We used to conceptualize the
geometry elements such as point, line, surface and space as having,
respectively, zeroed, one, two and three dimensions. There is nothing wrong
with that as far as we are dealing with abstract objects such as a corner point
between a floor and two walls, meeting line between ceiling and wall, table's
surface or hall's spaciousness.
However, we cannot apply such a
concept for real bodies, whatever the size is. A grain of sand is not a
zeroed-dimensional object, but a three-dimensional cubic-like body, which has
small length, width, and thickness. A string is a three-dimensional long
cylindrical object having a small section. Similarly, a piece of paper is a
three-dimensional surface object whose thickness is very thin (Figure-1). Had
their thickness been reduced to zero, those objects would all have gone into
thin air.
Nature does not seem to give any
exception to natural bodies such as space or any other higher-dimensional
spacetimes. For their existence to have physical meaning, all those bodies
should have thickness.
It implies that space or
spacetime, whatever its dimensions, are always be embedded in an ambient
spacetime of at least one dimension higher. At the same token, the later is
also embedded in turn in another much higher manifold (Figure-2). This kind of infinite regress makes us
believe that nature is vast and infinite, not only its areas but also its
dimensions.
System
and Surroundings
Now, the formulation of the laws
of nature depends naturally on which system we choose. Suppose we want to
formulate physical laws within a system of an m-dimensional spacetime embedded
in N-dimensional ambient manifold, we
get, then, physical laws of a system having (N-m) extra dimensions. The
directions of these dimensions determine those of the spacetime’s thicknesses
pointing outwards away from it.
We can also describe the same
physical laws in a much simpler system where the same N-ambient space embedding
(N-1)-hypersurface, instead of an m-spacetime. The thickness of such a
hypersurface has the same direction as at the Nth dimension pointing
outward away from it.
The laws of nature in the first
system have very complex formulations and are difficult to resolve, as the
system has too many extra-dimensions and, hence, fewer symmetries. The laws of nature in the second system are
relatively more straightforward as the system has only one extra-dimension and
is highly symmetric.
We can best describe the laws of nature when the number of the dimensions of the ambient space embedding the system is large enough which "stretches" out the hypersurface to become completely flat and perfectly symmetric.
How do we determine the
dimensions of the ambient space (N) vis-a-vis that of the embedded spacetime
(m)? There is a minimum requirement for the number of the ambient space's
dimensions in order that the spacetime can be “properly" embedded in the
ambient space. The [non-flat] m-spacetime can be embedded in N-manifold only if
at least N = ½ m(m+1) 1). The metric tensor of the m-spacetime
dictates that the ambient space should have that amount of dimensions for all
of its components can be properly defined.
Multidimensional
Time
Now, what are these dimensions all about? As we have discussed previously, the spacetime is the physical manifestation of energy. In its original state, the spacetime was perfectly symmetric. All of its dimensions are indistinguishable, and they are all "temporal." When the respective energy segregates into the positive and negative energies, the [temporal] spacetime's dimensions along the interface [separating those opposing energies] are transformed into spatial dimensions.
For the
classical 4-spacetime, the energy segregation transforms three of the
spacetime's temporal dimensions along the interface into spatial (Figure-3). In
a 6-spacetime, the energy’s segregation transforms the spacetime's five
temporal dimensions along the interface into spatial dimensions. The same case
also prevails for the 10-spacetime., where nine temporal dimensions along the
interface become spatial.
The temporal
dimensions t1, t3 and t7 related to the 4-, 6- and 10-spacetimes,
respectively, are different from each other. It is against the mainstream
premise which tacitly asserts that there is only one temporal dimension in
nature.
Based on the rule we have, a 4-spacetime requires a 10-ambient space
for the physical laws to have solutions. However, as we have in this case 3
spatial dimensions and seven [imaginary] extra-temporal dimensions, the
physical laws we get would be very complicated. It is imperative, therefore, to
have the same laws applied to a system consisting of a 10-ambient space
embedding 9-hypersurface, which are simpler as we have only one imaginary
temporal dimension on top of the nine real ones.
It is more or less what physicists have done in developing the string theory, except that the extra-dimensions were assumed to curl into tiny loops. Also, the temporal dimension of the system was assumed to be the same as that of ordinary time. Such wrong assumptions have been put forward because mainstream physics holds the premise that time is one-dimensional as previously mentioned.
The relativity theory should rigorously hold the equivalence of space
and time dimensions. The spatial and temporal dimensions should be transferable
to each other depending on the system they become part. The extra dimensions
are undetectable not because they curl into tiny loops but because they are
temporal.
Supermanifold and Supersymmetry
Generators
Physicists have many problems with their mathematical propositions as
they used to conceptualize the spacetime as a standalone basis. Under such a
concept they have taken the part of the reality out of the system. Such as is
the case of the Big Bang theory, which is entirely Platonic, a system without
any geometrical thicknesses, surrounding, nor even 3-space.
A reader of the Scientific
American2) once asked: "Where is the universe expanding
to?" The authoritative answer from
the expert was: "... the universe's expansion does not push it into new
territory - rather the spacetime grid itself is expanding". The issue has arisen again and again since
the Big Bang theory was put forward, as only a few people were satisfied with
such an explanation. The excellent answer should be that the universe is
expanding to at least the 10-dimensional ambient space, and not into nothing.
To make their model closer to the
reality, some physicists artificially introduced what they called supersymmetry
generators, replacing the thicknesses which they have “forgotten” to
incorporate in their mathematical model. They call this manifold having
thicknesses “Supermanifold”3). The physicists should put forward the
problems of embedding at the forefront of physical researches and develop a
more holistic model instead of a piecemeal one.
References:
1.
Sokolnikoff, L.S.: ”Tensor Analysis," Wiley
Toppan, Second Edition, New York, 1964, p. 205
2.
Kashlinsky, A.: "Where is the Universe
Expanding to?", Scientific American, (Ask the Experts Forum), May 2007, p.
104
3.
Penrose R.: "The Road to Reality,"
Vintage Books, London, 2005, p. 879
