Local and Global Flatness in Cosmology

We raise the question of how the curvature parameter k is related to the curvature of the universe. We also show that, for a cosmological model that can be interpreted geometrically as a pseudo-hypersphere with a time-dependent radius, the Einstein field equations are not sufficient to fully describe the model. In addition, the differential equation system of Bianchi identities is required to describe the temporal evolution of the universe. We discuss the facts using the example of the de Sitter universe, the subluminal universe and the h R ct = model by Melia. In particular, we discuss the formal differences between the two latter models and claim that both models are identical. We also examine the possibility of introducing non-comoving coordinates.


Introduction
In many papers on expanding cosmological models, the topic is introduced with findings on the curvature parameter k. An expanding model is based on the metric in the canonical form Here, ' r is the comoving radial coordinate of an observer participating in an expanding motion and Ω the solid angle. ' t is the cosmic time, which applies equally to all comoving observers and, at the same time, is the proper time of these observers. For 1 k = the underlying space should be positively curved and closed. For 0 k = the space is described as flat and 1 k = − negatively curved. The two latter universes are open, they exhibit infinite extension.
In an earlier paper [1], we showed that 0 k = does not necessarily mean that the universe described by the line element (1.1) is flat. We discuss this problem once again in Sec. 2. Sec. 3 deals extensively with the two versions of the de Sitter universe and its inconsistencies. In Sec. 4, we extend the considerations to the subluminal universe and to Melia's model in Sec. 5. We also discuss the 3-dimensional Ricci scalar and how meaningful the relation 3 0 R = is for 0 k = . In Sec. 6, we explore the possibilities of finding coordinate systems for non-comoving systems.
Furthermore, we will use the following variables: R radius of the universe, K scale factor, H Hubble parameter, , , B C U curvature quantities, D tidal forces.

The Curvature Parameter
In our paper [1], we examined in detail the free fall in the Schwarzschild field, with the intention of extending the associated methods to expanding cosmolog- Comparison with (1.1) shows that the curvature parameter of the metric is 1 k = , Schwarzschild geometry thus builds on a positively curved space. Furthermore, (2.3) formally corresponds to the line element of the de Sitter universe.
We will build on this.
Lemaître used a coordinate transformation to transform the Schwarzschild metric into the form As in the cosmological models, K is referred to here as a scale factor. The line element is of type 0 k = . The new coordinate system ( ' i ) accompanies a free-falling observer family. ' t is the common time for all observers and ' r the comoving radial coordinate. From the metric (2.4), we learn that 4'4' 1 g = .
This means that there is no gravity present in this system. To get more insight There is no doubt that there has been no change in the curvature of space due to the motion of the free-falling observers. 0 k = does not mean that the underlying space is globally flat, but rather that it is only locally flat for the free-falling observers. This consideration is missing in papers which deal with cosmological models that expand in free fall.

The Two Versions of the de Sitter Universe
De Sitter designed a static cosmological model with a metric in the form (2.3). Its metric is of type 1 k = and can be interpreted as a metric on a 4-dimensional pseudo-hypersphere embedded in a 5-dimensional flat space. The pseudo-hypersphere has the time-independent radius R . A transformation given by Lemaître [2] [3] transforms this metric into the form (2.4) with the scale factor ' e ψ = K via: It is of type 0 k = . Other models, the anti-de-Sitter model, the Lanczos and the Lanczos-like model have similar characteristics. These models are grouped into the de Sitter family. The behavior of these models in transformations from comoving to non-comoving coordinates has been extensively studied by Florides [4]. We [5] [6] [7] have complemented the Lemaître coordinate transformations using Lorentz transformations.
Since the scale factor over ' ' t ψ = R is time-dependent, the dS metric is considered in the form (2.4) as the metric of an expanding universe. However, this interpretation leads to contradictions. First of all, this view violates the principles of the general theory of relativity: A coordinate transformation cannot change the physical content of a theory. All possible coordinate systems are equal, and the choice of a particular coordinate system is usually a matter of utility.
The conservation law leads to another discrepancy. If one has redefined the cosmological constant that is unpopular with many authors using This also makes it clear that neither a coordinate transformation nor a Lorentz transformation can change the geometric base structure of the space. 0 k = in (3.1) thus results from Einstein's elevator principle and cannot be considered as a criterion for the flatness of the space. The question of how the de Sitter model is to be understood in its two versions was widely discussed among German physicists at the time. Finally, they turned to the great mathematician Klein [11].
His detailed answer ended the discussion. It is not known whether Klein's authority or the argumentative content of his work was the decisive factor. However, we cannot find any link between the geometries of the hyperspheres or their space-time slices and Klein's statements. We have not found any work that responds to Klein's publication.
The geometric structure of the pseudo-hypersphere may best expressed with the metric in the form of From it one takes the differential of the proper time Thus, the relative velocity is geometrically determined. At a pole arbitrarily fixed with 0 r = , it has the value 0 v = and, on the equatorial spherical surface, the value 1 v = , which is the value of the speed of light in the natural system of measurement. Thus, this horizon is also a cosmic horizon. In [12], we have shown that the observers' recession velocity can only reach the speed of light asymptotically. This means that in the dS universe, the basic laws of special relativity are not violated.

