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Thread: Expansion and the speed of light

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    Expansion and the speed of light

    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 ein[(x+y)/x], so the equation simplifies further to
    dp=(de+v*t)[1-(c/v)*ein[(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)*ein(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)*ein[(1+He*t)]
    1=(c/v)*ein(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)*ein[(x+y)/x]
    =c*(Te+t)*ein[(Te+t)/Te]

    Now, Te+t=Tr=1/Hr=dr/v=(de+v*t)/v, so
    dt=c*(de+v*t)/v*ein[(de+v*t)/(v*Te)]
    =c*(de+v*t)/v*ein[(de+v*t)/de]

    Since we are solving for dt=dr=de+v*t, then
    (de+v*t)=c*(de+v*t)/v*ein[(de+v*t)/de]
    which reduces to 1=(c/v)*ein[(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 moving through space instead of with space 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.
    Last edited by grav; 2006-Dec-09 at 09:30 PM. Reason: A limit for the value of z with a constant speed of light is so far undetermined.

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    Ratiio Meaning

    Quote Originally Posted by grav View Post
    ..., the ratio of de to dr is extremely close to 1 for the galaxies we can observe anyway.

    ...
    grav,

    I have not read this post too thoroughly, yet, but want to make a clarification, in plain English:

    The above ratio means that for all intents and purposes the light travel time is determined by the instantaneous distance between the emitter and receiver at the moment the photon is emitted.

    Space expansion has little to no effect upon the photon's travel time between the emitter and receiver ... but the expansion will have an effect upon the length of time for the photon if it is reflected back towards the source ... since the return distance is now longer.

    Is this a correct summary of your calculations - for the general reader?

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    Grav, what do you mean by the expression "ein[(x+y)/x]"? At first I thought this was an expontential, but now I'm not so sure.
    Conserve energy. Commute with the Hamiltonian.

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    Quote Originally Posted by Squashed View Post
    grav,

    I have not read this post too thoroughly, yet, but want to make a clarification, in plain English:

    The above ratio means that for all intents and purposes the light travel time is determined by the instantaneous distance between the emitter and receiver at the moment the photon is emitted.
    Yes. It means that the distance that the galaxies move away from each other during the light travel time isn't much different than if space were static, and that the time for light to travel from one to the other isn't that much different either.

    Quote Originally Posted by Squashed
    Space expansion has little to no effect upon the photon's travel time between the emitter and receiver ... but the expansion will have an effect upon the length of time for the photon if it is reflected back towards the source ... since the return distance is now longer.
    It has very little effect, yes, and the time when reflected back will be longer because it is travelling a further distance also. But if a pulse is sent from each toward the other at the same time, then the travel time will be the same in both directions.

    Quote Originally Posted by Squashed
    Is this a correct summary of your calculations - for the general reader?
    Yes it is.

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    Quote Originally Posted by Grey View Post
    Grav, what do you mean by the expression "ein[(x+y)/x]"? At first I thought this was an expontential, but now I'm not so sure.
    I believe I originally wrote it similar to an exponential in the other thread, but it means that if z=ein[(x+y)/x], then ez=(x+y)/x.

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    Quote Originally Posted by grav
    I believe I originally wrote it similar to an exponential in the other thread, but it means that if z=ein[(x+y)/x], then ez=(x+y)/x.
    Ah, then you're inventing notation for something that already exists. The inverse of an exponential (and I now realize that the subscript "in" probably meant "inverse") is called the natural logarithm, and is written ln(x) or ln x.
    Conserve energy. Commute with the Hamiltonian.

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    Quote Originally Posted by Grey View Post
    Ah, then you're inventing notation for something that already exists. The inverse of an exponential (and I now realize that the subscript "in" probably meant "inverse") is called the natural logarithm, and is written ln(x) or ln x.
    Yes, I suppose you're right. I must have started using it that way for my own convenience long ago, probably so I could immediately identify it as a logarithm for base e. I like to see things spelled out like that. Now that I've gotten used to using it that way, I'm wondering how many times I may have seen In(x) and not even recognized it right away. I guess I'll start using it the correct way from this point on. Thanks.

