Exormetism

Exormetism is a novel theory that can explain the red shifts of the galaxies that are moving from each other. This SATOCONOR.COM page is also an unique because it is the first time that third party software is on production at the site! The architect and programmer of the software is the mentioned author of this Exormetism article. This software visually simulates the movement and creation of the galaxies in a simplified 2 dimensional flat Universe. (See: Download and Explanation and the end of this article.)

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P.C. Karagiorgis ‘Exormetism:

A Steady-State Theory’ (2007)

Full paper

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Exormetism: A Steady-State Theory

Unlike antigravity, that just balances gravity, for the universe to be Euclidean,

the inferable dark force is clearly fundamental, as the law of its action implies.

By Panagiotis C. Karagiorgis

For comments: karagiorgis@tellas.gr

The parameters entering into this theory have been estimated from the redshifts of all galaxy clusters listed in the NASA Extragalactic Database (NED). September 9, 2003.

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Abstract:

A basic principle of the relativity theory states that all laws of physics remain unaltered, at all possible times and places in the universe. These precise laws, as we know them, hold neither inside black holes, nor during the first moments of a primitive big bang, which can only be thought of, as having occurred “elsewhere”, not in the real universe.

Virtually, the big-bang hypothesis suggests that, a tiny bubble of space, which first appeared from nowhere, was rapidly concurred by a sudden, ultra-condensed, tremendous flow, justifying an expansion so fast, that we are still wondering, billions of years after that violent incident, if the inflation of such a bubble will ever stop… accelerating. Ordinarily, we should be wondering, instead, where the principal laws came from. Since never changing, these laws were present within the initial vacuum. Then, how could a tiny ball contain the energy required for such a big bang? Yet, the so-called theory will continue to be popular, until a purely scientific explanation of the universal expansion is seriously attempted. Developed under a relativistic scope, the mathematical “Exormetism” is a novel theory based on few cosmological principles, including Fred Hoyle’s continuous creation. A mainly mathematical approach, attempting to draw quantitative results from such principles isn’t found, to my knowledge, anywhere in the literature.

The idea, inhere, is that repulsive gravity dominates the intergalactic space, which is forced thereby to be curved negatively, while positive gravity prevails in the neighborhood of galaxies, quasars, black holes, or any objects of considerable mass. Should we consider the average density of mass in space (great-scale homogeneity), the outcome would be a gravitational equilibrium, asserting that the universe is Euclidean (and, therefore, infinite). The respective proof is solidly based on a most general version of the cosmological principle, and implements formally the theory of relativity. This theory, in fact, dissuades the night-sky paradox (Olbers’ paradox), which would apply in geometrically infinite space, if the universe was static, or even if the rate of its expansion was so slow, that one could ignore both Doppler’s effect and time dilation (between subsequent light emissions, by the remote sources). In particular, the power radiated to Earth, by a source moving fairly fast, away from the observer, is just a fraction of the power that would be transmitted to Earth from a resting source of the same size, at the same distance. This is why the total power radiated to our planet by the entire universe, as factually calculated in the theory of exormetism, is perfectly finite, and this makes the night sky look so dark!

Considering the universal expansion, totally linear space-time implies that, besides negative gravity, which fills the intergalactic space, there also exists some other universal force, effectively carrying the galaxies away, and away from each other. The subsequent expansion, however, could eventually reduce the average mass density, thus working in favor of a repulsive field, since deforming hyperbolically the geometry of space. This undesired evolution is, amazingly, being suspended by the balancing consequences of another cosmological process:

As we already assumed, negative curvature characterizes extensive regions of ultra-low mass density. In such regions, creation of new particles may just occur, as a reaction of space to a certain level of hyperbolic deformation. Wherever this occurs, the mass density increases, on the expense of field-energy being released locally. This eternal process is such to retain steady the gravitational equilibrium in the universe. What acts rather as a catalyst, is a dark force resulting from accumulative pressure, exercised by the energy of vacuum, over a long distance. This energy is ethereal (mass-less) field-energy and prevails in areas negatively curved, ought to the absence of ordinary matter. Beyond a critical point, the hyperbolic deformation of space relaxes by turning part of that energy into particles of matter, so as to keep its concentration perfectly controlled.

Fig 1: Distribution of all known galaxy clusters (2003) along the axis of redshift ratios.

1. Introduction

1.1. Announcing Exormetism.

Merely introduced as the 5^th force in nature, F_ε=Њ·m_o·υ_ε (where υ_ε is velocity, m_o the mass at rest, and Њ is Hubble’s constant) is directly proportional to the Newtonian “horme” J_o=m_o·υ_ε (“ορμή”=momentum, in Greek) of a body being pushed away by this force alone. As a matter of fact, both of the vectors F_ε and J_o point steadily to the invariant direction of the velocity υ_ε, and adjust simultaneously their measures |F_ε| and |J_o| to the instantaneous |υ_ε|.

The force F_ε was initially called “exormetic”, because it can be extracted, directly, from (“εξ”: ex) the momentum J_o (“ορμή”: horme). However, the formally arising composite word, in its Greek version, makes more sense as “thrusting out”, which happens to describe rather well the fundamental force being introduced by the theory of “exormetism”. Since causing the phenomenon of recession, that force will be called, after all, exormetic, as if this term was synonymous to “remitting away”.

Normally, the value of J_o is less than the relativistic momentum J=m·υ, where m=m_o/Ö(1-υ²/c²), while υ is generally different from the exormetic υ_ε. Indeed, υ=υ_ε if that body, of instantaneous velocity υ, was never (in the infinite past) influenced by forces of any type, other than F_ε. It is also worth mentioning, that even under these conditions, a hypothetical observer permanently attached to that body, will assert that he never experienced the slightest acceleration!

The outcome is that force F_ε acts on a certain body only with respect to an observer located sufficiently far from that body. What generates this “paradox” is the natural cause of appearance of the exormetic force, namely the continuing creation of particles in the cosmos. This steady “rain” of particles, falling allover the infinite ocean of the universe, starts flowing, as being blown away from the observer –so as to keep unaltered the average depth (denoting mass density), while this also applies to any other observer –since conditions are the same everywhere (by the principle of relativity). Everyone around is, obviously, flowing away from the observation point, as being blown with strength proportional to a slowly increasing velocity. However, our observer is quite sure –judging by his own experience– that nobody, among the others around, feels a slight pressure ought to the action of some dark force!

Having served the purpose to justify the term “exormetism”, F_ε and υ_ε will be symbolized hereon, without their indices (i.e. as F and υ, respectively). Hence, the formula F_ε=Њ·m_o·υ_ε, which gave this theory a name, will be appearing as: F=Њ·m_o·υ.

1.2. The Concept of Signed Gravity.

The law of exormetism, as described above, requires the existence of antigravity, for universal gravity to be balanced, by a force of similar nature. This introduces the concept of “signed gravity”, which makes it rather unnecessary for antigravity to be theorized as a separate fundamental force. On the other hand, the need for gravity to become signed changes the conception of its nature, from “convexity of space”, to “degree of curvature”, which generally admits signed values.

It must be pointed out that, positive gravity represents low-dense field-energy, while antigravity corresponds to field-energy of density higher than a critical point. It is not wise to describe ordinary gravity as “negative energy”; this would complicate things, since the equivalence E=m·c², would lead to “negative mass”, which makes no sense. By this theory, however, ordinary energy (or mass) competes, for vital space, with its “ethereal” equivalent, which is field-energy (mass-less energy). A high concentration of ethereal energy arises in areas of low mass-density, controlled by negative gravity; and, a high mass density makes the ethereal energy poor, in favor of positive gravity.

1.3. The Principles of Exormetism.

Pr1. The Extended Cosmological Principle

The Hubble’s constant Њ , the velocity of light c , and the average resting-mass density ρ , are all absolute constants, throughout the space-time. In addition, the universe is, roughly, homogenous and isotropic, at all times.

Notes: Њ and c are not mentioned in the, so-called, “Perfect cosmological principle”.

The big-bang hypothesis is contradicted by either of the above two versions.

Pr2. The Principle of Relativity

All observation points in space-time, and all motion conditions, are equivalent, as far as the fundamental laws of nature are concerned.

Note: Again, the initial conditions of a big bang do not comply with this principle.

Pr3. The Principle of Ethereality

Creation of new particles occurs, on the expense of “saturated field-energy”, as a reaction of space to its extreme deformation, in extensive areas of ultra-low mass density. Field-energy is ethereal (mass-less), and its concentration varies, from zero, where the highest attractive gravity occurs (on the boundary of every black hole), to a positive maximum, representing full-strength antigravity (where generation of new particles starts happening). Thereby, at a certain concentration of “ethergy” (ethereal energy), both gravitational forces vanish, and the space becomes, locally, Euclidean.

