Why the Grand Relativity Theory? (Part III)

One of the significant physicists' misconceptions about nature is the uniqueness of time. When physicists encounter higher-multidimensional surroundings in their theory, they instinctively assign the extra-dimensions (beyond the ordinary four) as spatial. This premise, regarding the inequality of space and time footing, is evidence against the relativity principle.

The string theory, which had its origins in experimentally observed features of the strong force, requires the existence of six compactified extra spatial dimensions (a) and the four known spacetime dimensions. This presumption leads the theory to grave difficulties as it should deal with myriad different kinds of Callabi-Yau tiny manifolds or other similar bizarre things.

On the other hand, the grand relativity theory holds that the extra dimensions are temporal; thus, circumventing such complexities. Therefore, we may regard a system such as our world consisting of a 3D-space (hypersurface) embedded in 10D-manifold in which all extra dimensions are temporal ^b). Under the grand relativity theory, we have every right to transform it into, for instance, 9D-hypersurface embedding in the same 10D-manifold, by turning some of the temporal dimensions into spatial (Figure-1). The latter is exceedingly simpler than the former in terms of mathematical formulations, and yet it gives us the same solutions.

Physicists are used to oversimplifying their physical model using geometrical objects, such as 0D-point, 1D-line, and 2D-plane. However, one should be extremely cautious of using such objects, having no thickness at all as he or she probes more in-depth into the quantum realm.  At the quantum level, the range of actions could be concise, which might be approaching the object [quantum] thickness (at the range of 10^-33 to 10^-17 cm) ^c) that one deliberately ignores. It is no wonder that with such zero depth for their physical objects or models, physicists are confronting many irritating infinity problems.

The grand relativity theory requires any physical objects or hypersurfaces having thicknesses, the number of which depends on the number of dimensions of the embedding manifold ^d) (Figure-2). However, physicists obliged to do a similar way by incorporating such forgotten thickness into their model called supersymmetry generators e).

In Brane theory, physicists assign some space and time's dimensions on and along its surface while off of it spatial. They certainly make a significant confusion as to the brane, like hypersurface embedding in a higher dimensional spacetime, should have solely spatial dimensions on and along its surface and temporal dimension[s] off of it.

The hypersurface or brane is the loci of things that co-occur if not at lower temporal dimensions it would certainly so at higher temporal dimension (Figure-3). 

The hypersurface tends to flatten out as it has higher dimensions. But how high should they be? Mathematically, a spacetime may embed an n-dimensional hypersurface properly only if the former has at least ½ n(n+1) dimensions. Our 4D-world, for example, requires a sufficient ample ambient space i.e., 10D-manifold, for having a complete degree of freedoms without being constrained at whatever directions.

It turns out that some theories, such as Supergravity, require an embedding spacetime of even higher dimensions. The Supergravity demands an eleven-dimensional ^f) embedding spacetime, but still, nobody can fully renormalize the Supergravity.

Notes:

a.  Physicists are assumed to be curled up into tiny loops as nobody ever directly experiences them.

b.   It can be expressed mathematically as Octonion, a Hypercomplex consisting of one real and seven imaginary variables. The real part is the spatial variables' function, while the seven imaginary parts represent seven different temporal dimensions.

c.  No real particle smaller than the hypersurface's thickness, except virtual particles perpetually emerge and submerge across the depth. This thickness size, which becomes the minimum size of the real particle is what physicists call hierarchy problem.

d.  Analogically under 3D-ambient space, a point has three thicknesses, string two thicknesses (cross-section), and plane one thickness. Otherwise, they would be evaporating into thin air.

e.  Mathematically physicists may express such a framework in terms of Superalgebra equation whose ordinary and super parts are sometimes called body and soul, respectively ¹. It is equivalent to Octonion Hypercomplex with its real and imaginary parts. Amazingly, this is a proper way to describe a subtle structure such as [higher dimensional] soul embedding a body, contrary to what most people think the soul is inside the body.

f.  It is a sort of pseudo dimension representing the tip of the 11+ "iceberg" dimensions. The ancients (among other Empedocles: 490-430 BC) described the realms consisting of earth, water, air, and fire analogous to a changing phase from ice, water, vapor, and steam.  We may interpret that earth representing energy/material stable things in 3D-space, water representing energy in 4D - 10D-spacetimes, the air in 11D - 55D-spacetimes and fire in 56D - 1540D-spacetimes, the dimension ranges of which are speculated under the ½ n(n+1) rule, if we may do so. This metaphor shows us that the energy associated with particular spacetime is hotter as the spacetime dimensions become higher. The reality beyond those dimensions is far from our wildest imagination and concern.