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    In the thread, "Big Bang Blown?", I mistakenly used the formula t=e(Hd/c)/H instead of t=e(Hd/c)/He, which gave a limit to the value of z that we should observe as z=1. If the limit for the value of z was actually z=1, then the universe would would have originated at a size that is half as great as it is today. But that would mean that when the age of the universe becomes twice as great, it will have expanded to twice the present size, and we would then be saying that it originating at half that value, putting it back to the present size before expansion even occurred, which is obviously incorrect. In fact, any limit to the value of z would present the same paradox.

    Let's find the correct limit for the value of z for a constant speed of light. I cannot seem to be able to determine any precise calculations that would tell us that offhand, so I will find it the old fashioned way, through trial and error. The limit to z will occur at the "edge" of the observable universe, which for the model for a constant speed of light, lie at the peak value for an observable distance where the light is just now reaching us within the time allowed for the age of our universe. I will assume this peak to lie at d=c/H first, and see what we get, by comparing values for just over and under that.

    c=3*10^8, H=2*10^-18, de=d/e(Hd/c), c/H=1.5*10^26

    d, de
    ----------
    1.499*10^26, 5.518190391*10^25
    1.5*10^26, 5.518191618*10^25
    1.501*10^26, 5.518190392*10^25

    Well, it looks like it worked out the first time, and the limit of observable distance is then d=c/H. The expansion would have then begun at an original size of d/e=d/2.71828. The limit for z would be z=1.71828, and we have the same dilemna. We should not be able to see redshifts greater than that, and a limit for z is a paradox in itself.

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    Now let's try it for a speed of light that increases in direct proportion to the expansion. Here are the calculations for a pulse of light that is travelling away from an emitting galaxy.

    de'=ce*ti*(de+v*ti)/de

    ce'=ce*(de+v*ti)/de
    de''=(de'+ce'*ti)(de+v*2ti)/(de+v*ti)
    =ce*ti*(de+v*2ti)/de+ce*ti*(de+v*2ti)/de
    =ce*2ti*(de+v*2ti)/de

    ce''=ce'(de+v*2ti)/(de+v*ti)
    de'''=(de''+ce''*ti)(de+v*3ti)/(de+v*2ti)
    =ce*2ti*(de+v*3ti)/de+ce*ti*(de+v*3ti)/de
    =ce*3ti*(de+v*3ti)/de

    As one can see, the integration becomes simply dt=ce*(y*ti)*(de+v*(y*ti))/de, where t=y*ti, so dt=ce*t*(de+v*t)/de. Since dt is the distance travelled by the pulse, which equals the total distance travelled from one galaxy to another during that time, then dt=dr=de+v*t.

    So de+v*t=ce*t*(de+v*t)/de
    1=ce*t/de
    t=de/ce

    Interestingly, the time of travel for the pulse is exactly the same as if it had only travelled the original distance at the original speed of light to begin with. And since distance scales and the speed of light increase in proportion, then t=de/ce=dr/cr=d/c, which would make it appear as if light has simply travelled the current distance at the current speed, as if no expansion were taking place.

    Now, since for a constant expansion, dr=de+v*t=de+He*de*t=de*(1+He*t), then

    dr/de=1+He*t
    dr/de=1+He*(de/ce)

    Also, since the speed of light is proportional to the expansion, then ce=c*(Te/T)=c*(H/He), so ce*He=cr*Hr=c*H. The equation now becomes

    dr/de=1+He*de/ce
    =1+He*de/(c*H/He)
    =1+He^2*de/(c*H)
    dr=de+He^2*de^2/(c*H)
    where de*He=dr*Hr=H*d and d=dr here, so
    d=de+H^2*d^2/(c*H)
    =de+H*d^2/c
    de=d-H*d^2/c
    de=d*(1-H*d/c)

    We can easily see here that the greatest distance that can currently exist between two galaxies is c/H, otherwise we will get an original separation with a negative value. For d=c/H, de=0, as it should be.

