Turkish
Journal of Physics, 2012, Vol. 36, No 3, P. 461 –
464.
Scale
Dimension as the Fifth Dimension of Spacetime
Sergey G. Fedosin
Perm, Perm Region, Russia
Abstract
The
scale dimension discovered in the theory of infinite nesting of matter is
studied from the perspective of physical implementation of well-studied
four-and n-dimensional geometric objects. Adding the scale dimension to
Minkowski four-dimensional space means the necessity to use the
five-dimensional spacetime.
Keywords: scale dimension; multi-dimensional space;
theory of infinite nesting of matter.
Intensively developed in recent years the theory of infinite nesting of
matter, which studies the entire hierarchy of space systems, faced the
necessity of introducing scale dimension into the everyday life of science. The
purpose of this article is to analyze the properties of the new dimension and
its relationship with geometric theory of n-dimensional
spaces.
The number of basic dimensions in physics is determined by the number of
degrees of freedom or independent variables that set the location of a physical
body and its elements, considered as points in a given frame of reference. The
number of dimensions or degrees of freedom gives the dimensionality of the
spacetime used. By adding the scale dimension to the four-dimensional
spacetime, we obtain the five-dimensional manifold, which includes the usual
spacetime. According to the order of historical understanding this can be
written as (3 +1 +1)-space, where in the first place the spatial dimensions are
reflected, and then the dimensions of time and scale. In terms of geometry it
is convenient to consider the axes of the all dimensions perpendicular to each
other.
In contrast to geometry, physics does not deal with mathematical points,
but with specific material objects. Infinite division of these objects in parts
does not lead to the emergence of mathematical points, but instead new objects
of the smaller size appear. Although these objects seem more and more similar
to the points, they contain an infinite number of matter carriers and the
smallest field quanta. If the motion inside of material objects is regarded as
motion along the scale axis, then this axis is a special dimension of space.
During the motion along the scale axis transition between the scale levels of
matter takes place and we can see that the objects of lower levels become parts
of the objects of higher levels of matter. As a result the scale dimension can
be represented as the dimension characterizing the complexity of space systems,
in the sense of their composition of lower order systems.
The theory of infinite nesting of matter and the idea of the scale
dimension acquired their modern form owing to the works by Robert Oldershaw [1]
(self-similar cosmological model), Sergey Sukhonos [2-3] (scale similarity of
space systems) and Sergey Fedosin [4-5] (similarity of matter levels).
Spatial relations are perceived by the observer through the shape of
objects and their configuration relative to each other, and temporal relations
are determined as the changes of shapes and configurations, the angular and
linear shifts. Owing to the fact that objects at different levels of matter
evolve and change relatively independently from each other and the speed of
processes and the characteristic sizes and masses of objects differ
significantly, the scale dimension is considered as a new dimension of
spacetime. At the same time shapes and configurations of many three-dimensional
objects at different levels of matter in many respects are similar to each
other. This is fixed by relations of similarity for various physical quantities
and SPФ-symmetry,
which states the similarity of physical laws at different levels of matter [4].
In the field of research that uses large-scale
transformations and spatial dimensions, the scale dimension can be studied by
geometrical methods, and thus it is in the scope of geometry. In
four-dimensional spacetime as the fourth dimension not the time t itself is taken, but the product 
, where c
is the speed of light (since for the exact spacetime measurements
electromagnetic waves are commonly used). Section of such
spacetime that is perpendicular to the time axis at some time point, defines
the hyperplane in the form of three-dimensional geometrical space at a given time.
If in the three-dimensional space an object moves to the future upcoming events
through the point of the present time, in the four-dimensional spacetime the
movement from time 1 to time 2 is seen as a trace increasing in size from point
1 to point 2 that is associated with the object. The possibility of observing
such a trace in the coordinate system means that the motion can be described as
a set of three-dimensional images of an object at different times.
The introduction of the scale dimension takes into account that at
different levels of matter the rates of time flow, regarded as the speeds of
typical processes for similar objects, differ from each other. However, it is
possible to introduce the total coordinate time, based for example on periodic processes
in an electromagnetic wave. In this case, the hyperplanes which are
perpendicular to the scale axis will determine the single four-dimensional
spacetime at different levels of matter. By analogy with the four-dimensional
Minkowski spacetime, now we can introduce the five-dimensional coordinate
system and write down for each world event its vector as follows: 
, where 
is a
three-dimensional radius-vector, 
− a
coordinate along the scale axis.
