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).

How does E = mc2 work at Planck level

We do not need to go to a faraway land only to get to the frontier of the material world we live. We are indebted to Planck (1900) who enabled us to discover this frontier beyond which nothing exists but energy. Everybody can reach this boundary just right from where he or she stands on, going deep inward into the center of micro-realm, to the Planck level at the distance of 10^-33 cm or period of 10^-44second below which neither space nor time has any meaning. 

Planck himself was not aware of the broader physical meaning of his finding. We can derive those values from his constant, h = 6.6 x 10^-34 joule-seconds, the measure of a unit of action which is a little bit weird and uncommon. So, why then we are facing such an odd unit? What Planck had discovered was not merely a measure of a unit of action which is an abstract thing but the real size of four-dimensional discrete energy (quantum). Energy is like water phase consisting of discrete four-dimensional "molecules," the smallest constituent of the underlying four-dimensional sea of energy.

"Shortly" after Planck momentous discovery, Minkowski (1908) revealed the four-dimensionality of the macro-world. The unification of these theories would lead us to conclude that the four-dimensionality of the spacetime is just the manifestation of the degree of freedom of energy which itself is four-dimensional. We have, then, a very consistent geometrical pattern of the four-dimensional world, both microscopically and macroscopically, from the tiniest thing like four-dimensional quanta to the largest ones such as the four-dimensional spacetime as a whole.

However, still, nobody grasps the critical implication of this quanta's four-dimensionality. The fundamental particles, the smallest building blocks of the universe, which are the derivative of these quanta as dictated by the energy equation E= mc² should also be four-dimensional. As discrete both spatially and temporarily, these little kinds of stuff are short-living.  The persistence of everything we see around us is just the appearance of the underlying perpetual creation and annihilation of fundamental ephemeral things, an action that takes place at the Planck level.

As these ephemeral kinds of stuff made the macro-world, the same thing would happen in the macro-realm. At any instant, our physical space appears and disappears in the same rhythm with what happens in the micro-realm.   How could this happen? The notions of creation and annihilation are just the terms we use to describe the transformations of energy into matter and vice versa.  Down to the Planck level the energy equation, E= mc² is highly reversible in which matter and energy flashing to and fro at the pace equal to the speed of light.

The appearance and disappearance of the whole thing from the space as a whole down to the minuscule particles at the deepest level of reality give us a feeling that something passes by. So this something is what we call time.   The past, present, and future are just a series of continuous process of formation and dissolution just like a single water wave moves across the surface of the water; the peak of the wave is the now, the flat horizon behind it is the past and ahead is the future. It is the true nature of time that people are searching for a long time. Had everything persisted there would be no time at all.

Now what we perceive as the same persistent particle is a series of similar but different particles appear and disappear rapidly at the speed of light. The constant c is nothing but the rate of the constant creation and annihilation of the quanta, the building blocks of the universe. Among this process what we can directly measure is the rate of the continuous creation and annihilation of the photon which is equal to the speed of light. It is an underlying fact why nothing can move faster than the light; otherwise, there would be something running ahead of its very own existence.

How this creation and annihilation physically take place? The relativity energy equation E² = m²c⁴ + p²c² implicitly shows that the energy inherently consists of the opposites: the positive and negative energies. As they are repulsive, they both tend to segregate separating from each other. At Planck level, the interplays of these two (the exchange of 4D-electric currents?) across the interface (thin 3D-surface) create quantum sparks ("quarks"), analogous to the sparks appear and disappear on our TV screen. These quantum sparks are what we perceive as fundamental particles perpetually appear and disappear at a gigantic 3D- TV screen -- our precious physical space.