Cosmological Model with A Nonhomogeneous Cosmic Time
Kang Li${}^{a}$
The principle of consistent relativity for establishing of the dynamics equation is firstly introduced in \cite{chen} and then we can discuss the dynamics law of the universe by use of the method of inertial reference frame. We strictly refer to the establishing of the geometrized equation of the gravity on the solar system scale, and gravitational physics effect opened up form the Schwarzschild space-time. Then the metric of the gravity-geometrized space-time is restated. We introduce two geometrical variables $\{b(t),a(t)\}$ into the cosmological metric according to the varying gravity in the universe. For $b(t)$ is the gravitational time dilation factor, the running of $b(t)$ symbolizes the nonhomogeneous evolution of the cosmic time ($b(t)dt$) within the same time interval ($dt$) for the present observer rest on the Earth. On this base, we investigate the corresponding fundamental equations in our cosmological model with the nonhomogeneous cosmic time, and lastly, the $chain$ $condition$ between the geodesics of an instantaneous metric solution and the distribution of the gravitational matter in the next moment is also carried out. \\ PACS number(s): 98.80.Cq, 11.10.Kk
hep-th0411047.tex
\documentclass[prl,nofootinbib]{revtex4}
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\usepackage{latexsym}
\def\be{\begin{equation}}
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\def\bea{\begin{eqnarray}}
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\def\btheta{\tilde{\theta}}

\begin{document}
\title{Cosmological Model with A Nonhomogeneous Cosmic Time  }

\author{Kang Li${}^{a}$}
\email{kangli@hztc.edu.cn}

\author{ChiYi Chen${}^{b,c}$}
\email{chenchiyi@center.shao.ac.cn}



\affiliation{${}^a$Hangzhou Teachers College, Hangzhou 310036, PR
China} \affiliation{${}^b$Shanghai Astronomical Observatory,
Chinese Academy of Sciences, Shanghai 200030, PR China}
\affiliation{$^{c}$Graduate School of Chinese Academy of Sciences,
Beijing 100039, China} \affiliation{${}^d$National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, PR
China}

\begin{abstract}
The principle of consistent relativity for establishing of the
dynamics equation is firstly introduced in \cite{chen} and then we
can discuss the dynamics law of the universe by use of the method of
inertial reference frame. We strictly refer to the establishing of
the geometrized equation of the gravity on the solar system scale,
and gravitational physics effect opened up form the Schwarzschild
space-time. Then the metric of the gravity-geometrized space-time is
restated. We introduce two geometrical variables $\{b(t),a(t)\}$
into the cosmological metric according to the varying gravity in the
universe. For $b(t)$ is the gravitational time dilation factor, the
running of $b(t)$ symbolizes the nonhomogeneous evolution of the
cosmic time ($b(t)dt$) within the same time interval ($dt$) for the
present observer rest on the Earth. On this base, we investigate the
corresponding fundamental equations in our cosmological model with
the nonhomogeneous cosmic time, and lastly, the $chain$ $condition$
between the geodesics of an instantaneous metric solution and the
distribution of the gravitational matter in the next moment is also
carried out.
\\
PACS number(s): 98.80.Cq, 11.10.Kk
\end{abstract}


\maketitle
\section{INTRODUCTION}
In Ref.\cite{chen}, the authors have pointed out that the most
essential conception for the "inertial reference frame" is that the
dynamics relativity must be maintained from the aspect of symmetry
and causality. Once it has been done in all inertial reference
frames, we would have the same form of the dynamics law in these
reference frames. Furthermore, we can make a adjustment of the
causalities on the both sides of the dynamics equation in any
inertial reference frame among of them, then first law of motion
would be automatically carried out in all the inertial reference
frames.

In the spirit of above proposition, we have discussed in detail the
nature way to realize the dynamics relativity from the aspect of
symmetry and causality, and proposed the principle of consistent
relativity for establishing the dynamics equation. The start point
of this proposition is based on such a fact that our observation of
gravity executed on all objects in celestial system is local, which
also is the origin of relativity. In fact, the influences of the
gravity from out-space of the system on the inner relative motion
can be almost erased by the global motion of the system. Thus, in
any gravity-free system, the count of gravitation interaction should
be treated as a relative physical quantity, to all the observers in
this system.

It is well known that the kinetic quantity appearing in dynamic
equation must be a relative physical quantity. Therefore, to keep
the invariant form of the dynamic equation, we need to start from
the idea of symmetry on the relativity in both sides of dynamic
equation. From the viewpoint of symmetry, we know that the speed of
the observed particle is relative, and the accelerated speed of the
observed particle also is relative; thus, to keep the invariance of
the dynamic equation, the force to be counted also is relative.
While in fact, there exists such a celestial system with natural
superiority, which means the relativity in different level of
gravity-bounded system can be counteracted globally and symmetrical
with high precision, the essential reason owe to the equivalence of
gravitational mass and inertial mass.

In Ref.\cite{chen}, the author realized the definition of a kind of
natural inertial reference frame through the introduction of the
gravity relativity, and proposed that the minimal gravity-free
observer can construct the inertial reference frame naturally. The
relativity symmetry can be realized, only for the dynamic equation
established on this kind of observer. Because we can find physical
inertial reference frame for all systems to be observed, but the
system of gravity to be counted must be consistent with the
selection of its inertial reference frame, thus we can call it as
the principle of consistent relativity in dynamics.