Notes:

a) Gravity appears only near comparatively low concentrations of ethergy, this flowing, when remote, towards the observer (contrary to ordinary matter), and uniting in regions of increasing antigravity, as if aiming to create matter.

b) Antigravity is repulsive, and equivalent to hyperbolic deformation of space. (On the contrary, gravity is attractive, and quite equivalent to convex space.)

c) Elimination of particles occurs where the convexity of space reaches a certain limit. In the neighborhood of black holes (massive areas of collapsed matter), the presence of existing ethergy is insignificant. (On the opposite extreme, creation of particles occurs whenever condensation of ethergy takes place, in highly deformed areas, dominated by maximal antigravity.)

Pr4. The Principle of Spatial Expansion

According to observational evidence, the galaxies and other remote sources keep flying away from our galaxy and away from each other. The recessing velocity υ, at a distance r, depends on Hubble’s constant Њ: υ/r ®Њ (theoretically), as r®0₊ .

1.4. Arguing on a Pending Theorem.

It will be proven, eventually, that the equilibrium of gravity/antigravity is a consequence of the preceding 4 principles. Until then, we will assume that the universal expansion (stated in Pr4) is not caused by prevailing (on the average) antigravity, which is the only alternative to the existence of a 5^th fundamental force, as indirectly implied by the following two arguments:

1^st. The possibility of an expansion ought to randomly scattered, in space-time, explosions, in a theoretically convex, or even Euclidean universe, can be ruled out, by the observed rarity and ineffectiveness of such explosions, seeming only to affect the motion of galaxies being directly involved.

2^nd. Alternative repulsion, resulting from pressure caused by radiated particles, including photons, is just too weak to account for the observed recession of galaxies.

2. General Implications

2.1. Relating Velocities with Distances.

Consider, in a single direction from our point of view, at distances r-υ_A·δt/2 and r+υ_B·δt/2, where r>>0 and δt®0₊, two typical moving points, respectively A and B, having remitting velocities υ_A and υ_B. Then: υ^-2=c^-2+(Њ·r)^-2,

where υ=(υ_A+υ_B)/2 and Њ>0: υ/r®Њ as r®0₊ (when r>>0 is certainly rejected).

Proof

The moving point A, as it was selected, represents all matter in the volume bounded by two expanding spherical surfaces, both centered at our observation point, the inner one starting with radius r-υ_A·δt, and the other having, initially, radius r. The volume of such a thin shell, is then:

V_A=4·π·(r-υ_A·δt/2)²·υ_A·δt@4·π·r·(r-υ_A·δt)·υ_A·δt . (1)

Because point A is accelerating outwards, it will have reached the average velocity υ>υ_A, by the time it develops the instantaneous υ_B>υ. This limited time, which is δt=(υ_A·δt/2+υ_B·δt/2)/υ , will be just enough for that expanding shell to increase its thickness, as enough new particles will be created in that remote mass, for its volume to grow by: δV=V_B-V_A@4·π·r·((r+υ_B·δt)·υ_B-(r-υ_A·δt)·υ_A)·δt Þ

ÞδV/δt@4·π·r·(r·δυ+(υ_A²+υ_B²)·δt)@4·π·r·(r·δυ+2·υ²·δt) , (2)

where δυ=υ_B-υ_A, and δV/δt denotes the rate at which the current average volume V=(V_A+V_B)/2@4·π·r²·υ·δt grows.

(2)Þ(δV/δt)/V@δυ/(υ·δt)+2·υ/rÞd(lnV)/dt=(dV/dt)/V=dυ/dr+2·υ/r , (3)

where V=V_o·Ö(1-υ²/c²) reflects the relativistic contraction of distances, by which the respective local volume V_o must be such that:

d(lnV)=(dV/V_o)/Ö(1-υ²/c²)=d(lnV_o)-(υ/c²)·dυ/(1-υ²/c²) . (4)

Besides, dt=dђ/Ö(1-υ²/c²) , (5)

where ђ is the local time ordinate (a notation preferred to t_o, when υ is non-constant) increasing with –but not as fast as– the relativistic time t : υ=dr/dt . (6)

By (5), relation (4) becomes: d(lnV)/dt=Ö(1-υ²/c²)·d(lnV_o)/dђ-(dυ/dt)·υ/(c²-υ²) ,

which, by (6), leads to: d(lnV)/dt=Ö(1-υ²/c²)·d(lnV_o)/dђ-(dυ/dr)·υ²/(c²-υ²) , (7)

By (3) and (7), we have: Ö(1-υ²/c²)·d(lnV_o)/dђ=(dυ/dr)/(1-υ²/c²)+2·υ/r , (8)

where d(lnV_o)/dђ=3·Њ , (9)

since υ²/c²®0 , dυ/dr®Њ , and υ/r®Њ , as r®0₊ .

Obviously, (8) is equivalent, because of (9), to the following differential equation:

3·Њ·(1-υ²/c²)^3/2=2·(1-υ²/c²)·υ/r+dυ/dr , (10)

where r=r(υ): r(0)=0 is the initial condition, for any Њ>0 , after which:

If dr/dυ=(1-υ²/c²)^-3/2/Њ "υÎ[0,c) (11)

Ûr=r(υ)=(υ/Њ)/Ö(1-υ²/c²)+r_c , for a constant r_cÎÂ: r(0)=0 , i.e. for r_c=0

Ûr=(υ/Њ)/Ö(1-υ²/c²) "υÎ[0,c) , (12)

then (12)Þr=r(υ) , (13)

and (10) Û3·Њ·(1-υ²/c²)^3/2=2·(1-υ²/c²)·Њ·Ö(1-υ²/c²)+Њ/(1-υ²/c²)^-3/2,

Û3=2+1 , which is true "υÎ[0,c) , and for any Њ>0 .

Hence, (12) is a solution of (10).

Assume that q=q(υ) "υÎ[0,c) is another solution of (10), whereby:

$υ_oÎ[0,c): υ_o=max{υ_όÎ[0,c): q(υ)=r(υ) "υÎ[0,υ_ό]} .

Then, $ε>0: (dq/dυ-dr/dυ)·(q(υ)-r(υ))>0 "υÎ(υ_o,υ_o+ε) ,

which implies, because of (13), that:

2·(1-υ²/c²)·υ/q(υ)+dυ/dq¹2·(1-υ²/c²)·υ/r(υ)+dυ/dr=3·Њ·(1-υ²/c²)^3/2, according

to (10), contradicting the assumption that q=q(υ) is a solution of (10) "υÎ[0,c) .

In conclusion, (12) is the only solution of (10), which proves that:

(10)Û(12)Ûυ^-2=c^-2+(Њ·r)^-2 "r>0 (and υ=0 for r=0), given Њ>0 (14)

(whereby, υ/r®Њ as r®0 , and υ®c as r®+¥) .

2.2. The Law of Exormetism.

In the case of a gravitational equilibrium, the exormetic (=remitting away) force is just F=Њ·m_o·υ , where: υ is the exormetic velocity of a source, m_o is its mass at rest,

and Њ is Hubble’s constant.

Proof

To express all quantities as functions of a common variable, we consider the Doppler effect on light, by which the red-shift ratio, here, is: μ=Ö(1+υ/c)/Ö(1-υ/c) . (15)

By (15), we have: υ=c·(μ²-1)/(μ²+1) . (16)

By differentiating (16), we get: dυ=4·c·μ·dμ/(μ²+1)². (17)

Additionally, it is implied by (16) that: Ö(1-υ²/c²)=2/(μ+1/μ) . (18)

By replacing (18), and υ from (16), in (12), we get: r=(μ-1/μ)·c/(2·Њ) . (19)

By differentiating (19), we obtain: dr=(1+1/μ²)·dμ·c/(2·Њ) . (20)

By the definition of instantaneous velocity: dt=dr/υ . (21)

By the definition of acceleration, (21) becomes: dυ/a=dr/υ . (22)

Now, (22) is solved as for a, by use of (17) and (20): a=8·Њ·υ/(μ+1/μ)³. (23)

If m_o is mass at rest, by (18), the relativistic mass is: m=m_o·(μ+1/μ)/2 , (24)

while the longitudinal mass m_l=m_o·(1-υ²/c²)^-3/2 is: m_l=m_o·(μ+1/μ)³/8 . (25)

The exormetic force, causing acceleration a, is then: F=m_l·a=Њ·m_o·υ , (26)

by the theory of relativity, which completes the proof.

