Hubble’s law *v = Hr* suggests that since in spacetime *r = ct,* where* t* is time past, and since *v = dr/dt* or *dt = dr/v,* there’s a legitimate acceleration observable:

*a = dv/dt = dv/(dr/v) = v*dv/dr = Hr*d[Hr]/dr = H*^{2}*r.*

So Hubble’s law, *v = Hr,* is equivalent to: *a = H*^{2}*r*. This means that the receding universe has outward force by Newton’s second law *F = ma = mH*^{2}*r*. This outward force is not as easy to evaluate as appears at first glance, because the density we’re concerned with increases with distance *r* from us, because we’re looking back in time to more compressed eras of the big bang.

Since volume of universe is (4/3)π*R*^{3}, since there is no gravitational retardation on observed expansion (i.e., radial distances of most distant matter are proportional to time after big bang *t,* not to *t*^{2/3} which was incorrectly predicted by the Friedmann-Robertson-Walker metric until 1998 observations disproved it), expansion is *R = ct, *and age of universe is *t =* 1*/H* rather than *t* = (2/3)/*H* which was based on the Friedmann-Robertson-Walker metric before 1998 (*this is irrespective of whether the failure of the Friedmann-Robertson-Walker metric is due to the lack of between rapidly receding masses in the universe due to redshift of force causing quantum gravity exchange radiation, or whether the lack of gravitational retardation is caused by dark energy producing an acceleration which cancels out gravitational acceleration over such distances*), so density varies in proportion to *t*^{–3}*,* and for a star at distance *r,* its absolute time after big bang will be *t – r/c* (where *t* is our time after big bang, about 15 Gyr), so the relative density of the universe at its absolute age corresponding to visible distance *r,* divided by the density at 15 Gyr, is [(*t – r/c*)/*t*]^{-3 }= (1* – rc*^{-1}*t*^{-1})^{-3} = (1* – rc*^{-1}*H*)^{-3}, and this is the factor needed to multiply up the nearby density to give that at earlier times corresponding to large visible distances.* *

*What we are interested in is the equal inward reaction force to this outward big bang force of mass receding in spacetime. *The inward reaction force of radiation pressure arises from the 3rd law of motion, and allows us to calculate gravitation. The inward gauge boson radiation from masses which are receding very rapidly and are located at very early times after the big bang, however, is redshifted and thus has little energy. By Planck’s law, *E = hf*. Hence, the gravity causing exchange radiation from distant receding masses which is redshifted in frequency *f* carries less energy as observed by us, due to redshift. The relative frequency change due to redshift is:

1 + *z = f _{emitted}/f_{observed}*

*= R _{now}/R_{then}*

= *ct/[c(t – r/c)] = t/(t – r/c)* = (1* – rc*^{-1}*t*^{-1})^{-1} = (1* – rc*^{-1}*H*)^{-1} where *z* is redshift. The relative observed frequency from radiation emitted by receding matter at radius *r* is: * f _{observed }= f_{emitted} *(1

*– rc*

^{-1}

*H*), so the relative observed energy using Planck’s law

*E = hf*is:

*E _{observed }= E_{emitted} *(1

*– rc*

^{-1}

*H*).

Therefore evaluation of *F = ma = mH*^{2}*r *to find the inward force carried by gauge boson radiation requires using calculus to take account of the way that two correction factors derived above work (for density as a function of radial distance in spacetime, and for redshift energy depletion of gravitons as a function of the radial distance at which the mass is receding from us). Since spherical volume is the integral between 0 and *r* of the product of spherical surface area and radial element of thickness *dr* (i.e., the integral of 4π*r*^{2}*dr *is the volume 4π*r*^{3}/3), it follows that mass is the integral ò 4π*r*^{2}r *dr* over the range 0 to* r. *Hence:

*F = ma = mH*^{2}*r*

*= *ò(4π*r*^{2}r )(1* – rc*^{-1}*H*)^{-3}(1* – rc*^{-1}*H*)*H*^{2}*rdr,*

where r is the density of the universe at our time after the big bang (15,000 million years or whatever), (1* – rc*^{-1}*H*)^{-3} is the dimensionless density correction factor for spacetime (derived above), and (1* – rc*^{-1}*H*) is the dimensionless gauge boson redshift correction factor (also derived above), because the inward force is carried by the momentum of radiation, *p = E/c = hf/c,* and radiation that is redshifted in wavelength is also reduced in frequency, which reduces its energy and thus its momentum and the force it can impart in our reference frame.

