Fünfdimensionale Physik

Universe without expansion and big bang

Dipl.-Phys. Roland Alfred Sprenger, Germany, 24.07.2024




 
 

Contents


The standard model of cosmology is confronted with a model of a universe which is curved within a five-dimensional spacetime. The redshift of the spectra of galaxies is explained by the curvature; the expansion of the universe and the big bang do not exist in this model. According to this the universe is a four-dimensional hypersphere within the five-dimensional spacetime. Its radius is calculated from the distance of standard candles. 
 
Paragraphs:
1. Criticism of the standard model of cosmology
2. Explanation of the redshift by curvature
3. Derivation of formula
4. The radius of the universe
5. Further calculations
6. Alternative to accellerated expansion
7. Advantages of the curvature hypotheses over the big bang theory
8. Summary and outlook
Attachment

 

 

1. Criticism of the standard model of cosmology


The widely recognized big bang theory has some weaknesses and contradictions. The most serious weakness is that the so-called inflation has to be postulated, because in relatively close globular star clusters stars have been found which are older than the farthes galaxies which arose only a short time after the Big Bang. Therefore is assumed that space expanded in the first 10-34 s with multiple speed of light, quasi as a continuation of the Big Bang, and as unexplained as this one itself. A mysterious inflaton field is cited as its cause. 
 
According to general relativity for the expansion of space itself no energy is required as at this bodies stay at their place if their own movement is refrained from. But this then is independent of whether the expansion is uniform or accelerated as for example in the final phase of the inflation. It then appears contradictory that as cause of the currently, in 1998 discovered, accelerated expansion a likewise mysterious so-called dark energy is postulated. 
 
The question also arises where the per se finite energy of the big bang comes from, respectively if the energy, the big bang and the whole universe emerged from “nothing”. But the explanation of the emergence of “something” from “nothing” what a big bang challenges us to do, is impossible. For both notions are absolute opposites, dialectical terms. “Something” is defined as “not nothing” and vice versa. The question what the course of the Big Bang is is not yet answered. In the macrocosmos still each incidence has a cause.
 
Considering the past of the universe and continuing its contraction linearly up to a dimensionless point and postulating a singularity as it has been done before the introduction of the inflation and in the end has been maintained is rather arbitrary. If the expansion currently is accelerated it probably was that in the past too. At this the shrinking to a singularity is only one of several possibilities, for example of an oscillation or an asymptotic progression out of a finite initial size (without inflation) which could as well be very large. Infinite small objects are no objects of physics. With the singularity in which in form of energy the nearly infinite big mass of all billions of galaxies is contained a mysterium has been implemented into physics. But apparently the mentioned mysteries satisfy a need of the religious kind. That might be the main reason for the emergence and modification of the Big Bang theory.
 
Recently, in 2025, with the James Webb Space Telescope observations were made which contradict the Big Bang theory, namely bigger early galaxies than predicted by the theory and oxygen in the presently farthest galaxy Jades-GS-z14-0 although heavy elements could not have developed that early. 
 
The cosmic microwave rediation is considered as a proof of the standard theory of cosmology. Its redshift shows a polarisation with more and less redshift [1] from which the velocity of the galactic center in relation to the cosmic microwave background has been calculated as 552 km/s and that one of the lokal group of galaxies as even 627 km/s. These are  inexplicably high amounts. Furthermore the cosmic background radiation is a reference object in relation to which for each object in the universe the direction and the amount of its velocity can be indicated clearly. These absolute figures contradict the previous relativity of space. The rest system of the cosmic background radiation is excellent before all other reference systems and defines a resting, absolute space.
 
The big bang theory is so mysterious, imperfect and contradictory that other attempts to explain the redshift and other models of the cosmos are justified.
 
