In this thread, we discussed some considerations about the time of travel for light in an expanding universe. In it, I derived some calculations for a constant rate of expansion and a constant speed of light. It is calculated for two otherwise stationary galaxies, or points in space, in respect to each other, with a pulse of light travelling between them at a constant speed, but while all distance scales between them are expanding in the meantime, so that the galaxies separate over time. This thread is a rewrite of all relevant calculations I have made so far. In the calculations, values that vary will be followed by an 'e' or 'r' to denote whether it is the value at the time of emission of the pulse or when received. Values that are not followed by one of these two notations are either the present day values or are considered a constant. The value for 'd', however, which is the current distance from an observer to an emitting galaxy, is not the observed distance, which probably lies somewhere between de and dr, and is yet to be determined. But as we shall see, the ratio of de to dr is extremely close to 1 for the galaxies we can observe anyway.

We'll start by subtracting the distance travelled by light during a small time interval 'ti', and count them using the variable y, so that t=y*ti. So d'=de-c*ti. Then we'll multiply this by the ratio space has expanded during this time, so that d'=d'*(de+y*v*ti)/(de+(y-1)*v*ti), where v=He*de and remains constant for the two galaxies. Then d' becomes d and we start over. This shows the distance between the pulse and the second galaxy diminishing with time. Here are the calculations that are derived from this.

d'=(de-c*ti)(de+v*ti)/de

=(de+v*ti)[1-c*ti/de]

=(de+v*ti)[1-c*ti*(1/de)]

d''=(d'-c*ti)(de+v*2ti)/(de+v*ti)

=(de-c*ti)(de+v*2ti)/de-c*ti*(de+v*2ti)/(de+v*ti)

=(de+v*2ti)[1-c*ti/de-c*ti/(de+v*ti)]

=(de+v*2ti)[1-c*ti*(1/de+1/(de+v*ti))]

d'''=(d''-c*ti)(de+v*3ti)/(de+v*2ti)

=(de+v*3ti)[1-c*ti*(1/de+1/(de+v*ti))]-c*ti*(de+v*3ti)/(de+v*2ti)

=(de+v*3ti)[1-c*ti*(1/de+1/(de+v*ti)+1/(de+v*2ti))]

So basically, after so many time intervals, it becomes

dp=(de+v*t)[1-c*ti*(1/de+1/(de+v*ti)+1/(de+v*2ti)+...+1/(de+v*t)], where dp is the final distance between the pulse and the receiving galaxy. Figuring for constant expansion, de=v*Te, where Te=1/He, and we'll make Te=x*ti, so de=v*x*ti. Also, t=y*ti. The equation now becomes

dp=(de+v*t)[1-c*ti*(1/(v*x*ti)+1/(v*x*ti+v*ti)+1/(v*x*ti+v*2ti)+...+1/(v*x*ti+v*y*ti)]

=(d+v*t)[1-(c/v)(1/x+1/(x+1)+1/(x+2)+...+1/(x+y))]

The smaller we make ti, the more precise this becomes, and we get very large values for x and y that are in proportion, and then the series 1/x+1/(x+1)+1/(x+2)+...+1/(x+y) converges on e_{in}[(x+y)/x], so the equation simplifies further to

dp=(de+v*t)[1-(c/v)*e_{in}[(x+y)/x]

Since (x+y)/x=(de/(v*ti)+t/ti))/(de/(v*ti))=1+vt/de=1+He*de*t/de=1+He*t, the equation now becomes

dp=(de+v*t)[1-(c/v)*e_{in}(1+He*t)]

Now, since dp is the final distance from the pulse to the receiving galaxy and we want to know the time it takes to get there, then dp=0. So

0=(de+v*t)[1-(c/v)*e_{in}[(1+He*t)]

1=(c/v)*e_{in}(1+He*t)

e^{(v/c)}=(1+He*t)

e^{(He*de/c)}=(1+He*t)

t=(e^{(He*de/c)}-1)/He

t/(de/c)=(e^{(He*de/c)}-1)/(He*de/c)

Since H decreases with time while d increases in proportion for otherwise stationary points in a universe expanding at a constant rate, and since the relative speed between two galaxies experiencing a constant rate of expansion is also constant, then v=He*de=Hr*dr=H*d, so

t/(de/c)=(e^{(H*d/c)}-1)/(H*d/c) using current values.

