__COSMOLOGY OF A BINARY UNIVERSE__

__Part 1:__

__ The Origin, Properties and Thermodynamic
Evolution of a Universal Wave and Force Field__

__Contents of Part 1__

1. Introduction

2. The current cosmology of
general relativity

3. Properties of a Universal Wave
and Force Field (UF)

3.1. Dynamics and
Thermodynamics

3.2. Longitudinal
and transverse waves

3.3. Gravitational force

4. Cosmological origin, nature and thermodynamic evolution of the UF during
an interrupted Carnot Cycle

4.1. Initial postulate
of the existence of a cosmic fluid

4.2. Cavitation or rupture of the cosmic fluid

4.3. Isothermal expansion of vapor filled cavity to
critical size

4.4. Adiabatic
collapse and compression of the cosmic vapour to form baryonic matter

4.5. The Big Bang

4.6. Adiabatic
expansion through cosmic time after the Big Bang during an interrupted Carnot
Cycle

4.7. Critical
conjunction point C*

4.8. Current
accelerated isothermal expansion

4.9. Summary of Part 1

References

Appendix A: Thermodynamic Properties of an Isothermal Gas
Law for the Chaplygin/Tangent Field

Appendix
B: A New Explanation for the
Anomalously Weak Tensile Strengths of Liquids and Solids

Appendix C: Physical Derivation of the Schrödinger
Equation of Quantum Mechanics from the Concept of a Universal Wave Field

Appendix D: Philosophical Caveat

[Part II : The Origin of
Matter, ‘Dark’ Matter and ‘Dark Energy’ is in preparation].

** Note** In this monograph, in all pressure- volume
equations unit mass is assumed, so that v and ρ then refer to specific volume and specific density. Also, mass in our present system is taken to
be dimensionless; this convention simplifies the notation, and is physically
supported in the theory, since it is agreement with our definition of mass as a

__1.(I) Introduction__

Current cosmologies have from
the Newtonian dynamical cosmology of classical physics which dealt with mass
particles and forces, and then from general relativity which introduced its
field theory of gravitational force based on mass-induced distortions in the
geometry of a postulated but non-observable space-time continuum. This general
model had been adjusted periodically with advances in quantum mechanics and
particle physics to become the Standard Big Bang Cosmology with considerable
success.

The basic physical facts
about the universe on our present knowledge are that it is about 13.7 billion
years old, spatially flat, and not only expanding but doing so at an unexplained
accelerated rate. It is roughly made up of ordinary matter (4%) exotic dark
matter (20%) and dark energy 76%.

However, in the current
theory there are many limitations. Specifically, we do not know the origin of ordinary
matter. We do not know the nature or origin of either the dark matter, or the
so-called dark energy. We do not know how to integrate gravitation and general
relativity with quantum theory. We do not know how radiation physically travels
in space nor how gravitational force is transmitted through space. We do not
know how the Big Bang’s enormous energy and temperature came about. We
currently do not know for certain how inflation came about nor even if it is
real. We do not know why the cosmos is accelerating in its expansion. General
relativity is apparently powerless to answer these questions so that there is a
need for some more comprehensive and basic explanation [1].

Here in Part I we shall
present the physical theory of a Universal Wave and Force field (UF), related
to the Chaplygin/Tangent gas of aeronautics and astrophysics, which offers new
answers to most of the above questions.
We then present a cosmological Carnot
cycle for this UF, starting in the pre-Big Bang era, continuing with explanations
for the emergence of ordinary matter and
radiation after the Big Bang and leading up to the evolutionary cosmic
expansion of the cosmos since then.

In Part II, currently under
preparation, we shall deal with the origin of matter in more detail and explore
the emergence of the dark matter and dark energy of the universe.

__2 (I). General Relativity Cosmology __

The present standard
cosmology is based on the theory of general relativity, and so we first must
briefly examine its formulation, assumptions and limitations before presenting
our new cosmology. The current
theory models a unitary universe made up
of galaxies and interstellar gas clouds as an incompressible, hydrodynamic
assemblage having pressure p and density
ρ within a geometrical space-time continuum in which the masses and mass
particles generate the distortions of
the metric from which gravitational force emerges.

The Friedmann formulation of
general relativity [2,3] is as follows:

(2/R) (d^{2}R/dt^{2}) + (1/R^{2})
(dR/dt)^{2} = 8πG p/c^{2} − k/R^{2}

(dR/dt)^{2} = (8π/3) GρR^{2} −k

where R is the radius of the cosmos at any time t, G
is the universal gravitational constant, and k is a constant related to the
metric which describes the geometry of the space- time continuum.

If we introduce the Hubble
relationship V = HR, (where V is velocity of expansion) we get an alternative
Friedmann formulation

H^{2} 8π/3 Gρ = − k/R^{2}.

The Friedmann equation is
readily compared to the various Newtonian dynamical formulations [3]. For
example, if p = 0 we get the Newtonian model

2/R(d^{2}R/dt^{2}) = GM_{o}/–kR
= 4/3 π GρR^{2} −k.

In this classical model, if k
is 0 or negative then the radius of the cosmos
R increases indefinitely. If k is positive, then R increases to some
maximum size R_{max}. Also, we
are restricted here to the cases where V <<< c and p <<< ρc^{2}. In
general relativity, such as with the Friedmann formulation, there is no such
restriction.

From the Friedmann model
other general relativity formulations, such as
those of Einstein,
Eddington-Lemaitre, de Sitter, Einstein-de Sitter etc., are readily derived
[2,3].

The assumptions of the
general relativity cosmology are (1) the universe is homogeneous and
isotropic,(2) there exists a universal geometric construct called the
space-time continuum, which has the physical property of sustaining
stress. In the limit of finite, uniform,
relative velocities the laws of special relativity apply, while in the limit of
very small velocities the equations reduce to those for Newtonian motion.

A basic postulate derived
from the special theory also applies, namely that uniform motion (i.e.
force-free motion) through space cannot be
detected experimentally (the speed of light being the same for all
inertial observers). The
experiments that are used to attempt to
detect motion through space involve various types of optical instrumentation
such as interferometers and oscillators, and the special relativity proposition
is equivalent to saying that light does not travel through space in any
physically real medium.

Such a proposition is today
confronted with the generally small but
universal physical effects of both uniform and accelerated motion that are
readily detectible with modern optical and resonant instrumentation.( www.energycompressibility.info) However, the experimental confirmation of various physical effects of
motion through space are met either by
dismissing them as being statistically insignificant, or by making successive
adjustments to the Einstein cosmological constant, which is an arbitrary energy term added to his field
equations to account for such things as dark energy and acceleration in the rate of cosmic
expansion.

General relativity offers no
explanation for the existence of matter, nor for the quantum nature of gravity.
It has no explanation for the Big Bang singularity, nor does it extend before
the instant of that event. Its explanation for the transmission of
gravitational attraction through space is based on the assumption of the
existence of the space-time continuum construct which allows no direct experimental verification of its reality.

To repeat, its basic tenets
are its assumptions of general invariance and the impossibility of detecting
any absolute uniform motion through space; i.e. there is assumed to be no
universal underlying physically real medium, only its postulated,
geometrical, space- time continuum.

Another important technical limitation
of general relativity and its cosmology is that it is a differential field
theory, and so solutions to its differential equations of motion require a
knowledge of the appropriate boundary
conditions. In the case of local
applications, the boundary conditions can often be specified or assumed so that
solutions can be obtained. In the case
of cosmology, however, the boundary conditions are those of the entire cosmos
and are essentially unknown – indeed, perhaps unknowable. This is a fundamental
limitation of all differential field theories when applied to the entire
cosmos. Another drawback with differential field theories is that they have
difficulties in dealing with singularities. Physical singularities such phase
changes and the Big Bang tend to become infinities in the differential field
theories. We now describe the physical basis for a new alternative cosmology
which, being based on an integral equation of state, avoids this problem..

__3.0 (I). PHYSICAL PROPERTIES OF A UNIVERSAL
WAVE AND FORCE FIELD (UF)__

__3.1 (I).
Dynamics and thermodynamics of the Chaplygin/Tangent gas __

To describe the
motions of any compressible fluid continuum, three basic equations are needed:

1.__Euler’s classical
hydrodynamic equation of motion:__

For 1-dimensional
flow this equation is

∂u/∂t + u ∂u/∂x +
v ∂u/∂y +w ∂u/∂z = −(1/ρ) ∂p/∂x

where the term on the right hand side is called the __pressure gradient force .__

2. __The equation of
continuity, or conservation of mass:__

∂ρ/∂t + div (ρ**w**) = 0.

3**. Equation of State:** relating pressure p
to specific density ρ ( or to its reciprocal, the specific volume v = 1/ρ)

(a) For real
physical gases undergoing __adiabatic____
__motions__ (__ i.e. no heat flow, dQ = 0) the general equation of state
is :

pv^{k} = constant.

The wave speed c is
given by c^{2} = kpv. Here **k**, the ratio of the specific heats (k =
c_{p}/c_{v} ), is the adiabatic exponent (sometimes also
denoted by γ). The adiabatic exponent ratio k is related to **n** the partition parameter- roughly
equivalent to the number of ways the energy of the system is divided- by k =
(n+2)/n; n = 2/(k−1).

