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1.3.2 Photon Trajectories and Expansion Ratio

Visualizing the trajectory of photons passing through the continuum is complicated by the fact that the velocity of photons (speed of light) is increasing as the universe expands. This phenomenon is explained by the diagram, which is a plane section through the hyper-sphere at two different times during its expansion. This plane section contains three points; the point from which photons are emitted from a distant star at an earlier time, the point at which they are detected on planet Earth at a later time, plus the center point of the hypersphere. If the distant star is at a separation angle, \(\theta\), from Earth, and photons are not deflected by the gravity of stars along their path, then their entire trajectory will be in this plane.

In the diagram, the inner circle represents the universe when the photons were emitted, and the outer one represents the universe at the later time when the photons are detected. Photons traveling between these two locations, must do so within the continuum which is expanding according to an exponential law. If the continuum through which the photon travels is free of matter, (which is largely the case) then it can be considered to have the properties of the ideal universe, and expansion of its radius, \(R\), is governed by the relationship \(R = R_o\ e{Ht}\), derived in an earlier section.

Photons were emitted from this distant star when it was at point \(S_e\), the Earth was at point \(E_e\), and the radius of the universe was \( R_e\). These photons are detected, \(\Delta t\) seconds later, when the distant star is at point \(S_o\), the Earth is at point \(E_o\), and the radius of the universe is \(R_o\). Because light travels at an increasing velocity, photons emitted by the star travel through the continuum at speeds increasing from \(c_e\) at emission to \(c_o\) at detection.

The following relationships can be established:

\(R_o = R_e\ e^{H\Delta t}\)
\(r_o = r_e\ e^{H\Delta t}\)
\(c_o = c_e\ e^{H\Delta t}\)

The common factor, \(e^{H\Delta t}\), is the expansion ratio which represents the increase in the radius of the hyper-sphere during the time period, \(\Delta t\). It also governs the increase in arc distance and expansion velocity, so that we have \begin{align} \frac{R_o}{R_e} = \frac{r_o}{r_e} =\frac{c_o}{c_e} = e^{H \Delta t} = expansion\ ratio \nonumber \end{align}

The observer on earth detects photons from the star and assumes they have traveled from the star in a straight line. In reality they have traveled along a curved trajectory through the curved volume of the energy continuum. The diagram shows only a single photon trajectory from the distant star. In reality, there are an infinite number of such trajectories radiating in every direction. On Earth, the Astronomer receives photon trajectories from other stars, from every direction in which he may point his telescope.

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