Posted Feb. 28/2020

__Part X__

__The Big Bang: Visible
Matter and DarkMatter
__

__Compressible
Flow and Thermodynamic Equations of State:__

__Exploring the
Possibility that Visible Matter May Have Originated at a Big Bang Embedded __

__Compressible Shock __

__and__

__The Dark Matter at a Subsequent Inflation Embedded Rarefaction
Shock __

__ Summary __** **

**In physical cosmology, the
universe is known to contain about 4.9% visible matter, 27% dark matter and 68%
dark energy. The visible matter is thought to have originated at the Big Bang
explosion some 13.9 billion years ago. **

**Here, we explore the
possibility that the concept of the compressibility of energy and its
associated shock wave forms might supply
a unified physical framework capable of theoretically uniting at least two
of disparate main components of our universe and describing their emergence in a
similar process at or close to the Big
Bang. **

**Visible matter is
presented as originating at an intense compression shock associated with the Big Bang. **

**The origin of the dark
matter and its nature are currently almost
unknown, but here we explore its possible origin in a rarefaction shock developing from and embedded in the super-inflation
that immediately followed the Big Bang. **

** Our approach
is tentative, even sketchy, but seems adequately supported to warrant further
examination. **

** [The dark energy is
considered to constitute a vacuum energy density required to explain the known
various quantum wave fields. It is,
however, thought to have emerged when
the universe was about half its present age, that is at about 7 billion years
before present. It was introduced to explain an observed acceleration of the
rate of expansion of the universe beginning at that time. From this point of
view it would not be associated with the Big Bang, but may be associated with a
Vacuum Energy Density, or with some later cosmological event, or may even eventually
be seen as unnecessary to postulate at all.]**

__Contents__

__1 Introduction__

__2. Compressible Flow, Waves, Shock Waves, Shock
Energy Condensations and Rarefactions__

__3. Evidence for he Origin of Baryonic Matter by Energy
Compressibility in Shock Wave Condensations in Linear Accelerators__

** **

**4.**__ Origin of our Visible
Matter Via the Big Bang at Possible Strong and Weak Compression Shocks, and the
Origin of the ‘Dark Matter’ at the Big Bang Inflation as Strong and Weak Rarefaction Shocks__

**A New Big Bang Proposal**

**Proposed Elliptical Equation of State for Dark Matter**

**How A Physical Compression Shock Model followed by a Rarefaction Shock Model Might Fit into the Standard Model of The Big Bang**

**5. Some Relevant Further Questions to be Addressed**

__6. Conclusions__

__ __

__Appendix A: Outline
of Compressible Fluid Flow__

__References__

** **

__1. Introduction __

In a previous Page [ www.energycompressibility.info
‘ *Part VII: Equations of State for Cosmic Fields’*] we explored the cosmological
application of thermodynamic equations of state. Specifically, we proposed the hyperbolic equation of state [ pv = constant] for our visible matter in the cosmos, the elliptical equation [ p^{2}/b^{2} + v^{2}/a^{2}
= a^{2 }b^{2} \ for the
dark matter, and the linear equation [ p = ± Av ] for stable waves of both types and possibly
also for quantum electromagnetic waves. [ Fig. 1.].

As we shall show, the elementary particles of our visible universe can be described by compression shocks in the hyperbolic state [pv = constant]. The evidence for this is that it fits the Standard Model ‘s data for elementary particles formed in linear accelerators.

We here propose that __the dark matter’s elliptical state, should
produce rarefaction forms, and that this rarefaction may explain both the dark matter’s invisibility
and its non-interaction with other forms of matter, except by way of
gravity.__

The linear equation of state is taken as describing sets of linear waves both of compression and of rarefaction. These may be associated with electromagnetic waves.

__ __Figure 1:
Equations of State for Visible Matter, Dark Matter and Linear Waves

The ‘dark matter’ of the universe constitutes about 28% of the total mass, which is over five times as much as that of our ordinary visible matter of the Hubble cosmos [ 4.9 % ] . Its nature is still almost unknown.. It gravitates, which is why we know it exists, but otherwise it apparently interacts very little, if at all, with the other constituents of the cosmos..

The concept of the ‘dark’ energy is mainly needed for its negative
pressure which would account for the
increased rate of expansion of the universe recently discovered. It acts through gravity and constitutes about 68% of the mass energy of the entire
cosmos. It is a successor to Lambda (Λ), the cosmological constant of
general relativity, and to the vacuum energy density or e quantum vacuum
energy, which are all necessary (1) to counteract the inward pull of gravity
and so prevent a collapse of the cosmos,
and (2) to account for the acceleration in the rate of expansion
observed since the 1990’s.. The dark energy is thus the latest in a succession
of adjustments to the basic standard model pf physical cosmology.

[The dark energy is considered to
constitute a vacuum energy density required to explain the known various
quantum wave fields. It is,
however, thought to have emerged when
the universe was about half its present age, that is at about 7 billion years
before present. It was introduced to explain an observed acceleration of the
rate of expansion of the universe beginning at that time. From this point of
view it would not be associated with the
Big Bang, but may be associated with a Vacuum Energy Density, or with some
later cosmological event.]

We shall take the origin of the Hubbble cosmos as being well described by the Big Bang Theory. In this, the visible or Hubble universe is described as originating from a state of extreme compaction of energy followed by a very short inflationary interval in which , as the universe cooled by the immense expansion, the elementary particles of matter formed. In the far slower expansion during the 13.9 billion years which followed the Big Bang and the brief inflation, the astronomical universe of gas, nebulae, galaxies and stars gradually emerged.

