Stationary Universe Model (SUM)

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The Stationary Universe Model (SUM) has been developed as work in progress by the physicist Peter Ostermann since 2001 from the presuppositon of an eternal, infinite universe on base of the original field equations of Albert Einstein [1] [2] without cosmological constant and physically formulated as a reasonable alternative [3] [4] [5] to the numerically utmost successful big bang model. It has been presented at the 12. Marcel Grossmann Meeting [6] [7] in Paris (MG12 2009, and at the DPG-Frühjahrstagung [8] 2007 Heidelberg before, first pre-prints are at arXiv.org). In comparison with the due to excellent apparent confirmations for a long time prevailing standard model (Concordance Model) in form of the inflationary Lambda CDM cosmology is SUM on base of a new line element representing a concept, which in contrast to the outdated Steady State Theory seems compatible to current observational facts. Despite amazing unexpected successes the model doesn't reach the highly developed stand with regard to various details of the CCM, but is still in an initial phase; it is an encouragement in the sense of clearing up: "sapere aude!" (dare to think) and seems almost a miracle, that on basis of Einstein's equations the idea of an infinite stationary universe turns out to imply clear indication that individual cosmoses are of finite dimensions in space and time. It is in particular this conclusion that arises from the interplay of local special relativity (macroscopically representing quantum mechanics) and universal general relativity (representing gravitation). The unexpected feature is that – describing our evolutionary cosmos as part of a stationary multiverse – the same mathematical model is bringing ideas of various cultural areas to mind about existence of an eternal universe and the creation of cosmoses. While these ideas seem implausibly unbalanced in the prevailing western single-bang standard approach, the new concept SUM, if understood in sense of a Tao Cosmology, seems naturally incorporated into the picture of a multiverse as an oscillation of forces, which might philosophically also be named Yin and Yang. Preliminarily this tries to be a mainstream-breaking attempt to free human spirit from all dogmatic chains unfortunately limiting ubuntu science so far.

After the science-fiction breaking Hossenfelder wake-up call [9] , the central question reamains: What does the dilemma of today's mainstream physics mean in reality? Conclusions from SUM – completely new in contrast to the fictitiously asssumed big bang – allow for a solution in principle of the most puzzling questions of today's Lambda-CDM Concordance Cosmology, which after a paradigm shift to SUM might be resolved at one blow:

  • matter-antimatter baryon asymmetry – is a natural fact in a stationary universe without need for justification.
  • cosmological redshift without universal spatial expansion due to another kind of ordinary gravitational redshift
  • 'dark energy' – homogeneous distribution of e.g. neutrinos (or other WIMPs) filling the gap to critical density
  • dark matter – neutrinos (thermalized in parts)
  • SNeIa magnitude vs. resdshift measurements ...
  • requiring two Hubble 'constants' (the local and the universal one) instead of an accelerated expansion ('Hubble trouble')
  • Planck spectrum from a black-body background of redshifted microwave radiation emitted within a non-expanding multiverse
  • the law of entropy restricted to evolutionary processes (without conflict against any laboratory experience – time after time allowing for 'primordial' nucleosynthesis in 'multi bang' processes of re-creation)
  • SUM may describe a local-bang 'multiverse' (which is just another word for actually one universe with multiple cosmoses).

Team leader and Nobel Laureate Adam Riess' comment on the 'Hubble trouble': "The tension seems to have grown into a full-blown incompatibility between our views of the early and late time universe" (July 12, 2018 at <nasa.gov/feature/goddard/2018/hubble-and-gaia-team-up-to-fuel-cosmic-conundrum>). “At this point, clearly it’s not simply some gross error in any one measurement. It’s as though you predicted how tall a child would become from a growth chart and then found the adult he or she became greatly exceeded the prediction. We are very perplexed [...] The tension could thus provide evidence of physics beyond the standard model (or unaccounted systematic errors) [...] These recent results clearly motivate a detailed study of possible extensions of the Lambda-CDM model and an inspection of the current cosmological data sets, checking for inconsistencies."

According to recently found limitations of proper length and proper time the Hubble radius is setting an uppermost limit for the validity of any approximate SRT concepts and processes transferred to cosmology. Thus the new cosmological model SUM – distinguishing our evolutionary cosmos from the stationary background universe, what also solves Immanuel Kant's first antinomy of space and time – does neither include an expansion of space nor an end of the universe in bleak emptiness as nowadays usually otherwise predicted. It starts from the physical fact that nothing arises from nothing. The entirety of all is regarded as a consistent reality and in sense of stationary universe - not at all static - distinguished from our evolutionary cosmos. Therefore both its age and its extension are infinite, but this does not apply to our cosmos as part of it.

What in view of the FLRW singularity is otherwise called 'age of the universe', now in view of SUM turns out to be rather the maximum age of macroscopic structures. Seemingly opposite observations of e.g. oldest galaxies cannot convince of a singular origin. This in analogy to the commonplace experience that the existence of people with each member not older than about one hundred years does not prove this individual maximum lifetime to be the age of the whole population. Therefore – in contrast to the natural search for the vital history of our cosmos – it does not make sense to search for a continuous history of the entire universe.

In spite of both presupposing in common the perfect cosmological principle the model SUM with its statistically constant values of redshift is fundamentally different from the Steady State Theory of an expanding universe [10] [11]. The mathematically exactly derived values of redshift correspond to the statistically constant Euclidean distances in universal coordinates which are also called comoving coordinates because of historical grounds. After a preliminarly purely mathematical derivation of a stationary background radiation with Planck spectrum without reduction to a fictional big bang) a previous version of SUM [12] has been developed to Model of a Stationary Background Universe Behind Our Cosmos (2013) [13]. The underlying articles are at "independent-research.org" ready for download.


