Abstract

From the "Decreasing Universe" model,1,2 establishing the contraction of space tissue when exposed to a gravitational field, we derive the Hubble Law and, from there, we will explain the effects "Dark Energy" and "Dark Matter".

Introduction

We know from observation that galaxies seem to move away in an accelerated way. This observation was synthesized in the famous "Hubble Law."3. To explain this accelerated scape an entity was proposed that became known as "Dark Energy."4 Subsequently, it was also observed that stars orbit galaxies at a much higher velocity than calculated. As if there was much more matter inside the galaxies than the one observed. To explain this phenomenon was proposed another entity that became known as "Dark Matter."5 However, despite many efforts, no other evidence of both "Dark Energy" and "Dark Matter" has been found. So we are proposing a new explanation that replaces these two "dark" entities with another hypothesis: The "Decreasing Universe" Hypothesis.1 By reducing the number of hypotheses - from two to one - we will be in agreement with the "Ocam Razor", beyond which, as we shall see, we can also derive the "Hubble Law".

In the "Decreasing Universe" model1 it is established that the gravitational field causes a space contraction that can be detected by an observer who is not subjected to such field. In this way, all objects within this space are also contracted, especially instruments of these observers. Particularly our planet is subjected, in a greater or lesser degree, to various fields: The gravitational field of the Earth itself, the Sun, the Moon, the distant galaxies, and on. If, for example, we have a measuring instrument as a scaler with the length "L" (=1 meter), then, from the point of view of an observer who does not suffer gravitational influence, it will notice that this scaler, with the time, will decrease in size.

Of course, for observers subject to these there will be no change, since all space and everything that is immersed in it is contracting at the same time so that it will have no change in the measurements made by them. For example, here on Earth, a table with 2m length, after millions of years, will continue to measure 2m, as the table shrinks in the same proportion as the measuring scaler and, locally, no difference can be observed.

Outer space

However, in the intergalactic space the gravitational field is practically null and, therefore, this space does not suffer the same contraction that we here on Earth are undergoing. Thus, in the absence of a considerable gravitational field in intergalactic space, the space between us and a distant galaxy will not contract in the same proportion as our own terrestrial space is contracting.

Consider, for example, the enormous time a photon, emitted by a distant galaxy, takes to reach us. In this long period of time, which may be billions of years from the emission of the photon until it reaches our planet, our space - and our scalers - will be reduced in size compared to the original size they had when this photon was emitted under the view of an observer who is not subject to such a gravitational field. This reduction of our local space and the size of our 'scalers' will cause us to distance ourselves to the star larger than it was at the time the photon was emitted (even if its actual distance did not change in that period).

Defining some concepts

Let's call of "Local Space" (=“LS”) the region of space that is subject to a non-negligible gravitational field and thus suffers spatial contraction.

Let's call of "Local Observer" (=“LO”) the observer who belongs to a “LS” and therefore subject - him and his instruments - the spatial contraction. For example, the planet Earth is an “LS” and we are “LO”.

Let's call of "Outer Space" (=“OS”) the region of space that is subject to a very weak and despicable gravitational field.

Let's call of "Sidereal Observer" (=“SO”) those observers located in this spatial region. For example, observers in the intergalactic region would be an “OS”.

To clarify ideas, we may think that observers in the “OS” (=“SO”) play a role similar to an observer in an inertial frame6 as opposed to observers located in the “LS” that would play an analogous role to observers in a non-inertial referential.

Exemplifying the concepts

Consider, for example, at an arbitrary initial instant any t0 in the “LS”, a scaler of length L0 that an “LO” uses to make its measurements.

Suppose that at this moment t0 an “SO”, in intergalactic space, take this measure of this same L0 scaler as the standard measure for your own measurements.

Then, at time t0 both observers (“LO” and “SO”) will consider the pattern L0 of the same size. However, the “LS” will continue to contract in relation to the “OS”. The “LO” will not notice the variation of the scaler L0 because both its measuring scaler and everything in his “LS” decrease in the same proportion. However, the “SO”, after a time "t" ("t">"t0") will see the scaler of the "LO” decrease to a smaller size "L".