Here we outline
a proposal ** that the dark matter of
the universe emerged in the Big Bang** along with the visible matter, the visible matter emerging
in a compressible energetic flow governed by hyperbolic compression in the
formation of baryons and leptons, while the dark matter is here theorized to
have formed via an intense rarefaction
shock occurring in an elliptical state expansion and rarefaction flow in the
inflationary period following the Big Bang compression.

Many questions arise, some of which we shall
attempt to show can be explained by compressibility concepts. For example : Why does the dark matter not interact with our
visible or ordinary matter, nor with electromagnetic radiated waves? Why do the three states depicted in Fig. 1 at
their common tangent point, not interact
there? Again, can the compression shock proposed as forming the visible
universe’s elementary particles of matter be identified with the Big Bang’s __ intense compression__ state? And can the
proposed rarefaction shock for the formation of the dark matter be identified with the termination of the Big Bang __inflationary period__?

Since our answers will involve waves and shock
waves, we shall briefly outline some of these effects.

__2. Compressible Flow, Waves, Shock Waves, Shock
Energy Condensations and Rarefactions__

There are two possible types of stable pressure waves,
namely __compression __waves and __rarefaction__ waves. If these are of
low amplitude, they are stable and are called ** acoustic type waves**. If they are of finite amplitude they
are unstable, giving us, in the hyperbolic case of our visible matter, compression
waves which grow quickly to form compression shocks. In the elliptical state case
they appear as rarefaction waves, growing quickly to form rarefaction shocks.

This prompts us to ask: What happens if rarefaction waves of one state encounter
compression waves of another state, say elliptical waves ( rarefaction)
encountering hyperbolic (compression) waves) ?
An intuitive answer is that they might act to nullify one another. If so, does this
explain why the dark matter (elliptical and rarefied) does not apparently
interact with our visible world matter (hyperbolic and compressed)? Or is the
answer that they are just too dissimilar to interact ( except through gravity).

First, for clarity, we shall briefly review some pertinent aspects
of compressible flow theory.

** **

__Compressible Flow__

Compressible fluids can
undergo increases in pressure ( compression) or decreases in pressure (
rarefaction or expansion). These pressure changes are accompanied by reciprocal
changes in specific volume ( volume per unit mass).

The thermodynamic variables
of pressure p, specific volume v = 1/ρ,
and temperature T are interrelated, and are given by an ** equation of state** ( e.g. pv = RT, the common hyperbolic gas law
in our visible matter world).

__Waves in Compressible Flow__

Pulses or changes in
flow rate in compressible fluids cause pressure fluctuations of compression or
rarefactions which result in traveling disturbances called pressure waves. These can be of either compression or rarefaction
depending on the physical nature of the compressible fluid. Waves in material
gases of visible matter are always compression while rarefaction waves are damped and die out.

If the wave amplitude
is ‘infinitely’ small, the waves are
stable and are clled ‘acoustic waves’. If the wave amplitude is finite
to begin wit,h then the waves grow rapidly to form shocks or shock waves.

The criterion for wave
behavior is the curvature of the equation of state d^{2}p/dv^{2}
on the pv diagram. If the curvature is positive [ d^{2}p/dv^{2} ›
0] as in the hyperbolic equation of state of material gases then hyperbolic compression waves are non-linear and steepen to shocks
and rarefaction waves die out.

If the curvature is
negative [d^{2}p/dv^{2} ‹ 0], as in the elliptical dark matter, then their rarefaction waves are
non-linear and steepen with time to form
rarefaction shocks while compression waves die out.

[Note: In all cases, if
the pressure disturbances are of infinitesimally low amplitude then stable
waves of either compression or
rarefaction called acoustic waves can
persist without growing to shocks].

In the case where the
curvature is zero [d^{2}p/dv^{2} = 0], the pv curve is a straight line and waves of
both compression or rarefaction, and of any amplitude large or small, are all
stable and can propagate unchanged. Growth to shock waves is not possible in the
linear case.

__ Shock Waves __

With both
hyperbolic and elliptical ( Fig.1)
equations of state, **finite amplitude perturbations** grow to form **shocks**. These are highly concentrated states through which, along
the directions of flow, the pressure p,
tempersture T and specific volume v, change . The magnitude of these changes in these state variables through
the shock discontinuity is directly related to shock strength.

Shocks can be
perpendicular to the direction of
flow ( i.e. ‘normal’ shocks0 or
oblique to the flow ( oblique shocks).

For more detail on
shocks see Appendix A : Outline of Compressible
Flow Physics.

All finite amplitude,
compressive waves are non-linear and grow in amplitude with time to form shock
waves. These shocks are discontinuities in flow, across which the
flow variables p, ρ, V, T and c change abruptly. (Note p = pressure).

_{ }

__ Normal
Shocks __

V_{1} > V_{2}
(8)_{
}

p_{1},
ρ_{1}, T_{1}
< p_{2}, ρ_{2, }T_{2}
(9)
_{ }

_{ }

_{}

__Entropy Change Across Shock:
__

∆S = S_{1} – S_{2} = −
ln(ρ_{02}/ρ_{01})
(10)

__Maximum Condensation Ratio:__

ρ_{1}/ρ_{2} = [n+1]^{1/2}
= V_{max}/c*
(11)

__ Oblique Shocks __

If the discontinuity is
inclined at angle to the direction of the oncoming or upstream flow, the shock
is called **oblique**.

**Oblique Shock **

V_{1N} > V_{2N}

p_{1}ρ_{1}T_{1} < p_{2}ρ_{2}

Since the flow V is
purely relative to the oblique shock front, the shock may be transformed to a
normal one by rotation of the coordinates, and the equations for the normal
shock may then be used instead.

__Equations of State__

These refer to
equations linking the thermodynamic state variables of pressure, p, specific volume (v = 1/ρ) and
temperature T.

