Sebn-ut Kamuas Tutamen
139 McIntyre St.
Savannah, GA 31401

by Sebn-ut Kamuas Tutamen

The astronomical pursuits of this century has brought the modern astronomer face to face with strange new possibilities for forms of matter and space. The theorists who created the formulae leading to the modern body of astrophysical theories like Scharzchild, De Sitter, Cantor, and Einstien and the modern theorists, Hawkings, Pendling, Davies and Chandrasekhar, along with a host of others, have all brought the theoretical structure of the universe to a highpoint of possibilities for the existence of exciting new forms of matter. Of particular importance has been the evolution of the theories of gravitation, and their role in the prediction of the collapse of stars, and the curvature of space. At this, one of the great thresholds of discovery in astronomy, we encounter our greatest theoretical stumbling blocks; the meaning of infinity and its importance in relation to the structure of the universe. I would like to address this particular problem here, by offering a hypothetical explanation of how and why infinity plagues our mathematical calculations in the study of the gravitational fields.

The origin of the concept of infinity can be traced to the Egyptian mathematics and philosophy. It has been proven that infinity does exist in our system of logic in the philosophies of the Greeks, particularly Zeno, and the mathematics of Georg Cantor and Karl Weirstrass. The latter proofs have shown the unattainable nature of infinity; the fact that its realization is beyond our grasp. [1 Paul Davies, "The Edge of Infinity", New York, 1981, 22-24.] Modern mathematics has come to, essentially, the same conclusions.

What has progressed, however, is our understanding of the nature of the infinite. We can now apply infinity to limited mathematical uses. However, these applications have not been satisfactory, nor adequate, for the needs of modern theorists in their formulae. The appearance of infinity in an equation halts mathematical progress.

There are equations where the infinite values can be replaced by finite values, called renormalizable equations. However, the opposite type, the non-renormalizable equations, leaves one at a virtual standstill in theoretical progress. This is the case with the theories of quantum gravity, which are rendered meaningless by the reappearance of a string of infinities. [2 Ibid., 42.]

The theorem that I am about to explain will increase the possibility of renormalization of the equations accompanying the quantum theories of gravity, or provide a foundation for the formulation of a new set of equations to describe the phenomenae.

Matter and Space

Consider the nature of matter and space. Matter is thought to be finite in its quantity and internal structure. On the other hand, the nature and quantity of space is still vague; hinging on being unknown. Even within this diametric condition between the two things, a concrete relationship of matter to space can be observed. We can see that matter nullifies space in an event that is either occupation or displacement. It does so in measureable, finite quantities. To do this, the matter must "nullify" an amount of space equal to its volume. Therefore, a particular volume of matter nullifies an equal volume of space. This can be expressed mathematically as equation (1). Equation (1)' shows that the ratio of the volume of matter to the volume of space is equal to unity.

(1) Vm = Vs
(1)' Vm / Vs = 1

Unoccupied space is called free space. This is the type of space that is found inside of an absolute vacuum, and this is the quantity of space that we refer to whenever we use the term free space. Knowing that matter is absent from free space, and constructing a mathematical model from equation (1)', we get equation (2).

(2) 0 / Vs = 1

An inversion of space and matter can be derived mathematically from equation set (1)

1 / Vs = 1 / Vm

and, when the space is free

(2)' 1 / Vs = infinity , Vm = 0

This says that there are an infinite number of fractional parts into which the space can be divided, because no matter how large a volume of space you begin with, a unit fraction of it is infinite, and thereby makes the entire quantity infinite.

The proof of this has been done using the length of a quantity of space, and then subjecting it to repeated divisions. These divisions result in smaller portions of the length of the space being created by each division, but still there is always some space remaining, because the divisor is infinite in scope. Since this is true, it follows that our mathematics reveals an interesting characteristic of calculations involving free space. That is, that the ratio 0 / 0 is equal to unity for free space, and not zero, as it usually is in mathematics.

Using the reciprocal of formula (1)', we get the relation,

(2)'' Vs / Vm = 1.

This relation is still unity, but when the space is free space we have the relations Vs / 0 = 1 or infinity = 1. This type of result is useless to us at this point. If we multiply Vs / 0 = 1 by the equation 0 = 0, the result obtained in the free space domain will be Vs = 0. Substituting in equation (2) : 1 / 0 = infinity : This is true by definition. So when calculations concerning free space are made, the product of the volume of space divided by 0, and 0, gives the volume of space times unity in the form; (0/0) * Vs. What this states mathematically is that when the volume of matter, which is zero, enters free space, the volume of space equals, or becomes, zero (eq. 1). The space is nullified by matter. So as our volume of matter goes from 0 towards infinity, our volume of space increases proportionally. Hence-

(2)''' limVm->inf Vs = Vm.

The volume of space is directly proportional and equal to the volume of matter.

Equation (1) forces the volume of space to be zero when we require free space and we have discovered that under this condition once Vs > 0, we are not expressing the volume of free space, but we are expressing the volume of matter in terms of space, and the volume of space in terms of matter, which follows from equation (1). The two are merged, and occupy the same volume of free space at the same instant. A type of space has been derived, then, that I shall call matter-space. I shall show through mathematics and deductive reasoning that matter-space possesses the quality of space transforming to matter and matter transforming to space.