The stationary line element

Fig. W1. – The assignments ([math]\Omega_\mathrm{M}[/math], [math]w_\mathrm{M}[/math], [math]\Omega_\mathrm{\Lambda}[/math]) = (0, 0, 1), (0.1, 0, 0.9), (0.27, 0, 0.73), (1,–1/3 , 0) , (0.6, 0, 0.4), (1, 0, 0), i.e.: Steady-state Theory [math]a_\mathrm{SST}(t') = \mathrm{e}^{Ht'}[/math] [upper grey solid line], this model discussed as a possible option in the past; a first alternative to [math]a_\mathrm{CCM}(t')[/math] with higher value [math]\Omega_\mathrm{\Lambda}[/math]) [blue broken line]; today's Concordance Model [math]a_\mathrm{CCM}(t')[/math] [blue solid line]; the stationary ultra-large scale universe [math]a_\mathrm{SUM}(t') = HT'[/math] = [math]1+Ht'[/math] [red straight line]; a second variant to [math]a_\mathrm{CCM}(t')[/math] with lower value [math]\Omega_\mathrm{\Lambda}[/math]) [lower blue broken line]; the Einstein-de-Sitter model [math]a_\mathrm{EdS}(t') = [/math] [math](1 + \mathrm{3/2}Ht')^\mathrm{2/3}[/math] [lower grey solid line, favored before the SNe-Ia breakthrough]. In contrast to all other comparable values the CCM best-fit parameter [math]\Omega_\mathrm{\Lambda}[/math]) = 0.737 (blue solid line) seems determined by the condition, that its line should meet the SUM scalefactor (red) at its boundaries, i.e. at HT' = -1 exactly and at Ht' ≈ 0 approximately today. – SUM's unexpected local character as pseudo-proper FLRW form is concluded according to the corresponding sections above.

In contrast to static the word stationary means, that all things considered remain at large the same, though single components may permanently change. The concept of SUM can be based on a deductive development different from any inductive conclusion. Two postulates are sufficient to derive SUM's stationary line element:

  • Postulate I – With respect to sufficiently large scales the universe is stationary, homogeneous and isotropic.
  • Postulate II – Except for local deviations the universal speed of light is c* = c .

From these postulates - both extraordinarily simple - SUM's line element dσ*SUM of a spatially flat non-empty universe results uniquely to be

[math] \mathrm{d}\sigma_\mathrm{SUM}^{\ast} = \mathrm{e}^{Ht^{\ast}}\mathrm{d}\sigma_\mathrm{SRT}^{\ast} [/math].

Here H stands for a true Hubble constant (which in contrast to CCM's time-dependent conventional Hubble parameter H0 of the big bang cosmology is called significant; the abbreviation dσ*SRT stands for the line element of special relativity, where the additional symbol '[math]\,[/math]*[math]\,[/math]' is indicating at each occurance the feature, that corresponding values are related to universal coordinates (besides comoving coordinates l[math]\,[/math]* also the 'conformal time' t[math]\,[/math]*).

According to SUM the universal coordinates temporarily match the proper lengths and the proper times of special relativity again an again. From the line element's explicit quadrat written out is the constancy of the universal speed of light obvious because of c* ≡ dl[math]\,[/math]*/dt * = c (for dσ*SUM = 0).

Redshift and the significant Hubble constant

Fig. W2 - 'Hubble Trouble' - SUM's prediction of the Hubble contrast which is completely incomprehensible in the framework of Lambda-CDM 'big-bang' cosmology: Two different values for the Hubble 'costant' [math]H_{\mathrm{local}}[/math] = [math]H_{\mathrm{CCM}}[/math] and [math]H_{\mathrm{universal}}[/math] = [math]H_{\mathrm{SUM}}[/math]. -- Top panel (a)}: The blue solid line does represent the real values [math]z_{\mathrm{observed}}[/math] of the SNe-Ia maesurements,the red boken SUM line is peliminarily neglecting possible 'peculiar flows' or local inhomogeneities. The maximum deviation [math]\delta z[/math] ≈ 0.002} ≈ 600 km/s/c within [math]z[/math] < 0.027 does correspond too a maximum contrast [math]H_{\mathrm{local}}[/math] / [math]H_{\mathrm{universal}}[/math] - 1} of about 9% at this distance while [math]H_{\mathrm{universal}}[/math] ≈ 67 km/s/Mpc. -- Bottom panel (b)}:Within [math]r^* \lt 110[/math] Mpc the blue line corresponds to [math]H_{\mathrm{local}}[/math] ≈ 73 km/s/Mpc, while the mean value in the transition zone (up to [math]z[/math] ≈ 0.13) is about [math]H_{\mathrm{trans}}[/math] ≈ 70 km/s/Mpc. The difference leads to [math]H_{\mathrm{local}}/H_{\mathrm{trans}}[/math] - 1 ≈ 4.7% up to [math]H_{\mathrm{local}}/H_{\mathrm{universal}}[/math] - 1 ≈ 8.9% thus approximately corresponding to the range of the local Hubble contrast reported by Jha, Riess & Kirshner to be 6.5% ± 1.8%.}

It is simply wrong to claim that the cosmic redshift proves any expansion. The universal redshift of distant galaxies is understood in the SUM framework not as a Doppler effect but in sense of an extended Einsteinian gravitational redshift. This corresponds exactly to the well-known fact that the redshift once measured by Pound and Rebka is not at all caused by a flight between emitter at the top of the Jefferson tower and a receiver on the ground (even Edwin Hubble [14] had already considered such an alternative explanation instead of a fictive mysterious Doppler effect which would be out of place in this context).

The only difference in comparison with an ordinary gravitational redshift is that in this case the potential takes a time-dependent form, which due to its sign always affects as a red shift. According to the general definition of the redshift parameter z = λ[math]\,[/math]observed / λ[math]\,[/math]emitted – 1 do result with

[math] z_{\mathrm{SUM}} = \mathrm e^{Hl^{\ast} / c} - 1 [/math]

which like the significant Hubble constant in the SUM frameword are independent of time. Such values apply to all cosmic structures, which by commonly accepted presupposition are statistically at rest with respect to universal coordinates (x*, y*, z*). Thhe independence of time applies to all other quantities which are actual functions of z. In particular the constant universal distances

[math] l^{\ast} = \tfrac {c}{H} \ln {(1+z)} [/math]

are real measurands in addition to the local SRT proper lengths due to the constant values of redshift of corresponding objects [15]. Contrary to big bang theory the universal distance l* is uniquely maesurable - though indirectly - in the SUM framework.

Universal time t[math]\,[/math]* and the limits of proper length and proper time

Because of the exponential mean time scalar eHt *, all corresponding relative temporal changes depend on differences t * – tR*</sup> according to eHt */ eHtR* = eH(t * – tR*) solely, where tR*</sup> is a respective reference point of universal time. Therefore no special fixation of the universal time scale t * is preferred. This fundamental feature is what allows arbitrarily to set tR*</sup> = 0 for arbitrary complexes of observation. The same fact applies to all other coherent events, in particular also to spontan emission and subsequent absorption of photons in distant galaxies.