On this base, we are going to demonstrate that the geometric
equivalence of gravity in solar system meets the principle of
consistent relativity, which means that the count of relative
gravity in solar system is consistent with the choosing of mass
center of solar system as the origin. In this regard, the
cosmological metric should be established on the inertial reference
frame of universal background. The main purpose of this paper is to
discuss the evolution of universe and its observation theory by use
of the method of inertial reference frame with the gravity physics
disclosed in the Schwarizild case.

\section{Gravitational Physics In The Schwarzschild Case}
As it is well known, the gravity testing of Einstein's field
equation have achieved great successes on the solar scale.
Therefore, the establishment of the cosmological metric, besides
resorting to the Cosmological Principle, must be also resorting to
the physics in the Schwarzschild metric on the solar scale. Above
all, the scope of the interaction under the consideration in these
gravity testings is the gravitational interaction in the inner
region of the sun system, hence the mass center of the sum system
should be chosen as the origin of the inertial reference, and
actually we have so done it before. That is to say, we have a
background inertial reference on the solar scale,
\begin {eqnarray}
ds^2=-dt'^2+d\vec{x'}^2.
\end {eqnarray}
Firstly, it is introduced in mathematics because all the space-time
intervals in the inner region of the sun system should be measured
by the observer rest on the Earth with their proper scales.
Secondly, the metric is mathematically homogeneous and isotropic
which has just corresponded to a physical assumption: the proper
scale of the space-time on different space points would be equal to
each other if it is considered only in the background space-time
where doesn't exist any gravitational interaction. Then we can
geometrize the gravity in the inner region of the sum system on the
basis of this inertial metric (1) and we directly write down its
correct form---the Schwarzschild metric,
\begin {eqnarray}
ds^2=-B^2(r')dt'^2+A^2(r')dr'^2+r'^2d\theta'^2+r'^2sin^2\theta'
d\varphi'^2.
\end {eqnarray}
It is the spherical reference frame with the origin at the mass
center of the sum system, which can be seen from the theoretical
analysis of solar gravity experiments. In addition, $t',r'$ are
still coordinates of the background reference frame (1). Therefore,
after geometrized, the gravity effect is completely demonstrated by
the difference of the proper scale of the space-time on different
space points under the solar gravity. The strength of gravity is
equivalently geometrized into the actual proper space-time scale
measured by the observer test on the Earth at the same moment. In
this regard, as well as we have been identifying $dt'$ as the
reading of proper clock on the observer rest on the Earth, we can
identify $B(r')dt'$ as the reading of the proper clock on different
space points (Both are read out at the same moment). As we know,
$B(r')<1$ in the Schwarzschild case, which has just meant that the
running of the proper time is absolutely decelerated under the
action of the gravity. Furthermore, $B(r')$ is decreasing with the
decrease of $r'$ where the gravity becomes more and more strong. To
the observer rest on the Earth, he will have $B(r'_1)dt'<B(r'_0)dt'$
at the same moment if $r'_1<r'_0$. That is to say, the stronger the
gravity is, the longer the period of proper events behaves (keep
mind that is relative to the same observer). Then the gravitational
redshift in the Schwarzschild case is theoretically carried out.


The above discussion that the gravity equals the change of the
space-time interval of the unit proper events is one explanation of
the gravitational redshift in the Schwarzschild case. The essential
reason of the redshift of a photon is the difference of the total
energy gained from the same kind of energy transition under
different gravity field. the motion would not result in the
redshift, once the photon is generated. That is to say, in the first
scenario of the gravity geometry theory, the discrepancy of the
gravitational space-time scale is a true phenomenon in physics,
which not only take effect on the motion of the observed particle,
but also take effect for all the proper events.