 

2. Explanation of the redshift by curvature


The articles “Relativistic addition of velocities in a five-dimensional spacetime” [2] and “Relativistic velocities and dimensions” [3] give hints on the existence of a fifth dimension of spacetime. In the here constituted hypothesis is asserted, that the three-dimensional universe is curved in the further dimension, namely first with constant curvature everywhere. Then it corresponds to a spherical surface; one can call it hypersphere. It is assumed that the redshift results from this curvature. Then the expansion of space is not needed for the explanation of the redshift; it does not exist in this model. Furthermore, is assumed that light spreads straight in this five-dimensional spacetime. By this results a bigger wavelength i.e. a redshift in the three-dimensional universe, see fig. 1.
 
In order to display the spreading of the light a time-axis is needed. If the three spatial dimensions are reduced to one as usual, in order to be able to display the time-axis and the w-axis of the fifth dimension too, then the concentric spherical waves around the light source become circular waves. In fig. 1 they lie on a cone shell the axis of which passes through the light source in the point A orthogonally to the x-w-plane. On this cone shell the way of the light particles from point A (with negative t-coordinate) to the observer in the coordinate origin O is part of a spiral. The orthogonal projections of the wave length on the spiral onto the x-w-plane are longer than their projections from the straight connection between A and O. Because the universe as assumed does not expand the radius of the hypersphere is constant. The circle with the arc b in fig. 1 has the same radius at all times. It is the orthogonal projection of the cylinder with the radius R and the central axis through the point M on the x-w-plane and represents the universe curved in the four-dimensional hypersphere.
                                                             
Fig. 1: Projection of the straight spreading of the light of a light source in point A to the observer in the origin of the x-w-t-space; λ not true to scale

 

3. Derivation of formulas
 

Given respectively measurable are the red-shift independent luminosity distance b of relatively close galaxies and the frequencies  f0 and f from which arise the wavelengths λ0 and λ . From this can be calculated the respective angles ζ , the radius R of the hypersphere and with that one then the distances b of far galaxies too. 

The redshift is defined as                          z = Δλ / λ0 = (λ - λ0) / λ0 = λ / λ0 - 1 .

It follows                                  λ0 / λ = 1 / (z + 1)                                                                                         (1)

As the wavefronts at the observer, at the origin, in the range of some lightyears are nearly plane respectively straight in fig. 1 and the arc there yet barely differs from the x-axis, there actually is located a right triangle in which applies in a very good approximation                                                                                                                
                                 cos⁡ η = λ0 / λ .

The angle between a tangent and a secant that emanates from the point of contact is always half of the centre angle of the secant, so                                                                                     η = ζ / 2   .                                                              
 
                                 cos η = cos (ζ / 2)                                                                         

It follows                                  ζ / 2 = arccos (λ0 / λ)

and                                  ζ = 2 arccos (λ0 / λ)   .                                                                (2)

From the definition of the angle follows                  R = b / ζ                                                                                                         (3)
 
 

4. The radius of the universe


From the redshift can be calculated with (1) and (2) the centre angle ζ to the location of a galaxy. 

In the articles about galaxies in the internet, especially at Wikipedia, their distances always are stated as “hubble-distance”, that is considering the assumed expansion of space. Only in a few cases a distance independent of redshift is stated, i.e. a distance that is based on a measurement of standard candles – in most cases type Ia supernovae or sometimes cepheides – in the relevant galaxies, that is without consideration of an expansion. The hubble-distance therefore in general is larger than the redshift independent distance. 

In table 1 from the redshifts and the redshift independent distance statements for four galaxies each time radius R is calculated, from which then the average is formed. 

                                                
Table 1: Computation of the radius R of the universe in R4
 
As mean with scatter results                            R = (263 ± 21) ∙ 106  Ly .

The maximum distance of a galaxy then is            (826 ± 66) ∙ 106  Ly             .
 
