_{1}

The metrics of gravitational and cosmological models are brought into canonical form in comoving coordinates. The FWR curvature parameter
*k* is read from this and it is shown that
*k*=0 does not correlate to a flat model, but for a spatially positively curved geometry in which reference systems which are in free fall exist. This also corresponds to Einstein’s elevator principle. Moreover, we will show that our subluminal cosmos is associated with the
*R*
_{h}=
*ct* model of Melia, assuming that
*k*=0 is related to a free-falling system in the sense described above.

One of the main features of general relativity is the identification of the gravitational forces with the effect of the curvature of space on observers, but also the possibility of “transforming away” the gravitational effect, that is to keep an observer without force by an appropriate choice of a reference system. This does not mean that one can eliminate the curvature of space by an observer transformation, but only that one can annul the effect of the curvature of space for certain observers. This is the case for a freely falling observer in the Schwarzschild field of a stellar object. Such an effect can also be expected for cosmological models which expand in free fall.

The effect has become known in the literature as Einstein’s elevator. We will recall this effect for the case of the Schwarzschild field, but we will also show that it is useful to consider it for cosmological models with position-independent spatial curvature and obeying the cosmological principle. We will discuss the elevator principle when referring to the de Sitter cosmos and to a subluminal model. We will resort to earlier results [

The introduction of the elevator principle will be crucial for the structure of the universe. It determines whether a metric with the curvature parameter

The line element of the Schwarzschild field in the standard form is

with ^{1}-curvature parameter

To bring the Schwarzschild metric into the canonical form, we should remember that the radius of curvature

Thus we have

and the Schwarzschild metric in canonical form

It is now similar to the de Sitter metric which we will discuss in the next Section. By comparison with the FRW standard form

we find

From (2.1) or (2.4) we calculate the components of the Ricci-rotation coefficients^{2}. The radial and the two lateral components are

The geometric quantity

Lemaître has found a coordinate transformation associated with a freely falling observer. The metric in these coordinates is

Herein

It is noteworthy that the space-like components of the lateral field quantities

are precisely those that one would expect for a flat geometry in polar coordinates. But it would be premature to call the geometry flat. The basic geometric structure of a model cannot be modified by a coordinate transformation. We want to get to the bottom of the matter.

From the coordinate transformation

For the lateral field quantities one obtains with

from (2.6) first the components

which a free falling observer would measure. However, since the velocity of a freely falling object in the Schwarzschild field is coupled to the angle of ascent

one has for the Lorentz factor and the metric factor according to (2.6)

This means that the geometric quantity

Special attention should be paid to the conversion

The

We recognize that in the freely falling system the spatial components of the

This means that the gravitational force

is Einstein’s elevator equation and

the Friedman equation for free fall in the Schwarzschild field in tetrad form.

Although the metric which relates to the free fall is of the type

We have dealt with the problem so minutely because in cosmological models the problem is the same and the formal treatment does not differ much from what has just been put forward.

By de Sitter [

It is the line element on a pseudo-hyper sphere with the time-independent radius

it can be brought into the canonical form

We read from this metric

with the definitions

By Lemaître [

Therein

From the metric we derive the field quantities

which differ from those of the freely falling Schwarzschild system only in the time-like components, i.e. the tidal forces. The Schwarzschild geometry is parabolic, the dS geometry spherical. The Schwarzschild reference system contracts, the dS reference system expands. The geometry only seems to be flat. The same arguments as in the Schwarzschild geometry speak against the flatness of space, if the metric is written in the expanding form (3.6).

The field quantities in (3.8) could also have been derived with the Lorentz transformation

whereby for the variables

applies. We also find just like in (2.15)

(3.10) is the Einstein elevator equation for the dS-Universe. The reference system established with the Lorentz transformation (3.9) expands in freefall. The form parameter read from the metric (3.6) is

In a previous paper [

The model is based on the de Sitter universe. The geometry is a pseudo-hyper sphere and its radius a function of time. Thus, the model expands and is positively curved and closed. The equator of the pseudo-hyper sphere is the cosmic horizon. For galaxies drifting apart due to expansion, the highest attainable velocity is the velocity on the horizon, namely the velocity of light. The model is subluminal; there are no superluminal values for the recession velocities of galaxies.

Since this subluminal model is an extension of the dS cosmos, we will start from the dS metric as seed metric, but the radius of curvature of the pseudo-hyper sphere will be time-dependent. Thus, a new quantity enters into the theory

Again, the system

All three field quantities seem to be flat. Since the subluminal cosmos expands in freefall, the supposed flatness of space is due to the elevator effect.

One obtains the field quantities of the non-comoving system with a Lorentz transformation of the type (3.9). The relative velocity and the Lorentz factor are defined in the same way as in the dS cosmos

For the lateral field quantities

and the Lorentz term

one now gets

Therein

The Friedman equation for the subluminal model

is a subequation of Einstein’s field equations. With

The expansion in this model is constant. In a cosmos expanding in freefall comoving observers are not exposed to acceleration according to Einstein’s elevator principle.

Let us consider this interesting result from a different perspective. If an observer does not perform an individual motion one has

At the equator

whereby the definition of the velocity

No signal beyond the horizon can reach an observer at

If one completely evaluates Einstein’s field equations, one has for the pressure, the matter density, and the equation of state of the cosmos

Remarkably, these results are identical with those that Melia [^{3}. Melia derives his model in the comoving coordinate system, the underlying metric is of type

that the universe described by this metric expands in freefall. Here, the scale factor is proportional to the cosmic time

with

and we calculate from this the components of the Ricci-rotation coefficients

However, these quantities are identical with those of (4.3) which we have derived for our subluminal model. Thus, despite

On the other hand the model of Melia is complemented by a full covariant field structure by our subluminal model. We have formulated the problem with field quantities which are closely related to the geometrical quantities and quite clearly reflect the prevailing conditions in the underlying pseudo-hyper sphere. We do not limit ourselves to discuss the Friedman equation, but we have presented the field equations and their subequations in a clear structure in both reference systems, the comoving and non- comoving systems. By this method one gains a deeper insight both into the physical and in the geometrical structures.

The standard model of cosmology allows galaxies to recede faster than light and make galactic island formation possible, i.e. without any causal connection and any exchange of information between galaxies. The standard model allows the fundamental laws of special and general relativity to be invalidated. The standard model is not an exact solution of Einstein’s field equations: into the pressure-free Friedman cosmos pressure is inserted by hand. We have therefore raised the question as to whether it is possible to formulate a model that is an exact solution of Einstein’s field equations, includes pressure, and respects the laws of special and general relativity. We have found the subluminal model, initially only with the intention to show the theoretical possibility of such a model without regard to conformity with astrophysical data.

Surprisingly, some of our results have been consistent with those of the

Burghardt, R. (2016) Einstein’s Elevator in Cosmology. Journal of Modern Physics, 7, 2347-2356. http://dx.doi.org/10.4236/jmp.2016.716203