Space-time curvature

P. K. Anastasovski and D. B. Hamilton, Space-Time Curvature Around Nucleons, U.S. Department of Energy Website http //www.ott.doe.gov/electromagnetic/papersbooks.html. 

Recognition of space-time curvature as the decisive parameter that regulates nuclear stability as a function of the ratio, Z/N, with unity and the golden mean, r, as its upper and lower limits, leads to a consistent model for nucleogenesis, based on the addition of -par tides in an equilibrium chain reaction. This model is also consistent with the limitations imposed by the number spiral. 

From a chemical point of view the most important result is that number theory predicts two alternative periodic classifications of the elements. One of these agrees with experimental observation and the other with a wave-mechanical model of the atom. The subtle differences must be ascribed to a constructionist error that neglects the role of the environment in the wave-mechanical analysis. It is inferred that the wave-mechanical model applies in empty space Z/N = 0.58), compared to the result, observed in curved non-empty space, (Z/N = t). The fundamental difference between the two situations reduces to a difference in space-time curvature. 

Extrapolation of the hem lines to Z/N = 1 defines another recognizable periodic classification of the elements, inverse to the observed arrangement at Z/N = t. The inversion is interpreted in the sense that the wave-mechanical ground-state electronic configuration of the atoms, with sublevels / < d < p < s, is the opposite of the familiar s < p < d < f. This type of inversion is known to be effected under conditions of extremely high pressure [52]. It is inferred that such pressures occur in regions of high space-time curvature, such as the interior of massive stellar objects, a plausible site for nuclear synthesis. 

Scharzschild uses spherical coordinates t, r, d,

[Pg.327]

The gravitational field that exists in the nebula causes a characteristic shift for each metal, which cannot be explained by assuming a constant red-shift for all metals. No other satisfactory explanation of the observed effect has been reported. Where others look at a time-dependent fine-structure constant or variable c or h to account for the modified spectra, we ascribe the observation to a simple response to space-time curvature. 

Relativity theory has equally dramatic implications on the nature of the vacuum, which is shown not to be a void, but a medium that supports wave motion and carries electromagnetic fields. A new perspective on the nature of the vacuum is provided by the principle of equivalence. Space-time curvature can be described mathematically by a Riemann tensor, which the principle implies, should balance the gravitational field, which is sourced in the distribution of matter. This reciprocity indicates that Euclidean space-time is free of matter, which only emerges when curvature sets in. This is interpreted to mean that the homogeneous wave field of Euclidean vacuum generates matter when curved. Like a flat sheet that develops wrinkles when wrapped arormd a curved surface, the wave field generates non-dispersive persistent wave packets in the curved vacuum. 

The shift of electronic energy levels under pressure predicts a redshift in all spectra eminating from regions of high space-time curvature. This includes all galaxies and quasars and their immediate environments. Consistent interpretation of observed redshifts on these grounds would drastically modify the picture currently based on cosmological Doppler shifts. 

On the topic of Chemical Cosmology not many concepts are as relevant as gauge invariance - the most direct manifestation of space-time curvature. In an attempt to unify the electromagnetic and gravitational helds the idea of gauge transformation was hrst proposed by Herman Weyl (1920) as a space-time dependent change of scale, S dx, on displacement from point... 

The periodic table of the elements is a subset of a more general periodic function that relates all natural nuclides in terms of integer numbers of protons and neutrons, the subject of elementary number theory. The entire structure is reproduced in terms of Farey sequences and Ford circles. The periodicity arises from closure of the function that relates nuclear stability to isotopic composition and nucleon number. It is closed in two dimensions with involution that relates matter to antimatter and explains nuclear stability and electronic configuration in terms of space-time curvature. The variability of electronic structure predicts a non-Doppler redshift in galactic and quasar light, not taken into account in standard cosmology. 

Detailed solution of the field equations of projective relativity is not known, but has been shown to give a unfied description of gravity, electromagnetism and wave mechanics, in which the golden ratio occurs as a descriptor of space-time curvature. 

The involution that occurs in projective geometry defines conjugate regions with time inversion and conjugate forms of matter. The function that describes the observed periodicity of atomic matter is of the same projective form and varies with local space-time curvature. This variation shows that spectroscopic analysis of light waves, stretched between sites of different curvature, must be frequency shifted, as observed. 

Other radii that could be considered are those related to the space-time curvature in general relativity theory. If the electron is viewed as a micro-universe with a rest mass mo uniformly distributed within a 3D sphere of radius rc, then the space-time inside the electron would be endowed with a Gaussian 2D curvature increasing with the mass-energy density po, according to the formula [18,19]... 

Another point of view is to consider the space-time curvature induced by the rest mass mo of the electron outside a volume of radius rq. According to general relativity theory, the curvature radius Rg around the electron would be given by [18,19] ... 

Fig. 8 Variability of the periodic table of the elements depends on space-time curvature as shown in the frame on the left. The triangular segment defines the field of stability. The symmetriceil version on the right is conveniently mapped to the surface of a Mdbius band as in Fig. 9, but resolution is only possible in 4D projective space...

A plot of V vs —1/ 2 should be linear, with equal slope and intercept. The most reliable and accurate data available fail this test statistically [24], The discrepancy is not large but significant and reminiscent of the spectroscopic red shift measured in galactic light. The common origin of these discrepancies cannot be a Doppler effect and most likely is due to space-time curvature. 

Here is the first inkling about the universal importance of the golden ratio. If it measures the relationship between underlying space and tangent space, it is not surprising for it to show up in the apparent structure of so many objects from atoms to galaxies. It could even be interpreted as a measure of space-time curvature. 

The demonstration [1] that both Lorentz transformation and quantum spin are the direct result of quaternion rotation implies that aU relativistic and quantum structures must have the same symmetry. This is the basis of cosmic self-similarity. The observation that the golden mean features in many known self-similarities confirms that r represents a fundamental characteristic of space-time curvature. The existence of antimatter and the implied CPT symmetry of space-time favors... 

It is also known as self-similarity, a concept which is intimately related to the golden ratio, and known to operate on a cosmic scale. Our observations may therefore be rationalized by considering elementary matter as the product of large-scale space-time curvature, as described by the golden ratio. We reach the provocative conclusion that a construct, which is entirely governed by the properties of the golden ratio and number theory, predicts the electronic configuration of all atoms, without reference to any chemical know-how, as a basis of a chemical theory. 

The geometrical definition of both k and r depends on space-time curvature, and since both e and T represent converging number sequences, the identity e " = — 1 cannot hold universally, unless the number system also adapts. 

The irrational number, known as the golden ratio, is said to be the most irrational of them all. Like other irrationals, it also occurs as the limit of a regular series of rational fractions, in this case the Fibonacci fractions. In nature, it occurs as the convergence limit of the mass fractions of stable nuclides, Z/(y4 — Z). As a clue to its physical meaning, it is noted that the stability of nuclides depends on their space-time environment [4]. In regions where space-time curvature approaches infinity, the mass ratio Z/ A — Z) 1. In the hypothetical situation of zero curvature, matter does not exist. It is inferred that in an intermediate situation of curvature, conducive to the development of biological life, the mass ratio Z/(A — Z) z. 

It is no accident that both wave motion and the fundamental theory of chemistry are best described in terms of natural numbers. However, conventional wave mechanics in three dimensions offers only a partial elucidation of the periodic table of the elements. On the other hand, a detailed reconstruction, also of the more general periodicity of stable nuclides, derives directly from elementary number theory. It shows, in addition, how the periodic function responds to the state of space-time curvature and identifies the golden ratio as a possible parameter that links perceptions in tangent space to the simation in curved space-time. 

---