*F **= *ò(4π*r*^{2}r )(1* – rc*^{-1}*H*)^{-3}(1* – rc*^{-1}*H*)*H*^{2}*rdr*

*= *ò(4π*r*^{2}r )(1* – rc*^{-1}*H*)^{-2}*H*^{2}*rdr*

*= *ò (4 π *r *r ) [ { 1/(*H**r*) }* – *(1/*c*) ]^{-2}*dr*

*= *4 π r *c*^{2 }ò *r** *[ {*c*/(*H**r*) }* –* 1 ]^{-2 }*dr.*

This seems to be in error, because contributions tend towards infinity when *r *approaches the value upper limit of integration *R = c/H.* Hence, the redshift effect by itself doesn’t seem to be enough to cancel out the effect of rising density at the greatest distances. One possible ‘error’ above is that I used the Friedmann-Robertson-Walker result (I’ve already stated above why the Friedmann-Robertson-Walker solution of general relativity for cosmological purposes is wrong; it excludes vital quantum gravity dynamics such as redshift of gauge bosons exchanged between receding masses in an expanding universe, but I don’t think that it is wrong in so far as describing redshift by stretching of wavelengths over the expanded size of the universe, which looks logical to me so was used in the analysis above) to evaluate the energy redshift of gauge boson radiation from our reference frame, 1 + *z = f _{emitted}/f_{observed }= R_{now}/R_{then }*, when I could have alternatively used the relativistic doppler effect which for motion along a radial line of sight which is: 1 +

*z = f*[(1 +

_{emitted}/f_{observed }=*v/c*)/(1 –

*v/c*)]

^{1/2}

*=*[(1 +

*Hr/c*)/(1 –

*Hr/c*)]

^{1/2}. This does change the situation:

Instead of *E _{observed }= E_{emitted} *(1

*– rc*

^{-1}

*H*), the correct redshift factor is

*E _{observed }= E_{emitted} *[(1 +

*Hr/c*)/(1 –

*Hr/c*)]

^{-1/2}

= *E _{emitted}* [(1 –

*Hr/c*)/(1 +

*Hr/c*)]

^{1/2}.

This changes *F = *ò(4π*r*^{2}r )(1* – rc*^{-1}*H*)^{-3}(1* – rc*^{-1}*H*)*H*^{2}*rdr *to:

*F = *ò(4π*r*^{2}r )(1* – rc*^{-1}*H*)^{-3}[(1 – *Hr/c*)/(1 + *Hr/c*)]^{1/2}*H*^{2}*rdr*

*F = *ò(4π*r*^{2}r )(1* – Hr/c*)^{-5/2}(1 + *Hr/c*)]^{-1/2}*H*^{2}*rdr*

I’m suspicious that isn’t going to give a finite result either, and on the basis of all available evidence at present the relativistic doppler shift formula seems more likely (based on SR) to be in error than 1 + *z = f _{emitted}/f_{observed }= R_{now}/R_{then }*. The real error is probably just the omission of a correction factor for another implicit assumption in the analysis above: the analysis assumes 100% of the universe is receding matter at all times, but actually

*at very early times after the big bang, the ratio of mass-to-energy was extremely small, and the universe was radiation dominated.*Such radiation recedes from us at the velocity of light, so it isn’t accelerating away from us, thus it doesn’t have any outward force and can’t send any reaction force to us by gauge boson radiation (Newton’s 3rd law); only receding mass can do that. So the problem will give a finite answer when this is properly incorporated.

So the additional correction factor needed inside the integral is the ratio of matter density to matter plus radiation density as a function of radius *r*. There is a nice clear explanation at the page here which shows that whereas the mass density in an expanding universe is just mass/volume, i.e. it is inversely proportional to the cube of the radius (or to the time, since* t = R/c*) after big bang, i.e. r ~ *t *^{-3}, *the radiation density falls more quickly* because *the average energy per photon diminishes in addition to the divergent spreading, due to the expansion of the universe and the resultant ‘redshift’ and frequency degradation, which reduces the energy per photon* (true regardless whether this is merely stretching of photons over expanding spacetime, or whether it is due to the relativistic doppler effect), so since the mean energy per photon falls inversely with the increasing size of the universe (see discussion above) for the radiation dominated universe r ~ *t *^{–3}* t *^{–1}* ~t *^{–4}*.*

‘The energy densities of radiation and matter are about equal at the temperature of the transparency point, about 3000 K. At much lower temperatures, the energy is dominated by matter.’ So 50% of the mass-energy of the universe was matter when the cosmic background radiation was emitted at 3,000 K, around 300,000 years after the big bang.

