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My book Gravitation and the Electroform Model: From General Relativity to Unified Field Theory derives physical cosmology on the basis of my equations for unified quantum field theory, which I obtained after ruling out classical General Relavitity based on some internal non-self-consistency problems. I concluded that the observation that very distant and quite evolved spiral galaxies seem to move away at velocities close to the speed of light at the outer limits of observation can be understood on the basis of observational time-delay τ = L/c in the measurements,
where L is the distance to the observed galaxy.
As we shall see, based on relatively nearby measurements, we arrive at a value for the Hubble constant such that is the equation that describes the velocity with which distant galaxies recede at the distance L. Here H is the Hubble constant H = (2/3)/t, where t is the age of the universe in seconds. This formula will be derived below from the equations of the expanding universe, and can be shown to be true for many different mathematical models of the expanding universe. Now the velocity v of a receding galaxy should be before we include the observational time-delay accounting for the distance to the observed object, τ = L/c. Inserting this observational time-delay, we find At the limits of observation possible with our telescopic equipment, we find distant whole spiral galaxies in well-formed condition receding with a velocity approaching the speed of light. Then so that ct - L = (2/3)L, or ct = (5/3)L, hence L = (3/5)ct, where again t is the age of the universe. Thus we expect to find distant galaxies racing away at light speed when L = (3/5)ct = Lmax, the absolute maximum depth of observation with telescopic equipment. This is evidently well after the era of galaxy formation, which takes place at time tgalaxy << (2/5)t. In my book Galaxy Formation, I show how this happens when galaxies form as a consequence of the pressure jog induced by the phase transformation He++ -> He+ in the cooling Big Bang fireball. If we substitute numerical values for the Hubble constant H and the age of the universe in seconds t into from L = (3/5)ct we can compute the distance to the edge of the visible universe in light-years or Mpc : Lmax = 2399 Mpc = 7.821 billion light-years. Derivation of H(t) = (2/3)/t Our argument above depends on our formula H(t) = (2/3)/t for the Hubble constant we expect to measure from relatively nearby observations prior to introducing the observational time-delay τ = L/c to obtaining the Hubble constant we actually expect to measure, where t is the age of the universe in seconds and L is the distance to the observed galaxy in meters. One begins by modeling the universe during the matter-dominated era when it is just coasting outward as a universe of dust of uniform density &ro;. The equations must be approximately Newtonian, so that We notice by substitution that r = a t2/3 is an exact solution: This will fit our model for Newtonian expansion if Thus a = [(9/2)GM]1/3 will provide the model for a universe of mass M. Since r(t) = at2/3, we must have v = dr/dt = (2/3)at-1/3 = (2/3[at2/3]/t = [(2/3)/t)]r = Hr, Consequences of Ignoring Observational Time Delay If we ignore observational time delay L/c when we compare observations with experiment, our differential equations above lead us to v = H(t)L = [(2/3)/t]L, or L = (3/2)vt. Suppose v = c. Then L = (3/2)ct, where t is the age of the universe and L > ct. In other words, then we would NEVER see a galaxy moving away with a velocity close to light speed, which contradicts our observations. Suppose ct = 13.5 billion light years. Then L = (3/2)ct is a point 20.25 billion light-years away well outside a sphere expanding with the speed of light at ct = 13.5 billion light-years. However, all of the material contents of the expanding universe should be inside a sphere of 13.5 x 109 LY in size. It is interesting to compare predictions of the position of L for various values of v/c = α. Let observational time-delay be ignored. Then Lig(α) = (3/2)αct. Let observational time-delay be included, as above. Then we find Lob(α) = (3α/(2 + 3α))ct. I might add that I've seen it argued that H must be constant all the way out to the limits of observation where Big Bang material must stop if v < c, and that H = 1/t is more like it, since then v = HL = L/t and thus Lprim = αct yields a primitive's view of the universe in which he can see all the way back to the beginning of time. However, this primitive point of view does not include observational time-delay due to the finite speed of light and is not credible. I think a number of fellows do this anyway, and therefore we read reports that they saw all the way back to 13.5 billion light-years in the past, which is mistaken! Let ct = 13.5 billion light-years. Then we obtain the following table:
Lob(α), including observational time-delay L/c, is thought to be the correct distance. The point of view that H is constant all the way out to v = c was argued at one time by some men from the school of cosmology at Princeton, and is the primary competitor working against the correct viewpoint according to me. Here we have my viewpoint in the 2nd column and the source of odd reports in the news in the 3rd column. To me, it seemed that the Princeton school fellows making do with H = 1/t were basing their viewpoint on their measurements of constant H from a set of nearby galaxies and the easy result that for v < c all the material contents of the entire universe could be inside the special relativity limit. Also, putting it that way left some room for improvement by bright graduate students. I think it is possible to compute all the observational details without assuming a cosmological constant exists or otherwise introducing complexities into the fundamental laws of Nature.
