Dark Matter without Matter

We design a cosmological model that expands at a speed less than that of free fall and which allows accelerations of the recession velocity. In addition, the underlying geometry of the model can be adjusted in such a way that attractive forces arise in the cosmos, forces whose sources are not matter. This could explain dark matter as a property of space and one could also address the question of why galactic systems are not subject to expansion.


Introduction
Careful measurements have proven that there must be attractive forces in the galaxies that cannot be explained by the action of mass of the stellar objects. It has been suggested that these forces emanate from matter which does not emit light and therefore one has called the source of these forces as dark matter. Despite 20 years of searching, one has not found matter, and other explanations have not been satisfactory.
We can imagine that the forces do not come from localizable objects, but stem from the geometric structure of space. This concept is in perfect harmony with the principles of general relativity. Gravity, tidal force, and also rotational forces are usually described by the properties of space. In addition, pressure and mass density are derived from geometrical quantities.
Our new approach is based on mathematical methods already hidden in Schwarzschild's paper on the interior solution (1916). We have worked out these methods in [1] and [2] [3] and have also successfully applied them to the gravitational collapse [4] of the interior Schwarzschild solution. Now we want to apply the procedure to a cosmological model. We call our model P-model. This model

Preliminary Remarks
The dS model in its static version is based on a pseudo-hyper sphere with the ra- (2.10) The structures (2.8) and (2.10) can be used for spherically symmetric systems, static, expanding, or collapsing systems. If we reinterpret the cosmological constant in the field equations of the dS model, the stress-energy-momentum tensor has the form In one of our papers [5] we have critically examined the dS cosmos concerning its interpretation. Our P-model is based on the dS model. The above structures are the source for further considerations.

The Static P-Model
As a first step towards a generalized model which admits attractive forces, we examine a static cosmological model by introducing the projector technology borrowed from the interior Schwarzschild solution. We do not attribute physical significance to the model. It aims to explain the use of the projectors which we introduce in order to transform repulsive forces into attractive forces. It is the starting point for a more sophisticated model.
We start from the previously described dS model. But now we put with reference to (2.7). P is a space-dependent function which we investigate as follows. We call it as a projector.
The radial field quantities (3.1) are treated according to Therein the first line of (2.8) was used. To get a typical Friedman equation, we put zero the last brackets in the above expression and we get the important rela- and it has to be examined whether the pressure gradient leads to a contradiction.
we see immediately with (3.7) that (3.8) is satisfied. Thus, the introduction of the function P has proved to be useful and opens the way to cosmological models in which the pressure is position-dependent.

The Expanding P-Model
In this section we want to show that the static P-model can be extended to an expanding one. We address the problem without reference to conventional cosmology. We avoid using an expanding metric and the scale factor. Nor do we refer to a comoving coordinate system, but we use orthogonal local reference systems. We start from the static de Sitter model. Its metric is the seed metric for our expanding model. In this case, the radius of curvature of the universe will be time-dependent. Then at any moment the expanding cosmos is a snapshot of the dS-cosmos. The coordinate system of the dS-cosmos is completely sufficient to carry out simple mathematical operations. We develop the main part of the theory in the tetrad calculus. This provides reference systems in which the quantities of the model have physically interpretable components. We use comoving and non-comoving reference systems. Between them a Lorentz transformation mediates, which includes as a parameter the rate of expansion of the cosmos.
In an earlier paper [5] we have extended the dS model by introducing a second velocity. We again apply this method, but we abandon the constraint . const = R which applies to the dS cosmos in order to obtain a genuine expanding model. T is the proper time of an observer who participates in the expansion of the cosmos. The pseudo-hyper sphere with the time-dependent radius R forms the skeleton of the cosmological model. r, as in previous models, is the radial coordinate of the non-comoving observer. R v is, like in the 'expanding' dS cosmos, the speed of a fictitious freely falling observer. E v and 'R are to be understood in analogy to the definition of R v , where 'R is another time-dependent parameter. Thus, we can assume that in a fictive cosmos, which is preliminary to the physical cosmos, E v has a similar geometric meaning as R v has in the dS cosmos. The validity range of the quantities is

