#  Space-Time Entropy, Space of Singularities
and Gravity Origin: a Case Study

Fayçal BEN ADDA111Community College of Qatar, Doha, Qatar. Email:
<EMAIL_ADDRESS>
###### Abstract

A new definition of entropy is introduced using a model that simulates an
expanding space-time compatible with the fundamental principle of cosmology.
The entropy is obtained by mean of a state function that measures the
variation of the space-time normal curvature, from a highly compressed space
to a lower compressed space. The defined entropy leads to work out a new
understanding of the earliest conditions that last for a period estimated to
380 000 years after the Big Bang. It leads to understand via a short period of
inflation the process that generates the uniform distribution of matter and
energy at the surface of the last scattering. It involves gravitational
singularities in a process of gradual decompression propitious to the
incubation of matter recombination, and it allows to trace back gravity
origin.

Keywords: Curvature, Entropy, Singularity, Black holes characteristics,
Gravity

PACS: 97.60.Lf , 98.80.Bp , 98.80.-K , 04.30.-w , 45.20.D-

AMS: 28D20 , 83C57

###### Contents

  1. 1 Introduction
  2. 2 Change of the state function and entropy generation
    1. 2.1 Quantification adjustment
    2. 2.2 Modeling Adjustment
    3. 2.3 Space-time state function
    4. 2.4 Quantified entropy generation and entropy generation
    5. 2.5 Properties of the space-time state function
  3. 3 Asymptotic estimation, Space of asymptotic basic elements
    1. 3.1 Set of asymptotic expanding basic spheres
    2. 3.2 Asymptotic expanding space-time ${\mathcal{E}_{r}}$
    3. 3.3 Singularity, space of singularities
  4. 4 Interpretation of the earliest conditions and gravity origin
    1. 4.1 Period of the initial space ${\mathcal{E}_{i}}$
    2. 4.2 Period of the space of singularities ${\mathcal{E}_{r}}$ and gravity origin
      1. 4.2.1 Short period of inflation
      2. 4.2.2 Gravity origin
      3. 4.2.3 Transformation of the space-time of singularities ${\mathcal{E}_{r}}$
    3. 4.3 The period of primordial space-time ${\mathcal{E}_{1}}$
  5. 5 Conclusion

##  1 Introduction

The analysis of the Planck’s CMB image has revealed that the distribution of
mass/energy density of the universe is estimated today to 68,3 $\%$ of dark
energy, 26,8 $\%$ of dark matter and 4,9$\%$ of normal matter (European
Spacial Agency ESA /Planck). However, our understanding of the components of
the distribution remains unknown to satisfactory define, in particular if many
phenomena in relation with our universe are still unknown including the nature
of the dark energy, the nature of the dark matter, the characteristics of the
inflationary period, the universe phase transitions, the asymmetry in the
average temperatures on opposite hemispheres of the sky (from CMB), and the
earliest conditions of the universe expansion.

The oldest light propagation in the universe, known as the cosmic microwave
background (CMB), is actually related to a non accessible young universe of
380 000 years old. A critical period is missing for the comprehension of the
earliest conditions that generate the distribution of matter at large scale as
well as its transformation together with the space-time expansion. During the
380 000 years, light was trapped within an opaque environment that left no
traceable direct evidence to understand what happens in the earliest phase of
the universe transformation. Many hypothetical attempts to explain the
earliest conditions after the Big Bang have been elaborated and the list is
not limited to the following selection
([7],[8],[9],[10],[13],[15],[11],[21],[22],[23]).

To access the earliest conditions of the universe expansion, we introduce a
new entropy that associates a state function sensitive to the variation of the
space-time curvature, provided that this variation is directly related to its
expansion. For this purpose we use a model introduced in [2] that simulates an
expanding space-time compatible with the fundamental principle of cosmology.
The space-time is simulated by an accumulation of an infinite family of equal
open balls endowed with a discrete simultaneous expansion in all directions.

An illustration of a ball packing in one layer is given in Fig.2, and a volume
filled with a ball packing is illustrated in Fig.2, where the interstices are
covered by open stacked balls of different sizes to take into account the
density of the space-time (as illustrated in Fig.6 or in Fig.6). The
homogeneity and the expansion can be verified in all directions due to the
simultaneously expansion of the stacked open balls. Meanwhile, uniformity is
only verified within a limited number of orientations (eight orientations in
Fig.2). Indeed, the open ball packing conveys a relative anisotropic expansion
referred to certain orientations with different geodesics. A more stable ball
packing is illustrated in Fig.4 and Fig.4 with interstices covered by the
Apollonian gaskets of Fig.6. This ball packing verifies homogeneity and
expansion in all directions thanks to the simultaneous expansion of the open
balls. However, the uniformity of the space with respect to the geodesics is
also verified within limited directions (ten symmetric orientations). Debates
about homogeneity and isotropy of the universe can be found in the following
selected references [33], [19], [26], [24], [25], [36], [37].

Nevertheless, the non uniformity in the above examples is referred to the non
existence of identical geodesics in all directions. However since geodesics
are related to the space-time geometry, and the geometry of our universe
remains invisible to observation (only matter can be seen), then the
consideration of matter distribution within the surface of those expanding
stacked open balls may approximate homogeneity to satisfactory needs. An
example of illustration of the distribution of matter as dots on the surface
of a portion of stacked open balls at large scale conveys the idea of
homogeneity, isotropy and expansion. Indeed, the simultaneous expansion of
balls in the illustration Fig.8 increases the distance between dots in all
direction and simulates the space expansion. Meanwhile the observation of dots
without geometry (illustrated in Fig.8) reflects isotropy and homogeneity of
matter distribution regardless of the direction of observation, and simulates
the expansion of the space where the dots are plotted.

Figure 1: Illustration of one layer of $8\times 8$ equal ball packing, where
each interstice is enclosed between four stacked balls: the uniformity is only
verified within four orientations.

Figure 2: Illustration of a space of 8 stacked layers as illustrated in Fig.2
of same size, where the interstices are enclosed between three or four stacked
balls: the uniformity is only verified within eight orientations.

Figure 3: Illustration of one layer of $7\times 6$ equal ball packing, where
each interstice is enclosed between three stacked balls: the uniformity is
only verified within six orientations.

Figure 4: Illustration of a space of 6 layers as illustrated in Fig.4 of same
size, where each interstice is enclosed between three stacked balls: the
uniformity is only verified within ten orientations. This packing model is
more stable than the one illustrated in Fig.2.

Figure 5: Illustration of stacked balls of different sizes to fill the
interstice between three stacked equal balls, called Apollonian gasket. The
biggest ball is located in the center of the interstice. The smaller the
number of stacked equal balls that enclose the interstice is, the more stable
the ball packing is.

Figure 6: Illustration of stacked balls of different sizes to fill the
interstice between four stacked equal balls. The biggest ball is located in
the center of the interstice.

The plan of this paper is summarized as follow: in section 2 the change of the
state function and entropy generation is introduced, as well as its
properties. In section 3 we use asymptotic estimation to define singularities
and the space-time of singularities. In section 4 an interpretation of the
space-time earliest conditions and gravity origin are presented if matter is
considered, including a description of the initial space before the expansion
event, the period of the earliest inflation, the period of the space-time of
singularities transformation, gravity origin, and the period of primordial
space-time when light was released for the first time. In section 5 the
conclusion.

##  2 Change of the state function and entropy generation

It is known that the change in the state function (entropy) can be determined
at thermal equilibrium in terms of the changes of system’s energy and
temperature as a heat (Clausius’ entropy [6]), or it can be determined for an
isolated system on the total number of micro-states (Boltzmann’s entropy [4],
[5]), or it can be determined via a probability distribution of the system
(Shannon’s measure of information [34], [35]), or in a generalized quantum
gravity entropies such as ([27], [28], [29]). Nevertheless, the different
entropy’s definitions mentioned above concern matter characteristics,
transformations or distributions, despite the fact that the universe is made
of the space-time (the container) and of all form of matter, energy and
radiation (the content).

To provide a state function sensitive to the space-time transformation
consistent with the fundamental principle of cosmology, we use the model of
space-time introduced in [2]. This model is defined by an accumulation of an
infinite number of equal open balls (called basic elements and quantified in
the subsection below). The simultaneous expansion of all basic elements of the
model simulates the space-time expansion and warranties its homogeneity and
isotropy at large scale. Meanwhile, taking into account the density of the
space-time, all the interstices are covered by a family of stacked open balls
of different sizes known as Apollonian gasket and illustrated in Fig.6. The
open balls of different sizes that form the Apollonian gasket within each
interstice warranties the non homogeneity and the anisotropy of the space-time
at small scale. The initial space of the model is taken to be a limit space as
the radii of all balls tend to zero simultaneously. This model simulates the
space-time expansion and leads to the following outcomes:

  * •

it leads to understand what happens to the geodesics as the space-time expands
via simultaneous expansion of its basic elements. The new result asserts that
there exists an infinite number of transversal oscillating paths of least time
between any two distant locations, and the straight line geodesics do not
exist between any two distant locations of the expanding space-time. The local
curvature of each path of least time is equal to $1\over r$, where $r$ is the
radius of the expanding equal open balls. As the radii of the stacked open
balls increase simultaneously, the space-time expands and the local curvature
of each path of least time decreases. The local curvature $1\over r$ of each
path of least time corresponds to the normal curvature of each boundary of the
expanding open balls. This normal curvature is sensitive to the space-time
expansion and will be called the normal curvature of the space-time.