The another explanation of the gravity geometry theory still not
capable being ruled out is that the motion of any photon in gravity
field results in the redshift. In this scenario, what can be showed
in the Schwarzschild metric is nothing but the kinematical
equivalence of the gravity on the background of vacuum space-time in
the investigating of the motion of a gravity-free particle. It is an
equivalent description in mathematics and only take effect on the
motion of the particle. But in fact the true space-time with the
gravity field is still homogeneous and isotropic, which means that
the space-time scale correspond to local proper events at different
space points are uniform. However, there exists one point different
with Newtonian mechanism and Special relativity that the motion of
any particle in the gravity field is also obeying the geometrical
variation principle. No matter of what explanation is, they would be
equivalent with each other on the investigation of the motion of
particles, which is just the physics concerned in this paper.
Considering that the principle of space-time interval invariant is
consistent with the proper energy conservation law in Special
relativity, we prefer to investigate the gravitational redshift of
the photon in the Schwarzschild case by use of a simulated approach
of the proper energy conservation law in gravity field. First, the
kinematics theory is pure of geometry and the essence of the motion
would not be changed by performing a general coordinates
transformation. Therefore, it is always available to redefine the
scale of Schwarzschild space-time obtaining a mathematical flat
space-time,
\begin {eqnarray}
ds^2=-dT^2+dR^2+R^2d\theta^2+R^2sin^2\theta d\varphi^2.
\end {eqnarray}
In this mathematical flat space-time, we may regard its coordinates
are equivalently measured by the physical events occurred on
different space points (first kind of explanation), so kinematical
quantities described by this set of coordinates can be named as the
event reading. The radial velocity of the particle satisfies
\begin {eqnarray}
V=\frac{dR}{dT}=\frac{a(r')dr'}{b(r')dt}=\frac{a(r')}{b(r')}v.
\end {eqnarray}
As long as the motion of the gravity-free particle is always to obey
the same principle for different observers, the traditional mass
transformation caused by the speed can still be quoted here,
\begin {eqnarray}
m=\frac{m_0}{\sqrt{1-\frac{V^2}{c^2}}}=\frac{m_0}{\sqrt{1-\frac{a^2(r')v^2}{b^2(r')c^2}}}.
\end {eqnarray}
We assume that the light source is moving apart from the observer in
the speed of $v$ and the frequency of a photon, when emitted, is
$\nu_0$, the frequency when received by the earth observer is $\nu$.
We may take such a equivalence for the photon,
\begin {eqnarray}
h\nu&=&mc^2\\
h\nu_0&=&m_0c^2\\
p&=&mv.
\end {eqnarray}
here $m_0$ is the kinematical mass of the photon from the viewpoint
of the light source. In general, as long as the rest mass of the
particle equals with each other in different background of gravity,
the proper energy conservation law is in the similar form of Special
relativity,
\begin {eqnarray}
-m^2_0c^4=-m^2_1c^4+p^2_1c^2=-m^2_2c^4+p^2_2c^2.
\end {eqnarray}
In the spirit of the geometrical equivalence for the gravity, the
kinematical description of the gravitational potential energy may be
written down as
\begin {eqnarray}
\frac{1}{2}mv^2=\frac{1}{2m}p^2\Leftrightarrow\frac{GMm}{r'}.
\end {eqnarray}
Substituting (10) into (9), we may propose the following proper
energy conservation law for gravity-free particle,
\begin {eqnarray}
-m^2_0c^4+\frac{2GMm^2_0}{r'_0}c^2=-m^2c^4+\frac{2GMm^2}{r'}c^2.
\end {eqnarray}
Therefore,
\begin {eqnarray}
h^2\nu^2+h^2\nu^2_0\frac{2GM}{c^2r'_0}=h^2\nu^2_0+h^2\nu^2\frac{2GM}{c^2r'}.
\end {eqnarray}
Then we have
\begin {eqnarray}
\frac{\nu^2_0}{\nu^2}&=&\frac{1-\frac{2GM}{c^2r'}}{1-\frac{2GM}{c^2r'_0}}=\frac{b^2(r')}{b^2(r'_0)},\\
\frac{\nu_0}{\nu}&=&\sqrt{\frac{1-\frac{2GM}{c^2r'}}{1-\frac{2GM}{c^2r'_0}}}.
\end {eqnarray}
After that, the exact expression of the redshift in the
Schwarzschild is recovered. The kinematical equivalence of the
gravitational potential energy takes the following form,
\begin {eqnarray}
p^2c^2|_{(\mbox{r'������������})}=\frac{2GMm^2}{r'}c^2=(1-b^2(r'))m^2c^4.
\end {eqnarray}

In addition, it should be noted that the physics demonstrated by the
change of the coordinates scale on the inertial reference frame (2)
is only the kinematical equivalence of the gravity. Therefore, to
practical transformation of the space-time interval in gravity
field, we must take further consideration of the effect Lorenz
transformation, besides if gravity takes effect. In a similar way,
the counting of the effect of Doppler is also required in the
calculating of practical redshift of a photon.