The Subluminal Model
The dS universe dealt with in the last section is not particularly suitable for the adaptation of astrophysical data. Nevertheless, it is significant for historical reasons. It has been instrumental in driving research into expanding cosmological models and is still the starting model for new expanding approaches. It has also been criticized that in expanding universes whose metric is known in comoving systems and to which a mass distribution can be assigned, no forces are acting on the masses. The expansion in free fall is responsible for the missing forces and consequently the common cosmic time for all observers. In [12], we envisaged an extended dS model in which the observers drift apart more slowly than in free fall and recognized forces acting on such observers. This model is only of mathematical importance, but the presented technique may be useful for building more sophisticated models.
Another, rather promising attempt was a model [13] that builds on the dS universe, but drops the condition . const = R . We have called it a subluminal model because it definitively rules out that the recession velocity of the galaxies exceeds the speed of light. The subluminal universe is positively curved and closed. It has the position-independent pressure The subluminal model therefore differs significantly from the FRW standard model in which the pressure is inserted by hand and is therefore not an exact solution to Einstein's field equations. Since the Einstein field equations do not fully determine FRW models, it is necessary to introduce numerous parameters, namely, the Ωs and the deceleration parameter. These quantities must then be filled using astrophysical data. The subluminal model needs only one parameter, the radius of curvature of the universe, or the scale factor. The Friedman equation takes the simple form . The expansion rate of the model is constant.
For models that build on a pseudo-hypersphere with a non-constant radius, Einstein's field equations are insufficient to determine all the quantities of the model. The metric on a surface will determine the properties of that surface, but it will not be able to predict the change in the curvature of that surface. This is what the contracted Bianchi identities (4.1) The system (II) leads to the conservation law || 0 n m n T = . This is often used in the literature to establish an outstanding relation to variables. However, little reference is made to the above considerations. For models with constant R , the conservation law is trivial. Therefore, there is no need to use the system (II) to complement such a model.
The subluminal model has a geometric horizon, namely the equatorial surface of the hypersphere. As with the dS universe, it is determined by the relation (3.4) and, at the same time, it is the cosmic horizon.
However, we have shown in [13] that this expression can be translated to Here, 1 dx and dT are the proper length and proper time of a non-comoving observer. Thus, the recession velocity is defined independently of the coordinates and is also the velocity used in the Lorentz transformation, which transforms the non-comoving system into the comoving system.
In the introduction we explained, with the aid of the well-known Schwarzschild model, why gravity cannot be experienced in a free-falling elevator; we then transferred the problem to cosmic free-falling observers. We now want to address the problem in greater mathematical depth by borrowing a quantity from the Ricci-rotation coefficients that is closely related to the curvature of the space-like greater circles of the pseudo-hypersphere.

R. Burghardt Journal of Modern Physics
The static dS metric is of type (2.3) and is the seed metric for the subluminal model. From this metric, using the standard technique of the tetrad representation, we obtain the above-considered quantity 1 cos , 0, 0, 0 , 1, 2,3, 4 Here η is the polar angle of the pseudo-hypersphere and After a Lorentz transformation from the static system into the comoving system, this variable takes the form Here, according to (3.4), sin v η = is the relative velocity between the two systems and The spatial part of the quantity B is and for a time-like slice on the pseudo-hypersphere, so for a pseudo-circle From (4.3) it can be seen that even in the free-falling system, the space curvature is still present via the geometric term cosη , but is compensated by the kinematic term α . If one writes all components of the quantity B in the 5-dimensional embedding space of the pseudo-hypersphere, one has with the local extra dimension 0' This quantity can hardly be assigned to a flat space. Since all of the above expressions can be deduced directly from the type 0 k = metric, one will not be able to assume that 0 k = inevitably leads to a flat space. The new forces are the tidal forces, which act on the observer in Einstein's elevator but are too weak in Earth's proximity to be perceived by observers. Misner, Thorne, and Wheeler [14] derived these forces in their textbook with the aid of the geodesic deviation from the Riemann curvature tensor and Sharan [15] also illustrated them in his textbook. However, in an early article [16], we deduced the tidal forces directly from Einstein's field equations. We now want to transfer the process to the cosmological problem.
We summarize the three new components in (4.4), (4.5) and (4.6) to a quantity 2 D αβ remain. After a short calculation, one obtains We note that the 3-dimensional Ricci scalar 3 R does not vanish. Finally, for the Einstein tensor one has It can be seen from the above system of equations that curvature effects can also be described in the freely expanding system with the Einstein field equations.