    Now let's find the ratio of the original and final wavelengths during the time of travel, whose redshift should match the rate of expansion of the universe. If a galaxy originally at a distance of de from another emits a pulse of light every interval of time of tp, then

    Te+tp=Te', so 1/He+tp=1/He',
    and t=de/ce and t'=de'/ce', so
    tp'=(t'-t)+tp
    =(de'/ce'-de/ce)+tp
    =(de'*He'/(c*H)-de*He/(c*H))+tp
    =(de'*He'-de*He)/(c*H)+tp
    =(de*He-de*He)/(c*H)+tp
    =0+tp
    =tp

    Now we have tp'=tp, so the frequency remains the same, and it would appear at first that the light is not altered, but to continue, fe=1/tp and fr=1/tp', and ce=fe*we and cr=fr*wr, for the frequency and wavelength of light, so

    tp=tp'
    1/fe=1/fr
    we/ce=wr/cr
    wr/we=cr/ce=Tr/Te

    The ratio for the original and final wavelengths are indeed the same as that for expansion, but now we have to wonder if the energy of light should then be measured by the frequency or wavelength over time, since one changes and the other does not, unless the value for Planck's constant, h, changes with the age of the universe as well for E=h*f. In any case, I would venture as far as to say, then, that the calculations for any reasonable cosmological model for expansion might well produce equal proportions of redshift to expansion for said model. So both models work out in this respect, then, but with some major overall differences.

    Both are the same in respect to correct proportions of redshift to expansion, and both have a limit of observation of c/H, but this model for a speed of light that increases in proportion to the expansion does not carry the paradoxial situation that a constant speed for light does. At the limit of observation, c/H, the time required for the light to reach us within the age of the universe puts the initial distance at zero for T=0. This remains the same regardless of the age, unlike the first model. Also, the value of z, then, has no upper limit, which resolves the paradox. It seems this model has just one upped the constant speed model, and it looks like the Big Bang is back in business. [EDIT-On second thought, if the same galaxies exist at the limit of the observable universe during expansion regardless of its age, as they should, then how could the light received from them always take an amount of time to reach us that is equal to the entire age of the universe. That would either mean that all of the light we ever recieve from them was emitted at the same time, T=0, or that light coming from that limit will never quite reach us to begin with, as a limit in itself; probably the latter. I will have to think on it some more.]

    Another interesting thing about this model is that with a steadily increasing speed of light, one would not know the difference between galaxies moving through space and with it. The measurements in both cases are identical (except for a measure of the speed of light as the universe ages), which may be an additional point for this model as well.
    Last edited by grav; 2006-Dec-25 at 11:26 PM.

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    Well, at this point it seems that a speed of light that increases in direct proportion to the expansion is indeed the best candidate for a cosmological model for constant expansion after all. I originally thought that this would only serve to decrease our limit of observation even further than c/H, but all it really did was to bring all distance scales that were over the peak in the constant light speed model to within the actual limit of observation as determined by the present age of the universe, as it should be. So it describes redshifts that are in proportion to the expansion, allows for a Big Bang singularity, and any paradoxes now appear to be resolved. It also describes a time of travel for light that is consistent with the current distance and speed of light at the time of observation, so the only real difference between galaxies moving through space and with space is a measure of the speed of light as determined by the age of the universe. The question now is how does the expansion of the universe directly affect the speed of light, so that it also increases in proportion?

    The only real problems at this point are those that conflict with direct observation. As far as I know, we can currently see past a distance of c/H, or about 1.5*10^26 meters or just under, which we should not be capable of doing. Also, the redshift to distance relationship is z+1=d/de=1/(1-Hd/c), so z=(Hd/c)/(1-Hd/c). This means that the redshift should increase faster with distance than the direct redshift to distance ratio, z=Hd/c, not decrease as is observed, so that the expansion of the universe would then appear to be decelerating, not accelerating, which also conflicts with observation.

    I guess the next step would be to find a relationship between an acceleration of expansion and the speed of light that accomplishes the same things this model does, this one being an inertial expansion where the acceleration is zero, which can then be set to match the observations once the value of some control variable is determined, similar to that of the cosmological constant. I still say the tired light model is the best for all of this, however, and would eliminate the need for expansion altogether, but I will continue along this avenue for now just to see where it might lead.

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    An edit here. It appears that if z+1=ed*H/c for a constant expansion and speed of light, then the universe would appear to decelerate in comparison to a direct redshift to distance ratio. This is because if wavelengths are expanded exponentially with distance, then so are they also with time, making it appear that expansion was faster in the past (with greater time). Tired light theory would also decelerate, then, since it matches the same formula.