The double-velocity ansatz is
With the ansatz (4.1) we have established the method of double velocities. The

Lorentz transformation
mediates between the observer system m' comoving with the expansion of the The composition of the speeds can be taken from the drawing Figure 1. The system m'' is a fictitious reference system in a cosmos that can be represented by a pseudo-hyper sphere with the radius R . At any given moment, a snapshot of the cosmos corresponds to the static dS cosmos. This enables us to use the coordinate system of the dS cosmos for basic mathematical operations.
The physical system m' is the reference system which is in the empirical space and follows an expansion that is less than that of the free-fall and m the non-comoving reference system. In order to describe all states of motion, we complement (4.2) with Furthermore, the Lorentz relations apply.
With the definitions in (4.1) one obtains for the recession velocity From (4.1) it can be seen that R v accepts the value 1 at the equator of the pseudo-hyper sphere. That is the speed of light in the natural measuring system. In [5] we have shown that the velocity can only reach asymptotically the speed of light, i.e. it can come infinitely close to the value 1 only after an infinitely long time. Furthermore, we can see from Einstein's addition law of velocities (4.3) that the recession velocity is lower than the one of free fall Thus, we have envisaged a cosmological model whose rate of expansion is slower than the ones of the FRW models and whose rate of expansion can be manipulated with the parameter 'R . Since the recession velocity does not reach the speed of light and definitely cannot exceed it, there is no violation of the special theory of relativity and no violation of causality nor is there any galactic island formation.
For the further development of the theory some basic relations are necessary.
We are already familiar with referring to the geometry of the hyper sphere and the Lorentz transformations In calculating the changes of the two velocities R v and E v , and finally of the recession velocity v, we must proceed with care. Both quantities contain, apart from the radial coordinate, the variable quantities R and 'R . Therefore we define the two quantities (4.10) With this and with (4.5) and (4.9) we get From the above relations, it is obvious that the change of the two speeds has two causes. First, the speeds depend on the position. The recession velocity and thus the redshift of the light of a galaxy is higher, the farther the galaxy is dislocated from the observer. Secondly, the recession velocity depends on the expansion of the cosmos, i.e., on the quantities m R and ' m R . So we can expect accelerations of expansion, which is suggested by recent observations. We will take a closer look at this problem later.
Now the necessity arises to develop the properties of the quantities (4.10) from the geometric and the kinematic structure of the model. A look at Figure 1 shows us that for an observer with the velocity E v the system m' is the rest frame. Therefore we demand that in analogy to the non-comoving observers of the dS cosmos [5] For the velocity R v , similar constraints would bring us back to the dS cosmos.
In order to determine the quantity m R from the structure of the model, further considerations are necessary. First of all, it is evident that the comoving observers cannot ascertain any position dependency of R . The model is homogeneous and its curvature is everywhere the same for ' .
We do not derive our model by solving Einstein's field equations under specific conditions, but we make assumptions about a particular geometry, we derive from them all necessary quantities. Then we check, whether they satisfy Einstein's field equations. Since a homogeneous world model is based on a spherical space, some of the required quantities are already familiar to us. Thus, we know the expressions for the two lateral field quantities B and C in the system at rest We do not derive the field quantities of the comoving system from an expanding metric, but with a Lorentz transformation from the metric of the static seed system. With (4.14) From these relations we realize that the spatial components of these quantities do not appear to be flat. This was the case for the FWR models and the simple subluminal model, i.e. for all models that expand in free fall. For the special case 0 The spatial parts C α α = of the above equation now appear to be flat and correspond to the quantities of a free-falling system, as one can easily convince oneself. As we have repeatedly stated, this does not mean that the space is flat, but according to Einstein's elevator principle [6], no forces act on 'U , which is closely related to the quantity 4' R , can however be deduced without further calculation, if we demand that the expansion in all three spatial directions is equal. Thus, after looking at (4.14) one has Therein P is the projector already used in the last section, which will be discussed in more detail below. Knowing applies.
we get back the dS model. Then one has 4' 0 For the two systems m' and m one finally has Now we are also able to specify completely the velocity changes. If we substitute the above relations into (4.11) we get 1,0,0,0 ,0,0, 1 , ,0,0, constituents of these Ricci-rotation coefficients, which transform from the non-comoving system m to the comoving system m' according to The last term in the above relation is called the Lorentz term. This formula which is fundamental in the tetrad calculus must be generalized to the 1, = = = = P P P P P. However, it can be seen that this modified transformation law has no effect on the lateral field quantities B and C. They transform homogeneously and are not changed by the projector. They are given in the comoving system by (4.14). Thus, we can concentrate on the quantity U.
If the Ricci-rotation coefficients of the seed model are designated by Since the Lorentz transformation is to be interpreted as a pseudo rotation, which takes place in the [ With a look at (4.2) we calculate with (4.26) and evaluate this with the relation With (4.20), (4.12) we get We define three quantities where G describes the change of R v , l the change of E v , and f the change of R .
Finally, one has In (4.15) we have deduced that the quantity 4' 'U must have the same value as the quantities 4' B and 4' C in order to correctly represent the expansion scalar.
Exactly this value would have to be calculated with a Lorentz transformation from the static auxiliary system. We therefore rely on the relation (4.27) and we calculate the desired quantity using 1 ' ' , ,  ' ' , , Thus, we have completely calculated the quantity Rather new to FRW models is the radial component in 'U. Similar to the static dS cosmos a radial force occurs which can be attractive or repulsive depending on the sign of P . We realize that with the projector P one can manipulate the structure of the geometry. We will come back to this later.
After we have succeeded in deriving a complete set of field quantities from the geometry, we want to check whether these quantities satisfy the field equations and the conservation laws and which equation of state results for the cosmos.
It can be seen that we will come across a term | 1' P in the progress of the calculations. The analogy to (3.3) ( ) will prove. After some calculation we find the Friedman equation   So far, we have worked very successfully with the function P without specifying it. From (4.43) one recognizes that P is closely connected with the equation of state of the cosmos. By restructuring one gets It can be seen that the function P allows a class of different cosms, depending on the choice of the equation of state. We will discuss some specific cases later on.
In addition, we want to note that the P-geometry can also be derived from the freely falling system of the seed metric. One has for the inhomogeneous trans-