\section{COSMOLOGICAL MODEL WITH A NONHOMOGENEOUS COSMIC TIME }
In cosmology, the object we deal with is all the ordinary matter and
the all gravitational interaction in the universe. In the spirit of
the symmetry of the relativity, the background space-time on which
we could geometrize all the gravity in our practical universe must
be the space-time of an entirely empty universe. Obviously, the
background space-time to the cosmological gravity is just the
Minkowski space (conversely, which may be regarded as the definition
of the Minkowski space ). To the present observer rest on the Earth,
the Minkowski metric as a background can also be available just as
an instantaneous observer in the entirely empty universe. Because
the observer rest on the Earth still is in gravity free at the large
scale. Consequently, performed by an instantaneous rescaling, the
metric of the background space-time with the scale of the present
observer rest on the Earth can be written as
\begin {eqnarray}
ds^2=-dt^2+d\vec{x}^2.
\end {eqnarray}
The next step is to constitute a geometrized space-time model
according to the Cosmological Principle. Therefore, we can
establish a connection between the comoving coordinates of the
geometrized metric and the background reference frame of the
present Earth observer to describe the evolution of the
gravitational matter in the background space-time. On the other
hand, as it is well known, a stationary gravity field must result
in the discrepancy between the clocks rest on different space
points. Along this way, it is reasonable to extrapolate that the
running of the clock rest on the same space point will also be
nonsynchronous (with the time) due to the varying gravity executed
on it. Therefore, the geometrized metric of the cosmological
gravity can be given by
\begin {eqnarray}
ds^2=-b^2(t)dt^2+a^2(t)[\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2sin^2\theta
d\varphi^2].
\end {eqnarray}
Where $\{t,r,\theta,\varphi\}$ still are the background coordinates
of the inertial observer of universe, and they no longer denote
commoving coordinate, but the distance variance between the
commoving points can be fully represented by the dynamic law. In
addition, $a(t)$ has just characterized the homogeneousness and
isotropy of the space and the proper time scale $b(t)$ bears the
effect of gravitational time dilation. For $t$ is the background
time of the inertial reference frame (16), the running of $b(t)$
symbolizes the nonhomogeneous evolution of the cosmic time $b(t)dt$
within the same time interval $dt$ for the present observer rest on
the Earth. In addition, it can be checked and approved that the
world line of the commoving coordinates point (
$(r,\theta,\varphi)=const.$) is just the one kind of the geodesics
\cite{bondi} in the space-time with metric (17). What's more,
according to solution of Schwarzschild metric, the solution of
cosmological metric should be executed under the expression of
inertial observer in universe. According to the line element (17),
the non-zero components of its Ricci tensor are given by
\begin {eqnarray}
R_{00}&=&-3\frac{\ddot{a}}{a}+3\frac{\dot{a}\dot{b}}{ab};\\
R_{ij}&=&\frac{1}{b^2}(2\frac{\dot{a}^2}{a^2}+\frac{\ddot{a}}{a}-\frac{\dot{a}\dot{b}}{ab})g_{ij}+\frac{2k}{a^2}g_{ij}.
\end {eqnarray}
To be consistent with the symmetries implicated in the metric (17),
the total stress-energy tensor should be diagonal and may be simply
realized by that of a perfect fluid
$<T^{\mu}_{\nu}>_{grav}=diag\{\rho,-p,-p,-p\}$. Then, the
fundamental equations of the cosmology may be taken as
\begin {eqnarray}
\frac{\dot{a}^2}{a^2b^2}+\frac{k}{a^2}&=&\frac{8\pi G}{3}(\rho+\Lambda);\\
\frac{1}{b^2}(\frac{\ddot{a}}{a}-\frac{\dot{a}\dot{b}}{ab})&=&-\frac{4\pi
G}{3}(\rho+3p)+\frac{8\pi G}{3}\Lambda.
\end {eqnarray}
It is well known, in Einstein's field equation, there are ten
independent components in the whole tensor equation, which is just
corresponding to the 10 freedom numbers in the symmetric metric.
Just as the ${g^{\mu\nu}}_{;\mu}\equiv 0$ is nothing but a
definition on the affine connection and do not change the number of
the freedom of the symmetric metric, the conversation law
${T^{\mu\nu}}_{;\mu}=0$ is not capable influencing the completeness
of the gravitational field equation. The completeness of the above
equations can be carried out due to a fact that the introduction of
the inertial reference frame in our theory has been actually in
equivalence to the introduction of the coordinates conditions.
Therefore, the $2$ geometric variables which we have introduced from
previous physical considerations just correspond to the $2$ freedom
numbers of the perfect fluid model of the ordinary matter in a
homogeneous isotropic universe, and the equivalence between the
gravity and the geometrization language is naturally carried out in
such a metric form (17).


According to the assumption of cosmological homogeneous isotropy and
the Poincare relativity of universal background observer, we can
representatively choose an adjacent commoving system that consists
of the earth and a certain light source, where the propagation law
is applicable to all local regions in the universe. To simplify the
mathematic representation, we assume that commoving points move
along radial direction. Because currently existed universe is in
expanding, the initial speed of commoving points must be not zero.
Then we can denote the initial speed of the commoving point
including the earth at the moment $t_e$ of decoupling from other
interaction to be $v_e$; The initial speed of the light source
relative to inertial observer of universe is $v_s$ , when it start
to be in gravity free at $t_s$. Similar to the derivation of the
mathematical flat time-space from the redefinition of the
coordinates scale in Schwarzschild case, the redefined speed of the
cosmological space-time characterized by Eq.(17), which satisfies
the Lorenz transformation, can be expressed as follows:

\begin {eqnarray}
V_e(t)&=&\frac{a(t)dr}{b(t)dt}=\frac{a(t)}{b(t)}v_e(t);\\
V_s(t)&=&\frac{a(t)dr}{b(t)dt}=\frac{a(t)}{b(t)}v_s(t)
\end {eqnarray}
From the aspect of theory, the motion of the gravity-free particle
should be obtained according to the geometrical variation principle,
no matter the first explanation or the second explanation of the
gravity geometry theory. But, there exist difficulty for directly
applying this principle as for the non-stationary cosmological
metric. Thus, we hope to realize the geometrical discussion of the
dynamics law of cosmological commoving points, in resorting to the
geometrical language of invariant space-time interval adopted by
special relativity. At random time $t$, no matter the commoving
particle with mass is accelerated in any way, according to the
proper energy conservation of gravity-free particle in gravitational
geometry, we have the following relation
\begin {eqnarray}
-m^2|_{(\mbox{IS})}c^4+p^2c^2|_{(\mbox{IKEGF})}
=-m^2c^4+p^2c^2|_{(\mbox{KEGF})}.
\end {eqnarray}
here $IS$ denotes "initial speed" and $IKEGF$ denotes "the
kinematical equivalence for the gravity field around the particle at
the initial moment ". And $KEGF$ denotes "the kinematical
equivalence for the gravity field around the particle at the moment
t". In further study, after referring to the connection between the
change of the gravity potential energy in the Schwarzschild
space-time and the gravitational red-shift of photon, we obtain the
following relation for the gravity-free point of the earth.