In contrast to the respective local quasi-proper time t ' = T ' – TH (where TH = 1/H is the Hubble time) the universal time t * has neither a beginnig nor an end. According to the SRT line element remaining locally valid in any place and ever again (dσ2SRT = c2dt[math]\,[/math]2SRT – dl[math]\,[/math]2SRT) the intervals of proper length and proper time are defined always together. From SUM's line element the mathematical approximations are easily concluded yielding

[math] \mathrm{d}t_{\mathrm{SRT}}[/math][math]\mathrm e^{Ht^{\ast}}\mathrm{d}t^{\ast} [/math] ,
[math] \mathrm{d}l_{\mathrm{SRT}}[/math][math]\mathrm e^{Ht^{\ast}}\mathrm{d}l^{\ast} [/math]

to be temporarily valid within locally restricted regions

[math] l^{\ast} \ \stackrel{!}{\lt } \ R_{\mathrm{H}} [/math]

of the infinite Euclidean space (dl 2 = dx[math]\,[/math]2 + dy[math]\,[/math]2 + dz[math]\,[/math]2), where RH = c/H is the Hubble radius. The same applies to each SRT concept at all. If, however, there was perfect equlity in the above approximations with equal signs '=' instead of approximate signs, the each cosmological line element would be nothing but that of SRT itself – whose Riemann, Ricci, or Einstein tensors and therefore the entire universal mass energy density would mathematically vanish to zero. The last inequality above results after a double coordinate transformation t * = ln[math]\,[/math](HT ')[math]\,[/math]/[math]\,[/math]H and l * = l '[math]\,[/math]/[math]\,[/math](HT '), where T ' corresponds approximately to a local proper time, which is limited according to the above inequality respectively.

The FLRW form of the SUM line element

If according to t * = ln[math]\,[/math](HT ')[math]\,[/math]/[math]\,[/math]H only the universal time t * was transformed, it would follow the stationary line element in a Friedmann-Lemaître-Robertson-Walker form (FLRW form) with the simplest scale factor a[math]\,[/math]SUM = HT ' yielding

[math] \mathrm{d}\sigma_\mathrm{SUM-FLRW}^{\ast 2} = c^2\mathrm{d}T^{\,\prime\,2} - (HT^{\, \prime})^2 \mathrm{d}l^{\ast 2} [/math]

without changing any real physical facts. It is easily verified, for example, that from SUM's exact Hubble relation between redshift z and universal distance l[math]\,[/math]* above holds in its time-independent form, too. It is of decisive importance that the singularity quasi-proper time T ' = 0 of an apparent 'beginning' applies only to locally restricted areas and does not at all represent an age of the entire universe. It means a maximum lifetime of spatially restricted (evolutionarry) structures bound to l[math]\,[/math]* < RH instead (while T ' = 0 means t * = –oo with respect to the universal time coordinate).

Energy density and the negative gravitational pressure

The SUM component G00 of the covariant Einstein tensor - obviously corresponding to the universal energy density - is temporally constant. On the other hand the mixed tensor component G00, which would apply at application of gravitational equations on local perfect fluids, seems to depend on universal time t[math]\,[/math]*. But that such a dependence over arbitrarily large periods would be unrealistic again - opposed to locally restricted proper times t - follows from the fact, that any special reference point of universal time t[math]\,[/math]* does not exist. The SUM line element implies a negative gravitational pressure of –1/3 the critical density at each arbitrarily chosable zero point of universal time t0* = 0. In contrast to ordinary particles in a box (which due to their positive pessure would immediately diffuse without its walls), galaxies within a limited region of the stationaty universe would mass together due to their mutual attraction if there was no negative gravitational pressure (caused by the gravitational potential of matter and energy outside).

SUM result of two different values ('Hubble Trouble') for the local and the universal Hubble constant

Fig. W3 - Left panels (a)-(e): Comparing the SUM magnitude-redshift prediction (for [math]\kappa[/math] = 0) with the SNe-Ia data and the CCM-prediction, there is a straightforward SUM agreement on large universal scales z > 0.1 where the universe may be rightly regarded homogeneous and isotropic. The red SUM-line coincides almost completely with the blue CCM-line (though of a 9 % higher Hubble constant). -- Intermediate right panel (b): Full scale compatibility with the High-z Supernova Search Team's (HZT) Riess et al. 'gold' data, if given a local Hubble contrast analogous to W2. -- Lower right panels (c),(d),(e): Also in these panels again including the data of the Supernova Cosmology Project (SCP) as well, the broken straight lines are determined by the method of least quadratic deviations and should ideally prove congruent with the respective z-axis.

In both panels of Fig. W2 the solid blue lines may represent the real SNe-Ia observations, the broken red lines respectively below do represent straight SUM. A maximum deviation [math] \mathrm \delta z [/math] ≈ 0.002 corresponds to a maximum Hubble contrast of ≈ +9%. With [math]H_{\mathrm{universal}}[/math] ≈ 67 km/s/Mpc e.g. this would mean [math]H_{\mathrm{local}}[/math] ≈ 73 km/s/Mpc within [math] \mathrm{r^{\ast}}[/math] < 110 Mpc, i.e. within [math]z[/math] < 0.027 (while the mean value in the transition zone is about [math]H_{\mathrm{trans}}[/math] ≈ 70 km/s/Mpc).

Now recently there has been reported another "local value" [math]H_{\mathrm{0}}[/math] = 73.2 km/s/Mpc with an uncertainty of only 2.4% [16] [17] as well as 71.9 km/s/Mpc ±3.8% (approximately corresponding to 72.8 km/s/Mpc ±3.3% in Bonvin et al. [18].

Close to Freedman's value of 72 km/s/Mpc [19] , but in clear contrast to 67 km/s/Mpc predicted by the Lambda-CDM cosmology from the new Planck high-redshift measurements [20] – or approximately also the 68 km/s/Mpc of Cheng Cheng & Qing-Guo Huang [21] – this remarkably means a Hubble contrast of about +9% again, the latter almost perfectly matching the original SUM prediction of 2007 in "Indication from the Supernovae Ia Data ..." {s. above as well as also Fig. W2}. Apparently the authors of the new report presuppose the 'curved' shape of a Lambda-CDM Hubble diagram (without explicit justification) and therefore, of course, cannot find any difference between the local and global value of the Hubble constant. This seems to be also the reason that the Hubble contrast previously reported by Jha, Riess, & Kirshner [22] is no more discussed. There, however, it convincingly read: "... the feature is present in the Hubble flow SN sample, and this has important implications for using SN Ia as tools for precision cosmology."