It is interesting to compare formula (19) with: r*=υ/Њ=(c/Њ)·(μ²-1)/(μ²+1) , which is in accordance with the big-bang hypothesis. We have: r*/r=2/(μ+1/μ) ,

implying that: μ·r*/r®1 as μ®1 , while: μ·r*/r®2 as μ®+¥ .

For every μ>1, it can be proven that μ_r=μ·r*/r =μ_r(μ): 1<μ_r<2 .

It follows that the longest distances, of all to date recorded, have been systematically underestimated, since they resulted from calculations respecting the big bang. This is, probably, what makes most quasars resemble to be remote in space, sources from a distant past, having existed for a while, after the believed beginning of time.

An idea favored by the theory of exormetism, is that quasars could be exceptionably bright sources, rather uniformly distributed in space-time, only extremely rare (Appendix A).

2.3. About Size, Mass, and Time - Calculation of the Time Integrals.

It is clear that the universal expansion does not reduce the average mass density ρ (which would contradict Pr1), because natural particles are constantly being created, according to Pr3 (elimination of old particles, in black holes, is comparatively rare).

The creation of particles, however, does not occur inside or near remote bodies and galaxies, because the presence of ordinary gravity is forbidding. On the other hand, intergalactic matter, already existing in the neighborhood, is actually influenced by this gravitational field, and a certain amount keeps “falling” inside, until it unifies completely with the attracting source. This process is regarded as being irrelevant to the theoretical size, or mass, of a typical body. Relativity formally applies, as on time:

Replace υ from (16), and dr from (20), in (21):

dt=(1/(2·Њ))·((μ+1/μ)²/(μ²-1))·dμ . (27)

Integrate (27), and use (19) to extract: t=r/c-(1-1/μ+ln((μ+1)/(μ-1)))/Њ+τ , (28)

where τ is the constant of integration.

By (5) and (18), the local-time differential will be: dђ=(2/(μ+1/μ))·dt , (29)

Replace dt from (27), in (29), to obtain: dђ=(1/Њ)·((μ+1/μ)/(μ²-1))·dμ . (30)

Integrate (30), in order to provide: ђ=ln(r·Њ/(2·c))/Њ+τ , (31)

where the constant of integration is identical to the τ used in relation (28),

for the difference Δt=t-ђ to be: Δt=r/c-(1-1/μ+ln((μ+1/μ+2)/4))/Њ , (32)

so that, Δt: Δt®0, as μ®1 (and r®0).

2.4. A Theorem Stating the Gravitational Equilibrium.

The observed expansion (see Pr4) is not caused by universally prevailing antigravity, and, consequently, the space is, on the average, Euclidean (hence, infinite).

Proof

The conclusion of this theorem is based on a couple of reasonable arguments, already submitted in “Arguing on a Pending Theorem”. As for its main statement, this will be proven by contradiction, through the following analysis:

If antigravity prevails, then the space is, theoretically, hyperbolic. (33)

The volume of any spherical shell, defined by r³r_i: r£r_o , for given r_i,r_o: 0<r_i<r_o ,

is greater in a hyperbolic, than in the Euclidean space (say, V_h>V_E).

By working on spherical volumes, in both spaces, we get: V_h/V_E®¥ as r_i®¥ . (34)

In Euclidean space, the exormetic velocity υ=υ(r)®c as r®+¥ , for any Њ>0 . (35)

By (9), the local rate of spatial expansion is: (dV_o/dђ)/V_o=d(lnV_o)/dђ=3·Њ , (36)

even in a hyperbolic space, since any curved space behaves, locally, as Euclidean.

By (34) and (36), the generalization of (35), in an infinite universe, takes the form:

$ç>0: υ=υ(r)®ç=ç(Њ,R^-2)£c as r®+¥ , for any Њ>0, R^-2£0 . (37)

In particular, if R^-2<0, then: υ=c·sinAtan(3·(Њ/c)·R₂·(sinh(r/R₂)-r/R₂)/(cosh(r/R₂)-1))

and ç=((3·Њ·R₂)^-2+c^-2)^–1/2, where R₂=Ö(-R^-2)/2 . (38)

Proving these last formulae may (harmlessly) be omitted, since there is no reasonable doubt, that in this case, the universe is hyperbolic and, therefore, supposed to satisfy:

υ=υ(r)®ç=ç(Њ,R^-2)<c as r®+¥ , for any Њ>0, R^-2<0 , (39)

from which it follows, by (5), that: dђ/dt®Ö(1-c_R²/c²)>0 as r®+¥ . (40)

In a hyperbolic space, when light is emitted around, by a remote source, it is expected to reach, eventually –ought to (39) and (40)– any observer in the universe, no matter how far from us, or from that source, that observer may instantly be. (41)

Consider two remote sources, A and B, in opposite directions, with respect to our observation point P, so that light emitted by A passes from P, on the way to B.

Let B be much nearer to P, than A, so that light emitted by A, from distance r_A, finally reaches B, when its increasing distance from P, gets equal to r_A.

If μ_A is the red-shift of that light, when it passes from P, then we may analyze μ_A as:

μ_A=μ_a·μ>1 , (42)

where μ>1 is only Doppler, while μ_a>1 is due to an increasing gravitational potential.

Eventually, in the distant future, that same light catches up with B. Over there, a hypothetical observer, moving with B, would then realize the total red-shift as being:

μ_t=μ², (43)

because: the distance r_A of the emitting source from P, at the time of emission, equals the observer’s (r_A) at B, when he receives that light. Hence, there can be no energy gains or losses, arising from different gravitational potentials. Indeed, the considered potentials must be the same, for the universal isotropy and relativity to be respected.

Now, assume that the observer at B, receives, simultaneously with that light (coming from A, via P), a beam directly emitted from P. It follows from (42) and (43), that this beam will be red-shifted by: μ_B=μ_t/μ_A=μ/μ_a>1 . (44)

But we may, also, analyze μ_B as: μ_B=μ_b·μ>1 , (45)

where μ_b=1/μ_a<1 arises, inevitably, because of a decreasing gravitational potential.

By Doppler, considering (39), it is: μ=Ö(1+υ/c)/Ö(1-υ/c)<μ_o , (46)

where μ_o=Ö(1+ç/c)/Ö(1-ç/c) is finite. (47)

By (44) and (46), we realize that: 1<μ_a<μ<μ_o , (48)

Set, next, μ_α: μ_a®μ_α as r_A®+¥ .

Then, by (39), υ®ç , hence by (46), μ®μ_o , and by (42), μ_A®μ_α·μ_o>μ_o as r_A®+¥ , (49)

while, by (44), μ_B®μ_o/μ_α<μ_o as r_A®+¥ . (50)

On the other hand, it is obvious that: μ_A=μ_A(r_A) and μ_A(0)®1 , (51)

and we know that: μ_B=μ_B(r_A) and μ_B(0)®1 . (52)

By (49) and (51), 1<μ<μ_α·μ_o represents the full range of red-shifts affecting light we (theoretically) receive, from the entire universe.

By (50) and (52), 1<μ<μ_o/μ_α represents the full range of red-shifts, at which the light emitted by our Milky Way, is expected to reach other galaxies in the infinite space.

But (48)Þμ_a>1Þμ_α·μ_o>μ_o/μ_α , meaning that the “maximums” of μ, in the above two statements, are not equal. This inequality, taken as an addendum to statement (41), is certainly in contradiction with Pr2 (the relativity principle). Consequently, we must reject prevailing antigravity, by exempting (33), which resulted to inconsistency.

Having stated, by contradiction, the gravitational equilibrium, we may now safely conclude that the universe is, on the average, Euclidean (hence, infinite). (53)

2.5. On the Cosmological Factor Λ.

To keep the tradition alive, we must set Λ=Λ(μ): F=Λ·m·c²·r Þ Λ=F/(m·c²·r) . (54)

By replacing in (54), F, m, r, and υ, from (26), (24), (19), and (16), respectively, we get:

Λ=(Њ/c)²·4/(μ+1/μ)², (55)

which is, clearly, non-constant.

However, (26) implies that: F=(Њ/c)²·m_o·c²·υ/ЊÞ F=Λ_o·m_o·c²·r*, (56)

where r*=υ/Њ is a distance estimation by the big-bang, and Λ_o=(Њ/c)² is a constant.

In fact, when r®0 Þ υ®0 Þ m® m_o and, by (12), r/r*®1 . Hence: Λ®Λ_o as r®0 .

For comparison reasons, consider Λ*=Λ*(μ): F*=Λ*·m·c²·r₁ÞΛ*=F*/(m·c²·r₁) , (57)

where F* is the remitting force at distance r₁=r₁(A) , A being the age of the universe.

