “Dark Energy” (In fact, the Phys-
ics Nobel for 2011 was due to such an interpretation):
4π4π
=(3)= 3
33π
ee
SS pp






4
(134)
The extra faintness is explained by assuming that the
galaxies are further away than expected (in a decelerat-
ing universe); i.e., by considering and which is
possible for a fine tuned positive
>0S

3
4πp
 (135)
But if indeed =0
, then BBM cannot explain a
supposed . Nonetheless, it has been claimed that if
“Cold Dark Matter”, another crucial ingredient of BBM,
will have pressure, BBM would be able to explain
>0S

Copyright © 2011 SciRes. IJAA
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194
>0S
 even if [28]. However, this suggestions is
patently false because
=0
3p
p
includes contribution
from both visible matter and DM, and any increase in the
value of matter pressure , can only give .
Therefore, the faintness of distant supernovae need not
imply that they are further away; and the faintness could
be due to unknown systemetic effects [29].
<0S

The fact that the BBM cannot explain the physical
universe is apparent from the fact, for the BBM [30]
00
2
=expe
ee
p
gp
(136)
And in oder that the FRW metric indeed has ,
one must have
00 =1g
2=0
e
ee
p
p
(137)
Since <
ee
p
, it is necessary here that
=8π=0
e
pp . Then, it is immediately seen from
Equation (134) that, BBM model must (actually) have
4π
=
3
e
S
S


0
(138)
Thus contrary to the popular belief, the BBM actually
cannot explain any accelerating universe! Further
=8π=0
e
pp would imply =8πp. And if we
would rule out such a fine tuning of matter pressure, we
should have =8π=0p for the BBM irrespective of
the result obtained in the context of ESU. And with
, there can be neither any matter pressure nor any
microwave background radiation[30].
=0p
12. Why the Big Bang Model Could Be
Vacuous
We have already pointed out that the total energy associ-
ated with the FRW metric is zero; and this is not because
of any negative self-gravitational interaction. The latter
can reduce total energy only when 00 00,
but for the FRW case, 00 00 [26,27]. On the
other hand field energy density . Thus vanishing
of total energy indicates
=(,)<ggrt
=1
0
1
=()ggt
0
0
t
=0
e
. We probed this impor-
tant question independently and found that FRW metric
should intrinsically correspond to =0
e
in order that
total energy is conserved [31]. In fact there can be a sim-
plified way to confirm this:
Long back Kriele showed that there cannot be any
trapped surface for a spherically symmetric homogene-
ous perfect fluid [32]. This seems to be a special case of
the no trapped surface theorem obtained by demanding
that timelike worldlines associated with material parti-
cles must always remain time like [33], i.e.,
2
2(,)
<1
GMr t
Rc (139)
For the FRW case, one has and
=()RrSt
3
4π
(,)= ()
3e
R
M
rt t
(140)
so that one should have
22
2
8π() ()<1
3e
GrtSt
c
(141)
And in order that does not blow up either in
future or past, one needs to have
2
eS
2
()()=
et Stconstant
(142)
Again for this, first, it is necessary that =0
and
=
e

:
2
() ()=tStconstant
(143)
But the FRW model generally obeys
so that one should have
2=t constant
()St t
(144)
And it is only for the Milne model (with =1k
), one
obtains ()St t
. But the Milne model is empty with
e
==0

! Since we have already found , we
eventually obtain the vacuum EOS for the FRW metric:
=0
e
p
=p=p0
ee