    Of course, the only thing that could actually produce a direct redshift to distance relationship is galaxies that move apart inertially through static space. So if we compare instead to the formula of a constantly expanding one, or z+1=ed*H/c, then tired light would match, but still not account for an apparent accelerated expansion. It would neither accelerate nor decelerate. The formula for a constant expansion with a speed of light that varies with it, however, which is z+1=1/(1-Hd/c), and is the only one that appears to be self-consistent for constant expansion, would also appear to decelerate in comparison, only slightly for short distances, but greatly at larger ones.

    I am coming to the conclusion at this point that any actual acceleration would require some real additional pressure, such as dark energy. I need to find the precise formula Permutter is comparing to when he says that the expansion of space appears to be accelerating.

  12. #12
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    Varying Speed of Light

    Quote Originally Posted by grav View Post
    Well, at this point it seems that a speed of light that increases in direct proportion to the expansion is indeed the best candidate for a cosmological model for constant expansion after all. ...
    Grav, if the speed of light is constantly increasing then wouldn't that mean that the size of atoms is constantly increasing also?

    I view the internal workings of atoms as interactions between various energy packets that are mediated by photons and so if the speed of the photons was slower in the past then in order to keep the synchronicity between the energy packets then the whole assembly must contract.

    It would be like a system of dominoes whereby one interaction causes another interaction which causes a third interaction which continues in this manner until ultimately the final interaction occurs with the beginning component to start the whole process over again - if the timing gets thrown off at some point then the integrity of the whole assembly is disrupted and the particle "decays" or self-destructs. The result of the destruction is that subatomic particles and/or photons are thrown off at various tangents to carry away their respective momentums that used to interact in a balanced and symbiotic manner to produce what we call an atom.

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    Quote Originally Posted by Squashed View Post
    Grav, if the speed of light is constantly increasing then wouldn't that mean that the size of atoms is constantly increasing also?

    I view the internal workings of atoms as interactions between various energy packets that are mediated by photons and so if the speed of the photons was slower in the past then in order to keep the synchronicity between the energy packets then the whole assembly must contract.

    It would be like a system of dominoes whereby one interaction causes another interaction which causes a third interaction which continues in this manner until ultimately the final interaction occurs with the beginning component to start the whole process over again - if the timing gets thrown off at some point then the integrity of the whole assembly is disrupted and the particle "decays" or self-destructs. The result of the destruction is that subatomic particles and/or photons are thrown off at various tangents to carry away their respective momentums that used to interact in a balanced and symbiotic manner to produce what we call an atom.
    Frequencies stay the same while wavelengths increase, so the energy of light becomes E=hf=h(c/w), where the speed of light and the wavelength increase together, so the energy does not change. You make a good point, though. Take a black hole, for example. If light becomes trapped near the event horizon, and its wavelengths still continue to increase with time, then would the entire photosphere also expand or would light just travel around it faster? Probably the latter, since the speed of light would be increasing, but then the Schwarzschild radius should decrease with time as the speed of light increases, unless a measure of mass increases with its square. In other words, as the speed of light increases, it should steadily escape the black hole's grip.

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    Speed of Gravity Increases

    Quote Originally Posted by grav View Post
    Frequencies stay the same while wavelengths increase, so the energy of light becomes E=hf=h(c/w), where the speed of light and the wavelength increase together, so the energy does not change. You make a good point, though. Take a black hole, for example. If light becomes trapped near the event horizon, and its wavelengths still continue to increase with time, then would the entire photosphere also expand or would light just travel around it faster? Probably the latter, since the speed of light would be increasing, but then the Schwarzschild radius should decrease with time as the speed of light increases, unless a measure of mass increases with its square. In other words, as the speed of light increases, it should steadily escape the black hole's grip.
    If the speed of light increases then so would the speed of gravity and so the two would offset and the blackhole would remain the same which means the light will not "steadily escape the black hole's grip".

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    Cosmic Microwave Background Radiation

    Quote Originally Posted by grav View Post
    ...

    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.