Let us call "TJ" (Jocaxian Time) the time period ( Δt=t t 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaamaabmaaba GaeyiLdqKaamiDaiabg2da9iaadshacqGHsislcaGG0bWaaSbaaSqa aiaaicdaaeqaaaGccaGLOaGaayzkaaaaaa@423C@ necessary for an “SO” see the space (and the scaler) of the OL contracting at half size it had at time t0 That is, to shrink itself to a size L= L 0 /2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaadYeacq GH9aqpcaWGmbWaaSbaaSqaaiaaicdaaeqaaOGaai4laiaaikdaaaa@3E86@ at time t 0 + T j . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaacshada WgaaWcbaGaaGimaaqabaGccqGHRaWkcaWGubWaaSbaaSqaaiaadQga aeqaaOGaaiOlaaaa@3EF9@

Before we go on let's make some simplifications.

Some simplifications

Before we continue, we will consider that:

If the gravitational field is constant, the time required for the “LS” (and all that is contained in it), to contract to half its size, called TJ, measured by an “SO”, will also be constant. We will also consider that the galaxies, from the point of view of an “SO” are not necessarily rapidly moving away from each other. For simplicity we will calculate the effects of "Dark Energy" and "Dark Matter" only as a result of our gravitational contraction, keeping constant its distances (from the point of view of an “SO”). Also we will not consider the effect of time dilation7 due to the gravitational force in the “LS” in relation to the “OS”.

Local space contraction formula (“LS”)

We can mathematically translate the concepts we saw above into the following formula:

L(t)= L 0 / 2 Δt/ Tj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaadYeaca GGOaGaaiiDaiaacMcacqGH9aqpcaWGmbWaaSbaaSqaaiaaicdaaeqa aOGaai4laiaaikdadaahaaWcbeqaamaalyaabaGaeyiLdqKaamiDaa qaaiaadsfacaWGQbaaaaaaaaa@4542@ (E1)

(Formula of space contraction from the point of view of an “SO”)

Where:

L (t) = Measure of L0 in the “LS” by a "Sidereal Observer".

t 0 =Initial time( arbitrary ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaacshada WgaaWcbaGaaGimaaqabaGccqGH9aqpqaaaaaaaaaWdbiaadMeacaWG UbGaamyAaiaadshacaWGPbGaamyyaiaadYgacaqGGaGaamiDaiaadM gacaWGTbGaamyza8aadaqadaqaa8qacaWGHbGaamOCaiaadkgacaWG PbGaamiDaiaadkhacaWGHbGaamOCaiaadMhaa8aacaGLOaGaayzkaa aaaa@5194@

L 0 =Length measured in t= t 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaacYeada WgaaWcbaGaaGimaaqabaGcqaaaaaaaaaWdbiabg2da9iaadYeacaWG LbGaamOBaiaadEgacaWG0bGaamiAaiaabccacaWGTbGaamyzaiaadg gacaWGZbGaamyDaiaadkhacaWGLbGaamizaiaabccacaWGPbGaamOB aiaabccacaWG0bGaeyypa0JaamiDamaaBaaaleaacaaIWaaabeaaaa a@510B@

Tj = Jocaxian Time

Δt=t t 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiabgs5aej aadshacqGH9aqpcaWG0bGaeyOeI0IaamiDamaaBaaaleaacaaIWaaa beaaaaa@40AA@

Note that for an “LO”, L=L0 (always!), that is, the size of the scaler does not change with time in the “LS”.

At each "TJ" period of time, our space (and our scales) are contracted in half (from the point of view of an "SO").

If we define:

F j ( Δt )= 2 Δt/ Tj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOramaaBaaaleaacaWGQbaabeaakmaabmaabaGaeyiLdqKa amiDaaGaayjkaiaawMcaaiabg2da9iaaikdadaahaaWcbeqaamaaly aabaGaeyiLdqKaamiDaaqaaiaadsfacaWGQbaaaaaaaaa@45A5@ (E1-B)

(Jocaxian Factor).