According to a simple big-bang model, that will be discussed elsewhere, there is no reason for a remitting force to exist, because υ, i.e. the particular velocity of a specific source, never changes (unlike its condition), and r₁=υ·Α , which explains why, today, the universe has age A@1/Њ . This leads to F*=0 , and considering (57), Λ*=0 , as well. In fact, the absence of F* implies the existence of weakening antigravity, to equalize the presence of gravity, in a relativistic universe of immutable total mass (Û energy).

By this last principle, in the refined model of big bang, the Newtonian “constant” G increases according to A², for the gravity law to function properly, when r₁/A is tiny.

2.6. Population of Substantive Entities, Emitting Light Red-Shifted Less than μ.

The term “substantive entity” means a considerably massive, or significantly active, concentration of matter, having any form, luminosity, or structure (even collapsed, as in black holes), and occupying a definite space, apart from other such concentrations.

The population of any specific, well defined, class of substantive entities, in a sphere of radius r=r(μ) , is theoretically proportional (~) to: Ñ~(μ⁴-μ^-4)/4-ln(μ²) .

Proof

If we set χ=ln(μ⁴)Ûμ⁴=exp(χ), and since Ñ=Ñ(χ): Ñ(0)=0 , it suffices to show that:

Ñ~(sinh(χ)-χ)/2Û4·(dÑ/dχ)~2·cosh(χ)-2~μ^-4-2+μ^-4=(μ²-μ^-2)²·(dχ/dμ)·dμ/dχ Û

Û4·(dÑ/dμ)~(μ²-μ^-2)²·d(ln(μ⁴))/dμ=(μ²-μ^-2)²·4·μ³/μ⁴~4·(μ²-μ^-2)²/μ=4·(μ⁴-1)²/μ⁵Û

ÛdÑ/dμ~(μ⁴-1)²/μ⁵. (58)

By relativity, the population density, say p_ο , rises as in: dÑ~r²·dr·p_ο/Ö(1-υ²/c²) . (59)

By (18) and (59): (58)Û(μ²-1)²·(μ²+1)/μ⁴~p_ο·r²·dr/dμ=(μ-1/μ)²·(1+1/μ²)·p_ο·c³/Њ³ Û

Û1~p_ο·c³/Њ³, which is always true (trivial proportionality),

as is, accordingly, its equivalent: Ñ~(μ⁴-μ^-4)/4-ln(μ²) . (60)

Note that, the just proven relation is not referring to the observable entities, which are the brightest, among the physically uncovered sources, as appearing in our times.

2.7. The Coverage C=C(μ), and its Density C’=dC/dμ.

We intend to show that the sky is totally covered by substantive entities, according to a particular distribution C=C(μ). Let us begin with analyzing the density of coverage: C’=dC/dμ~(c/Њ)²·n’/r², where n’=dn/dμ , while n=n(μ) is the apparent population, of not covered by others (in the way) entities, regardless of brightness, in a sphere of radius r=r(μ), centered at the point of observation. The delaying view, imposing (62), below, leads inevitably to: C=1-exp(K·(1-μ²))/μ^2·K, where K>0 is a universal constant. Although its value is still unknown, assuming arbitrarily that K=0,03 serves our purpose rather well, at least in demonstrating graphically some theoretical results.

Proof

What we have to prove is equivalent to: ln(T)=ln(1-C)=K·(1-μ²-ln(μ²)) , (61)

where T=1-C is the proportion of the sky left uncovered by entities appearing closer than r . By (16), and due to homogeneity: dn/dÑ=(1-C)·(1+υ/c)=T·2/(1+μ^-2) , (62)

which is multiplied by (58), to provide: n’=dn/dμ=2·T·(μ⁴-1)·(μ²-1)/μ³. (63)

Besides, it is known that the solid angle, under which a certain object is observed, is inversely proportional to the square of its distance. So, all entities being at distance r , contribute in the total coverage, just by: dC~(c/Њ)²·dn/r²~dn/(μ-1/μ)², (64)

according to (19).

By (63) and (64), we extract dT/dμ=d(1-C)/dμ=-dC/dμ~-2·Τ·(μ+1/μ) ,

which leads to: dT/T~-2·(μ+1/μ)·dμ Þ d(lnT)~-2·(μ+1/μ)·dμ . (65)

By integrating (65), we have: ln(T)~-μ²-2·ln(μ)+1 , (66)

where the constant of integration was set to +1 , so that T=1 for μ=1 .

But, (66) ascertains us that there is a positive constant, say K>0, for which (61) is true.

In other words, we have proven that: C=1-T=1-exp(K·(1-μ²))/μ^2·K, (67)

which leads, through (61), to ln(1-C)®^–¥ as μ®+¥ , that is: C®1 as r®+¥ . (68)

Therefore, the sky is totally covered by substantive entities.

By (67), the density of coverage is, simply: C’=dC/dμ=2·K·(1-C)·(μ+1/μ) . (69)

3. A Prospective Comparison

3.1. Definition of a Refined Big-Bang Model.

There is a remarkable coincidence concerning the Hubble’s constant, estimated to be 71,4 km·s^-1/Mpc@0,073 GY^-1=Њ, contrary to “other evidence” suggesting that the age of the universe, say A, is today A=A_o@13,7 GY. The relation A_o·Њ@1 makes it hard for the traditional theory to explain why observation does not conform with A·3/2@1/Њ , as it should (since 1/Њ is Hubble’s time, at present). Things may get even worse, in the future, if the universe is presently accelerating, meaning that υ₀=υ₀(Α) for any fixed distance r₀, eventually encountered by substantive entities of remitting velocity υ₀. To diverge just negligibly from the expected values of υ₀, concerning the recent past and the near future, we first introduce a refined, seemingly consistent, big-bang model, based on υ₀(A)=r₀/A "A³r₀/c . A fact following from this relation, is that every concrete entity has a constant velocity υ=r₁/A , where r₁=r₁(A) is its true distance from the observer, as occurring when A is the exact age of the universe.

We have chosen this big-bang version, because of its simplicity, to compare its results to those obtained by exormetism. There is, though, one more issue to settle, before we adopt this model. The theoretic distribution of entities (of the same age), in space, is not demanded to be uniform, but only to appear as such, from the observer’s point of view.

To comply with the above demand, the real population density, p₁, and the apparent one, p₂, must be two smooth functions p₁=p₁(A,r₁) and p₂=p₂(A,r₂), where 0<r₁<c·A and 0<r₂<c·A/2, each decreasing with time, while respecting the here introduced version of the cosmological principle:

$f₂=f₂(A): 1<f₂·p₂(A,r₂)<2 "(0<)r₂<c·A/2 . (70)

Over and above this first constraint, the refined-model definition implies, theoretically, that p₂=p₁·(1+υ/c)⁴, hence:

(70) Û p₁~(c·A+r₁)^-3Ù p₂~(c·A)^-2/(c·A-r₂) "(0<)r₂<c·A/2 .

3.2. Results and Predictions Based on the Big Bang.

Theorem 1

Starting with the coverage C* and its density C’*=dC*/dμ , we aim to prove that:

C*=1-exp(4·K·ln((1+μ^-2)/2)/(Њ·A)²) ,

and C’*=dC*/dμ=8·K·(1-C*)·(Њ·A)^-2/(μ³+μ) .

Proof

When superimposed by C*=C(μ)*: C(1)*=0 , the above relations are equivalent, since the derivative of C*=1-exp(4·K·ln((1+μ^-2)/2)/(Њ·A)²) Û (71)

happens to be, exactly: Û C’*=dC*/dμ=8·K·(1-C*)·(Њ·A)^-2/(μ³+μ) . (72)

To prove (72), we first realize that all uncovered entities, at distance r₂=A·υ/(1+υ/c) , are, say: dn*~(1-C*)·p₂·r₂²·dr₂~(1-C*)·((c·A)^-2/(c·A-r₂))·(A·υ/(1+υ/c))²·A·dυ/(1+υ/c)², by implementing p₂, as it resulted from (70). Then, according to (16) and (17), we get:

dn*/dμ~(1-C*)·(dυ/dμ)·υ²/(c+υ)³~(1-C)·(μ/(μ²+1)²)·((μ²-1)/(μ²+1))²/(μ²/(μ²+1))³,

yielding: n’*=dn*/dμ~(1-C*)·μ^-5·(μ²-1)²/(μ²+1) , (73)

which, by the way, implies that the population of all entities (including the covered), being “virtually” closer than r₂(μ), is: N*~2·(((6-μ^-2)·μ^-2-5)/8-ln((1+μ^-2)/2)) . (74)

Now, since the solid angle of observation is inversely proportional to r₂², we have:

C’*~(c/Њ)²·n’*/r₂²~(c/Њ)²·n’*/(A·υ/(1+υ/c))², which, by (73) and (16), leads to:

C’*=K_p·(1-C*)·(Њ·A)^-2/(μ³+μ) (75)

where the constant K_p must be such that C’*/C’®1 as μ®1 , just because, for short distances, the coverage and its density, as calculated by the two alternative theories, should be today, one by one, approximately the same. Given that A·Њ=1 , presently, by (75) and (69), we get: (K_p·(1-C*)·(Њ·A)^-2/(μ³+μ))/(2·K·(1-C)·(μ+1/μ))®1 as μ®1 , where (1-C*)/(1-C)®1 as μ®1 . Then, K_p=8·K , in (75), which proves (72) and (71).

