1.3.3 Photon Transit Time
At each point in the trajectory of a photon from a distant star to a telescope on planet Earth, a photon moves at an increasing velocity through the continuum which itself is expanding. During its transit, the velocity of the photon within the continuum will have the same expansion ratio as the expansion velocity of the continuum, that is, \(c(t) = HR(t) \). This means that, at each point, photon trajectory is always at an angle of 45 degrees to both the continuum and the radius vector.
The component of photon velocity within the curved continuum is, effectively, an angular velocity, and since photon velocity is, \(c(t) = HR(t)\), the angular velocity, \(c/R= H\), the Hubble's constant, with units of radians per second.
The total photon velocity is angular velocity times radius, \(H R(t)\), and since the angular separation between Earth and the distant star is, \(\theta\), then the transit time of a photon passing between them is \(\Delta t = \theta /H\), so that \(\theta = H \Delta t = \theta\). Hence, expansion ratio can be written, \(e^{H\Delta t} = e^{\theta}\), and we have that \begin{align} expansion\ ratio = \frac{R_o}{R_e} = \frac{r_o}{r_e} =\frac{c_o}{c_e} = e^{H \Delta t} = e^{\theta} \nonumber \end{align}
A consequence of the photon's constant angular velocity is that the transit time of a photon passing between two galaxies, \(\Delta t = \theta /H\), depends only on their angle of separation.
This applies to photons which arrived in the past and those which will arrive in the future. This seeming paradox is resolved when we consider that even though photons which arrived in the past traveled a shorter distance, they also traveled at a slower average speed. Likewise, photons which will arrive in the future must travel a longer distance, but will do so at a higher speed.
This result resolves the confusion surrounding the idea that there is a 'horizon' beyond which photons from distant stars can never reach the Solar system. This is supposed to occur when recession velocity is equal to the speed of light. However, we have shown that photons from galaxies separated from the solar system by any angle will always reach us, given enough time. There is, however, a maximum value of distance between any two points on the line-of-sight circle, and this is when their angle of separation is \(\pi\) radians or 180 degrees. This corresponds to a distance of \(\pi R_o\) which, in the current era, is about \(4\ 10^{26}\) meters, or 13,300 Mpc. The transit time for a photon to traverse this distance is \(1.37\ 10{18}\) seconds or about 43 billion years.
Photons arriving at earth from stars exactly \(\pi\) radians away, will arrive at earth at exactly the same time and from exactly opposite directions. Since these two trajectories are exactly opposite, they may be represented on the same plane section. Such trajectories will have the same transit time, so that they provide two views of the star at the same point in time.
Photons from stars with less than \(\pi\) radians separation will also have two possible trajectories to earth, one short and the other long going in the opposite direction from the star. They will have different transit times and photons emitted at the same time will not arrive in the Solar system at the same time. The photons which do arrive at the same time will have been emitted at different times in the past and therefore they will show the star at two different points in time. It is not impossible that photons may traverse the universe more than once, however, such photons arriving at earth may have such long wavelengths that they are probably undetectable. There is a maximum wavelength at which a photon can exist as a particle, and this is discussed in a later chapter on the structure of photons.
The constancy of photon transit time between two galaxies with a constant separation angle means that it can be used as a proxy for distance. For example, the photon transit time from the Andromeda galaxy, at a distance of 780 kpc and an angle \(1.9\ 10^4\) radians, is \(83\ 10^{12}\) seconds, or about 2.6 million years. This will not change in the future even though the photons must travel further, they will do so at a greater speed.
Constant transit time between any two points in the universe provides a second method of recording distance in addition to angle of separation.
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