(b) For __isothermal____ motions__ in a
real gas ( i.e. no temperature change, dT = 0) the equation of state becomes

pv = RT
or p/ρ = RT.

Real gas
equations of state all lie in Quadrant I of the pressure –volume field of
Figure 1 below.

**Figure 1.
Pressure-volume relationships in compressible fluids **

Quadrant 1: Real gases and Tangent gas (exotic)

Quadrant IV: Chaplygin gas and
Tangent gas (exotic gases)

__3.1.1. Adiabatic Equations of State for
Compressible Fluids where k____= −1: The Chaplygin/Tangent gas __

For real gases
and fluids the adiabatic exponent k = c_{p}/c_{v} is always positive in the adiabatic
equations of state. However, if k is, instead, taken as being a negative
number then the properties of the
resulting theoretical fluid change radically. In 1901 a Russian aerodynamicist,

Within the last
five years, however, the cosmological problem raised by an unexplained
acceleration in the expansion of the universe
has been cited by some cosmologists [6,7,8] as indicating that a fluid called the
Chaplygin gas may exist physically as an exotic universal “cosmic fluid” which
is the seat of the so-called ‘dark
energy’ of the universe, presently calculated to comprise about 76% of the
total ‘matter’ of the cosmos.

The **Chaplygin gas** has the adiabatic
equation of state

pv^{-1}= p/v = p ρ = constant, or

p = −Av = −A/ρ

where p is the
pressure, v = 1 /ρ is the specific
volume, ρ is the density per unit
mass of fluid and A is a positive
constant. This equation plots with negative slope dp/dv on the pressure-volume
diagram ( Fig. 1 above) and its pressure
p is always negative. This possibility of a negative pressure is the attractive
feature for the present day cosmologists who are concerned with the apparent
accelerated expansion of the universe, and the Chaplygin gas is increasingly
being proposed as a physically real exotic cosmic fluid to address this cosmological
problem. Its properties are quite
bizarre compared to our real world gases. The Chaplygin gas lies entirely in
Quadrant IV of Figure 1.

While the success
of the Chaplygin gas in rescuing general gravitation and superstring theories
of gravity from their current difficulties is problematical, it also has
properties pointing to a much broader universal wave and force field
existing in Quadrant I, and it is this we investigate here.

A closely related
fluid to the Chaplygin gas is called the ** Tangent Gas [**5**]** which has an equation of state identical to the Chaplygin gas
except for the addition of a constant B. It may lie in either Quadrant 1
or Quadrant IV of Figure 1. Its equation of state is

pv^{-1} = constant, or

p = −Av + B = −A/ ρ +B.

In this form the pressure
p is positive for values of Av less than B. The relationship became very useful
to aerodynamics in the 1940’s. As a tangent curve to the adiabatics and
isotherms for ordinary atmospheric air,
the “tangent gas” provided a linear relationship between pressure and
density, which, for small variations of these variables, gives a useful approximation to the more cumbersome, exact,
non-linear thermodynamic equations. It was only very recently during the
present examination of the Chaplygin gas that the special properties of the
tangent gas as a fundamental cosmological fluid became apparent.

**Figure 1.
Pressure-volume relationships in compressible fluids **

Quadrant 1: Real gases and
Tangent gas (exotic)

Quadrant IV. :Chaplygin gas
and Tangent gas (exotic gases)

__3.1.2.
A New Isothermal Equation of
State for the Chaplygin/Tangent gas __

Recently [21] a
new isothermal equation of state has been derived for this peculiar fluid
field, as follows:

The
generalized adiabatic equation of state
for the fluid with k = −1 is pv^{k } =constant, and so this becomes

p v^{-1} = p/v = const. = ± A, or

p = ± Av

We now must
choose the sign before the positive numerical constant A, and this will determine the slope of the
equation of state in the p-v field. In the case of the Chaplygin gas the sign
is chosen as negative so as to give p = −Av = −A/ρ, and for the Tangent gas as p =
−Av + B = −A/ ρ +B , mainly because this makes them both agree
with all real gases in having negative
slope **−** dp/dv on the p-v
diagram and also guaranteeing a
positive wave speed with c^{2} = +dp/dρ.

However, there is
also no apparent reason to completely reject the alternative choice of +A for
the equation of state, that is, to chose a positive slope for dp/dv. This would
give the **isothermal equation of state** for the fluid:

p =+ Av through the origin and

p = +A/ρ
–B in Quadrant I.

These two new
isothermal curves with positive constant A are strictly orthogonal to the
adiabatic Chaplygin and Tangent gas curves. They constitute an **isothermal equation of state **for the
fluid with k = −1, since p/v
=+A, and, if the positive constant A is set equal to a __constant
temperature__ + A =RT ,we would then have p/v = RT= constant which is
certainly an isothermal relationship.

We then have :

Real gases ( k ≥
1) : pv>^{1} = constant = +A
is the general adiabatic equation of state

Real gas ( k = 1)
: pv^{+1} = constant. = +A =
RT is the isothermal gas or the equation of state for any ideal gas with constant temperature T

Exotic gases ( k
= − 1) :

a)
** Chaplygin gas: ** Adiabatic equation of state p = −Av = −A/ρ

Isothermal equation of state p
= Av = vRT; p/v = RT

b) ** Tangent Gas**
Adiabatic
p = −Av + B

Isothermal
p = +Av − B

__We now propose to treat these exotic states, not as
separate physical entities or ‘gases’ but instead as simply the
adiabatic and isothermal equations of state of one single, universal
compressible field ( UF).__

** **

__Figure 2.
Equations of State for the Universal Field (k = − 1; pv ^{-1} = const.__

For a more
complete report on the theoretical derivation of the isothermal gas see Appendix A: Thermodynamic
Properties of an Isothermal Gas Law for the Chaplygin/Tangent Field

__3.1.3.
The Energy Equation__

An additional
useful relation for any compressible field is the energy flow equation relating
compressive wave speed c to __relative__ flow velocity V:

c^{2} = c_{o}^{2}
– V^{2}/n

where n (
=2/k – 1) is the number of ways the energy of the system is partitioned. If we divide through by the square of the
static wave speed c_{o}^{2} we get the wave speed ratio

c/c_{o} = [ 1 – V^{2} / n
c_{o}^{2} ]^{1/2}

which ( when n = 1 ) is formally identical to the
Lorentz/Fitzgerald contraction factor of special relativity theory.

For unsteady or
pulsed flow, the energy equation
becomes c^{2} = c_{o}^{2}
– V^{2}/n – 2cV/n where the
additional, or ‘pulse’ term, 2cV/n is of
great importance in quantum phenomena.

__3.2. (I) A Universal Wave and Force Field: Longitudinal and Transverse waves __

As
stated, the evidence seems to be that, instead of there being
three separate exotic “gases”( Chaplygin gas, Tangent gas and Orthogonal or isothermal
gas), there is only a single,
compressible, fluid entity or field, namely a universal field (UF) having the usual adiabatic and isothermal
equations of state.

We now explore
the proposition that there exists this single Universal Field (UF) having compressible properties
of the adiabatic Chaplygin/Tangent gas, and the new isothermal orthogonal gas,
which supports stable waves uniquely obeying the classical wave equation for
both transverse and longitudinal vibrations. It will then be shown that its transverse waves correspond to Maxwell’s
electromagnetic waves of light, while the longitudinal vibrations correspond to
waves which transmit gravitational force through space.

** 3.2.1. Equations of State** relating pressure , p, to specific
density ρ ( = 1/v) and where n =−1; k = (n+2)/n = −1:

**Adiabatic: ** p
= −Av + B = −A/ρ
+B (/Tangent Chaplygin gas)

**Isothermal: **p= +Av −B ; p/v = RT
(Orthogonal Gas)

** 3.2.2. Energy Equation** : ( relating wave energy c

c^{2} = c_{o}^{2} –
(1/n) V^{2}

and, with n = −1 we have

c^{2} = c_{o}^{2} + V^{2}.

If we divide
through by the static wave speed c_{o}^{2} we get

c/c_{o} = [ 1 – V^{2} / n
c_{o}^{2} ]^{1/2}

and with n = − 1 we now have

c/c_{o} = [ 1 + V^{2} /c_{o}^{2}
]^{1/2}.

__3.2.3.
Force in the Universal Field (UF)__

In the UF the force is given by the Euler
equation which, for 1-dimensional flow, is

∂u/∂t + u ∂u/∂x +
v ∂u/∂y +w ∂u/∂z = −(1/ρ) ∂p/∂x.

__3.2.4.
Longitudinal Wave Motion in the UF____ __

The Universal
Field is unique in that (1) it is the only field in which the Classical Wave
Equation is strictly valid and which therefore can transmit stable
longitudinal waves, of either
condensation or rarefaction, of any amplitude,
and (2), as we shall see in Section 3.2.9 below, it is the only tenuous __fluid__
which can support and transmit __transverse__ waves.

In general, the
adiabatic speed of sound waves c in a fluid is related to the pressure p and
density ρ by the equation

c^{2} = (dp/dρ)_{s}.

From this, the
adiabatic ( i.e. no heat flow, ∆Q = 0) speed of sound c in a perfect
gas is also given by

c^{2} = kpv = kp/ρ =kRT.