Now the occurrence of two different values for the Hubble constant seems another unexpected coincidence between the contradictory Lambda-CDM framework and a consistent SUM concept. This proves that there may be also other surprising features which are still to be found. The other way round, with the clear SUM relations on hand it seems even possible to determine the peculiar energy distribution in our probably anisotropic 'local' cosmic environment [math]z[/math] < 0.1 with help of high precision measurements of apparent SNeIa luminosities.

Comparison with the Supernova-Ia data

To explain the Supernova-Ia data [23] [24] on sufficiently large universal scales the model SUM does not need any 'dark energy' and thus is also different from the Lambda-CDM model as standard model of today's cosmology. Deviations in the region z < 0.1 are here reduced to a local Hubble contrast as in order of magnitude actually obseved. The universal SUM distance modulus results from the stationary line element as usual yielding

[math] (m-M)_\mathrm{SUM} = 5 \log{[(1+z) \ln{(1+z)}} + 25+ 5 \log{(\tfrac{R_\mathrm{H}} {\mathrm{Mpc}})} [/math],

where the apparent magnitude is represented by m and the absolute brightness of Supernovae type Ia standard candles by M. The MG12 document Indication from the Supernovae Ia Data of a Stationary Background Universe [25] demonstrates the immediate agreement of SUM's distance modulus with the SNeIa data for universal scales z > 0.1 (if at all beyond such scales - in order of magnitude corresponding to that of the Sloan Great Wall - the universe can be rightly regarded to be homogeneous und isotropy). There in a number of systematically successive illustrations it is shown why to overcome the expected predictions of the Einstein-de-Sitter model and that of the outdated Steady-state theory a strange concept of a 'dark energy' representing a fictitious cosmological constant seemed necessary to explain the new observational data. Based on Einstein's original gravitational equations without such a constant of his "biggest blunder", however, the SUM line element had been immediately confirmed by the new Supernova measurements [26]. Even taking into account an infinite number of stars, due to redshift and time dilation there results a realistic finite value for the low brightness (darkness) of the night sky. Olbers' paradox is solved here without any hypothesis concerning a 'big bang' or any spatial expansion.

Alternative to a material contribution of 'dark energy'

In addition to the inhomogeneous contribution of indirectly meausurable dark matter there should be also a roughly homogeneously distributed transparent one, which in accordance with SUM is not even detectable by the effect of gravitational lensing. This homogeneous part - in the Lambda-CDM framework ascribed to dark energy could fill the gap between that amount of the matter actually observed today and the critical density required for a flat universe without spatial curvature. At the same time there is no expansion of space and therefore no need for any additional energy to accelerate the fictive universal expansion.

It is simply a wrong statement to read: "The measured [math]H_0[/math] is also highly inconsistent with the simplest inhomogeneous matter models invoked to explain the apparent acceleration of the universe without dark energy" [27], s. both panels b) of Fig. W3 on the left and on the right instead. That popular fallacy seems to rely on the erroneous presupposition of the fictive big-bang framework as a certain scenario. In fact this premature statement of Riess et al. 2011 is easily disproved. The refutation of that claim does not even require any own special astrophysical expertise, since their publicly available and Nobel awarded data seem rightly unimpeachable. Each ubuntu child may be able to debunk it as superstition (unfortunately almost soundig like a fake) by straighforwardly plotting those authors' own SNeIa 'gold' data in the universal redshift range z > 0.1. </span>

If, however, dark matter was built in parts of thermalized neutrinos – as suggested by SUM and thereby possibly solving the most irritating 'dark energy' puzzle of modern cosmology – then it might be possible to disprove the big-bang origin of the CMB directly by identifying hDM particles emitting the CMB or by detecting any such photons within shielded cavities (today assumed instead to come from z > 1000). A preliminary assessment would yield about 10 locally emitted hDM photons a year within a 1000-m[math]^3[/math] tank (in rough order of magnitude; ideally such a 'surrounding detector' would have to be cooled down below 2.7 K). The unavoidable thermal radiation emitted from any respective measuring device, however, might make a clear classifcation of single photons most likely difficult if not impossible in that frequency range.

Planck spectrum of redshifted microwave background radiation and a homogeneous part of 'dark matter'

Fig. W4 - The bold solid black line shows the total CMB spectrum for [math]\kappa[/math] = 2 as actually observed. The bold broken red line shows the emission of the hDM radiation exemplarily in a local sphere of [math]\Delta r^*[/math] = 100 Mpc as calculated. In addition, here the thin red solid lines show respective parts coming from within z = Z in this case from bottom to top by Z = 0.1, 0.2, .. 1.0 respectively.

In the famework of SUM the microwave background radiation (CMB) is explained in parts as thermal radiation of 'dark matter'. This concept may solve two fundamental problems at the same time: (a) there is no macroscopic distribution of matter without temperature or without thermal radiation; and (b) any stationary background radiation must have origined within the infinite universe. In an initially purely mathematical derivation it has been possible, to compose a perfect Planck spectrum out of redshifted radiation without the hypothesis of a 'big bang' [28] [29].

At a statistically mean universal temperature the inhomogeneities of that cold thermal radiation seem to reflect the obviously existing acoustic oscillations of matter, which had formed also here over astronomical periods of time by the interplay of gravitation and radiation pressure. The chance alternatively to explain the CMB anisotropies within SUM is obvious from the comparison of different illustrations as e.g. Figure 14-e of N. A. Sharp[30] and Figure 7(b) of C. L. Bennett[31] or Figure 5 of R. Piffaretti [32]. In contrast to the concept of an etrenal infinite universe itself, the preliminary explanation of the Planck spectrum seems falsifiable with help of the Sunyaev–Zel'dovich effect.

In any case it is no longer possible to take the sheer existence of the CMB as a certain proof for a big-bang origin of the entire universe.

The alternative Sunyaev-Zel'dovich effect

Fig. W5 - The CMB parts [math]\rho _Z^*[/math] coming from behind [math]z = Z[/math] according to SUM. Thin curved solid red lines are shown here from top to bottom for [math]Z[/math] = 0.1, 0.2, .. 1.0.

The originnal effect of Rashid Sunyaev and Yakov Zel’dovich describes a slight distortion of the CMB background radiation by inverse Compton scttering of galaxy clusters. Different from the Cosmological Concordance Model (CCM) framework, according to its preliminary explanation in SUM [33] [34] that effect [35][36] should be gradually weakened with increasing distances. At redshift z > 1 it should approximately vanish except for statistical fluctuations. Remarkably the corresponding Planck-catalogues [37][38] do contain predominantly observations up to this limit.

