The Radiative Era.
The idea is that the so-called "constants" of physics behave like absolute constants during the matter era, but vary significantly during the radiative era. This may seem very artificial, but this idea could solve the problem of the homogeneity of the early Universe, as recently emphasized by several authors, including Magueijo (1999), although it was discovered by the author 13 years earlier, at the end of the 1980s ([44], [45], [46]) and later developed ([4] and [47]). First, note that the choice of the time parameter t remains arbitrary: it represents only "the way we imagine things have unfolded." Absolute time has no meaning in cosmology. No phenomenon "exists" if there is no observer in the Universe to look at it, to compare a sequence of events to its own temporal flow. At present, everything is compared to the observer's time, the way it is lived. But the past and future depend on the way it is imagined, since it cannot travel to the past or future. The past and future are nothing more than images we construct. We will say these images are correct if they agree with particular local phenomena we call "observations," "measurements."
Consider the "constants of physics." They have been discovered quite recently: these include the speed of light c, the gravitational constant G, the Planck constant h, the masses of particles, the elementary electric charge e, the vacuum permittivity ε₀, and others. Laboratory measurements do not reveal any significant change. However, researchers have tried to study the impact, over long periods, of a variation of these constants on various cosmic phenomena. But they have modified these constants one by one, independently of each other. Under these conditions, it can be shown that any slight variation of an isolated constant leads to contradictions with observational data. What about joint variations? Surprisingly, one can conceive of a joint variation of all the constants, which could not be detected in the laboratory, since the measuring instruments are built from the fundamental equations of physics. If this gauge process leaves these equations invariant, it becomes impossible to detect the variation of any constant, since the instruments and the constants they are supposed to measure undergo parallel drifts. Imagine you want to measure the length of a metal table using a metal ruler, both at room temperature. If the length of the table seems constant over time, you cannot claim it does not vary, because the table and the ruler could simply expand equally in case of a temperature change.
So let us look for such a fundamental gauge process. Consider, for example, the field equation, which includes the Einstein constant. We assume that the divergence of this equation is zero, which, in the Newtonian approximation, corresponds to the conservation of matter and energy. If this is not the case, we must introduce a source term. According to this hypothesis, the Einstein constant c must be an absolute constant. Does this imply that G and c must be absolute constants? Absolutely not. It only implies that:
As introduced for the first time in 1988, we assume that all kinds of energies are conserved, but
that masses, electric charge, etc., are not. This gives, for example:
In physics, all students know the method called dimensional analysis. Given a physical problem governed by an equation (or a system of equations), we construct lengths, times, and characteristic numbers from the constants and experimental data. We now consider that everything in the equation can vary, including the "constants." We then express everything in dimensionless form. Take, for example, the Boltzmann equation:
We introduce a characteristic length scale R and a characteristic time scale T:
The equation becomes then:
We observe that the Schwarzschild radius varies as the scale factor R. In summary, we obtain:
We see that the Jeans length Lⱼ varies as R, while the Jeans time tⱼ varies as T. R and T are related by a relation that evokes a Friedman model. But if we look more closely and interpret this as a gauge relation, it means that Kepler's laws are also invariant:
By introducing pressures (as energy densities), we obtain the gauge variations of these parameters, and we observe that the corresponding energies are conserved (in this model, all forms of energy are conserved during the radiative era). We have thus determined how the speed of light c varies with energy density when radiation dominates.
Consider now the Schrödinger equation:
Introduce a dimensionless expression of the potential and transform this equation:
Thus, the energy remains unchanged by this gauge process. The Planck constant h increases with T, as Milne had first conjectured [48]. The characteristic lengths:
vary as the spatial scale factor R, while the Planck time tₚ varies as the temporal scale factor T. From then on, the evolution during the radiative era can be interpreted as a gauge process. This challenges the notion of the "Planck barrier." Does the "pre-quantum era" have a real meaning?