    ...
    Grav, what implications does this have for the CMB radiation because the CMB radiation supposedly originates from when the universe was very small but because of DA enlargement it now surrounds us - it seems to me that your calculations discount this explanation of the CMB radiation?

  16. #16
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    Quote Originally Posted by Squashed
    Originally Posted by grav
    Frequencies stay the same while wavelengths increase, so the energy of light becomes E=hf=h(c/w), where the speed of light and the wavelength increase together, so the energy does not change. You make a good point, though. Take a black hole, for example. If light becomes trapped near the event horizon, and its wavelengths still continue to increase with time, then would the entire photosphere also expand or would light just travel around it faster? Probably the latter, since the speed of light would be increasing, but then the Schwarzschild radius should decrease with time as the speed of light increases, unless a measure of mass increases with its square. In other words, as the speed of light increases, it should steadily escape the black hole's grip.
    If the speed of light increases then so would the speed of gravity and so the two would offset and the blackhole would remain the same which means the light will not "steadily escape the black hole's grip".
    I hadn't thought about that.

    Originally Posted by grav
    ...

    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.

    ...
    Grav, what implications does this have for the CMB radiation because the CMB radiation supposedly originates from when the universe was very small but because of DA enlargement it now surrounds us - it seems to me that your calculations discount this explanation of the CMB radiation?
    What you just quoted was for a constant rate of expansion with a constant speed of light. That cosmological model does not work as far as I can tell. A speed of light that varies with the expansion does, however. For this, the energy of photons remains the same regardless of the redshift due to expansion, but the volume of space still increases with time. So for an unchanging number of photons in an original volume of space, the total energy remains the same, but the number per volume varies with (de/dr)^3 as the universe expands. The energy density would then also change with (de/dr)^3, so the temperature of the CMB would fall with (de/dr)^(3/4). For this cosmology model, there is a limit to the distance galaxies can travel within the age of the universe, c/H, but there is no limit on the redshift observed.

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    It's been a while and I still haven't posted the calculations for an accelerated expansion, so here they are.

    de=ce*ti*(de+v*ti+a*ti^2/2)/de

    ce'=ce*(de+v*ti+a*ti^2/2)/de
    de'=(de+ce'*ti)*(de+v*2ti+a*(2ti)^2/2)/(de+v*ti+a*ti^2/2)
    =ce*ti*(de+v*2ti+a*(2ti)^2/2)/de+ce*ti*(de+v*2ti+a*(2ti)^2/2)/de
    =ce*2ti*(de+v*2ti+a*(2ti)^2/2)/de

    ce''=ce'*(de+v*2ti+a*(2ti)^2/2)/(de+v*ti+a*ti^2/2)
    =ce*(de+v*2ti+a*(2ti)^2/2)/de
    de''=(de'+ce''*ti)*(de+v*3ti+a*(3ti)^2/2)/(de+v*2ti+a*(2ti)^2/2)
    =ce*2ti*(de+v*3ti+a*(3ti)^2/2)/de+ce*ti*(de+v*3ti+a*(3ti)^2/2)/de
    =ce*3ti*(de+v*3ti+a*(3ti)^2/2)/de

    So dt=ce*(y*ti)*(de+v*(y*ti)+a*(y*ti)^2/2)/de
    =ce*t*(de+v*t+a*t^2/2)/de

    and since dt=dr=de+v*t+a*t^2/2, then
    dt=ce*t*(de+v*t+a*t^2/2)/de
    =ce*t*dr/de

    1=ce*t/de
    t=de/ce, the same as with constant expansion.

    Now, john hunter has done some interesting work which relates to this accelerating expansion also, which I have commented on and run through some calculations for as well. So instead of reproducing them here, I will simply provide a link to that thread.

  18. #18
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    Okay. I have found the limiting distance for a constant rate of expansion with a speed of light that varies with it to be c/H, but that does not account for the amount of time that has elapsed before the galaxy emitted the light to begin with. So let's do that. Te is the age of the universe at the time the light was emitted, t is the time of travel from the emitted galaxy to the receiver, and Tr is the present age of the universe. So Te+t=Tr. Also, as found before, He*de=Hr*dr=v and t=de/ce=dr/cr. We now get...