Now in the statistics of the Sunyaev-Zel'dovich effect with the Planck 2015 data on hand, there has come a chance to decide whether or not the CMB once originated after a 'big bang', or whether, the other way round, the CMB is emitted from 'dark' matter within a non-expanding background universe.

Fig. W6 - The realistic SZ effect among other CMB distortions like e.g. re-shifting inhomogeneities. -- Upper panel (a): Isothermal CMB fluctuations of order [math]y[/math] ≈ 10^{-4} are plotted in faint red, while the thin curved blue and grey lines show changes of the local SZE. -- Bottom panel (b): This highly important Figure demonstrates a possibly resulting realistic SZ signal as bold red line ([math]X_{\mathrm{back}}[/math][math]-5 \cdot 10^{-5}[/math]) where SUM's isolated frequency shift according to the broken green line seems largely compensated by such a random 'back'-ground inhomogeneity (lower intensities might be understood as lower y's).

Other observational facts and open questions

In addition to those observational facts of universal redshift and the apparent SNe-Ia magnitudes, which usually are claimed to be fundamental pillars of the big bang theory, but on the other hand most naturally to describe in the SUM framework, other important features seem to be alternatively explainable - though at least rudimentarily so far.

  • Nucleosynthesis - In a stationary universe the respective parts of all material components is determined according to the laws of quantum mechanics by the requirement that in 'original processes' they be restored at the same rates as they have disappeared before in gravitational centers of extreme strength. The big bang model does presuppose in no place that outside a corresponding area of extreme temperature and density no other such events may happen (multi bangs). It even cannot be ruled out that the release of matter in form of jets - incluasive plasma bubbles - provides a permanent restoration of primordial cores and her components.
  • Quasar and older galaxy dstributions at great distances – Since in the years about 1960 it has been discovered, thet the quasar 3C273 was no nearby star, but at a redshift of z = 0,158 in the region of distant galaxies, there have been later observed such bright quasi-stellar objects (QSOs) also in very much larger distances. In the meantime there have been discovered quasars up to redshifts of at least z = 7,1. It follows, that the nearest of them may be located today in cosmic neighborhood. Even a final conclusion that QSOs are only at distances above average would be selfunderstood, if the evolution of our 'local' cosmos had started from a multi bang event (compatible with SUM). This could also be a reason that most distant galaxies may look younger. On the other hand the measured frequency scale of quasars seems to be at first reduced (and then limited) at large scales z > 2-3 by effects of self selection like the so-called Malmquist bias, while all objects below a weak brightness limit remain unconsidered.
  • Principal SUM differences from a 'big bang' origin out of nothing – Tough various fundamental observationall facts known as 'pillars' of big bang cosmology seeem numerically to be brillantly confirmed, there actuallly remain several important questions. A false vacuum of quantum fluctuations[39] could not exist without an energy density, if our evolutionary cosmos had originated according the big bang theory out of that an essence. Even such a tohu-va-bohu would require a physical description as a chaotic background by Einstein's graviational equations. Except of the SUM framework no such solution seemms to exist. Indeed here do not arise any correspondihg questions of the Lambda-CDM Concordance Cosmology or they appeear in completely different light.
  • Restriction of the law of entropy to evolutionary processes - At gravitational processes of new recreation, possibly in super massive objects like active galaxy nuclei (AGNs), hypernovae or sources of gamma-ray bursts, the entropy has to temporarily decrease in local areas of an eternal universe to remain stationary in total. To this effect it requires a restriction of the law of permanent increase of second law|entropy, without a risc to contradict any experimental experience made by living beings ever. The equirement of a temporarily local restriction of entropy is inevitably necessary for every stationary concept at all (otherwise there would arise that strange heat-death of the universe as already discussed long ago). Even if such a restriction may sound improbable, however, it is less improbable than an origin of the entire universe from this nothing.

To overcome fundamental fine tuning problems of the big bang theory, which are related to spatial flatness of the universe, the non-existence of magnetic monopoles or its fictive horizon, there has been invented a phase of cosmic inflation. This should have been driven by a scalar inflaton field, which never has been observed, though. Also the fact of the universal matter-antimatter asymmetry is significantly incompatible to experiental experience in today's predominant framework of the current Cosmologial Concordance Cosmology. Just as little it is understood from the original big bang theory, that after a unversal origin from nothing there could be consistent steady laws of nature at all (and not all events do happen only chaotically to this day). Irritating is also an appeal to an imperfect cosmological principle arbitrarily excluding only time from the otherwise pefect universal symmetry. The biggest of all appears in the coincidence that the span of time t0 usually called age of te universe should approximately only today equal the reciprocal value of the conventional Hubble parameter 1/H0.

According to SUM the anthropic principle does not only apply to the stationary universe itself, but to all local evolutionary cosmoses possibly therein. By the adjustment of the big bang concept having been extended in the meantime - from originally only one big bang universe via additional parallel universes up to one multiverse now all-embracing again - cosmology finally seems fürther to develop to the SUM concept with his natural distinction of the stationary universe from evolutionary multi bang or local quasi bang cosmoses. In the context of the alleged universal age and assumed boundaries it has to be remarked that corresponding ideas have been extended again and again, since Immanuel Kant recognized in star nebulae, even observable at those times, possible other Milky Ways.

The reconcilableness of relativity theory (RT) and quantum mechanics (QM)

Six years after the intermediate completion of general relativity (GR) Albert Einstein has confronted in Geometrie und Erfahrung [40] two mathematically possible interpretations and thereby explicitly admitted Poincaré's claim of understanding the non-euclidean geometry [41] ("Sub specie aeterni in my opinion Poincaré is right with this interpetation"). Therefore it is possible to ascribe the non-euclidean geometry of GR not to space and time itself, but to gravitationally affectable unit sticks, clocks, and to all other real physical objects and otherwise to keep euclidean geometry for the one complete description of all natural event. This turns out to work with the help of Nathan Rosen's bi-metric relativity [42]. On the other hand such a chance would even explain why in spite of big efforts like in string theory, loop quantum gravity no concept seems to have convincingly succeeded to quantize spacetime and thus to basically reconcile the conventional relativity theory with quantum mechanics in a unified theory of gravitation and quantum mechanics. According to SUM, however, seems possible as a matter of principle to allow valid statements in future to otherwise mere mathematical singularities of GR or at least protect from misinterpretations. Besides the black holes this also concerns the singularity theorems of Hawking und Penrose, where in the absence of a consistent unified theory the detailed quantum structure of matter had to be neglected so far.