To conclude, we must deal with the Maxwell equations.
Continue to apply this kind of "generalized dimensional analysis." We obtain:
In order to maintain the structure of atoms during the evolution process, we assume that the fine structure constant is an absolute constant, which provides the complete solution:
We obtain easily:
Thus, if cosmic evolution is identified with a gauge process during the radiative era, all characteristic lengths vary as R (above, the Bohr radius), all characteristic times vary as T, and all energies remain constant.
All constants, as well as spatial and temporal scales, are involved in this gauge process, which can be described by choosing any one of them. We can take T as our time parameter t.
Then, the variation of the constants during the radiative era as a function of the radiation pressure pᵣ:
If we assume that the values of the constants depend on the radiation pressure, and introducing a critical pressure p_cr (to be defined), we can write:
G₀, m₀, h₀, c₀, e₀ correspond to the current values. We assume that these critical conditions are reached for a value t = t_cr of the chosen time parameter.
which corresponds to Figure 16.
Fig. 16: Variation of the constants during the radiative era.
t ≫ t_cr corresponds to the matter era.
The Homogeneity of the Universe.
Any model requires observational confirmation. Figure 17, on the left, illustrates the classical paradox of the homogeneity of the early Universe. "Classical explanation": the "inflation theory," which is based on many strong assumptions. Today, some are beginning to consider a model with variable constants, including a secular variation of c, which they call "VLS" (for "variable light speed"). In fact, I developed this idea as early as 1988 [44]. Thanks to the suggested variation of c, mentioned in the previous section, the horizon varies as R(t), thus ensuring homogeneity at all times.
Fig. 17: The horizon, according to the standard model and according to the present model.
When the adverb "before" fails.
As mentioned above, a time parameter corresponds to an arbitrary choice, devoid of intrinsic meaning. In the standard model, if we consider the distant past of the Universe, the temperature increases and the velocities of the particles tend toward c. All particles become relativistic, which raises a question: "How to build a clock, and with what material?" When looking at a clock, what do we see? The movement of a hand: one turn corresponds to a minute or an hour. One turn of the Earth around the Sun corresponds to a year. Whatever we call it, this 360° rotation has an undeniable physical reality: it is an undeniable event. Similarly, we can consider a reference system composed of two masses m orbiting around their common center of gravity: we can call it our "elementary clock." In a thermodynamic equilibrium gas, the available energy is distributed between translational kinetic energy, rotational energy, and vibrational energy. A pair of particles orbiting around their common center of gravity is conceivable if the system's energy is comparable to that of free particles, which move freely. In a system with variable constants, this becomes possible. Thus, we can count the number of turns using the time parameter t, which has no real meaning: it is only a chronological marker.
Fig. 18: The elementary clock.
What does this mean? According to this description of the Universe, an infinite number of "elementary events" have occurred in the past. If this clock corresponds to a measurement of time, then the past is infinite and the time parameter t is just a fiction. Let's take an image: suppose you visit an editor and say: "I want to publish a thick book of two inches." This depends on the thickness of the pages. You could trick the editor by using pages whose width tends to zero, while pretending to read "the first pages": although the overall thickness of the book appears finite, it tells an infinite story. The right question the editor should ask you is: "How many characters, sentences, words, letters does your book contain?" Each letter of your book can be compared to an "elementary event." Like your book, titled "History of the Universe," which, when going back in time, reveals an infinite number of "elementary events," it has... no beginning, and you will never be able to read the author's preface. Moreover, as shown in reference [4], the number of turns of our elementary clock corresponds to the entropy per baryon. log t is also called "conformal time." Indeed, if this is chosen as the new time parameter, the metric becomes conformally flat:
In the previous section, we established that the Planck time varies as the time parameter t. This means that when we go back to the so-called "initial singularity (t = 0)," the Planck time decreases. What does this mean? I don't know. Anyway, this model does not solve all the problems: we do not address the strong and weak interactions. It is only another point of view on what we call "time."
