1.2.2 Radius Vectors, Angles of Separation and Arc Distance
The concept of the universe as an energy continuum in the shape of an expanding hyper-sphere correlates perfectly with the measurements made by Astronomers. For a hyper-sphere to expand it is only necessary that its notional radius, R, increases. This allows the continuum to increase its volume while retaining its hyper-spherical shape. To understand how the relationships between objects (stars and galaxies) change as the hyper-sphere ex-pands it is instructive to analyze the analog situation on the surface of an expanding sphere. Each point on the surface of an ordinary sphere can be represented by the angle of its radius vector from the radius vector of some convenient reference point. It is clear that this separation angle will remain constant even as they expand and the arc distance between them increases.
The rules for radius vectors in a hyper-sphere are the same as those for the ordinary sphere outlined above. The separation angle between the radius vectors of two stars will remain constant even as their vectors expand along with the whole hyper-sphere. However, the arc distance, \(r\), between the two stars will increase as their radius vectors expand. Hence, if the separation angle between two stars is \(θ\), then the length of the arc separating them, \(r\), is simply \(r = Rθ\), where \(R\) is the length of radius vectors, and \(θ\) is in radians. It's important to note that distances between objects in the hyper-sphere are arc segments because, in curved volume, these are the equivalent of straight lines.
The diagram shows a plane section through the energy hyper-sphere of the universe. It is well known that a ’straight’ line in a hyper-sphere extended indefinitely will join up with its origin. This is analogous to a ‘straight’ line drawn on the surface of an ordinary sphere. In both cases the line is constrained to follow the curvature of the surface or volume in which it exists. Taking a plane section through a hyper-sphere which includes its center point, a ‘straight’ line will appear as a circle.
We can appreciate this if we consider the mathematical expression representing a hyper-sphere. A hypersphere in four dimensions is \(R^2=W^2+X^2+Y^2+Z^2\). Consider the plane \(Y= 0\), and \(Z= 0\) cutting through it. This leaves \(R^2=W^2+X^2\), which represents a circle of radius = \(R\).
It's clear that the length of the arc segment separating a star from Earth, \(Rθ\), is also the variable, \(r\), in Hubble’s Law, and the recession velocity, \(v_r\) is the velocity with which the star moves away from Earth. We have that separation distance \(r=Rθ\). By Hubble’s Law
$$\frac{\partial r}{\partial t} = Hr = \frac{\partial R}{\partial t}θ = HRθ\nonumber$$
so that $$\frac{\partial R}{\partial t} = HR\nonumber$$
Again, this is a differential equation with the exponential solution \(R(t) =R_0e^{Ht}\),where \(R_0\) is the radius of the hyper-sphere at some time designated to be \(t = 0\).
In the diagram, each of the two circles represents a plane section cut through the continuum at different moments in time. On this scale, a star can be represented as a single point on the circle with a radius vector, \(R(t)\). As the continuum expands, the star is carried along with it so that its radius vector increases according to \(R(t) =R_0e^{Ht}\).
For two stars whose radius vectors are separated by the angle,\(θ\), the distance between them is arc length, \(r=θR(t)\), and since the angle \(θ\) remains constant, \(r\) increases with the same exponential law as \(R\).
This is Hubble’s law and it applies to all distances in the continuum since any distance or length may be represented as an arc segment. In the present era, the radius of the hypersphere continuum is about \(1.22 \times 10^{26}\) meters. In terms of parsecs, this is about 4,200Mpc.
As an example consider the near-by galaxy Andromeda now at a distance of \(24.2 \times 10^{18}\) km, which corresponds to an angle \(1.9 \times 10^{−4}\) radians, about 0.01 degrees. Assuming that this angle is fixed, then one billion years ago the distance to Andromeda would have been \(22.5 \times 10^{18}\) km, and in one billion years time, Andromeda will be at a distance of \(26.0 \times 10^{18}\) km.
Given the hyper-spherical shape of the continuum, no galaxy may be separated from the solar system by an angle more than 180 degrees, or \(π\) radians. This corresponds to adistance of \(πR_0\) - about \(4.1 \times 10^{26}\) meters, or 13,200 Mpc.
One of the most distant galaxies yet discovered is GN-z11, in the Ursa Major constellation, which is estimated to be at a distance of 9,300 Mpc. This corresponds to an angleof about 2.2 radians, or 127 degrees. More recently, the Webb telescope has located a galaxy at a distance equivalent to an angle of about 156 degrees.
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