Corollary

For the density of population n’* to be directly comparable to n’ (the respective result, by exormetism), as provided by (63), we multiply by 8 the right-hand member of (73), and replace C* from (71), in (73), in order to obtain:

n’*=n’(μ)*~8·exp(4·K·ln((1+μ^-2)/2)/(Њ·A)²) ·μ^-5·(μ²-1)²/(μ²+1) , (76)

Accordingly, (74) takes the form: N*~16·(((6-μ^-2)/μ²-5)/8-ln((1+μ^-2)/2)) . (77)

The integral population n*, being defined by:

òn’(μ)*·dμ =n(μ)*: n(μ)*®0 as μ®1 , (78)

can only be numerically approximated, since its analytic formula is still unknown.

Theorem 2

Let y*=ln((C’*/n’*)/μ²) be expressing the luminosity of a typical entity, at distance r₂=A·υ/(1+υ/c) . By retardation and Doppler, just 1/μ² of the initial power is left to fall in a solid angle C’*/n’*. We will prove that:

y*=y(μ)*=ln(K/(Њ·A·(μ-1/μ))²) . (79)

Proof

By (72) and (76), we have: y*=ln(K·(μ⁵/μ)/(Њ·A·(μ²-1)·μ)²)=ln(K/(Њ·A·(μ-1/μ))²) ,

which is identical to (79).

Result 1

Let μ be specific, and assume that y*=E(Y*), where Y*=Y(μ)* is a possible luminosity, following a normal distribution of standard deviation σ, it being independent from μ, unlike the average y*=y(μ)*. The ability to distinguish an entity of luminosity Y*, at distance r₂=A·υ/(1+υ/c), depends on the sensitivity of the observational instruments used to perceive the particular Y*. Let Ϋ be the threshold of luminosity, over which a source becomes detectable. Τhe number of observable entities, say κ’*=κ’(μ)*, will be κ’*<n’*, while, on the contrary, their average luminosity will rise to w*=E(Y*>Ϋ)>y*. Evidently, Y*=Y(μ)* implies that w*=w(μ)*. We obtain κ’* and w*, for various values of μ , from tables of the standard normal distribution, by using linear interpolation.

Result 2

The incoming electromagnetic power, reaching us at a specific red-shift μ , and being denoted by dP*, where P*=P(μ)*, is proportional, by (76) and (79), to the quantity:

dP*~exp(y*)·dn*=8·K·(μ/(Њ·A))²·exp(4·K·ln((1+μ^-2)/2)/(Њ·A)²)·μ^-5·dμ/(μ²+1) .

Hence: P’*=dP*/dμ~8·K·(Њ·A)^-2·exp(4·K·ln((1+μ^-2)/2)/(Њ·A)²) ·μ^-3/(μ²+1) . (80)

Τhe (converging) integral of that power: P*=òP’(μ)*·dμ =P(μ)*: P(μ)*®0 as μ®1 , (81)

can only be numerically approximated, since its analytic formula is still unknown.

3.3. Population of Uncovered Entities being Theoretically Red-Shifted Less than μ.

The integral population n, is just defined by: òn’(μ)·dμ =n(μ): n(μ)®0 as μ®1 , where n’=n’(μ), was provided by (63). If T is replaced, in (63), as obtained from (67), we get:

n’=n’(μ)~2·(μ-μ^-3)·(μ²-1)·exp(K·(1-μ²))/μ^2·K, (82)

the derivative of which is:

n”=dn’/dμ=(6·μ²-2+(2-6/μ²)/μ²-4·K·(μ²-μ^-2)²)·exp(K·(1-μ²))/μ^2·K. (83)

Now, pick a step for μ , say h , so little, that (for any μ>0) the value of n’=n’(μ) is well approximated through a cubic curve, inclined by n_k”=n”(μ_k) and n_k+1”=n”(μ_k+1), at (μ_k , n_k’) and (μ_k+1 , n_{k+1’), correspondingly, where k=[(μ-1)/h], μk}=1+k·h, μ_k+1=μ_k+h, n_k’=n’(μ_k), and n_k+1’=n’(μ_k+1).

Set n₀=0 and n_k=h·((n_k‘-n_k”·h/6)/2 + _j=1Σ_k-1 n_j’) "k>0 , so that n_k@n_k=n(μ_k) "k³0, since by (82) and (83), n₀’=n’(1)=0=n”(1)=n₀”. We’ll show, "k³0: μÎ[μ_k , μ_k+1] , that:

n=n(μ)=n_k+((((n_k”+n_k+1”-2·(n_k+1’-n_k’)/h)·(μ-μ_k)/4 +

+n_k+1’-n_k’-(n_k+1”+2·n_k”)·h/3)·(μ-μ_k)/h²+n_k”/2)·(μ-μ_k)+n_k’)·(μ-μ_k) @n=n(μ) . (84)

Proof

When superimposed by n(μ_k)=n_k , the approximate –to n=n(μ)– function n=n(μ) , as defined in (84), is reliable, if and only if, its derivative n’=dn/dμ approximates well n’=n’(μ) "μÎ[μ_k,μ_k+1] . Once, in its turn, n’(μ_k)=n_k’ is set as a prerequisite, n’=n’(μ) is close enough to n’, if and only if: n”=dn’/dμ@n”=n”(μ) "μÎ[μ_k,μ_k+1] . (85)

The derivative n’=dn/dμ , according to the formula of n=n(μ) , appearing in (84), is:

n’=n’(μ)=(((n_k”+n_k+1”-2·(n_k+1’-n_k’)/h)·(μ-μ_k) +

+3·(n_k+1’-n_k’)-(n_k+1”+2·n_k”)·h)·(μ-μ_k)/h²+n_k”)·(μ-μ_k)+n_k’. (86)

The 2^nd derivative of n(μ) , denoted by n”=n”(μ)=dn’/dμ , and arising from (86), is:

n”=(3·(n_k”+n_k+1”-2·(n_k+1’-n_k’)/h)·(μ-μ_k)+

+6·(n_k+1’-n_k’)-2·n_k+1”-4·n_k”)·h)·(μ-μ_k)/h²+n_k” (87)

Notice that, n”(μ_k)=n_k” and n”(μ_k+1)=n_{k+1”, both, follow from (87), which represents a
fairly short parabolic segment, well adjusted to confine, from above, a
vertical strip of area equal, exactly, to
h·(nk+1}’-n_k’) . Because n’(μ) is reasonably smooth, according to (82), and since h is sufficiently small, n’’=dn’/dμ does not vary considerably for μÎ(μ_k,μ_k+1).

Consequently, the parabolic approximation n”(μ)@n”(μ) "μÎ[μ_k,μ_k+1] is quite satisfactory. Besides, it follows from (86) that n’(μ_k)=n_k’ , while the formula of n=n(μ), in (84), verifies that n(μ_k)=n_k , proving, in fact, that n(μ)@n(μ) "μÎ[μ_k,μ_k+1] .

3.4. Population of “Virtual Entities” being Hypothetically Red-Shifted Less than μ.

The population of all entities (including the covered) virtually present in space-time, is only visualized by the observer, as being:

N=òN’(μ)·dμ =N(μ): N(μ)®0 as μ®1 ,

where N’=N’(μ)~n’/(1-C)~2·(μ-μ^-3)·(μ²-1) . (88)

In fact, N can be analytically expressed: N=N(μ)~(μ⁴+3)/2-μ²-μ^-2-ln(μ²) . (89)

Proof

Proportionality (89), since conforming to N(1)=0 , results from its derivative, that is:

N=N(μ): dN/dμ~4·μ³/2-2·μ+2/μ³-2·μ^/μ²=2·(μ³-μ+μ^-3-1/μ) =

=2·(μ·(μ²-1)+(1-μ²)/μ³)=2·(μ²-1)·(μ-1/μ³) ~N’=N’(μ) ,

according to (88), which, by equivalence, proves (89).