The P-Model in the Non-Comoving System
The cosmologists endeavor to find field quantities and field equations also for the non-comoving system. Although we believe that a representation in the non-comoving system does not provide any new insights, we nevertheless show that such a representation is possible and that the P-model proves to be extremely robust to transformations.
First, the meaning of a non-comoving observer must be clarified. In the literature, one finds the demand . r const = , i.e. the fixing of the radial coordinate r of the higher-dimensional embedding space. However, it is obvious that such an observer does not comove with the expansion but still moves with respect to an adjacent observer, e.g. to an observer located on the arbitrarily chosen pole of the pseudo-hyper sphere.
For the distance of the non-comoving observer from the pole one has specifically In an expanding cosmos its radius R depends on time, and its polar angle η also changes if one holds the radial coordinate r tight at a fixed position. Thus, the condition . r const = is quite artificial for an expanding cosmos.
In order to complement the P-model in the sense above, field quantities and field equations still have to be revealed in the non-comoving system. The Lorentz transformation is given by (4.2). The lateral field quantities transform homogeneously and can be taken from (2.7). However, more attention is given to the radial field quantities which transform inhomogeneously according to Comparison with (4.35) shows that the U-equation is form invariant. With (5.3) we recognize that the radial force U of the non-comoving system does not reduce to that of the static auxiliary model. That would only be the case if The lateral field equations again are formulated with the graded derivative.
According to (5.3) these equations now contain terms with |m R ( ) ( ) From this one gains , .
Now all that one has to do is to show how the quite different expressions in  Thus, it can be seen that the stress-energy-momentum tensor (5.6) takes the expected values in the non-comoving system.