For example, we have
the Ideal Gas Law as the hyperbolic
equation of state for visible matter[ pv = const. = RT]; and the
elliptical equation iof state [p^{2}/b^{2}
+ v^{2}/a^{2} = a^{2}b^{2} = const] which we here
propose to describe the dark matter of the cosmos.

__ ____3. Evidence for the Origin of Baryonic Matter by Energy
Compressibility in Shock Wave Condensations__

__We propose that : All elementary particles of matter (with
the possible exception of the neutrino) are condensed energy forms produced under__

The forms are given in
terms of a simple, integral number **n** ( n = degrees of freedom of the compressible
energy flow, which is roughly the number
of ways the energy of the system is divided)..
The experimental values of the ratio of the masses to one another are
then related to the maximum theoretical
compression ratio for each compression shock. ( Eq. 16 below). The observed fit is to within 1%.

__A. Formation of Baryons
and Heavy Mesons) in a Linear Accelerator__

__Maximum Compression Ratio__

**m _{b}/m_{q} = V_{max}/c*
= [n+1]^{1/2 }**

m_{b} is the
mass of any hadron particle, m_{q}
is a quark mass, V_{max} = c_{o} n^{1/2} is the escape
speed to a vacuum; that is, it is the maximum possible relative flow velocity
in an energy flow for a given value of n, the number of degrees of freedom of
the energy form, This is a
non-isentropic relationship which __ corresponds physically to the maximum possible
strong shock. __

.

Experimental
verification values for this hadron
mass- ratio formula are given in Table A
below. (See Appendix B: The Production of Visible Matter at a Strong Condensation
Shock in a Linear Accelerator )

** **

** Table A) Hadrons (Baryons and Heavy Mesons**

**-------------------------------------------------------------------------------------------- **

**n n +1 [n+1] ^{1/2 }Particle Mass (m_{b}) Ratio to**

**
( MeV) quark mass **

**_____________________________________________**

0 1 1 quark (ud) 310 MeV 1

quark
(s) 505

1

2 3 ** 1.73 ** eta
(η) 548.8

3

4

5 6
** 2.45** rho (ρ) 776

6

7

8 9 ** 3** proton
(p) 938.28

neutron
(n) 939.57 __3.03__

Λ (uds) 1115.6 ** 2.97** (2)

Ξ^{o}
(uss) 1314.19 ** 2.99** (3)

9 10
** 3.16** Σ

10 11
** 3.32** Ω

Note: Average quark
mass is 310 MeV; (2) Average quark mass
is (u + d+ s)/3 = 375 MeV (3) Average
quark mass is (u+s+s)/3 = 440 MeV; (4)
Average quark mass is 505 Mev.

__Comparing column three,
the maximum shock compression [n+1] ^{1/2 }], to the final
column “Ratio to quark mass” we see that
they closely agree, so that the proposed
origin of hadrons by strong shock compression theory expressed in Equation 16 is verified.__

__B. Origin of Leptons,
Pion and Kaon__

**m _{L}/m_{e}^{- } = k/α^{2}
= [(n+2)/n]/α^{2} = {(n+2)/n] x 137**
(17)

where α =
1/11.703 = [1/137]^{1/2 }is the
fine structure constant of the atom , and k is the adiabatic exponent or ratio
of specific heats, k = c_{p}/c_{v} = [(n+2)/n].

Because of the presence of k, this equation
for the mass of the leptons is
thermodynamic and quasi-isentropic.

__We propose that the leptons are formed via the weak shock
option( i.e. they involve the reduction in strength of the fine structure
constant [1/137] ^{1/2} __

The experimental
verification for the lepton mass ratio formula of Eqn. 17 is given in Table B
below.

**
Table B) Leptons, Pion and Kaon**

**a**

N k = (n+2)/n Particle Mass Ratio Ratio

(MeV) to x 1/137

Electron

__________________________________________________________

1/3 ** 7** Kaon K

2 ** 2** Pion
π

4 ** 1.5** Muon
μ 105.66 206.77

- - Electron 0.511 1

__Clearly, column 2 values for k ≈ m _{l}/m_{e}
(1/137) closely match column 6 for the
mass ratio reduced by 1/137, thus verifying
Equation 17 and the theory that the leptons are formed by weak shock
condensation. . __

__Summary __

The problem of the
origin of the observed mass-ratios of the elementary particles of matter to one
another has here been explained by the
compressible flow expressions to within about 1% of the experimentally observed
values. This grounds the creation of matter in the strong compressible shock for
the baryons, and in the weak shock option for the electron and leptons.

The principle of the compressibility of energy
flow, therefore, would seem to underlie the emergence of material particles in the visible
universe from some underlying energy
field or continuum such as a modified
general relativity field.

The above data are
those of the standard model of elementary particle formation as verified in
high energy accelerators. Next we propose that this shock compression process
for particle formation in accelerators may fit into the existing Big Bang model as well.

__4. Origin of Visible Matter of the Cosmos Via
the ____Big Bang____ in Strong and Weak ____Compression____ Shocks, and the Origin of the
‘Dark Matter’ Via the Subsequent Inflation ____ ____in ____ ____Strong and Weak ____Rarefaction____ ____Shocks__

__ ____A New Big Bang____ Proposal__

The Big Bang theory for the origin of the material, visible
universe of baryons and leptons is acknowledged to be a very successful explanation.

The nature and origin of other constituents of the cosmos,
such as the dark matter and dark energy, are today still open questions.

