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.

Energy, Spacetime and Riemann Sphere

The ultimate goal of the special relativity theory, if we are aware of, is the unification of the spacetime and energy. The union of these two would reveal the reality of energy which differs entirely from what we used to think about. Energy is a substance whose geometrical structure is reflected by the dimensionality of its spacetime.

Our ordinary [four-dimensional] spacetime is the physical manifestation of the energy whose vibrations are four-dimensional. As a [three-dimensional] human being we can only perceive this pure energy through its superficial three-dimensional aspect. Many underlying higher-dimensional ambient spaces which physicists have discovered, as they are adventuring deeper and deeper into the micro-world realms, are nothing but the reflections of various hyper-vibrations energies.

The relativity theory demonstrates that energy as a whole is inherently composed of positive and negative energies, as implicitly expressed in the relativistic energy equation E² = m²c⁴ + p²c². The segregation of its opposite parts that resulted in the split of energy created a 3-hypersurface (space) between the two halves of its separated respective spacetime. The world was not created through the "big" tiny bang bursting out of nothing.

In the quantum realm, energy as a wave can be described proportionally to its frequency as expressed by the Planck equation, E=hv.  We can express the energy, not only microscopically but also macroscopically, in term of the wavefunction whose character is completely holistic and non-local. However, the more natural way for us to grasp, is to express it in a periodic function such as a Fourier series ^a) or its complex form: the Laurent series ^b). 

In the relativistic framework, we map events in a complex plane where we express time in term of an imaginary variable. In such a plane, we have a Laurent series written as:

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

which is a wave function expressed as the sum of its positive frequency (F⁺(z)) and negative frequency (F^–(z)). Physically, the constant term c₀ represents the real space at t=0 (the present now). This c₀, wholly or partially, becomes either the part of F⁺(z) or F^–(z) or maybe both. However, as there is only positive energy, at least in this corner of the world where we live in, the space (c₀) should be part of the positive ocean of energy, F⁺(z), and on the other side of the world it should be part of negative ocean of energy, F^-(z).

Geometrically, we can figure out this more clearly in terms of the Riemann sphere, the stereographic projection of the complex plane. On this sphere, we can transform the real axis (X) of the complex plane into a circle located on the equator, while the imaginary time coordinate (it) by the longitudinal circles. The positive frequency F⁺(z) extends holomorphically into the southern hemisphere, and the negative frequency F^–(z) extends holomorphically into the northern hemisphere (Figure 1). 

From this description, we have a complete similarity between that of purely mathematical geometry with the relativistic (Minkowski) spacetime geometry (Figure 2 A and B). The real circle on the Riemann sphere represents the 3-hypersurface (space) in the Minkowski 4-spacetime which separates it into two 4-halves. The positive energy segregates to one half, and the negative energy segregates to the other half of the spacetime. We can These two halves are respectively represented by the southern and northern hemispheres on the Riemann sphere. The real circle on the Riemann sphere and the hypersurface in the Minkowski spacetime, each represents the loci of all matters which may exist in the whole universe.

With this simple mathematical representation, we can have a deeper insight into how nature may work. As we have figured out previously, the real circle in the Riemann sphere represents the space, and the imaginary circle represents the time path. As time passes by along the imaginary circle, space (real circle) is rotating accordingly along the time direction (imaginary circle). As time flows resemble electric current, space is also rotating on its plane following the right-hand rule. The rotation of this space as a whole rotates, in turn, all celestial bodies from the super-galaxies down to galaxies, solar systems, and the earth.

Notes:

References:

1. Penrose, R.: “The Road to Reality”, Vintage Books, London, 2005, pp. 157-162.

IT’S NOW OR NEVER

    For St. Augustine who knew about time until someone asked for him to explain ¹
    For Albert Einstein who was worried about the problem of Now
    For him who felt so sad that the experience of Now could not be grasped by science ²
    For Eckhart Tolle who claimed of getting power from the Now ³

    Also,… for Elvis Presley who used to sing the song

What is Time? What is Now?

The "now," the present time, is when everything is there
The past time is when everything was there but is not there anymore
The future time is when everything will be there but is not yet there

The present is when everything is present
The past is when everything was present but is not present anymore
The future is when everything will be present, but is not yet present

The present is when everything does exist
The past is when everything did exist but does not exist anymore
The future is when everything will exist but does not yet exist
 Everything does exist simultaneously in a very brief moment of now ^a)
Nothing does exist in the past and the future
Everything does exist just only now, right now, in this very brief moment of now

When now becomes the past, everything is annihilated
When the future becomes now, everything is created ^b)
Nothing persists; everything is created and annihilated perpetually and orderly
Space is nothing but the simultaneous presence of everythingSpace is the totality of now
Now and space are indistinguishable, the two different aspects of the same thing ^c)

Now separates the past from the futureSpace separates the past from the future Space is there to establish cosmos out of chaos ^d).

Technical Notes:

a)      The relativity theory denies the existence of the absolute simultaneity of now. What we are talking about in this case is within the framework of multiple extra-temporal dimensions (as against the concept of curled extra-spatial dimensions), where the term of “now” here is similar but in a higher temporal dimension. We can always find a higher temporal dimensional where everything happens simultaneously (a completely flat higher-dimensional space).

b)     We view it from human perspectives and languages. It turns the reality entirely upside-down.  The perpetual creation and annihilation do not happen in time. On the contrary, the act of perpetual creation and annihilation which makes time appears to pass by. Enough for wishful time travel debates and no more chronology protection is required.

c)      The "now" and space is the smallest unit of [four-dimensional] time. Space, which can be regarded as a 3-hypersurface, having a thickness (in the direction of the fourth dimension) of 10^-33 cm. This thickness is also representing the duration of fundamental time unit (10^-44 second) which we call Now.

d)     The interplay between the positive and negative energies perpetually creates a dynamic ephemeral space. A spacetime without space (universal "now") is chaotic.  Under the current light-cones framework, we cannot single out the simultaneity of events from the chaos of coexistence and succession of things.

References:

1.   Russell, B.:” History of  Western Philosophy," Routledge, London, 1996, p. 352

2.   Barbour, J.:" The End of Time," Phoenix, London, 2001, p. 143

3.   Tolle, E.:" The Power of Now," New World Library, Novato, California, 1999