Friday, January 21, 2011

The Cosmic Inflation Never Happened

The Big Bang theory holds the premise that the universe originated from a singularity which came into being out of nothing through a single massive explosion. The concept of the minuscule size of such primeval singularity born from the thought projection of the current universe's expansion backward far in time.

The theory has at least two dubious primary grounds. The first one is the speculative concept of nothingness. The Big Bang theory presumes, violating the first law of thermodynamics, that energy (and matter) was created out of nothing. This idea came from the mindset that the creation of the universe (4-spacetime) was the beginning of everything. Notwithstanding, the theory takes for granted that the quantum fluctuation which stimulated the primeval explosion held in the nothingness before such creation.

The second speculative ground is about the size of the universe which can be shrunk indefinitely backward in time from the current size into a singularity. Close to the moment of creation, the size exponentially shrunk about 1060 smaller just within 104 seconds, from 10-33 to 10-37 second posterior to the explosion, the rate of which was exceedingly faster than the speed of light1.

As we have elucidated so far, there can be no such thing as nothingness. The energy, as the only reality in nature, can neither be created out of nothing nor destroyed into nothing. The universe was born as the result of the interplay between the opposite (positive and negative) energies that created the universe and everything within, not out of nothing.

We can mathematically describe energy in its pure condition as waves' spectrum of different frequencies and amplitudes expressed in terms of Fourier series or its complex form, the Laurent series:

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


It is a wave function expressed as the sum of its positive frequency (F+(z)) and negative frequency (F(z)).

Globally, we can depict this wave function in terms of Riemann sphere, the positive frequency F+(z) extends holomorphically into the southern hemisphere, and the negative frequency F(z) extends holomorphically into the northern hemisphere, where the equator represents the real coordinate and the longitudinal circles its imaginary time coordinate.

The domain of the positive and negative frequencies, however, does not fully extend to the poles, as the Riemann sphere has an annulus of convergence which excludes the domain around the zero points (singularity) as well as the infinity (Figure-1)

 This pure mathematical analysis indicates that the split of the energy cannot create a stable interface (hypersurface) from the beginning when the energy started to split up to a certain period where it reaches the minimum size (represented by the Riemann sphere's inner ring of convergence).  The interface created in this period would instantly dissolve into energy.

It is only after reaching this limit that the hypersurface comes into being where it stabilized until it reaches its maximum size (represented by the Riemann sphere's outer ring of convergence).

The doomsday comes when the hypersurface reaches its maximum size.  At this particular time, the hypersurface becomes extremely unstable that makes it break down into pieces dissolving back into pure energy.  We illustrate these phenomena in Figures-2 and 3A.

This cosmology scenario avoids the need of either the concept of singularity to represent the beginning of the creation or the big crunch at the end of the universe's life as well as the everlasting expansion where the universe has no dead end.


This cosmology theory also excludes the need of the concept of cosmic hyper-inflation in the early period of the creation (Figure-3B), as the baby universe was born in an exceedingly larger size than that of the singularity which the Big Bang theory presumed.

The interplay between the positive and negative energies generates quantum fields across through the interface (hypersurface) a), located in between the two, perpetually creates quantum sparks (fundamental particles), the building block of the universe. As the split (hypersurface) area is enlarging with time, new matters are created in the expanding horizon keeping the average matter density per area almost constant.


This cosmology concept resembles the one of the continuous creation, steady-state expanding universe put forward by Fred Hoyle2.


Notes:

a) We use the split area, interface, hypersurface, and space interchangeably.

References:

1.  Guth, A.: "The Inflationary Universe," Basic Books, New York, 1997
2.  Gregory, Jane: "Fred Hoyle's universe," Oxford University Press, New York, 2005