\begin {eqnarray}
p^2c^2|_{(\mbox{KEGF})}&=&(1-b^2(t))\frac{m^2_0}{1-\frac{V^2_e(t)}{c^2}}c^4.\\
p^2c^2|_{(\mbox{IKEGF})}&=&(1-b^2(t_e))\frac{m^2_0}{1-\frac{V^2_e}{c^2}}c^4.\\
m^2c^4|_{(\mbox{IS})}&=&\frac{m^2_0}{1-\frac{V^2_e}{c^2}}\cdot
c^4=\frac{m^2_0}{1-\frac{a^(t)v^2_e}{b^(t)c^2}}\cdot c^4.
\end {eqnarray}
From the kinematics theory of the gravity-free point, we obtain the
speeds of earth and light source $v_e(t)$ and $v_s(t)$ at the moment
$t$,
\begin {eqnarray}
\frac{a(t)}{b(t)}v_e(t)=V_e(t)=\sqrt{c^2-\frac{b^2(t)}{b^2(t_e)}(c^2-V^2_e)}=\sqrt{c^2-\frac{b^2(t)}{b^2(t_e)}(c^2-\frac{a^2(t)v^2_e}{b^2(t)})}\\
\frac{a(t)}{b(t)}v_s(t)=V_s(t)=\sqrt{c^2-\frac{b^2(t)}{b^2(t_s)}(c^2-V^2_s)}=\sqrt{c^2-\frac{b^2(t)}{b^2(t_s)}(c^2-\frac{a^2(t)v^2_s}{b^2(t)})}.
\end {eqnarray}
Assuming the distance between light source and the earth to be $L_0$
at $t_0$, and to be $L$ at $t$, then the following relation can be
obtained
\begin {eqnarray}
\nonumber L_0&=&L+\int^{t_0}_{t}v_s(t)dt-\int^{t_0}_{t}v_e(t)dt\\
\nonumber&=&L+\int^{t_0}_{t}\frac{b(t)}{a(t)}\sqrt{c^2-\frac{b^2(t)}{b^2(t_s)}(c^2-\frac{a^2(t)v^2_s}{b^2(t)})}\cdot
dt\\
&&-\int^{t_0}_{t}\frac{b(t)}{a(t)}\sqrt{c^2-\frac{b^2(t)}{b^2(t_e)}(c^2-\frac{a^2(t)v^2_e}{b^2(t)})}\cdot
dt.
\end {eqnarray}
This is just the kinematics law assumed to be equivalent to the
geodesic motion of the gravity-free particle in the homogeneous
expanding universe.

We assume that the light source emits a light signal at $t$, which
is received by the observer in earth at $t_0$. According to the
non-stationary cosmological metric, the propagation of the light
signal satisfies the zero geodesic principle. We can directly write
down the propagation distance $D_H$ of the photon from $t$ to $t_0$,
\begin {eqnarray}
\int^{t_0}_{t}\frac{b(t)}{a(t)}\cdot
cdt=\int^{r_0}_{r}\frac{dr}{\sqrt{1-kr^2}}:=D_H.
\end {eqnarray}
According to previous analysis of distance between the light source
and the earth observer, we obtain
\begin {eqnarray}
D_H=L-\int^{t_0}_{t}\frac{b(t)}{a(t)}\sqrt{c^2-\frac{b^2(t)}{b^2(t_e)}(c^2-\frac{a^2(t)v^2_e}{b^2(t)})}dt.
\end {eqnarray}
The minus in above equation results from definition of positive
direction for the coordinates of the universal inertial reference
frame, here the speed of the earth is assumed to be toward the light
source.

After the kinematics law of the commoving point related with
observation theory is presented, we can discuss the $chain$
$condition$ between the geodesics of an instantaneous metric
solution of $t$ and the distribution of the gravitational matter
in the next moment $t+dt$. First, in a matter-dominated and
homogenous isotropic universe, according to the energy
conservation principle, the energy of the matter contained in the
range $L_0$ at $t_0$ should equal to that in the range $L$ at $t$,
\begin {eqnarray}
\rho_{t_0}\cdot L^3_0=\rho_{t}\cdot L^3.
\end {eqnarray}
Second, we can directly apply Eqs.(30)-(32) of the observation
theory, and then
\begin {eqnarray}
L&=&\int^{t_0}_{t}\frac{b(t)}{a(t)}\cdot
cdt+\int^{t_0}_{t}\frac{b(t)}{a(t)}\sqrt{c^2-\frac{b^2(t)}{b^2(t_e)}(c^2-\frac{a^2(t)v^2_e}{b^2(t)})}\cdot
dt;\\
L_0&=&\int^{t_0}_{t}\frac{b(t)}{a(t)}\cdot
cdt+\int^{t_0}_{t}\frac{b(t)}{a(t)}\sqrt{c^2-\frac{b^2(t)}{b^2(t_s)}(c^2-\frac{a^2(t)v^2_s}{b^2(t)})}\cdot
dt;
\end {eqnarray}
After that, the connection between the instantaneous space-time
geometry in the homogenous expanding universe and the matter
evolution at the next moment satisfy the following condition
\begin {eqnarray}
\rho_{t}=\rho_{t_0}\cdot
\frac{L^3_0}{L^3}=\rho_{t_0}[\frac{\int^{t_0}_{t}\frac{b(t)}{a(t)}\cdot
cdt+\int^{t_0}_{t}\frac{b(t)}{a(t)}\sqrt{c^2-\frac{b^2(t)}{b^2(t_s)}(c^2-\frac{a^2(t)v^2_s}{b^2(t)})}\cdot
dt}{\int^{t_0}_{t}\frac{b(t)}{a(t)}\cdot
cdt+\int^{t_0}_{t}\frac{b(t)}{a(t)}\sqrt{c^2-\frac{b^2(t)}{b^2(t_e)}(c^2-\frac{a^2(t)v^2_e}{b^2(t)})}\cdot
dt}]^3.
\end {eqnarray}
Therefore, Eqs.(20)-(21) and (36) actually constitute a complete
equations group to solve the evolution law of the non-stationary
universe. Here, $\{t_e,v_e\}$ and $\{t_s,v_s\}$ correspond to the
initial speed of gravity-free points at the initial moment, which
can be looked as the initial condition of the cosmology.


