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COSMOLOGY

By Charles Rhodes, P.Eng., Ph.D.

PURPOSE:
This web page compares conflicting data from different astronomical measurements. This data prompts us to question our understanding of physical reality on the scale of inter-galactic distances.
 

TWO OBSERVERS:
Consider two observers, one at position "a" and one at position "b" where positions "a" and "b" are separated by many light years.

Both observers use an atom such as cesium with a well defined energy transition in a field free vacuum as a standard of time. This energy transition has a corresponding frequency F due to the relationship:
F = DeltaE / h
where h is the Planck constant.

For observer "a" an element of elapsed time is:
dT = (dN / Fa) where dN is a number of cycles of frequency Fa (typically 1).

Observer "a" assumes that the speed of light C is everywhere constant and uses it to convert the standard unit of time into a standard unit of distance.

Then for observer "a" an element of distance dX is:
dX = C dT = C (dN / Fa)

For observer "b" an element of elapsed time is:
dT = (dN / Fb) where dN is a number of cycles of frequency Fb (typically 1).

Observer "b" assumes that the speed of light C is everywhere constant and uses it to convert the standard unit of time into a standard unit of distance.

Then for observer "b" an element of distance dX is:
dX = C (dN / Fb)

Now assume that someone travels in a spaceship from point "a" to point "b". This space traveller uses the same cesium clock apparatus to determine how far he has travelled. To the space traveller the distance D is given by:
D = Integral from a to b of dX
= Integral from a to b of [C dN / F(N)]
where we allow for F to change along the spaceship path. Note that N is the number of cycles (or wave fronts) transversed along the space ship path from point "a" to the space ship location. Thus the change in F along the space ship path is:
Integral from a to b of dF
= (Fb - Fa).

This change in F is necessary because the separation distance between points "a" and "b" may change while the spaceship is travelling. Also energy wells at points "a" and "b" may change while the space ship is travelling causing acceleration and/or deceleration along the transit path.

The elapsed time in transit experienced by the space traveller is:
(Tb - Ta) = Integral from Ta to Tb of dT
= Integral from Na to Nb of [dN / F(N)].

Today astronomers are able to accurately measure Fb / Fa, where Fb is the frequency of a photon received from a distant star.

However, there are several elements of potential confusion in determination of:
(Fb / Fa),
one is the doppler effect, the second is special relativity and the third is general relativity. There might also be another yet to be discovered element of confusion.

DOPPLER EFFECT:
Usually:
(Fb / Fa) < 1
and is known as a red shift.

For a some stars:
(Fb / Fa) > 1 and is known as a blue shift.

For a century this spectral shifting has been primarily attributed to the doppler effect which is due to a differential velocity V of point "b" with respect to point "a" along the path between point "a" and point "b".

If points "a" and "b" are at a constant separation:
Fb = Fa = C / Lamda
where:
C = speed of light
and
Lamda = wavelength

However if the separation between points "a" and "b" is increasing at speed V then the apparent frequency at point "b" is:
Fb = (C - V) / Lamda
= (C - V) / (C / Fa)
= Fa [1 - (V / C)]
where C = speed of light and V can be either positive or negative corresponding to a increasing or decreasing distance between point "a" and point "b".
 

SPECIAL RELATIVITY:
There is a second correction factor due to special relativity which affects the perception of time. The correct expression taking special relativity into account is:
Fb = Fa [1 - (U / C)^2]^0.5 (1 - (V / C))
where U is the velocity of point "a" in the frame of reference of point "b". Special relativity assumes that neither point "a" nor point "b" are accelerating.

For the special case of U = V:
Fb = Fa [1 - (V / C)^2]^0.5 [(1 - (V / C))^2]^0.5
= Fa [(1 - (V / C)) (1 + (V / C))]^0.5 [(1 - (V / C))^2]^0.5
= Fa {(1 - (V / C))^3 (1 + (V / C))}^0.5

For (V / C) << 1 this expression simplifies to:
Fb = Fa [1 - (V / C)]
 

GENERAL RELATIVITY:
General Relativity can cause the fraction (Fb / Fa) to decrease if point "a" is located deeper in a gravitational potential energy well with respect to point "b". Such a potential energy well can exist close to a gravitational black hole.
 

THE PROBLEM:
Circa 1970 a US astronomer named Vera Rubin discovered a major problem with spiral galaxies. They seem to defy the laws of physics. The issue can be described as follows:

The doppler effect indicates that the tangential velocity V of stars on the radial arms of a spiral galaxy is almost constant from near the galaxy hub out to the tips of the radial arms where they cease to be visible.

A spiral galaxy can be modelled as a spoked (armed) wheel with a hub radius Ro, a hub thickness To and a hub mass per unit volume Rhoo. Then the hub mass Mo is given by:
Mo = Rhoo Pi Ro^2 To

A star of mass Ms at radius Ro from the galactic center held in place by gravity should satisfy the Newtonian force balance equation:
G Mo Ms / Ro^2 = Ms V^2 / Ro
or
G Mo / Ro = V^2

Let Mr be the galactic mass inside radius R. Note that along the galactic arms Mr increases with radius R. Then for any peripheral arm star where R > Ro:
G Mr / R = V^2
or
dMr / dR = [V^2 / G]

Thus along the galactic arms where V = constant:
dMr / dR = [V^2 / G] = constant.