We can rewrite (E1):

L= L 0 / F j ( Δt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamitaiabg2da9maalyaabaGaamitamaaBaaaleaacaaIWaaa beaaaOqaaiaadAeadaWgaaWcbaGaamOAaaqabaGcdaqadaqaaiabgs 5aejaadshaaiaawIcacaGLPaaaaaaaaa@4326@ (E2)

We can also rewrite the same Jocaxian Factor (Fj) in a more friendly way:

F j ( Δt )=exp( ln(2)* Δt/ T j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOramaaBaaaleaacaWGQbaabeaakmaabmaabaGaeyiLdqKa amiDaaGaayjkaiaawMcaaiabg2da9iGacwgacaGG4bGaaiiCamaabm aabaGaciiBaiaac6gacaGGOaGaaGOmaiaacMcacaGGQaWaaSGbaeaa cqGHuoarcaWG0baabaGaamivamaaBaaaleaacaWGQbaabeaaaaaaki aawIcacaGLPaaaaaa@4DFD@ (E3)

“Dark Energy” effect

Of course, if intergalactic space does not contract, and if our 'scale of measurement' decrease in size, then this intergalactic space should seem to us larger, in the same proportion as our ‘scale of measurement’ contract itself.

If, for example, at t=t0, we measure the distance to a galaxy "X" with our scale of length L0 as being after a time "Tj" our rule will be measuring half of its initial size L0 and therefore, when we (“LO”) measure the distance to that galaxy, we will measure it as being D=2* D 0 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamiraiabg2da9iaaikdacaGGQaGaamiramaaBaaaleaacaaI Waaabeaakiaac6caaaa@3F43@

We should note that a measure within our “LS” sizes do not change, as everything decreases along with our scaler and rulers, but the “OS” does not contract like our “LS”. So we will have this illusion that the galaxy "X" is moving away from us. This is what we can call the "Dark Energy Effect".

Apparent distance formula

The measured distance is inversely proportional to the length of the measurement pattern. We can synthesize this idea mathematically with the following formula (See the Appendix A):

D( Δt )= D 0 * F j ( Δt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamira8aadaqadaqaa8qacqGHuoarcaWG0baapaGaayjkaiaa wMcaa8qacqGH9aqpcaWGebWaaSbaaSqaaiaaicdaaeqaaOGaaiOkai aadAeadaWgaaWcbaGaamOAaaqabaGcpaWaaeWaaeaapeGaeyiLdqKa amiDaaWdaiaawIcacaGLPaaaaaa@4803@ (E4)

(Distance formula with the Jocaxian Factor).

Where:

"t0" is the time at which the photon was emitted by the galaxy.

"t" is the time at which Earth received this photon.

"D0" is the distance we would measure, from Earth to galaxy at time

"D( Δt )" MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaaiOiaiaadseapaWaaeWaaeaapeGaeyiLdqKaamiDaaWdaiaa wIcacaGLPaaacaGGIaaaaa@3FCB@ is the distance we measured from Earth to a galaxy after time.

"Δt"=t t 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaaiOiaiabfs5aejaadshacaGGIaGaeyypa0JaamiDaiabgkHi TiaadshadaWgaaWcbaGaaGimaaqabaaaaa@4215@ Time period

As F j ( Δt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOramaaBaaaleaacaWGQbaabeaak8aadaqadaqaa8qacqGH uoarcaWG0baapaGaayjkaiaawMcaaaaa@3FA6@ grows exponentially with time (E3) the Earth's distance from the galaxy will also appear to increase exponentially with time.

Hubble’s Law

With the formula of the apparent distance (E4) we can calculate the apparent distance speed:3

V=d[ D( Δt ) ]/dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOvaiabg2da9iaadsgapaWaamWaaeaapeGaamira8aadaqa daqaa8qacqGHuoarcaWG0baapaGaayjkaiaawMcaaaGaay5waiaaw2 faa8qacaGGVaGaamizaiaadshaaaa@45FF@ (E5-A)