 
In the standard model the self-movements of the stars, galaxies and clusters of galaxies overlap with the general expansion of the universe. In the model shown here the expansion is substituted by a general curvature. Although it is basically static there still are the self-movements which by the Doppler effect contribute to the wavelength change too. In cosmic proximity the Doppler effect may predominate the effect of curvature so that a blueshift might occur as for example at the Andromeda Galaxy with the redshift z = -0.001001. Therefore, the galaxy NGC 3034 was not used for the calculation of the average radius because of its relative proximity of 11.5 ∙ 106 Ly and the resulting minimal radius R = 156 ∙ 106 Ly . At the galaxy NGC 4340 on the other hand with randomly also R = 156 ∙ 106 Ly the hubble distance with 40 ∙ 106 Ly is smaller than the luminosity distance of 60 ∙ 106 Ly which was determined with standard candles.
 
The curvature of the four-dimensional spacetime by big masses as well overlaps with the postulated general curvature. According to the present observations the universe on a large scale seems flat though. But that will be caused by the circumstance we three-dimensional beings are not experienced with and therefore do not have imagination of a fourth spatial dimension and hence interpret indirect evidence of it – like the redshift and the addition of relativistic velocities – based on three spatial dimensions. In the outlook (paragraph 7) is predicted which future observations should contradict to that.
  
In paragraph 2 a constant curvature of the universe was assumed. If future observations respectively measurements should contradict to this the form of the universe could deviate from a hypersphere. But the combination of curvature in a fifth dimension and expansion is not taken into consideration in this hypothesis.

 

5. Further calculations

 
With the calculated radius R now the distances b of galaxies can be determined, that are so far away that their distances cannot be calculated from the measurements at type Ia supernovae. For example, results from the galaxy GLASS 12 discovered in 2022 with the James Webb Space Telescope with z = 12.4 and the hubble distance 13.6 ∙ 109 Ly from (1) and (2) the angle ζ = 2.992 = 171.4°. From (3) then follows  b = 787 ∙ 106 Ly , which is only 5 % less than the expected maximum distance of galaxies of  826 ∙ 106  Ly.  

The hubble distance is 17 times bigger because of the underlying expansion.
 
The circumstance that xA , the x-component of the point A for ζ > 90° decreases again means that xA is the distance in an only imagined, purely mathematical, uncurved three-dimensional space. It does not differ perceptibly from the real curved outer space in distances of some million lightyears but in larger distances the difference becomes fundamental.  
 
 
As well of interest are the light path s in R^4 and the coordinates of the points M and A (see fig. 1). In table 2 are listed 4 of the earliest discovered and thus closest galaxies and 4 of the just recently discovered most distant galaxies. At the first ones the calculation of s and wA using the mean R would lead to incorrect deviations. Therefore, hereinafter for this case formulas are derived that depend of b and ζ . For distant galaxies b is not known; therefore, the respective formulas depending on R are derived.
 
The distance b of a far galaxy results from the definition of an angle to
 
                                                                                              b = R ∙ ζ .                                                                  (4)

If the y – and z – components of the light path are 0 like in fig. 1 then in the triangle with the corner point M and the side lengths R, R and s applies
                                                             s2 = R2 + R2 - 2 R R cos ⁡ζ = 2 R2 - 2 R2  cos ⁡ζ = 2 R2  (1 - cos⁡ ζ)   .

So then applies                                               s = R √(2 - 2 cos⁡ ζ)                                                          (5)

For the maximum angle  ζ = π  follows s = 2 R    .

With   R = b / ζ   follows                                               s = (b / ζ) √(2 - 2 cos⁡ ζ)                                                          (6)
 
If the light path s is specified in Lightyears and the runtime t in years the measures of s and t are equal because 1 lightyear is the distance which light passes in vacuum in 1 year. 