\section{REVISION OF THE OBSERVATIONAL THEORY}
 We know that one of the theoretical bases of observational cosmology
 is Hubble theorem, the relationship between the metric geometry and the observed photon's red-shift.
Without explicitly solving the fundamental equations Eqs.(20-21) for
the dynamics of the expansion, it is still possible to understand
the kinematic effects (see details in \cite{kolb}) from the
gravity-geometrized metric model (17). First, we assume the absolute
luminosity of the light source is ${\cal{L}}$, and luminosity
distance can be defined by its observed light flux. The absolute
luminosity is defined as the energy emitted from the light source
per unit time of the reference frame rest on the light source. The
light flux is defined as the energy received by the detector on the
earth per area per unit time. According to the conservation of the
light's sphere energy propagating in three dimensional space. We
have
\begin {eqnarray}
D^2_L:=\frac{\cal{L}}{4\pi\cal{F}}
\end {eqnarray}

 Now we turn to the kinematical problem of light propagation. Because the light source and the earth
 can be treated as the gravity-free commoving point, propagation distance of the light signal received
 by the observe on earth at the moment after $t_0$ can be obtained by taking into
 consideration of the respective displacement of the light source and the earth observer relative to the
 background space-time.
\begin {eqnarray}
\nonumber \int^{t_0+\delta t_0}_{t+\delta
t}c\frac{b(t)dt}{a(t)}&=&D_H+[-v_e(t_0)\delta t_0+v_s(t)\delta
t]\\
\nonumber &=&D_H+
[-\frac{b(t_0)}{a(t_0)}\sqrt{c^2-\frac{b^2(t_0)}{b^2(t_e)}(c^2-\frac{a^2(t_0)v^2_e}{b^2(t_0)})}\delta
t_0\\
&&+\frac{b(t)}{a(t)}\sqrt{c^2-\frac{b^2(t)}{b^2(t_s)}(c^2-\frac{a^2(t)v^2_s}{b^2(t)})}\delta
t].
\end {eqnarray}
Where $-v_e(t_0)\delta t_0$ is the displacement of the earth
observer relative to background space-time within $\Delta t_0$ at
$t_0$, similarly, $v_s(t)\delta t$ is the displacement of the light
source relative to background space-time within $\Delta t$ at $t$ .
Substituting Eq.(31) into Eq. (38), we have
\begin {eqnarray}
\nonumber \int^{t+\delta
t}_{t}c\frac{b(t)dt}{a(t)}+\int^{t_0}_{t+\delta
t}c\frac{b(t)dt}{a(t)}&=&\int^{t_0}_{t+\delta
t}c\frac{b(t)dt}{a(t)}+\int^{t_0+\delta
t_0}_{t_0}c\frac{b(t)dt}{a(t)}\\
&&-[-v_e(t_0)\delta t_0+v_s(t)\delta t].
\end {eqnarray}
Applying the median theorem, we have
\begin {eqnarray}
(c\frac{b(t)}{a(t)}+v_s(t))\delta
t=(c\frac{b(t_0)}{a(t_0)}+v_e(t_0))\delta t_0.
\end {eqnarray}

 In the case of non-stationary metric, the time interval of the same waveband
 at $t$ will be not consistent with that at $t_0$, its connection is given
 by Eq.(40). Similarly, we take the event reading of the light signal emitted at $t$
 as 1, namely $b(t)dt=1$, the lasting time is $dt=1/b(t)$, according to previous obtained propagation
 law, the observer at $t_0$ can detect the light signal in the time interval of the $dt_0=\frac{cb(t)/a(t)+v_s(t)}{cb(t_0)/a(t_0)+v_s(t_0)}dt$.
  Hence the event reading the same light signal at $t_0$ is $b(t_0)dt_0=\frac{c/a(t)+v_s(t)/b(t)}{c/a(t_0)+v_s(t_0)/b(t_0)}$.
  Considering the local event speed reading of the
  light source at $t$ is $V_s(t)$, and the local event speed reading of earth observer at $t_0$ is $V_e(t_0)$. Therefore,
  the Lorenz transform which is caused by these local event speed is $\gamma=\frac{1}{\sqrt{1-[V_s(t)-V_e(t_0)]^2}}$.
  The ultimate frequency shift of a light
  signal emitted from the light source and received by the earth observer
  is,
\begin {eqnarray}
1+z=\frac{a(t_0)b(t_0)}{a(t)b(t)}\cdot \gamma
\cdot(\frac{cb(t)+v_s(t)a(t)}{cb(t_0)+v_e(t_0)a(t_0)}).
\end {eqnarray}

