Expanding Space, Convergence and Hyperfunction

So far we have shown that the process of the spacetime splitting can be described purely from mathematical analysis. We transformed the complex plane into a Riemann sphere to get a more pictorial representation of the entire spacetime, including the infinities. The real space axis (X) on the complex plane is now represented by the real equatorial circle while the imaginary time axis (it) of the complex plane by the imaginary longitude circle of the sphere (Figure-1).

Since the energy in its purest nature can be described as a wave, mathematically it can be best formulated in the form of Laurent series f(z) expressed as a sum of its positive frequency part and the negative frequency part:

                      f(z) = F⁺(z) + c₀ + F^–(z)

When it is mapped on the Riemann sphere ^a), the positive frequency F⁺(z) extends holomorphically into the southern hemisphere, and the negative frequency F^–(z) extends holomorphically into the northern hemisphere.

The constant c₀ is represented by the real equatorial circle which is the shared boundaries of those opposing frequencies.

This is how the energy split is described mathematically into its positive and negative energies. Physically, we can explain such energy split through the division of the spacetime into two halves. One half, the southern part of the spacetime, represents the positive energy zone and the other half, which is the northern part, the negative energy zone.  The interface between those two opposing zones is the space where all material things, the super-galaxies, a cluster of galaxies, galaxies, stars, solar system, and the earth are located.

So far we have assumed that the size of the real equatorial circle on the Riemann sphere or the space in the spacetime is not changing in time.  The whole spacetime is completely split instantly which makes the universe static. In term of the Riemann sphere the real circle, which physically represents the space, immediately gains the infinite equatorial size. 

In reality, there is no such a static universe as supported by the astronomical observations (among other the Hubble's redshift)^b). In line with that, space should be gradually expanding in time ^c), depicted by an expanding interface in the spacetime (Figure-2B). In term of the Riemann sphere, this is represented by an expanding real circle (Figure-2A).

Now we come to an essential question as to whether the expanding real circle is starting from the beginning as a point (singularity) or from certain definite size? As we remember, the Riemann sphere has a kind of annulus of convergence which excludes a particular area surrounding the zero point and that of the infinity (Figure-3).

The real circle starts to appear only after gaining a certain size. This means that space begins to materialize just after achieving a certain minimum size, below which nature abhors the existence of such space and all materials within. Also, the annulus of convergence also indicates that space will disappear entirely after gaining a certain maximum size for the space to exist.

Does nature allow a partial spacetime split? According to excision theory, the hyperfunction of the shared boundaries (the real circle of the Riemann sphere) is independent of the particular choice of the region of the f(z) as shown in Figure-4.

 

This means that the real circle (space) ^d) can take any size as far as it is within the annulus of convergence. Space appears only after it gains a sufficiently large size and expands afterward up to a maximum allowable size for the space to exist. After that space dissolves completely into pure energy. Nature keeps the physical world to take place outside the vicinity of the singularity and infinity.

Notes:

a)     As the spacetime is the structural quality of energy, the Riemann sphere becomes the domain of the energy wave function.

b)     The interpretation of which gave rise to the Big Bang theory

c)     According to the Big Bang [and the general relativity] theory, it is the spacetime instead of space which is expanding.

d)     The real circle on the Riemann sphere or the physical space within the spacetime framework is not necessarily flat and may be curved more or less significant.

References: 

1. Penrose, R.: "The Road to Reality," Vintage Books, London, 2005, pp. 159, 172-175.