. Does this result lead to any contra-
diction with Equation (137)? The answer is “no” because
a 00 form could be anything including 0. One may
also point out that the result =0
e
for the FRW model
is not rigorous because we did not offer any proof for the
usual relation . Such an objection would
be valid, and thus now we focus on the de-Sitter case
with
2=t
constant
2
=8π
ec
. Now Equation (141) will reduce to
22
1()<1
3rSt (145)
In order that above constraint is satisfied, one must
have =0
. Thus, in a rigourous manner, we should
have =0
in order to satisfy the condition of “no
trapped surface” in the de-Sitter model.
So for the time being, let us ignore the possibility that
the FRW metric subtly represents a vacuum spacetime
and which is the reason that there is no density or pres-
sure gradient and there is an universal Newtonian time
despite the supposed presence of self-gravity. And even
if one would ignore the proof that , there are al-
ready many suggestion that and “Dark Energy”
could be illusions created by an inhomogeneous lumpy
universe which is significantly different from the sim-
plistic BBM [34-38]. However if there would be inho-
mogeneity in a monotonous and continuous manner,
there would be a preferred centre of the universe in vio-
lation of the “Copernican Principle” of no unique centre.
=0
Copyright © 2011 SciRes. IJAA
A. MITRA195
On the other hand, the discrete fractal models could still
satisfy the Copernican Principle.
13. Fractal Model
In a fractal distribution, the number of galaxies increases
with as [39-41]
R
(< )
D
NRR (146)
and the number density of galaxies varies as