A positive wave
speed c, therefore requires that
dp/dρ must be positive. Since v = 1/ ρ, we see that dp/dρ
and dp/dv must have opposite signs. Real
gases, and the adiabatic equation of state for the UF (Tangent gas) all have a negative slope for dp/dv on the p-v
diagram and therefore have positive
adiabatic wave speeds.

For compressive
waves, the classical or exact wave equation, expressed in terms of the wave
function ψ for amplitude, is [9]

Ñ^{2} ψ^{ } =^{ }1/c^{2} ∂^{2}ψ/∂t^{2}
[ 1 + Ñψ ]^{(k }^{+ 1)}

or, for one
dimensional motion in the x-direction

∂^{2}ψ //∂x^{2} = [(1/c^{2}) ∂^{2}ψ /∂t^{2} ]/ [ 1 + ^{ }∂
ψ /∂x ]^{ 1+k}.

This exact
equation means that, for real or material gases ( k > 0), compressive waves
are always unstable and grow with time . For very small amplitude waves
however, the term in the denominator involving 1 +∂ ψ /∂x approximates to
unity, and the equation simplifies to become the classical wave equation of
very low amplitude sound waves (acoustic waves)

Ñ^{2 }ψ = (1/c^{2 })∂^{2}ψ/∂t^{2}

^{ }

which has the general solution

ψ = ψ_{1(}x – ct) +
ψ_{2} ( x – ct)

In the case of the
UF, however we see that, since k = – 1, the exponent ( k + 1) of
the denominator term becomes zero so that the term itself becomes unity ,
thereby automatically reducing the equation to the simple classical wave
equation, but without any of the approximation needed for real gases such as
air.

The UF is unique
in that it alone is automatically exact for waves of any amplitude, large or
small, that is to say it is no longer
limited to infinitesimal waves as is the case with real gases.
The UF is therefore unique among gases,
since in all real gases finite amplitude
waves always either steepen or die out, and only sound waves of infinitely low amplitude (i.e. acoustic
waves) can persist as stable waves.

The natural representation of the solutions to the
classical wave equation

ψ = ψ _{1}x – ct) +
ψ _{2}( x = ct)

is on the (x,t), or space-time diagram. Figure 3 shows the characteristic lines
representing two families of left-running and right-running waves on the
space-time or ‘physical plane’ diagram.

**Figure 3. Space-time( physical plane) Plot of 1-dimensional**

**Wave Characteristic Lines **

The fact that
these characteristic solutions of the classical wave equation ( i.e. of the UF)
are linear and can be superimposed directly relates waves in the UF to quantum physics and wave mechanics.

__3.2.5.
Wave speeds in the UF__

For isothermal motions
( i.e. constant temperature, ∆T = 0), we have the isothermal
(Newtonian) speed of sound waves in a
perfect gas

c^{2} = pv = p/ρ = RT ; c = [pv]^{1/2}.

Now in the UF where k = −1 we have:

**a)** The Tangent ( i.e. adiabatic, constant
heat , ∆Q = 0 ) Equation of State
:

p = −Av + B

The general adiabatic sound speed equation is c^{2} = kpv, therefore, in the Tangent case with k
= − 1, we must have

c^{2} =k pv = k [−Av^{2} +^{-}Bv] = +Av^{2
}− Bv

and the sound wave speed c is positive as it should be.

**b)** Our
isothermal equation of state is:
p = +Av − B

The wave speed
in an isothermal gas is given by
c^{2} = pv, and therefore we must have

c^{2} = pv = v[+Av − B]
= +Av^{2 }− Bv

Therefore, since the right hand side of
the equation is positive, the isothermal
state also appears to have a positive wave speed so long as B is less than A .
However the usual derivation of c^{2} = dp/dρ gives us dp/
dρ= d(+Av –B) / dρ = d(+A/ρ − B) / dρ = −A/ρ^{2}
= c^{2 } so that here c^{2}
is negative and the wave speed c involves the imaginary i, which suggests
damped oscillations only, instead of waves. The problem appears to centre
around the required negative slope of the curve on the p-v diagram for a positive
wave speed in Quadrant 1.

__3.2.6. Isentropic ratios ( c, p, ρ,
T) and relative motion (V)__

The general
isentropic ratios relating pressure, density and temperature for any ideal gas,
are ( using n = 2/(k-1))

c/c_{o} = [p/p_{o}]^{1/(n+2)} = [ρ_{o}/ρ]^{1/n}
= [T/T_{o}]^{1/2}

In the UF where k = − 1 we see that n =2/(k – 1)
also is equal to − 1, and
these isentropic ratios then become

c/c_{o} = p/p_{o} = ρ_{o}/ρ
= v/v_{o }= [T/T_{o}]^{1/2}

^{ }

but, while this
is so for the isothermal gas, it is not so for the adiabatic or Tangent gas because of the negative slope of the latter equation of
state on the p-v diagram. Careful analysis is needed on this point. For the Tangent gas the isentropic ratios
are

p/p_{o} = [−Av +B] / [−Av_{o}
+B] which for the special case where B = A = 1 is approximately given by

p/p_{o} = ρ/ρ_{o} = v_{o}/v_{ }

The energy
equation for relative motion V with k = n = − 1 becomes

^{ }

c/c_{o} = [ 1 + (V_{ }/c_{o})^{2}
]^{1/2}

Therefore, we
also have

p/p_{o} = [1+ V^{2}/c_{o}^{2} ]^{1/2}

p/p_{o
}−1 = ∆p/p_{o}
= [1+ V^{2}/c_{o}^{2} ]^{1/2} − 1

∆ρ/ρ = [1+ V^{2}/c_{o}^{2}
]^{1/2} − 1

relating pressure
pulses ∆p, and corresponding density pulses ∆ρ to relative
fluid motions V in the UF, for example
to oscillations of an electric charge or accelerations of a material particle.

The various
possible types of wave motions can then be investigated by imposing (a)
condensation pulses ( +∆ρ), (b) rarefaction pulses (−∆ρ) and (c) density
oscillations ( ± ∆ρ) as in Fig. 4.

__Figure 4. Density Perturbations and Wave
Motion in the Universal Field__

** **

**Figure 5. Various Wave Motions in the UF **

__3.2.7. Stagnation Values of Pressure, Temperature and Density
for the UF__

In any
compressible field the basic initial values of interest are often those when
there is an adiabatic reduction to a state of no fluid flow ( V = 0). These are
called the stagnation values and are designated as p_{o, } T_{o}
and ρ_{o}^{ }. Their numerical values for the UF
remain to be determined.** **

So far we have
not distinguished between longitudinal and transverse wave motions in the UF.
Clearly there is no problem with longitudinal waves; they are uniquely
supported in the UF, and, moreover, they are not restricted to low amplitude
acoustic type waves as in real gases where k is positive.__ __We shall now
present evidence that ** transverse**
waves are also uniquely supported in the UF, and that they in fact correspond
to Maxwell’s electromagnetic waves which transmit light and other radiation
through space.

Real gases, being
tenuous fluids, can only support longitudinal waves, that is to say, waves in
which the density variations ±∆ρ
are along the direction of wave propagation. They cannot support transverse
waves in which the density variations would be transverse to the direction of
wave propagation. It was this inability of a tenuous medium to transmit the
transverse waves of light which led to the demise of the old luminiferous ether
concept. __We now ask: What is the ____evidence for transverse fluid waves ____in the Universal Wave Field
(UF____ )____ with its mutually orthogonal adiabatics
and isotherms?__

__3.2.8. Evidence for transverse waves in a
tenuous fluid__

We consider a
simple pressure pulse ( ±∆p) in the UF as in Fig.6 below:

__ __

**Figure 6. A
pressure pulse ( ±∆p) in the Orthogonal
Environment of the UF**

** **

The initial or stagnation
state is designated as p_{o}._{ }When the pressure pulse ( +∆p) is imposed from outside in some way, the UF
must respond thermodynamically in two completely orthogonal, and hence two
completely isolated ways, namely, by (1) an adiabatic stable wave along the
adiabatic( TG) and (2) by an isothermal stable pulse along the isotherm (OG).

_{ }

Spatially, the pressure
disturbance ( +∆p) must propagate in the direction of the initial
impulse. But, since the two components of the pulse are orthogonal, they must
still remain completely independent and physically isolated. The only way possible for this to take place
is for the two mutually orthogonal components to also be __transverse__ to
the direction of propagation of the two pressure pulses. This requires an axial wave vector V in the direction of propagation ( say z),
and with the two pulses orthogonally disposed
in the x-y plane. i.e. TG x OG = V which is reminiscent of the Poynting
energy vector S = E x B in an electromagnetic wave.

** **

A wave of
amplitude ψ traveling in one direction (say along the axis x) is represented by the unidirectional wave equation

dψ/dx = 1/c dψ/dt

__3.2.9.
Maxwell’s electromagnetic waves__

Here, however, in
the case of our adiabatic and isothermal pressure pulses we have two coupled
yet isolated unidirectional waves, and this reminds us of Maxwell’s coupled
electromagnetic waves for E and B, as follows

dE_{y}/dx = (1/c) dB/dt and dB_{y}/dx = (1/c)
dH/dt

where c is the
speed of light, E is the electric intensity and B is the coupled magnetic
intensity.