    Te+t=Tr and
    1/He+de/ce=1/Hr
    de/(He*de)+de*Tr/(cr*Te)=1/Hr
    de/v+de*He/(cr*Hr)=1/Hr
    de/v+v/(cr*Hr)=1/Hr
    de=v/Hr-v^2/(cr*Hr)

    So since de is the observed distance, Hr is the inverse of the present age of the universe, and cr is the present locally measured speed of light, then

    d=(1-v/c)(v/c)(c/H)

    Now, this seems to reach a peak the same way as that for a universe with a constant speed of light did. To find that peak, we can solve for v with

    v/c=[1-sqrt(1-4H*d/c)]/2

    We see here that we should not observe a distance further than 4H*d/c=1 or d=c/4H, which is a quarter of the age of the universe in light years, or about 3.5 billion light years total. Of course, we can see much further than that. The greatest speed we should observe a galaxy moving away from us is half of the current local speed of light. But there is something much more than this. Another paradox. If the furthest distance we can observe is d=c/4H, then when the age of the universe doubles, H will half and c will double, so we should observe up to four times a greater distance, but the size of the universe will only double. When the galaxies move ten times their current separations, we will see one hundred times further. At some point, then, we should see further than the actual size of the universe. Whether this means there will be a cutoff point and we would observe an edge or just the same thing over and over or something, I don't know, but this should have been happening from the get-go anyway, every time its age doubles and so forth, so it makes no sense, really.

    Now, there may be a limited number of ways around this. One, an accelerating universe might automatically rectify the situation, but I doubt it. I'll have to run some calculations and see. Second, as far as seeing past 3.5 billion light years or so, it may be that our way of measuring distances may be off. That is, the galaxies may actually be much closer than we measure, but our luminosity to distance formulas may be off. Redshift itself, for instance, will throw off a direct drop in luminosity per square of the distance relation, making galaxies appear much further than they really are. It would also explain why far away galaxies appear further along than they should be as we are really looking back in time, why the expansion appears to be accelerating, help explain Olber's paradox, and so forth. But it would also depend on the frequency measured, since the Planck distribution per frequency is different depending on this also. Luminosity for frequencies originally below the distribution curve might actually increase with distance before they begin to fall as it moves through the distribution curve since the photon densities for the various frequencies will not change. As well as this, normally pure relativity would not play a part, since we are using our own clocks and rulers, and contraction of galaxies that are moving away from us or their own time dilation relative to us would not matter, especially since their size tangent to the line of motion would not change and so forth, so that they would not appear to magnify or shrink to appear closer or further away, but it may have some effect on luminosity. I'll have to look into that.

    Finally, if all of this does turn out to contradict itself all the way through, as it so far appears to be, then there simply may not have been a Big Bang after all. The universe may actually be infinite and static, and the galaxies already spaced out throughout the universe. So what is this accelerated expansion we are seeing? Well, let's see. We are seeing an accelerated expansion that can be traced back to somewhere around 13 billion years ago, about the time the galaxies and stars came to be. What is causing this acceleration? Well, an actual acceleration would require a force. Force comes from pressure. So some additional pressure may have come into existence about the same time the stars began to burn and the galaxies in the region began moving apart. I wonder what kind of pressure that could be.

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    I posted this in another thread and thought I should add it to this one.

    For a universe which is expanding at a constant rate (no acceleration), the relevant velocity would be exactly in proportion to the time dilation observed, which are both then also in proportion to the redshift, relating a direct velocity to distance ratio for Hubble for any observed distance, the same as in Euclidean geometry, regardless of whether the distances might be different in reference to the time of expansion than with Euclidean, since the velocity and redshift also differ in exact proportion. If the expansion is accelerating, however, the relevant acceleration also varies with the time of expansion, so things become slightly different. I hadn't noted that in my other thread, so I guess I'll have to go back to that.

