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Chapter 2: Curvature and Gravity

2.1 Shape of the Energy Continuum

2.1.1 Introduction

In the previous chapter it was proposed that our universe is a three dimensional continuum of energy shaped like a hyper-sphere. In its ideal form, this energy continuum has a perfect hyper-spherical shape and hence its curvature, \(\kappa(t)\) has the same value at each point, and the energy density, \(\rho(t)\), is uniform throughout. A suitable definition of curvature will be given in the following sections.

The volume expansion of this ideal universe is exponential such that \(Vol(t) =Vol(o)\ e^{+3Ht}\), where \(Vol(o)\) is the volume of the continuum at a time chosen to be time = 0, and H is the Hubble constant. If the total energy, \(E_u\), of the continuum is constant (conserved) over time, then its energy density must decrease exponentially such that energy density \(\rho(t) = \rho(o)\ e^{-3Ht}\), where \(\rho(o)\) is the value of energy density at a time chosen to be zero. The total energy in the continuum is then given by, \(E_u=Vol(o) \times \rho(o)\), which is assumed to be constant, so that there is conservation of energy.

Since it is impossible to visualize a hyper-sphere, the three dimensional, curved volume of the continuum is analyzed by extrapolating the mathematics of lower dimensional analogues - circles and spheres. By these means, curvature may be visualized in a simplified form to provide insight into the large scale, 'straight' line distances and photon trajectories between galaxies in the curved volume of the energy continuum.

The word 'shape' is usually used to describe the curved surface of a three dimensional object. In this work, 'shape' is also used to describe curved volume. In this sense it may be described in terms of gradient and curvature at each point within the volume. These properties cannot be seen, or visualized, however they are assumed to exist by an extrapolation of geometry from one and two dimensions to three.

If the shape of the universe is thought of as being separate from its size, its clear that the shape of the ideal continuum is constant, so that by converting all distances and lengths to angles, these angles will remain constant over time. Hence, the 'shape' of the energy hyper-sphere will not change as its size increases. Since the rate of length expansion, \(e^{Ht}\), is very small, being about one part in a million every 12,000 years, it is small enough to be ignored in most calculations concerning the current era.

Expansion velocity may be regarded as the velocity of each point in the ideal continuum, relative to the center of the hyper-sphere and perpendicular to all three axes of the three dimensional 'hyper-surface'. This ensures that, in the ideal continuum, adjacent points move apart at the same rate - analogous to points on the surface of an inflating balloon. Expansion velocity turns out to be the speed of light, \(c\), which itself is subject to increase, as an expansion acceleration, which is shown to be equal to \(Hc\), where \(H\) is the Hubble constant.

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