Multidimensional Time and Hypercomplex Numbers

Long before physicists embarked on the study of higher-dimensional spacetimes, 19^th-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,

where x, u, v and w are real numbers. This 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 c_i of the respective temporal dimensions t_i 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 c_i of the respective temporal dimensions t_i 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

Multidimensional Time and Hypercomplex Numbers