Joint Gravitational Instabilities.
In section 3, we presented a model of a galaxy confined by its "repulsive twin matter" environment. This work was semi-empirical. In the present section, we present an exact spherically symmetric solution. If we start from the coupled field equations, we assume they are divergence-free:
From these equations, we can derive the Euler equation. The method is entirely similar to that applied to the Einstein equation.
Combined with the Poisson equation:
The classical perturbation method leads to two coupled Jeans-type equations, Lⱼ and Lⱼ being the characteristic Jeans lengths.
A stationary spherically symmetric solution, with the initial conditions:
On Figure 19, the typical numerical solution.
Fig. 19: Joint gravitational instabilities.
Formation of a matter cluster surrounded by a repulsive twin matter environment.
Note (May 23, 2007):
The general shape of the curves depends on the initial conditions. The chosen conditions are arbitrary and correspond to equal densities and equal thermal velocities in the two "folds." Nevertheless, we observe an interesting feature. On Figure 19 bis, we can trace the direction of the gravitational field:
Fig. 19 bis: "Dark matter halo" effect
The gravitational field induces a gravitational lensing effect, which measures the gravitational field, regardless of its source. In our theory, ordinary matter - that of our "fold" - contributes its own. The "twin matter" also contributes (it behaves like matter with negative mass).
If we assume that the strong lensing effects observed are due to a mysterious "dark matter," then, from the measured gravitational field and the distribution of visible matter, we can calculate the distribution of this dark matter, if it actually exists. On Figure 19 bis, we observe an inversion of the direction of the gravitational field, accompanied by a corresponding variation of the local gravitational lensing. According to the matter + dark matter model, we could calculate the distribution of dark matter that would produce the observed lensing effects. By examining the top image of Figure 19 bis, we could deduce that this cluster is surrounded by "a hollow shell of dark matter." The bottom image suggests this conclusion.
As we know, the Hubble Space Telescope recently discovered a "dark matter halo." See the following figure.
Fig. 19 ter: The "dark matter halo" discovered by the Hubble Space Telescope
In order to maintain the structure of atoms during the evolution process, we assume that the fine structure constant is an absolute constant, which leads to the complete solution:
We obtain easily:
As can be seen, during the radiative era, if cosmic evolution is identified with a gauge process, all characteristic lengths vary as R (above, the Bohr radius), all characteristic times vary as T, and all energies remain constant.
All constants, as well as the spatial and temporal scales, are involved in this gauge process, which can be described by choosing any one of them. We can take T as our time parameter t.
Then, the variation of the constants during the radiative era, as a function of the radiation pressure p:
r:
If we assume that the values of the constants depend on the radiation pressure, introducing a critical value p
cr
to be defined, we can write:
G
o
, m
o
, h
o
, c
o
, e
o
denote the current values. We assume that these critical conditions are reached for a value t = t
cr
of the chosen time parameter.
which corresponds to Figure 16.
Fig.
16:
Variation of the constants during the radiative era.
t
t
cr
corresponds to the matter era.
The Homogeneity of the Universe.
Any model requires observational confirmation. Figure 17, on the left, illustrates the classical paradox of the homogeneity of the early Universe. "Classical explanation": the "inflation theory," which is based on strong assumptions. Today, some are beginning to consider a model with variable constants, including a secular variation of c. They call it "VLS" (for "variable light speed"). In fact, I developed this idea as early as 1988 [44]. Thanks to the suggested temporal variation of c - which, as we saw in the previous section, makes the horizon vary as R(t) - homogeneity is ensured at all times.
Fig.
17:
The horizon, according to the standard model and according to the present model.
When the word "before" fails.