This implies that the fraction of the universe’s mass-energy which is present as mass at time *t* is *f = (matter density)/(matter density + energy density) = t* ^{-3 }/ (*t* ^{-3}* + t* ^{-4 }) where time is measured in units of 300,000 years (so that at unit time *f = *0.5).

To express *f = t* ^{-3 }/ (*t* ^{-3}* + t* ^{-4 }) with radial spacetime distance from us *r *as the variable, we employ the definition: *t* = (*H* ^{-1}* – r/c*^{ })/(unit of time, i.e., 300,000 years expressed as seconds is the unit equal to 9.5*10^{12 }seconds) = 1.1*10^{-13 }(*H* ^{-1}* – r/c*^{ }).

Hence, *f = t* ^{-3 }/ (*t* ^{-3}* + t* ^{-4 }) *= *1/(1 + ^{}*t*^{ }^{-1 }) = 1/[1 + {1.1*10^{-13 }(*H* ^{-1}* – r/c*^{ })}^{-1 }].

Including this fraction of term in

*F = ma = mH ^{2}r*

*= *ò(4π*r*^{2}r )(1* – rc*^{-1}*H*)^{-3}(1* – rc*^{-1}*H*)*H*^{2}*r *[1 + {1.1*10^{-13 }(*H* ^{-1}* – r/c*^{ })}^{-1 }]^{-1 }*dr*

*= *4 π r *c*^{2 }ò *r** *[ {*c*/(*Hr*) }* –* 1 ]^{-2 }[1 + {1.1*10^{-13 }(*H* ^{-1}* – r/c*^{ })}^{-1 }]^{-1 }*dr.*

This is a sophisticated model which takes account of all the known physics likely to influence the output and will probably be useful. As you can see from the derivation above, I’m using the expanding spacetime model for the redshift effect, instead of using the relativistic doppler equation.

The assumption that we can use the fraction of the energy of the universe in mass (rather than that which is redshifted in energy) looks superficially like an error, because it looks as if the mass density falls by the inverse cube of time anyway: but we are allowing for conservation of mass-energy by doing so. When the energy density was extremely high at early times, masses would have been smaller because they had lower velocities: mass varies with velocity. This affects the total kinetic energy of matter.

So the *fall in the average energy of radiation* (due to redshift caused by cosmic expansion) is *accompanied by an increase in the effective kinetic energy of matter (and because mass increases with velocity as relativistic velocities are approached, this effect increases the amount of mass in the universe),* due to the increasing velocity gained with distance as observed in spacetime as Hubble’s law *v = Hr.* This suggests that quite apart from pair production creating matter from energy in the first second of the big bang, and nuclear fusion in the first minute, there is also an increase in masses from any given reference frame in spacetime, which is caused by increasing velocities, which (by the relativistic mass increase with increasing velocity) increase mass.

The Hubble law would not be expected to hold if we were not looking back to earlier eras when seeing greater distances: if somehow force fields and light could propagate instantly, the universe would appear very different and it is possible that the Hubble law would then be in error because there would be no delay times and no information on earlier eras of the big bang. However, this is not the case. The Hubble law is empirical, and it’s an empirical fact that we are looking back in time with increasing distance; spacetime.

It appears that the physical mechanism connecting the cosmic expansion (with increasing kinetic energy of matter, and increasing mass as matter speeds up) and the fall in the average radiation energy due to redshift, is that *force-causing gauge boson radiation exchange is causing the cosmic expansion and Hubble law. *

There is a severe mainstream ignorance of all this, caused by the false belief that general relativity (which is excellent on small scales, and works by energy-conservation – see previous two posts – for the bending of light by stars, the procession of the perhelion of Mercury, gravitational redshift of gamma rays moving upwards, etc.) applies to *cosmology* (which it doesn’t, because it doesn’t include vital physics of quantum gravity which become important on large scales, such as redshift and energy degradation of exchange radiation between receding masses in the actual, expanding, universe we live in.