Maxwell’s E and B
vectors are also orthogonal to each another and transverse to the direction of
positive energy propagation.

**Therefore, we have established in outline
a two component wave system in the
Universal Field (**k = −1)
**which formally corresponds to the E and
B two component orthogonal system of Maxwell for electromagnetic wave
propagation through space in a continuous medium. **His equations for E and B
are

Curl E
= ∂E_{y}/∂x = −(1/c) ∂B/∂t

Curl B = ∂B_{y}/∂x = −
(1/c) ∂E/∂t

If we now
designate our Tangent gas as A ( for Adiabatic) and our Orthogonal gas as I (
for Isothermal) then our analogous wave equations would be

Curl A = ∂A_{y}/∂x = −
(1/c) ∂I/∂t

Curl I = ∂I_{y}/∂x
= − (1/c) ∂A/∂t

The two systems
are formally identical. Therefore, we propose that the medium in which
Maxwell’s transverse electromagnetic waves travel through space is to be physically identified as a
Universal Compressible Field (UF) having the above described thermodynamic
properties for adiabatic and isothermal motions initiated in the UF by imposed
pressure pulses ( presumably by accelerated motions of electric charges). The __compressibility__ of the UF now
properly accounts on physical grounds for the finite electromagnetic wave speed
(speed of light), and in addition, __wave motions in this tenuous fluid medium
are transverse,__ as required by the observations..

It is possible to
reduce Maxwell’s two equations UF equations to a symmetrical single wave
equation

∂^{2}**E**/∂x^{2 }=^{
}(1/c^{2}) ∂^{2}**E**/∂t^{2}

∂^{2}**B**/∂x^{2 }=^{
}(1/c^{2}) ∂^{2}**B**/∂t^{2}

and similarly
with **A and I ** for our Adiabatic/Isothermal
coupled wave in the UF:

∂^{2}**A**/∂x^{2 }=^{
}(1/c^{2}) ∂^{2}**A**/∂t^{2}

∂^{2}**I**/∂x^{2 }=^{
}(1/c^{2}) ∂^{2}**I**/∂t^{2}

This is not
surprising since the UF with its k = −1 thermodynamic property is the
unique compressible fluid which
automatically generates the classical wave equation with its stable, plane
waves. The formal agreement of the UF theory with Maxwell is striking.

Instead of taking our initial external perturbation as a pressure pulse ( +∆p) we could more realistically from the physical
standpoint take it to be a __density condensation__ (s
= ( ρ – ρ_{o })_{ }/ ρ_{o} = +∆ρ/ ρ_{o}). This
will now result in a positive pressure pulse
(+∆p) appearing in the adiabatic
(TG) phase of the UF __but a negative
pressure pulse ( −∆p) in the isothermal or orthogonal
perturbation component (OG)__ . This perturbation is represented by the two
orthogonal sets of arrows on the pv diagram, one corresponding to +∆p and
the other set corresponding to − ∆p. As the wave progresses the two
orthogonal vectors also rotate.

__Figure 8. The physical ambiguity which
results from a pressure/density perturbation in the Orthogonal UF __

Therefore, an
oscillating density perturbation ( ±∆ρ) results in an axial wave
vector having ** two mutually orthogonal
components ( adiabatic and isothermal ) in a density perturbation wave. ** This appears to correspond formally to the
Maxwell electromagnetic wave system with its two mutually orthogonal vectors
for electric field intensity

**We have thus established a case for the
compressible UF being a cosmic entity
which transmits transverse electromagnetic waves through space**. ** **A
necessary next step will be to examine the UF in relation to all the
multifarious established facts relating to electromagnetic radiation.. These
must include the nature of electric charge, electrostatic fields, the
compressed fields of moving charges and resulting magnetic fields, etc. etc.
Preliminary work has indicated that this additional reconciliation will be
successful.

__3.2.10. Polarization and Spin __

Since
we are dealing here with two linked mutually orthogonal states, all the formal
requirements of electromagnetic polarization and spin are automatically
satisfied. Various other physical
details remain to be examined by specialists.

__3.2.11. Electromagnetic Wave Quantization: Photons
and the Orthogonal State Waves__

Perhaps
a bit simplistically, we could just consider each individual axial vector wave
as single basic wave entity or quantum entity and then build up more
complicated energetic states by superposition of the basic linear waves. That
is to say, we could consider each individual perturbation a quantum. But we
would still have to explain the basic energy quantum relation e = hv.

However, the quantization can perhaps also be seen on
a more physical basis, if we set
the UF’s stagnation or rest pressure p_{o}
at some small value very close to p = 0,
say at p_{o} = 6.673 x10^{-11 }kilopascals. This at once makes
the maximum allowed value of any negative pressure perturbation −∆p
equal to 6.67 x10^{-11} kpa as
well, because, if only symmetrical pressure perturbations ±∆ρ are allowed to transmit waves, that
is, if only equal amplitude perturbations are permitted , then the waves would be quantized at the maximum
amplitude set by −∆p equal to
6.67 x10^{-11} kpa,( so
long as p = 0 is the lowest pressure permitted, i.e. so long as negative absolute values of the pressures
are excluded).

This quantization procedure is also applied to gravitation in a
succeeding section.

** 3.2.12. Special Relativity: ** The question of
whether the UF proposal reintroduces a cosmic “medium” into space will also
eventually come up, although it is more
a matter for experiment to settle than for theory. Perhaps all that need be said at this point
is that when the relativity of motion is investigated with respect to the new
UF field proposal, the laws of compressible flow must be applied. In
compressible flow all velocities are physically purely relative, and, as shown
above the compressible energy equation yields the Lorentz/Fitzgerald transformation
in a more general form than special relativity, and now on physical grounds. In
the matter of the central problem of relativity, which involves the direct
mathematical addition of material source velocities to the velocity of light,
the addition formula from compressible theory containing the energy partition
parameter n must be used, instead of the failed classical direct addition of
velocities that led to special relativity being introduced in the first place.

To
repeat for clarity in this matter, the new addition rule will involve (c ±V/√n) where n = 2/(γ–1),
instead of the old classical (c ± V).
This is because it derives from the kinematic energy flow equation

c^{2} = c_{o}^{2}
– (1/n) V^{2}

and
we get the ratio of wave speeds

c/c_{o} = [ 1 – V^{2} / n
c_{o}^{2} ]^{1/2}

which sets the correct
velocity addition formula for compressible flows. We see that it involves the
introduction of the energy partition parameter **n** ( n = 2/k – 1). This formula is just the familiar
Fitzgerald/Lorentz contraction factor with the addition of the parameter n. In
most cases this addition of n greatly reduces the expected fringe shifts and
oscillation changes that are theoretically predicted by the old failed
classical formula and it bring them into line with the magnitude of the
perturbations that are observed in
experiments designed to detect uniform or absolute motion through
space. For details, see a current review
of the observational data of the Michelson-Morley and later experiments at www.energycompressibility.info
/__Appendix A:
Relativity and Results of Michelson-Morley Type Experiments.__

Some of the
formulae of the Lorentz transformation, and hence of special relativity, are
directly related to the above formulae for compressible flow for the special
case of n = +1 ( k = 1.67) in experiments such as those on the apparent increase of mass with velocity in
accelerators. However, in general, the value of n for any given experiment is not unity,
but must be properly determined or
estimated, since an experiment involves not just the wave transmitting field
but the interaction of the material apparatus and all relative motions and
accelerations. For the Michelson-Morley apparatus, for example, the value of n seems to be about 9, for the atmosphere it is 5, for combustion gases it is close to 9, and
for high speed accelerators it is close to 1.

But in
compressible flow in the UF, Lorentz invariance becomes an approximation which
is valid only for low fluid velocities. For higher relative velocities near the
critical wave speed ( c* , Mach 1) the kinematic energy equation must be
applied, since in compressible flow the wave speed c is a physical variable for
both compressive waves and transverse ( electromagnetic ) waves.

** 3.2.13. Summary of Transverse Waves in the UF. ** Maxwell’s electromagnetic field equations
involve two mutually orthogonal field vectors

Finally, with the
UF taken as a unique, compressible fluid medium which transmits transverse
electric and magnetic waves, then the question naturally arises: Can the compressible UF also account for the
unique nature of, and the spatial transmission of, the **force of gravity**? This is
discussed next.

For a more detailed treatment of some aspects of this section see ( www.energycompressibility.info
and www.shroudscience.info at the Page entitled Properties of a universal wave field.

__3.3 (I) Gravitational Force and the
Universal Field__

__3.3.1.
Characteristics of Gravitation__** : ** The principal characteristics of
gravitational force to be properly
accounted for in a new theory are (1) its exclusively attractive nature
for mass, (2) its extreme weakness relative to the electromagnetic force, and
(3) its 1/r

Any new,
physically based theory attempting to explain the nature and behaviour of gravitation
will obviously have to account for the three physical characteristics set out
above. It will not have to meet the general relativity postulates, but it will
eventually have to explain why the latter theory does make successful
predictions which offer corrections to the Newtonian predictions, such as the
advance in the perihelion of Mercury, and various gravitational lensing effects
on light waves.

