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# Mass and Total Energy of Moving Bodies In a Quantified Expansion

Fayçal Ben Adda  New York Institute of Technology

###### Abstract

In this paper, we present a new formulation of Lorentz transformations using a
metric that quantifies the space expansion. As a consequence, we sort out that
the limiting velocity of moving bodies is decreasing together with the space
expansion. A new adjustment of relativistic laws is added to incorporate the
non static nature of space-time. The conservation of the physical laws at each
step of the quantified expansion allows the obtaining of new formalisms for
the physical laws, in particular when an object starts moving under any force,
its total energy, momentum and mass are directly affected by the expansion of
the space. An example of inelastic collision is studied and several
conclusions derived, specially the example of fission of atoms leads to clear
correlation between liberated energy and universe expansion, it turns out that
the liberated energy is increasing together with the universe expansion.

##  1 Introduction

It is known that the special relativity theory is the physical theory of
measurement in inertial frame of reference proposed by Albert Einstein in
1905, after substantial contributions of Hendrik Lorentz and Henri Poincaré
([6],[7]). This theory gave us $E=mc^{2}$ which led to the atomic bomb and
unlocked the secret of the stars. At that time the expansion nature of the
universe was not discovered yet, and this information was not included in the
formulation of the special relativity theory. In 1915 Albert Einstein proposed
the general relativity theory, in which the expansion of the universe was not
the main objective. He proposed that gravity, as well as motion, can affect
the intervals of time and of space. The general relativity theory gave us: The
space warps, the Big Bang, and Black Holes. The metric
$ds^{2}=c^{2}dt^{2}-a(t)(dx^{2}+dy^{2}+dz^{2})$ of the space-time proposed in
general relativity does not allow the incorporation of the expansion in the
the Lorentz transformation (since the increasing function of time $a(t)$, that
characterizes the expansion nature of the space, omits the linearity in the
considered transformations ), and then it does not permit to perceive if the
laws of nature proposed by the special relativity are affected or not together
with the universe expansion. Many essays have been achieved in this purpose in
the last hundred years, which can be found easily in the literature. The only
cited papers in this article are those who have a clear and direct
relationship with this kind of approach. Given that the expansion was studied
in the general relativity theory, it seems counter nature to discuss universe
expansion in the special relativity theory. However, discussing the effect of
the expansion on the laws of nature in the special relativity is different,
and might entail some disturbing new concepts which will challenge deeply our
deep-rooted understanding of the relativistic laws of nature. In this paper we
will explore the incorporation of a special expansion in the special
relativity theory that preserves the notion of inertial frame, and we will
rewrite and interpret the relativistic laws of nature in the light of this
special expansion.

###  1.1 Plan

In this article we introduce a practical quantification of the space expansion
in our universe using a metric with a numerical product that quantifies the
expansion of the space step by step (we will call it universe expansion), and
maintains the notion of inertial frame on it. In the second section we rewrite
the Lorentz transformation equations using the new metric to obtain a family
of Lorentz transformation equations $(T_{n})_{n\geq 0}$, where each
transformation $T_{n}$ represents the Lorentz transformation equations at the
step n of the space expansion. This family of transformation leads to a
decreasing limiting velocity of moving bodies from one step to another, and
then the physical laws are affected by the universe expansion, followed by
conclusions derived concerning energy, masses, and momentum in an expanding
space. In section three, the composition of velocity is derived using
$T_{n}$-Lorentz transformation equations to discuss the invariance of the new
limiting velocities of matter. The rest mass of photon is then discussed in
section forth and new quantum formalism is introduced. We conclude this
section by discussing an inelastic collision in which we find that the
conversion of the rest-energy into kinetic energy is affected by the universe
expansion as well as the fact that the total liberated energy by fission of
atoms is increasing naturally together with the universe expansion. We
conclude by some comments related to the notion of time and proper time.

###  1.2 Expansion quantification

From the rate at which galaxies are receding from each other in our universe,
it can be deduced that all galaxies must have been very close to each other at
the same time in the past. Meanwhile galaxies are receding from each other
what really happen to the physical laws, are they affected by the process of
expansion or not? In the purpose to investigate the status of the physical
laws together with the universe expansion we propose the following
quantification of the universe movement: Suppose that we can quantify the
whole universe movement, from the beginning of the universe expansion where
matter was together until present time with current picture of matter
distribution. Suppose that we can subdivide the whole process of expansion
from the beginning until present time into n steps, in which we approximate
the rate of change of the distance between any two separated events from one
step to another together with the universe expansion as follow:

$\bf{\mathcal{Q}}antification$

${\mathcal{Q}}$: if the distance between two events at the step n is equal to
$L_{n}$, and at the step (n+1) is equal to $L_{n+1}$ then
$L_{n+1}=a_{n+1}L_{n}$ for all $n\geq 0$, where $a_{n}$ is a sequence such
that $a_{0}=1$, $a_{n}>1$ $\forall n\geq 1$, and $\prod_{i=0}^{n}a_{i}$
converges.

For this quantification the step 0 represents the beginning of the expansion
of our universe (once $L_{0}\neq 0$) and the space-time is defined as the set
of events together with the notion of squared interval $S_{0}^{2}$ defined
between any two events. An event $A$ in a reference frame $\mathcal{S}$ is
fixed by four coordinates $x_{0}$, $y_{0}$, $z_{0}$ and $t_{0}$, where
$(x_{0},y_{0},z_{0})$ are the spatial coordinates of the point in space when
the event in question occurs at the step 0, and where $t_{0}$ fixes the
instant when this event occurs. Another event $B$ will be described within the
same reference frame by four different coordinates $x_{0}+\Delta x_{0}$,
$y_{0}+\Delta y_{0}$, $z_{0}+\Delta z_{0}$, and $t_{0}+\Delta t_{0}$. The
points in space where these two events occur are separated by the distance
$L_{0}$ given by $L_{0}^{2}=\Delta^{2}x_{0}+\Delta^{2}y_{0}+\Delta^{2}z_{0}$.
The moments in time when these two events occur are separated at the step 0 by
a time interval given by $\Delta t_{0}$. The squared interval $S_{0}^{2}$
between these two events is given as function of these quantities, by
definition, through the generalization of Pythagorean Theorem

$S_{0}^{2}=c^{2}\Delta^{2}t_{0}-L_{0}^{2}=c^{2}\Delta^{2}t_{0}-(\Delta^{2}x_{0}+\Delta^{2}y_{0}+\Delta^{2}z_{0})$
(1)

where $c$ is the maximum speed of signal propagation at the beginning of the
universe expansion (the maximum speed for the transmission of information). At
the step 1 of the universe expansion, the event $A$ in the reference frame
$\mathcal{S}$ is fixed by four new coordinates $(x_{1},y_{1},z_{1})$ and
$t_{1}$, where $(x_{1},y_{1},z_{1})$ are the spatial coordinates of the point
in space when the event in question occurs at the step 1, and where $t_{1}$
fixes the instant when this event occurs, meanwhile the event $B$ is described
within the same reference frame by four different coordinates $x_{1}+\Delta
x_{1}$, $y_{1}+\Delta y_{1}$, $z_{1}+\Delta z_{1}$, and $t_{1}+\Delta t_{1}$.
The points in space where these two events occur are separated by the distance
$L_{1}$ given by $L_{1}^{2}=\Delta^{2}x_{1}+\Delta^{2}y_{1}+\Delta^{2}z_{1}$.
While the universe expands from the step 0 to the step 1 at a rate such that
$\Delta x_{1}=a_{1}\Delta x_{0}$, $\Delta y_{1}=a_{1}\Delta y_{0}$ and $\Delta
z_{1}=a_{1}\Delta z_{0}$, then we have $L_{1}=a_{1}L_{0}$. The moments in time
when these two events occur are separated at the step 1 by a time interval
given by $\Delta t_{1}$. The squared interval $S_{1}^{2}$ between these two
events is given as function of these quantities, by definition, through the
generalization of Pythagorean Theorem