As mentioned above, a time marker corresponds to an arbitrary choice. It has no intrinsic meaning. In the standard model, if we consider the distant past of the Universe, the temperature increases and the velocities of the elements tend toward c. All particles become then relativistic, which raises a question: "How to build a clock, with what material?" When looking at a clock, what do we look at? At the rotation of a hand. One turn corresponds to a minute or an hour. One turn of the Earth around the Sun corresponds to a year. Whatever we call it, this 360° rotation has a real physical meaning: it is an undeniable event. Similarly, we can consider a reference system composed of two masses m orbiting around their common center of gravity. We can call it our "elementary clock." In a thermodynamic equilibrium gas, the available energy is distributed between translational kinetic energy, rotational energy, and vibrational energy. A pair of particles orbiting around their common center of gravity is conceivable if the system's energy is comparable to that of free particles, which move freely. In a system with variable constants, this is possible. Thus, we can count the number of turns, using the time marker t, which has no real meaning: it is only a chronological marker.
Fig
.18:
The elementary clock.
What does this mean? According to this description of the Universe, an infinite number of "elementary events" have occurred in the past. If this clock corresponds to a measurement of time, the past is infinite and the time marker t is just a fiction. Let's give an image: suppose you visit an editor and say: "I want to publish a thick book of two inches." This depends on the thickness of the pages. You could deceive the editor by using pages whose width tends to zero, while pretending to read "the first pages." Although the overall thickness of the book appears finite, it tells an infinite story. The right question the editor should ask you is: "How many characters does your book contain? How many sentences, words, letters?" A letter of your book can be compared to an "elementary event." Like your book, titled "History of the Universe," which, when going back in time, reveals an infinite number of "elementary events," it has... no beginning, and you will never be able to read the author's preface. Moreover, as shown in reference [4], the number of turns of our elementary clock corresponds to the entropy per baryon. Log t is also called "conformal time." Indeed, if this is chosen as the new time marker, the metric becomes conformally flat:
In the previous section, we established that the Planck time varies as the time marker t. This means that when we go back to the so-called "initial singularity (t = 0)," the Planck time decreases. What does this mean? I don't know. Anyway, this model does not solve all the problems. We do not address the strong and weak interactions; it is only another point of view on what we call "time."
Joint Gravitational Instabilities.
In section 3, we presented a model of a galaxy confined by its environment of repulsive twin matter. This work was semi-empirical. In the present section, we present an exact spherically symmetric solution. If we start from the coupled field equations, we assume they are divergence-free:
From such equations, one can deduce the Euler equation. The method is entirely similar to that applied to the Einstein equation.
Coupled with the Poisson equation:
The classical perturbation method leads to two coupled Jeans-type equations, Lj and Lj being the characteristic Jeans lengths.
A stationary spherically symmetric solution, with the following initial conditions:
On Figure 19, the typical numerical solution.
Fig.
19
: Joint gravitational instabilities.
Formation of a matter cluster surrounded by a repulsive twin matter environment.
Note (May 23, 2007):
The general shape of the curves depends on the initial conditions. The chosen conditions are arbitrary and correspond to equal densities and equal thermal velocities in the two folds. Nevertheless, we observe an interesting feature. On Figure 19 bis, we can trace the direction of the gravitational field:
Fig 19 bis: "Dark matter halo" effect
The gravitational field induces a gravitational lensing effect, which constitutes a measurement of the gravitational field, regardless of its source. In our theory, ordinary matter - that of our "fold" - contributes its own. The "twin matter" also contributes (it behaves like matter with negative mass).
If we choose to consider that the strong lensing effects observed are due to a mysterious "dark matter," then, from the measured gravitational field and the distribution of visible matter, we can calculate the distribution of this dark matter, if it actually exists. On Figure 19 bis, we observe an inversion of the direction of the gravitational field, accompanied by a corresponding variation of the local gravitational lensing. According to the matter + dark matter model, we could calculate the distribution of dark matter that would produce the observed lensing effects. By examining the figure at the top of Figure 19 bis, we could deduce that this cluster is surrounded by "a hollow shell of dark matter." The image at the bottom suggests this conclusion.
