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.

Energy, Spacetime and Riemann Sphere

Ephemeral Fundamental Particles

We are used to perceiving matter as a tangible thing, however small it is. We believe that when we divide a chunk of matter into its smaller parts we will eventually unable to divide them any further. The classical reason for this is because they have no more internal structure to be split.

We have taken for granted that the smallest particle is persistent, continuously at rest or moving around in space, just like an ordinary billiard ball. This continuity and deterministic features of nature hold not only as the foundation of the classical but the relativity theory as well.

The quantum theory which mostly deducted from experiments holds the contrary. To take into account this paradox, we put forward a new paradigm: “when we divide a chunk of matter into its smaller parts we will eventually unable to divide them any further because they are so small and extremely unstable that they immediately disappear into "thin air".

The smallest part of matter, the fundamental particle" is ephemeral. The persistence of a particle in the macro realm is nothing but the appearance of the perpetual creation and annihilation of fundamental particles. In the deepest level of reality (Planck distance) matter is created and annihilated to and fro energy, in accordance with the famous Einstein’s equation E = mc2. This is the underlying reality of the discontinuity and probabilistic features of the quantum realm.

The Energy and its Polarity

The special relativity theory concerns the fundamentals namely space, time, energy, and matter which basic nature has puzzled people for thousands of years. The special relativity theory has classified and unified those fundamentals into two distinct entities: the spacetime and energy. The space and time have been unified into a four-dimensional entity called the spacetime while the energy and matter into a single energy substance from which matter is derived, as expressed in the equation E = mc2.

And the special relativity theory ends here. People presume that the spacetime is like a container and energy something that fills it. This may lead to the misconception that the spacetime could be empty devoid of energy. Once again a new paradigm shift is called for:” The spacetime and energy are inseparable just as the spherical form and water substance in a water drop to be. The spacetime is just the geometrical aspect of energy. Contrary to what Minkowski presumed, spacetime does have no existence on its own.

Now, here we have energy as the one and only independent reality in nature from which everything else is derived. It is primordial; neither can be destroyed nor created, the only existence in nature. Another important aspect of energy is its dimensions. The spacetime’s dimensions are just the reflection of the degree of freedom and the potency of energy. The higher the energy potency the bigger the number of the spacetime’s dimensions to be. This is the fact behind the existence of higher dimensional worlds which some physical theories have revealed. Under the special relativity theory, as the spacetime is four-dimensional the energy is four-dimensional accordingly.

Another important principle that the special relativity theory reveals is polarity energy. The energy as a whole consists of positive and negative energies as expressed in the relativity energy equation E2 = m2c4+p2c2. Surprisingly, nobody has paid enough attention to this energy polarity, which in reality is the basic prerequisite of the creation of the universe.

Creation by Separation

The energy as a whole tends to separate into its positive and negative components. As they segregate from each other, a hyper-interface is created in between the two; analogous to the interface between oil and water that we can observe in our daily life. As the energy and its opposite substances are 4-dimensional, this hyper-interface is 3-dimensional (space).

How do the two opposite energies split? The split does not take place instantaneously across the whole spacetime, but gradually stretching out across the spacetime just like a crack propagating in the middle of a glass sphere. This represents the expanding universe as an alternative against the Big-Bang theory. The repulsive force of the opposite energies that stretches out the 3-interface (space) is the external energy that we call the dark energy.

Probabilistic and Non-Locality Features of Nature

The interplay between the positive and negative energies gives rise to quantum fields piercing through and igniting quantum-sparks (“quarks”) at the 3-interface analogous to the pixels appearing and disappearing at a gigantic plasma TV screen. This is the deepest reality of fundamental particles.

Quanta are created and annihilated to and fro the energy in accordance with the energy equation E=mc2. There is no persistent fundamental particle exists in nature. What we perceive as a persistent particle in the macro realm is, in reality, the appearance of a series of different particles subsequently created and annihilated one after another. It is analogous to the flame of a candle which is never the same and always being different entities at any instant.

This is the underlying reality of the discontinuity and probabilistic features of quantum mechanics. Another important feature, the non-locality, is nothing but the manifestation of the perpetual particles’ creations which are taking place simultaneously.

References:

Herbert, Nick:" Quantum Reality," Anchor Books, New York, 1985

Ridley B.K.: "Time, Space and Things," Cambridge University Press, Cambridge, 2002

LINKS:

Multidimensional worlds:http://multidimensionalworlds.com/

Ephemeral Fundamental Particles