$S_{1}^{2}=c^{2}\Delta^{2}t_{1}-L_{1}^{2}=c^{2}\Delta^{2}t_{1}-a_{1}^{2}\Big{(}\Delta^{2}x_{0}+\Delta^{2}y_{0}+\Delta^{2}z_{0}\Big{)}$
(2)

where $c$ is the maximum speed of signal propagation at the beginning of the
universe expansion, etc.. At the step n of the universe expansion, the event
$A$ in the reference frame $\mathcal{S}$ is fixed by four coordinates $x_{n}$,
$y_{n}$, $z_{n}$ and $t_{n}$, where $(x_{n},y_{n},z_{n})$ are the spatial
coordinates of the point in space when the event in question occurs at the
step n, and where $t_{n}$ fixes the instant when this event occurs, meanwhile
the event $B$ is described within the same reference frame by four different
coordinates $x_{n}+\Delta x_{n}$, $y_{n}+\Delta y_{n}$, $z_{n}+\Delta z_{n}$,
and $t_{n}+\Delta t_{n}$. The points in space where these two events occur are
separated by the distance $L_{n}$ given by
$L_{n}^{2}=\Delta^{2}x_{n}+\Delta^{2}y_{n}+\Delta^{2}z_{n}$. While the
universe expands from the step (n-1) to the step n at a rate such that $\Delta
x_{n}=a_{n}\Delta x_{n-1}$, $\Delta y_{n}=a_{n}\Delta y_{n-1}$ and $\Delta
z_{n}=a_{n}\Delta z_{n-1}$, then we have

$L_{n}=a_{n}L_{n-1}=a_{n}a_{n-1}L_{n-2}=\dots=(\prod_{i=1}^{n}a_{i})L_{0}.$
(3)

The moments in time when these two events occur are separated at the step n by
a time interval given by $\Delta t_{n}$. The squared interval $S_{n}^{2}$
between these two events is given as function of these quantities, by
definition, through the generalization of Pythagorean Theorem

$S_{n}^{2}=c^{2}\Delta^{2}t_{n}-L_{n}^{2}=c^{2}\Delta^{2}t_{n}-(\prod_{i=1}^{n}a_{i})^{2}L_{0}$
(4)

which gives

$S_{n}^{2}=c^{2}\Delta^{2}t_{n}-(\prod_{i=1}^{n}a_{i})^{2}\Big{(}\Delta^{2}x_{0}+\Delta^{2}y_{0}+\Delta^{2}z_{0}\Big{)}$
(5)

where $c$ is the maximum speed of signal propagation at the beginning of the
universe expansion. The above metrics, from the beginning of the universe
expansion (step 0) until the present time, quantify the expansion of the
space-time step by step. Indeed, a ball ${\mathcal{B}}_{0}(0,r)$ defined by

$x_{0}^{2}+y_{0}^{2}+z_{0}^{2}\leq r^{2}$ (6)

will increase if its radius increases, and we have for $a_{i}>1,\quad
i=1,2,..,n$, $\forall n\in\hbox{\tenbb N}$,

${\mathcal{B}}_{0}\Big{(}0,r\Big{)}\subset{\mathcal{B}}_{1}\Big{(}0,(a_{1})r\Big{)}\subset\dots\subset{\mathcal{B}}_{n}\Big{(}0,(\prod_{i=0}^{n}a_{i})r\Big{)}.$
(7)

However, for all $n\in\hbox{\tenbb N}$, the sphere

${\mathcal{S}}_{n}(0,(\prod_{i=0}^{n}a_{i})r)\subset{\mathcal{B}}_{n}(0,(\prod_{i=0}^{n}a_{i})r),$

is an expanding sphere given by

$x_{n}^{2}+y_{n}^{2}+z_{n}^{2}=(\prod_{i=0}^{n}a_{i}^{2})r^{2}=(\prod_{i=0}^{n}a_{i}^{2})(x_{0}^{2}+y_{0}^{2}+z_{0}^{2})$
(8)

where $\prod_{i=0}^{n}a_{i}>1$ is an increasing and bounded product. The
infinitesimal space-time interval at the step n, $\forall n\in\hbox{\tenbb
N}$, between two neighboring events within a locally inertial frame metric
takes then the form

$d\sigma^{2}_{n}=c^{2}dt_{n}^{2}-\Big{(}\prod_{i=0}^{n}a_{i}\Big{)}^{2}(dx^{2}+dy^{2}+dz^{2}),$
(9)

in which $n$ represents the expansion step of the physical universe. In such
locally inertial reference frame the laws of the special relativity apply at
each step. We propose to rewrite the physical laws in each space-time step by
step to investigate any possible evolution in the laws of nature. The metric
(9) can be written as

$d\sigma^{2}_{n}=c^{2}dt_{n}^{2}-(dx_{n}^{2}+dy_{n}^{2}+dz_{n}^{2}),$ (10)

with $dx_{n}=(\prod_{i=0}^{n}a_{i})dx_{0}$,
$dy_{n}=(\prod_{i=0}^{n}a_{i})dy_{0}$, and
$dz_{n}=(\prod_{i=0}^{n}a_{i})dz_{0}$.

##  2 Lorentz transformations equations

Let us consider an expanding space $\mathcal{E}$ in which the proper distance
between any distant points is increasing following the above quantification
$\mathcal{Q}$ such that at small scale as well as at the subgalactic scale the
fundamental forces of nature maintain the form of matter we have today. Let us
consider that the expansion is totally modeled by an adequate numerical
quantification represented by the product $(\prod_{i=0}^{n}a_{i})$ introduced
above (that is to say if the position of a body at rest or the location of an
event is expressed locally via three space coordinates $(x,y,z)$ in an
inertial frame at the step 0, then at the step 1 its location at rest is
expressed in the same frame by $(a_{1}x,a_{1}y,a_{1}z)$, at the step 2 is
expressed in the same frame by $((a_{1}a_{2})x,(a_{1}a_{2})y,(a_{1}a_{2})z)$,
and at the step n its location at rest is expressed by
$((\prod_{i=0}^{n}a_{i})x,(\prod_{i=0}^{n}a_{i})y,(\prod_{i=0}^{n}a_{i})z)$,
where n represents the expansion step of the space movement quantification).
Let us consider two Cartesian frames $S(x,y,z,t)$ and
$S^{\prime}(x^{\prime},y^{\prime},z^{\prime},t^{\prime})$ in $\mathcal{E}$
such that at each expansion step the equations of Newtonian mechanics hold
good. Suppose that an event has Cartesian coordinates $(x,y,z,t)$ relative to
$S$, and $(x^{\prime},y^{\prime},z^{\prime},t^{\prime})$ relative to
$S^{\prime}$ at the same step of expansion. In general for a linear
transformation of the Cartesian frames we have

$\left\\{\begin{array}[]{ll}x^{\prime}=&a_{xx}x+a_{xy}y+a_{xz}z+a_{xt}t\\\
y^{\prime}=&a_{yx}x+a_{yy}y+a_{yz}z+a_{yt}t\\\
z^{\prime}=&a_{zx}x+a_{zy}y+a_{zz}z+a_{zt}t\\\
t^{\prime}=&a_{tx}x+a_{ty}y+a_{tz}z+a_{tt}t\end{array}\right.$ (11)

where the coefficients $a_{ij}$ depend on the movement. The general properties
of homogeneity and isotropy of the space with the choice that the frame
$S^{\prime}$ moves in the x direction of $S$ with uniform velocity $v$
(without losing generality), such that the corresponding axes of $S$ and
$S^{\prime}$ remain parallel throughout the motion having coincided at
$t=t^{\prime}=0$ will give

$\left\\{\begin{array}[]{ll}x^{\prime}=&a_{xx}(x-vt)\\\ y^{\prime}=&a_{yy}y\\\
z^{\prime}=&a_{zz}z\\\ t^{\prime}=&a_{tx}x+a_{tt}t\end{array}\right.$ (12)

Suppose that a light signal is emitted from the common center of the two
frames at $t=t^{\prime}=0$. Since the space $\mathcal{E}$ is expanding, then
the observer in the frame $S$ will observe that the light travels in all
direction in an expanding sphere, at the step n, given by

$\Big{(}\prod_{i=0}^{n}a_{i}\Big{)}^{2}(x^{2}+y^{2}+z^{2})=c^{2}t^{2}$ (13)

and from the frame $S^{\prime}$, the observer perceives the same

$\Big{(}\prod_{i=0}^{n}a_{i}\Big{)}^{2}(x^{\prime 2}+y^{\prime 2}+z^{\prime
2})=c^{2}t^{\prime 2}$ (14)

and we have

$\displaystyle\Big{(}\prod_{i=0}^{n}a_{i}\Big{)}^{2}(x^{2}+y^{2}+z^{2})-c^{2}t^{2}=\Big{(}\prod_{i=0}^{n}a_{i}\Big{)}^{2}(x^{\prime
2}+y^{\prime 2}+z^{\prime 2})-c^{2}t^{\prime 2}.$ (15)

Substituting formula (12) in (15) gives
$\prod_{i=0}^{n}a_{i}^{2}(x^{2}+y^{2}+z^{2})-c^{2}t^{2}=\prod_{i=0}^{n}a_{i}^{2}\Big{(}(a_{xx}(x-vt))^{2}+(a_{yy}y)^{2}+(a_{zz}z)^{2}\Big{)}-c^{2}(a_{tx}x+a_{tt}t)^{2}$,
which yields the following system