Numerical hints to the existence of 24 elementary spin-1/2 torsion particles

Though of unprecedented numerical success in describing the observational facts of modern cosmology, there is another strange hint that the inflationary Lambda-CDM big-bang model might fail. This is because of an apparent materialization of an antisymmetric torsion tensor [math]T_{ikl}[/math]. The universe seems constituted of 24 elementary spin-½ particles which are 6 leptons + 3 colors * 6 quarks. These curling structures, behaving as 'whirl' particles, may represent exactly the 24 components of a real torsion tensor which naturally include 6 'temporal' + 3 * 6 'spatial' constituents

[math]T^i_{kl}[/math] = [math]T^0_{\alpha\beta}+T^\gamma_{\alpha\beta}[/math],

what seems to be more than a mere coincidence [here Latin in­dices i, k, l ... = 0, 1, 2, 3 in contrast to Greek spatial indices, here [math]\gamma[/math], [math]\alpha[/math], [math]\beta[/math] ... = 1, 2, 3 only]. In addition, of the 6 'lepton'-components in [math]T_{\alpha \beta}^0[/math] there may be 3 'electric' + 3 'magnetic' (according to the assignment in the electromagnetic field strength tensor), thus reflecting three [math]e, \mu, \tau[/math] particles plus three respective [math]\nu _e, \nu _\mu, \nu _\tau[/math] neutrinos. As has been shown by Landau & Lifshitz [43] long time ago, however, the physical existence of a non vanishing torsion tensor would contradict Einstein's equivalence principle. This principle is underlying the literally geometric interpretation of his gravitational equa-tions, while in view of SUM the geometric approach fails in reducing physics to exclusively Riemannian properties of non-Euclidean space and time as also indicated by the existence of tetrads [44]. Therefore not only a microscopic violation of the fundamental equivalence principle would contradict the whole spacetime concept where today's Concordance(Consensus) Model of cosmology is relying on. In view of extended elementary spin-½ torsion structures (in most situations identifiable and acting as wholes) also Heisenberg's uncertainty principle can be understood in contrast to the strange behavior of 'point' particles otherwise unrealistically presupposed so far. In any case, contrary to its historical reception, quantum mechanics may be understood as theory of extended whirl structures of variable shape (i.e. as theory of possible 'torsion particles'). A first deductive attempt to extended structures (outlined by [45].[46] seems to explain Bohr's energy-frequency formula and to imply Heisenberg's uncertainty relations in accordance with approved principles of relativistic physics. Thus this feature is shown to be anything but an incomprehensible surprise after all.

Particles like electrons and protons as well as their constituents are neither real mass points without any extensions nor one-dimensional 'strings', nor two or higher-dimensional 'branes', but they are three-dimensional deformable structures with particle parameters in form of several characteristic constant integrals pertaining to rest mass, charge, and spin among others. This concept, though, does not deny fundamental achievements of mathematical abstractions (like in particular the concept of 'point masses' in Newtonian mechanics), of course. Also the quantum mechanical result that particles do not have an unambiguous momentum is only a natural statement in view of interacting extended structures, where a possibly varying momentum density is self-evident. On the other hand, in spite of unavoidable uncertainties due to relative inner motions, the total momentum of a free particle can be exactly determined. While inner details may prove strange, the natural laws behind should be clear. In contrast to solid bodies, remarkable characteristics of torsion structures are a completely different steadiness and their temporally dissolved identities. It is obvious that a theory of elementary whirl particles subdivides kinematics and dynamics of existing structures from a theory of production and transformation ('Erzeugung und Verwandlung' in Einstein's words). Contrary to naive point-particle models, the new concept allows a fundamentally simple understanding of transformations. While concerning free motion of whole objects, only kinematics may be of interest, in particle physics inner forces play the decisive role. Even the indistinguishability of elementary particles of same kind – otherwise a complete mystery – is no longer unintelligible as well as interference and diffraction phenomena.

The torsion model is independent of the question whether such particles may exist as material objects in vacuum or in form of whirl structures in a continuously extended medium. Nature may show both aspects (like spiral nebulae in a background of dark matter, for example). It seems an evident chance that:
– Elementary particles are whirl structures.
– Due to the conservation law of angular momentum the tiny extended free torsion structures are in parts steady for cosmologcal periods of time.
– Torsion particles are subject to processes of formation and decay.
– In transitional phases whirl structures lose their identity.
– On one hand torsion structures may be approximately described as particles.
– In other situations torsion structures may be approximately described as waves.
– Detailled velocities of parts of torsion structures are realized at the same time as the motion of their respective center of gravity. Natural relations of uncertainty follow from that.
Thus the elementary particles, which are assumed to constitute the entire universe, are essentially different from those eternal solid 'atoms' of the pioneering antique philosophers Leucippus and Democritus. Since it is clear that only at the price of unavoidable uncertainties torsion structures can be dealt with as extensionless point particles, a complete relativistic mechanics has to contain a future consistent formulation of quantum theory. As already addressed in SUM14, an appropriate basis will be Rosen's bi-metric relativity [47][48] after fixation to the (preferred) universal frame.

History, SUM precursors, various related attempts

It is impossible to do cosmology without appropriate principles which – besides the indispensable compatibility to observational facts – should fulfill the criteria of simplicity, adequacy and clarity. In the absence of such criteria not even the decision between a heliocentric and a geocentric conception of our planetary system would be possible within GR because of legitimate mutual coordinate transformations.

Albert Einstein establishing relativistic cosmology has started from the presupposition of an eternal universe [49]. Hiis attempt failed because he had searched a purely static solution, whose line element - as opposed to that of Friedmann as well as to about a hundred years later now to the line element of the stationary solution SUM - does not contain in the corresponding gravitational potentials gik any coordinate of time. This also applies to the original form of De-Sitter's model, which however allowed a transfer into the later Steady-state Theory by a mere coordinate transformation.