As we know, the Hubble Space Telescope recently discovered a "dark matter halo." See the following figure.
Fig 19 ter: The "dark matter halo" discovered by the Hubble Space Telescope in May 2007, as "deduced from calculations."
Interestingly, this halo is centered on the visible galaxy cluster.
We think it does not correspond to a flat structure, but to a spherically symmetric structure.
We predict that such structures should soon be discovered. In any case, the "halo" will be centered on the cluster, so that astrophysicists will admit that it is not a halo, but "a kind of hollow structure."
The halo structure could be considered the result of an ancient encounter ("resembling a smoke ring").
Suppose my prediction is confirmed. If astrophysicists have to admit that these observations correspond to a spherically symmetric structure, how would they model this hollow shell of dark matter?
If this hypothesis is confirmed, it could provide the necessary elements to choose between the matter + dark matter model and the twin universe model.
Confinement of Spherical Galaxies.
In section 7, figure 11, we indicated that the field of a hole in a uniform negative energy matter is equivalent to the field created by an equivalent sphere filled with positive energy matter and surrounded by vacuum. It is now appropriate to justify this statement. Recall how the link with the Poisson equation is established in classical theory (see, for example,
[ 52 ]
).
This gives (a):
Then (b) is written. Using (d) and (c), equation (b) identifies with the Poisson equation. However, we immediately notice that the perturbed metric given corresponds to stationary conditions. This is only conceivable if the zero-order solution (the Lorentz metric) corresponds to a vacuum universe, in which no gravitational force or pressure is exerted.
There is therefore a link between the field equations and the Poisson equation. However, if the Universe is assumed to be non-empty and homogeneous, this method is no longer valid, because we can no longer refer to a stationary metric. What is the impact? We cannot define a gravitational potential in a homogeneous universe filled with constant density matter. If we examine the Poisson equation (e), written in spherical coordinates, and if we assume that
r
is constant, we obtain the spherical solution (f), and the corresponding gravitational field is (g). Isn't it surprising to find a non-zero gravitational force directed towards an arbitrary coordinate center and tending to infinity with the radial distance? Explanation: this pseudo-solution is incorrect, because the Poisson equation does not exist in a stationary homogeneous universe. The field is zero everywhere, which seems more physical.
Fig
. 20
: Spherical hole in a matter distribution with constant density
and the associated gravitational potential.
Figure (b) shows the gravitational field outside and inside a sphere filled with matter of constant positive density (like Earth). In (c), the associated gravitational potential. If we reverse the arrows of (b), we obtain the field associated with a sphere of negative mass. If we add this to (a), we obtain a homogeneous and unlimited region filled with negative mass, with a zero field; thus, (a) represents the field inside a spherical cavity, which is not zero. Thus, we obtain a confinement effect, and the field intensity is maximum at the inner boundary. This explains why spiral galaxies retain their arms and why the decrease in gas density in the disk is so abrupt in the periphery.
Summary of the article ---
The radiative era.
.