Friday, January 7, 2011

Multidimensional Time and Hypercomplex Numbers

Long before physicists embarked on the study of higher-dimensional spacetimes, 19th-century mathematicians had firmly established the geometry concept of multidimensional metric manifolds. Many of these concepts were straightforward generalizations of ideas on the properties of surfaces embedded in the three-dimensional Euclidean manifold.
To simplify things, the mathematicians have introduced a multidimensional surface-like concept called hypersurface for modeling multidimensional space embedded in a higher multidimensional ambient manifold. A flat m-hypersurface can be appropriately embedded in an (m+1) space, but matters become more complicated when one comes to consider curved hypersurfaces. A curved m-dimensional hypersurface requires an ambient space whose dimensions are at least equal to or greater than ½m (m+1) 1.
Accordingly, a 4-dimensional curved spacetime requires at least a 10-dimensional ambient space. The spacetime's point position and, hence, the curvature of the spacetime is completely defined through a collection of numbers associated with the coordinate system set up in such 10-ambient space which we are more familiar with as the metric tensor's independent components of such 4-spacetime.
So what is so startling about it is when we explore the micro realm we would be confronting with the same 10-dimensional ambient space. Alas, in the later development, such as in that of the superstring theory, physicists made a blunder as they wrongly assumed the curly nature of the extra dimensions of such ambient space,  which made them going nowhere


The same fate happened to Big-Bang theory as physicists firmly exclude the existence of the universe's surrounding spaces. In doing so, physicists throw away the more significant part of the system, and this might be the reason why the theory incorporates only five percent of the total mass and energy that it actually should be.

Now the only option to cope with this impasse is jumping off the ship and abandon not only about the curly nature of the extra dimensions but also the one-dimensionality of time.

As the last article has deliberated,  those multiple temporal dimensions are the results of a series of successive symmetry breakings occur which had created different worlds, each of which had its respective temporal dimension (Figure-1).
Quaternion and Octonion
Now, how do we describe the structure and the geometry of such multiple temporal dimensions? To do this, we need to build a coordinate patch within such ambient space framework. To start with, let us deal with our 3-dimensional physical space embedded, as it should be, in a 6-ambient space. In such a case, we assign a coordinate patch consisting of three real space coordinates x1, x2, and x3 and three imaginary time coordinates whose basis ij and k.
If we denote x= x(x1, x2, x3), then we can define any world point in such 3-physical space as:

q = x + ui + vj + wk,
expression is found to be nothing but the quaternion; a generalized complex number discovered a long time ago by Hamilton who established the geometry and the algebraic structure of this quaternion in 1843.
If we express the time variables u, v and w proportionally to the speed of light ci of the respective temporal dimensions ti then we can write:

q = x+ ic1t1 jc2t2 kc3t3,

This quaternion describes a general vector within a 6-dimensional space expressed as a function of space and time coordinates. Quaternions, therefore, describe a 6-dimensional vector space over the real numbers, depicting the dynamical geometry of 3-space embedded in 6-ambient space.
Similarly, we can define the 4-spacetime whose ambient space is ten dimensional through a coordinate patch consisting of three real space coordinates and seven imaginary time coordinates.
Again if we assign a space coordinates as x= x(x1, x2, x3) and i, j, k, l,m, n, and o denote independent imaginary numbers as the coordinate basis representing seven different time coordinates, then we can define any point located at the 3-space in such coordinate patch as:
q = x + ai + bj + ck + dl + em + fn + go
, where x, a, b, c, d, e, f  and g are real numbers. Graves and Cayley had already discovered this expression, known as double quaternion or octonion, long time ago in 1845, although they did not know about the physical implication of it.
If we express the time variables a,b,c ... g proportionally to the speed of light ci of the respective temporal dimensions ti then we can write:
q=x+ ic1t1 jc2t2 kc3t3 +lc4t4 mc5t5 nc6t6 oc7t7
Octonions form a 10-dimensional vector space over the real numbers, depicting a 3-physical space embedded in 10-dimensional ambient space.
In a later development, the original notions of quaternion and octonion are further modified and generalized through what so-called Clifford and Grassmann algebras applied to any higher dimensions framework which is found to have powerful implications in modern physics.
Many mathematicians and physicists wrongly perceived the quaternions and octonions as respectively describing 4-dimensional and 8-dimensional spacetime (having both one-dimensional time), which is inappropriate.
Penrose2 regarded Hamilton's 22 year-devotion in his life in attempting to develop the quaternion calculus resulted in relative failure. On the contrary, we regard the Brougham Bridge's stone carved with the Hamilton fundamental equation would become a momentous milestone of the application of the hypercomplex calculus on the geometry of multidimensional time in both macroscopic and microscopic realms.
References:
1.    Sokolnikoff, L.S: "Tensor Analysis," Wiley Toppan, Second Edition, New York, 1964,   p. 205.

2.    Penrose, R.: "The Road to Reality," Vintage Books, London, 2005, p. 201