  * •

it allows to determine matter distribution at large scale using the infinity
of paths of least time and to deduce an optimal shape of the real space-time
where matter resides. The model indicates that the distribution of matter is
located within the boundaries of the expanding open balls (as illustrated in
Fig.8). Since matter exists within a continuum space-time of four dimensions,
then the boundary of each open ball illustrates a reduced form of a bent
continuum space-time of four dimensions, boundary of an expanding object of
five dimensions. The open balls in the model illustrate a reduced form in
dimension of the expanding object of five dimensions.

  * •

it raises a fundamental distinction between recession movement of matter (the
content) and space-time expansion (the recipient), and indicates that the
infinite family of stacked equal open balls that simulates in the model the
simultaneous expansion of the space-time represents approximatively 74$\%$ of
the space-time (Kepler conjecture [14]), meanwhile the interstices represents
26$\%$.

Moreover it is known that in geometrical optics the interpretation of the
interference pattern has failed from its beginning because it is not possible
to reproduce interference patterns using the straight line paths. However the
use of an infinite number of oscillating paths of least time in phase
introduced in [2] reproduces the interference patterns observed in the Young’s
double slit experiment [38] of 1802, and provides an adequate interpretation
of all its characteristics [1]. In addition it has also led to a new
development about the Van Der Waals torque, as well as a detailed description
of the Casimir attraction/repulsion’s unknown mechanism in nanotechnology [1].
For the large scale, the model has also led to obtain a compatible
interpretation of the cosmological data of supernovae (SNe) of type Ia [3],
that was the outstanding discovery of the Nobel laureates of 2011
([30],[31],[32]) conveying a shocking reverse of understanding of the state of
the universe expansion.

Thus the compatibility of the model with the recent cosmological data (SNe)
for large scale as well as the new development in geometrical optics for the
small scale justifies its use to construct a new entropy sensitive to the
variation of the normal curvature of the space time. Nevertheless, some
notations in relation with the needed formalism from the quantification of the
model in [2] will be presented below.

###  2.1 Quantification adjustment

The transformations of the space-time are quantified in [2] by the sequence
${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 0}$: the initial space-time is
denoted by ${\mathcal{E}_{0}}$ and the present space-time is denoted by
${\mathcal{E}_{n}}$. If we denote $T$ the age of the expansion of the space-
time from the initial space-time ${\mathcal{E}_{0}}$ to the present space-time
${\mathcal{E}_{n}}$, and $n$ the integer that represents a subdivision of $T$
into $n$ equal periods of time $T_{s}$. Minor changes of notation will be
taken into account as follow:

  1. 1.

The primordial space-time will be denoted ${\mathcal{E}_{1}}$. It will
represent in this quantification the surface of the last scattering from where
the cosmic microwave background (CMB) comes. The primordial space-time
${\mathcal{E}_{1}}$ appears 380 000 years after the Big Bang, when the
universe changes from opaque to transparent and releases light to travel for
the first time to reveal the cosmic microwave background for observation.

  2. 2.

The initial space will be denoted ${\mathcal{E}_{i}}$ instead of
${\mathcal{E}_{0}}$ to avoid confusion. The initial space ${\mathcal{E}_{i}}$
represents an approximation of the space before the Big Bang. The basic
elements of the initial space ${\mathcal{E}_{i}}$ are the limit of the basic
elements of the primordial space-time ${\mathcal{E}_{1}}$ as their tiny radii
tend to zero simultaneously. The earliest transformations and conditions will
be determined using the characteristics of the limit.

  3. 3.

The space-time expansion is quantified using a subdivision of the age $T$ of
the space-time expansion starting from the initial space ${\mathcal{E}_{i}}$
to the present space-time ${\mathcal{E}_{n}}$ after $n$ discrete expansions.
The period of time $T_{s}$ represents the elapsed time between any two
successive expansions of the space-time. The ratio of $T$ to $T_{s}$ is the
integer $n={T\over T_{s}}$ that represents the number of subdivisions of the
age of the space-time expansion. The bigger the number $n$ of subdivisions of
$T$ is, the smaller the period of time $T_{s}$ between two successive
expansions is. Conversely, the smaller the number $n$ of subdivisions of $T$
is, the bigger the period of time $T_{s}$ between two successive expansions
is. Thus the space-time expansion from the primordial space time
${\mathcal{E}_{1}}$ to the present space-time ${\mathcal{E}_{n}}$ will be
quantified by the sequence ${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$. The
earliest conditions of the space-time expansion between the initial space
${\mathcal{E}_{i}}$ and the primordial space-time ${\mathcal{E}_{1}}$ will be
quantified separately.

###  2.2 Modeling Adjustment

In this model the expansion of the space-time is generated by the simultaneous
expansion of the stacked open balls of same size. The boundary of each open
ball simulates a reduction in dimension of the bent space-time where matter
resides. To involve in a state function a variable normal curvature dependent
of the expansion of the space-time, the boundaries of the stacked open balls
are a good print of the expansion of the space-time. Therefore the boundary of
each open ball will be considered rather than the open ball, and we have the
following:

1\. The initial space ${\mathcal{E}_{i}}$ is the space whose basic elements
are limit entities with a topology close to that of a point. The basic element
of ${\mathcal{E}_{i}}$ are defined by the limit as the radius of the basic
elements of the primordial space-time ${\mathcal{E}_{1}}$ tends to zero.

2\. The primordial space-time ${\mathcal{E}_{1}}$ is the first space-time
accessible for cosmological observation at the step 1. The space-time
${\mathcal{E}_{1}}$ is defined by an infinite family of stacked spheres
(rather than stacked open balls) with identical tiny radius $r\in]0,1[$, the
smallest size of sphere that physics can decern from a dimensionless point.
These identical spheres define the basic elements of the primordial space-time
${\mathcal{E}_{1}}$.

3\. The space-time ${\mathcal{E}_{n}}$ is the space-time ${\mathcal{E}_{1}}$
after $n$ simultaneous expansion of its basic elements. It is defined by an
infinite family of stacked basic elements with identical size, quantified by
stacked spheres of same size with radius given by

$R_{n}={r\Big{(}\prod_{j=1}^{n}a_{j-1}\Big{)}}$ (1)

where the step-expanding parameters $(a_{n})_{n\geq 0}$ form a numerical
sequence that quantifies the discrete expansion of the space-time for all
$n\geq 0$, and satisfies for $a_{0}=1$ the following conditions:

i) for all $n\geq 1$,

$a_{n}>1,$ (2)

ii) for all $n\geq 1$,

$a_{n+1}<a_{n},$ (3)

iii) the product

$\prod_{1}^{n}a_{j-1}\qquad\hbox{is convergent.}$ (4)

The normal curvature of each basic element is defined for all $n\geq 1$ by

$\delta_{n}(r)={1\over R_{n}}={1\over r(\prod_{j=1}^{n}a_{j-1})}.$ (5)

The parameter $a_{j}$ is called the $j^{th}$ quantified expanding parameter of
the space-time between the steps $j$ and $j+1$, and the product
$\prod_{j=1}^{n}a_{j-1}$ is called the expanding parameter of the space-time
until the step $n$.

4\. The interstices: the discrete expansion of the space-time
${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$ simulates an homogeneous,
isotropic and expanding space-time that expands via simultaneous expansion of
its basic elements. Thus as the stacked equal spheres expand, they create an
increasing uncovered narrow space between them called the interstice. To
simulate a fully covered dense space by spheres, the interstices are filled
with Apollonian gaskets of stacked spheres of different sizes. This process is
iterated as the stacked spheres of different sizes expand and create different
interstices in turn. Therefore to conserve the density of the space as it
expands, all increasing interstices are in turn filled with stacked spheres of
different sizes. These repeating imbedded interstices, filled with stacked
spheres of different sizes are generated by the space-time expansion and
convey the non homogeneity and the anisotropy of the space-time locally.