3
<D
nRR
(147)
Thus, in principle, for a fractal distribution over suffi-
ciently large scale, one may have ,
0n0
. Most
of the fractal patterns however tend to have a preferred
center and thus not suitable for “Copernican Principle”.
But there could be some fractal structures like Levy Dust
which may not have any preferred center.
Indeed many cosmologists believe that within the
patch of the observed universe, galaxies are distributed in
fractal like pattern atleast on scales Mpc [42-45],
and one may recall here a succint comment by Wu, La-
hav & Rees [45]
10
The universe is inhomogeneous—and essentially
fractal—on the scale of gala xies and clusters of galaxies,
but most cosmologists believe that on larger scales it
becomes isotropic and homogeneous.
This review written in 1999 however concluded that
galaxy distribution is homogeneous on larger scales. But
now it is seen that galaxies are distributed in a roughly
fractal pattern with dimension even on
Mpc scale [46,48-52]. Note, many of the objections
against the fractal model have already been addressed to
[53].
2.2D100
14. Conclusions
Since cosmological constant , so-called “Dark
Energy” too is expected to be absent. Simultaneously it
has been found that the Big Bang Model intrinsically
corresponds to zero pressure and zero temperature [30].
Note this result is in perfect agreement with the fact that
an ideal Hubble flow implies smooth radial motion with
no collision amongst the test particles. Thus even if one
would assume , in view of Equation (138), the
BBM cannot explain any cosmic acceleration. Note,
Equation (138) can be obtained by purely Newtonian
gravity! But when can a GR result be exactly synony-
mous with a purely Newtonian result? To appreciate this,
have a close look at GR Poission equation (124).
=0
=0
Assume gravitation is extremely weak:
00 2
12gc
 (148)
where
is the Newtonian potential and 2
21c
.
Then by further stting , one can obtain the ap-
proximate Newtonian Poission equation:
=0p
24πG
 (149)
But by starting from GR equation, one would never
obtain an exact Poission’s Equation 2=4πG
if
0
. Thus, GR can yield an exact Newtonian result
only when the gravitational potential =0
! This sug-
gests that though, mathematically, one can conceive of
global clock synchronization and set 00 , physically,
gravitation tends to manisfest itself with non-synchroni-
zation of clocks! In other words, a model assuming
global synchronization of clocks could be vacuous with
zero matter density! Thus one expects all dust models to
correspond to
=1g
=0
. And indeed, the famous Oppen-
heimer -Snyder dust collapse has explicitly been found to
correspond to =0
[54]. The Schwarzschild “Black
Hole” exact solution too is illusory because the integra-
tion constant involved there is actually zero implying
true black holes have zero gravitational mass; and the
so-called “Black Hole Candidates” must be something
else[55]. Similarly, it has been found that a uniform den-
sity sphere cannot undergo any adiabatic collapse at all
and hence scores of exact solutions indicating collapse
must actually correspond to =0
[56]. In general,
there cannot be any adiabatic gravitational evolution at
all, and which shows that all exact solutions indicating
evolution must tacitly correspond to =0
[26]. Even
the famous interior solution for a static uniform density
spherical star is illusory in the sense that it actually cor-
responds to =0
[18]. This may be suggesting that
self-gravity manifests itself not only through pressure
gradient and non-syncronization of clocks, but also
through density inhomogeneity.
Thus it is indeed possible that FRW metric too subtly
corresponds to a vacuum solution just like its static
counterpart. In general, though, “exact GR solutions”
may often be physically meaningful for the static cases,
they need not be so for the complex non-static cases; the
complexities of a physical system, such as unknown and
evolving equation of state of matter, unknown radiation
transport properties, unknown evolution of shear and
dissipation, may rarely allow physically meaningful
non-static exact solution s. For instance, it has been found
that celebrated non-static Kerr solution is an illusion be-
cause the integration constants, namely the rotation pa-
rameter and mass m are actually zero:
[57,58].
a==0am
Even if this suggestion that FRW metric too is tacitly a
vacuum solution, would be ignored at this moment, there
is now firm conclusion that galaxies are distributed in a
fractal pattern atleast on scales ~100 Mpc in violation of
Copyright © 2011 SciRes. IJAA
A. MITRA
196
the assumptions of CDM model [46-52]. In fact the
latest Sloan Digital Sky Survey results confirm that the
observed universe is indeed lumpy on the largest scale
[59], and Thomas et al. too have rightly questioned the
physical reality of “Dark Energy” [59]. The recent ob-
servation that there is no hint of non-baryonic “Cold
Dark Matter” in two dwarf galaxies Fornax and Scul-
ptor raises serious questions about the validity of the
CDM model [60].
Note that it is possible that the distant Type 1a super-
novae may have different luminosities compared to local
ones because of different metallicities or other evolu-
tionary effects. And even if they would be standard can-
dles without appreciable cosmic evolution, there is some
chance that their faintness could be due to opacities of
distant inter-galactic medium. For example, it has been
pointed out that Lyman Alpha clouds might introduce
some non-transparency for optical emission from distant
universe [61]. It has been also argued that the atmos-
phere of planets could be additional sources of opacities
for very distant supernovae lights [62]. Also, there could
be a fundamental non-accuracy in estimating the precise
luminosity distances of distant supernovae because of
missing gaps and extrapolations in the cosmic distance
measurement ladders. Further, consider the fact that we
are able to “see” galaxies and measure their redshifts