V=d[ exp( ln( 2 )*Δt/Tj  ) ]/dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOvaiabg2da9iaadsgapaWaamWaaeaapeGaamyzaiaadIha caWGWbWdamaabmaabaWdbiaadYgacaWGUbWdamaabmaabaWdbiaaik daa8aacaGLOaGaayzkaaWdbiaacQcacqGHuoarcaWG0bGaai4laiaa csfacaWGQbGaaeiiaaWdaiaawIcacaGLPaaaaiaawUfacaGLDbaape Gaai4laiaadsgacaWG0baaaa@5044@ (E5-B)

V=( ln( 2 )/Tj )*D( Δt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOvaiabg2da98aadaqadaqaa8qacaWGSbGaamOBa8aadaqa daqaa8qacaaIYaaapaGaayjkaiaawMcaa8qacaGGVaGaamivaiaadQ gaa8aacaGLOaGaayzkaaWdbiaacQcacaWGebWdamaabmaabaWdbiab gs5aejaadshaa8aacaGLOaGaayzkaaaaaa@49B7@ (E6)

(Distant galaxies’ apparent distance speed formula.)

But Hubble's law is exactly like this:

V= H 0 *D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOvaiabg2da9iaadIeadaWgaaWcbaGaaGimaaqabaGccaGG QaGaamiraaaa@3EB4@ (E7)

(Hubble’s Law)

Where (H0= Hubble’s constant and D is the distance from the galaxy)

As E6 = E7, now we can determine Tj:

T j =ln( 2 )/ H 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamivamaaBaaaleaacaWGQbaabeaakiabg2da9iaadYgacaWG UbWdamaabmaabaWdbiaaikdaa8aacaGLOaGaayzkaaWdbiaac+caca WGibWaaSbaaSqaaiaaicdaaeqaaaaa@4370@ (E8)

Substituting (E8) into (E3) we will have:

F j ( Δt )=exp( H 0 *Δt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOramaaBaaaleaacaWGQbaabeaak8aadaqadaqaa8qacqGH uoarcaWG0baapaGaayjkaiaawMcaa8qacqGH9aqpcaWGLbGaamiEai aadchapaWaaeWaaeaapeGaamisamaaBaaaleaacaaIWaaabeaakiaa cQcacqGHuoarcaWG0baapaGaayjkaiaawMcaaaaa@4A1A@ (E9)

(Jocaxian Factor in terms of the Hubble constant)

What provides us:

D( Δt )= D 0 *exp( H 0 *Δt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamira8aadaqadaqaa8qacqGHuoarcaWG0baapaGaayjkaiaa wMcaa8qacqGH9aqpcaWGebWaaSbaaSqaaiaaicdaaeqaaOGaaiOkai aadwgacaWG4bGaamiCa8aadaqadaqaa8qacaWGibWaaSbaaSqaaiaa icdaaeqaaOGaaiOkaiabgs5aejaadshaa8aacaGLOaGaayzkaaaaaa@4B5A@ (E10)

(Apparent distance formula in terms of the Hubble constant)

If we want to calculate the real distance from Earth to the galaxy, using the measurements that our scales had at the time the photon was emitted (at t = t0) then:

∆t = D0/c (E11)

(Time for a photon emitted from the galaxy to reach us, where c = speed of light)

From (E11) and (E10) we will have:

D= D 0 exp( D 0 * H 0 /c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamiraiabg2da9iaadseadaWgaaWcbaGaaGimaaqabaGccaWG LbGaamiEaiaadchapaWaaeWaaeaapeGaamiramaaBaaaleaacaaIWa aabeaakiaacQcacaWGibWaaSbaaSqaaiaaicdaaeqaaOGaai4laiaa dogaa8aacaGLOaGaayzkaaaaaa@4779@ (E12)

(Apparent distance of a galaxy depending on the actual distance)

Some Values

As H 0 =2.2e18 s 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamisamaaBaaaleaacaaIWaaabeaakiabg2da9iaaikdacaGG UaGaaGOmaiaadwgacqGHsislcaaIXaGaaGioaiaadohadaahaaWcbe qaaiabgkHiTiaaigdaaaGccaGGSaaaaa@4567@ we can replace it in (E8) and find Tj

Tj = 3.15E17 s = 10 billion years

That is, the Jocaxian Time, the time necessary for our space to contract in half, is 10 billion years.