$\left\\{\begin{array}[]{ll}\prod_{i=0}^{n}a_{i}^{2}\
=&(\prod_{i=0}^{n}a_{i}^{2})\ a_{xx}^{2}-c^{2}a_{tx}^{2}\\\ \quad
1\qquad=&a_{yy}^{2}\\\ \quad 1\qquad=&a_{zz}^{2}\\\
-c^{2}\qquad=&(\prod_{i=0}^{n}a_{i}^{2})\ a_{xx}^{2}v^{2}-c^{2}a_{tt}^{2}\\\
\quad 0\qquad=&(\prod_{i=0}^{n}a_{i}^{2})\
a_{xx}^{2}v+c^{2}a_{tx}a_{tt}.\end{array}\right.$ (16)

The resolution of the system (16) leads to the Lorentz transformation
equations $T_{n}$ with a new limiting velocity

$T_{n}:\left\\{\begin{array}[]{ll}x^{\prime}=&\displaystyle{x-vt\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}}\\\ y^{\prime}=&y\\\ z^{\prime}=&z\\\
t^{\prime}=&\displaystyle{t-(\prod_{i=0}^{n}a_{i}^{2}){vx\over
c^{2}}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}}\end{array}\right.$ (17)

###  2.1 First consequence: limiting velocity

The first immediate consequence from the Lorentz transformations in an
expanding universe is the following: There exists a limiting velocity for
moving bodies given by

$v_{l_{n}}=\displaystyle{c\over\prod_{i=0}^{n}a_{i}},$ (18)

which depends on the fossil light velocity (velocity of light at the beginning
of the universe expansion) and the expanding parameter of the universe.
Indeed, $\prod_{i=1}^{n}a_{i}$ is an increasing and bounded product greater
than one, then the limiting velocity $v_{l_{n}}$ is decreasing along with the
space expansion of the universe. It turns out that the limiting velocity was
maximal at the beginning of the expansion of the universe (step 0) and was
equal to $c$, that is why we call it the fossil velocity of light. Thus the
limiting velocity of moving body, including the light, will decrease in an
expanding universe. However, the physical meaning of the equations (17)
obtained in respect to moving rigid bodies and moving clocks remain the same
as for the classical Lorentz transformations (deformability and lose of
synchronization, time transformations, etc) except the fact that the expansion
of the universe will be involved in all formulation of the physical laws as we
will see later on.

###  2.2 Other Immediate consequences

####  2.2.1 Mass and Total Energy

As a consequence of these new transformations and taking into account the
dynamic of the universe, we propose to adjust the Einstein’s equation relative
to the mass-energy as follow: The total energy of moving body at the step n of
the space expansion is given by the law

$E_{n}(v)=m_{n}(v)\ \Big{(}{c\over\prod_{i=0}^{n}a_{i}}\Big{)}^{2}$ (19)

and the body’s rest-mass energy at the step n of the space expansion is given
by the law

$E_{n}(0)=m_{0}\ \Big{(}{c\over\prod_{i=0}^{n}a_{i}}\Big{)}^{2},$ (20)

where $m_{0}=m_{n}(0)$ represents the mass of a body at rest relative to the
observer in an expanding universe at the step n. The relative mass at the step
n is given by the law

$m_{n}(v)={m_{0}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}}.$
(21)

Indeed, using the Newtonian formula $\displaystyle{dE_{n}(v)\over dt}=F.v$,
the equation (19), and the fact that the Newton’s law remains valid if it is
written as

$F={d\Big{(}m_{n}(v)\ v\Big{)}\over dt}$ (22)

give the following:

${d\Big{(}m_{n}\ {c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}\Big{)}\over
dt}={d(m_{n}\ v)\over dt}v={d(m_{n}\ v_{x})\over dt}\ \ v_{x}+{d(m_{n}\
v_{y})\over dt}\ \ v_{y}+{d(m_{n}\ v_{z})\over dt}\ \ v_{z},$ (23)

on the other hand we have

${d\Big{(}m_{n}\ {c\over\prod_{i=0}^{n}a_{i}}\Big{)}^{2}\over dt}={2m_{n}\
c\over\prod_{i=0}^{n}a_{i}}\ {d\Big{(}m_{n}\
{c\over\prod_{i=0}^{n}a_{i}}\Big{)}\over dt}=2m_{n}\ {d\Big{(}m_{n}\
{c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}\Big{)}\over dt}$ (24)

and

${d(m_{n}\ v_{s})^{2}\over dt}=2m_{n}\ v_{s}{d(m_{n}\ v_{s})\over
dt}\qquad\hbox{for}\quad s=x,\ y,\ z,$ (25)

which yields in (23)

$\displaystyle{1\over 2m_{n}}{d\Big{(}m_{n}^{2}\
{c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}\Big{)}\over dt}={1\over 2m_{n}}{d(m_{n}\
v_{x})^{2}\over dt}+{1\over 2m_{n}}{d(m_{n}\ v_{y})^{2}\over dt}+{1\over
2m_{n}}{d(m_{n}\ v_{y})^{2}\over dt},$ (26)

then

${d\Big{(}m_{n}^{2}\ {c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}\Big{)}\over
dt}={d\Big{(}m_{n}^{2}(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})\Big{)}\over
dt}={d(m_{n}^{2}\ v^{2})\over dt}$ (27)

with $v^{2}=v_{x}^{2}+v_{y}^{2}+v_{z}^{2}$. By integration we obtain

$m_{n}^{2}\ {c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}=m_{n}^{2}\ v^{2}+k.$ (28)

For $v=0$, and $m_{n}(0)=m_{0}$, we obtain $k=m^{2}_{0}\
\displaystyle{c^{2}\over\prod_{i=0}^{n}a_{i}^{2}},$ and then

$m_{n}^{2}\ {c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}=m_{n}^{2}\ v^{2}+m^{2}_{0}\
{c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}$ (29)

which gives equation (21). On the other hand the law (21) leads to the law
(19), indeed, the work of displacing a body by a distance $dl$ at the step n
is

$Fdl={dP_{n}(v)\over dt}\ dl=dP_{n}(v)\ {dl\over dt}=v\ dP_{n}(v)$ (30)

where $v$ is the velocity vector. If this work serves to increase the energy
$E_{n}(v)$ of the body at the step n, then

$dE_{n}(v)=v\ dP_{n}(v)=v\ d\Big{(}m_{n}(v)\ v\Big{)},$ (31)

then

$dE_{n}(v)=m_{n}(v)\ v\ dv+v^{2}\ dm_{n}(v).$ (32)

Since $m_{n}(v)$ is given by (21), we have

$dm_{n}(v)=\displaystyle{m_{n}(v)\ (\prod_{i=0}^{n}a_{i}^{2})\ v\ dv\over
c^{2}\Big{(}1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}\Big{)}},$ (33)

by substitution we obtain

$dE_{n}(v)=m_{n}(v)\ v\ dv+\displaystyle{m_{n}(v)\ (\prod_{i=0}^{n}a_{i}^{2})\
v^{3}dv\over c^{2}\Big{(}1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}\Big{)}}.$ (34)

then

$dE_{n}(v)={m_{n}(v)\ v\ dv\
\over\Big{(}1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}\Big{)}}.$ (35)

which gives together with formula (33)

$dE_{n}(v)={c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}\ dm_{n}(v).$ (36)

that is to say the law (19). The universe dynamic does not affect the mass of
a given object at rest. However, object at rest has a rest energy $E_{n}(0)$
given by (20) in which the universe movement is manifested. When the object
starts moving under any force, its total energy, momentum and mass are
directly affected by the expansion parameters of the universe. If the speed
$c$ is the fossil velocity of light, then the term
$\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$ in laws (19), (20), and (21)
represents the current experimental velocity of light, of course if we
consider that the present time corresponds to the step n of the universe
expansion, and it constitutes a limiting velocity for any motion or transfer
of interaction at the step n of the universe expansion.

####  2.2.2 The relativistic kinetic energy

For small values of $\displaystyle(\prod_{i=0}^{n}a_{i}){v\over c}$, the total
energy (19) goes over into the classical expression of the kinetic energy as
shown by

$E_{n}(v)={m_{0}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}}\
\Big{(}{c\over\prod_{i=0}^{n}a_{i}}\Big{)}^{2}$ (37)

which gives

$E_{n}(v)\simeq m_{0}\Big{(}{c\over\prod_{i=0}^{n}a_{i}}\Big{)}^{2}+{1\over
2}m_{0}v^{2}+\dots$ (38)

The first term represents $E_{n}(0)$ the energy at rest, and the second term
represents the Newtonian kinetic energy. The relativistic kinetic energy
$E_{c_{n}}$ or the motion energy at the expansion step n can be defined by
subtracting the rest energy $E_{n}(0)$ from the total energy $E_{n}(v)$

$E_{c_{n}}=E_{n}(v)-E_{n}(0)$ (39)

the substitution of formula (19) and (20) in the last equality gives

$E_{c_{n}}=m_{0}\Big{(}{c\over\prod_{i=0}^{n}a_{i}}\Big{)}^{2}\Big{(}{1\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}}-1\Big{)}$ (40)

and using relative mass (21) we obtain

$E_{c_{n}}=(m(v)-m_{0})\Big{(}{c\over\prod_{i=0}^{n}a_{i}}\Big{)}^{2}$ (41)

The kinetic energy of a body is not only related to the increase of masses
when they are moving as asserted by the special relativity but also related to
the universe expansion.

