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, which is the main focus of the general relativity theory, 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 Rm in 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 don’t 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 really 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, a spacetime as a system is depicted as a hypersurface embedded in a higher dimensional metric manifold (hyperspace) representing it’s surrounding. It is conventionally defined 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 perpetual 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 important in revealing the quantum phenomena. The thickness of our 3-dimensional space, for example, was found to be equal to fundamental Planck distance of 10-33 cm or is equal to the Planck instant of time which is 10-44 second. 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 is located in between two immiscible “fluids”, which we refer as the opposite 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 phenomenon (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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