Hyperdimensional Geometry and the Nature of Physical Spacetime

In local radial gravitational force fields there is no viable flow of the force itself, and if divergence of a vector is constructed within the field, then it actually requires time-based radial gravitational potentials rather than the usual distance-based radial potentials. When cast in structural relationships of multispatial hyperspace, the new multispatial representation of the usual, Newtonian radial gravity term suggests thus that the allegedly missing mass-energy of the known universe may be the result of inappropriate accounting methods rather than cosmic mass deficiency, as some used to believe. More detailed analysis of local radial gravitational field reveals that the allegedly universal gravitational constant hides fairly compound functional of locally constant value and therefore the radial force depends on more variables than are explicitly displayed in the Newton's inverse square law of gravitation.

Cantor and generalized contunuum hypotheses are shown to be possibly factually false. Two new hyper-continuum hypotheses are posted to challenge pure mathematics and stir investigations of the abstract multispatial hyperspace, which is important for physical applications. Unlike the Cantorian hypotheses, however, the new ones conjecture existence of certain distinct, mixed kind of hyper-continuum that may serve as the manifold over which dual pairs of linear vector spaces with quite unrestricted abstract operations could be effectively constructed.

Division by zero is proposed in order to open new horizons for abstract mathematics and physics. Although the formerly forbidden operation may be of little practical value, the feasibility of unrestricted operations is needed for meaningful deployment of linear vector spaces, whose duals are inverse transposes of their primary vector spaces. Both quantum mechanics and pan-geometry of multispatial hyperspace operate on vectors that belong to mutually dual linear vector spaces. The division by zero also may remove many years old paradox related to multiplication of zeros and makes it possible to introduce the concept of bigroup as an additive and multiplicative group at the same time. This operational extension to algebraic groups makes thus an infinite-dimensional, fractal Cantorian spacetime the preferred, invertible, operationally unrestricted, abstract mathematical infrastructure for physics.

Newton’s hesitation to publish his Principia was presumably caused by his inability to explain why volumes were irrelevant to radial action of gravity, which appeared somewhat incomplete without possible nonradial effects. His doubts about completeness of his gravitational theory are justified by results from several 20th century experiments, which suggested presence of a nonradial impact of gravity in addition to the radial attractive force that he has introduced. Existence of nonradial effects is implied by mathematical and physical preconditions, if total energy is to be conserved, and it has already been confirmed by experiments. Nonradial potentials do not affect directly the radial Newtonian potentials, which generate the radial force of gravitational attraction between bodies, but they decrease energy of the rays that run across gravity fields. The nonardial potentials also contain repulsive effect of gravity that becomes dominant at very large distances where the usual attractive radial force practically vanishes. They provide thus global mechanism for an accelerated expansion of the universe and so they explain the lack of superclusters in it. They also imply gravitational contraction, yet another new long-range effect that may explain formation of galaxies.

Elie Cartan has proved that highest dimensionality of any simple geometric space is three and that an exterior differentiation of a 3D+ geometric object gives bivector, which may correspond to some two 2D surfaces as if the 3D+ geometric object comprised two 3D objects. Since one cannot increase the dimensionality of a 3D space even though more than four independently varying physical magnitudes do exist, then an expansion of dimensionality requires a multispatial hyperspace that contains many simple geometric 3D spaces. Presence of such a hyperspace prompts for an entirely new concept of vectors with an isometric operation of vector multiplication of traditional vectors (3-tuples). This new operation on 3-vectors implies presence of a 3D mass-based linear vector space and consequently thus a 9D geometric hyperspace for classical mechanics alone. Also an outline of entirely new, synthetic approach to physics and mathematics is introduced. This synthetic approach can be used to design a computer-aided knowledge extracting system, which could generate entirely new scientific knowledge.

This paper deals with a particular type of quadro-cubic Cremona transformations and its associated fundamental pencil-space-time. The transformation is generated by a homaloidal web of quadrics which share, apart from a real line, three isolated real points of which one falls on the line in question. It is shown that the corresponding pencil-space-time exhibits the same dimensionality (4) and signature (3+1) as that characterizing a generic case, yet its spatial dimensions are not all equivalent; one of them is found to stand on a slightly different footing than the other two. A brief account of possible Riemannian, Cantorian space $\text{E}{}^{\mathrm{(\infty )}}$and heterotic string space-time parallels of this intriguing spatial anisotropy is given.