####  2.2.3 The relativistic momentum and energy

The use of equation (21) in the equation of motion (Force is equal to the rate
of change of momentum) remains valid. However, here the momentum is now at the
step n given by:

$P_{n}(v)=m_{n}(v)\ v={m_{0}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}}\ v.$ (42)

The equation $F=\displaystyle{dP_{n}(v)\over dt}$ determines the motion of a
body acted on by any force, but now correctly takes expanding effects into
account. Clearly, the form of the momentum and force equation is very similar
in relativity theory and in our approach; the effective mass $m_{n}(v)$
depends on the speed of motion of the body relative to the observer and the
n-th step expansion parameter $\prod_{i=0}^{n}a_{i}$ according to the formula
(21), while in relativity theory it is independent of the universe expansion.
The crucial feature is that the effective mass $m_{n}(v)$ diverges as $v$
tends to the limiting velocity
$v_{l_{n}}=\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$ and so the momentum
$P_{n}(v)=m_{n}(v)v$ also diverges. As for relativity theory, the energy and
momentum are related. Indeed, using formula (19) and (42) we obtain the
following relation

$P_{n}(v)\ c=E_{n}(v)\ {v\over c}\ \prod_{i=0}^{n}a_{i}^{2}.$ (43)

##  3 The composition of velocities

Like for special relativity we will use the classical definition of velocity.
If we call $\vec{u}$ the velocity of an object in the frame $S$, its
components with respect to the three axes are $u_{x}={dx\over dt},\
u_{y}={dy\over dt},\ u_{z}={dz\over dt}$. The same velocity is measured in the
frame $S^{\prime}$ as $\vec{u^{\prime}}$ with components given by
$u^{\prime}_{x}={dx^{\prime}\over dt^{\prime}},\
u^{\prime}_{y}={dy^{\prime}\over dt^{\prime}},\
u^{\prime}_{z}={dz^{\prime}\over dt^{\prime}}.$ For simplicity and without
losing generality, we will consider only that the velocity $\vec{u}$ is
parallel to the x-axes. A simple differentiation of the equations of Lorentz
transformations $T_{n}$ gives

$\left\\{\begin{array}[]{ll}dx^{\prime}=&\displaystyle{dx-
vdt\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}}\\\
dy^{\prime}=&dy\\\ dz^{\prime}=&dz\\\
dt^{\prime}=&\displaystyle{dt-(\prod_{i=0}^{n}a_{i}^{2}){vdx\over
c^{2}}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}},\end{array}\right.$ (44)

then we have

$\left\\{\begin{array}[]{ll}\displaystyle{dx^{\prime}\over
dt^{\prime}}=&\displaystyle{dx-vdt\over dt-(\prod_{i=0}^{n}a_{i}^{2}){v\over
c^{2}}dx}\\\ \displaystyle{dy^{\prime}\over
dt^{\prime}}=&\displaystyle{dy\over dt-(\prod_{i=0}^{n}a_{i}^{2}){v\over
c^{2}}dx}\Big{(}1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}\Big{)}^{1\over
2}\\\ \displaystyle{dz^{\prime}\over dt^{\prime}}=&\displaystyle{dz\over
dt-(\prod_{i=0}^{n}a_{i}^{2}){v\over
c^{2}}dx}\Big{(}1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}\Big{)}^{1\over
2}\\\ \end{array}\right.$ (45)

which gives

$\left\\{\begin{array}[]{ll}u^{\prime}_{x}=&\displaystyle{u_{x}-v\over
1-(\prod_{i=0}^{n}a_{i}^{2}){v\over c^{2}}u_{x}}\\\
u^{\prime}_{y}=&\displaystyle{u_{y}\over 1-(\prod_{i=0}^{n}a_{i}^{2}){v\over
c^{2}}u_{x}}\Big{(}1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}\Big{)}^{1\over 2}\\\ u^{\prime}_{z}=&\displaystyle{u_{z}\over
1-(\prod_{i=0}^{n}a_{i}^{2}){v\over
c^{2}}u_{x}}\Big{(}1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}\Big{)}^{1\over 2}.\\\ \end{array}\right.$ (46)

Since $\vec{u}$ is parallel to the x-axes, then the law of composition of
velocity is given by

$u^{\prime}={dx^{\prime}\over dt^{\prime}}={u-v\over
1-(\prod_{i=0}^{n}a_{i}^{2}){v\over c^{2}}u}$ (47)

and inversely

$u={dx\over dt}={u^{\prime}+v\over 1+(\prod_{i=0}^{n}a_{i}^{2}){v\over
c^{2}}u^{\prime}}$ (48)

which has the form of the relativist law of composition of velocity with the
appearance of the term $(\prod_{i=0}^{n}a_{i})$ that characterizes the
expansion of the universe at the step n. In the extreme case where an observer
in the frame $S^{\prime}$ is emitting light signals in the direction of
x-axis, since the frame is in an expanding universe at the step n, then the
velocity of these signals in the frame $S^{\prime}$ is given by
$u^{\prime}=v_{l_{n}}=\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$ which gives

$u={v_{l_{n}}+v\over 1+(\prod_{i=0}^{n}a_{i}^{2}){v\over
c^{2}}v_{l_{n}}}={{c\over\prod_{i=0}^{n}a_{i}}+v\over
1+(\prod_{i=0}^{n}a_{i}){v\over c}}$ (49)

which gives $u={c\over\prod_{i=0}^{n}a_{i}}=v_{l_{n}}$ and then the limiting
velocity of light $v_{l_{n}}$ at the step n is invariant, that is to say the
velocity of light ”measured” in a moving frame at the step n, appears to be
equal to $\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$ in any direction. The
speed of light in vacuum, at the step n of the universe expansion, will be
found the same (equal to $v_{l_{n}}$) by any two observers in uniform relative
motion and this is true for the whole process of universe expansion, meanwhile
this speed is decreasing together with the universe expansion. For $n=0$ the
Lorentz transformation $T_{0}$ gives $c$ as an invariant limiting velocity of
matter, meanwhile $T_{1}$ gives $\displaystyle{c\over a_{1}}$ as the invariant
limiting velocity of matter, and so on, until the Lorentz transformation
$T_{n}$ which gives $\displaystyle{c\over(\prod_{i=0}^{n}a_{i})}$ as the
invariant limiting velocity of matter. It turns out that the limiting velocity
of moving bodies is decreasing together with the universe expansion and that
the velocity of light in empty space is independent of the velocity of its
source at each step of the universe expansion despite of its decreasing nature
together with the universe expansion. In the whole process of universe
expansion the velocity of light remains independent of the reference frame of
the observer. The formula (19) can be justified exactly as the famous formula
$E=mc^{2}$ (Einstein [4]), by using the new composition of velocity provided
by the Lorentz transformation $T_{n}$ and the momentum conservation at the
step n.