**Therefore, we here are going to tentatively
explore the possibility that the addition to the Big Bang explosion process of the
compressible fluid flow, flow waves and shock wave processes, for both compressive and
a succeeding rarefaction, may provide
for the origin of our visible matter and
for the origin of the dark matter as well. **

**r**

In short, if compressible flow process can be integrated
into the Big Bang event, then shock wave condensation process can account for the
visible natter, and so also, we now propose, the subsequent rarefaction flow and rarefaction
shock process could account for the emergence of the dark matter, for its observed ‘dark’ and non-interaction properties and
possibly for its relative abundance aa well

__A. Origin of Visible Matter__

We have shown above how **visible matter**’s elementary particles, baryons and leptons, can be explained
as emerging from the reaction of a liner accelerator by envisaging a __compressible flow’s strong and weak compression
shocks. __

__We have now extended this
approach to the origin of matter in the Big Bang’s intense explosion and the
subsequent inflationary expansion.__

__Maximum Strong Shock Compression Ratio__

**m _{b}/m_{q} = V_{max}/c*
= [n+1]^{1/2 }**

m_{b} is the
mass of any hadron particle, m_{q}
is a quark mass, V_{max} = c_{o} n^{1/2} is the escape
speed to a vacuum; that is, it is the maximum possible relative flow velocity
in an energy flow for a given value of n, the number of degrees of freedom of
the energy form, This is a
non-isentropic relationship which __ corresponds physically to the maximum possible
strong shock. __

If the visible matter equation of state is taken
as the Ideal Gas Law [ pv = constant],
then we note that, the
corresponding degrees of energy , freedom become theoretically infinite [ n =
∞ ]. Thus the maximum condensation
ratio in this fluid would be , not infinite, but, practically speaking, enormous. This enormous condensation ratio
for the strong shock would correspond well to the enormous condensed state
required at the initial instant of the
Big Bang Theory. Thus, shock compressibility
theory and Big Bang theory are in
agreement on this requirement.

__B. Origin of Dark Matter__

Because , as we have shown, the dark’ matter’s elliptical
equation of state requires rarefaction waves,
**its’ particles’ must then be
rarefied energy forms instead of the condensed energy forms of our ordinary
visible matter.**

We now propose that the **dark matter** may have originated via the alternative
strong and weak __rarefaction
shocks____which could have
occurred in the expansion or rarefaction
period following on the Big Bang.
__

That is to say, the visible matter formed first at
compressive shocks associated with the initial high pressure explosive phase of
the Big Bang; subsequently the rarefied,
elliptical dark matter was formed in the
inflationary expansion /rarefaction phase of the Big Bang.

**It would seem that this dark
matter proposal of ‘rarefaction forms’ instead of our visible matter’s ‘condensed
forms’ ,could now account for (1) its
invisibility (2) its weak or
non-existent interaction with visible matter and with electromagnetic radiation.
(3) its relative abundance may be related.**

In addition, it may also provide some insight into the reason for the dark matter’s relative abundance being some 5
times greater than our visible cosmic matter**, namely 27% of the total mass energy of the cosmos versus only 4.9% for our
visible matter. **One may ask questions such as: Q. Is its abundance related to the much larger
volume of space available for any cosmic
event in the expansion after the Big Bang as compared with the tiny volume
assigned to thr actual Big Bang event ‘s highly condensed state.

taking place

__I__

__Proposed
Elliptical Equation of State for Dark Matter__

__ ____How A Physical
Compression Shock Model followed by a Rarefaction
Shock Model Might Fit into the Standard
Model of The Big Bang__

** First, a caveat: O**ur proposal ,which follows, that compressible shock phenomena may be , intimately
involved in the standard Big Bang cosmology,
is necessarily tentative and suggestive only

Its justification appeals to the correlation evidence given
above for shock compressions being involved in elementary particle production
processes in linear accelerators. __We
note that the detailed complexities of
such applications are a matter for the specialists.__

__1. __The ‘super-strong strong shock would fit in at the current
initial maximum density and tempersture Planck epoch. The baryon and lepton matter would emeege or
form there.

**2. The inflationary epoch would
seem to fit in with post strong shock,
‘down stream’, expansion flow.**

**3. A super rarefaction weak shock
episode might then terminate the inflationary expansion epoch. The seeds of the
dark matter ( which we postulate to be rarefaction forms) would emerge or form in this expansion. **

This Big Bang intense compression is immediately followed by an intense
inflationary expansion or rarefaction which could bring about an intensely strong rarefaction shock. ** In such a rarefaction shock, immediately
following the Big Bang condensation, we speculate that the dark matter forms**. [ Since the volume enclosed and the surface
area of the rarefaction shock are larger than the big bang compression sphere,
therefore the amount of dark matter formed by this succeeding rarefaction, one
could speculate, could be greater than
the visible matter formed from the big
bang condensation. Conceivably then this “ big bang” double process of of compression and subsequent rarefaction may helpful
in an explanation for the observed 25%
to/4.9% ratio of dark matter mass to visible
universe matter seen at the present day.

These proposals are clearly suggestive only, and a convincing
account would require the expertise and
cooperation of compressible flow physicists with cosmologists.

__Maximum Compression Ratio__

**m _{b}/m_{q} = V_{max}/c*
= [n+1]^{1/2 }**

m_{b} is the
mass of any hadron particle, m_{q}
is a quark mass, V_{max} = c_{o} n^{1/2} is the escape
speed to a vacuum; that is, it is the maximum possible relative flow velocity
in an energy flow for a given value of n, the number of degrees of freedom of
the energy form, This is a
non-isentropic relationship which __ corresponds physically to the maximum possible
strong shock. __

If the visible matter equation of state is taken
as the Ideal Gas Law [ pv = constant], then we note that, the corresponding degrees of energy , becomes
theoretically infinite [ n = ∞ ].
Thus the maximum condensation ratio in this fluid would be, certainly not infinite
but, practically speaking, speaking, certainly enormous. This enormous
condensation ratio for the strong shock would correspond well the enormous condensed state required at the
initial instant of the Big Bang Theory. Thus, shock theory and Big Bang theory are in agreement on
this requirement.