In the lower right triangle in fig. 1 applies                                sin⁡ ζ = xA / R   ,                                             

so                                                       xA = R ∙ sin ζ .                                                  (7)
 
With (4) follows                                       xA / b = (sin ζ) / ζ ,

so                                                       xA = b ∙ (sin ζ) / ζ                                                                  (8)
 

In the right triangle in fig. 1 with the vertices A and M the short cathete has the catheter length  R ∙ cos ⁡ζ . The w-coordinate of A then has the absolute value 
                                                                              R - R ∙ cos⁡ ζ = R (1 - cos⁡ ζ) = (b / ζ ) (1 - cos⁡ ζ)

i.e.                                                     wA = - (b / ζ) (1 - cos⁡ ζ)   .                                                                            (9)
 
With 4 follows                                                              wA = - R ∙ (1 - cos⁡ ζ)     .                                                (10)
 
 
In the order of the coordinates x, y, z, w, t the points A and M have the coordinates                                                    

A ( b ∙ (sin ζ) / ζ  | 0 | 0 | - (b / ζ) (1 – cos ζ))

or                                            A ( R ∙ sin ζ | 0 | 0 | - R ∙ (1 - cos⁡ ζ) )

and                                            M ( 0 | 0 | 0 | - 263 ∙ 106  Ly ) .
 

In the following table the data calculated with (4) till (10) are indicated.
                                                 
                                                
Table 2: Centre angles, distances and coordinates of some of the closest and most distant galaxies

 

6. Alternative to accellerated Expansion

 
Already in fig. 1 is visible that the stretching of the wavelength of light is not constant. Therefore the reshift z cannot depend linearly on the distance b of the observer in the coordinate origin. Hereinafter the function z(b) is derived.

From the definition of redshift                                          z = Δλ/λ0 = (λ - λ0)/λ0 = λ/λ0 - 1
follows with   cos⁡ η = λ0/λ                          z = 1/cos⁡(ζ/2) - 1
and with    ζ = b/R                          z = 1/cos⁡(b/2R) - 1  ,                                           (11)
i.e. a nonlinear function z(b) the graph of which  is depicted in fig. 2 and fig. 4 (in the attachment) for the above calculated radius R. 
 
                                              
Fig. 2: The red graph depicts z(b) for R = 263 ∙ 106 Lj, the blue line a linear approximation which results from the Hubble-constant H0 (see attachment). The approximation more or less fits within the borders from 0 to 55 ∙ 106 Ly for b and 0 to 0,0055 for z, which corresponds roughly with the numbers in table 1.
  
So in this curvature model the redshift developes overproportionally with the distance. This corresponds to an accellerated speed of expansion vH  in the expansion model. For with the formula used there               
                                                                                             vH ≈ c ∙ z 
and equ. 11 follows                                                              vH ≈  c ∙ [1/cos⁡(b/2R) - 1] ,
 
a likewise overproportional dependance of the expandance speed on the distance. Its graph runs like that one of the function z(b) and can be approximated by a parabel with the equation   v = 0,564 ∙ b2 in the range from 0 to 200 ∙ 106 Ly (see  attachment, fig. 3). Possibly the accellerated expansion of the standard model is a wrong interpretation of the curvation effect which manifests in the function z(b).

As Hubble's measurements of distances were that wrong that for H0 resulted a value eight times as big (560 km/s/Mpc, Hubble and Humason [4]) as the actual value it is imaginable that the nonlinear dependance of the redshift on distance was not realised and as a first approximation was set linear,  corresponding to the blue line in fig. 2.


7. Advantages of the curvature hypothesis over the big bang theory


The handicaps of the big bang theory, the standard model of cosmology, stated in the introduction do not exist in the curvature hypothesis. The postulated “inflation” of the universe with more than light speed is omitted in the curvature hypothesis as well as the velocities higher than light speed of very distant galaxies with a redshift >1 which are supposed to come about by the expansion.
 
Together with the expansion also disappears the accelerated expansion and thus the postulate of a “dark energy”. As explained in paragraph 6 instead of accelleration curvature should cause the overproportional increase of redshift.
 
As well unnecessary are explanations for the observation that the galaxies and galaxy clusters do not expand together with space and that no gravitational time effects occur by the expansion.
 