We can now find the contraction rate of our space for every billion years:

( Tx ) 10  =2Tx=exp( ln( 2 )/10 )=7% MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaamaabmaaba aeaaaaaaaaa8qacaWGubGaamiEaaWdaiaawIcacaGLPaaapeWaaWba aSqabeaacaaIXaGaaGimaiaabccaaaGccaqG9aGaaGOmaiabgkDiEl aadsfacaWG4bGaeyypa0JaamyzaiaadIhacaWGWbWdamaabmaabaWd biaadYgacaWGUbWdamaabmaabaWdbiaaikdaa8aacaGLOaGaayzkaa Wdbiaac+cacaaIXaGaaGimaaWdaiaawIcacaGLPaaapeGaeyypa0Ja aG4naiaacwcaaaa@53C3@

That means:

For every 1 billion years our space (and our scalers) is contracted 7% of their original size.

It is interesting to note that this value (7%) corresponds exactly to the contraction rate calculated from the "Redshift" of the Galaxy NGC3034.2 Currently the apparent distance of the 'NGC3034' is about 11E6 light years, (or 1E23 meters).

Applying (E12) to the galaxy NGC3034 and knowing that H 0 /c=8E27  m 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamisamaaBaaaleaacaaIWaaabeaakiaac+cacaWGJbGaeyyp a0JaaGioaiaadweacqGHsislcaaIYaGaaG4naiaabccacaWGTbWaaW baaSqabeaacqGHsislcaaIXaaaaaaa@455D@ we will have the following equation for the distance to the galaxy NGC3034:

1E23= D 0 *exp( D 0 *8E27 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaaGymaiaadweacaaIYaGaaG4maiabg2da9iaadseadaWgaaWc baGaaGimaaqabaGccaGGQaGaamyzaiaadIhacaWGWbWdamaabmaaba WdbiaadseadaWgaaWcbaGaaGimaaqabaGccaGGQaGaaGioaiaadwea cqGHsislcaaIYaGaaG4naaWdaiaawIcacaGLPaaaaaa@4AFA@ (E13)

(Equation of the real distance of the Galaxy NGC3034)

Using a solver8 we will obtain for the real distance:

D0 = 9E22 that is, this galaxy is about 10% closer to Earth than it appears to be.

Dark matter

Dark Matter5 can also be observed being an effect of our spatial contraction.

As nomenclature, we will suppress the subscripts "obs" of the measurements observed here from Earth. So we'll simplify:

Vobs=V; (Rotation Speed Observed)

Dobs=D; (Distance Observed)

Robs=R; (Radius Observed)

Wobs=W; (Angular Speed Observed)

Mobs=M; (Mass Observed)

Considering the illustration: star orbiting a distant Galaxy. (Figure 1)

Figure 1 From earth we observe a star rotating a distant galaxy.

Suppose that from Earth we now observe a star circling the periphery of a galaxy that is at an observed (apparent) distance "D" from our planet. According to Figure 1 above, the observed “R” radius of the galaxy will be proportional to this distance:

R=sen( α )*D MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOuaiabg2da9iaadohacaWGLbGaamOBa8aadaqadaqaa8qa cqaHXoqya8aacaGLOaGaayzkaaWdbiaacQcacaWGebaaaa@432E@ (E14)

(Orbit radius as a function of observed distance and angle)

As we saw earlier, at the time the photon was emitted, the real distance would be D0, so:

R 0 =sen( α )* D 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOuamaaBaaaleaacaaIWaaabeaakiabg2da9iaadohacaWG LbGaamOBa8aadaqadaqaa8qacqaHXoqya8aacaGLOaGaayzkaaWdbi aacQcacaWGebWaaSbaaSqaaiaaicdaaeqaaaaa@4504@ (E15)

(Real orbit radius as a function of actual distance and angle)

From (E9) and (E11) we can define the Jocaxian Factor of the Galaxy:

F J =exp( D 0 * H 0 /c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOramaaBaaaleaacaWGkbaabeaakiabg2da9iaadwgacaWG 4bGaamiCa8aadaqadaqaa8qacaWGebWaaSbaaSqaaiaaicdaaeqaaO GaaiOkaiaadIeadaWgaaWcbaGaaGimaaqabaGccaGGVaGaai4yaaWd aiaawIcacaGLPaaaaaa@46C6@ (E16)

(Jocaxian Factor of the Galaxy)

So, we will have:

R 0 =R/ F J MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOuamaaBaaaleaacaaIWaaabeaakiabg2da9iaadkfacaGG VaGaamOramaaBaaaleaacaWGkbaabeaaaaa@3FBC@ (E17)

(Real Radius as a function of the Jocaxian factor of the Galaxy)

If M is the observed mass of the galaxy where the star orbits, and V is its tangential velocity observed from the Earth, and G is the Gravitational constant of the Galaxy, we will have:9

V 2 =M*G/R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOvamaaCaaaleqabaGaaGOmaaaakiabg2da9iaad2eacaGG QaGaam4raiaac+cacaWGsbaaaa@4049@ (E18)

(Equation of Velocity as a function of mass and radius)

In terms of the angular velocity, we have:

V 2 = W 2 * R 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOvamaaCaaaleqabaGaaGOmaaaakiabg2da9iaadEfadaah aaWcbeqaaiaaikdaaaGccaGGQaGaamOuamaaCaaaleqabaGaaGOmaa aaaaa@40B0@ (E19)

(Equation of Velocity as a function of angular velocity and radius)

From (E18) and (E19) we derive:

W 2 =MG/ R 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaam4vamaaCaaaleqabaGaaGOmaaaakiabg2da9iaad2eacaWG hbGaai4laiaadkfadaahaaWcbeqaaiaaiodaaaaaaa@4086@ (E20)

(Equation of the angular velocity as a function of Mass and Radius)

From (E20), at t = t0, we have:

W 0 2 = M 0 G/ R 0 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaam4vamaaBaaaleaacaaIWaaabeaakmaaCaaaleqabaGaaGOm aaaakiabg2da9iaad2eadaWgaaWcbaGaaGimaaqabaGccaWGhbGaai 4laiaadkfadaWgaaWcbaGaaGimaaqabaGcdaahaaWcbeqaaiaaioda aaaaaa@4356@ (E21)

(Equation of the angular movement in function of the Real Radius and Real Mass)

The angular velocity does not change with the observed distance, since the time interval between two emitted photons is the same interval when they arrive to Earth. So:

W = W0  (E22)

(The angular speed is the same for an “LO” as for a “SO”)

From (E19), E (21), E (22) we find:

V 2 =( M 0 G/ R 0 3 )*  R 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOvamaaCaaaleqabaGaaGOmaaaakiabg2da98aadaqadaqa a8qacaWGnbWaaSbaaSqaaiaaicdaaeqaaOGaam4raiaac+cacaWGsb WaaSbaaSqaaiaaicdaaeqaaOWaaWbaaSqabeaacaaIZaaaaaGcpaGa ayjkaiaawMcaa8qacaGGQaGaaeiiaiaadkfadaahaaWcbeqaaiaaik daaaaaaa@4747@ (E23 )

Using (E17):                         

V 2 =( M 0 F j 3 )*G/R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamOvamaaCaaaleqabaGaaGOmaaaakiabg2da98aadaqadaqa a8qacaWGnbWaaSbaaSqaaiaaicdaaeqaaOGaaiOramaaBaaaleaaca WGQbaabeaakmaaCaaaleqabaGaaG4maaaaaOWdaiaawIcacaGLPaaa peGaaiOkaiaacEeacaGGVaGaaiOuaaaa@45E1@ (E24)

(Equation of the tangential velocity as a function of the Barium Mass and Apparent Radius)

Comparing (E24) with (E18) we conclude that:

M= M 0 * F j 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamytaiabg2da9iaad2eadaWgaaWcbaGaaGimaaqabaGccaGG QaGaaiOramaaBaaaleaacaWGQbaabeaakmaaCaaaleqabaGaaG4maa aaaaa@40C0@ (E25)

(Apparent mass as a function of actual mass)

From Earth we observe a mass M for the galaxy larger than the mass at t = t0.