__3.3.2.
Exclusively attractive forces in the Universal Field__

In any
compressible fluid, force is given by the Euler equation , which for 1-dimensional flow is

F = ∂u/∂t + u ∂u/∂x
+ v ∂u/∂y +w ∂u/∂z = −(1/ρ) ∂p/∂x,

where the term on the right hand side is called the __pressure
gradient force.__

In any
compressible fluid medium, waves set up local transient pressure gradients and
so forces arise. Most waves in real gases and fluids are pressure oscillations
(±∆p) and so they do not exert any net directional force on a material
object they encounter. However, in the
UF special types of waves can occur which can be either exclusively pressure
compressions (+∆p) thereby exerting a net repulsive force on any material
body in their path, or they can be
exclusively pressure rarefactions (−∆p) which would then exert __exclusively attractive force.__

Consider Figure 9 where an isothermal condensation pressure pulse
(**+∆ρ; +∆ρ) ** imposed on the initial stagnation pressure p_{o
} in the UF will produce a
rarefaction pressure pulse (**−∆ρ) ** in
the isothermal mode of response. Consequently, a source of pressure
condensations will produce a train of isothermal rarefaction pressure pulses as
a response in the UF, and these pulses will travel spherically outwards through
space.

When these rarefaction waves eventually impact a material body (mass)
they will exert a net attractive force on it. This mechanism, therefore, in
simple outline is formally equivalent to the force of gravity being produced by
a rarefaction pressure gradient force.

__Figure 9.____ ____Positive density pulse ( +∆ρ) of any magnitude produces a quantized
gravitational isothermal rarefaction pulse (−∆p _{g} ) of constant magnitude p_{g} =6.67 x
10^{-11} kilopascals__

__3.3.3.
Extreme Weakness of the Gravitational Force and the Universal Field__

The three main
forces in nature are the strong nuclear force, the electromagnetic force and
the gravitational force. The nuclear force is very much the strongest. The
electromagnetic force is about 1/137 th of the nuclear force. The force of gravity, however, is
extraordinarily weaker that the other two, being only about 10^{-40} th. the strength of the
electromagnetic force. Thus we have

F_{s} / F_{e/m} = 1/137 ;
and F_{g} / F_{e/m} = 10^{-40}.

We have shown
above that a wave train exclusively made up of pressure rarefactions would
explain the attractive nature of gravity. Now we must explain how these
rarefaction waves can all be so extremely weak.

Consider, again,
a positive density pulse (+ ∆ρ) imposed on the UF. (Fig. 9).
Adiabatically this will result in a pressure increment (+∆p) , but in the isothermal mode this
will give a pressure rarefaction (−∆p).

We must now consider
whether a sufficiently large negative
pressure pulse (−∆p)
can transmit in the isothermal mode right through the p = 0 expansion and on into the negative pressure
region of Quadrant IV. It appears that it cannot, because of the fact that a
negative pressure will entail a negative temperature, and quantum theory and
experiments all show that temperatures
of opposite sign are not equal in magnitude [a, b, c] Thus the isothermal
condition will be discontinuous on the v-axis at the p = 0 point. Physically this appears to require that the
negative pressure pulse must terminate at this point. The consequence then is
that all negative isothermal pressure
pulses will be truncated to some maximum amplitude.

(−∆p) _{isothermal} =
( p_{o} – 0) = p_{g} – 0
= constant = 6.67 x 10^{-11 } kilopascals.

Therefore, if the
initial pressure/density perturbation is inserted into the UF at its static
pressure p_{g} = 6.67 x 10^{-11} kpa, then the induced negative isothermal pressure
rarefaction (−∆p ) will be (a) extremely weak __and always of the same maximum amplitude
of 6.67 x ^{10-}11 kpa. __no matter what the amplitude of the
initiating positive density pulse (+ ∆ρ).

Again, we have F_{g} = − v dp/dx = − 1/ ρ
dp/dx . Therefore, for unit mass at unit
distance we have F_{g} = − ∆p = G = 6.673 x 10^{-11}. Thus the basic UF stagnation pressure p_{o}
seems to correspond numerically to G, the gravitational constant, as we
have just postulated (p_{o} =
6.673 x10^{-11} kilopascals).

On this physical
model, gravitational force is carried through space isothermally by a train of
rarefaction pressure pulses (−∆p), all physically constrained to
have the same invariant maximum amplitude regardless of the amplitude
of the initiating positive density pulse (+ ∆ρ).

Then to show that
our gravitational formulation gives the required weakness for the gravitational
force**,** consider

F_{ig} = (G m_{1} m_{2})
/ r^{2}

and the Euler
pressure gradient force

F_{p.g} = −1/ρ (dp/dx)
= −v (dp/dx)

If we now take dp
= lGl and, realizing that the equation is for unit mass ( m = 1) , we get (at
unit distance dx = 1) the force

on a unit
mass F_{g } = − (1) 6.67 x 10^{-11 } ^{-27}
kg) we would then have

F_{g }^{ }= 1.67x10^{-27} [ 6.67 x 10^{-11}]
= 1.1x10^{-37 }

which reconciles the new UF force formulation with the
Newtonian predictions.

__3.3.4. A Unique Wave Speed Assciated with Gravitational Waves__

In general, in the UF the wave speed, given by c^{2}
=(3 x 10^{8})^{2 } = ∆p
/∆ρ holds, and the wave speed c is the speed of light in space.
However, consider what happens when in the generation of a gravitational wave
pulse as just described above, the
reduction in p in the isothermal mode approaches the zero pressure point. At this point, if the wave pressure action
then cannot continue on into negative values of p in Quadrant IV as required by
the magnitude of the initiating positive pressure pulse , then we have the UF fluid approaching a state where v = 1/
ρ = **constant** and so any
additional ∆ρ needed must become zero. Bu**t then the gravitational wave speed must approach
infinity since c ^{2} =dp/ 0 must then equal infinity.**

**As to the probable magnitude
of this new gravitational pulse wave speed we propose the following: ** The ratio of the gravitational
constant to that of the Planck constant is

G/h = 1.04 x 10 ^{23} = constant

If mass is taken as
dimensionless ( as we do everywhere in this theory on the grounds that mass is
a condensation of energies and so can be
consider a ratio of energy before and after elementary particle formation ) then the dimensions of the ratio G/h
are those of velocity or
speed [l t^{-1}].

It thus seems possible that the ‘reflection’ of the
positive pressure waves of Quadrant I at
the p = 0 boundary or discontinuity may generate an additional transient,
extremely fast UF wave as well as propagating
the basic electromagnetic or gravitational pulses that it reflects and
sustains. This new wave seems more likely to relate to the spread of quantum
information through space rather than to the transfer of energy.

In summary, we have exclusively attractive gravitational
waves traveling at the speed of light speed of light, but, as each wavelet
reaches the zero pressure point ( p = 0) it emits a secondary wave which
spreads spherically at quasi-instantaneous speed.

For a variety of reasons we also associate this secondary quasi-instantaneous
pulse radiation with quantum wave information spread.

__3.3.5. Quantum Information and the UF__

While this very
large and complex subject has not yet been reexamined in great detail, there
are some aspects which already indicate
that compressible flow and the UF concept are intimately related to quantum phenomena, just as has been shown
above for electromagnetism and gravitation.
Some of these will be presented
Part II.

For a more detailed treatment of this section see (a www.energycompressibility.info)
and (www.shroudscience.info and open
the page entitled __Properties of a Universal Wave field__

** 3.3.6. Summary.** We have shown that the UF can support and
transmit stable rarefaction waves which
(1) exert exclusively attractive force on masses and (2) have the necessary
extreme weakness. This meets the general requirements for them being
gravitational waves and for the UF
therefore being the physical seat of universal gravitational force.

__ __

There are, of course, many other aspects of gravitation we
have not considered here These also will have to be considered in light of
the theory, but they are not essential
for this general presentation.

Having shown that
the UF is the source of radiation and gravitation, we are now in a position to
proceed to examine its thermodynamics and show that its Carnot cycle forms the
basis for a new cosmology. Then in Part II ( in preparation) we show that the
UF Carnot cycle explains (1) the origin of the Big Bang, (2) the nature and
origin of ordinary baryon matter, (3) the nature and origin of dark matter, and (4) the
nature of dark energy so as to thereby
construct a new general cosmology.

__4.0 (I) COSMOLOGICAL ORIGIN AND
THERMODYNAMIC EVOLUTION OF THE UNIVERSAL FIELD
(UF) DURING A CARNOT CYCLE __

__4.1. Cosmic Origins: Initial postulate of the existence of a
cosmic fluid__

The proposed new
cosmological model has one physical assumption, namely that at an
initial cosmic time t_{initial}
there existed a physical **cosmic fluid continuum** having pressure
p_{L}, density ρ_{L}
, temperature T_{L},
energy ε_{L} and tensile strength ∆p_{L}.

This postulated cosmic fluid
has an equation of state of the general form
pv^{k} = constant.

__4.2
Initial Event: Cavitation or rupture of the cosmic fluid to form a
vapour filled Universal Field (UF)__

We now envisage the emergence and evolution of
the Universal (vapour) Field (UF from the cosmic fluid as follows:

At initial time t_{R}
a ** rupture** of the cosmic fluid
is postulated to have occurred at one
point in the cosmic continuum (or possibly at many).