As the curvature hypothesis describes a basic static cosmos without expansion there is no beginning and no end in time. Thus misses the intellectual problem having to explain the existence of the universe in contrast to its non-existence. For the notions existence and non-existence are only applicable to parts of the cosmos, not to the entirety. Only the parts arise and disappear and thereby are definable as notions and by that depictable i.e. existent. Expressed the other way round there is no difference between existence and non-existence for the cosmos. This idea had already the antique Greek philosopher Parmenides of Elea (515 – 470 B. C.). The idea of Heraklit of Ephesus (535 – 470 B. C.) that in the world only an eternal change (panta rhei) takes place also implies that there was no emergence from nothing and thus is an intellectual tradition for the curvature hypothesis.

 

8. Summary and outlook

 
By the concept of a five-dimensional spacetime [2], [3], [5], [6] the redshift of the galaxies can be explained more problem-free than by the big bang theory.
 
Based on the curvature hypothesis can be predicted that at further progress in space-based telescope technology finally galaxies will be observable that are located at the opposite point of the hypersphere of universe. At observations in any directions the same formations of galaxies – just turned – must appear. 
 
An additional prediction is that before this region does not exist a dark zone, for this scenario is based on the expansion or compression in the big bang theory. Consequently can be predicted that the therefrom resulting expected limits of the redshift of  z = 30  for first stars and z = 1100 for the beginning of the dark age (background radiation z = 1089) shall be exceeded. The limit of z = 20 for the end of the dark age from which result a maximum angle of  ζ = 3.05 = 175° and a maximum distance of 802∙106  Ly  nearly is already reached (172.5° and 792∙106  Lj ). At measurements of z > 20 the big bang theory would start to waver.
 
In the curvature hypothesis from (2) follows for the maximum angle ζ = π 
                                                         π = 2 arccos⁡ (λ0 / λ)
then                                                  λ0 / λ = 0
and with (1)                                          1 / (z + 1) = 0
which becomes true only for z towards infinity. Therefore very big redshifts are to be expected which at most will reach a metrological limit in the spectroscope.

References


[1] 2022 April 03 - CMB Dipole: Speeding Through the Universe ;
     picture of the cosmic background radiation and video with explanation

[2] www.zenodo.org ; DOI 10.5281/zenodo.10966257 ;
     www.roland-sprenger.de: Relativistic addition of velocities in a five-dimensional spacetime
 
[3] www.zenodo.org; DOI 10.5281/zenodo.11086318 ;                      
     www.roland-sprenger.de: Relativistic velocities and dimensions

[4] www.astro.uni-bonn.de/~deboer/
      hubble/Hubble.html
  
[5] T. Kaluza: Zum Unitätsproblem der Physik. In: Sitzungsberichte Preußische
     Akademie der Wissenschaften, 1921, S. 966–972, archive.org.   
        
[6] Klein, O. Quantentheorie und fünfdimensionale Relativitätstheorie. Z. Physik 37,
     895–906 (1926). https://doi.org/10.1007/BF01397481. 
     EOS | Quantum Gravity in the First Half of the Twentieth Century | Oskar Klein (1926):
     Quantentheorie und fünfdimensionale Relativitätstheorie (edition-open-sources.org)
 


Attachment


Calculation of the linear approximation function in the interval 0 to 55 ∙ 106 Ly
 
From the approximational equation   c z = H0 D 
With the distance D of the observed galaxy and the present Hubble-constant H0 ≈ 71 (km/s)/Mpc   = 21,8 ( km/s)/(106 Ly)  follows with the here  chosen denotation for the distance b (bow length)                 z = (H0/c)  D = 7,26 ∙ 10-5 ∙ b / (106 Ly)  .

 
Nearly quadratic development of vH from 0 to 200 ∙ 106 Ly

                                            
Fig. 3: The graphs of the function  v = 300000 [1/cos⁡(b/526) -1]  (red, b in 106 Ly, R = 263 ∙ 106 Ly) and of the approximation parabel      v = 0,564 ∙ b2 (black) match well in the intervall from 0 to 200 ∙ 106 Ly.

 

Development of z(b) in its whole domain

                                               
Fig. 4: The Graph of z(b) (red) from 0 up to the maximum distance of galaxies    π R = 826 ∙ 106  Lj