Scale dimension allows us to understand and prove the possibility of
existence of real four-dimensional bodies in five-dimensional spacetime. For
constructing such bodies the method of induction may be used, based on the
following reasonings:
– One edge is a line
segment bounded by two points (vertices);
– Three edges in a
plane give a triangle with three vertices, in the form of a face;
– Six edges with
four vertices in three-dimensional space give a tetrahedron (triangular
pyramid) with four faces;
– Ten edges with
five vertices, with ten faces and the five tetrahedrons in four-dimensional
space represent an object called pentachoron (4-simplex).
For a n-dimensional tetrahedron or a
simplex with increasing n-dimensionality of space to one, the
number of vertices increases, too, the number of vertices equals k = n + 1. Each edge is a connection between two arbitrary vertices of the
tetrahedron, so that the number of edges depends on the number of vertices
according to the formula for combinations of two with the k vertices. To determine the number of faces of n-dimensional tetrahedron in the form of
triangles we need to find combinations of three with the k vertices. In general, the formula for r combinations of k
elements is expressed by means of factorials and has the form:
.
(1)
Equation (1) allows us to find the number of edges, faces, r-polyhedrons as part of n-dimensional tetrahedron.
For a n-dimensional cube
(hypercube), we can assume that its number of vertices, depends on the
dimensionality of space n according
to the formula: 
, where dimensionality of the point space is
zero, the linear space has
dimensionality 1, the flat space has dimensionality 2 etc. Consequently, a
tesseract (a tetracube) as a hypercube in four-dimensional space must have 16
vertices, and according to (1) 120 edges. If we remove from this number the
internal edges between the vertices of different faces, as well as the diagonal
edges lying in the plane of faces, then only 32 external edges would remain,
which are responsible for the convexity of a tesseract as a geometric figure.
Besides a tesseract has 8 external three-dimensional cubic faces and 24
external two-dimensional square faces which limit it in the four-dimensional
space. The common name for n-dimensional bodies with flat sides is the
polytope, for four-dimensional bodies the term polychoron is accepted, whereas
for three-dimensional bodies the term polyhedron is used, and for
two-dimensional bodies the term polygon is used.
In space of n dimensions it is
possible to construct a variety of figures, and find their properties,
including the n-dimensional angles,
areas and volumes. In particular, the volume of the n-dimensional sphere can be expressed by means of its radius R and the gamma function, according to
[6]:
.
This formula was known already in 19th century in the writings of the
Swiss mathematician Ludwig Schläfli [7].
Currently, the branch of mathematics which studies geometric objects in
space of n dimensions is developed so
well that all possible objects are divided into classes according to symmetry
properties [8]. There are online resources, demonstrating the projection of
multidimensional bodies on a plane or in three-dimensional space [9-10],
obtained by computer modeling.
From the mathematical point of view, any space of smaller
dimensionality, which is part of the space of higher dimensionality, is a
hyperplane on which it is possible to make projections of points of n-dimensional body. If a simplex or a
tesseract is oriented strictly along the fourth spatial dimension, then at each
point of a three-dimensional hyperplane perpendicular to the axis of the fourth
dimension, there will be a tetrahedron, or a cube, respectively. Similarly, if
we dissect a tetrahedron or a cube perpendicular to their bases, then on the
section plane (two-dimensional hyperplane) there will be a triangle, the size
of which depends on the cut-point, or a square. While the four-dimensional body
is passing through the three-dimensional space one would expect the sudden
appearance and disappearance of the projections of this body and changing of
their size. As in everyday life such examples are not commonly found, it is
seen that the fourth spatial dimension does not reveal itself directly.
From the stated above it follows that the scale axis, discovered in the
hierarchy of space systems is a physical embodiment of the fourth spatial
dimension [11]. A set of three-dimensional objects at different levels of
matter represents a four-dimensional body oriented along the scale axis, so
that each cross section of this axis represents a three-dimensional hyperplane,
which may contain corresponding three-dimensional objects.
In contrast to the mathematical idealization, the objects of a natural
system because of their limited total number are located on the scale axis with
unequal probability, but unevenly, grouped into separate levels of matter. In
addition, physical properties of objects’ substance are changed at different levels of matter what
is usually ignored in geometry. At those levels of matter, where gravitational
forces prevail, the dominating space objects have spherical shape.
Any set of objects, consisting of spherical galaxies, globular star
clusters, individual stars, planets, nucleons, etc., forms a definite single
four-dimensional spherical body, stretching along the scale axis as along the
fourth spatial coordinate. This four-dimensional body is discovered not by its
projections as the result of its passing through our three-dimensional world,
but in the opposite situation, when the observer himself is moving relative to
this body and examining its components in various hyperplanes. Thus, the scale
dimension confirms the possibility of real existence of four-dimensional
geometric objects in nature. This enables the application of mathematical
methods for n-dimensional spaces for
the study of composite physical objects at different scale levels of matter.
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Source:
http://sergf.ru/mi5en.htm