The Topology of Space-Time

This observation is no proof of the topology of space-time but it shows how matter and antimatter can coexist, without mutual annihilation, in projective space-time. 

The topology of space-time is not as arbitrary as often made out. The theory of relativity unequivocally specifies a manifold of more than three equivalent coordinate axes with positive curvature. Infinite universes with negative or zero curvature are therefore excluded immediately. This limitation was recognized early on by Chfford (1955)  

Like ignoring the irregular sinface features on a planet, symmetry suggests that space-time must be of constant positive curvature in the large. Only 

Another way of looking at four-dimensional projective space is by adding a point at infinity on each coordinate axis of Minkowski space. Any displacement implies a change in all coordinates. The only difference between stationary antipodal points is an inversion of local chirality, which includes the direction of time flow. There is no possibility of communication between such points which move apart on their respective world lines. 

This is the topology visualized for hypothetical matter-free space-time. The proposition is neither in conflict with general relativity nor with Einstein s strict interpretation of Mach s principle. Superimposed on the uniform curvature of the real projective plane is the local gravitational field that 

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Anticipating the final conclusion that matter and energy are special configurations of space-time, the investigation starts with the topic of relativity, the only theory that has a direct bearing on the topology of space-time and which demonstrates the equivalence of energy and matter and a reciprocal relationship between matter and the curvature of space. 

One number theoretic concept with an obvious link to geometry and to the topology of space-time is the golden ratio, obtained as a root of the quadraric equation... 

In quantum mechanics almost everything is uncertain. So why should the topology of space-time be fixed "... 

On weighing the impartial evidence it becomes obvious that standard cosmology underestimates the age of the world by the same margin that it overestimates the size of the universe. Both factors relate to the Doppler interpretation of redshifts, which should be re-examined, and to geometry and the topology of space-time. The obvious alternative to an infinite Euclidean universe is a world closed in space-time. As originally pointed out by Weyl (1922)(p.278) ... 

Before one can seriously consider an alternative plasma cosmology the ideas of Alfven and others need to be integrated with a sensible alternative to universal expansion and the topology of space-time. Instead of chasing after non-baryonic dark matter the role of hydrogen in that regard should be explored and the interaction between matter and antimatter, an important argument in the current theories (Lerner, 1991), must be rationalized. 

General relativity goes no further than this. The extent of curvature or the topology of space-time depends on additional observations and criteria. Assumptions to simplify the mathematics, but not supported by testable data, must inevitably produce spurios results. Examples of such simplifications include the assumption of universal time or Euclidean space-time. Einstein s thought experiment which established the non-Euclidean nature of space-time also predicts the nature of the curvature as positive, which excludes hyperbolic space. 

The findings reported here provide new evidence for the unity of micro- and macrophysics and refute the perception of separate quantum and classical domains. The known universe exists as a four-dimensional space-time manifold but is observed in local projection as three-dimensional Euclidean tangent space that evolves in universal time. The observable world, at either micro- or macroscale, can be described in either four-dimensional (nonclassical) or in classical three-dimensional detail. The descriptive model may change, but the reality stays the same. This realization is at the root of self-similarity between large and small. The symmetry operator, which reflects the topology of space-time, is the golden logarithmic spiral. 

Covalent interaction in diatomic molecules depends on the golden mean t, the interatomic distance d and the radius ratio x r /r2 of the constituent atoms, as summarized in Figure 5.6. The golden mean is a universal constant that matches the geometry and topology of space-time, the radius ratio is a known function of atomic number and dl relates to the optimal wave-mechanical distribution of valence-electron density in the diatomic system. 

Multiple images occur, not only because of light rays that circle the cosmos, but also as a consequence of multiply-connected topology, that causes a single object to be visible in different directions. If our proposition, of a black hole as a hole in the vacuum interface, holds, each such object increases the connectivity of space-time and further multiplies the possibility of multiple imaging. 

The subsequent discovery (Boeyens, 2003) of the grand periodicity of atomic matter put these speculations into sufficient perspective to allow definite conclusions about the projective topology of space-time and the universe. In the final analysis, all conclusions reached in this work can be reduced to the gauge principle, as summarized in Appendix B. Some readers may like to set the scene by reading this appendix before the main text. 

Time reversal transformation, t - — t This is like space inversion and most likely space-time inversion is a single symmetry that reflects the local euclidean topology of space, observed as the conservation of matter. 

Any astronomically measured frequency shift consists of several components, including the chemical shift, described here. Other contributions include relativistic gravitational redshifting, a distance-dependent redshift caused by the topological curvature of space-time, and a Doppler shift where the source is in relative motion. 

The common imponderable is the topology and dimensionality of space-time, not very likely to be resolved in the foreseeable future. The smart way out is to look for the simplest solution to the problem - a closed universe with an illusionary size, suggested by an infinitude of multiple images of celestial objects. This way many exotic objects such as black holes, supernovae, quasars, radio sources and the like may be located either in the Milky Way or its immediate environs of sibling galaxies. 

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