##  4 Lorentz transformations and rest-mass of photon

The quantification of the expansion introduced above leads us to set up
different steps-Lorentz transformations $(T_{n})_{n\geq 0}$ that permit to
quantify the physical laws evolution together with the universe expansion, and
derive a relationship between the space-time views obtained by any two
observers in an expanding universe. The result obtained unifies the
fundamental special-relativity results of time dilation, length contraction,
and the relativity of simultaneity, in a single relation for different steps
of quantified expansion. However, from these different Lorentz
transformations, we sort out that the limiting velocity $v_{l_{n}}$ of matter
is decreasing from one step to another including the velocity of light. The
velocity of light at the beginning of the expansion of the universe (step 0)
was constant and equal to $c$ an unknown value, and the $T_{0}$-Lorentz
transformation equations are given by

$(T_{0})\left\\{\begin{array}[]{ll}x^{\prime}=&{x-vt\over\sqrt{1-{v^{2}\over
c^{2}}}}\\\ y^{\prime}=&y\\\ z^{\prime}=&z\\\ t^{\prime}=&{t-{vx\over
c^{2}}\over\sqrt{1-{v^{2}\over c^{2}}}}\end{array}\right.$ (50)

this transformation $(T_{0})$ leads to the relativistic formulation of the
physical laws

$\left\\{\begin{array}[]{ll}E_{0}(v)&\hbox{the total energy at the step 0}\\\
E_{0}(0)&\hbox{the rest-energy at the step 0}\\\ E_{c_{o}}(v)&\hbox{the
kinetic energy at the step 0}\\\ P_{0}(v)&\hbox{the momentum at the step 0
}\\\ m_{0}(v)&\hbox{the relative mass at the step 0 }\\\
m_{0}(0)=m_{0}&\hbox{the rest-mass at the step 0 }\end{array}\right.$ (51)

On the other hand, at the step 1 the $T_{1}$-Lorentz transformation equations
become

$(T_{1})\left\\{\begin{array}[]{ll}x^{\prime}=&{x-vt\over\sqrt{1-a_{1}^{2}{v^{2}\over
c^{2}}}}\\\ y^{\prime}=&y\\\ z^{\prime}=&z\\\ t^{\prime}=&{t-a_{1}^{2}{vx\over
c^{2}}\over\sqrt{1-a_{1}^{2}{v^{2}\over c^{2}}}}\end{array}\right.$ (52)

from which we sort out that the matter has a new limiting velocity
$v_{l_{1}}=\displaystyle{c\over a_{1}}$. At this step all the equations of
Newtonian mechanics hold good, and the formulations of the physical laws are
obtained under some changes relative to the new limiting velocity. Similar
formulations of the physical law equations (51) are introduced with the new
limiting velocity and denoted by

$\left\\{\begin{array}[]{ll}E_{1}(v)&\hbox{the total energy at the step 1}\\\
E_{1}(0)&\hbox{the rest-energy at the step 1}\\\ E_{c_{1}}(v)&\hbox{the
kinetic energy at the step 1}\\\ P_{1}(v)&\hbox{the momentum at the step 1
}\\\ m_{1}(v)&\hbox{the relative mass at the step 1 }\\\
m_{1}(0)=m_{0}&\hbox{the rest-mass at the step 1 }\end{array}\right.$ (53)

At the step n, the $T_{n}$-Lorentz transformation equations become

$(T_{n})\left\\{\begin{array}[]{ll}x^{\prime}=&\displaystyle{x-vt\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}}\\\ y^{\prime}=&y\\\ z^{\prime}=&z\\\
t^{\prime}=&\displaystyle{t-(\prod_{i=0}^{n}a_{i}^{2}){vx\over
c^{2}}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}},\end{array}\right.$ (54)

in which the limiting velocity of matter reaches the value of
$v_{l_{n}}=\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$, and where all
formulations of the physical laws are possible to obtain under some changes
with the hypothesis that the equations of Newtonian mechanics are valid at
this step of the universe expansion (conservation of momentum, conservation of
energy, etc). The new limiting velocity
$v_{l_{n}}=\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$ at the step n induces
some change in all formulations (as regards both mechanics and electro-
magnetism in inertial systems), including the Maxwell equations, to obtain
that the physical law remains the same under the $T_{n}$-Lorentz
transformations. The formulation of the physical law equations at the step n
with the new limiting velocity $v_{l_{n}}$ are denoted by

$\left\\{\begin{array}[]{ll}E_{n}(v)&\hbox{the total energy at the step n}\\\
E_{n}(0)&\hbox{the rest-energy at the step n}\\\ E_{c_{n}}(v)&\hbox{the
kinetic energy at the step n}\\\ P_{n}(v)&\hbox{the momentum at the step n
}\\\ m_{n}(v)&\hbox{the relative mass at the step n }\\\
m_{n}(0)=m_{0}&\hbox{the rest-mass at the step n }\end{array}\right.$ (55)

where the quantities $E_{n}(v)$, $E_{n}(0)$, $E_{c_{n}}(v)$, $P_{n}(v)$ and
$m_{n}(v)$ take the form (19), (20), (40), (42), and (21) respectively under
the $T_{n}$-Lorentz transformation for all $n\geq 0$. The Lorentz
transformations $(T_{0})$, $(T_{1})$,$\dots$, and $(T_{n})$ have the same
mathematical form, and all the physical quantities are affected by the space
expansion from one step to another, except the rest-mass $m_{0}$ which is
invariant under universe expansion. Since the light has no rest frame, and
since the formalism must be the same for all matter, the quantity $m_{0}$ in
the above formalism represents in reality the mass of matter which is
invariant under universe expansion, we will call it the invariant mass. The
family of transformation $(T_{n})_{n\geq 0}$ leads to fatal variation of the
limiting velocity $v_{l_{n}}$ of moving bodies according to the universe
expansion. This has interesting consequences. In the expansion process the
rest energy at the step n can be evaluated compared to the rest energy at the
beginning of the universe expansion. Indeed, the rest energy at the step n is
given by the law

$E_{n}(0)=m_{0}\Big{(}{c\over\prod_{i=}^{n}a_{i}}\Big{)}^{2}=m_{0}c^{2}\Big{(}{1\over\prod_{i=0}^{n}a_{i}}\Big{)}^{2},$
(56)

and using the rest energy notation we obtain

$E_{n}(0)=E_{0}(0){1\over\Big{(}\prod_{i=0}^{n}a_{i}\Big{)}^{2}},$ (57)

which means that the rest energy of matter is decreasing along with the
universe expansion. However, the relative mass of matter is increasing
together with the universe expansion, indeed, from the law (21) at the step n
and the same law at the step n+1 we sort out that

$m_{n+1}(v)\sqrt{1-(\prod_{i=0}^{n+1}a_{i}^{2}){v^{2}\over
c^{2}}}=m_{n}(v)\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}$ (58)

and we obtain for all $v<{c\over\prod_{i=0}^{n+1}a_{i}}$

${m_{n+1}(v)\over m_{n}(v)}={\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}\over\sqrt{1-(\prod_{i=0}^{n+1}a_{i}^{2}){v^{2}\over c^{2}}}}>1$ (59)

which gives

$m_{n+1}(v)>m_{n}(v)>\dots>m_{0}(v),$ (60)

this means that the relative mass of matter in our universe is increasing
along with the universe expansion as well as the momentum, meanwhile the rest
energy is decreasing. It is possible to determine the evolution of energy,
momentum and relative mass together with the universe expansion by using the
invariant mass $m_{0}$, indeed, it is not difficult to write all these
quantities at the step n function of their values at the step 0 (beginning of
the expansion) and we have the law of evolution of the relative mass together
with the universe expansion given by

$m_{n}(v)=m_{0}(v){\sqrt{1-{v^{2}\over
c^{2}}}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}},$ (61)

meanwhile the evolution of momentum together with the universe expansion is
given by the law

$P_{n}(v)=P_{0}(v){\sqrt{1-{v^{2}\over
c^{2}}}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}}$ (62)

and the total energy evolution together with the universe expansion is given
by the law

$E_{n}(v)=E_{0}(v){1\over\prod_{i=0}^{n}a_{i}^{2}}{\sqrt{1-{v^{2}\over
c^{2}}}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}}.$ (63)

However, the evolution of the physical laws from one step to another are given
by the following formulas:

$m_{n}(v)=m_{n-1}(v){\sqrt{1-(\prod_{i=0}^{n-1}a_{i}^{2}){v^{2}\over
c^{2}}}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}},$ (64)

the evolution of the momentum from one step to another is given by

$P_{n}(v)=P_{n-1}(v){\sqrt{1-(\prod_{i=0}^{n-1}a_{i}^{2}){v^{2}\over
c^{2}}}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}}$ (65)

and the evolution of the total energy from one step to another is given by the
law

$E_{n}(v)=E_{n-1}(v){1\over\
a_{n}^{2}}{\sqrt{1-(\prod_{i=0}^{n-1}a_{i}^{2}){v^{2}\over
c^{2}}}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}}.$ (66)

It turns out that the concept of expansion is a real source of energy, and
this will be analyzed later on with a concrete example.