On the other hand, the absolute luminosity $\cal{L}$ appeared in the
luminosity distance expression ${{d}_L}^2=\frac{\cal{L}}{4\pi
\cal{F}}$ of astronomical observation theory should be described by
unit event reading of time $b(t)dt$ (first kind of explanation).
Here we assume the effect of the gravity is really originated from
the change of the space-time interval of proper events at the points
of different matter density, thus we must apply a global coordinate
transformation for introducing the physical quantity of event
reading of time. We can define
\begin {eqnarray}
 -b^2(t)dt^2\longrightarrow-dT^2,
\end {eqnarray}
Because here we consider only the kinetic law, the above mathematic
transform wouldn't change the nature of kinematics(Strictly
speaking, in the processing of the astronomical data from the
supernova, what should be put into analyzing together are those
detected $on$ $the$ $same$ $time$ by the observer rest on the Earth
because that the redefinition of (42) is available only for once in
the same processing). After that, the metric (17) immediately
recovers to following form
\begin {eqnarray}
ds^2=-dT^2+a^2(T)[\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2sin^2\theta
d\varphi^2].
\end {eqnarray}
Thus we can apply the mathematic techniques of conventional
observation theory. According to Eq.(42), after we make
corresponding coordinates transform for the frequency shift in
Eq.(41) we can obtain
$1+z=\frac{a(T_0)}{a(T)}\cdot\gamma\cdot(\frac{c+v_s(T)a(T)}{c+v_e(T_0)a(T_0)})$.

In addition, considering the physical requirement included in the
definitions of light flux and absolute luminosity, we have
\begin {eqnarray}
 D_L^2=D_{H}^2(1+z)^2.
\end {eqnarray}
That is to say, redshift equal to the effect that the light energy
passing through two different sphere of light source is reduced to
$(1+z)^2$ times per unit time, where one factor $(1+z)$ comes from
the decrease of the photon energy caused by red-shift, the another
factor $(1+z)$ comes from the decrease of the photon number detected
per unit time caused by the increase of the corresponding interval
background time for a photon propagating from the light source to
the earth.

According to the mathematical method of traditional observation
theory, if we define
$1+z'=\frac{a(T_0)}{a(T)}=(\frac{c+v_e(T_0)a(T_0)}{
c+v_s(T)a(T)})\cdot(\frac{1+z}{\gamma})$, we can easily obtain
\begin {eqnarray}
\nonumber D_H=\int^{T_0}_{T}\frac{dT}{a(T)}&=&a^{-1}(T_0)H_0^{-1}[z'-\frac{1}{2}(1+q_0)z'^2+(\frac{1}{3}+\frac{5}{3}q_0-\frac{j_0}{6}+\frac{q_0^2}{2})z'^3\\
\nonumber&&+(-1-\frac{21q_0}{4}+\frac{5}{6}j_0+\frac{s_0}{24}+\frac{5}{12}q_0j_0-\frac{15}{4}q_0^2-\frac{5}{8}q_0^3+1+2q_0\\
&&-\frac{j_0}{6}+\frac{q_0^2}{2}-3(1+\frac{q_0}{2})(\frac{q_0}{6}+\frac{1}{3}))z'^4+...]
\end {eqnarray}
Here cosmological parameters: Hubble factor H, decelerating factor q
et al, satisfy
\begin {eqnarray}
H:&=&\frac{{\dot{a}}(T)}{a(T)};\\
q:&=&-\frac{{\ddot a(T)}}{a(T) H^2}\\
j:&=&\frac{a^{(3)}(T)}{a(T)\cdot H^3}\\
s:&=&\frac{a^{(4)}(T)}{a(T)\cdot H^4}
\end {eqnarray}
Substituting Eq. (45) into Eq. (44), we obtain
\begin {eqnarray}
{H}_0{D}_L=a^{-1}(T_0)\frac{1+z}{1+z'}\cdot
[z'+\frac{1}{2}(1-{q}_0)z'^2+\frac{1}{6}({q}_0+3{q}_0^3-1-{j}_0)z'^3+...].
\end {eqnarray}
Because the practical observation is compared with the earth
observer, if cosmological metric (17) of universal inertial observer
is scaled by the standard rule of the earth, it automatically
require that $a(T_0)=1$, after substitution of the relation between
$z'$ and the practical red-shift $z$, we have
\begin {eqnarray}
\nonumber
{H}_0{D}_L&=&a^{-1}(T_0)\cdot\gamma\cdot(\frac{c+v_s(T)a(T)}{c+v_e(T_0)a(T_0)})\cdot
\{[\frac{(c+v_e(T_0)a(T_0))}{(c+v_s(T)a(T))}\cdot(\frac{1+z}{\gamma})-1]\\
\nonumber &&+\frac{1}{2}(1-{q}_0)[\frac{(c+v_e(T_0)a(T_0))}{(c+v_s(T)a(T))}\cdot(\frac{1+z}{\gamma})-1]^2\\
&&+\frac{1}{6}({q}_0+3{q}_0^3-1-{j}_0)[\frac{(c+v_e(T_0)a(T_0))}{(c+v_s(T)a(T))}\cdot(\frac{1+z}{\gamma})-1]^3+...\}.
\end {eqnarray}
Here, the relation between the red-shift of specific galaxy and its
initial speed has been given by Eq.(41). The distribution of initial
speed $v$ of commoving points is the initial condition of homogenous
cosmology.