The whole idea that general relativity (see previous posts on this blog) can be applied to the universe as a whole is a farce and a fraud: because gravity becomes negligible on very large scales, the universe is flat on such large scales and only curved on small scales, i.e., near masses. Therefore, the universe is shaped like a simple spherical expanding fireball, not like a ‘boundless’ hyperspace where straight lines curve back on themselves and so eventually return where they began. Spacetime is not boundless like that, because it isn’t curved by gravitation on large scales. The coupling constant for gravity falls with distance between masses on large scales in this universe, because masses are receding and the gauge boson exchange radiation is redshifted, losing energy. Only crackpot ‘mainstream’ morons don’t grasp the underlying physics.

Ignoring these quantum gravity implications, yes, the universe would be curved on large scales and the radius of curvature *R* would be elliptic for uniform positive curvature of 1/*R*^{2}, so straight lines would return to their origin after transversing a path length of π*R,* and the volume of space would be π^{2}*R*^{3} instead of 4π*R*^{3}/3. Nice idea, but wrong by experimental determination which discredits the idea that gravitation is slowing down expansion on cosmological distance scales.

Focussing on the force causing exchange radiation, it doubtless contributes to the Hubble acceleration just as air molecules pumped into a balloon cause it to expand (in the process of the balloon inflating, the air molecules lose some energy as they cause expansion by doing work against the elastic of the balloon, so in some simple ways this analogy holds). Perhaps a better analogy is fireball expansion. It’s quite clear that the ‘cosmological principle’ (nowadays interpreted to mean that Copernicus somehow disproved absolute coordinate systems for the universe, instead of absolutely replacing Ptolemy’s system with the solar system) attributed to Copernicus is an abuse of science, because evidence for the solar system has absolutely nothing to do with the question of whether it is possible or not to give our position within the universe. The fact that the universe is isotropic, i.e. similar in all directions, does not prove that such would be the case everywhere in the universe. Anyone in science who asserts otherwise is a charlatan, because they are asserting a belief as if it had factual evidence. (There are plenty of ‘mainstream’ charlatans.)

What values to use for Hubble parameter *H* and local (time of 13,700 million years) density r? The WMAP satellite in 2003 gave the best available determination: *H = *71 +/- 4 km/s/Mparsec = 2.3*10^{-18} s^{-1}. Hence, if the present age of the universe is *t = 1/H* (as suggested from the 1998 data showing that the universe is expanding as *R ~ t, *i.e. no gravitational retardation, instead of the Friedmann-Robertson-Walker prediction for critical density of *R ~ t*^{2/3 }where the 2/3 power is the effect of curvature/gravity in slowing down the expansion) then the age of the universe is 13,700 +/- 800 million years. As for r, most sources are based on the critical density in the speculative, unphysical and crackpot ‘mainstream’ Lambda-CDM (cold dark matter) model of cosmology which uses a cosmological constant powered by ‘dark energy’ to explain the lack of observable retardation on the recession rates of distant supernovae. The critical density model is completely false because the galaxies aren’t being slowed down due to a *lack of gravity* at extreme distances caused by *redshift and hence energy loss of gravity force causing exchange radiation in quantum gravity, *not by a ‘dark energy’ epicycle popping up to cause just enough repulsion to cancel out gravity over such distances!

Hence, if we want to know the value of r at our present time after the big bang, we should ignore the crackpot ‘mainstream’ estimate of approximately of r = (3/8)*H*^{2}*/* ( π *G*) = 9.5*10^{-27 }kg/m^{3 }and instead work out an estimate of r from observational evidence. The Hubble space telescope was used to estimate the number of galaxies in a small solid area of the sky. Extrapolating this to the whole sky, we find that the universe contains approximately 1.3*10^{11 }galaxies, and to get the density right for our present time after the big bang we use the average mass of a galaxy at the present time to work out the mass of the universe. Taking our Milky Way as the yardstick, it contains about 10^{11 }stars, and assuming that the sun is a typical star, the mass of a star is 1.9889*10^{30 }kg (the sun has 99.86% of the mass of the solar system). Treating the universe as a sphere of uniform density and radius *R = c/H,* with the above mentioned value for *H* we obtain a density for the universe at the present time (~13,700 million years) of about 2.8*10^{-27 }kg/m^{3}. So we have a good estimate of the Hubble parameter and age of universe, and a rough estimate for the density of the universe, based on observations. These data can be used in the model above.