5\. The normal curvature of the expanding space-time
${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$ is approximatively determined by
the normal curvature of the expanding basic elements of the space-time.
Indeed, since all basic elements are identical, then the normal curvature of
the space-time is given by the normal curvature of one basic element which is
a sphere with radius defined by (1) for all $n\geq 1$. Thus since all paths of
least time on a sphere are great circle arcs of radius (1), then the normal
curvature of a sphere is given by the normal curvature (5) of the great circle
arc for all $n\geq 1$. The greater the radius of the basic element is, the
lower the normal curvature of the space-time is (as illustrated in Fig.9).

The normal curvature of the spheres in the interstices related to the non
homogeneity and the anisotropy of the space-time is not considered since it
depends on the normal curvature of the basic elements of the space-time.
Therefore, the definition of the space-time state function will be related
only to the normal curvature of the expanding basic elements of the space-time
that provides homogeneity, isotropy at large scale and generates the space-
time expansion.

Figure 7: Illustration of a portion of stacked equal balls on which matter is
represented by dots. Since matter exists within a continuum space-time of four
dimensions, then the boundary of each open ball illustrates a reduced form of
a bent continuum space-time of four dimensions, boundary of an expanding
object of five dimensions. An open ball in the model illustrates a reduced
form in dimension of the expanding object of five dimensions. The objects of
five dimensions cannot be visualized; nevertheless this model allows to
approximate the visualization of the homogeneity and isotropy.

Figure 8: The invisibility of the geometry of the stacked equal balls means
that the illustration Fig.8 is the same illustration as Fig.8 without
visualization of the boundaries of the stacked balls. Matter represented by
dots appears to be held in the space without pillars (the spheres) that one
can see. Homogeneity and isotropy are approximately verified within this model
since there is no privileged direction revealed by observation.

Figure 9: Illustration of the expansion of one basic element quantified from
step 1 to step $n$. The greater the radius of the basic element is, the lower
the normal curvature of the space-time is.

###  2.3 Space-time state function

Using the normal curvature (5) of the stacked spheres of the expanding space-
time ${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$, we define the space-time
state function as follow:

###### Definition 1

Consider the space-time $\mathcal{E}=({\mathcal{E}_{n}})_{n\geq 1}$ where
${\mathcal{E}_{1}}$ is the primordial space-time. For an arbitrary tiny radius
$r\in]0,1[$ and an arbitrary finite positive constant k, we define the entropy
of space-time $\mathcal{E}$ at the Step n for all $n\geq 1$ as

$S_{n}(r)=-k\ \ln(\delta_{n}(r)),$ (6)

where $\delta_{n}(r)={1\over R_{n}}$ is the normal curvature of the basic
elements of radius $R_{n}={r\prod_{1}^{n}a_{j-1}}$ of the space-time
${\mathcal{E}_{n}}$ at the step $n$.

The function (6) is a state function because it is given by an integral that
depends only on the lower limit 1 and the upper limit $\delta_{n}(r)$. The
lower limit is given by the normal curvature of a unit sphere. The upper limit
is given by the normal curvature of an expanding basic element of the space-
time ${\mathcal{E}_{n}}$, quantified by the radius given by (1) for all $n\geq
1$. Indeed, the function (6) can be written as

$S_{n}(r)=-k\int_{1}^{\delta_{n}(r)}{1\over t}\ dt=-k\
\Big{(}\ln(\delta_{n}(r))-\ln(1)\Big{)}=-k\ \ln(\delta_{n}(r)).$ (7)

###  2.4 Quantified entropy generation and entropy generation

Using the above state function given by (6), the quantified change of the
state function together with the space-time expansion, denoted by $\Delta
S_{n}$, is given by

$\Delta S_{n}=S_{n}(r)-S_{n-1}(r)$ (8)

where $S_{n}(r)$ is the state function of the space-time $\mathcal{E}_{n}$ at
the Step $n$, and $S_{n-1}(r)$ is the state function of the space-time
$\mathcal{E}_{n-1}$ at the Step $n-1$ for all $n>1$.

Notation

a) We call”quantified entropy generation between the steps $n-1$ and $n$” of
the space-time $\mathcal{E}=({\mathcal{E}_{n}})_{n\geq 1}$ and we denote it by
$\Delta S_{n}=S_{n}(r)-S_{n-1}(r)$, the quantified change of the state
function of successive expansions of the space-time $\mathcal{E}$ for all
$n>1$.

b) We call ”space-time entropy generation between the steps 1 and n” of the
space-time $\mathcal{E}=({\mathcal{E}_{n}})_{n\geq 1}$ and we denote it by
$\Delta\mathbb{S}_{n}=S_{n}(r)-S_{1}(r)$, the change of the state function
between the primordial space-time $\mathcal{E}_{1}$ (at the Step 1) and the
present space-time $\mathcal{E}_{n}$ (at the Step $n$ for all $n>1$).

Thus for all $n>1$ the quantified entropy generation between the steps $n-1$
and $n$ of the space-time $\mathcal{E}=({\mathcal{E}_{n}})_{n\geq 1}$ is given
by

$\Delta S_{n}=k\ \ln({\delta_{n-1}(r)\over\delta_{n}(r)})$ (9)

where $\delta_{n}(r)={1\over R_{n}}$ is the normal curvature of the basic
element of the space-time $\mathcal{E}_{n}$ at the Step $n$, and using (9) we
have

$S_{n}(r)=S_{n-1}(r)+k\ \ln({\delta_{n-1}(r)\over\delta_{n}(r)}).$ (10)

###### Remark 1

i) The arbitrary positive constant $k$ in the equality (9) can be considered
as the Boltzmann constant if needed. The state function (6) is defined by an
increasing function of $\delta_{n}(r)$. Indeed, using (5) the normal curvature
of the space-time ${\mathcal{E}_{n}}$ for all $n>1$ is a decreasing function
of $n$ (for an arbitrary constant $r\in]0,1[$), and we have

$\forall n>1,\qquad\delta_{n}(r)<\delta_{n-1}(r),$ (11)

which makes (6) an increasing function of $\delta_{n}(r)$ as $n$ increases.

ii) The quantified entropy generation (9) between the steps $n-1$ and $n$ can
be written using integral notation as follow

$\Delta S_{n}=k\ln\Big{(}{\delta_{n-1}(r)\over\delta_{n}(r)}\Big{)}=-k\
\int_{\delta_{n-1}(r)}^{\delta_{n}(r)}{1\over t}\ dt.$ (12)

###  2.5 Properties of the space-time state function

It is known that the normal curvature measures the local degree of deformation
of a curve (or a surface). It is a quantity that measures the speed at which
the graph of a given curve (or surface) deviates from the direction of the
tangent line in the neighborhood of local points. Since the basic elements of
the space-time $\mathcal{E}_{n}$ are expanding spheres, then the equality (8)
for all $n>1$ verifies the following:

###### Property 1

Let $(S_{n})_{n\geq 1}$ be the sequence of state functions of the space-time
${\mathcal{E}}=(\mathcal{E}_{n})_{n\geq 1}$, then the entropy generation
between the steps 1 and n of the space-time ${\mathcal{E}}$ verifies for all
$n>1$ and $r\in]0,1[$

$\Delta\mathbb{S}_{n}=S_{n}(r)-S_{1}(r)=k\ln({\delta_{1}(r)\over\delta_{n}(r)}).$
(13)

Proof: Using the space-time entropy generation between successive steps (8)
for all $n>1$, we obtain

$\begin{array}[]{cc}\Delta S_{n}=&S_{n}(r)-S_{n-1}(r)\\\ \Delta
S_{n-1}=&S_{n-1}(r)-S_{n-2}(r)\\\ \vdots\quad\vdots&\vdots\\\ \Delta
S_{2}=&S_{2}(r)-S_{1}(r)\end{array}$ (14)

by adding the $(n-1)$ equalities (14), it gives

$\Delta S_{n}+\Delta S_{n-1}+\ldots+\Delta S_{2}=S_{n}(r)-S_{1}(r)$ (15)

and using (9), the above equality is simplified to

$S_{n}(r)=S_{1}(r)+k\ln({\delta_{1}(r)\over\delta_{n}(r)}).$ (16)

The result can also be obtained directly by substituting (6) in the definition
of $\Delta\mathbb{S}_{n}$.

###### Property 2

The quantified entropy generation (9) between the steps $n-1$ and $n$ of the
space-time ${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$ verifies

$(i)\ \forall n>1,\quad\Delta
S_{n}=0\quad\Longleftrightarrow\quad{\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq
1}\quad\hbox{is a static space-time},$ (17) $(ii)\ \forall n>1,\quad\Delta
S_{n}>0\quad\Longleftrightarrow\quad{\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq
1}\quad\hbox{is an expanding space-time},$ (18) $(iii)\ \forall
n>1,\quad\Delta
S_{n}<0\quad\Longleftrightarrow\quad{\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq
1}\quad\hbox{is a contracting space-time}.$ (19)

Proof: (i) For all $n>1$, we have $\Delta S_{n}=0$, then using (9) we have
$k\ln\Big{(}{\delta_{n-1}(r)\over\delta_{n}(r)}\Big{)}=0$, which gives using
the normal curvature (5)

$k\ln\Big{(}{R_{n}\over R_{n-1}}\Big{)}=0,$ (20)

that is to say for all $n>1$, $R_{n}=R_{n-1}$, which characterizes a static
space-time. The converse is evident.