  • Steady-state Theory: Under the name Steady State Theory, which tried to describe a steady expansion, in the year 1948 a model was presented (due to differing observational facts later failed in various versions) which in view of its authors because of permanent new creation of matter from nothing - in clear contrast to SUM - should have been regarded to be filled with constant matter density. Only recently it became known that already Einstein essentially anticipated and rejected the concept of this theory. A new C field introduced by Sir Fred Hoyle may be regarded as precursor of the scalar field of today's inflationary Lambda-CDM model. In spite of its name Steady-state Theory the redshift values of individual galaxies do not result steady as constants (except of their peculiar motions), but corresponding to the titles of those articles (explicitly refering to an expanding universe) would continuously increase.
  • Coasting Cosmology: A model of a more general form of the linee element than that of SUM shown above - but including a scale factor HT - has been discussed at its time as a universe of coasting expansion [50], before in the year 2011 a closely related concept has been discussed once more in the context of today's big bang Lambda-CDM cosmology [51]. Both models are on principle different from SUM (the fundamental consequence of constant values of redshift from sources in constant universal distances was originally not even mentioned there).
  • Chaotic Inflation: The physicist Andrei Linde has developed the concept of chaotic inflation[52][53] and thereby overcame the theoretical fixation on one single big bang from nothing. On the other hand there is dealt with parallel universes, where each of them had origined with own inflation and even possibly own laws of nature. This clearly contradits SUM's fundamental presupposition of many cosmic areas within one universe of everywhere the same laws of nature (multiverse). In that context - though of completely different mathematics - also appeared a title 'Stationary Universe Model'[54] for the first time. With regard to a universe eternally situated in change there obviously always had been correspondig ideas, which seem related to cycle of growth and decay in eastern religions.

Also in view of Epicurus there exists an infinite number of worlds in infinite large space, what has ben reported by Lucretius in On the Nature of Things. According to Rhazes - in sense of timeless laws of nature - there shall be an eternal matter from atoms, an absolute and eternal universal time (different from relative 'proper' time) as well as an absolute and eternal space being unchanging. Similar concepts of space and time are found later from Newton. Nikolaus von Kues has thought about a multiplicity of worlds in a single infinite universe. The according to modern natural science revitalized idea of an eternal infinite universe, however, is existing at the latest since Thomas Digges. His contemporary Giordano Bruno defended the idea of "many worlds ", what is shown in particular from his document of 1584 About the infinite, the universe and the worlds (choice of words of this title corresponds also to SUM's distintion of "Universe" from " cosmoses"). He had the idea that this eternal infinite universe was interwoven by the same divine pulse. His thinking had become influenced by the predecessors mentioned above. In the context of SUM it has now been shown that corresponding concepts are not only compatible to Einstein's equations, but seem to follow from them. On the one hand, although this new model is hardly realized till now by mainstream-cosmology, but on the other hand the current Cosmological Concordane Model (CCM) including its highly speculative phase of inflation (founded by no experimental experience but indispensable there) seems to strike increasing scepticism with quite a few people.

The idea leading to SUM, which is shown to be the only arguable solution of Einstein's original equations without cosmological constant, is that no universal horizons must limit physical reality. Beyond local applicability any 'proper' SRT concepts will prove overstrained in the conventional GRT framework. This insight goes back to a seminal analysis ""Die Einweg-Lichtgeschwindigkeit auf der rotierenden Erde und die Definition des Meters"" of Einstein's ideas [55] Therefore SUM demands a scientific discussion instead of an endless sham fight against the outdated historical attempt of the SST. In contrast to apparent attempts based on premature fictions of a big bang and accelerated expansion of space it is a matter of course that cosmology is not to teach nature, but to learn from and to describe the universe in awe and with due modesty.

Literatur

  • Peter Ostermann, "SUM – Model of a Stationary Background Universe Behind Our Cosmos", digIT Verlag, 2014, ISBN 978-3-941550-25-4.
  • Steven Weinberg, "Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity", Wiley, New York 1972, ISBN 0471925675.
  • Fred Hoyle, Geoffrey Burbidge, and Jayant V. Narlikar, "A Different Approach to Cosmology", Cambridge University Press, 2000, ISBN 0-521-66223-0.
  • Collected Papers of Albert Einstein (CPAE) – Einstein Papers Project

