Dear Remi, i agree totally. But My question was how to interprate the fourth plot, not if the graph fits in our todays model. The Problem is that in the other Plots (graphical)  the time-arrow is the y-abscissa. If we let that be the same in the fourth graph (and i cant think of a reason why not doing so), then the plot assumes that above asymptotical limit the Expansion goes ahead but  reverses its direction as a function of the y-abscissa (time). I'm asking for a possible mathematical formulation of the plot, not if its realistic or "nuts". 

In flat space the volume inside a closed surface is exactly the quantity suggested by geometry of the surface. In a closed space the volume inside a closed surface is always less than geometry would predict from the surface, by an amount determined by the curvature. This is the exact meaning of the Ricci Tensor in Einstein field equations.For an open space the volume inside is always more than tis predicted from geometry of an enclosing surface. 

For the graphs the same principle applies to the dark surface surrounded by a closed line. In the first three graphs the white spaces have no significance to the contents of the dark surface.

In the 4th graph you probably need more than 4 dimensions to avoid complex representations in 4D, because the enclosed white spaces acquire a significance to the contents of the black spaces. Physical significance is that more than 4 dimensions are needed to contain some of the proposed physical systems like time reversal, quantum tunneling, and worm holes. 

Is it a nuisance if I ask the question whether one of the authors/readers interested in the present fundamental question might feel like writing down for me just for the fun of it, the "global-c transform" of the field equations.

Maybe even posing the question in this way is flawed. So I should perhaps more properly ask whether some of you could reformulate my question in a form which makes enough sense to make it operational in the first place.

My belief that this playful thing to do may have further merits has nothing to do with the playful question posed here.

Thank you.

dimensions are nothing but abstract information needed for orientation. there is only an aleatory number of locating parameters, called dimensions. more specifications (alias dimensions) could be indicated just to cover more phenomena (velocity, acceleration- variations of other degrees than 1and 2, discontinuities, inflations, implosions; even a rotation of the universe is presumed (s. Gödel`s solution for Einstein`s equations)!). more: curvature implicates an increasing of the number of (so-called) dimensions. as locating implies nothing else but to enumerate the required properties to locate something in time and space, the number of dimensions becomes irrelevant; one can use as many dimensions as necessary required, or even (redundantly) more (kaluza-klein, superstrings in 11 or in 26 dimensions). and: locating particles should be made in concordance with Heisenberg‘s uncertainty principle (measuring/observation influences the particles).

 A ``big crunch'' doesn't mean that time is reversed. Time continues to flow in one direction. 

The three possibilities refer to homogeneous expansion, incidentally. It is, however, possible to relax the conditions that lead to FLRW. 

@Stam Nicolis:

In case of the first scenario (Closed Universe) the Big Crunch doesn't mean that time is reversed - never claimed that. But: its not about this approach, but the fourth graphical description in the Discussion Poster. How to interpret the line going back along the y-abscissa, which in all other approaches indeed is the time arrow. 

I don`t think that time flows in ONLY 1 direction! what would You say about kgm/s²- i.e. Square Seconds. Seconds at the power of 2 is still 1-dimensional!??

One can draw pictures-but whether they mean anything relies on equations. So, as a spacetime diagram, the fourth diagram doesn't make sense, that's all, absent further information.

The others aren't, necessarily correct, either, however, because spacetime diagrams rely on light cones-how they behave indicates the presence, or not of event horizons.

It suffices to draw the Carter-Penrose diagrams for the different metrics-that's what they were invented for, in the first place. 

 I don't know the math behind it, but I would probably describe the shape as a hollow torus in space-time.  It seems that some matter is attracted to the center of the torus, meeting at a point, which would have to be a black hole.  

I wonder what phenomena would explain why the matter outside the torus is causing it to move away from the black hole...  Is this meant to be the radius of its event horizon?  The event horizon would expand as more matter falls into the center, and if somehow matter were to leave, that would cause the event horizon to shrink again.  

But I guess it can't be that because this would show the event horizon existing outside the infalling matter, unless this is a new concept which models both the inside and outside of the black hole simultaneously.

but the implications of the fourth diagram seems to be a black hole - white hole phenomenon.  Matter near the black hole is falling in until it forms a singularity.  after a period of time in the reference frame of the surrounding matter, it undergoes another "Big-Bang" type event.  

But outside the torus, is a steady-state universe...  Where matter just exists continuously from the bottom to the top of the diagram.

in an expandable infinite unlimited set there is no infalling matter.

begin with a moving Point which describes a line, then draw the perpendiculars to that line, then the perpendiculars to the created plane, and after that plot the perpendiculars to the prism (that becomes difficult, but makeable), aso... note: in Gravitation perpendicular means orthonormal.

Aleksandar, if I understand your question, either k=0 or k>0 or k

if time flows in one direction, how do you Interpret square seconds?!