because they are radiating. But in the standard cosmolo-
gies, galaxies are considered as non-radiating neutral
“dust” particle. Consequently, one should leave here an
open window for some hitherto unknown physics or sur-
prise for understanding this new phenomenon. In general,
there could be unknown systemetic effect behind the
interpretation of extra faintness of distant Type 1a su-
pernovae [29].
And if the universe would indeed be unbounded and
infinite hierarchial fractal, then the basic assumption
behind the formulation of Hubble’s law, i.e., strict ho-
mogeneity, gets invalidated. In such a case, the universe
need not be expanding and the observed redshits could
be of non-Doppler origin. Recall long back Ellis [63]
argued that observations really cannot rule out a static
but inhomogeneous universe. Photons propagating over
cosmic scales might undergo energy losses by yet un-
known feeble quantum electrodynamical interaction with
the quantum vacuum [64]. Photons might also lose en-
ergy by interacting with plasma permeating the whole
cosmos [65]. It is interesting to note that the plasma red-
shift proposed by Ari Brynjolfsson can simulate a value
of “Hubble Parameter” Km/s/Mpc by considering
a mean cosmic free electron density of cm–3
[64-68]. The only “new physics” involved with his red-
shift mechanism is to include an electron-electron colli-
sion term in an otherwise standard quantum derivation of
photon-electron plasma interaction cross-sections. Thus,
the excess dimming at high redshifts could simply be a
consequence of the Compton scattering that accompanies
the redshifting mechanism in the hot, sparse electron
plasmas that fill intergalactic space.
75
0.0002
In Brynjolfsson’s cosmology, the universe is not ex-
panding, and there is no time dilation affecting super-
nova light curves, and it claims to account for the red-
shift/SNe Ia luminosity data.
Though the BBM has had many successes to its credit
and it is the most developed cosmology, it fails to satis-
factory explain many cosmological facts [69]. And irre-
spective of the presence or absence of an appropriate
static model for the universe, there are innumerable
problems with the BBM. In particular,
BBM may be failing the angular size test; whereas a
static model fits data better [70].
The age of extremely red and massive galaxies at
very high redshift may be contradicting the
CDM
cosmology [71]. And most importantly,
The Gamma Ray Bursts having zs as large as 9
do not show signs of supposed space-expansion re-
lated time dilation [72,73].
In fact a static model of universe (which must be en-
tirely different from a vacuous ESU) may fit at least
some of the cosmological observations better [74,75]
without any exotic assumptions.
Also, in an infinite and eternal universe, part of the
starlight may get thermalized to generate the observed
microwave background radiation, as was first suggested
by Sir Fred Hoyle. However, there could also be alterna-
tive explanations for the same. For instance, the micro-
wave background radition might be due to superposition
of redshifted quiescent surface glow of the Eternally Col-
lapsing Objects, the so-called “Black Holes” [76].
Note, when acoustic wave packets traverse a disper-
sive medium for very long duration, they develop “red-
shift” [77]. And if the wave packet will have a Gaussian
shape, it maintains this shape despite central energy loss;
and further, the attendant redshift obeys the Hubble’s
Law [77]. Recall, all emitted electromagnetic lines are
actually narrow “wave packets”, and their propagation in
the cosmos is likely to be dissipative. At a classical level,
one can think that the cosmos with matter and plasma
behaves as a “dissipative medium”; and on a Quantum
Electrodynamic level, one may take the QED vacuum as
a “dissipative medium”. Can the electromagnetic wave
packets too undergo red-shift similar to the acoustic
waves? Or can the electromagnetic wave packets spread
in the momentum space like the “de-Broglie matter
waves” and somehow simulate the Hubble’s law?
In any case, from a purely observational view point,
one can say that [78] “it is impossible to conclude either
Copyright © 2011 SciRes. IJAA
A. MITRA197
way whether the Universe is expanding or static. The
evidence is equivocal; open to more than one interpreta-
tion. It would seem that cosmology is far from a preci-
sion science, and there is still a lot more work that needs
to be done to resolve the apparent evidences.”
15. Endnote
A very old version of this paper was submitted to Phys.
Rev. D. in 2008. And though the assigned referee did not
point out any technical error, he would not accept the
result claiming it would be against observation!
Simulataneously one moderator of arXiv.org also quietly
shifted it from astro-ph to phys-gen section; and in frus-
tration, I did not look at this manuscript for 3 years. Only
this yesr, another version was submitted to J. Cosmology
& Astroparticle Physics. The referee here accepted that
the proofs claiming are correct. But he suggested
that the revised version should (1) not only cite and high-
light the reference [28] for explaining the result,
but (2) all explanations in the light of inhomogeneity,
fractal strcture etc. should be avoided. While the revised
version conceded the first part of this arm-twisting, it
could not yield to the latter part. Then this paper was
rejected by JCAP even after accepting that the proof
was correct. Note here that the suggestion by
Kleidis & Spyrou that an increase in the value of for
the DM can make even when is com-
pletely wrong (see Equation [134]) even though it is
published in A & A! Yet Dr Spyrou killed this paper in
order to highlight his completely wrong idea by sup-
pressing all competing correct ideas! However, the refe-
ree of IJAA found this paper so important that the proc-
essing charges were kindly waived off. Thanks IJAA.
=0
=0
>0
S

=0
=0
=0p
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