** **

__5. Some Specific
Further Relevant Questions to be Addressed__

1. Does the proposed
compressive shock fit time- wise before,
or in, or after the Big Bang? Can the
new proposal solve this ?

2. At what stage after
the Big Bang would the rarefaction shock and dark matter formation best fit?

3. Can we now better
explain the 5 to 1 rstio of dark matter
to visible matter, e.g. 27%/4.9%.

4. Does the dark energy fit in here at all? It is usually invoked to explain an observed increased rate of expansion of the
cosmos about

half way in time between the Big Bang and
the present time which is to say around 7 billion years age. Currently an
energy density amounting to around 65%
of the total is required [ 65%27%4.9%

5. Compressible flow
theory requires a flow of “something” in which waves and shocks form. Is this ‘something’ a ‘vacuum energy density’
?

**6 . Conclusions**

We have proposed that imposition of compressible flow theory into the Big Bang hypothesis produced many
points of explanation of current cosmic cosmological mysteries, for example:

The immense original matter
energy density of the Big Bang is in
agreement with a hyperbolic ideal fluid
strong shock

compression.

The emergence of the elementary
particles of matter of quantum theory is compatible with the same strong shock process.

The dark matter can b envisaged
as an elliptical rarefaction wave flow occurring in conjunction with the period
of hyperinflation following the Big
Bang. In addition ‘elementary particles
of dark matter can be predicted to form
from a rarefaction shock process occurring
in this standard hyperinflationary episode.

Because compressible theory predicts dark matter would be rarefaction in nature, this this would
intuitively also be expected not to
interact with compressed visible matter
and radiation. This non-interaction is
what is observed and which at present cannot be explained.. Explaining this mystery
would be
string point in favor of the new compressible approach.

The observed relative amounts of visible (4.9%) and dark matter (28 %) can
possibly also now be explainable.

. The baryons and leptons of visible matter can be related to condensation
shocks in a compressible flow.

First, this suggests that the
origin of visible matter in the Big Bang condensation nay have involved the
shock compression process.

Second, if the dark matter has an
elliptical equation of state, then its forms will be
rarefied and would form rarefaction
shocks. This further suggests that the
rarefied dark matter may have firmed in such rarefaction shocks set off
by the sudden cosmic __inflationary stage__ which immediately followed the
Big Bang condensation.

The lack of interaction of the dark
matter with visible matter may be due to the fact that it is a rarefied form,
whereas visible matter is made up of condensed forms.

The invisibility of the dark matter may be explained in a similar way by
noting that rarefied matter may not interact with compressible leptons such as
electrons and photons.

Dark matter forms and rarefied
waves should interact with each other.

The degree of condensation or
compression of the Big Bang event was enormous.
This is matched by compressible theory in the case of the Ideal Equation
of State [pv = const.] where the maximum condensation of its str0ong shock is ** infinite**. We have,

Maximum Condensation ratio is: **m _{b}/m_{q}
= [n+1]^{1/2}**

^{ }

^{ }And so, in
pv = const we have k = 1 and therefore n, the energy division becomes
infinite.[ n = 2 / (k-1 = 2 / 0 =
∞ ] Therefore, the maximum condensation
ratio also becomes infinite.

Thus, the Big Bang’s extreme condensation state is compatible with
compressible theory for the case of The Ideal
Equation of State. This agreement seems quite remarkable, and it
provides powerful support for the compressible theory’s application to Big Bang
theory.^{
}

__References__

1. Power, Bernard A., Unification of Forces and
Particle Production at an Oblique Radiation Shock Front. *Contr. Paper N0. 462. American Association for the Advancement of Science, Annual
Meeting, *

2. .---------------, Baryon Mass-ratios and
Degrees of Freedom in a Compressible Radiation Flow. *Contr.
Paper No. 505. American Association for the Advancement of Science, Annual
Meeting, *

3.
.---------------, Summary of a Universal Physics. Monograph (Private
distribution) pp 92. Tempress, __Appendix B: Summary of a
Universal Physics”)__

**4. **. Shapiro, A. H. *The Dynamics and Thermodynamics of
Compressible Fluid Flow. * 2 vols.
John Wiley and Sons,

5.
Courant, R. and Friedrichs, K. O. (1948). Supersonic Flow and Shock
Waves. Interscience,

6. Lamb,
Horace., *Hydrodynamics *6^{th} ed.

7. Chaplygin,
S., Sci. Mem.

8. Tsien, H.
S. *Two-Dimensional Subsonic Flow of Compressible Fluids,* J. Aero. Sci. Vol. 6, No.10 (Aug., 1939), p.
399.

9. Bachall,
N.A., Ostriker, J.P., Perlmutter, S., and P.J. Steinhardt. The Cosmic Triangle:
Revealing the State of the Universe. *Science*,
**284,** 1481 1999.

10.
Kamenshchick, A, Moschella, U., and V. Pasquier. An alternative to
quintessence. *Phys. Lett. B ***511**, 265, 2001.

11. Bilic, N.,
Tupper, G.B., and R.D. Violier. Unification of Dark Matter and Dark Energy: The
Inhomogeneous Chaplygin Gas. *Astrophysics
, astro-**ph/0111325*. 2002

** **

__Copyright, Bernard A. Power, Feb. 2020 __

__Back to Top__

__Appendix A__

__2.0 Outline of Compressible
Fluid Flow__

The following is a
listing of some relevant basic compressible flow principles. For more complete treatment see texts on compressible fluid flow or
gas dynamics.[3,
4,5,6.7.8].