Corollary

It is implied, by (89) and (60), that 1<N(μ)/Ñ(μ)<2 "μ>1 , because:

N’(μ)/Ñ’(μ)=2·μ²/(μ²+1)=2/(1+μ^-2)>0 "μ³1 , while this ratio increases with μ , which implies that N(μ)/Ñ(μ) is also increasing, "μ³1 , starting with:

N(μ)/Ñ(μ)®N’(μ)/Ñ’(μ)®1 as μ®1 ; while, by (89) and (60), N(μ)/Ñ(μ)®2 as μ®¥ .

3.5. Theoretic Luminosity of a Typical Entity, at Red-Shift Ratio μ.

Let y=ln((C’/n’)/μ²) be expressing the theoretic luminosity of a typical entity, at distance r=r(μ). By retardation and Doppler, just 1/μ² of the initial power is left to fall in a solid angle C’/n’. We will prove that: y=y(μ)=ln(K/(μ²-1)²). (90)

Proof

By (69), (82), and (67), we have: y=ln(K·(μ+1/μ)·μ^-2/((μ⁴-1)·(μ²-1)/μ³)) =

=ln(K/(μ²-1)²) , which is identical to (90).

3.6. Apparent Luminosity of a Typical Entity

and Apparent-Population Density of Uncovered Entities, at Red-Shift Ratio μ.

Let μ be specific, and assume that y=E(Y), where Y=Y(μ) is a possible luminosity, following a normal distribution of standard deviation σ, it being independent from μ, unlike the average y=y(μ). The ability to distinguish an entity of luminosity Y, at distance r=r(μ), depends on the sensitivity of the observational instruments used to perceive the particular Y. Let Ϋ be the threshold of luminosity, over which a source becomes detectable. Τhe number of observable entities, say κ’=κ’(μ), will be κ’<n’, while, on the contrary, their average luminosity will rise to w=E(Y>Ϋ)>y. It is evident, since Y=Y(μ), that w=w(μ) . We obtain κ’ and w, for various values of μ , from tables of the standard normal distribution, by using linear interpolation. (91)

To give an example, on how κ’ and w are actually calculated, say x=(y-Ϋ)/σ=-7,487 .

Then, by linear interpolation, η=η₀+(η₁-η₀)·λ , where λ=(x-z₀)/s (s=15/4096 is given) ,

z₀=-7,489013671875 , η₀=0,000000000000003 , and η₁=0,000000000000004 . In this case, λ=1,6496/3 , hence η=1,06496·10^-14/3 , providing κ’=η·n’. Now, we apply, once more, linear interpolation, to obtain w=y-σ·(index₀+(index₁-index₀)·λ), for σ=1,5 (as likely to be), index₀=-7,4945068359375 , and index₁=-7,49267578125. Since y=Ϋ+σ·x=Ϋ-11,2305 , we have w=Ϋ+0,00975 , which verifies that w>y . Returning to Table 1, below, in the formula of β, the notation x_^ points to a preceding x , one row above the current.

Table 1: Implies the number κ’ of observable entities and their average luminosity w.

3.7. Apparent Population of Uncovered Entities, being Red-Shifted Less than μ .

The integral population κ , is just defined by: òκ’(μ)·dμ =κ(μ): κ(μ)®0 as μ®1 , where κ’=κ’(μ), is evaluated according to (91). Let κ”(μ)=dκ’/dμ be the derivative of κ’(μ). By its nature, κ’³0: κ’(μ)<n’(μ) "μ>1 , while n’(1)=n”(1)=0 , by (82) and (83) , as we have already seen. What obviously follows is that κ”(1)=0 . Unfortunately, for μ>1 , there is no accurate formula providing κ”(μ) . However, a good approximation, say κ”=κ”(μ) "μ>1: μ=1+k·h , where k=[(μ-1)/h], and h is the step used in (84), can serve, as well, our purpose to estimate κ(μ) , for whichever μ>1 . Based on polynomials of the 4^th degree, for μ_k=1+k·h and k³0 , we define, say, κ_k”=κ”(μ_k) , formulated as:

κ₀”=0 , κ₁”=(κ₁’+κ₂’-κ₃’/9)/(2·h) ,

and κ_k”=(8·(κ_k₊₁’-κ_k_-1’)-κ_k₊₂’+κ_k_-2’)/(12·h) "k>1 , (92)

where the notation κ_j’=κ’(μ_j) "j³0 was adopted.

Next, we may set κ₀=0 and κ_k=h·((κ_k‘-κ_k”·h/6)/2 + _j=1Σ_k-1 κ_j’) "k>0 , as we did to approximate n=n(μ) by n=n(μ) . Evidently, κ_k@κ_k=κ(μ_k) "k³0 . We intend to show, "k³0: μÎ[μ_k , μ_k+1] , that: κ=κ(μ)=κ_k+((((κ_k”+κ_k+1”-2·(κ_k+1’-κ_k’)/h)·(μ-μ_k)/4 +

+κ_k+1’-κ_k’-(κ_k+1”+2·κ_k”)·h/3)·(μ-μ_k)/h²+κ_k”/2)·(μ-μ_k)+κ_k’)·(μ-μ_k) @ κ=κ(μ) . (93)

Proof

When superimposed by κ(μ_k)=κ_k , the approximate –to κ=κ(μ)– function κ=κ(μ) , as defined in (93), is reliable, if and only if, its derivative κ’=dκ/dμ approximates well κ’=κ’(μ) "μÎ[μ_k,μ_k+1] . Once, in its turn, κ’(μ_k)=κ_k’ is set as a prerequisite, κ’=κ’(μ) is close enough to κ’, if and only if: dκ’/dμ@κ”=κ”(μ) "μÎ[μ_k,μ_k+1] . (94)

The derivative κ’=dκ/dμ , according to the formula of κ=κ(μ) , appearing in (93), is:

κ’=κ’(μ)=(((κ_k”+κ_k+1”-2·(κ_k+1’-κ_k’)/h)·(μ-μ_k) +

+3·(κ_k+1’-κ_k’)-(κ_k+1”+2·κ_k”)·h)·(μ-μ_k)/h²+κ_k”)·(μ-μ_k)+κ_k’. (95)

The 2^nd derivative of κ(μ) , denoted as κ”=κ”(μ)=dκ’/dμ , and arising from (95), is:

κ”=(3·(κ_k”+κ_k+1”-2·(κ_k+1’-κ_k’)/h)·(μ-μ_k)+6·(κ_k+1’-κ_k’)-2·κ_k+1”-4·κ_k”)·h)·(μ-μ_k)/h²+κ_k” (96)

Notice that, κ”(μ_k)=κ_k” and κ”(μ_k+1)=κ_{k+1”, both, follow from (96), which represents a
fairly short parabolic segment, well adjusted to confine, from above, a
vertical strip of area equal, exactly, to
h·(κk+1}’-κ_k’) . Because κ’(μ) is reasonably smooth, according to (91), and since h is sufficiently small, κ’’=dκ’/dμ does not vary considerably, for

μÎ(μ_k,μ_k+1). Consequently, the parabolic approximation κ”(μ)@κ”(μ) "μÎ[μ_k,μ_k+1] is quite satisfactory. Besides, it follows from (95) that κ’(μ_k)=κ_k’ , while the formula of κ=κ(μ) , in (93), verifies that κ(μ_k)=κ_k , proving, finally, that κ(μ)@κ(μ) "μÎ[μ_k,μ_k+1] .

3.8. The Incoming “Photonic Power” P=P(μ), and its Density P’=dP/dμ .

The incoming electromagnetic power, reaching us at a specific red-shift μ , and being denoted by dP , where P=P(μ) , is proportional, by (82) and (90), to the quantity:

dP~exp(y)·dn=2·K·exp(K·(1-μ²)) ·(μ²+1)·dμ/μ^2·K+3.

Hence: P’=dP/dμ~2·K·exp(K·(1-μ²)) ·(μ²+1)/μ^2·K+3. (97)

The (converging) integral power P, is òP’(μ)·dμ =P(μ): P(μ)®0 as μ®1 , and can only be approximately calculated, by exploiting its 2^nd derivative P”=P”(μ)=dP/dμ , as resulting from (97): P”=-2·K·exp(K·(1-μ²)) ·(μ²+3+2·Κ·(μ²+1)²)/μ^2·K+4. (98)

For μ_k=1+k·h , where h is the chosen step and k=[(μ-1)/h], we set, as usual, P_k=P(μ_k), P_k’=P’(μ_k), and P_k”=P”(μ_k), noting that P₀’=-4·K and P₀”=-8·K·(2·K+1) . Then, for P₀=0 and P_k=h·((P_k‘-4·K-(P_k”+8·K·(2·K+1))·h/6)/2 + _j=1Σ_k-1 P_j’) "k>0 Þ P_k@P_k=P(μ_k) "k³0.