Then the effect "Dark matter" will be the difference of M with the real mass M0:

Dark Matter=M M 0 = M 0 *( exp( 3* D 0 * H 0 /c )1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamiraiaadggacaWGYbGaam4AaiaabccacaWGnbGaamyyaiaa dshacaWG0bGaamyzaiaadkhacqGH9aqpcaWGnbGaai4eGiaad2eada WgaaWcbaGaaGimaaqabaGccqGH9aqpcaWGnbWaaSbaaSqaaiaaicda aeqaaOGaaiOka8aadaqadaqaa8qacaWGLbGaamiEaiaadchapaWaae WaaeaapeGaaG4maiaacQcacaWGebWaaSbaaSqaaiaaicdaaeqaaOGa aiOkaiaadIeadaWgaaWcbaGaaGimaaqabaGccaGGVaGaam4yaaWdai aawIcacaGLPaaapeGaai4eGiaaigdaa8aacaGLOaGaayzkaaaaaa@5A20@ (E26)

(Dark matter equation as a function of the Hubble constant and the actual distance)

Conclusion

If we adopt the "Decreasing Universe" where the gravitational field shrinks the space in which it crosses, we find that the accelerated separation of galaxies, often explained by the so-called "Dark Energy" is a kind of "illusion" resulting from this space contraction and, therefore, unnecessary. The "Dark Matter", on the other hand, can also be explained by the same effect of the gravitational contraction of our space, since the radius of the galaxies is observed as greater than it really is, consequently, the speed of translation of a star is seen as above-expected with the observed baryonic mass, providing the false impression that there is an extra, invisible matter responsible for the effect.

Appendix A

Derivation of the distance formula from the local contraction formula.

At t=t0 we will take as the measurement standard for both observers the measure "L0" for example, L0=1 meter. Thus, all distances will be taken as a number that multiplies the L0 pattern;

Then both observers, sidereal and terrestrial, measure the same distance to a given galaxy:

Distance=D0*L0   (A1)

(D0 is the distance that is measured to the galaxy by taking the measurement pattern L0)

After a time t (> t0), from the point of view of a sidereal observer, the terrestrial space shrank, and the rule tablet L0 decreased to L according to E1:

L= L 0 / F j ( Δt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamitaiabg2da9iaadYeadaWgaaWcbaGaaGimaaqabaGccaGG VaGaamOramaaBaaaleaacaWGQbaabeaak8aadaqadaqaa8qacqGHuo arcaWG0baapaGaayjkaiaawMcaaaaa@43F1@

(Spatial contraction formula (E2))

As in our hypothesis, from the point of view of the Sidereal Observer, the galaxy does not move away, the distance covered must be the same, that is:

Distance=D*L (A2)

(D is the measured Distance to the galaxy according to the L pattern, by the Terrestrial observer)

We must keep in mind that for the local terrestrial observer, L=L0, since he does not perceive his own contraction).

From (A1) and (A2) we have to:

D 0 * L 0 =D*L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamira8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGQaGa amita8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGH9aqpcaWGeb GaaiOkaiaadYeaaaa@4191@ (A3)

(From the point of view of the sidereal observer, the distance is not altered)

Using (E1), we have finally:

D= D 0 * F j ( Δt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMi=NPiFDYdXdHaVhbbf9v8qqaqFr0xc9pk 0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr 0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaa aapeGaamiraiabg2da9iaadseapaWaaSbaaSqaa8qacaaIWaaapaqa baGcpeGaaiOkaiaadAeapaWaaSbaaSqaa8qacaWGQbaapaqabaGcda qadaqaa8qacqGHuoarcaWG0baapaGaayjkaiaawMcaaaaa@4439@ (A4)

(Measured distance to the galaxy according to the terrestrial observer, as a function of time)

Acknowledgements

None.

Conflicts of interest

Author declare that there is no conflict of interest.

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Copyright (c) 2020 Joao Carlos Holland de Barcellos

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