A cosmology conserving momentum as required by Noether's Theorem is presented.

Cosmology according to Emmy Noether Jack Heighway Hubble’s discovery of the systematic cosmological red shift immediately suggested that galaxies were flying through space away from us and from one another – the further, the faster. This in turn suggested that the universe grew from an incredibly hot and dense condition (a singularity!) billions of years ago. Later, solutions of the field equations revealed that the galaxies were not actually moving through space, but rather that space itself was expanding. The conventional understanding holds that the wavelength of light is continually stretched in flight by the expansion of space. Thus in the present understanding, the redshift parameter, (Z+1) = λobs / λemit, shown by a galaxy is held to be proportional to the ratio of the scale of the present universe to that of the universe at the time of emission. Neither astronomers nor cosmologists seem to be concerned with the fact that this explanation of the cosmological redshift fails to conserve the momentum of the observed radiation, as Noether's Theorem demands in a spatially homogeneous universe. Observation shows that on a sufficiently large scale the universe is homogeneous and isotropic to a very high degree, and virtually every cosmological model has assumed this at the outset. Perhaps no one has even considered this problem since, as every observation of the red shift appears to have demonstrated, momentum seems to be manifestly not conserved. The photons of the cosmic microwave background radiation field that we detect today were born in a 3000°K hydrogen-helium plasma at ‘recombination’ time when the plasma first became transparent. Their wavelength has evidently increased by a factor of about 1000 – how is it possible that their momentum has not changed? The only possibility is that over the aeons, measuring instruments have changed, and are changing, decreasing their characteristic wavelengths – for example, diffraction gratings have shrunk and are shrinking. As we shall see, the only way this might occur is if all rest masses have been and are increasing in proportion to a(t) = A(t)/Ã, in which A(t) is the function that is presently interpreted as representing the increasing scale of the universe, and à is its present value. To understand this one needs to understand how the size of a physical object is determined by the rest mass of its constituent parts. The radius of the hydrogen atom, the Bohr radius, a0, is determined by three fundamental constants: Planck’s constant, h; the fundamental electric charge, e; and the rest mass of the electron, me. Explicitly, a0 = h2/e2me . Clearly, if the rest mass of the electron increases, the Bohr radius will shrink – and, since all physical objects must be governed by the same law, all rest masses must increase, and all objects, including diffraction gratings, will shrink to the same degree. To prove that momentum conservation requires rest masses to increase in proportion to a(t), one need only consider the motion of a test mass through space. In this case, there is an integral of motion 1 for the Robertson-Walker metric, namely, a(t)∙β (1–β2)−½ = constant. Since momentum is given by m*c β (1–β2)−½, it is clear that conservation of momentum requires that the true rest mass, m*, must be proportional to a(t), that is, m* = m·a(t) where m is the proper rest mass, a constant. A very elaborate calculation is required to solve the field equations for cosmology under the assumption that momentum is conserved.2 The surprising result is that a(t) turns out to be a simple exponential function of world time. But upon thoughtful reflection, it is obvious (in hindsight) that such must be the case, as will now be shown. Consider a source of monochromatic light and two observers lying on a straight line at distances d1 and d2 > d1 from the source. Generally, λobs = λstd a(tobs)/a(temit). Let λ1 & λ2 be the wavelengths observed at d1 & d2 , at times t1 & t2 . In the case considered, λ2 a(temit) = λstd a(t2) = λstd a(t1+(d2 –d1)/c). Momentum conservation implies that the wavelength of light remains constant in traveling between any two points. If that is so, the spectra measured at d2 will differ from that observed at d1 only as a result of the changes in the instruments at d2 caused by the general increase in rest masses during the time interval (d2 –d1)/c. And of course, the same is true for the time interval t1 = d1/c. Thus the function on the right hand side above must be the product of some function, u, of d1 /c and the same function of (d2 –d1)/c. We may then write a(d1 /c + (d2–d1) /c) = u(d1 /c)× u((d2–d1))/c). But when d2 = d1, we must have u = a. Thus the function, a, satisfies the relation a(x+y) = a(x) × a(y). Only the exponential function satisfies this functional equation, so a(t) = exp(– ω(tnow – t)), the form being chosen so that a(tnow) = 1. The frequency parameter, ω, is easily identified as the Hubble constant, H0. For nearby galaxies, Hubble’s ‘law’ implies λobs/λstd ≈ 1 + V/c = 1 + H0 D/c = 1+ H0 (tnow–temit). The exponential function gives, for nearby galaxies, λobs/λstd = 1/a = exp(ω(tnow–temit)) ≈ 1 + ω(tnow–temit),. Thus, ω=H0. Regarding the evolution of proper time, T, dT/dt = m*/m = a(t), hence 100()()tTatdtHataHT−−∞′′==⇒=∫. Thus the rest mass evolution function is a linear function of proper time, T. Proper time is that kept by physical clocks whose rates, looking backward in time, slow in proportional to a(t). World time, t, may be thought of as being defined with reference to any observable free electromagnetic radiation – in particular, the cosmic microwave background radiation itself – whose true frequency is constant. Paradoxically, it is quite clear that the universe is infinitely old in terms of world time, t, yet it is no less true that the universe suddenly came into existence with a ‘Big Bang’ some H0–1 seconds ago (~14 billion years) in terms of proper time, T. 1 Heighway J., Einstein, the Aether & Variable Rest Mass,, ISBN 978-1-61658-620-1, lulu.com, (2011), pp.77. 2 Ibid, pp.71-76