(ii) For all $n>1$, we have $\Delta S_{n}>0$, then using (9), it gives
$k\ln({R_{n}\over R_{n-1}})>0$, which gives, for all $n>1$,
$\ln(R_{n})>\ln(R_{n-1})$. That is to say, for all $n>1$, $R_{n}>R_{n-1}$,
which characterizes an expanding space-time. The converse is evident.

iii) For all $n>1$, we have $\Delta S_{n}<0$, that gives $-k\ln({R_{n}\over
R_{n-1}})>0$, which means that $k\ln({R_{n-1}\over R_{n}})>0$. Then, for all
$n>1$, $\ln(R_{n-1})>\ln(R_{n})$. That is to say, for all $n>1$,
$R_{n-1}>R_{n}$, which characterizes a contracting space-time. The converse is
evident.

###### Property 3

The quantified entropy generation between the steps $n-1$ and $n$ of the
space-time ${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$ measures the space-
time quantified expansion. In particular for all $n>1$

$\Delta S_{n}=k\ln(a_{n-1})$ (21)

with $a_{n-1}$ the $(n-1)^{th}$ space-time expanding parameter from the Step
$n-1$ to the Step $n$.

Proof: The substitution of the normal curvature (5) in the entropy generation
(9) gives

$\Delta S_{n}=k\
\ln\Big{(}{\delta_{n-1}(r)\over\delta_{n}(r)}\Big{)}=k\ln\Big{(}{R_{n}\over
R_{n-1}}\Big{)}=k\ln\Big{(}{r\Big{(}\prod_{i=1}^{n}a_{i-1}\Big{)}\over
r\Big{(}\prod_{i=1}^{n-1}a_{i-1}\Big{)}}\Big{)}=k\ln(a_{n-1}),$ (22)

which completes the proof.

###### Proposition 1

i) For $n>1$ the quantified entropy generation between the steps $n-1$ and $n$
of the space-time ${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$ for a short
period of time between successive steps of expansion verifies

$\Delta S_{n}\approx 0$ (23)

ii) For $n>1$ the quantified entropy generation between the steps $n-1$ and
$n$ of the space-time ${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$ for a
large period of time between successive steps of expansion verifies

$\Delta S_{n}>0$ (24)

Proof: i) The expanding parameter (4) of the space-time
${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$ can be written as

$\prod_{i=1}^{n}a_{i-1}=e^{\sum_{1}^{n}\ln(a_{i-1})},$ (25)

and as $n$ tends to infinity, the sequence (4) has to be convergent. Therefore
there exists a big $N>0$ such that for all $n-1\geq N$, $\ln(a_{n-1})$ tends
to 0, which gives for all $n-1\geq N$, the parameter $a_{n-1}$ tends to 1,
then we have for all $n-1\geq N$, the parameter $a_{n-1}\approx 1$ that gives
using (21)

$\Delta S_{n}=k\ln(a_{n-1})\approx 0.$ (26)

Since $n-1\geq N$, then $n$ is a big integer and represents the number of
subdivisions of the age $T$ of the expanding space-time ${\mathcal{E}}$ at the
step $n$. Therefore the bigger the number $n$ is, the shorter the period of
time interval between two successive expansions is. For this short period of
time, the quantified entropy generation (9) verifies $\Delta S_{n}\approx 0$.

ii) For all $n-1<N$, the integer $n$ is less than $N+1$ a big number and
represents the number of subdivisions of the age $T$ of the expanding space-
time ${\mathcal{E}}$ at the step $n$. Therefore, the smaller the number $n$
is, the larger the period of time interval between successive expansions is.
Thus for a large period of time interval between successive expansions such
that $n-1<N$, the $(n-1)^{th}$ quantified expanding parameter $a_{n-1}$ of the
space-time ${\mathcal{E}}$ verifies (2), that is to say for a small integer
$1<n-1<N$, we have $a_{n-1}>1$. Since $k$ is a positive number, then we have

$k\ln(a_{n-1})>k\ln(1),$ (27)

and using (21), the inequality gives

$\Delta S_{n}>0$ (28)

for a large period of time interval between successive expansions.

###### Proposition 2

If $(S_{n})_{n\geq 1}$ is the sequence of state functions of the space-time
${\mathcal{E}}=({\mathcal{E}}_{n})_{n\geq 1}$, then for all $n\geq 1$,

i) the entropy generation between the steps 1 and n of the space-time
${\mathcal{E}}$ due to its expansion from the primordial space-time
${\mathcal{E}_{1}}$ to the space-time ${\mathcal{E}_{n}}$ is increasing as $n$
increases, and we have

$\Delta\mathbb{S}_{n}=S_{n}(r)-S_{1}(r)=k\ \ln({\prod_{j=1}^{n}a_{j-1}}),$
(29)

where $a_{j-1}$ is the $(j-1)^{th}$ quantified expanding parameter of the
space-time.

ii) the quantified entropy generation between the steps $n-1$ and $n$ of the
space-time ${\mathcal{E}}$ due to a successive expansion is decreasing as $n$
increases and we have for all $n>1$

$\Delta S_{n+1}<\Delta S_{n}.$ (30)

Proof: i) Using the state function (6) and the normal curvature (5), we have
for all $n>0$

$S_{n}(r)=-k\ \ln\Big{(}\delta_{n}(r)\Big{)}=-k\ \ln\Big{(}{1\over
r(\prod_{1}^{n}a_{j-1})}\Big{)}=k\ \ln\Big{(}{r(\prod_{1}^{n}a_{j-1})}\Big{)}$
(31)

then

$S_{n}(r)=k\ \ln(r)+k\ \ln\Big{(}{\prod_{1}^{n}a_{j-1}}\Big{)}$ (32) $=k\
\ln(r)+k\ \sum_{j=1}^{n}\ln(a_{j-1}).$

Using (6) for $n=1$, it gives $S_{1}(r)=k\ \ln(r)$, then we have

$S_{n}(r)=S_{1}(r)+k\ \ln({\prod_{j=1}^{n}a_{j-1}}).$ (33)

since $\Delta\mathbb{S}_{n}=S_{n}(r)-S_{1}(r)$ for all $n>1$, then we have
(29), which gives

$\Delta\mathbb{S}_{n}=S_{n}(r)-S_{1}(r)=k\ \sum_{j=1}^{n}\ln(a_{j-1}).$ (34)

Using (2), we have for all $j-1>0$, $a_{j-1}>1$, which gives $\ln(a_{j-1})>0$,
therefore the entropy generation (34) is given by an increasing convergent
series with positive terms. Thus the entropy generation (34) of the space-time
${\mathcal{E}}$ is increasing as $n$ increases.

ii) From (21) we have for $i>0$

$\Delta S_{i}=k\ln(a_{i-1}),$ (35)

and the property (3) gives for all $i>0$

$a_{i}<a_{i-1},$ (36)

then

$\ln(a_{i})<\ln(a_{i-1}),$ (37)

and using the finite positive constant $k>0$ it gives

$k\ln(a_{i})<k\ln(a_{i-1}).$ (38)

The use of (35) (for index $i$ and $i+1$) in the inequality (38) gives for all
$i>0$

$\Delta S_{i+1}<\Delta S_{i},$ (39)

that is to say the quantified entropy generation $\Delta S_{n}$ is decreasing
as $n$ increases.

###### Remark 2

i) The approximation (23) means that the variation of space-time normal
curvature remains imperceptible for a short period of time $T_{s}$ between two
successive space-time ${\mathcal{E}_{i}}$ and ${\mathcal{E}_{i+1}}$ for all
$i\geq 1$. A short period of time $T_{s}$ is related to the age $T$ of the
space-time expansion at the step n by the equality $T=n\times T_{s}$. Since
$T_{s}={T\over n}$, then the period $T_{s}$ between two successive expansions
is short when n is a big integer. Meanwhile the period $T_{s}$ between two
successive expansions is large when n is a small positive integer, and in this
case the quantified entropy generation (24) is strictly positive.

ii) The quantified entropy generation $\Delta S_{n}$ between the steps $n-1$
and $n$ for all $n>1$ of the space-time
${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$ is decreasing as $n$ increases
(Proposition 2, ii)), meanwhile the entropy generation $\Delta\mathbb{S}_{n}$
of the space-time between the steps 1 and $n$ of the space-time
${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$ is always increasing
(Proposition 2, i)), with a decelerating increase since
$\Delta\mathbb{S}_{n}=\sum_{i=2}^{n}\Delta S_{i}$, and $\Delta S_{i}$ verifies
(39) .