The Model of Melia
In numerous papers 3  Both models describe the relation between the non-comoving radial coordinate r and the comoving ' where K is the time-dependent scale factor. We still have to show that the h R ct = model is compatible with the curvature of the pseudo-hypersphere. Melia has an extensive set of astrophysical data and has shown in some articles that this data can be best adapted to the h R ct = model, much better than to other FRW models. Thus, our subluminal model is well supported by Melia's data and analyses.
When developing our model, we did not envisage finding a model that closely relates to astrophysical data. Our goal was to provide mathematical foundations for a model that 1) is an exact solution to Einstein's field equations, 2) involves pressure, which is a result of this exact solution and is not inserted by hand, as is the case with numerous models, 3) does not allow superluminal speed and, 4) can be fully described geometrically.
The fact that this model is supported by astrophysical data was initially surprising to us, but it justifies our efforts. However, Melia's model also has an additional mathematical profile due to the subluminal model.

Coordinate Systems
Most cosmological models assume a metric written in comoving coordinates.

Journal of Modern Physics
This metric is also the natural framework for a model, because the rods and clocks associated with such a system are the ones we currently have available. The question is how to realize the latter in practice. The position of such an observer must be continuously recalculated and a fixation to the calculated point in space requires significant technical effort. This would be enough to calculate the Ricci-rotation coefficients for the non-comoving system. However, it is also quite inconvenient, since the new tetrads are still indicated in the comoving coordinate system ( ' i  [12]. Λs can indeed be found for this model. However, these do not fulfil the relations (6.2). Therefore, the coordinates are anholonomic, meaning that there are no coordinate lines. The Ricci-rotation coefficients can therefore not be calculated with the 4-bein alone, but must be complemented by the object of the anholonomity which one is accustomed to derive from 44 g or 4 4 e . Since 1 U is not a gradient, it is not possible to go in the opposite direction and derive the metric coefficient 44 g from (6.4). The quantity m F prevents this from being possible, wherein said quantity was obtained from the expansion of the universe. It is only if one switches off the expansion ( ) 0 = F that the whole expression is reduced to the known dS quantity Û , which can be derived from 44 g . Thus, to a non-comoving observer cannot be assigned a fully non-comoving coordinate system. It should also be remembered that in (6.4) we are looking for the quantity 44 g , which is a solution to the differential equation system I. However, the expression containing the quantity F is a solution to the differential equation system II.
The search for the lapse function 44 g is probably historical. Even in the Schwarzschild model, the metric coefficients 44 g were used to calculate the gravitational redshift and/or time dilation. Recalling our discussion of free fall in the Schwarzschild field, we find that the ratio of the proper time of the free-falling observer and that of the static observer No general method is known from the literature with which one could determine for which model static coordinates are possible. Investigations in this direction have been undertaken by Mitra [17] and Gautreau [18] [19], among others. In his papers, Melia has also tried to bring FRW metrics into the Schwarzschild form.
Apart from some marginal notes, we cannot contribute anything to this. It could be that Florides [4], with his six models, has already exhausted all the possibilities.
The subluminal model was developed by the simple generalization On the other hand, one tries to set up a metric for a surface in non-comoving coordinates which describes not only the properties of the surface but also the temporal change of this surface. This attempt is reminiscent of the German story of Baron Münchhausen, who pulls himself out of the swamp by his own braid.
The properties of the surface would have to be separated here, distinguishing between those belonging to system I and those belonging to system II.

Conclusion
In this paper, we have tried to establish a connection between our subluminal model and Melia's h R ct = model. We have argued that a cosmological metric with the curvature parameter 0 k = does not necessarily require global flatness of the universe, but rather a local flatness due to the free fall of the expanding universe. We have confirmed our point of view by gradually introducing curvature variables into the h R ct = model, bringing the h R ct = model into the formal vicinity of the subluminal model. The identity of both models is thus ensured.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.