- James A. Green, May 13, 2005 - 56 years old today. H2 = 8πρG/3 and Rmax, the expansion limit. Let d/dr(dr/dt) = -GM/r2. Multiplying both sides of the equation by dr/dt, (dr/dt)(d/dr(dr/dt)) = -(GM/r2)(dr/dt). Or d/dt( (dr/dt)2 ) = -2GM(dr/dt)/r2 = 2 (d/dt)GM/r. Integrating both sides of this equation over t, and including a constant of integration K, (dr/dt)2 = 2GM/r + K. If K = 0, then (dr/dt)2 = 2Gρ(4/3)πr2, so that H2 = [(dr/dt)/r]2 = 8πρG/3. To determine K, note that if we start dropping matter in from Rmax, then from (dr/dt)2 = 2GM/r + K, we find 0 = 2GM/Rmax + K, so that K = - 2GM/Rmax. Thus for K < 0, the universe is bound, never reaching any further out than Rmax. In reality, (dr/dt)2 = 2GM/r - 2GM/Rmax. Then (dr/dt)2 = 2G(4/3)πr2ρ - 2G(4/3)πr3ρ/Rmax. Thus H2 = (8πGρ/3)(1 - r/Rmax), where Rmax is given in meters, and H(r) = [(8πGρ/3)(1 - r/Rmax)]1/2. This equation might be used to determine Rmax if H(r) and ρ are both known, but the huge dimensions of Rmax make this impractical, as r/Rmax is locally small. Note H(r) is the locally observed Hubble constant. In general, we must include observational time-delay r/c, so that Hobserved(r, t) = H(t - r/c) = (2/3)/(t - r/c) = [(2/3)/t](1/(1 - r/ct)) = H(t)(1/(1 - r/ct)). Then c = Hobserved(r,t)v = (2/3)v/(t - r/c) when r = (3/5)ct, the observational limit. This is as far as telescopes can see, a point now about 8.1 billion light-years away where galaxies redden out, although near this depth of view back in time Δt = r/c they all seem to be closer together and rushing away more rapidly from each other, so that we see a number of colors in the Hubble Deep Field galaxies. Thus we may expect to measure something approximately like Hobserved(r, t) = H(r)[1/(1 - r/ct)] = [(8πGρ/3)(1 - r/Rmax)]1/2[1/(1 - r/ct)] when r < (3/5)ct. Usually and probably at this time Rmax >> ct > (5/3)r. Measurements of Hobserved(r,t) at the outer limits of observation may finally in principle determine Rmax. I note that ρ is measured by measuring H, and that the closure constant K < 0 for the universe is such that r/Rmax is locally small. The "missing mass" problem [8] in physical cosmology is that ρvisual < ρ(H), because it is hard to see all the mass visually. Some of it must be locked up in dark objects. Also, 1 - (3/5)3 = 1 - 0.216 = 0.784 of the total mass M must be invisible because of the cosmic censorship imposed by the limit of telescopic visibility at L = (3/5)ct, the Red Limit due to observational time-delay. You see less than 22% of the mass. See also Red Limit 2 with Galaxy M51 and Galaxy Formation "If the Theory of making Telescopes could at length be fully brought into Practice, yet there would be certain Bounds beyond which Telescopes could not perform. For the Air through which we look upon the Stars, is in a perpetual Tremor... The only Remedy is a most serene and quiet Air, such as may perhaps be found on the tops of the highest Mountains above the grosser Clouds." - Sir Isaac Newton, Opticks. |