###  4.1 Quantum formalism

The velocity of light was constant at the beginning of the expansion of our
universe and equal to $c$ (an unknown fossil value). What we measure today in
our experiment and what we call the velocity of light, corresponds in reality
to the velocity of light at the expansion step n, where its value is given by
$\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$. The light is described in quantum
mechanic as a quanta of zero mass where the relation between energy and
frequency is given by the Planck-Einstein’s relation $E=h\nu$ and where the
momentum is given by the relation

$P={E\over c}={h\nu\over c}={h\over\lambda_{0}}$ (67)

where $\lambda_{0}$ the wave length given by

$\lambda_{0}=cT_{0}={c\over\nu_{0}}.$ (68)

The rules of translation from corpuscular terminology to wave terminology, and
vice versa are based on the fact that an electromagnetic wave of length
$\lambda_{0}$ and intensity $I$ is considered as a stream of photons of
frequency $\nu_{0}$ and intensity $I=Nh\nu_{0}$ where $N$ is the number of
photons passing per unit time through unit areas, and the direction of motion
of the wave front is the direction of motion of the photons. In the purpose to
integrate the new formalism of expansion in quantum formulation we introduce
the following: We denote $\lambda_{0}$ the fossil wave length of the light at
the beginning of the universe expansion where the celerity of photon was $c$,
with $\nu_{0}=\displaystyle{c\over\lambda_{0}}$ the fossil frequency of the
light at that time. We denote

$\lambda_{n}=v_{l_{n}}T_{n}=\displaystyle{{c\over\prod_{i=0}^{n}a_{i}}\over\nu_{n}}=\displaystyle{c\over\nu_{n}\prod_{i=0}^{n}a_{i}}$
(69)

the current wave length of the light at the step n of the universe expansion,
where the celerity of photon is $\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$,
with $\nu_{n}=\displaystyle{c\over\lambda_{n}\prod_{i=0}^{n}a_{i}}$ the light
frequency at the expansion step n. Hence the wave length represents a
distance, therefore by using the property of the expansion of the space the
wave length expands and we have $\lambda_{n}=\lambda_{0}\prod_{i=0}^{n}a_{i}$
which leads to the following relation between the fossil light frequency and
the light frequency at the expansion step n

$\nu_{n}={c\over\lambda_{n}\prod_{i=0}^{n}a_{i}}={c\over\lambda_{0}\prod_{i=0}^{n}a_{i}^{2}}$
(70)

and then

$\nu_{n}=\nu_{0}{1\over\prod_{i=0}^{n}a_{i}^{2}}.$ (71)

From the relativistic relation between energy and momentum
$P_{o}(v)c=E_{0}(v){v\over c}$, we sort out the momentum of photon with
velocity $v=v_{l_{n}}$ given by

$P_{0}(v_{l_{n}})={E_{0}(v_{l_{n}})\over c}{1\over\prod_{i=0}^{n}a_{i}}\neq
0,$ (72)

thus $E_{0}(v_{l_{n}})\neq 0$ and well defined for $v=v_{l_{n}}$, then

$E_{0}(v_{l_{n}})={E_{0}(0)\over\sqrt{1-{1\over\prod_{i=0}^{n}a_{i}^{2}}}}\neq
0,$ (73)

which gives that the rest energy of a photon $E_{0}(0)$ is not equal to zero.
The substitution of (73) in the formula (72) gives

$P_{0}(v_{l_{n}})={E_{0}(0)\over
c\prod_{i=0}^{n}a_{i}}{1\over\sqrt{1-{1\over\prod_{i=0}^{n}a_{i}^{2}}}}.$ (74)

The precedent equality can be expressed using the rest energy (57) at the step
n and $v_{l_{n}}$ as

$P_{0}(v_{l_{n}})=\Big{(}{\prod_{i=0}^{n}a_{i}\over
c}\Big{)}\Big{(}{E_{0}(0)\over\prod_{i=0}^{n}a_{i}^{2}}\Big{)}{1\over\sqrt{1-{1\over\prod_{i=0}^{n}a_{i}^{2}}}},$
(75)

therefore the energy and momentum of photon are related by the law

$P_{0}(v_{l_{n}})={E_{n}(0)\over
v_{l_{n}}}{1\over\sqrt{1-{1\over\prod_{i=0}^{n}a_{i}^{2}}}}.$ (76)

The law (76) leads to the following approximation

$P_{0}(v_{l_{n}})\simeq{E_{n}(0)\over v_{l_{n}}}$ (77)

where $v_{l_{n}}={c\over\prod_{i=0}^{n}a_{i}}$ is the limiting velocity of
light measured at the step n. The relation (77) was introduced experimentally
by A. H. compton in 1923, [3].

###  4.2 Photon mass

Using the formulation given by Lorentz transformation $(T_{0})$, the relative
mass of moving body is given by

$m_{0}(v)={m_{0}\over\sqrt{1-{v^{2}\over c^{2}}}}\qquad\hbox{for}\quad\forall
v<c$ (78)

which gives

$m_{0}=m_{0}(v)\sqrt{1-{v^{2}\over c^{2}}}.$ (79)

then

$m_{0}^{2}=m_{0}^{2}(v)(1-{v^{2}\over c^{2}})\Longleftrightarrow
m_{0}^{2}c^{2}=m_{0}^{2}(v)(c^{2}-v^{2}),$ (80)

which gives

$m_{0}^{2}c^{4}=m_{0}^{2}(v)c^{4}-m_{0}^{2}(v)c^{2}v^{2},$ (81)

then for all $v<c$ we have

$m_{0}^{2}c^{4}=E_{0}^{2}(v)-P_{0}^{2}(v)c^{2}.$ (82)

Nevertheless, if the invariant mass $m_{0}$ of a photon was zero at the origin
of the expansion when its velocity was $c$, it will remain zero at the step n
when the light move at speed
$v_{l_{n}}=\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$ because of its natural
invariance under the universe expansion. If we suppose that the invariant mass
of a photon $m_{0}=0$ at the step n, $\forall n>0$, (since for $n=0$ the
formula (82) is not defined for $v_{l_{0}}=c$) from the universe expansion,
then the speed of light is given by $v=v_{l_{n}}$, and by substitution in the
the formula (82) we obtain

$0=E_{0}^{2}(v_{l_{n}})-P_{0}^{2}(v_{l_{n}})c^{2}.$ (83)

Using the value of $P_{0}(v_{l_{n}})$ given by formula (74), the equality (83)
becomes

$0=E_{0}^{2}(v_{l_{n}})-{E_{0}^{2}(0)\over{\prod_{i=0}^{n}a_{i}^{2}}}({1\over
1-{1\over\prod_{i=0}^{n}a_{i}^{2}}}).$ (84)

On the other hand we have (73), and then equality (84) gives

$E_{0}^{2}(0)({1\over{1-{1\over\prod_{i=0}^{n}a_{i}^{2}}}})={E_{0}^{2}(0)\over{\prod_{i=0}^{n}a_{i}^{2}}}({1\over
1-{1\over\prod_{i=0}^{n}a_{i}^{2}}}),$ (85)

which is equivalent to

$E_{0}^{2}(0)={E_{0}^{2}(0)\over{\prod_{i=0}^{n}a_{i}^{2}}}.$ (86)

Since the rest energy of a photon at the beginning of the universe expansion
(step 0) was not null, then formula (86) leads to the contradiction

$1={1\over{\prod_{i=0}^{n}a_{i}^{2}}}\qquad\forall n>0\quad\hbox{which is
impossible},$ (87)

then the invariant mass of photon $m_{0}\not=0$. Along the universe expansion
the invariant mass $m_{0}$ of matter remains the same, meanwhile the
associated rest energy decreases ($E_{0}(0)>E_{1}(0)>\dots>E_{n}(0)$), and the
momentum of matter increases. If the rest energy of photon was zero, it can
not decrease. Since $\forall v<c$, $E_{0}(v)\sqrt{1-{v^{2}\over
c^{2}}}=m_{0}c^{2}$, then we can evaluate the invariant mass $m_{0}$ for a
photon of velocity $v=v_{l_{n}}$. Indeed, we have

$m_{0}={{E_{0}(v_{l_{n}})\over
c^{2}}{\sqrt{1-{1\over\prod_{i=0}^{n}a_{i}^{2}}}}},$ (88)

the use of the formula (73) in the precedent equality gives

$m_{0}={E_{0}(0)\over c^{2}}$ (89)

and we have

$m_{0}={E_{0}(0)\over c^{2}}=\Big{(}{\prod_{i=0}^{n}a_{i}^{2}\over
c^{2}}\Big{)}\Big{(}{E_{0}(0)\over\prod_{i=0}^{n}a_{i}^{2}}\Big{)}$ (90)

using (57) and the velocity of light $v_{l_{n}}$ at the step n to obtain

$m_{0}={E_{0}(0)\over c^{2}}={E_{n}(0)\over v^{2}_{l_{n}}},$ (91)

which means that the ratio between rest energy and velocity of light is
invariant under universe expansion. The formula (89) means that the light has
non zero invariant mass. However, the invariant mass of a photon depends on
the fossil rest energy (the rest energy at the step n is impossible to
determine since the light is never at rest). We know only that the invariant
mass and the rest energy of a photon are not null, they might be small, very
small, but not null.