__2.1 Steady State Energy Equation __

c^{2} = c_{o}^{2} −
V^{2}/n
(1)

where c is the local
wave speed, c_{o} is the static [i.e at V = 0] or maximum wave speed, V
is the relative flow speed, n is the number of ways the energy of the flow
system is divided (i. e. the number of
degrees of freedom) of the system and [
n = 2/(k− 1), where k = c_{p}/c_{v}
_{ }is the’ adiabatic constant’ or ratio of specific heats.]

Here, ‘relative’ means
referred to any (arbitrarily) chosen physical flow boundary. The equation is for unit mass, that
is, it pertains to ‘specific energy’ flow.

The case where V = c = c* is called the critical state. The ratio (V/c)
is the Mach number M of the flow. The
ratio (V/c_{o}) is also a
quantum state variable. The maximum flow velocity V_{max} (when c =
0) is the escape speed to a vacuu:

V_{max }= √_{ }n c_{o}.

_{ }

__2.2 Unsteady State Energy Equation __

c^{2} = c_{o}^{2} – V^{2}/n
– 2/n (dφ/dt)
(2a)_{}

_{ }

where φ is a velocity potential, and dφ/dx = u is a
perturbation of the relative
velocity. Therefore, in three
dimensions, substituting V for u, we have dφ/dt = V (dx/dt) = Vc, and

c^{2} = c_{o}^{2} – V^{2 }/n − 2 (cV)/n
(2b)

__(See also sect. 3.2 below
for far-reaching implications of the 2cV interaction energy term in
quantum state physics).__

__2.3 Lagrangian Energy Function L __

L = (Kinetic energy) – (Potential energy) = ( c^{2}+
V^{2}/n) – c_{o}^{2} , and so, from (2b)

L = − (2cV)/n
(3)

**2. 4 Equation of State
for Compressible, Ordinary Matter
Systems **

The equations of state link the thermodynamic quantities of pressure
p, specific volume ( volume per unit mass v = 1/ρ] and temperature T. The basic
equation of state **for ordinary gases**
is the equilateral hyperbola of the Ideal Gas Law:

**pv = RT ; p/ρ = RT = constant**
(4)

Equation 4 is seen to
be isothermal (T= constant) . For **adiabatic changes** it becomes **pv _{}^{k} = constant**,
where the adiabatic constant k = c

Here, each point on the
curve presents the values for a particular
pressure and volume pair and
shows how the two relate to each other when one or the other is changed. In this hyperbolic equation, the product of
the two -- i.e. **the pv- energy** -- has a constant value as set out by the
equation of state.

Hyperbolic Equation of State (Ideal Gas Law)

Equations of State can be formed for gases, liquids or solids.
Here, we shall be concerned mainly with those for the highly compressible
states i.e. for gases.

__2.5 Waves and Flow__

__2.5.1 The Classical
Wave Equation __

∆^{2}ψ = 1/c^{2 }[∂^{2}ψ/∂t^{2}]^{
}_{ }^{ }(5)

where ∆^{2}
= ∂^{2}../∂x^{2 } + ∂^{2}..^{.}/∂y^{2}
+ ∂^{2}../∂z^{2};
**ψ** is the wave amplitude, that is, it is the
amplitude of a thermodynamic state variable such as the pressure p or the
density ρ. The local wave speed is
c.

The general solution of
(5) is

**ψ** = φ_{1} (x – ct) + φ_{2 }( x+
ct)
(6)

Equation 6 is a
linear, approximate equation for the case of low-amplitude waves
in which all small terms (squares, products of differentials, etc.) have been
dropped.

The natural graphical
representation of steady state compressible flows and their waves is on the
(x,t), or **space-time diagram. **

The classical wave
equation corresponds to isentropic conditions. It represents a stable,
low-amplitude wave disturbance, such as an acoustic–type wave.

__Unsteady State ( Accelerating) Compressible Flow __

In compressible flow
theory, **forces**, when present, introduce a curvature of the characteristic
lines for velocity on the space-time or xt-diagram. Space- time curvature thus
indicates velocity acceleration and the presence of force .

**(a) Straight characteristics and path lines show steady
flow and absence of force.**

**(b) Curved characteristics and path lines show acceleration
and presence of force**

In the case of
compressible flow and 3-space (x,y,z) ,
a curved path line dx/dt = v (path) may be ‘transformed away’ to a
straight line Lagrange representation [ dh/dt = 0].

[Note: In General Relativity the analogous
distortion of its 4-space (x,y,z,t) to obtain a force- free representation is a
**tensor distortion. **

However, it should be
noted that general relativity is a continuous field theory, and, as such,
excludes discontinuities or singularities such as shocks. Therefore, it appears
to be fundamentally incompatible with quantum physics.

On the other hand,
compressible flow as shown below in Section 3.1 on Visible Matter predicts
shock discontinuities as the physical mechanism for the emergence of the
elementary particles of matter by shock compression of an energy flow. Thus, compressible flow is compatible with
quantum theory whereas general relativity is not.]

__2.5.2 The Exact Wave Equation __

Ń^{2 }ψ^{ } =^{
}1/c^{2} ∂^{2}ψ/∂t^{2} [ 1 + Ńψ ]^{(k +
1)
}(7)

where k, the adiabatic
exponent, is k = c_{p}/c_{v} = ( n + 2) /n; and ( k + 1 ) = 2(
n + 1)/n = 2(c_{o}/c*)^{2}. Here, pressure is a function of density
only. This wave is isentropic, **non-linear, unstable, and grows to a
non-isentropic discontinuity called a shock wave.**

__2.5.3 Shock Waves __

All finite amplitude,
compressive waves are non-linear and grow in amplitude with time to form shock
waves. These shocks are discontinuities in flow, across which the
flow variables p, ρ, V, T and c change abruptly. (Note p = pressure,
ρ = momentum).

