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A Knot Theory of Physics, Spacetime in Co-dimension 2

A Knot Theory of Physics, Spacetime in Co-dimension 2 - A...

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A knot theory of physics, spacetime in co-dimension 2 C. Ellgen (Dated: January 23, 2009) Abstract Spacetime is assumed to be a 4-dimensional manifold embedded in a 6-dimensional space. The spacetime manifold can be knotted and those knots correspond to particles. Distortion of the space- time manifold around knots generates fields. Gravity, electrodynamics, and quantum properties result from the assumptions. * Electronic address: [email protected] ; www.knotphysics.net 1
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I. INTRODUCTION A. Purpose The purpose of this paper is to describe physics in terms of a single mathematical ob- ject, the spacetime manifold. The results of quantum field theory have already provided outstanding mathematical accuracy in predicting physical phenomena. This paper does not attempt to argue with the technique or formulation of quantum field theory. The purpose instead is to answer questions such as “what is a field?”, “what are mass and charge?”, and even “what is a particle?”. With these answers we will derive some principal results of physics. B. Motivation What line of reasoning leads to this theory? General relativity presumes that gravity is curvature of the spacetime manifold. Because the gravitational field is generated by mass, the next step is to describe mass in a way that leads to spacetime curvature. Next one would describe particles in a way that gives them mass. Given that description of particles, do any other fields arise without further modifications to the theory? Finally, how do the results of quantum mechanics follow? The first step is to describe mass. If mass generates spacetime curvature then perhaps mass is spacetime curvature. The equations of general relativity dictate that spacetime curvature propagates dispersively in space, in contrast to the nature of particles. To prevent that dispersion, the spacetime curvature of a particle would necessarily be restricted by topological constraints. Knots suit this constraint. Knots can only exist on an n-dimensional manifold embedded in an n+2-dimensional space. If the space is n+3-dimensional or higher, the knots spontaneously untie. If the space is n+1-dimensional or lower, the manifold cannot form a knot. Therefore a 4-dimensional spacetime manifold with knots is necessarily embedded in a 6-dimensional space. The rest of the paper will develop the notion of particles as knots, showing how that assumption gives rise to the known particles, fields, quantum mechanical results, and other aspects of physics. 2
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II. PARTICLES ARE KNOTS IN SPACETIME The spacetime manifold is a 4-dimensional manifold in a Minkowski 6-space and we require that the spacetime manifold cannot self-intersect. These two conditions allow the spacetime to be knotted. The most natural method of producing knots in the manifold is cobordism. However, if the manifold is everywhere Lorentzian then any cobordism is also a diffeomorphism, which implies that no topology change is possible. While we have assumed
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