##  3 Asymptotic estimation, Space of asymptotic basic elements

The space-time expansion from the primordial space-time ${\mathcal{E}_{1}}$ to
the space-time ${\mathcal{E}_{n}}$ is quantified by the sequence of the
expanding space-time ${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$. However,
the earliest period of expansion lasted for approximatively 380 000 years
after the Big Bang represents in this model the period of expansion between
the initial space ${\mathcal{E}_{i}}$ and the primordial space-time
${\mathcal{E}_{1}}$. An intermediary space-time that quantifies this period
can be found to understand the earliest transformation of the initial space
${\mathcal{E}_{i}}$. To cope with this unreachable period, the key is conveyed
by a limit approach that allows to build characteristics and properties for
better understanding. Indeed, from equality (5) we know that the normal
curvature of the space-time ${\mathcal{E}_{1}}$ is given for a tiny radius
$r\in]0,1[$ by

$\delta(r)={1\over r}.$ (40)

Then as all radii $r$ of the basic elements of the space-time
${\mathcal{E}_{1}}$ tend to zero simultaneously, the normal curvature
$\delta(r)$ of each basic element of the space-time ${\mathcal{E}_{1}}$ tends
to the limit

$\lim_{r\longrightarrow 0}\delta(r)=\lim_{r\longrightarrow 0}{1\over
r}=+\infty.$ (41)

From subsection $\S$ 2.2 the initial space ${\mathcal{E}_{i}}$ represents an
approximation of the space before the Big Bang. The basic elements of the
initial space ${\mathcal{E}_{i}}$ are the limit of the basic elements of the
primordial space-time ${\mathcal{E}_{1}}$ as their tiny radii tend to zero
simultaneously. The initial space ${\mathcal{E}_{i}}$ is then characterized by
a normal curvature close to plus infinity, and this characteristic leads to
others, but first we need to build the following:

###  3.1 Set of asymptotic expanding basic spheres

The divergence of the normal curvature $\delta(r)$ to plus infinity as the
radius $r$ tends to zero, means that for all $B>0$ a big real number, there
exists a small radius $r_{0}>0$, such that for all $r\in]0,r_{0}[$, the normal
curvature $\delta(r)>B$. This property leads to introduce a set of asymptotic
expanding spheres, denoted by $\mathcal{B}_{h}(c)$, defined by the disjoint
union of spheres $\mathbb{S}(c,r)$ with same center $c$, radius
$r\in]0,r_{0}[$ and a normal curvature close to plus infinity, as follow:

$\mathcal{B}_{h}(c)=\bigcup_{0<r<r_{0}}\mathbb{S}(c,r).$ (42)

The set of expanding asymptotic spheres $\mathcal{B}_{h}(c)$ represents all
continuous transformations of any limit basic element $c$ of the initial space
${\mathcal{E}_{i}}$ for all $r\in]0,r_{0}[$. The normal curvature of each
transformed sphere $\mathbb{S}(c,r)$ of $\mathcal{B}_{h}(c)$ remains close to
plus infinity. The set $\mathcal{B}_{h}(c)$ is an open ball without center,
defined by a disjoint union of imbedded spheres of radius $0<r<r_{0}$. Each
sphere is the expansion of previous spheres of smaller radii. All spheres of
$\mathcal{B}_{h}(c)$ are homotopic due to the continuous transformation by
expansion. The upper limit $r_{0}$ of the radius $r$ of $\mathbb{S}(c,r)$
depends on the choice of the big constant $B$.

###  3.2 Asymptotic expanding space-time ${\mathcal{E}_{r}}$

The spheres from the set $\mathcal{B}_{h}(c)$, with equal radius $r$ and for
all center $c$ basic element of the initial space ${\mathcal{E}_{i}}$, can be
used to define an asymptotic expanding space-time. Provided that it
incorporates all the transformations of each basic element of the initial
space ${\mathcal{E}_{i}}$ with a topology close to that of a point, to a basic
element of the primordial space-time ${\mathcal{E}_{1}}$ with a topology of a
sphere. Indeed, we define this asymptotic expanding space-time as follow:

###### Definition 2

We call asymptotic expanding space-time and we denote ${\mathcal{E}_{r}}$ for
$r\in]0,r_{0}[$, the space-time defined by the infinite family of accumulated
spheres $\mathbb{S}(c,r)$ of $\mathcal{B}_{h}(c)$, with equal radius $r$ for
all basic element $c\in{\mathcal{E}_{i}}$.

###### Property 4

The asymptotic expanding space-time ${\mathcal{E}_{r}}$, for all
$r\in]0,r_{0}[$, verifies the following properties:

  1. 1.

The normal curvature of the asymptotic expanding space-time
${\mathcal{E}_{r}}$, for all $r\in]0,r_{0}[$, is close to plus infinity.

  2. 2.

The entropy of the asymptotic expanding space-time ${\mathcal{E}_{r}}$, for
all $r\in]0,r_{0}[$, is close to minus infinity.

  3. 3.

The change of the state function due to a continuous expansion of an
asymptotic space-time ${\mathcal{E}_{r_{1}}}$ to an asymptotic space-time
${\mathcal{E}_{r_{2}}}$ for all $0<r_{1}<r_{2}<r_{0}$ is an indeterminate
form.

  4. 4.

The entropy generation of the continuous expansion of an asymptotic expanding
space-time ${\mathcal{E}_{r}}$, for all $r\in]0,r_{0}[$, to the primordial
space-time ${\mathcal{E}_{1}}$ is close to plus infinity.

Proof: 1\. The normal curvature of the asymptotic expanding space-time
${\mathcal{E}_{r}}$ is close to plus infinity since it is given by the normal
curvature of identical spheres of the open sets $\mathcal{B}_{h}(c)$ defined
by (42) for all initial basic element $c\in{\mathcal{E}_{i}}$.

2\. From the definition of entropy (6), the entropy $S_{h}$ of the asymptotic
expanding space-time ${\mathcal{E}_{r}}$ is close to minus infinity. Indeed,
for all $r<r_{0}$, $\delta(r)={1\over r}$ tends to plus infinity, which gives

$S_{h}=\lim_{\delta(r)\longrightarrow+\infty}-k\ln(\delta(r))=-\infty.$ (43)

3\. To evaluate the entropy generation of the expansion of the asymptotic
expanding space-time ${\mathcal{E}_{r_{1}}}$ to the asymptotic expanding
space-time ${\mathcal{E}_{r_{2}}}$ for all $0<r_{1}<r_{2}<r_{0}$, we use the
difference of their state functions denoted respectively $S_{h_{1}}$ and
$S_{h_{2}}$. Indeed, since $S_{h_{1}}$ and $S_{h_{2}}$ are given by (43) for
$r=r_{1}$, respectively $r=r_{2}$, and since for all $r<r_{0}$,
$\delta(r)={1\over r}$ tends to plus infinity, then we have

$S_{h_{2}}-S_{h_{1}}=(-\infty)-(-\infty),$ (44)

which is an indeterminate form.

4\. To evaluate the entropy generation of the expansion of the asymptotic
space-time ${\mathcal{E}_{r}}$, for all $r\in]0,r_{0}[$, to the primordial
space-time ${\mathcal{E}_{1}}$, we use the difference of their state functions
$S_{1}(r)-S_{h}$. Indeed, since $S_{1}(r)$ is negative and finite for all
$r_{0}\leq r\ll 1$, and $S_{h}$ is given by (43), then we have

$S_{1}(r)-S_{h}=S_{1}(r)-(-\infty)=+\infty,$ (45)

which means that the entropy generation after continuous expansion of the
asymptotic expanding space-time ${\mathcal{E}_{r}}$, for $r\in]0,r_{0}[$, to
the primordial space-time ${\mathcal{E}_{1}}$ is close to $+\infty$.

###### Remark 3

i) In Property 4, 3., the entropy generation is found to be an indeterminate
form, because the above entropies $S_{h_{2}}$ and $S_{h_{1}}$ approach minus
infinity, without being equal to minus infinity. That is why the changes of
the state function between two distinct asymptotic expanding spaces-time
${\mathcal{E}_{r}}$ is not zero, but unknown.

ii) The characteristics introduced in Property 4 convey divergence before the
appearance of the primordial space-time ${\mathcal{E}_{1}}$. This leads to
define a singularity, and a space of singularities.

iii) For all $r\in]0,r_{0}[$, the surface area to the volume ratio ${A\over
V}=3\delta(r)$ of each basic element of the asymptotic expanding space-time
${\mathcal{E}_{r}}$ is close to plus infinity.