_{ }

__2.5.3.1 Normal
Shocks __

V_{1} > V_{2}
(8)_{
}

p_{1}, ρ_{1},
T_{1} < p_{2}, ρ_{2, }T_{2}
(9)
_{ }

_{ }

_{ }

__Entropy Change Across
Shock: __

∆S = S_{1} – S_{2} = −
ln(ρ_{02}/ρ_{01})
(10)

__Maximum Condensation Ratio:__

ρ_{1}/ρ_{2}
= [n+1]^{1/2} = V_{max}/c*
(11)

__2.5.3.2 Oblique Shocks __

If the discontinuity is
inclined at angle to the direction of the oncoming or upstream flow, the shock
is called **oblique**.

Oblique Shock

V_{1N} > V_{2N}

p_{1}ρ_{1}T_{1} < p_{2}ρ_{2}

Since the flow V is
purely relative to the oblique shock front, the shock may be transformed to a
normal one by rotation of the coordinates, and the equations for the normal
shock may then be used instead.

__2.5.3.3 Strong and
Weak Oblique Shock Options: The Shock Polar __

For each inlet Mach
number M_{1} ( = V_{N1}/c), and turning angle of the flow
θ, there are **two physical options**:

1) the strong shock (
intersection S) with strong compression ratio and large flow velocity reduction (p_{2 }>> p_{1}; V_{2} << V_{1}, or

2) the weak shock
(intersection W, with small pressure rise and small velocity reduction.

Which of the two
options occurs depends on the boundary conditions: low back i.e. low downstream pressure favours the weak shock
occurrence; high downstream pressure favours the strong shock.

When the turning angle
θ of the oncoming flow is zero, the strong shock becomes the normal or
maximum strong shock, and the weak shock becomes an infinitesimal, low-amplitude,
acoustic wave.

__2.6 Types of Compressible Flow: __

a) Steady, subcritical
flow ( e.g. subsonic, V< c), governed by elliptic, non-linear, partial
differential equations.

b) Steady,
supercritical flow ( e.g. supersonic, V_{1}
> c) governed by hyperbolic, nonlinear, partial differential equations.

c) Unsteady flow (either subcritical or supercritical). These
are wave equations governed by hyperbolic, non-linear, partial differential
equations. They are often simplified to linear approximations, for example to
the classical wave equation (5); if of finite amplitude they grow to shocks..

The solutions to the
above hyperbolic equations are called characteristic solutions. If linear, they correspond to the
eigenfunctions and eigenvalues of the linear solutions to the various wave
equations of quantum mechanics ( Sect. 3.7), or, equally, to the diagonal
solutions of the matrix equation of Heisenberg’s formulation of quantum
mechanics.

__2.7 Wave Speeds __

c = [ c_{o}^{2} – V^{2}/n]^{1/2 } (steady flow)
(12)

c = c_{o}^{2} – V^{2}/n – [(2/n)cV
]^{1/2 }(unsteady flow) (13)

c^{2} = (dp/dρ)_{s}, where s is an
isentropic state.

Since V is relative, it
may be arbitrarily set to zero to give a stationary or “local” coordinate system
moving with the flow; this automatically puts c = c_{o} and transforms
the variable wave speed to any other relatively moving coordinate system.

The shock speed U is
always supercritical (U > c) with respect to the upstream or oncoming flow V_{1}.

__2.8 Wave Speed Ratio c/c _{o} and The
Isentropic Thermodynamic Ratios__

c/c_{o} = [**1
-1/n(V/c _{o})^{2}]^{1/2}**

__All the basic thermodynamic
parameters of a compressible isentropic flow are therefore specified by
the wave speed ratio c/c _{o}. . __

** 2.9 Relativity Effects in Compressible Flow: **In compressible flows all velocities [V or u] are relative only,
and, moreover, the wave speed c is a variable which is dependent on V and n; it decreases for larger
velocities V. and it reaches its maximum
value c

Interestingly, Equation (14)

c/c_{o} = [**1
-1/n(V/c _{o})^{2}]^{1/2}**

shows that the correction factor for the effect of flow
speed on wave speed c on the right hand side of the equation ** has the same form as the Lorentz Transformation of special
relativity**. If n = 1 the two correction factors become formally
identical.

The differences from special
relativity are that the wave speed **c** is now a variable and a function of
the flow velocity **V**, and that there is the energy partition constant **n**. Since the wave speeds are low ( c =
334 m/s for air at m.s.l.), the
‘Lorentz’ corrections for physical
compressible systems such as gases are
relatively large. Also, the flow speeds can exceed the wave speed (
supersonic flow), whereas in special relativity theory, the
wave speed c is a constant ( 3 x 10^{8}
m/s) which can never be exceeded.

Photon shocks are thus impossible in special relativity, whereas in
compressible flow they furnish a quantum physical theory for the origins of
matter itself via the formation of the
elementary particles of matter at
compression shock discontinuities.

__2.10 Wave
Stability __

Compression waves are
the rule in the baryonic physical world ( i..e. in Quadrant I on the
pressure-volume diagram) where density
waves are always compressive and all compression
waves of finite amplitude grow towards shocks.
Here, only **acoustic**
compression waves ( i.e. infinitely low-amplitude compressions) are stable.

Finite rarefaction
waves and rarefaction shocks are impossible in material gases; only infinitely low-amplitude rarefaction
waves can persist.

We shall see below
that, with elliptical equations of state ( dark matter) and linear equations of
state ( quantum radiation), stable, finite rarefaction waves do become
possible.