In most fluids rupture occurs at a tensile
strength that is many orders of
magnitude lower than the theoretical value [10,11,12,13]. In the course of
studying the properties of the UF [14 – 25] the reason for this anomalous
tensile strength was found to be described by a spherical compressible rupture
mode [24 ]. Our initial rupture flow is therefore taken to be radial (i.e.
spherical) and to obey the equation of
state or expansion law pv^{7} = constant (k = 7; n = 1/3) [24].

In accordance with fluid bubble
dynamics [10,11] this rupture event occurs when

∆p = const./ (∆V)^{7} = 2 σ /R_{C }

which
gives the relationship between the initial (radial) expansion ∆V needed for the rupture to form the void, the interfacial surface tension σ and
the rupture radius R_{C} .

__4.3. (I). UF Carnot Cycle Step #1: Isothermal expansion of vapour filled bubble
to critical sizeforming a Universsl Field__

The cosmic void or bubble next expands isothermally
according to the Rayleigh-Plesset equation to reach its stable radius R
[10,11].

(p_{B} – p_{L})/_{ }ρ_{L
= }∆p_{∞}/ ρ_{L } = R (d^{2}R/dt^{2}) + 3/2
(dR/dt)^{2} + 2σ /R ρ_{L}

At the stable or critical
size R_{C} this equation reduces to ∆p_{∞}. =
2σ /R_{C}.

__During its expansion the
cavity or bubble fills with vapour from the enveloping cosmic fluid continuum
to form a universal cosmic vapor field
(UF__) of radius R

** **

The UF has the
physical properties enumerated above in section 3, which include:

__Equations of State: __ We have shown above that in gas
dynamics there is the so-called Chaplygin gas with adiabatic equation of
state

hrough the origin :

pv^{-1} = p/ρ = constant ;or

p = −Av

In Quadrant I this becomes
the Tangent gas

**P = −Av +B**

The corresponding isothermal
equation of state [ 23] Appendix A is

pv^{-1} = const. or

in
Quadrant I

**p = +Av −
B**

These equations taken
together constitute the equations of state for our Universal Field.

In the case where the
isotherm starts at the point of origin
(p = v = 0) the isothermal equation reduces to

p = Av = A/ρ, so that its expansion obeys

p_{1}/p_{2} = v_{1}/v_{2}
= ρ_{2}/ ρ_{1 }which
is basically the inverse of that for the Chaplygin/Tangent gas.

The unusual properties of the
UF are re-stated as follows:. (1) It uniquely supports stable waves of any
amplitude both in compression and in rarefaction which obey the classical wave
equation; (2) Although it is a tenuous fluid or vapour, it uniquely supports
transverse waves which are identical to Maxwell’s electromagnetic equations; (3) Its compression waves are uniquely
attractive in nature and correspond to the requirements for a wave disturbance
carrying gravitational forces through space. This ‘graviton gas’ forms the
basis for a quantum gravity; (4) It supports waves corresponding to the de
Broglie wave/particle equation and thus relates to the quantum wave function
and the transfer of quantum information through the cosmos; (5) Its isothermal expansion is shown to be
the possible source and origin of the so-called ‘dark energy’, thus furnishing
a physical explanation for the observed increased rate of acceleration in the
expansion of the universe [6,7,8].

Its state equations are
graphed in figure 11.

** Figure
11. Equations of State for the Universal
Field (k = − 1; pv ^{-1} =
const.**)

__4.4. (I) UF Carnot Cycle Step #2. Adiabatic
collapse of the UF to a state of high energy and compression__

(A)__ Collapse: __The bubble filled
with the newly generated graviton gas field having reached the critical radius
now collapses. This bubble collapse generates an enormously increased
pressure, density and temperature [10,11
]. The pressure increase ratio is roughly proportional to the ratio of the
initial radius R

p = 100R_{C}/ R_{F}

This adiabatic compression of
the (UF) tangent gas obeys the compression ratio law

p_{1/}p_{2} =
[−Av_{1}+ B **]/[** −Av_{2}** +**B] , but, since A and B are not
known, no numerical solution can be
attempted without assumptions. If we assume
that A= B = 1 we get the approximation

p_{1/}p_{2} = [−v_{1}+1 **]/[** −v_{2}** +**1]

** **

__4.5. (I) UF Carnot zCycle Step #3: Isothermal further
compression towards the Big Bang: TheFormation of Matter__

__ __

In this step the UF
compresses isothermally following the same laws
as for Step 1 of the Carnot Cycle given in Section 4.3 above. A logical question here is: Why does the
expansion change from adiabatic to isothermal?

In Part II ( in preparation) we shall present the evidence
that ordinary particulate matter precursors e.g. quarks, **are
formed by compression ate the turning point to Step 3, ** and that they
then subsequently aggregate, as for example in the Standard Model of Big
Bang expansion, to form the atoms of our evolving physical world. It seems
reasonable to assume that it is this condensation or phase change that triggers
the change from Step 2’s adiabatic compression and volume collapse to Step 3’s
isothermal decompression and continued volume collapse leading to the Big Bang
state. We point out here that with many vapours,
condensation take place not upon expansion but upon compression and heating.
Many organic vapours, such as those of turpentine for example, condense upon
compression. The latent heat of condensation L offers a criterion as to which
process occurs. For example, the condition that the ratio of latent heat L to
temperature T is

L/T > (c_{p}−_{
}c_{v}) k/(k−1)

means that condensation in that particular vapour
takes place upon adiabatic expansion. Whereas with the condition

L/T
< (c_{p }−_{ }c_{v}) k/(k−1),

condensation would requires compression. The criterion
obviously is whether L is large or small compared to T. The latent heat of
condensation of water vapour, for example, is large ( 540 cal/g) and water vapour condenses to
liquid upon expansion of its vapour. A latent heat of around 100 cal/g, for an
essential oil such as pinene, would mean condensation from the vapour to liquid
upon compression. Consequently the proposal that World A particulate matter was
formed by compression just before the Big bang is physically possible. Evidence
that this is the case will be contained in Part II, currently in preparation
and will be posted on this website as ** Part II: Origin and Evolution of Matter, Dark Matter and Dark Energy**
when available].

We propose setting R_{F,}
the radius of the cosmos at the Big Bang compression ( the completion of Step
3) equal to the Planck length ( R_{Pl}
= 1.3x10^{-35} m). Then, if we surmise that the pressure at the end of
collapse is, say, the present day estimated value of the
quantum vacuum or 10^{121} kPa,
we get the size of the UF bubble before collapse to be about
10^{85 } m. This is 10^{59}
times the radius of the present day observable universe of ordinary matter
which is about 10^{26 }m^{ }._{ }

Continuing on in the Carnot
cycle to the enormous explosion of the Big Bang, we are presented with two choices :
(a) an adiabatic expansion of the
UF after the Big Bang instant or
(b) a
sudden constant-volume loss of pressure in the UF as a consequence of
the condensation of its quark plasma into the
substance of ordinary World A matter.

__4.6. (I) UF Carnot Cycle Step # 4: Two choices: (a) Adiabatic expansion of the UF following
the Big Bang or (b) Sudden constant volume pressure loss in the UF during the formation of particulate matter (quarks) by condensation __

In the general relativity model of cosmology, the post-Big
Bang expansion is modeled using the Friedman dynamic equation. Here, we also
could make the expansion from the Big Bang moment to the present cosmic time
with the Rayleigh-Plesset equation which
is formally quite close to the Friedmann equation. Both of these, however, are differential field
equations, and, as pointed out, they suffer from the drawback that they require
a knowledge of boundary conditions for a definite solution, while, of course,
the boundary conditions of the cosmos are unknown. Furthermore, they cannot really handle
singularities such as the Big Bang and the existence of the elementary
particles of matter, and so these components of the universe must be inserted ‘
by hand’ so to speak.

In
the present theory on the other hand we start not with differential equations
with undetermined solutions but with an **integral**
equation yielding a single definite solution.
Our integral field is an Equation of State of a compressible fluid
field, for example, the adiabatic equation for an ideal gas (pv^{k} =
const), or in the present case the Tangent gas with k = n = − 1 i.e pv^{-1}
= p/ρ = const.

The general
isentropic expansion ratios which relate pressure, density and temperature for
any ideal gas, are (using n = 2/(k-1)):

c/c_{o} = [p/p_{o}]^{1/(n+2)} = [ρ_{o}/ρ]^{1/n}
= [T/T_{o}]^{1/2}.

In the UF with n = − 1
we might therefore think that this would just become

c/c_{o} = p/p_{o} = ρ_{o}/ρ
= v/v_{o }= [T/T_{o}]^{1/2}

but we must realize that the
UF has two orthogonal equations of state, one with negative slope on the p-v
diagram, namely the adiabatic or tangent gas with positive wave speed, the
other the isothermal state with positive
slope and probably imaginary wave speed.

For the adiabatic expansion
of the UF( i.e. tangent gas) we must
use

p_{1}/p_{2} = v_{2}/v_{1}
= ρ_{1}/ρ ^{}

so that the pressure volume relationship is an inverse one.

For an isothermal expansion we have

p_{1}/p_{2} = v_{1}/v_{2.}
= ρ_{2}/ρ_{1}

where the pressure and volume
change in the same sense and it is the pressure-density relationship that is
now inverse.