The idea is that the so-called constants of physics behave like absolute constants during the matter era, but change drastically during the radiative era. It may look very artificial, but this idea may solve the problem of the homogeneity of the early universe, as recently pointed out by several authors, like Magueijo (1999), but was discovered by the author 13 years before, at the end of the eighties ( [ 44 ]
,
[ 45 ]
,
[ 46 ]
) and developed later ( [ 4 ]
and
[ 47 ]
). First, notice that the choice of the time marker t remains arbitrary. It is nothing but "the way we think the things happened". Absolute time has no meaning in cosmology. Any phenomenon does not "exist" if there is no observer in the Universe to look at it, to compare a succession of events to his proper time flow. At present time all is compared to the time of the observer, the way he lives. But past and future depends on the way he
imagines
it, for he cannot travel in past or future. Past and future are nothing but images we shape. We will say that these images are correct if they fit peculiar local phenomena, that we call "observations", "measurements". Consider the "constants of physics". They were discovered quite recently. They are the light speed c , the gravitational constant G, the Planck constant h , the masses of particles, the unit
electric charge e, the permittivity of vacuum
e o
, and some others. Measurements performed in labs show no significant change. People have tried to study the impact over large period of time of a change of these constants on various cosmic phenomena. But they moved these constants one after the other, independently. In such conditions one can show that any light variation of an isolated constant produces contradictions with observational data. But what about joint variations ? Surprinsingly we may conceive a joint variation of all the constant, which cannot be evidenced in lab, for the lab's instruments are built with the basic equations of physics. If this gauge process keeps these equations invariant, it will be impossible to evidence the variation of any constant, for the instruments and the constants they are supposed to measure experience parallel drifts. Imagine you want to measure the length of an iron table, with an iron scaled rule. Both are at room temperature. If the table's length is found constant in time, you cannot sware this length does not vary, for this table and your scaled rule may experience a room temperature variation and expand in the same way.
Let us search such basic gauge process. Consider for example the field equation, where we find the Einstein constant. We assume the divergence of this equation is zero which, in Newtonian approximation, corresponds to the conservation of matter and energy. If it is not, we must deal with source term. According to this hypothesis the Einstein constant
c
must be an absolute constant. Does it imply that G and c must be absolute constants ? Definitively not. It only implies that :
As introduced first in 1988 we assume that the energies, all kind of energies, are conserved, but
not the masses, electric charge and so on. This gives, for example
:
In physics all students know the technique called dimensional analysis. Given a physical problem, ruled by an equation, or a set of equations, we produce characteristic lengths, times and numbers, composed with constants and laboratory condition data. Now we consider that all what is present in the equation may vary, including the "constants". We put everything into a non-dimensional form. Consider for example the Boltzmann equation
:
We introduce a characteristic length scale R and a characteristic time scale
T:
The equation becomes
:
We see that the Schwarzschild length varies like the scale factor R. To sum up get
:
We see that the Jeans length L
j
varies like R, while Jeans time t
j
varies like
T.
Rand
Tare linked through a relation which evokes a Friedman model. But if one looks that closer and see it as a gauge relation, it means that the Kepler laws are also invariant :
By the way, introducing pressures (as energy densities) we get the gauge variations of
these parameters and see that the subsequent energies are conserved (in this model all kinds of energy are conserved during the radiative era). We have figured the way the speed of light c varies with the energy density when radiation dominates.
Now consider the Schrödinger equation
:
Introduce non-dimensional potential expression and transform this equation
:
As a result, the energy is unchanged by this gauge process. The Planck constant h grows with T, as conjectured first by Milne
[ 48 ]
. The characteristic lengths :
vary like the space scale factor R, while the Planck time t
p
varies like the time scale factor T. From this point of view, the evolution, during the radiative era is conceived as a gauge process. This makes the "Planck barrier" questionable. Does the "pre-quantic" epoch has a real meaning ? Now, to finish the job we have to deal with Maxwell equations.
Continue to perform that sort of "generalized dimensional analysis". We get :
In order to maintain the structure of the atoms during the evolution process we assume the fine structure constant is an absolute constant, which gives the whole solution :
We get easily :
As we can see, during the radiative era, is the cosmic evolution is identified to a gauge process, all characteristic lengths vary like R (above, the Bohr radius), all the characteristic times vary like T, all the energies are constant.
All the constants, space and time scales are involved in this gauge process, which can be described chosing any of them. We can take T as our time-marker t .
Next, the variation of the constants, during the radiative era, versus the radiative pressure p
r
:
If we assume that the values of the constants depend on the radiative pressure, introducing a critical value p
cr
, to be defined, we can write :
G o
, m
o
, h
o
, c
o
, e
o
correspond to the today's values. We assume that this critical conditions are achieved for a value t = t
cr
of the chosen time-marker.
which corresponds to figure 16.