Also, dr=de+v*t=de+He*de*t=(1+He*t)*de, so the equation then becomes

e^{(He*de/c)}=1+He*t

e^{(Hr*dr/c)}=dr/de

de=dr/[e^{(He*de/c)}]

=dr/[e^{(Hr*dr/c)}]

So if we make dr the current distance to a galaxy, d, and Hr the current value for H, we get

de=d/[e^{(H*d/c)}]

Now, the problem with this is that galaxies that are very far away must have started out extremely close to us for the amount of time it takes for light to get here. But we cannot see light from galaxies that take more time to reach us than the age of the universe, T=1/H. This means that there is a peak distance to how far we can see, or at least for the furthest possible current distance to an observable galaxy (in case the current distance and the distance we now observe them at are different). This implies a measurable size for the universe.

Here are the distances I get using the formula, do=d/[e^{(H*d/c)}]

d, de

-----------------------

10^24, 9.933*10^23

10^25, 9.355*10^24

10^26, 5.134*10^25,

10^27, 1.273*10^24

10^28, .11144 meters

As one can see, the values for the original distances reach a peak and then begin to overlap. This means that no galaxies can currently be further away than the peak value, which appears to lie somewhere around 10^26, or about (c/H) after all, depending on the field of view, and the amount of time that would be required to receive the light from greater current distances (above the peak), then, hasn't transpired because the time required to receive the pulse is greater than the age of the universe.

It also shows that the distance when the pulse was emitted and the current distance aren't too different, coming very close to a ratio of de/dr=1 for smaller distances (<c/H), unless one tries to go with the second set (above the peak), so the distance between two galaxies when a pulse of light was emitted and when it was received wouldn't be all that different, and we should not be seeing the galaxies very far away due to the expansion of space after the light was emitted when they were initially very close to each other, but when they were about the same as their current distance to begin with. And because the distance doesn't change much, neither should the field of view, and we are actually seeing the galaxies almost exactly as they were and at about the same distance as they were when the light was emitted.

Now, if we start with some really large distance and then backtrack to where the galaxies must have been when the pulse was emitted, then we will find that we can indeed place them very close for values "over the peak", where the current distance is extremely large (greater than c/H) and so it would take a very long time for the pulse to reach the other galaxy. The problem is that it also matches that of a very short distance in the first place. In other words, according to the equations, a galaxy that appears to be about 10^28 meters away now would have required a very long time for the pulse to reach us, putting it at an initial distance of just .11144 meters when emitted (see graph). But a pulse beginning at about .11144 meters away will also have travelled about .11144 meters in all when received. We get dual values for above the peak as below, but the difference is that the time required to travel the distance for values that lie above the peak are a time that is greater than the age of the universe. If T=1/H, which is the scenario we are using here, then one cannot yet receive light that is required to travel longer than for that period of time.

Here also are the calculations for the distance travelled by the pulse away from the emitting galaxy during expansion from the perspective of the time involved:

Te=1/He

d'=c*ti

d''=d'*(Te+ti)/Te+c*ti

=c*ti*(Te+ti)/Te+c*ti

=c*ti*(Te+ti)*[1/Te+1/(Te+ti)]

d'''=d''*(Te+2ti)/(Te+ti)+c*ti

=d'*(Te+2ti)/Te+c*ti*(Te+2ti)/(Te+ti)+c*ti

=c*ti*(Te+2ti)/Te+c*ti*(Te+2ti)/(Te+ti)+c*ti

=c*ti*(Te+2ti)*[1/Te+1/(Te+ti)+1/(Te+2ti)]

And so on. After many integrations, it becomes

dt=c*ti*(Te+y*ti)[1/Te+1/(Te+ti)+1/(Te+2ti)+...+1/(Te+y*ti)], where dt is the distance travelled.

Since y*ti=t and if we make x*ti=Te, then

dt=c*ti*(Te+t)[1/(x*ti)+1/(x*ti+ti)+1/(x*ti+2ti)+...+1/(x*ti+y*ti)]

=c*(Te+t)*[1/x+1/(x+1)+1/(x+2)+...+1/(x+y)]

For small ti, x and y are large, so

dt=c*(Te+t)*e_{in}[(x+y)/x]

=c*(Te+t)*e_{in}[(Te+t)/Te]

Now, Te+t=Tr=1/Hr=dr/v=(de+v*t)/v, so

dt=c*(de+v*t)/v*e_{in}[(de+v*t)/(v*Te)]

=c*(de+v*t)/v*e_{in}[(de+v*t)/de]

Since we are solving for dt=dr=de+v*t, then

(de+v*t)=c*(de+v*t)/v*e_{in}[(de+v*t)/de]

which reduces to 1=(c/v)*e_{in}[(de+v*t)/de]

and then becomes e^{(v/c)}=1+v*t/de=1+He*t

using the distance travelled, just like the what we found for the diminishing distance. They both give the same results.