Along the galactic arms:
dMr = [Rhoo 2 Pi R Fr Tr dR]
where:
Fr = (R / Ro) due to increasing space between the radial arms with increasing R;
Tr = decreases with increasing R.

Thus:
dMr = Rhoo 2 Pi R Fr Tr dR
= Rhoo 2 Pi R (R / Ro) Tr dR
= Rhoo 2 Pi Ro (R / Ro)^2 Tr dR

Try Tr = To (Ro / R)^2

Then along the galactic arms:
dMr / dR = Rhoo 2 Pi Ro To
which keeps (dMr / dR) constant along the galactic arms.

Recall that: dMr / dR = V^2 / G

Hence equating the two expressions for (dMr / dR) gives:
V^2 / G = 2 Rhoo Pi Ro To
= (Rhoo Pi Ro^2 To)(2 / Ro)
= 2 Mo / Ro
or
V^2 = 2 G Mo / Ro
or
V = [2 G Mo / Ro]^0.5

In astronomical observations: Ro = D Theta
where:
D = distance from Earth to the remote galaxy:
and
Theta = angle subtended by Ro as measured on Earth.

Thus:
V = [2 G Mo / D Theta]^0.5

For remote galaxies V is calculated from the observed red and blue doppler shifts, Mo is calculated from the observed amount of luminous matter in the galactic hub and the distance to galaxy D is obtained from "standard candle" distance calculations.
 

STANDARD CANDLE METHOD:
Let:
Is = solar irradiance
= sun's incident light intensity on Earth

I = observed light intensity from galaxy hub

N = number of stars in galaxy hub

Ds = Earth-sun distance

Ms = solar mass

Then:
I / Is = N Ds^2 / D^2
or
N = [D^2 / Ds^2] [I / Is]
and
Mo = N Ms
= [D^2 / Ds^2] [I / Is] Ms

Thus:
V = [2 G Mo / D Theta]^0.5
= {2 G [D^2 / Ds^2] [I / Is] Ms / D Theta}^0.5
= [1 / Ds] [2 G Ms / Is]^0.5 [(D I) / Theta]^0.5

The problem is that the measured [(D I)/ Theta] values are consistently two orders of magnitude smaller than the corresponding measured V values. If one believes gravitational theory there is a systematic problem with the measured [(D I) / Theta] values being too small and hence not indicating the full extent of the galactic mass. The present presumed explanation for this effect is that there is a lot of non-luminous galactic mass (dark matter) present in addition to the luminous mass.

Note that if there is photon scattering along the photon path through space the measured value of I will be too small but the corresponding value of D obtained by the standard candle method will be too large to the same extent, so photon scattering error in the (D I) product should cancel.
 

DIFFICULTIES:
The first difficulty is that when V is calculated one finds that V is too large with repect to the amount of observed luminous matter in the galaxies. It is presumed that the spiral galaxy arms are held in place by gravity. However, there is no visible or otherwise readily detectable source of the required amount of gravity. Scientists have tried very hard to find or identify the missing galactic matter without success. This missing non-luminous matter is referred to as "dark matter".

The second difficulty is that based on the doppler shift measurements and the best estimates of galactic distance it appears that other galaxies are receding from our own at an ever increasing rate. There is no physical explanation for this phenomena. This behaviour is explained by a theoretically permitted constant within the mathematical structure of General Relativity, but even Einstein did not believe that a non-zero value of this constant corresponded to reality. This constant corresponds to "Dark Energy" in a vacuum which forms positive rather than normal negative gravitational field energy.

The third difficulty is that there do not appear to be sufficient black holes or other large galactic mass sources that might explain the large observed red and blue shifts along the galactic radial arms.

The fourth difficulty lies in explaining how the universe came to exist in the first place. The explanation consistent with observed data suggests that the universe came into being at a point singularity and almost instantaneously expanded to a form close to its present geometry. There are further unexplained issues relating to the imbalance between matter and anti-matter.

There is no deep understanding of any of the aforementioned mechanisms, or even any assurance that they are not fictions concealing a fundamental lack of physical knowledge and understanding.

At the large hydron collider there has been an extensive search for particles that might contribute to dark matter, with no success.

An issue that needs further investigation is whether there might be something about the optics of inter-galactic space which causes Theta to grow over the photon path length. Perhaps when photons emerge from the gravitational potential well of a remote galaxy Theta is significantly magnified. If this is the case the apparent linear dimensions of many galaxies will be too large.

A fruitful line of investigation might be to model stable galaxy sizes from first principles. If calculated stable galaxy linear dimenstions are substantially smaller than apparent observed galaxy dimensions then the source of "dark matter" might simply be magnification of Theta by the galaxy gravitational potential energy well.

The gravitational potential energy well will act like a spherical graded index lens. The galaxy hub is at the center of this lens so it will appear enlarged to an external observer. If this enlargement is sufficient to account for the observed data the entire issue of "Dark Matter" will disappear.
 

This web page last updated July 26, 2018.

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