Fig
. 16 :
Variation of the constants during the radiative era.
t
t
cr
corresponds to matter era
The homogeneity of the Universe
.
Any model requires an observational confirmation. Figure 17, left, the classical paradox of the homogeneity of the early Universe. "Classical explanation" : the "Inflation Theory", requiring heavy hypothesis. Today, some people begins to think about a variable constants model, including a secular variation of c. The called it "VLS" : "variable light speed". In fact I developed this idea in 1988
[ 44 ]
. With the suggested time-variation of c , which with the precedent section the horizon varies like R(t) which ensures homogeneity at any time.
Fig.
17 :
The horizon, according to the standard model and to the present model.
When the adverb "before" fails.
As said above, a time marker corresponds to an arbitrary choice. It has no intrinsic meaning. In the standard model if we deal with the distant past of the Universe, the temperature rises and elements' velocities tend to c. All particles become relativistic, so that a question arises : "how to build a clock, with which material ?". When we look at a clock, what do we look at ? To the rotation of a needle. A turn corresponds to a minute, or hour. A turn of the Earth around the Sun corresponds to a year. Whatever we call it, this 360° rotation has a physically real meaning. It is an undeniable event. Similarly we can consider a reference system composed by two masses m orbiting around their common centre of gravity. We may call it our "elementary clock". In a gas at thermodynamic equilibrium the available energy is distributed in
translational energy, the rotational energy, vibrational energy. A couple of particles orbiting around their common centre of gravity is conceivable if the energy of the system is comparable to the energy of free particles, which cruise around. In a variable constant system this is possible. Then we can count the number of turns, using the time-marker t, which has no real significance : it's only a chronological marker.
Fig
.18 :
The elementary clock.
What does it mean ? According to this description of the Universe, an infinite number of "elementary events" occurred in the past. If this clock corresponds to a measure of time, past is infinite and the time-marker t is nothing but a fiction. Let's give an image. Suppose you visit an editor and say "I want to publish a two inches thick book". Depends on the width of the pages. You may deceive the editor if you use pages whose width tends to zero when trying to read "the first pages". Although the global width of the book seemed finite it tells a infinite story. The good question the editor must ak you is "how many types in your book, how many sentences, words, letters ?". A letter of your book can be compared to an "elementary event". As your book, called "Universe story", going towards the past, shows an infinite number of "elementary events", it has... no beginning and you will never succeed in reading the foreword of the author. By the way, as shown in reference
[ 4 ]
the number of turns of our elementary clock identifies to the entropy per baryon. Log t is also called "conformal time". In effect if chosen as a new time-marker the metric becomes conformally flat :
In the previous section we found that the Planck time varies like the time-marker t. It means that when one goes back to the so-called "initial singularity (t = 0)" the Planck time shrinks. What does it mean? I haven't the answer. Anyway this model doesn't clear up all problems. We don't deal with strong and weak interaction. It's only a different glimpse on what we call "time".
Joint gravitational instabilities.
In section 3 we have presented a model of a galaxy confined by its repulsive twin matter environment. This work was semi-empirical. In the present section we present a spherically symmetric exact solution. If we start from the coupled field equation, we assume that it is divergenceless
From such equations one can derive Euler equation. The method is completely similar to the one applying to the Einstein equation.
Coupled to Poisson equation :
The classical perturbation method gives two Jeans like coupled equations, Lj and
L j being characteristic Jeans lengths.
A steady-state spherically symmetric solution, with initial conditions :
On figure 19 the typical numerical solution.
Fig.
19
: Joint gravitational instabilities.
Formation of a clump of matter
surrounded by repulsive twin matter environment.