###  3.3 Singularity, space of singularities

Defining singularities when the geometry becomes unreachable is related to the
used theory and formalism of application. Indeed, singularities are seen as an
end of the space-time, a pathological set of points where our tools are not
defined. In mathematics, a singularity of a function refers to a point where
the function is not defined. In physics, singularities appear unavoidable and
carry a potential of great interest. In cosmology, a singularity refers to
collapse of matter in which the laws of physics become indistinguishable. A
gravitational singularity is a space location where the gravitational field
becomes infinite. In this framework singularities carry a potential of great
interest and are defined as follow:

###### Definition 3

We call singularity each sphere of $\mathcal{B}_{h}(c)$, for all basic element
$c$ of the initial space-time ${\mathcal{E}_{i}}$, with a normal curvature
close to plus infinity and a state function, defined by

$S_{r}(\delta(r))=-k\ \ln(\delta(r)),$ (46)

close to minus infinity, with $k$ an arbitrary positif constant and
$\delta(r)$ the normal curvature given by (40).

###### Remark 4

i) A sphere’s normal curvature $\delta(r)={1\over r}$ close to plus infinity,
means that for all $B>0$ there exists $r_{0}>0$ such that for all
$r\in]0,r_{0}[$, the sphere normal curvature $\delta(r)={1\over r}$ is close
to plus infinity.

ii) The open set $\mathcal{B}_{h}(c)$ given by (42) represents an archive of
the continuous transformation of one singularity (all imbedded spheres are
homotopic). Indeed $\mathcal{B}_{h}(c)$ is an open ball without center,
defined by a disjoint union of imbedded spheres of radius $0<r<r_{0}$. Each
sphere is a singularity characterized by a normal curvature close to plus
infinity and an entropy close to minus infinity.

###### Definition 4

We call space of singularities the space defined by an infinite accumulation
of identical singularities.

###### Remark 5

i) Since the basic elements of the asymptotic expanding space-time
${\mathcal{E}_{r}}$, for all radius $0<r<r_{0}$, are subsets of
$\mathcal{B}_{h}$ with a normal curvature close to plus infinity, and a state
function (46) close to minus infinity, then the asymptotic expanding space-
time ${\mathcal{E}_{r}}$ is a space-time of singularities for all radius
$0<r<r_{0}$.

ii) Based on (§2.2., 1.) each basic element of the initial space
${\mathcal{E}_{i}}$ is a limit entity as the radius of each basic element of
the primordial space-time ${\mathcal{E}_{1}}$ tends to zero. It gives to that
limit entity a topology close to that of a point, and indicates that the basic
elements of the initial space ${\mathcal{E}_{i}}$ are extremely contracted,
with a local entropy $S_{i}$ given by the limit

$S_{i}=\lim_{r\rightarrow 0}-k\ \ln(\delta(r))=-\infty,$ (47)

##  4 Interpretation of the earliest conditions and gravity origin

The above model presents the space-time of singularities ${\mathcal{E}_{r}}$,
for all $0<r<r_{0}$ with $r_{0}\ll 1$, as the first state of decompression of
an earliest extremely contracted limit space ${\mathcal{E}_{i}}$. It
introduces the space-time of singularities as the first expanding space-time
that quantifies the period of expansion between the extremely contracted space
${\mathcal{E}_{i}}$, and the primordial space-time ${\mathcal{E}_{1}}$ that
approximatively last for 380 000 years after the Big Bang. The consideration
of matter, energy and radiation within this model allows to deduce a possible
scenario for the earliest phase of the space-time expansion. It involves
extreme divergent conditions that incubate matter recombination and lead to
trace back the gravity origin. Indeed, the model allows to establish a
chronology of extreme transformation at the beginning of the space-time
expansion as follow:

###  4.1 Period of the initial space ${\mathcal{E}_{i}}$

The period of the initial space ${\mathcal{E}_{i}}$ is the period at which
matter, energy and radiation are assumed to be assembled before the Big Bang
(§2.2, 1.) (it is the backward period in time of the primordial space-time
${\mathcal{E}_{1}}$). The initial space ${\mathcal{E}_{i}}$ represents an
approximation of the space before the beginning of the expansion. It is a
space endowed with a local entropy close to minus infinity (47) and a normal
curvature close to plus infinity. It corresponds to an infinite space with
extremely contracted basic elements endowed with a topology close to that of a
point.

###  4.2 Period of the space of singularities ${\mathcal{E}_{r}}$ and gravity
origin

The period of the space of singularities is the closest period after the
beginning of the space expansion (the Big Bang), quantified by a space-time of
singularities ${\mathcal{E}_{r}}$ for all $0<r<r_{0}\ll 1$. This space-time of
singularities is characterized by a normal curvature close to plus infinity,
and an entropy close to minus infinity (Property 4, Remark 5). The
consideration of matter and energy within this model leads to understand the
major role of those earliest singularities for the recombination of matter and
all form of energy after the Big Bang.

For this purpose we assume that the expansion of the initial space
${\mathcal{E}_{i}}$ starts at the minimal time $t_{0}$, when the dismantlement
of matter at all scale begins via simultaneous expansion of the basic elements
of ${\mathcal{E}_{i}}$. Therefore the minimal time $t_{0}$ marks the beginning
of the period of the space-time of singularities that includes the following:

####  4.2.1 Short period of inflation

Observing matter receding from each other in all direction is an indication
that matter and all form of energy were assembled in the past within a volume
of an extremely contracted space. This space is described within this model by
the initial space ${\mathcal{E}_{i}}$. Thus any volume in the initial space
${\mathcal{E}_{i}}$ is described by an infinite accumulation of extremely
contracted limit entities with a topology close to that of a point. If all
extremely contracted limit entities expand simultaneously to a tiny spheres,
the initial space ${\mathcal{E}_{i}}$ expands instantaneously during a short
inflation. This short inflation is the first phase of the space expansion,
induced by a simultaneous decompression of extremely compressed dense basic
elements of the initial space ${\mathcal{E}_{i}}$. Any finite volume in the
initial space ${\mathcal{E}_{i}}$ expands via simultaneous expansion of its
basic elements to an infinite space of stacked tiny spheres.

Indeed the transformation of each basic element of the initial space
${\mathcal{E}_{i}}$ from an extremely contracted limit entity (with the
topology close to that of a point) to a tiny sphere induces the following:

  1. 1.

An instantaneous dismantlement of matter and energy. The density of the
initial space ${\mathcal{E}_{i}}$ induces matter dismantlement at all tiny
scales as all extremely contracted limit entities expand simultaneously to
tiny spheres pushing everything outside of its volume.

  2. 2.

An instantaneous short period of inflation. The short inflation induces an
extremely rapid recession motion of all amount of the dismantled matter and
energy from each other in all directions. It is a consequence of the
simultaneous expansion of the space-time’s basic elements from dimensionless
basic elements to tiny basic elements of three dimensions. This process
defines the mechanism that generates the instantaneous inflation.

  3. 3.

Any amount of the dismantled matter undertakes a local motion due of its
instantaneous dismantlement at all scales (it becomes in a state of motion).
This local motion is different from the recession motion of all amount of
dismantled matter due to space expansion.

  4. 4.

The simultaneous expansion of the extremely contracted basic elements
transforms the initial space ${\mathcal{E}_{i}}$ into the space-time of
singularities $({\mathcal{E}_{r})}_{r>0}$ for all $0<r<r_{0}\ll 1$. Any basic
element of the transformed space-time $({\mathcal{E}_{r})}_{r>0}$ for all
$0<r<r_{0}$ is characterized by a normal curvature close to plus infinity and
an entropy close to minus infinity. The singularities of
$({\mathcal{E}_{r})}_{r>0}$ for all $0<r<r_{0}\ll 1$ are called primordial
singularities.

  5. 5.

The motion of any amount of the dismantled matter within singularities induces
a centripetal acceleration directed toward the center of each singularity. The
induced centripetal acceleration is given by $\gamma_{r}=v^{2}{\delta_{r}}$,
where $v$ is the module of the local speed induced by matter dismantlement and
${\delta_{r}}$ is the singularity’s normal curvature. This radial acceleration
is responsible for the changes in the direction of the velocity of any amount
of the dismantled matter to follow the normal curvature of the space. Since
${\delta_{r}}$ is close to plus infinity, then any amount of the dismantled
matter experiences a radial acceleration close to plus infinity directed
toward the center of the singularity where it was distributed. As the effect
of gravity and acceleration are indistinguishable (principle of equivalence),
then any amount of the dismantled matter experiences a local centripetal
gravitational field close to plus infinity.

  6. 6.

Based on Newton’s second law, a force must cause this centripetal acceleration
of magnitude

$F=mv^{2}\delta_{r}.$ (48)

This force bends the motion of any amount of the dismantled matter to the
space normal curvature. It makes the curved motion possible in each
singularity. Without this force, the motion of any amount of the dismantled
matter would continue in straight line instead of following the normal
curvature of the space. It is a force that imposes a bent motion adapted to
the normal curvature of the space.

