Geometry of space

The Ricci tensor that represents the geometry of space is next equated with the so-called energy-momentum (stress) tensor of the matter field that defines the influence of matter and field energy... 

It is quite conceivable that the geometry of space in the very small does not satisfy the axioms of [Euclidean] geometry. .. The properties which distinguish space from other conceivable triply-extended magnitudes are only to be deduced from experience. 

The conscious final decision to take the risk, with the current sequence, should be read as a personal conviction that the beauty of chemistry can never be fully appreciated unless viewed against the background in which all matter originates - space-time, or the vacuum. Not only matter, but all modes of interaction are shaped by the geometry of space, which at the moment remains a matter of conjecture. However, the theory of general relativity points the way by firmly demonstrating that the known material world can only exist in curved space-time. The theory of special relativity affirms that space-time has a minimum of four dimensions. Again, spaces of more dimensions are conjectural at present. 

All solutions of Einstein s equations are conditioned by the need of some ad hoc assumption about the geometry of space-time. The only indisputably valid assumption is that space-time is of absolute non-euclidean geometry. It is interesting to note that chiral space-time, probably demanded by the existence of antimatter and other chiral forms of matter, rules out the possibility of affine geometry, the standard assumption of modern TGR [7]. 

The observation that bonds of all orders relate to the bonding diagram in equivalent fashion indicates that covalent bonds are conditioned by the geometry of space, rather than the geometry of electron fields, dictated by atomic orbitals or other density functions. The only special point related to electron density occurs at the junction of the attractive curves, where e = indicating that one pair of electrons mediate the covalent interaction. It is interpreted as the limiting length (dj) for first-order bonds. It is of interest to note that all known first-order bonds have d > d[. The covalence curve for the minimum ratio of x = 0.18 (for CsH) terminates at dl = d[. 

Bonds with r < dl < d[ become possible because of nuclear screening (increased bond order), which causes concentration of the bonding pair directly between the nuclei. The exclusion limit is reached at d = t and appears as a geometrical property of space. The distribution of molecular electron density is dictated by the local geometry of space-time. Model functions, such as VSEPR or minimum orbital angular momentum [65], that correctly describe this distribution, do so without dictating the result. The template is provided by the curvature of space-time which appears to be related to the three fundamental constants tt, t and e. 

Interatomic distances are determined by steric factors, of which the most important is the exclusion principle that depends directly on the geometry of space-time, observed as the golden ratio. Bond order depends on the ratio between the number of valence electrons and the number of first neighbours, or ligands, and affects interatomic distances by the screening of internuclear repulsion. 

Note that interpretations of the time-reversal experiments are only valid in strictly euclidean space-time. This condition is rarely emphasized by authors who state that all laws of physics are time-reversible, except for the law of entropy. Fact is that entropy is the only macroscopic state function which is routinely observed to be irreversible. One common explanation is to hint that entropy is an emergent property of macro systems and hence undefined for microsystems. Even so, the mystery of the microscopic origin of entropy remains. A plausible explanation may be provided if the assumed euclidean geometry of space-time is recognized as an approximate symmetry as demanded by general relativity. 

The broken symmetries of chapter 1, assumed responsible for shaping the physical world, refer to the symmetry of the vacuum and thus finally to the geometry of space-time. It is not immediately obvious that chemical theories could also be reduced to the same cause. While physics produced quantum theory and general relativity, the fundamental contribution from chemistry was the periodic table of the elements. Although the structure of individual atomic nuclei may be considered shaped by local space-time symmetry, the functional relationship between different nuclides needs further study. 

Chemical definition of the vacuum is a region of three-dimensional space devoid of matter. Chemical matter is the source of gravitational and electromagnetic fields and the removal of matter from some region of space does not prevent these fields from permeating the vacuum so created. There is only one way to obtain a field-free vacuum and that is by complete annihilation of chemical matter. It is not obvious what effect such annihilation would have on the vacuum. A partial answer to this question is provided by the theory of general relativity that outlines a reciprocal relationship between matter and the geometry of space-time. The implication is clear the physi-... 

In the course of this work it has been necessary to link the fundamental properties of numbers, photons, electrons, molecules and matter to the geometry of space-time, albeit without further insight. An effort will now be made to specify geometrical details consistent with experimental observation. 

We obtain the well-known 1/r dependence dne to the geometry of space as described by Eqnation 14. A3. 

All the observed phenomenas of the special relativity are necessarily placed in the scope of the pure geometry of space-time, and all the mathematical objects whose aim is to interprete these phenomenas are placed by the real formalism directly in this scope. 

Abstract. This chapter is devoted to the form of transition currents between two states. One can remark that, independent of the choice, real or complex, of the initial formalism, all that follows is placed in the real geometry of space-time. 

The next vital assumption, based on experimental observation, is the equivalence of inertial and gravitational mass. This means that a gravitational field must distort the Euclidean geometry of space and that the matter content of space must be in balance with the non-Euclidean geometry of space, known as its curvature. The field equations of general relativity summarize these assumptions mathematically. 

Theories like those of Lemaitre or Friedmann, which predict an expanding universe, are all based on forcing an affine metric, such as the Robertson-Walker metric, on the projective geometry of space-time. This has the effect of splitting local Minkowski space into separate space and time coordinates, without the natural complex relationship that ties space and time together. 

The foregoing is interpreted to mean that the projective model of space is closed by a single surface that corresponds to the ideal plane at infinity. In Euclidean geometry this plane appears curved. If we therefore assume that the structure of the cosmos is subject to mathematical analysis and that the mathematics applies without exception, it is a logical necessity that the geometry of space-time be projective. 

The demonstration in the theory of general relativity that gravitation reflects the geometry of space-time raised the reasonable expectation that all of physics could be reduced to a common geometrical principle. 

The principle is summarized by the statement that the geometry of space is determined by the matter distribution, or alternatively, that the inertia of one body is due to the presence of all other bodies in the universe. The principle is recognized in Einstein s equation by the relationship between which characterizes the matter-energy distribution and that specifies the geometry of space. 

Once this spatial decomposition has been stated, modeling the role of space consists of defining local variables that correspond to global state variables. Each local variable results from application of a spatial operator that depends on the geometry of space. The simplest is the Euclidean space, that is, ordinary space independent from time, also called flat space by opposition with curved space-time, which is the frame of the general relativity. 

To understand the motion of scroll wave filaments, we first review the differential geometry of space curves [22, 23]. We suppose that a space curve, in this case the filament of a scroll wave, is given by a position vector R(s) = (x(s),y(s), z(s)), 0 < s < L, where the independent variable s is taken to be arc length. To each point s on the curve R(s) we attach an orthogonal coordinate system defined by the unit vectors T(s), N(s), B(s), where... 

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