###  4.3 The conversion of the rest-energy into kinetic energy

In the purpose to sort out the real nature of energy gained by matter in an
expanding universe we introduce the following example of inelastic collision:
Consider two identical particles which move toward each other along straight
line, with equal speeds, and equal invariant mass $m_{0}$, they collide and
stick together in an expanding space at the step n. The conservation of
momentum at the step n gives:

$m_{n}(v)\ v-m_{n}(-v)\ v=M_{n}(V)\ V$ (92)

thus

$0=M_{n}(V)\ V$ (93)

which gives $V=0$, so the final object is at rest with mass $M_{n}(0)$. The
conservation of total energy at the step n from the universe expansion gives

$m_{n}(v){c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}+m_{n}(-v){c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}={M_{n}(V)c^{2}\over\prod_{i=0}^{n}a_{i}^{2}},$
(94)

since the final object is at rest (that is to say $V=0$), then it follows that

$m_{n}(v)+m_{n}(-v)=2m_{n}(v)=M_{n}(0),$ (95)

then we obtain

$M_{n}(0)={2m_{0}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}},$
(96)

the mass of final object is larger than the sum of the original masses. To
evaluate the nature of energy created with the increase of mass together with
the universe expansion from the step 0 to the step n we have to compare the
same collision at the origin of the expansion (step 0) and at the step n.
Indeed the universe expansion creates the increase of moving masses as

$M_{n}(0)-M_{0}(0)=\Big{(}{2m_{o}\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}}-{2m_{o}\over\sqrt{1-{v^{2}\over c^{2}}}}\Big{)}.$ (97)

For a small value of $(\prod_{i=0}^{n}a_{i}){v\over c}$ we can approximate
(97) as

$M_{n}(0)-M_{0}(0)\simeq 2m_{o}\ \Big{(}1+{1\over
2}(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}-1-{1\over 2}{v^{2}\over
c^{2}}\Big{)}$ (98)

which gives

$M_{n}(0)-M_{0}(0)\simeq{1\over
2}(2m_{o})(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}\Big{(}1-{1\over\prod_{i=0}^{n}a_{i}^{2}}\Big{)}.$ (99)

The gain of masses represents a rest energy at the step n given by

$\Big{(}M_{n}(0)-M_{0}(0)\Big{)}{c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}\simeq{1\over
2}(2m_{o})v^{2}\Big{(}1-{1\over\prod_{i=0}^{n}a_{i}^{2}}\Big{)}.$ (100)

This means that the rest energy gained by a moving particles together with the
universe expansion is equal approximatively to the sum of the classical
kinetic energy of the two particles factor the rate of increase of rest energy
due to the universe expansion, that is to say
$\Big{(}1-{1\over\prod_{i=0}^{n}a_{i}^{2}}\Big{)}$. Indeed, from formula (57)
it is easy to evaluate the variation of the rest energy between the beginning
of the universe expansion (step 0) and the universe expansion at the step n,
we obtain

$E_{0}(0)-E_{n}(0)=E_{0}(0)\Big{(}1-{1\over\prod_{i=0}^{n}a_{i}^{2}}\Big{)}$
(101)

from which it turns out that
$\Big{(}1-{1\over\prod_{i=0}^{n}a_{i}^{2}}\Big{)}$ represents the rate of
decrease of the rest energy together with the universe expansion. Since the
mass of final object is larger than the sum of the original masses, we can
approximate this excess of mass for a small value of
$(\prod_{i=0}^{n}a_{i}){v\over c}$, and compare it to the same value at the
beginning of the universe expansion. Indeed the approximation of the law (96),
for all $n\geq 0$, gives

$M_{n}(0)\simeq 2m_{0}+{1\over 2}(2m_{o})(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}$ (102)

which gives

$\Big{(}M_{n}(0)-2m_{0}\Big{)}\simeq{1\over
2}(2m_{o})(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}},$ (103)

and then we obtain

$\Big{(}M_{n}(0)-2m_{0}\Big{)}\simeq\Big{(}M_{0}(0)-2m_{0}\Big{)}\prod_{i=0}^{n}a_{i}^{2}.$
(104)

It turns out that the excess mass of the composite object at the step n of the
universe expansion compared to the mass of the same composite object at the
beginning of the expansion (step 0) is increasing together with the universe
expansion and then the kinetic energy brought in also increases. The lost
Kinetic energy has been converted to rest energy (mass). The classical
explanation for the loss of Kinetic energy attributes it to conversion into
thermal energy (heat): the final object will have a higher temperature or a
large internal energy. Now since the mass of the final object (after collision
formula (96) will have an increasing total mass, then this means that the
temperature of the final object will increase together with the universe
expansion. Since many evidences prove that early earth was molten from
bombardment of rocks and mini planets and others, then we can say that there
must exist a natural factor that contributes to the global warming of our
planet due to the universe expansion (the global warming of our planet is not
only attributed to human activities!).

###  4.4 Fission of atoms in an expanding space

Suppose that we have an atom of uranium with rest mass $M_{n}(0)=M$ measured
in an expanding universe at the step n, and suppose that something happens so
that the atom flies into two equal pieces moving with speed $v$, so that each
part has a relative mass $m_{n}(v)$ at the step n . Suppose that these pieces
encounter enough material to slow them up until they stop and then each part
will have an invariant mass $m_{n}(0)=m_{0}$. To reach its rest position each
part will give an amount of energy left in the material in some form,
whatever. The left energy for one part is then given by

$E_{left}=\Big{(}m_{n}(v)-m_{n}(0)\Big{)}{c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}.$
(105)

The conservation of total energy in an expanding universe at the step n gives

$E_{n}(0)=2E_{n}(v)\Longleftrightarrow M=2m_{n}(v),$ (106)

so the total liberated energy at the step n is given by

$E_{L_{n}}=2E_{left}=2\Big{(}m_{n}(v)-m_{0}\Big{)}{c^{2}\over\prod_{i=0}^{n}a_{i}^{2}},$
(107)

then

$E_{L_{n}}=\Big{(}M-2m_{0}\Big{)}{c^{2}\over\prod_{i=0}^{n}a_{i}^{2}},$ (108)

which can be written as

$E_{L_{n}}=E_{n}(v)-E_{n}(0),$ (109)

where $\forall v<\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$,
$E_{n}(v)=2m_{n}(v){c^{2}\over\prod_{i=0}^{n}a_{i}^{2}}$. The law given by
(108) for $n=0$ was used to estimate how much energy would be liberated under
fission in the atomic bomb since the mass of uranium atom was known as well as
the atoms into which it splits (iodine, xenon, and so on). However, it turns
out from the law (108) that the released energy when an atom of uranium
undergoes fission is correlated to the universe expansion which has
interesting consequences. As the universe expands we have the law (60), which
gives

$2(m_{n}(v)-m_{0})>2(m_{n-1}(v)-m_{0})>\dots>2(m_{0}(v)-m_{0}),$ (110)

and from (108) we have

$E_{L_{n}}\prod_{i=0}^{n}a_{i}^{2}>E_{L_{n-1}}\prod_{i=0}^{n-1}a_{i}^{2}>\dots>E_{L_{o}}.$
(111)

Using the law (101) we obtain

${E_{L_{n}}\over E_{n}(0)}>{E_{L_{n-1}}\over E_{n-1}(0)}>\dots>{E_{L_{o}}\over
E_{0}(0)},$ (112)

this means that the ratio between the total liberated energy by an atom of
uranium and its rest energy is increasing together with the universe
expansion. The total liberated energy by an atom of uranium (109) can be
approximated, indeed, we have

$E_{L_{n}}=E_{n}(v)-E_{n}(0)=E_{n}(0){1\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}}-E_{n}(0),$ (113)

then we have

$E_{L_{n}}=E_{n}(0)\Big{(}{1\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}}-1\Big{)},$ (114)

by using the law (57) that expresses the correlation between the rest energy
at the step n and the rest energy at the step 0, we obtain

$E_{L_{n}}=E_{0}(0){1\over\prod_{i=0}^{n}a_{i}^{2}}\Big{(}{1\over\sqrt{1-(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over
c^{2}}}}-1\Big{)}.$ (115)

For a small value of $(\prod_{i=0}^{n}a_{i}){v\over c}$, we obtain the
following approximation

$E_{L_{n}}\simeq E_{0}(0){1\over\prod_{i=0}^{n}a_{i}^{2}}\Big{(}{1\over
1-{1\over 2}(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}}-1\Big{)}.$ (116)

which gives

$E_{L_{n}}\simeq{E_{0}(0)\over\prod_{i=0}^{n}a_{i}^{2}}\Big{(}1+{1\over
2}(\prod_{i=0}^{n}a_{i}^{2}){v^{2}\over c^{2}}+({1\over
2})^{2}(\prod_{i=0}^{n}a_{i}^{2})^{2}{v^{4}\over c^{4}}-1\Big{)},$ (117)

then

$E_{L_{n}}\simeq E_{0}(0)\Big{(}{1\over 2}{v^{2}\over c^{2}}+({1\over
2})^{2}(\prod_{i=0}^{n}a_{i}^{2}){v^{4}\over c^{4}}\Big{)},$ (118)

it turns out that the total liberated energy by an atom of uranium is
increasing together with the universe expansion and we have for one atom of
uranium the following evolution of liberated energy from one step to another