Discussion of the Model
In the previous section we have set up a cosmological model with the help of a projector, which is formulated very generally. However, there is still the possibility of giving the model a specific appearance by specializing the projector. Now we assign some specific values to the projector. We recall that P is a function and a specification can only be made after the model has been worked out.
Case 1: According to (4.19) one obtains However, if one has 0 E v ≠ , one obtains the extended dS cosmos, which we have discussed in detail in [5]. This model is also static, i.e. has constant radius R of the pseudo-hypersphere. However, observer crowds drift apart at a speed less than that of free fall. forces act in the system in rest of this model, the comoving system is free of forces. The subluminal cosmos expands in free fall; the Einstein elevator principle applies [6]. In the non-comoving system the metric has a form from which can be read the curvature parameter 1 k = . This means that the space is positively curved and finite. After a transformation into the comoving system, the metric has the form 0 k = . This does not mean that the space is flat now. It rather means that the free-falling observers do not experience forces, they recognize the space to be locally flat, in the same way as the free-falling observers do in the Schwarzschild field.
In this model, the polar angle η on the pseudo-hyper sphere does not change in time for the comoving observer. With sin , d d ' According to [7] is in this case 1 = The pressure in this model is positive. By some authors the equation of state is attributed to an ultra-relativistic gas. What is interesting about this model is that in the comoving system, the system in which we live, an attractive force The change of the polar angle has two causes: the geometric structure of the model with the forces that occur, and the expansion of the cosmos. With (4.18) the above expression is reduced to Depending on the choice of P the cases discussed above are included in this relation. For 1 = P the change of η has as the cause the geometric structure on the pseudo-hyper sphere and for 0 = P no temporal change of η takes place.
Let us return to the problem of the recession velocity. Combining (4.11) with From this it is clear that only for 0 = P , i.e. for the subluminal model, is zero, the recession velocity is also temporally constant. For other values for P acceleration occurs. It turns out that both the generalized dS model and the P-model are exact solutions to Einstein's field equations that allow accelerations of the recession velocity. Furthermore, one gets for 1 = − P and from ( ) 2 6 1 R = + P R the relation 0 R = and thus also 0 T = in accordance with the equation of state 0 3 0 p µ − = .
By recent measurements of the redshift of galaxies one has found that the expansion of the universe is likely to be accelerated. We therefore want to work out how the acceleration for individual models in different frames of reference is obtainable. In the textbooks it is mentioned that the acceleration is not covariant, i.e. it does not transform like vectors between differently moving frames of reference. The problem is rarely discussed in more detail. But we can say that we have already outlined the problem, although it has not been explicitly stated. Accelerations are identical to forces on the unit mass. We have already discussed in detail how the forces behave in the various frames of reference. The forces are the field quantities and are components of the Ricci-rotation coefficients. The accelerations thus transform according to the inhomogeneous transformation law of the Ricci-rotation coefficients. With coefficients. In a more conventional way the above relation is written as The total acceleration of a galaxy relative to the non-comoving system is composed of a kinematic component resulting from the motion of the reference systems relative to each other

Conclusions
A strict geometric treatment of cosmology allows for the ability to modify the underlying geometry to reveal certain physical properties of the cosmos. Thus, the calculated recession velocity of the galaxies can be reduced, enforcing an acceleration of these and implementing attractive forces that can be attributed to dark matter. Such forces can also be responsible for ensuring that the internal structures of the galaxies are stable and not subject to the expansion of the universe.
Although the model proposed here is not very close to Nature, it contains mathematical methods that prove to be useful in the construction of cosmological models.
Because of the homogeneity of the universe, the new attractive forces are active throughout the universe and are not limited to galaxies, although this would be desirable. Here one could pursue an idea that was initiated by Einstein and Strauss [8] and was taken up by some other authors: The insertion of matter condensates into the homogeneous cosmic fluid.
In additions, the approach with 1 = − P seems to be too simple. A look at the interior Schwarzschild solution, from which we have borrowed the projector technology, shows that only a more complicated function for P leads to a func-

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