    Every point in the universe can be considered the center, according to the Bing Bang theory, so if we are stationary and a galaxy begins receding away from us from d=0 with an initial acceleration of a, then the relevant acceleration we would observe over time becomes ar=a/[1+(a*t/c)^2]^(3/2). The observed relevant velocity and observed distance travelled with an integral time of 'ti' then become

    ar'=a/[1+(a*ti/c)^2]^(3/2)
    v'=ar'*ti=a*ti/[1+(a*ti/c)^2]^(3/2)
    d'=v'*ti=a*ti^2/[1+(a*ti/c)^2]^(3/2)

    ar''=a/[1+(a*2ti/c)^2]^(3/2)
    v''=v'+ar''*ti=a*ti/[1+(a*ti/c)^2]^(3/2)
    +a*ti/[1+(a*2ti/c)^2]^(3/2)
    d''=d'+v''*ti=2*a*ti^2/[1+(a*ti/c)^2]^(3/2)
    +a*ti^2/[1+(a*2ti/c)^2]^(3/2)

    ar'''=a/[1+(a*3ti/c)^2]^(3/2)
    v'''=v''+a'''*ti=a*ti/[1+(a*ti)/c)^2]^(3/2)
    +a*ti/[1+(a*2ti/c)^2]^(3/2)
    +a*ti/[1+(a*3ti/c)^2]^(3/2)
    d'''=d''+v'''*ti=3*a*ti^2/[1+(a*ti/c)^2]^(3/2)
    +2*a*ti^2/[1+(a*2ti/c)^2]^(3/2)
    +a*ti/[1+(a*3ti/c)^2]^(3/2)

    For the velocity, this comes out to v=c * Sum[x/[1+(z*x)^2]^(3/2)], where x=a*ti/c, ti is very small, such that x<<1, and z=1 to t/ti.
    For distance, it is d=(c^2/a) * Sum[(t-z*ti)*x^2/[1+(z*x)^2]^(3/2)].
    I find these to be v=c*z*x/sqrt[1+(z*x)^2]=a*t/sqrt[1+(a*t/c)], as expected,
    and d=(c^2/a)*[sqrt(1+(z*x)^2)-1]=(c^2/a)*[sqrt(1+(a*t/c)^2)-1].

    Since we don't know the initial acceleration for any particular distance, let's just go ahead and pull that right on out of there by using y=z*x=at/c to our advantage here. We now get

    d/(c*t)=(1/y)*[sqrt(1+y^2)-1]
    d*y/(c*t)=[sqrt(1+y^2)-1]
    [d*y/(c*t)+1]^2=1+y^2
    [d*y/(c*t)]^2+2*d*y/(c*t)=y^2
    [d/(c*t)]^2*y+2*d/(c*t)=y
    y[1-(d/(c*t))^2]=2*d/(c*t)
    y=[2*d/(c*t)]/[1-(d/(c*t))^2]

    Next, we find that v=c*y/sqrt[1+y^2], and we can use these two equations to find a velocity to distance ratio, regardless of the acceleration. But first, we have to define a time for expansion. Of course, we want the full time of expansion, to get the present observations, so I initially tried t=1/H. This should give the v/d ratio of H locally. So for a distance of one meter (with H=2.1*10^-18 sec-1 and c=3*10^8 m/sec), we get

    d=1 m, so y=1.4*10^-26, and then v=4.2*10^-18 m/sec, so v/d=H=4.2*10^-18 sec-1

    Obviously, that's incorrect. So I then tried it for t=2/H. That comes to

    d=1 m, y=7*10^-27, v=2.1*10^-18 m/sec, v/d=H=2.1*10^-18 sec-1

    So right off the bat, we have just found the age of the universe in this particular scenario, and it is double that of the previous age for t=1/H, or about twenty-five to thirty billion years old. Now I want to do two things. I want to find the maximum observable distance, and I want to see if the value of the Hubble constant appears to drop at large distances.

    For d=10^26 m, y=.797720797, v=.623608017 c, H=1.870824*10^-18 sec-1 sec-1
    For d=2*10^26 m, y=2.745098039, v=.939597315 c, H=1.409396*10^-18 sec-1

    Yes, it drops, but of course I am using an acceleration after all. But as well as this, the maximum observable distance is now twice as great as for a simple c/H, but still d=c*t, however, but for twice the time as found with 1/H, and so becomes T=2/H.

    [EDIT-I'm still not sure if the time the light takes to reach us is included for the observable distance in this, however. I'm thinking it is, though, since that is partly what causes the time dilation in the first place.]

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