Any application of the
natural laws to a discrete portion of cosmos requires the definition of a
system and its surroundings1. A system, in this case, can be any
object, any region of space, or spacetime set apart mentally from everything
else, which then becomes the surroundings.
The general relativity
theory describes the universe, the four-dimensional spacetime as a whole,
without reference to the surroundings. The theory concerns mainly on the
intrinsic properties of the spacetime and tacitly considers the latter as a
system having no surrounding or a surrounding which is null and void. This
intrinsic geometry takes no account of the distinguishing characteristics of
spacetime as they might appear to an observer located outside the system.
Such a premise, i.e.
four-dimensional spacetime without surrounding, could be premature since at the
current stage of development the physicists have been forced to deal more with
higher and higher dimensional hyperspaces. The curvatures of the
four-dimensional spacetime, for example, could only take place if the spacetime
is embedded in a much higher dimensional hyperspace; otherwise, it could only
be flat.
The ten-dimensional
surrounding hyperspaces
The concepts of the
geometry of n-dimensional metric manifolds (hyperspaces) are straightforward
generalizations of ideas of the study of surfaces embedded in the
three-dimensional space. A generalization of the concepts of curvature and
torsions to curves embedded in the n-dimensional manifolds is direct and
straightforward, but matters become rapidly involved when one comes to consider
hypersurfaces2.
What we are interested
the most, in this case, is about the circumstances in which an m-dimensional
curved [Riemannian] hypersurface, Rm, can be embedded in the n-dimensional
[Euclidean] manifold. Concerning the
global embedding of the whole of Rmin E3 almost no
general results are known, however. It is possible to prove that a neighborhood
of Rm can be embedded in En if at least n =
½ m(m+1).
The curved
four-dimensional spacetime of the general relativity theory can only be
embedded at least in a ten-dimensional hyperspace. The ten independent
components of the metric tensor in such a system are nothing but the dimensions
of the surrounding hyperspace. The empty surrounding space (having zero
dimensions?) as assumed in the general relativity is an oversimplification of
the reality. We do not have to wait until the advent of the string theory only
to be aware of the requirement of such a higher dimensional ambient hyperspace.
Both macroscopically (the general relativity theory) and microscopically
(superstring theory) require at least ten-dimensional space to preserve the
proper applicability of physical laws. A question naturally arises: Do we need
to have hypothetical tiny curled extra-dimensions?
Hypersurface vs. hyperspace
It is often more
convenient to generalize the ideas of the study of surfaces embedded in the
three-dimensional space depicting a group of hyperspaces. As the generalization
of the idea, we depict a spacetime as a hypersurface embedded in a higher
dimensional metric manifold (hyperspace) representing its surrounding. We
conventionally define that in an n-dimensional framework, we have (n-1)
dimensional hypersurface embedded in n-dimensional surrounding hyperspace,
unless it is defined otherwise.
The advantage of using
such a model is that we can better describe the dynamic of the system, for
example, the rotation movement of the hypersurface around an axis located
across its surface describing a colossal cycle of closed time-like curves. We
can also describe the possible rotation movement of the hypersurface around an
axis normal to its surface to explain the constant rotation of the solar
system, galaxy, super-galaxy and so forth.
Another advantage we get
is that we can take into account the geometry element that hitherto overlooked,
i.e. the "thickness" of the space or hyperspace which is essential in
revealing the quantum phenomena. The thickness of our 3-dimensional space, for
example, was found to be equal to Planck distance of 10-33 cm
or is equal to the Planck instant of time which is 10-44second.
Nature abhors any object or shape to have zero thicknesses; otherwise, it will
evaporate into thin air. Space and hyperspace have no exception.
We should bear in mind
that what we are talking about the hypersurface here is not analogous to a
piece of paper floating freely in the air (as in the case of "brane"
theory), but more to an interface of an oil-water system. The hypersurface or
more precisely hyper-interface locates in between two immiscible
"fluids," which we refer as the opposing elements of the energy as a
whole: the positive and
negative energies.
Here again, we can study the hyper-interfacial tension of the system as related
to the gravitational constant (G).
References:
1.
Abbott & Van Ness: Thermodynamics, Schaum's outline series, Mc Graw
Hill Co., New York, 1967
2.
Sokolnikoff, I.S.: Tensor Analysis, John Wiley & Sons, Inc., New
York, Second Edition, 1964
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