  7. 7.

The existence of a centripetal force close to plus infinity within each
singularity traps any amount of the dismantled matter locally just after the
beginning of its dismantlement. The simultaneous expansion of the basic
elements of the initial space ${\mathcal{E}_{i}}$ into singularities
dismantles matter and energy at all scales, and traps locally any amount of
the dismantled matter and energy within an infinite family of stacked tiny
spheres. This process leads to an uniform distribution of the dismantled
matter and energy in the space-time of singularities during a fraction of the
first microsecond of expansion of the space ${\mathcal{E}_{i}}$.

Moreover, the centripetal force close to plus infinity within each singularity
traps any amount of the dismantled matter and energy and makes it experience
an infinite local centripetal compression propitious to matter recombination.
In particular it incubates the recombination of the elementary particles at
the first phase and later on the simplest atomic nucleus.

####  4.2.2 Gravity origin

Before the beginning of the expansion, matter and energy were assembled in an
initial space ${\mathcal{E}_{i}}$. The simultaneous expansion of the
dimensionless basic elements of ${\mathcal{E}_{i}}$ into basic elements of
three dimensions that pushes everything outside their volume, transforms the
initial space ${\mathcal{E}_{i}}$ into the space-time of singularities
$({\mathcal{E}_{r})}_{0<r<r_{0}}$, for an arbitrary tiny $r_{0}$ (§4.2.1, item
4.). This first transformation induces the following:

  * •

a non null momentum of any amount of the dismantled matter that defines its
state of motion. It is a consequence of its transformation from assembled
structures to a dismantled structures at all scales,

  * •

a local trap for any amount of the dismantled matter within singularities. It
is a consequence of its state of motion within an extremely curved space-time,

  * •

an intensely accelerated recession motion of any trapped amount of the
dismantled matter from each other in all directions. It is a consequence of
the short inflation period during the first microsecond of the expansion of
the initial space ${\mathcal{E}_{i}}$.

After the total dismantlement of matter and energy at all scales during the
instantaneous inflation period, gravity finds its origin revealed according to
the principle of equivalence between gravity and acceleration. Indeed, based
on §4.2.1 items 5. and 6., a centripetal acceleration close to plus infinity
appears within any singularity of the space-time
$({\mathcal{E}_{r})}_{0<r<r_{0}}$ where the dismantled matter and energy are
distributed in a state of motion. According to the Newton’s second law, a
force must cause this centripetal acceleration. A centripetal force within
each singularity of magnitude $F=mv^{2}\delta_{r}$, accelerates any amount of
the dismantled matter and energy by changing its velocity direction without
changing its speed, to make the bent motion adapted to the extremely curved
singularity. Therefore any trapped amount of the dismantled matter and energy
in a state of motion in the singularities experiences a radial gravitational
pull inward the geometrical center of each singularity. This radial
gravitational pull is responsible for the deviation of the motion of the
trapped amount of the dismantled matter, from linear motion to curved motion,
adhering to the extremely curved space-time.

The appearance of the centripetal gravitational force within each singularity
is an indication of a state of motion of any trapped amount of the dismantled
matter and energy in the singularity. It gives to the singularity the
characteristics of a gravitational singularity that include a normal curvature
close to plus infinity, a gravity close to plus infinity and an entropy close
to minus infinity. Without this local centripetal force close to plus
infinity, the local motion within each singularity of anything with mass would
be impossible. The centripetal gravitational force is a kind of frictional
force on any trapped amount of the dismantled matter from the expanding space-
time to impose a bent motion adhering to its curvature. It reveals the
resistance that any trapped amount of the dismantled matter offers to change
the state of its motion direction from rectilinear to curvilinear motion.
However, without the state of motion of the amount of the dismantled matter in
the expanding space-time, the radial gravitational pull would not exist. This
leads to assert the following: gravity exists because matter is never at rest
in a curved space-time.

####  4.2.3 Transformation of the space-time of singularities
${\mathcal{E}_{r}}$

Based on this model, the period of the space-time of singularities
$({\mathcal{E}_{r})}_{0<r<r_{0}}$, for an arbitrary tiny $r_{0}$, corresponds
to the period estimated to 380 000 years after the Big Bang. During this
period the basic elements of the space-time $({\mathcal{E}_{r})}_{0<r<r_{0}}$
are singularities of two kinds:

  * •

singularities without distribution of dismantled matter and energy. They are
characterized by a normal curvature close to plus infinity and an entropy
close to minus infinity

  * •

singularities with distribution of an amount of the dismantled matter and
energy in a state of motion. They are characterized by a normal curvature
close to plus infinity, an entropy close to minus infinity and a gravity close
to plus infinity. These singularities are called gravitational singularities.

The gravitational singularities are endowed with a centripetal gravitational
force close to plus infinity that traps everything inside including light. As
the space-time of singularities ${\mathcal{E}_{r}}$ continues to expand via
simultaneous expansion of its basic elements by increasing each radius $r$
toward $r_{0}$, the normal curvature $\delta_{r}={1\over r}$ of the space-time
of singularities $({\mathcal{E}_{r})}_{0<r<r_{0}}$ decreases gradually from
plus infinity. Therefore the centripetal gravitational force (48) within each
gravitational singularity reduces its extreme intensity gradually together
with the space-time expansion.

Thus the process of gradual decrease of the centripetal gravitational force
intensity induces a decompression of the gravitational singularities from plus
infinity (§4.2.1), which incubates the recombination of the dismantled matter.
In particular it recombines the elementary particles such as quarks, muons,
leptons, electrons, and positrons, followed by the recombination of the
composite particles such as protons, neutrons, other composite particles, and
later on the simplest atomic nucleus. The decompression of the gravitational
singularities continues at the rate of the space-time expansion until it
reaches the intensity that allows nuclei to trap electrons and later on to
form atoms. At this stage only simple atoms of neutral hydrogen are
recombined, the remaining intensity of the centripetal gravitational force
that allows electron to be trapped in the orbit of nucleus is insufficient for
nuclear fusion.

As the space-time expands, all singularities of the space-time
$({\mathcal{E}_{r})}_{0<r<r_{0}}$ loose simultaneously their extreme
characteristics gradually: the normal curvature of each singularity undergoes
a gradual decrease from plus infinity, inducing a gradual decrease of the
extreme intensity of the gravity from plus infinity and a gradual increase of
the entropy from minus infinity. This process of continuous changes of the
characteristics of the singularities lasts until the recombination of the
simplest neutral hydrogen atom and the release of the light to travel for the
first time as the radius $r$ of each singularity of ${\mathcal{E}_{r}}$
reaches the critical radius $r_{0}$. This makes the universe transparent and
reveals the surface of the last scattering from where the cosmic microwave
background comes. It indicates the end for the space-time of singularities
$({\mathcal{E}_{r})}_{0<r<r_{0}}$ that precedes the beginning of the
primordial space-time ${\mathcal{E}_{1}}$.

Moreover to conserve the density of the space-time of singularities
$({\mathcal{E}_{r})}_{0<r<r_{0}}$, as it expands, new singularities of
different sizes appear in the interstices of its stacked basic elements. The
new singularities of different sizes form an Apollonian gasket (illustrated in
Fig.6) within each interstice to fill the gap. Each singularity looses its
extreme characteristics as its radius expands to reach the critical radius
$r_{0}$. All the singularities of the interstices experience a transformation
similar to the transformation of the primordial singularities of
$({\mathcal{E}_{r})}_{0<r<r_{0}}$.

###  4.3 The period of primordial space-time ${\mathcal{E}_{1}}$

The period of primordial space-time ${\mathcal{E}_{1}}$ is the period that
comes after the end of the space-time of singularities
$({\mathcal{E}_{r})}_{0<r<r_{0}}$, approximatively 380 000 years after the Big
Bang. The primordial space-time ${\mathcal{E}_{1}}$ appears when the space-
time of singularities ceases to exist as the radius of each basic element of
${\mathcal{E}_{r}}$ reaches the critical radius $r_{0}\ll 1$. The release of
light to travel for the first time marks the beginning of this period, and
makes the universe transparent for observation. When gravity reaches an
intensity that allows the recombination of the simplest atoms of hydrogen and
releases light, matter and energy appear to be spread all around the previous
singularities’ location within an uniform distribution to form the primordial
observed space-time ${\mathcal{E}_{1}}$ (the CMB) at the Step 1.