Einzelnachweise

  1. A. Einstein, "Die Feldgleichungen der Gravitation", Sitz.ber. Preuß. Akad. Wiss., 25. November 1915, 844-847 – (reprint Doc. 25, CPAE Vol. 6)
  2. A. Einstein, "Die Grundlage der allgemeinen Relativitätstheorie", Ann. d. Phys. 49, 769-822 – (reprint Doc. 30 CPAE Vol. 6)
  3. P. Ostermann, "Ein stationäres Universum und die Grundlagen der Relativitätstheorie", arXiv:physics/0211054, 2002/04
  4. P. Ostermann, "Problems of single-bang cosmology from the perspective of the mathematically simplest alternative based on Einstein's equations"
  5. P. Ostermann, "New physics of an eternal infinite multiverse instead of today's singular big-bang cosmology", preprint 2018
  6. P. Ostermann, "Relativistic Deduction of a Stationary Tohu-va-Bohu Background Cosmology"; in: Damour Th., Jantzen R. T., & Ruffini R. (Eds.), Proc. MG12, W.Sci., 1408-1410, PDF article 2012; original Talk 2009
  7. P. Ostermann, "Indication from the Supernovae Ia Data of a Stationary Background Universe"; in: Damour Th., Jantzen R. T., & Ruffini R. (Eds.), Proc. MG12, W.Sci., 1373-1375, PDF article 2012; original Talk 2009
  8. P. Ostermann, "Das relativistische Modell eines stationären Hintergrunduniversums und die Supernova-Ia-Daten"; DPG-Vortrag/GR-205.2, 2007
  9. S. Hossenfelder: "Lost in Math", Basic Books, New York 2018
  10. H. Bondi & T. Gold, "The Steady-State Theory of the Expanding Universe" Monthly Notices, Royal Astronomical Society, vol. 108, 252-270, 1948
  11. F. Hoyle, "A New Model of the Expanding Universe" Monthly Notices, Royal Astronomical Society, vol. 108, 372-382, 1948
  12. P. Ostermann, "The Concordance Model - a Heuristic Approach from a Stationary Universe", arXiv:astro-ph/0312655v6, 2013 [with an error in equation (9)]
  13. P. Ostermann, "Model of a Stationary Background Universe Behind Our Cosmos", 2014 [equation (9) here ok]
  14. E.P. Hubble, "A relation between distance and radial velocity among extra-galactic nebulae", Proc. N. Acad. Sci. 15, 168-173, 1929
  15. P. Ostermann, "A Strange Detail Concerning the Conceptualization of the Hubble Constant"
  16. A.G. Riess et al., "A 2.4% Determination of the Local Value of the Hubble Constant", ApJ 826:56 (31pp), 2016, doi:10.3847/0004-637X/826/1/56, "arXiv:1604.01424"
  17. A.G. Riess et al., "New Parallaxes of Galactic Cepheids from Spatially Scanning the Hubble Space Telescope: Implications for the Hubble Constant", doi:10.3847/1538-4357/aaadb7, "arxiv:1801.01120"
  18. V. Bonvin et al., "H0LiCOW V. New COSMOGRAIL time delays of HE0435-1223: H0 to 3.8% precision from strong lensing in a flat ΛCDM model", "arXiv:1607.01790"
  19. W. L. Freedman et al., "Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant", ApJ, 553:47È72, 2001
  20. N. Aghanim et al., "Planck intermediate results. XLVI. Reduction of large-scale systematic effects in HFI polarization maps and estimation of the reionization optical depth", A&A 596, A107 (2016)
  21. Ch. Cheng and Q-G. Huang, "An accurate determination of the Hubble constant from baryon acoustic oscillation datasets", Sci China-Phys Mech Astron, 58 (2015), 599801
  22. S. Jha, A.G. Riess, & R.P.Kirshner, "Improved Distances to Type Ia Supernovae with Multicolor Light Curve Shapes: MLCS2k2", ApJ 659, 122-148, 2007 (s. a. weitere Literturangaben darin)
  23. A.G. Riess et al. (High-z Supernova Search Team), "New Hubble Space Telescope Discoveries of Type Ia Supernovae at z ≥ 1: Narrowing Constraints on the Early Behavior of Dark Energy", ApJ 659, 98-121, 2007
  24. M. Kowalski et al. (The Supernova Cosmology Project), "Improved Cosmological Constraints from New, Old, and Combined Supernova Data Sets", ApJ 686, 749-778, 2008
  25. P. Ostermann, "Indication from the Supernovae Ia Data of a Stationary Background Universe"; the original MG12-Talk (please use the PDF-full-screen mode)
  26. P. Ostermann, "The well-earned Nobel Prize for the wrong reason"
  27. A.G. Riess et al., "A 3% Solution: Determination of the Hubble Constant with the Hubble Space Telescope and Wide Field Camera 3", ApJ 730, 1-18+1, 2011 ( Zusatztext )
  28. P. Ostermann, "Model of a Stationary Background Universe Behind Our Cosmos", 2013 (chapter 2.8)
  29. P. Ostermann, "A microwave background of redshifted radiation within the stationary universe, (chapter 5 of "Problems ... " s. above)"
  30. N.A. Sharp, "The whole-sky distribution of galaxies", Astr. Soc. Pacific (ISSN 0004-6280), vol. 98, Aug. 1986, p. 740-754, 1986
  31. C.L. Bennett et al., "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results", ApJS 148, 1, 2003
  32. R. Piffaretti et al., "The MCXC: a Meta-Catalogue of X-ray detected Clusters of galaxies", A&A, 534, A109, 2011
  33. P. Ostermann, "Homogeneously distributed dark matter of second kind as an alternative to 'dark energy", Talk MG14 in Rome, also explaining the concept of 24 spin-1/2 torsion particles
  34. P. Ostermann, "The PLANCK 2015 model prediction mismatch of Sunyaev-Zeldovich cluster counts and a universal microwave background composed of redshifted radiation from 'dark' matter", Talk MG14 in Rome
  35. R. Lieu, J.P.D. Mittaz, & S.-N. Zhang, "The Sunyaev-Zel'dovich Effect in a Sample of 31 Clusters: A Comparison between the X-Ray Predicted and WMAP Observed Cosmic Microwave Background Temperature Decrement", ApJ 648, 176-199, 2006
  36. L.E. Bleem et al., "Galaxy Clusters Discovered via the Sunyaev-Zel'dovich Effect in the 2500-square-degree SPT-SZ survey", ApJS, 216, 27, 2015
  37. P.A.R. Ade et al. (Planck Collaboration), "Planck 2013 results. XX. Cosmology from Sunyaev–Zeldovich cluster counts", A&A 571, A20, 2014 article PDF 2013
  38. P.A.R. Ade et al. (Planck Collaboration), "Planck 2015 results. XXIV. Cosmology from Sunyaev-Zeldovich cluster counts", article PDF 2015
  39. V.F. Mukhanov & G.V. Chibisov, "Quantum fluctuations and a nonsingular Universe" (re-print), 1–8, 1981
  40. A. Einstein, "Geometrie und Erfahrung" (erweiterte Fassung des Festvortrags), Julius Springer, Berlin 1921
  41. H. Poincaré, "Wissenschaft und Hypothese", autor. dt. Ausg. von F. und L. Lindemann , Teubner 1904 (Das Original “La Science et l’Hypothèse” wurde vor 1905 von Einstein gelesen und, wie von seinem Freund Maurice Solovine in der 'Akademie Olympia' wochenlang diskutiert).
  42. N. Rosen, "Flat-Space Metric in General Relativity Theory", Ann. of Physics 22, 1–11, 1963, doi:10.1016/0003-4916(63)90293-8.
  43. L.D. Landau & E.M. Lifschitz: Klassische Feldtheorie; Lehrbuch d. theor. Physik, Bd. II, 12. Aufl., Berlin 1992
  44. P. Ostermann, "A natural vierbein approach to Einstein’s non-Euclidean line element in view of Ehrenfest’s paradox"; 2013
  45. P. Ostermann, "Skizze einer offenen Theorie von Elektrodynamik, Gravitation, Quantenmechanik"; 2006
  46. P. Ostermann, "Basic relations of a unified theory of electrodynamics, quantum mechanics, and gravitation"; MG11 Proceedings 2008
  47. N. Rosen, "General Relativity and Flat Space I/II", Physic. Rev. 57, 147-153, 1940, doi:10.1016/0003-4916(63)90293-8
  48. N. Rosen, "Flat-Space Metric in General Relativity Theory", Ann. of Physics 22, 1–11, 1963, doi:10.1016/0003-4916(63)90293-8
  49. A. Einstein, "Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie", Sitz. Preuß. Akad. Wiss., 142-152, 1917
  50. E.W. Kolb, "A coasting cosmology", ApJ 344, 543-550, 1989
  51. F. Melia F. & A.S.H. Shevchuk, "The Rh = ct Universe", MNRAS 419, 2579, 2012
  52. A. Linde, "Chaotic Inflation", Phys.Lett. 129B, 177–181, 1983
  53. A. Linde, "Prospects of Inflation", Phys. Scripta T117, 40–48, 2005
  54. A. Mezhlumian, "Stationary Universe Model: Inputs and Outputs", Stanford Pre-print, 1–11, 1994
  55. P. Ostermann, "Die Einweg-Lichtgeschwindigkeit auf der rotierenden Erde und die Definition des Meters"


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Kategorie: Kosmologie (Physik) - 04 August 2018 - ENGLISH