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 i, j 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.
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
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