Hypersurface, the Extrinsic View of the World

Any application of the law to a discrete portion of the universe or even something else more prominent such as the possible multiverses requires the description of a system and its surroundings. A system can be any region of space, the whole universe itself or any region of multiverses selected for study and set apart [mentally] from everything else, which then becomes the surroundings.

There are two ways on how we can geometrically describe the world, i.e. intrinsically or extrinsically. Let us take an example on how we describe a lower dimensional object such as a surface. We can describe the properties of surfaces without reference to the space in which the surface is embedded as intrinsic properties. We can imagine that in this particular case the properties of the surface are analyzed from two dimensional flat being living in such surface, whose universe is determined solely by [two] surface parameters. In this intrinsic geometry a pair of isometric surfaces, a cone, and cylinder, for example, are indistinguishable.

These surfaces appear to be quite distinct to an observer examining them [extrinsically] from a reference frame located in the space in which the surfaces are embedded. Geometrically, an entity that provides a characterization of the shape of the surface as it appears from the enveloping space is the normal line to the surface^[1].


Now let us turn to examine the concepts from the geometry of higher dimensional metric manifolds which are of our primary interest. Many of the concepts are straightforward generalizations of ideas introduced in the study of surfaces embedded in the three-dimensional [Euclidean] manifolds. For the visualization purposes, we wish to put forward the relative depictions of the n-dimensional world viewed intrinsically as standing alone n-hyperspace and viewed extrinsically as an n-hypersurface embedded in (n+1) or higher dimensional hyperspace (Figure-1).

An n-dimensional flat hypersurface can be entirely embedded in an (n+1) dimensional hyperspace, but as the hypersurface is curved, it needs much spacious ambient. Now, under what circumstances an m-dimensional variety (hypersurface) can be embedded in the n-dimensional Euclidean manifold (hyperspace)? The answer is that the m-hypersurface can be wholly embedded in hyperspace without restraint on whatever direction it might curve if and only if such a hyperspace has at least n = ½ m (m+1) dimensions. We call the latter as an allowable embedding hyperspace to a particular hypersurface.

It means that there are a total of [1/2m (m+1)-m] normal lines to the "surface" of such a hypersurface. In a dynamic condition where the hypersurface moves relative to the ambient hyperspace this-extra dimensions are identified as extra [temporal] dimensions, not the ones which curled in tiny looped as the string theory hypothesizes.

Einstein had developed the relativity theory both special and general relativity theories based on intrinsic geometry. It is the weakness of the relativity theory which ignores the surroundings representing more than 99% of the whole reality we are longing to recognize. No wonder the Big Bang theory, the derivation of the relativity theory, can only take into account 5% out of the total matter and energy affecting the known universe.

Intrinsically, we consider the independent variables of the metric tensor of Einstein four-dimensional spacetime as just mathematical variables having no physical significance. However, when we see the world extrinsically, those variables of the metric tensor are nothing but the manifestation of the underlying coordinates – the dimensions of the embedding hyperspace.


We can achieve the unification theory if we describe the world as a curves hypersurface at least in term of parameters of its minimum allowable embedding hyperspace. 

The concept of brane, the representation of the world as a hypersurface is already in the right track except for the concept of its ambient hyperspace. The brane is not like a piece of paper floating around in thin air, but more like an interface of higher dimensional watery like substances – higher dimensional of positive and negative pure energies (Figure-2).