As it turns out, dr/de=e^{(H*d/c)}is the same formula as that for tired light theory, as in my own cosmological model, since the wavelengths of light will also expand in proportion to space while in transit, so wr/we=dr/de=e^{(H*d/c)}. So the two theories are now mathematically the same in that respect. Also, since the redshifts are in proportion to e^{(H*d/c)}-1 instead of directly with H*d/c-1, so that it is not directly proportional to the distance after all, then this would account for an apparent acceleration of the expansion either way, when measured in respect to the ratio of redshift to distance.

We still have some problems with the expansion model, however, according to these calculations. We should not be able to observe galaxies that require more time for their light to be received than the age of the universe, and so we shouldn't see redshifts greater than that of a certain limit for z. Also, the current distance to galaxies is about the same as that when the light was emitted. Furthermore, galaxies that lie on the "edge" of the observable universe are moving with the expansion, and they will then remain on the edge, and we will never see further than this from Earth, regardless of the age of the universe, even though it would seem that as more time passes, we should see further. And so, by running this backwards, we can see that we having been receiving light from up to that same edge and from the same galaxies (or the areas where they formed) all the way back to T=0.

Now, this is all for a constant speed of light. If light speeds up, starting off very slowly, then the observable universe should be even smaller than what we actually observe, since the light from the current limit of observation will not have reached us within a time of T=1/H. If it slows down, beginning with some very high value, then the current distance to galaxies would be even closer to their observed distance for when the light was originally emitted.

So now let's find out if an observed redshift is the same as that for the expansion during the time of travel for a constant speed of light. In other words, if several pulses of light are emitted at regular intervals of time of tp by the emitting galaxy, so that the original wavelength is we=c/fe, where fe=1/tp, then wr/we should equal dr/de, if the redshift is to be determined by the expansion with the calculations in this model for a constant speed for light. We can easily imagine that if the original wavelength is we, then as light travels through space during expansion, the wavelength should expand at the same rate. First though, let's see what we get if the wavelength does not expand during travel, so that the galaxies, then, would be movingthroughspace instead ofwithspace as it expands.

We must say that the relativity of velocities for moving through space is still v, but not caused by an expanding universe, so the distance travelled by a pulse of light from a receding galaxy, from a static point relative to the other which we will consider stationary, is d1=de-c*tp for the first pulse during the time of tp since emission.

During this time also, the emitting galaxy has travelled further to a distance of dr=de+v*tp, where it will emit another pulse at this same distance of d2=de+v*tp. The distance between these two pulses is now the wavelength, or w'=d2-d1=(de+v*tp)-(de-c*tp)=(v+c)*tp. It is the wavelength produced due to the relative velocities between the source and observer. Now we get

w'/we=[(v+c)*tp]/we=[(v+c)*tp/(c*tp)=(v+c)/c=1+v/c

Since v=He*de=Hr*dr=H*d, we get w'/we=1+H*d/c=z+1, so z=H*d/c

This shows a direct relationship of redshift to distance as the expansion model is now presented, but this only works out, as I said, for galaxies moving through space, not with it. For galaxies moving with space, so that the distance between the pulses expands as well during travel, we find

T1=1/He, T2=1/He', and T1+tp=T2, so 1/He+tp=1/He'

Also t=(e^{(He*de/c)}-1)/He

and t'=(e[sup](He'*de'/c)-1)/He'

Since He*de=He'*de'=v=Hr*dr,

and the time between pulses as received is tp'=(t'+tp)-t, then

tp'=(t'-t)+tp

=(e^{(Hr*dr/c)}-1)(1/He'-1/He)+tp

=(e^{(Hr*dr/c)}-1)(T2-T1)+tp

=(e^{(Hr*dr/c)}-1)*tp+tp

=e^{(Hr*dr/c)}*tp

So tp'/tp=e^{(Hr*dr/c)}, and since dr/de=e^{(Hr*dr/c)}also, then tp'/tp=dr/de.

tp'=1/fr and tp=1/fe, therefore tp'/tp=fe/fr=(c/we)/(c/wr)=wr/we=dr/de, so yes, the wavelengths also expand directly with the expansion of space in accordance with the formula, dr/de=e^{(Hr*dr/c)}, for a constant speed of light.

However, just because the calculations for this model for a constant speed of light and a constant rate of expansion are in agreement with the expansion of the wavelengths during the time of propagation for the light, it doesn't necessarily mean that another model might not agree as well on its own terms, within its own parameters and involving different formulas. So next I will attempt an entirely different model based on a speed for light that increases in direct proportion to the age and size of the universe to see where that leads us as well.