In fact , "k³0: μÎ[μ_k , μ_k+1] Þ P=P(μ)=P_k+((((P_k”+P_k+1”-2·(P_k+1’-P_k’)/h)·(μ-μ_k)/4 +

+P_k+1’-P_k’-(P_k+1”+2·P_k”)·h/3)·(μ-μ_k)/h²+P_k”/2)·(μ-μ_k)+P_k’)·(μ-μ_k) @ P=P(μ) (99)

Proof by Reference

(99) can be proven similarly to the statements κ @κ , above, and n @n , elsewhere.

Fig 2: Comprehensive representation of quantities affected by the law of exormetism.

Fig 3: Comparison of calculated quantities to those resulting from a big bang (2003).

4. Complement and Conclusion

4.1. Estimations of Distances and Ages of Extragalactic Sources, according to

the Refined Big-Bang Model, versus Exormetism.

1^st Example

Let μ=1,0350 be the redshift ratio (wavelength received over wavelength emitted).

Then the velocity of the remote source is: υ=c·(μ²-1)/(μ²+1) . (1)

But, the refined model suggests that: υ/c=r/R , (2)

where r is the unknown instantaneous distance of the source, and R denotes the radius of the expanding space, at the time of emission, when the universal horizon, steadily recessing by the speed of light, was closer than today.

It follows from (2), because of (1), that: r=R·(μ²-1)/(μ²+1) , (3)

and, since the present age of the universe is A_o=13,69863 GY, that source emitted its light when the age of the cosmos was t=13,69863 GY-r/c=A_o-(R/c)·(μ²-1)/(μ²+1) . (4)

But, t is also the time in which the distance r was covered by the moving object, of constant velocity υ. Then, (2) implies that: t=r/υ=R/c , (5)

which, when combined with (4), yields: R=A_o·c·(μ²+1)/(2·μ²) . (6)

Now, (3) provides the distance: r=A_o·c·(1-μ^-²)/2=455,41 McY , (7)

(millions of light-years)

The age of the observed source is, then: t_o=t·Ö(1-υ²/c²)=(R/c)·Ö(1-r²/R²) ,

which simply leads to: t_o= A_o/μ = 13,23539 GY . (8)

(billions of years)

2^nd Example

Let the redshift ratio be μ=5,0500 .

Then, by using (7) and (8), we get: r=A_o·c·(1-μ^-²)/2=6,58074 GcY ,

and t_o= A_o/μ = 2,71260 GY .

The correct distance, according to the theory of exormetism, is: r_p=A_ό·r/t_o=μ·r , where A_ό is same with A_o , only thought of, here, as a constant, right after its inverse, that is Hubble’s Њ. Hence, in the above examples, r_p=471,35 McY when μ=1,0350 , and r_p=33,23274 GcY in the case of μ=5,0500 .

Regarding the average natural age of a source, as realized when its light was emitted, this average is the constant 1/(3·Њ), no matter the objective distance. To verify this, we agree to express the natural age as Δђ=ln(r_p/r_c)/Њ, where r_c is our distance from the exact location where a particular source, now appearing at distance r_p, was previously created.

Indeed, the expression Δђ=ln(r_p/r_c)/Њ arises from implementation of a formula providing the local time, according to the theory of exormetism:

ђ=ln(r·Њ/(2·c))/Њ+τ , where c, Њ, and τ, are all known constants. Hence ђ=ђ(r), which is used only to define the natural age, in the aim to prove the intended result.

To start with, let us adopt the notations: ђ_p= ђ(r_p) and ђ_c=ђ(r_c).

Then, Δђ=ђ_p-ђ_c=(ђ_p-τ)-(ђ_c-τ)

=ln(r_p·Њ/(2·c))/Њ-ln(r_c·Њ/(2·c))/Њ

=(ln(r_p·Њ/(2·c))-ln(r_c·Њ/(2·c)))/Њ

=ln((r_p·Њ/(2·c))/(r_c·Њ/(2·c)))/Њ=

=ln(r_p·Њ·2·c/(r_c·Њ·2·c))/Њ

Þ Δђ=ln(r_p/r_c)/Њ . (9)

Now, consider the differential of the “virtual population” (N), expressing, in a sense, the tiny number, dN, of clusters being hypothetically red-shifted “by μ₊”, just in the range between μ and μ+dμ .

According to the theory of exormetism, dN=2·(μ-μ^-3)·(μ²-1)·dμ , (10)

where μ=r·Њ/c+Ö((r·Њ/c)²+1), since r=(μ-1/μ)·c/(2·Њ)Ûμ²-2·(r·Њ/c)·μ-1=0<1£μ .

Hence, dμ=(1+(r·Њ/c)/Ö((r·Њ/c)²+1))·d(r·Њ/c)=(2·μ²/(μ²+1))·d(r·Њ/c). (11)

By replacing dμ, as from (11), into (10), we realize that:

dN=(4/μ)·(μ²-1)²·d(r·Њ/c)=16·(r·Њ/c)²·(r·Њ/c+Ö((r·Њ/c)²+1))·d(r·Њ/c). (12)

Consider, next, the known relation υ^-²=c^-²+(Њ·r)^-², which may take the form:

υ/c=(r·Њ/c)/Ö((r·Њ/c)²+1), where υ=dr/dt is the actual recession velocity.

Ought to the delay of light, the apparent time is t_a=t+r/c , which implies, that:

Њ·(t_a-τ)=Њ·(t-τ)+r·Њ/c , (13)

where τ is the known time constant.

It also follows, that dt_a=dt+dr/c=dr/υ+(υ_a/c)·dt_a ,

where υ_a=dr/dt_a is the apparent velocity of recession.

Hence, 1=υ_a/υ+υ_a/cÞυ_a=c/(c/υ+1)Þυ_a/c=1/(1+Ö((r·Њ/c)²+1)/(r·Њ/c)), which gives:

υ_a/c=(r·Њ/c)/(r·Њ/c+Ö((r·Њ/c)²+1))=(r·Њ/c)·(Ö((r·Њ/c)²+1)-r·Њ/c). (14)

Then, (12) leads to:

dN=(4/μ)·(μ²-1)²·d(r·Њ/c)=16·(r·Њ/c)²·(r·Њ/c+Ö((r·Њ/c)²+1))·(υ_a/c)·d(Њ·(t_a-τ))

ÞdN=16·(r·Њ/c)³·d(Њ·(t_a-τ)). (15)

It follows that, in a moment represented by d(Њ·(t_a-τ)), and at a distance expressed as r·Њ/c , the quantity dN increases by an infinitely small number of clusters, which is:

d₂N=16·((r·Њ/c+d(r·Њ/c))³-(r·Њ/c)³)·d(Њ·(t_a-τ))

=16·(r·Њ/c)³·((1+d(r·Њ/c)/(r·Њ/c))³-1)·d(Њ·(t_a-τ))

=16·(r·Њ/c)³·(1+3·d(r·Њ/c)/(r·Њ/c)-1)·d(Њ·(t_a-τ))

=48·(r·Њ/c)²·d(r·Њ/c)·d(Њ·(t_a-τ))

=48·(r·Њ/c)²·(υ_a/c)·d²(Њ·(t_a-τ))

Þ d₂N=48·(r·Њ/c)³·(Ö((r·Њ/c)²+1)-r·Њ/c)·d²(Њ·(t_a-τ)). (16)

Now, we are ready to formulate the distribution density of the locations, where the clusters presently occurring at a distance r_p, were initially created. Because of (15) and (16), the probability density of the variable r_c·Њ/c , that represents the above locations, is:

dP/d(r_c·Њ/c) =(48·(r_c·Њ/c)³·(Ö((r_c·Њ/c)²+1)-r_c·Њ/c)/(16·(r_p·Њ/c)³)·d(Њ·(t_a-τ))/d(r_c·Њ/c)

=3·(r_c·Њ/c)³·(Ö((r_c·Њ/c)²+1)-r_c·Њ/c)/((r_p·Њ/c)³·υ_a/c)

ÞdP/d(r_c·Њ/c)=3·(r_p·Њ/c)^-3·(r_c·Њ/c)². (17)

In conjunction with (9), (17) implies that, the average natural age, of all clusters now occurring at a certain distance r_p+, i.e. between r_p and r_p+dr_p, is equal to:

E(Δђ)=3·((r_p·Њ/c)^-3/Њ)·ò(r_c·Њ/c)²·ln((r_p·Њ/c)/(r_c·Њ/c))·d(r_c·Њ/c),

where the integral is considered definite, from r_c·Њ/c=0 up to r_p·Њ/c .