Remark ( 2007, may 23 ) :
The general feature of the curves depends on the initial conditions. The chosen conditions are arbitrary and correspond to the equal mass densities and equal thermal velocities in the two folds. Anyway, we find an interesting feature. On figure 19 bis we can plot the gravitational field direction :
Fig 19 bis : "Dark matter halo" effect
The gravitational field induces a gravitational lensing effect. This last is a measure of the gravitational field, whatever is the source of this field. In our theory, the ordinary matter, the one of our "fold", brings its own contribution. The "twin matter" brings its contribution too ( it behaves like negative mass material ).
If we choose to consider that the observed strong lensing effects are due to some mysterious "dark matter", given the gravitational field and the distribution of visible matter one can compute the distribution of this dark matter, if it does exist. On figure 19bis we observe an inversion of the direction of the gravitational field, which goes with a corresponding variation of the local gravitational lensing. Following the model of the matter plus dark matter we could compute the distribution of dark matter which would give the corresponding lensing effects. Considering the figure at the top of Fig 19 bis we would deduce that this cluster is surrounded by "an hollow shell of dark matter". The picture at the bottom evokes such conclusion.
As we know, the space Hubble telescope has recently discovered a "halo of dark matter". See the next figure.
Fig 19 ter : The "halo of dark matter" discovered by the Hubble space telescope in 2007, may. As "deduced from computation".
Surprisingly this halo is centered on the visible galactic cluster.
We think that it does not correspond to a plane structure but to a spherically symmetric structure
.
We predict that similar structures should be discovered soon. In all cases the "halo" will be centered of the cluster, so that the astrophysicist will admit this is not a halo but "some sort of hollow structure.
The halo structure could be considered as the result of an old encounter ( looking "like a smoke ring" ).
Suppose my prediction would be confirmed. If astrophysicists have to admit that these observations correspond to spherically symmetric structure, how will they modelize this hollow dark matter shell ?
If it is confirmed, this could bring the elements to make a choice between the matter plus dark matter model and the twin universe model.
The confinement of spheroidal galaxies .
In section 7, figure 11, we said that the field due to a hole in a uniform negative energy matter was equivalent to the field created by an equivalent sphere, filled by positive energy matter and surrounded by void. It has now to be justified. Let us recall how the link to Poisson equation is built in classical theory ( see for example
[ 52]
).
This gives (a) :
The one writes (b). With (d) and (c) the equation (b) is identified to Poisson equation. But notice immediately that the given perturbed metric corresponds to steady-state conditions. This is conceivable only if the zero order solution (the Lorentz metric) corresponds to an empty universe, where no gravitational force and no pressure are acting.
Then there is a link between the field and the Poisson equations. But is the Universe is supposed to be non-empty and uniform this method does not hold any longer, for we cannot refer to steady state metric. What is the impact ? We cannot define a gravitational potential in an uniform universe, filled by constant density material. If we look at the Poisson equation (e), written in spherical coordinates and if we suppose
r
is a constant, we find the spherically solution (f) and the corresponding gravity field is (g). Isn't surprinzing to find a non-zero gravitational force, pointing towards an arbitrary centre of coordinates and tending to infinite with radial distance ? Explanation : this pseudo-solution is not correct, for Poisson equation does not exist in an steady state uniform universe. The field is zero everywhere, which looks more physical.
Fig
. 20
: Spherical hole in a constant density twin matter distribution
and associated gravitational potential.
The figure (b) shows the gravitational field around and inside a sphere filled by constant positive density material (like the Earth). In (c) the associated gravitational potential. If we reverse the arrows of (b) we get the field associated to a sphere filled by negative mass. If this is added to (a) we get an uniform and unbounded region, filled by negative mass, with a zero field, so that (a) figure the field inside a spherical cavity, which is non-zero. We get a confining effect and the intensity of the field is maximum at the internal border. This explains why the spiral galaxies keeps their arms and why the decrease of the gas density of the disk is so abrupt at periphery.
Paper's Summary ---