The newly recombined matter within the space-time ${\mathcal{E}_{1}}$
experiences a centripetal gravitational force proportional to the space-time
normal curvature of magnitude given by Newton’s second law
$F_{1}=mv^{2}\delta_{1}(r)$, to the extent that for all ${n\geq 1}$ the
centripetal gravitational force is given by $F_{n}=mv^{2}\delta_{n}(r)$ within
each basic element of the space-time ${\mathcal{E}_{n}}$ where matter is
distributed. This gravitational force experiences a very slow decrease at a
rate set by the expansion rate of the space-time. The centripetal
gravitational force within each basic element remains invariant for a short
period of time since the normal curvature is imperceptible (Remark2, i)), and
decreases for a large period of time as the space-time ${\mathcal{E}_{n}}$
expands.

The interstices of the accumulated basic elements of the space-time
${\mathcal{E}_{1}}$ are the locations where new singularities of different
sizes appear to fill the gaps. Each singularity lasts for a period
approximatively greater than or equal to 380 000 years. The expansion of the
different singularities of the interstices of the space-time
${\mathcal{E}_{1}}$ depends on the rate of expansion of its basic elements. As
the primordial space-time ${\mathcal{E}_{1}}$ expands, one can deduce briefly
what will happen to the first recombined matter as the space-time expands
(more details will be the subject of a separate paper). The major changes of
matter transformation starting from the space-time ${\mathcal{E}_{1}}$ are
generated by the singularities of the interstices at the rate of the space-
time expansion. The appearance of the first stars, the recombination of
heavier atoms and the formation of big structures are the product of the
gravitational singularities of the interstices. Indeed, the singularities of
the interstices of the primordial space-time ${\mathcal{E}_{1}}$ carry a
potential of great interest as the space-time expands for two main
characteristics:

  * •

The first main characteristic of the interstices resides in the continual
appearance of other imbedded interstices between any three expanding stacked
singularities of different sizes of the space-time ${\mathcal{E}_{1}}$. These
repeating appearances generate an iteration of imbedded interstices for each
$n\geq 1$. Each interstice is a nursery of increasing stacked singularities of
different sizes.

  * •

The second main characteristic resides in the behavior of the interstices when
matter is trapped inside their singularities. Indeed, each basic element of
the space-time ${\mathcal{E}_{1}}$ touches twelve neighboring basic element,
which creates eight interstices, filled with singularities of different sizes
that surround each basic element of the space-time ${\mathcal{E}_{1}}$. Each
interstice becomes filled with stacked gravitational singularities if anything
with mass is trapped inside it. Therefore each interstice is endowed with a
variety of extreme decompressions generated by the gradual decrease of the
gravitational forces of extreme intensities (close to plus infinity). This
makes each interstice a family of incubators that traps and transforms the
first recombined matter. Moreover, the distribution of the interstices with
gravitational singularities in the space-time induces a variation in the
intensity of the gravitational field of the space-time. These variations are
propitious to change with time any uniform distribution of matter and energy
in the space-time ${\mathcal{E}_{1}}$ into lumpy and condense under the
extreme gravitational pull of the gravitational singularities of the
interstices. Indeed, with their different extreme intensities of compression,
the gravitational singularities of the interstices present adequate conditions
to trigger the incubation of grouped stars, the recombination of heavier
particles, and thus they are the appropriate locations for the accumulation of
clouds of gas, dust of matter, and grouped stars to form galaxies and the big
structures we know today.

If the freshly recombined neutral hydrogen atoms are trapped by the
singularities of the interstices of the expanding space-time
${\mathcal{E}_{1}}$, the different extreme compressions inside those
gravitational singularities are propitious for the recombination of heavier
elements, as well as to overcome the Coulomb barrier. If they trap the neutral
hydrogen, the gravitational singularities of ${\mathcal{E}_{1}}$ trigger the
proton-proton chain needed for the nuclear fusion of neutral hydrogen atoms at
the first stage to produce helium, energy, radiation, and give birth to the
first big number of stars in the universe to end the period of expansion of
the space-time ${\mathcal{E}_{1}}$ in darkness.

##  5 Conclusion

A new definition of entropy is introduced using a model that simulates an
expanding space-time compatible with the fundamental principle of cosmology.
This entropy is obtained by mean of a state function that measures the
variation of the space-time normal curvature consequence of its expansion. It
leads to work out a new insight that unravels our understanding of the
earliest conditions that last for a period estimated to 380 000 years after
the Big Bang. This most mysterious period of the standard model of cosmology
remains not accessible to direct observation and subject to hypothetical
approaches or speculations.

The model used in this investigation provides a reduced schematic description
of a phenomenon that account for the properties of the space-time expansion
compatible with observation. It allows to investigate the space-time
characteristics and to approach the earliest conditions that incubate matter
and energy recombination after their total dismantlement. The earliest phase
and conditions are found to be described by a space-time of singularities
$(\mathcal{E}_{r})$, for $r\in]0,r_{0}[$ endowed with basic elements of
extreme characteristics in terms of normal curvature close to plus infinity,
and entropy close to minus infinity. As the topology of each basic element of
the space-time of singularities approaches the topology of that of a point,
the model describes an extremely contracted initial space $\mathcal{E}_{i}$
where matter and energy are supposed to be assembled before the Big Bang
event.

The simultaneous decompression of extremely compressed dense basic elements of
the initial space $\mathcal{E}_{i}$ generates the space-time of singularities
$(\mathcal{E}_{r})$ for $r\in]0,r_{0}[$, after a short period of inflation.
The presence of any amount of the dismantled matter in a state of motion
within those singularities transforms them into gravitational singularities,
characterized by a normal curvature close to plus infinity, a gravity close to
plus infinity and an entropy close to minus infinity. This extreme environment
traps everything with mass including light. It prevents any observation of the
earliest extreme conditions that incubate matter recombination during a period
quantified by the space-time $(\mathcal{E}_{r})$ for $r\in]0,r_{0}[$. This
period is estimated to approximatively last for 380 000 years after the Big
Bang.

The state of motion of the dismantled matter within an extremely curved
singularity gives birth to an infinite gravitational attraction directed
toward the center of the singularity, a center that represents an external
location equidistant from all positions in the singularity. This extreme
gravitational force is a frictional force on the dismantled matter in a state
of motion exerted by the normal curvature of the space-time of singularities
to adjust its direction to the motion within the curved space. The centripetal
acceleration exerted to anything with mass in a state of motion within each
singularity of the space-time $\mathcal{E}_{r}$ divulges the origin of a
gravity with an extreme intensity that compresses any trapped thing with mass.
The gradual decrease of the intensity of the gravitational attraction induces
a decompression at the rate of the space-time expansion that transforms each
gravitational singularity into an incubator for matter recombination.

In particular the gradual decompression maintains suitable conditions for the
recombination of elementary particles at the first stage, followed by the
recombination of the composite particles and later on the recombination of the
simplest atomic nucleus. This process goes on at the rate of the space-time
expansion, until the intensity of the decompression allows nuclei to trap
electrons in their orbits and form the first simple neutral atoms of hydrogen.
This process suggests that the amount of liberated energy we measure today
when nucleus parts are disassembled by nuclear fission might be traced back,
using the principle of conservation of energy, to the earliest gravitational
extreme pressure that was born within the gravitational singularities of the
young expanding space-time.

This case study allows to:

  * •

study the change of a state function depending on the normal curvature of an
expanding space-time compatible with the fundamental principle of cosmology,
using an expansion quantified by a sequence of space-time
${\mathcal{E}}=({\mathcal{E}_{n}})_{n\geq 1}$ that expands via simultaneous
expansion of its basic elements. The state function measures the local
transformation of the space-time from a highly compressed space to a lower
compressed space.

  * •

justify the existence of an instantaneous inflation that generates
approximatively the uniform distribution of matter and all form of energy
observed in the surface of the last scattering (CMB). This inflation is the
first transformation of an initial extremely contracted space
$\mathcal{E}_{i}$ to the expanding space-time of singularities
$(\mathcal{E}_{r})$ for $0<r<r_{0}$.

  * •

quantify, characterize and provide an interpretation of the hidden earliest
phase closest to the beginning of the space expansion, that lasts
approximately for 380 000 years.

  * •

trace back the earliest conditions propitious to the incubation of matter and
energy recombination after their dismantlement by the Big Bang. It allows to
understand the environment and process that lead to the birth of the first
stars.

  * •

assert that gravity exists in a curved expanding space-time because matter is
never at rest.

Within this model the energy that generates the simultaneous expansion of the
stacked equal spheres is unknown. But it induces a simultaneous decompression
of the extremely contracted dense basic elements of the initial space
$\mathcal{E}_{i}$ that approximates the space before the Big Bang. The densest
model of stacked equal spheres to cover a volume is found to represent
approximatively 74$\%$ of the volume, which suggests that taking into account
the density of the space-time, the family of stacked equal balls responsible
for the expansion of the space-time within this model represents
approximatively 74$\%$, and this estimation does not change with time.

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