__2.11 Elliptical
Equations of State and Rarefaction Waves __

**II**

+p -v

+v

-p

__Elliptical
Equation of State favours Rarefaction Waves and Shocks__

**The
criterion for wave behaviour [4]** is the curvature ( dp/dv) of the isentropic equation of state:

1.
If d^{2}p/dv^{2} is > 0 (i.e. the hyperbolic curve) compression waves form and steepen, while rarefaction waves flatten
and die out. Only compression waves of
infinitely small amplitude (“acoustic” or “sound” waves” ) are stable.

2.
If d^{2}p/dv^{2} is <
0 , ( i.e. the elliptical curve) rarefaction waves form and steepen, while compression waves flatten and die out.

.

We therefore see that ordinary gases are hyperbolic and favour
compression waves and compression shocks. No real gas is known whose equation of state is elliptical and favours
rarefaction waves; ** but we are now proposing that it is this elliptical relationship
which furnishes the Equation of State
for the Dark Matter of the
universe. (** Section 3.3 below).

** We
are also proposing that the third option , namely the linear equation of state, fits the Quantum state and the Electromagnetic
field. **(
Section 3.2 below).

__The Origin of
Visible Matter at a Strong Compressive Shock in a LinearAccelerator__

**3.1
Ordinary Visible Matter**

Ordinary matter
consists of elementary particles held together as atoms and molecules by
electromagnetic forces. Atomic and
molecular matter can exist in three states as gas, liquid and solid. Gases are
highly compressible and all the laws of compressible flow apply to them.
Liquids and solids range from slightly compressible and distortional to completely
incompressible. All three physical
states can support waves. .

First, we examine the
question of ** What is Matter?** Here we shall show that

__ A Theory of the
Origin of Baryonic Matter: Energy Compressibility in Shock Wave Condensations__

__We propose that : All elementary particles of matter (with
the possible exception of the neutrino) are condensed energy forms produced __

The forms are given in
terms of a simple, integral number **n** ( n = degrees of freedom of the compressible
energy flow, which is roughly the number
of ways the energy of the system is divided)..
The experimental values of the ratio of the masses to one another are
then related to the maximum theoretical
compression ratio for each compression shock. ( Eq. 16 below). The observed fit is to within 1%.

__A. Origin of Hadrons (Baryons and Heavy Mesons)__

__Maximum Compression Ratio__

**m _{b}/m_{q} = V_{max}/c*
= [n+1]^{1/2 }**

m_{b} is the
mass of any hadron particle, m_{q}
is a quark mass, V_{max} = c_{o} n^{1/2} is the escape
speed to a vacuum; that is, it is the maximum possible relative flow velocity
in an energy flow for a given value of n, the number of degrees of freedom of
the energy form, This is a
non-isentropic relationship which __ corresponds physically to the maximum possible
strong shock. .__

Experimental
verification values this hadron mass-
ratio formula is given in Table A below.

**
**

**
Table A) Hadrons (Baryons and
Heavy Mesons)**

**-------------------------------------------------------------------------------------------- **

**n n +1 [n+1] ^{1/2 }Particle Mass (m_{b}) Ratio to**

**
( MeV) quark mass **

**_____________________________________________**

0 1 1 quark (ud) 310 MeV 1

quark
(s) 505

1

2 3 ** 1.73 ** eta
(η) 548.8

3

4

5 6 ** 2.45** rho
(ρ) 776

6

7

8 9 ** 3** proton
(p) 938.28

neutron
(n) 939.57 __3.03__

Λ (uds) 1115.6 ** 2.97** (2)

Ξ^{o}
(uss) 1314.19 ** 2.99** (3)

9 10
** 3.16** Σ

10 11
** 3.32** Ω

Note: Average quark
mass is 310 MeV; (2) Average quark mass
is (u + d+ s)/3 = 375 MeV (3) Average
quark mass is (u+s+s)/3 = 440 MeV; (4)
Average quark mass is 505 Mev.

__Comparing column three,
the maximum shock compression [n+1] ^{1/2 }], to the final
column “Ratio to quark mass” we see that
they closely agree, so that the proposed
origin of hadrons by strong shock compression theory expressed in Equation 16 is verified.__

__B. Origin of Leptons,
Pion and Kaon__

**m _{L}/m_{e}^{- } =
k/α^{2} = [(n+2)/n]/α^{2} = {(n+2)/n] x 137**
(17)

where α =
1/11.703 = [1/137]^{1/2 }is the
fine structure constant of the atom , and k is the adiabatic exponent or ratio
of specific heats, k = c_{p}/c_{v} = [(n+2)/n].

Because of the presence of k, this equation
for the mass of the leptons is
thermodynamic and quasi-isentropic.

__We propose that the leptons are formed via the weak shock
option( i.e. they involve the reduction in strength of the fine structure
constant [1/137] ^{1/2} __

The experimental
verification for the lepton mass ratio formula of Eqn. 17 is given in Table B
below.

**
Table B) Leptons, Pion and Kaon**

**a**

N k = (n+2)/n Particle Mass Ratio Ratio

(MeV) to x 1/137

Electron

__________________________________________________________

1/3 ** 7** Kaon K

2 ** 2** Pion
π

4 ** 1.5** Muon
μ 105.66 206.77

- - Electron 0.511 1

__Clearly, column 2 values for k ≈ m _{l}/m_{e}
(1/137) closely match column 6 for the
mass ratio reduced by 1/137, thus verifying
Equation 17 and the theory that the leptons are formed by weak shock
condensation. . __

__Summary __

The problem of the
origin of the observed mass-ratios of the elementary particles of matter to one
another has here been explained by the
compressible flow expressions to within about 1% of the experimentally observed
values. This grounds the creation of matter in either the strong compressible shock
for the baryons, or in the weak shock option for the electron and leptons.

. The principle of the
compressibility of energy flow, therefore, would seem to underlie all material
particles and the whole material universe.

__Copyright, Bernard A. Power, June 2019 __

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