__a) Choice #1: Adiabatic expansion of the UF
after the Big Bang__

The adiabatic expansion ratios for the UF are p_{1}/p_{2}
= [−Av_{2} +B]/[ −Av_{1} +B].

Since A and B are presently
unknown we cannot use this formula to make numerical estimates to compare with any
known thermodynamic values. But, if we were to assume, for example, that A = B
= 1 we could get an approximation p_{1}/p_{2} = [v_{2}+1]/[v_{1}+1],
and therefore, the expansion from the
Big Bang (p,v)_{N} to the present time (p,v)_{N} would be approximately
p_{B} = p_{N} [v_{N}] , since v_{B} is
so very small that v_{B}+1 is
effectively equal to 1.

With this assumption and
taking ** **p_{B}= 10^{121} and v_{N }= 9.2 x10^{78} m^{3}
we get p_{N} =1.1x 10^{42} kPa which makes little physical
sense.

Even if for v we compute the
__specific__ volume v = 1/ρ from an assumed density of 1 atom per cubic
meter, which is a currently used
theoretical value, i.e. v_{N} = 1/ρ_{n} = 6x10^{26},^{
} then we get p_{N}
= 1.64x10^{94 } kPa. These two
values give us pv =10^{121} kPa by definition, but the UF pressures
cannot be assessed against present physical data. We therefore tentatively conclude that the
expansion in the UF from the time of the Big Bang to the present time, while it
may have been adiabatic, does not yield any obviously simple correlations with known physical data and so is
problematical.

__b) Choice #2: Sudden constant-volume pressure loss in the
UF during the formation of particulate matter ( quarks) by condensation
followed by an Isothermal expansion to the present day.__

wever, instead of the above adiabatic expansion of the UF after the Big
Bang Figure 15 , we can envisage an alternative
expansion which has the interesting quality of providing a physical explanation
for ** inflation. **This step would be

This collapse of
UF pressure would then release the enormously compressed plasma of World A matter into the sudden massive
expansion proposed by ** inflation
theory [28] **as an explanation for the observed, overall, quasi
-uniformity of the cosmos in contrast to the presence of the galaxies and
stars.

Our proposed UF pressure drop is taken as occurring at constant volume from the Big Bang value
(10^{121} kPa) down to the value p_{o} = 6.67x10^{-11}
kPa ( Fig. 16), which is our postulated stagnation pressure of the UF for reason discussed above
in Section 3.3 on gravitational force.

The expansion
ratios for the isothermal state in UF taken
through the origin are, as given above

p_{B}/p_{N} = v_{B}/v_{N}.

Using this ratio
we can expand the UF isothermally from v_{B} =9.26x10^{-105} m^{3}
and p_{o} = 6.67x10^{-11} kPa to estimated present day values
of v and p as follows:

__Initial UF
values after inflation__: v_{o}
= v_{B} = 1.08x10^{-104} ( Planck volume)

p_{B} = 6.67x10^{-11} ( assumed)

UF values after
isothermal expansion: Volume(actual) =
9.2x10^{78} m^{3( }_{ }(Calculated
from radius of cosmos at 13.7 billion years )

Calculated UF Pressure Now: p_{N} = p_{B}v_{N} /v_{B}
= 6.67x10^{-11} x 9.2 x10^{78} /9.26x10^{-105 }= 6.6x10^{172}

^{ }

Or, using __specific
volume__ Now v_{N} = 6 x 10^{26 } = we get

Calculated
UF pressure Now: p_{N} = p_{B}v_{Nsp}**/v _{B} **= 6.67x10

^{ }

^{ }In effect this would mean that the UF, since inflation, has
expanded isothermally to around its
original Big Bang pressure. This is a rather unexpected finding.

We find that the hypothesis of an isothermal
expansion of the UF down to the present day, following the Big Bang
inflationary adjustment, is roughly
supported by the data. ^{ }

The next question
is How would such a UF catastrophic pressure drop in inflation and subsequent
expansion affect the companion field of our ordinary World A matter of atoms
and molecules embedded in it?

Figure 16 shows
one possibility, namely a smooth adiabatic expansion of World A matter from the
Big Bang down to the present time.

There is however
the more likely possibility that the UF inflation also caused a brief
inflationary expansion of World A ordinary matter as well. This was followed by
an adiabatic smoother expansion to the present day condition as depicted in
Figure 17 below:

__Figure 17. World A expansion from Big Bang with initial inflationary phase followed by
adiabatic expansion__

__4.7 (I) Critical Point Intersections of the
UF and World A Expansion Curves____ __

We have several points not yet addressed. (1) There is the
formation of our physical World A of condensed matter particles – atoms and
molecules- in the Big Bang and subsequent expansion to the present time which
is examined in Part II.

(2) There is the fact that at the

(3) There is the
possibility of different
phase transformation from World A
matter to World B matter ( A- to- B transformation ) taking place at the
Critical Point p* (Fig. 16) which may be related to the problem of the nature
and occurrence of the so-called ‘dark matter’ in our cosmos which is currently
thought to make up some 24% of the total mass-energy. This possibilitywill be
explored in Part II to follow.

__ ____4.8. (I) Current accelerated
isothermal expansion of the UF: Dark Energy ?__

If the UF, at the Critical Point p* (Fig. 17)
were to continue in an isothermal
expansion it would, in effect be retracing its original path in Step 1. This
increase in UF energy might explain or be related to the problem of the
recently observed accelerated expansion of the cosmos and the so-called ‘dark
energy’ which has been proposed as a solution [6,7,8].

This completes our present exploratory cosmology for the Binary
Universe of World A plus the UF, so far
as the Carnot Cycle of the UF is concerned.

__Fig. 18. Summary of UF and World A
evolutionary possibilities __

__ and Critical Point
Intersections__

__4.9.(I)
Summary__

Based on a
standard physics of compressible fluid flow, the concept of a binary universe
consisting of ordinary matter and the UF has provided a number of fundamental
new insights that argue for its validity. It is based on only one physical
assumption.

Starting
with this single unifying postulate of the existence of a compressible fluid field, the origin and evolution of a
Binary Cosmos has been presented, first the
emergence of the of the tenuous vapour field of the UF in a fluid
rupture and isothermal Carnot expansion, then its adiabatic collapse generating
the enormous concentration of energy at the Big Bang,
and forming the baryons of ordinary matter, then
the Big Bang expansion and evolution of atomic matter, and then either
(a) a continuation of the UF Carnot
cycle in an adiabatic expansion towards the present cosmic time, or, more
likely, (b) a sudden inflationary episode followed by an isothermal
re-expansion.

The
dynamic and thermodynamic properties of
the UF - a development of the
Chaplygin/Tangent gas – have been explored. It is seen to constitute a unique
wave and force field with the ability to account for the transfer of
electromagnetic radiation through space and to explain the origin and transfer
of gravitational force through space.

The
present cosmos is envisaged as being a Binary Universe, a co-existing dual
entity consisting of ordinary matter and the co- evolving dynamic UF wave and
force field. In Part II (under preparation) the origin of matter, the origin of
dark matter and the nature and evolution of the dark energy will be examined.

The field equation
of the UF is an integral equation of state which permits a unique solution with
boundary conditions being the total dynamic energy ( enthalpy) of the universe.

It provides an
explanation for the origin of both electromagnetic radiation and gravitational
force and their transmission across cosmic and local space.
Moreover, this concept is verifiable experimentally with optical and other
means.

It provides [Part
II, in preparation] an explanation for the origin of ordinary matter as being a
condensed energy state arising out of an
enormous compression of energy
which started the Big Bang, and which, in its subsequent expansion and
evolution after the Big Bang, accommodates the successful Standard Model of
particle physics.

It provides a physical
basis for the proposed inflation of the early universe immediately after the
Big Bang.

It provides [Part
II, in preparation] an explanation for the nature and origin of dark matter
within the same theoretical framework of compressible fluid flow as being a
rarefied, energetic, gravitating
substance related to and generated from ordinary matter.

It provides an
explanation for the ‘dark energy’
postulated to explain the recently observed increased rate of cosmic
expansion,.

It provides a
unification of quantum physics and gravitational force.

The ability of a
theory to provide a linked explanation for so many basic, disparate, physical
phenomena from the single physical postulate of the reality of an
energetic compressible fluid medium is obviously a strong argument in favour of
its general validity and its being worth detailed theoretical examination and
further experimental verification.__ __

__References and Notes__

1.
Turner, M. S., Quarks and the Cosmos. *Science*,
**315**, 59,

2.
Bergmann, Peter G. *Introduction to the
Theory of Relativity* .

3.
McCrea, W. H., Cosmology. *Reports on
Progress in Physics*. Pp 321-363, Vol
XVI, 1953.

4.
Chaplygin, S., *Sci. Mem. **Moscow** **Univ.** Math.Phys*. 21, 1 1904.

5.
Shapiro, A. H. *The Dynamics and
Thermodynamics of Compressible Fluid Flow. * 2 vols. John Wiley and Sons,

6.
Bachall, N.A., Ostriker, J.P., Perlmutter, S., and P.J. Steinhardt. The Cosmic
Triangle: Revealing the State of he Universe. *Science*, **284,** 1481 1999.

