which means that the last part of Eq. (156), in the case of just electromagnetism, is connected with q or the ”charge” of elecrtomagnetism or the coupling constant of electro- magnetism plus the ”number” of photons. In the same way, we must treat all the other terms, the d.energy term will involve the ”number” of vaccua (as we will see in section
(15) related with Higg’s boson) and the d.matter term should involve the coupling con- stant or the d.matter ”charge” and the number of dark particles. It is very interesting the presence of λ in both the ordinary matter and dark matter terms. The equation Eq.(156) is the simplest analogue of the first Friedmann equation, as it comes from our theory. Velocity V is reffering to the velocity as it looks from R8 space-time, while v is the projection to our 4-d usual space-time.
13.2 Relation to current Cosmology
The current cosmological model (FRW) depends on general relativity, as it formulated in our usual 4-d space-time, fper a specific choice of metric. There is no use to refer to the success of general relativity, but it is certain, that it does not tell us about dark matter and dark energy, as even for dark energy, the cosmological constant is still added ad-hoc. Today, in order to do Cosmology, we do not only need to define dark matter and dark energy but at the same time to fill our framework with extra inputs such as proper distance-time, comoving distance-time, comoving frames, peculiar velocities etc, in a pure Copernican-Newtonian way,as these inputs are not derived directly from general relativity or geometry itself. We need to imagine observers that travel along with expansion, local observers and we need some kind of geoemtry to connect them,as comoving and proper distances are not the same concept of distance as the concept of distance in special relativity. In our oppinion, it seems that we need a new type of special relativity or an extended special relativity, that could incorporate all these extra inputs and different type of observers and express them as naturally risen entities. Moreover, this extended special relativity and the extended general relativity should share a common ground. In this spirit, we consider that our presented extended special relativity (section 7) could serve us this way. In our oppinion, the current picture in Cosmology, remind us, two akward situations of the past, Ptolemy’s picture about Cosmos and Galilean relativity, where the main problem could be seen as: there is a bif difference between how things seem to us how things are actually are. In Galilean relativity pseudo forces and velocities existed, while in Ptolemy model objects of Cosmos where seem to follow peculiar orbits. We believe that through our model, we can propose a way to bridge the differences and manage to reach ”how things are actually are. In our consideration, the key entities that must be interpeted are
The second time T: The original time inC4isT=T+it. We can see that our usual time t is a periodical entity and we need periodical instruments to measure it, as clocks, orbits or even Cepheid stars. On the other hand T seems to be a straightful entity and we need some kind of ”sand timer”, in order to measure it. But, at the same time, we have already defined that T is some kind of ”cosmic time” as it is related with Cosmos’ radius. But we can imagine a ”sand time” suitable for our purpose. Imagine a population of many local observers, very close to each other, that formulates a big chain and that our ”sand timer” is the sum of the lenghts as it is measured by each one of them. This ”sand timer”, also defines cosmologicaly, the proper distance between two objects!
The velocity v: In the beginning, we start with a C4 space-time and a velocity of the typedS
dT, which is the velocity that a 4-d complex observer would measure. Unfortunately,
we are not 4-d complex observers, thus we embedded our usual 4-d space-time inR8 in order to see, how usual 4-d observers, observe what happens in C4 space-time. As a result, the velocity V. tell us what is the velocity od 4-d complex observers, in terms of our usual 4-d real observers. Alternatively, S is the lenght of a 4-d complex observer, as it looks to us the 4-d real observers, with respect to our ”local” time t! The advantage is, that in the case that our Cosmos in not actually described by a 4-d real model, we could see how our Cosmos realyy looks. In the case, that our Cosmos is really described
by a 4-d real model, the hypothesis of a 4-d complex one will fall. In the first case, we actuallyu consider that 4-d complex observers, can see properly Cosmos, while our usual 4-d real observers, experience Ptolemy-Galilean effects. Our extended special relativity will bridge the differences
The velocity ∂T
∂t:AsT=T+itthe following relation holds
∂T
∂t
=
∂T
∂t
−i→
∂T
∂t
=
∂T
∂t
+i
(174)
This velocity yell us that the rate of T and T with respect to t, are different to an imaginary constant. Now, we must understand, what really T is. Let us consider the simplest case, where we have only dark energy, thus Eq.(171) will be
V
2=v
2+
∂T
∂t
2(175)
ordS
dt
2=
dR
dt
2+
∂T
∂t
2(176)
If ∂T∂t = 0, then the lenght S of the 4-d complex or 8-d real observers, will be equal to the
lenght R of our usual 4-d real obserbers, and as expected nothing new is defined. But, if ∂T
∂t 6= 0, then this term would seem to a 4-d real observer as a ”Galilean” term, or a
”peculiar” velocity. But, through our extended special relativity this term is well defined and expresses a second type of energy or a new momentum (as we have already seen in section 7). As a result, usual 4- real observers, can not define properly dark energy with respect to their 4-d space-time, but they need auxiliary parameters or ad-hoc terms to the standard geometry of the 4-d space-time. In the simplest scenario, a 4-d complex or an 8-d real observer, can see Cosmos as a ”closed object”, whose velocity V, breaks into two velocities, a natural to us 4-d real observers v and a peculiar to velocity ∂T
∂t as