On the other hand, in our theory, what the Hubble's law carried out
from the supernova data and CMB \cite{perlmutter,WMAP,peebles} is
only the law of $\frac{d^2a(T)}{dT^2}$ rather than
$\frac{d^2a(t)}{dt^2}$. It should be noted that the Hubble parameter
in Eq.(51) doesn't denote the speed of commoving distance change.
The practical change of the relative distance between commoving
points satisfies Eq.(30). In addition, the cosmological parameters
in Eq.(51) are defined by the event reading of time, we must use the
representation of the background coordinate time of universal
initial observer to solve the geometrized equations (20)-(21) of
cosmic gravity. According to Eq.(42), the conversion relation
between the background coordinates reading and the event reading of
the function of cosmological scale factor satisfies,
\begin {eqnarray}
\frac{da(T)}{dT}&=&\frac{a(T_0)-a(T)}{T_0-T}=\frac{da(t)}{dt}\cdot\frac{dt}{dT}=\frac{da(t)}{dt}\cdot b^{-1}(t);\\
\nonumber \frac{d^2a(T)}{dT^2}&=&\frac{d^2a(t)}{dt^2}\cdot(\frac{dt}{dT})^2+\frac{da(t)}{dt}\cdot\frac{d^2t}{dT^2}\\
&=&\frac{d^2a(t)}{dt^2}\cdot
b^{-2}(t)+\frac{da(t)}{dt}\cdot(-b^{-3}(t)\frac{db(t)}{dt}).
\end {eqnarray}


\section{REMARKS}
In summary, we have discussed the geometrical equivalence of the
cosmological gravity resorting to the method of inertial reference
frame, which is carried out by the principle of consistent
relativity for constructing the dynamics equation \cite{chen}. And
we have introduced two independent geometrical quantities according
to cosmological principle and the hypothesis of perfect fluid. On
this basis, the fundamental equation for the geometry of cosmology
was given. While the universe is a non-stationary matter field, in
which the commoving points are in gravity free. Therefore,
instantaneous distribution of matter field is determined by
instantaneous space-time geometry, at the same time, instantaneous
space-time geometry would give the matter distribution at the next
moment according to geodesic motion of the gravity-free system.
After that, we discussed the motion of the gravity-free points in
cosmology by use of the proper energy conservation law in similar to
Special relativity. The advantage of this method is that the
amendment of mass caused by the change of velocity is taken into
consideration, is different to the traditional mechanical energy
conservation principle. Furthermore, the chain condition of the
matter field at the next moment and the metric geometry are given
mathematically. At last, we also discussed the function relation
between metric geometry and red-shift, which are the basis of
observation theory of cosmology. While the geometry quantity in the
metric derived in this paper no longer denotes the commoving
distance, and the practical commoving distance is not only related
with metric geometry, but also related with the initial condition of
the motion of specific galaxy.




\begin{thebibliography}{nn}
\bibitem{chen}  Chi Yi Chen and Kang Li, astro-ph/0502256.
\bibitem{bondi} H. Bondi, M.A. and F.R.S., "COSMOLOGY", The Cambridge University Press, 1960.
\bibitem{perlmutter} S. Perlmutter et al., Nature (London)
391,51 (1998); astro-ph/9712212; A.G. Riess et al.,
astro-ph/9805021.
\bibitem{WMAP} C. L. Bennett {\it et al}, {\tt astro-ph/0302207}, to appear in
ApJ; also see the WMAP homepage, {\tt http://map.gsfc.nasa.gov}.
\bibitem{peebles} P.J.E. Peebles and Bharat Ratra, Rev.Mod.Phys.75:559-606,2003 {\tt astro-ph/0207347}.
\bibitem{kolb} Edward W.Kolb and Michael S.Turner "The Early
Universe" , Addison-Wesley Publishing Company.
\end{thebibliography}
\end{document}

hep-th0411047.extracted_bib
\begin{thebibliography}{nn}
\bibitem{chen}  Chi Yi Chen and Kang Li, astro-ph/0502256.
\bibitem{bondi} H. Bondi, M.A. and F.R.S., "COSMOLOGY", The Cambridge University Press, 1960.
\bibitem{perlmutter} S. Perlmutter et al., Nature (London)
391,51 (1998); astro-ph/9712212; A.G. Riess et al.,
astro-ph/9805021.
\bibitem{WMAP} C. L. Bennett {\it et al}, {\tt astro-ph/0302207}, to appear in
ApJ; also see the WMAP homepage, {\tt http://map.gsfc.nasa.gov}.
\bibitem{peebles} P.J.E. Peebles and Bharat Ratra, Rev.Mod.Phys.75:559-606,2003 {\tt astro-ph/0207347}.
\bibitem{kolb} Edward W.Kolb and Michael S.Turner "The Early
Universe" , Addison-Wesley Publishing Company.
\end{thebibliography}