Hence, E(Δђ) =3·((r_p·Њ/c)^-3/Њ)·(ln(r_p·Њ/c)·(r_p·Њ/c)³/3-ò(r_c·Њ/c)²·ln(r_c·Њ/c)·d(r_c·Њ/c))

=(ln(r_p·Њ/c)-3·(r_p·Њ/c)^-3·((r_p·Њ/c)³·ln(r_p·Њ/c)/3-(r_p·Њ/c)³/9))/Њ

Þ E(Δђ) =1/(3·Њ)=4,56621 GY. (18)

The above result implies that the natural age of clusters and other objects, which we would see in the sky, if our observational instruments were ideal, follows a certain distribution, independently from the objective distance. The expectantly unique distribution function, of the “possibly observable” natural ages, admits the form:

P(Δђ<ђ_p)=1-exp(-3·Њ·ђ_p), " ђ_p³0 . (19)

To prove this, consider the respective density function, that is:

dP/d(Δђ)=3·Њ·exp(-3·Њ·Δђ)

Û dP/d(Њ·Δђ)=3·exp(-3·ln(r_p/r_c))

Û dP/d(Њ·Δђ)=3·((r_c·Њ/c)/(r_p·Њ/c))³. (20)

But, Њ·Δђ=ln(r_p/r_c)Þd(Њ·Δђ)=(r_c/r_p)·(-r_p/r_c²)·dr_c=d(-r_c)/r_c , where the negative sign appears, simply because dr_c and d(Њ·Δђ) have opposite directions. For the considered density function (20) to remain positive, after the transformation below, it is required to accept the fact that the relation d(Њ·Δђ)=dr_c/r_c must be used, instead, for this purpose.

Hence, (20)ÛdP/d(r_c·Њ/c)=3·(r_p·Њ/c)^-3·(r_c·Њ/c)², which is true according to (17), and yields the distribution function P(r_c<r)=(r/r_p)³, "(0£)r<r_p, once r_p>0 is given. By setting P=1/2 , we realize that the median value of the above distribution is: r_1/2=r_p/2^1/3. (21)

With respect to the mean value of the natural ages, as provided by (18), this exceeds significantly the median of their distribution, which is, according to (19):

ђ_p1/2=ln(2)/(3·Њ)=3,16506 GY . (22)

It is, so far, untested, if half of the clusters, or other distant objects, are older than ђ_p1/2 .

4.2. Instead of a Conclusion

No matter how convincing or promising the above mathematical approach to a steady-state cosmology may appear, there can be no final conclusions, because the existing observational evidence neither disproves nor verifies the theory of exormetism (nor the big-bang hypothesis, for that matter). There are only some quantitative results, which we have reached by processing together four cosmological principles, all being, perhaps, reasonable enough to be widely acceptable.

On the other hand, we deliberately didn’t discuss the nature of dark matter, or the background radiation, obviously because we aren’t convinced, for the time being, that we have satisfactory answers to these questions. The idea, for instance, that “the background radiation is only remnants from repetitive distortions, caused by intergalactic dust molecules, on light of extreme wavelengths (emitted from very remote areas of the universe)”, can be supported neither theoretically, nor observationally, and was therefore omitted altogether.

What remains to be added is that only through systematic observation, the theory of exormetism will be conclusively proven or disproved. Until then, we’ll be partial to a consistent thesis. Still, being consistent is not always the same as being correct!

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Wikipedia Background Information

Appendix A

Localization of the Observable Quasars

In order to approximate, theoretically, the apparent distribution of all detectable quasars in space, we need to assume that their luminosity, at any distance, is greater than a critical one, say ¥=¥(σ;μ), where the standard deviation σ is universal, as we have already accepted. To simplify the necessary calculations, we may, additionally, assume that ¥ is the maximal luminosity of listed galaxy-clusters, or other observable entities, being red-shifted as much as the quasars in question. Mainly because of this last assumption, the green curve, in the graph below, appears locally faulty, in the neighborhood of ξ@12,4 GcY. However, this curve is basically correct, in presenting the case of ¥=y(μ)+3,75·σ , and serves to estimate how much closer, the theorists of the big bang have misplaced the observed quasars in space and time. Notice that the erroneous red curve extends no further than ξ@c·A/2@6,85 GcY, where A is the believed age of the universe. Illustratively, the light coming from that distance was emitted when the universe was half its present age A , and traveled for another A/2 , to arrive here, presently. Nevertheless, the information transmitted by this light is as old as the hypothetical big bang; because whatever we see, today, at that distance, has always been moving away, almost as fast as light, which implies, by the theory of relativity, that the local time, over there, advanced insignificantly, since the big bang.

The formula giving φ’=φ’(ξ) is φ’@η(σ;Y>max{¥,Ϋ})·n’(μ)/c , where the functions η and n’(μ) are considered familiar, while ξ@r , for r=r(μ) being provided by the theory of exormetism. Τhe respective φ’*=φ’(ξ)* is just given by φ’*=φ’(ξ·μ/(Њ·A)) , whereas ξ@r₂ , for r₂=r₂(μ) being defined in the refined model of the big bang.

Fig 4: Alternative distributions of the observable quasars (2003), at various distances.

Quantitative Cosmology

Unlike what others may believe,

I know that God has never spent

a single moment of His time

to create the Cosmos –as they say.

The universe is not His work,

but His own body in a sense.

God’s spirit, on the other hand,

resides within the Cosmic Laws,

being the only constant ground,

on which the changing world flows…

Relativistic space and time

are linear-like, and infinite

in all directions, as implied

by four generic principles

established for a Steady State

to hold instead of the Big Bang.

The expanding fundamental force

called Exormetism, as from Greek,

applies on distant matter, while

it has no influence on light,

acts through the ether, and is caused

by pressure from the quantum void,

where young particles get born.

This is repulsive, as a blow

of rising strength, equal in fact

to Hubble’s constant multiplied

by the amount of mass at rest,

and the velocity presumed

after the redshift we observe.

Though anti-gravity exists

as massless energy, to cease

the gravitational effects

on the geometry of space,

it is that blowing force alone,

which drives the galaxies away,

their clusters fitting best the law

expressed above in formal terms.

Download two simulations of the Exormetism Universe (Sim2.zip)

Explanation Content Sim2.zip: Simulating the Motion of Galaxy Clusters, according to the Theory of Exormetism

Two programs have been developed, in the aim to represent the spatial expansion of a flat-shaped (Euclidean), steady-state universe, where new matter is continuously being created, gradually forming increasing entities and systems, all the way up to super-clusters of galaxies , which are almost uniformly distributed, throughout the infinite, relativistic space-time. In both programs, the outwards accelerating dots simulate existing (old or first appearing) galaxy clusters, while the reddish dot, in the center, represents the observer’s location. The white dots appear to be moving on the 2-d screen, according to the formulae ruling the motion of actual clusters, in 3-d space. All relevant 2-d population densities, were therefore adjusted to f.(x,y)=f(x,y,z)^(2/3), where f(x,y,z) are the respective 3-d population densities of the simulated clusters. The first program (apparent2.exe) shows the locations and velocities of receding clusters, just as the observer would see them (simultaneously), if his observational instruments were powerful enough. Ought to the fact that the speed of light is finite, notice that distant areas look too crowded, and the clusters, in there, seem moving slower than their distance would justify. The second program (inferable2.exe) shows the locations and velocities of receding clusters, that the above observer would only see (simultaneously), if the speed of light was, hypothetically, infinite. Hence, this simulation reveals the actual distribution and velocities of all clusters, at the time they occur, in (the observer’s) reality. A correct representation of this reality is only theoretically inferable, through the formulae of exormetism. Either program basically runs in a GUI window, but there is also a secondary text window (DOS/Console), were the user can watch the number of all dots (in the considered circular area) changing continuously, around a constant value, which is clearly higher in the case of “apparent2.exe”* than for “inferable2.exe”**, just as expected to be. The text window can be minimized or restored, but not closed, while the respective simulation is running. To terminate the program, the user must close the GUI window.

*/** Comments of the Chief-Editor Johan G. van der Galiën. You may need to Unblock the exe in the file Properties / General tab of Windows Explorer! These programs use the excellent Pseudo Random Number Generator: KeyMaker! Developed by the author Panagiotis Karagiorgis himself: One of the few algorithms that will pass all known randomness tests included the not yet published and marketed, but very discriminative, new version of RABENZIX. (See: Journal of RANDOMICS on the SATOCONOR.COM Homepage for the latest published version of RABENZIX.)