$E_{L_{n}}>E_{L_{n-1}}>\dots>E_{L_{o}},$ (119)

This has interesting consequences that will change our understanding of
gravity, the missing masses, or the estimation of stars energy and others. The
energy in our universe increases together with the universe expansion, and
this excess of energy is naturally due to the recession of galaxies from each
other. The increase of energy and mass in our universe compensates the
increase of distance between matters and might explain why the gravity exists
at long distance. Our estimation of stars energy, or galaxies masses are
erroneous if we omit to take into account the movement of matter due to the
universe expansion. We conclude by raising the fatal fact that there is no
stationary safe distance when an atom of uranium undergoes fission in an
expanding universe, since the liberated energy increases along with the
universe expansion, the security of nuclear reactor must be endlessly
reconsidered. It turns out that the tame of this energy is strewed with risk
and peril together with the universe expansion.

###  4.5 The experimental data and the new quantification

The family of transformations $(T_{0})$, $(T_{1})$, $(T_{2})$, $\dots$,
$(T_{n})$ is a mathematical formulation of the Lorentz transformation that
incorporates the given quantification of the space expansion. This family of
transformations $(T_{i})_{i\geq 0}$ is defined for all
$v<v_{l_{n}}=\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$, where $v_{l_{n}}$
represents the limiting velocity of moving bodies at the expansion step n.
From this limiting velocity it turns out that the velocity of light is
decreasing as the universe expands, and the limiting velocity $v_{l_{n}}$ was
maximal only at the step 0 (in the beginning of the universe expansion
$v_{l_{0}}=c$), however, at each step the velocity of light is constant in the
sense that it is independent of the reference frame of the observer. All
formulations and laws based on the assumption that the velocity of light in
empty space is independent of the velocity of its source remain valid for
matter and related physical phenomenon during the universe expansion step by
step, and the relativistic formulation of the physical laws remains valid
experimentally since the velocity of light at the step n is experimentally
given by $\mathcal{C}_{n}=2.99792458\times 10^{8}$ which represents in this
formalism $\displaystyle{c\over\prod_{i=0}^{n}a_{i}}$, that is to say

$\mathcal{C}_{n}=\displaystyle{c\over\prod_{i=0}^{n}a_{i}},$ (120)

where c is the fossil velocity of light (The fossil velocity of light could be
detected in our space). At each step of the universe expansion the measured
velocity of light remains invariant from one frame to another in uniform
relative motion. The meaning of this approximation based on the universe
quantification $\mathcal{Q}$ is that the velocity of light is constant locally
and variable globally with respect to the universe age, and what we mean by
constant locally is that the variation of the velocity of light is too small
to be detected in short time with respect to the universe age. All our
experimental tests and applications of the special relativity laws at present
time are and remain valid since the substitution of the experimental measure
of the light speed in the $T_{0}$-Lorentz transformation will represent
$T_{n}$-Lorentz transformations. Indeed, The velocity of light is locally
constant at the scale of normal time (using a short interval of time as unit)
and globally variable at the scale of cosmical time (using a large interval of
time as unit). This local and global behavior can be derived straight forward
from the quantification ${\mathcal{Q}}$. Indeed, the local behavior is reached
in the quantification ${\mathcal{Q}}$ if we use a big number of subdivisions:
The bigger the used number $n$ of steps is the shorter the time interval of
steps we obtain. Thus, as $n$ tends to infinity, we have $a_{n+1}\approx 1$ as
a consequence of the convergence of the product $\prod_{i=0}^{n}a_{i}$, then

$\prod_{i=0}^{n+1}a_{i}=(\prod_{i=0}^{n}a_{i})\
a_{n+1}\approx\prod_{i=0}^{n}a_{i},\qquad\forall n>A,$ (121)

where $A$ is a large positive real number. Hence the equations (121) and (120)
give

$\mathcal{C}_{n}\approx\mathcal{C}_{n+1}\qquad\forall n>A,$ (122)

therefore the velocity of light is almost constant for short period of time
(locally). However, the equation (122) is not valid anymore for the large
period of time. Indeed, the smaller the used number $n$ of steps is the bigger
the time interval of steps we obtain, and in that case we have $a_{n+1}>1$ for
all $n<A$ and

$\prod_{i=0}^{n+1}a_{i}=(\prod_{i=0}^{n}a_{i})\
a_{n+1}>\prod_{i=0}^{n}a_{i},\qquad\forall n<A,$ (123)

hence the equations (120) and (123) lead to the following inequality:

$\mathcal{C}_{n}>\mathcal{C}_{n+1}\qquad\forall n<A,$ (124)

that is to say the velocity of light is decreasing globally together with the
universe expansion from one step to another. The formalism obtained within
this work using a special expansion notion in the special relativity theory to
study the uniform motion of the observer relative to the source is based on
the validity of the special relativity theory step by step from the big bang
to nowadays. which leads to the conclusion that this new formalism is an
extension of the special relativity theory.

####  4.5.1 Time Space and Expansion

We close this article by some comments on the notion of time. The notion of
relative time as for special relativity remains valid with some adjustment,
from the metric (9) is clear that time is a required coordinate to specify a
physical event and that this quantity can not be defined without the notion of
space and the notion of expansion, the dynamic of the universe is involved in
the definition of time. The expansion of the universe affects the time and the
space. If we refer to the proper time $\tau_{n}$ as the time which is measured
by an observer in a reference frame at the step n of the universe expansion in
which the events occur at the same spatial point, then $dx=dy=dz=0$ and the
metric (9) gives

$d\sigma^{2}_{n}=c^{2}d\tau_{n}^{2},$ (125)

meanwhile in another inertial reference frame the same events will verify (9),
which gives

$c^{2}d\tau_{n}^{2}=c^{2}dt_{n}^{2}-\Big{(}\prod_{i=0}^{n}a_{i}\Big{)}^{2}(dx^{2}+dy^{2}+dz^{2}),$
(126)

from which we obtain

$({d\tau_{n}\over dt_{n}})^{2}=1-\prod_{i=0}^{n}a_{i}^{2}{v^{2}\over c^{2}}$
(127)

where $v^{2}=({dx\over dt_{n}})^{2}+({dy\over dt_{n}})^{2}+({dz\over
dt_{n}})^{2}$. It turns out that the rate of proper time for a system varies
with motion through space as for special relativity, and what is new here is
that the rate of the proper time varies together with the universe expansion
even if $v$ is constant. If we refer to the universe proper time $\tau$ as the
time which is measured by an observer in a reference frame at rest at the step
n of the universe expansion in which the events occur at the same spatial
point, it is not difficult to extract from the metric (9) the relationship
between proper time and coordinate time

$d\tau=dt_{n}\sqrt{1-\prod_{i=0}^{n}a_{i}^{2}{v^{2}\over c^{2}}}.$ (128)

The time depends on the observer and the universe expanding step. What is new
here is to associate space, time and expansion in the definition of event. Any
change of reference or steps affects all of them. If the proper time $\tau$ of
the universe is the same from one step to another and if the velocity $v$ and
$c$ are constant from one step to another then it turns out from the step
$n-1$ to the step $n$ that

$1={dt_{n}\over dt_{n-1}}{\sqrt{1-\prod_{i=0}^{n}a_{i}^{2}{v^{2}\over
c^{2}}}\over\sqrt{1-\prod_{i=0}^{n-1}a_{i}^{2}{v^{2}\over c^{2}}}}$ (129)

and then

${dt_{n}\over dt_{n-1}}={\sqrt{1-\prod_{i=0}^{n-1}a_{i}^{2}{v^{2}\over
c^{2}}}\over\sqrt{1-\prod_{i=0}^{n}a_{i}^{2}{v^{2}\over c^{2}}}}>1$ (130)

which allows us to compare the coordinate time between successive steps as

$dt_{n}>dt_{n-1}$ (131)

which means that the clocks at the step n will run slower than an identical
type of clocks at the previous step, the time will run slower together with
the universe expansion. One more thing, the product $\prod_{i=0}^{n}a_{i}$
which allows the quantification of the universe expansion was extracted from
the fractal manifold model [1]. More details about the natural construction of
the expanding parameter $\prod_{i=0}^{n}a_{i}$ and the universe expansion as
well as the universe shape can be found in [2].

## References

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