7.
Kamenshchick, A., Moschella, U., and V. Pasquier. An alternative to
quintessence. *Phys. Lett. B ***511**, 265, 2001.

8.
Bilic, N., Tupper, G.B., and R.D. Violier. Unification of Dark Matter and Dark
Energy: The Inhomogeneous Chaplygin Gas. *Astrophysic
, astro-ph/0111325*. 2002.

9.
Lamb, Horace., *Hydrodynamics. *^{th} edition. 1932 .

10.
Brennen, Christopher E., *Cavitation and
Bubble Dynamics*.

11.
Frenkel, J. *Kinetic Theory of Liquids*.

12.
Kittell, Charles. *Introduction to **Solid** **State** Physics*. , 6^{th}. ed. John Wiley & Sons
Inc.,

13.
Courant, R. and Friedrichs, K. O. (1948). *Supersonic
Flow and Shock Waves*. Interscience,

14.
Power, Bernard A., Some of the work
leading up to the present theory has appeared in connection with studies
into the scientific basis for the image
formation on the Holy Shroud of Turin. Some of these are as follows:

15. ---------------, Il Meccanismo di
Formazione dell’Immagine dela Sindon di Torino, *Collegamento pro Sindone*, Mgggio-Giugno, pp13-28, 1997, Roma.

16.---------------, Caratterizzazione di
una Lunghezza d’Onda per la Radiazione che Potrebe aver Creato I’Immagine Della
Sindone di Torino. *Collegamento pro
Sindone*, Roma. Novembre-Decembre, pp. 26-36, 1999.

17.---------------, An Unexpected
Consequence of Radiation Theories of Image formation for the Shroud of Turin. *Proc. Worldwide Congress Sindone 2000,*

18.---------------, Image Formation on the
Holy Shroud of *Collegamento
pro Sindone website* __(__www.shroud.it/) March 2002.

19.---------------, How Microwave Radiation Could Have
Formed the Observed Images on The Holy Shroud of *Collegamento
[ro Sindone Website*, Jan. 2003. (www.shroud.it/)--------------,
Shock Waves in a Photon Gas. *Contr. Paper
No. 203, American Association for the Advancement of Science, Ann. Meeting, *

20.---------------. Unification of Forces
and Particle Production at an Oblique Radiation Shock Front. *Contr. Paper N0. 462. American
Association for the Advancement of
Science, Ann. Meeting, *

21.---------------, 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, *

22.---------------,
*Summary of a Universal Physics*.
Monograph (Private distribution) pp 92. Tempress,

23.
----------------,Appendix A ( below) Thermodynamic properties of an isothermal
gas law for the Chaplygin/tangent field. 2006.

24. ---------------, Appendix B (below) A New Explanation for the Anomalous Weak
Tensile Strengths of Liquids and Solids. 2006.

25.----------------, Appendix C (below) A Derivation of the Schrödinger Equation
from the Concept of a Universal Wave Field. 2006.

26. ----------------, Appendix D (below) Philosophical and Theological Caveat. 2007.

27------------------, (a) www.shroudscience.info and open the page entitled __Properties of a Universal Wave Field__.

(b) www.energycompressibility.info

28. Guth, Alan.
*The Inflationary Universe.*

__Copyright Bernard A. Power,
April 2007__

__Appendix A__

**Thermodynamic properties of an isothermal
gas law for the Chaplygin/tangent field**

Bernard A. Power^{a)}

2006

An isothermal equation of state is
formulated to match the adiabatic Chaplyin/tangent gas, and it appears to be a
general gas law for a whole field
whose thermodynamic properties are very
unusual. In this field a working
substance expands with cooling and contracts with heating, and both processes
take place without any work being done. The 2^{nd} Law is inapplicable.

The theoretical
or exotic fluid known as the Chaplygin/tangent gas has an adiabatic equation of
state which has been widely studied and applied in aerodynamics and fluid dynamics for many years.^{1,2,3,4,5,6} However, the adoption by cosmology of the
negative pressure attribute of the Chaplygin gas as a possible solution to the
observed increased expansion of the cosmos
is only recent. ^{7,8,9}
A potential role as a universal cosmic fluid, however, would seem to
merit an examination of all aspects of
this field.

The equation of
state for the Chaplygin gas is the linear relationship p = − Av, (pv^{-1} = −A) ; it lies in
Quadrant IV of the p-v diagram where it
always has negative pressure. The
dynamically identical Tangent gas has an added constant and is also linear, with p = − Av + B, ( pv^{-1} = −A + B] ; in Quadrant I it has positive pressure (Fig.
1).

The
Chaplygin/tangent gas can also be described by the adiabatic equation of
state pv^{k } = −A, where A is a positive constant, k has the value of −
1, and where the minus sign preceding
the constant A provides a desired negative slope –dp/dv on the pressure-volume
diagram.

The proposed isothermal equation of
state for this field is p = +Av corresponding to the adiabatic Chaplygin gas, and p = +Av –B
corresponding to the adiabatic tangent
gas. If the
positive constant A is multiplied by a
constant__ __temperature T_{c}, we would then have p/v = A (T)_{c}
, p = vAT_{c}, and p = vAT_{c} − B, which provide the desired isothermal
relationship, provided that A takes on
the proper dimensions. The isotherms in
pressure and volume are seen to be
strictly orthogonal to the adiabatics of the Chaplygin and Tangent gas. (Fig. 1)

Since v =
1/ρ , these linear isothermal relationships in pressure and volume ( p =
vA T_{c} , etc ) can also be
written hyperbolically in pressure and specific density ρ as

pρ =
AT
(1)

The ideal gas law ( pv = RT) is hyperbolic in pressure and volume, but linear in pressure and density ( p = ρRT). The proposed new isothermal equation of state for the k = −1 field is the reverse, being linear in pressure and volume and hyperbolic in pressure and specific density.

It would appear
that the new relationship is not
so much just an isothermal counterpart to the Chaplygin/tangent
gas, but
rather that it plays the more fundamental role of being the general gas
law of the field, with the Chaplygin and tangent cases being simply the
exceptional modifications to the gas law that apply under the adiabatic
condition ( dQ = 0), just as is the case with real adiabatic gases which depart
exceptionally from the ideal gas law. The term
*universal field* ( UF)
would embrace both the isothermal and
adiabatic equations and where the “isothermal” equation we have derived would
become the basic gas law ( pρ = AT
)for the whole field.

The thermodynamic properties of this UF are very unusual because of (1) the direct proportionality of pressure and volume, (2 a negative specific heat either at constant volume or at constant pressure and (3) the positive slope +dp/dv of the isotherms on the pressure-volume diagram and its effect on expansion or contraction..

First, the isothermal pressure increase with volume expansion is the inverse of that for the ideal gas

Second the entropy
relationships in the UF are also unusual since we have k = c_{p}/c_{v} = −1,
so that either c_{p} or c_{v} must be negative. In the ideal
gas with
T and V as independent variables the entropy change is ∆S = c_{v} ln (T_{2}/T_{1})
+ R ln(V_{2}/V_{1}). And with T and P as independent variables,
it becomes ∆S = c_{p} ln (T_{2}/T_{1}) – R ln(P_{2}/P_{1})

In the UF we have
dS= c_{v} dT + (∂P∂T)_{v} and, from
P =AvT we have (∂P∂T)_{v} = Av^{2}/2 so that we obtain the
corresponding entropy changes in the
UF as
∆S = c_{v} ln (T_{2}/T_{1}) + Av^{2}/2
and ∆S = c_{p} ln(T_{1}/ T_{2}) ) −Av^{2}/2

(In the UF the
magnitude of the reversible heat Q_{r} needed to calculate the entropy
change ∆S is readily visualized on the pressure volume
diagram, since the Carnot cycle is then depicted simply as the rectangle formed
by the appropriate linear adiabatics and
isothermals, and Q_{r} is the
area enclosed).

Third, the
positive slope +dp/dv of the isothermals on the pressure-volume
diagram implies that there is no resistance to
pressure/volume increases or decreases, which would therefore take place at
ever increasing speed following any initial impulse.. This is analogous to
an expansion into a vacuum in the usual
analysis of an isothermal process in an
ideal gas where the work pdV is then zero
In the UF in an isothermal process it would seem that this work would
also necessarily approach zero. This would mean complete reversibility of all
UF processes, both adiabatic ( Chaplygin/tangent) and isothermal, with no
preferred direction for any physical
change. All complete cycles in the UF
would appear to be isentropic. Heat would
be convertible into internal energy but not into work.. The 2^{nd}
law of thermodynamics would not apply. Clearly the properties of this unusual
field should be further explored and critically evaluated.

__References__

1.. S. A.
Chaplygin, “On Gas Jets,” Sci. Mem. **21**, 1 (1904).

2. H.S. Tsien, “Two-Dimensional Subsonic Flow of
Compressible Fluids,”* J. Aeron. Sci .*

3. T. Von Karman, “Compressibility Effects in
Aerodynamics,” *J. Aeron. Sci*. **8, **337** (**1941).

4. A. H. Shapiro,* The
Dynamics and Thermodynamics of Compressible Fluid Flow.* 2 Vols.( John Wiley
& Sons, New York, 1953).

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

6. Horace Lamb, * Hydrodynamics*. 6^{th} ed . ( Dover Reprint, Dover Publications Inc.

7.
N.A. Bachall, J.P. Ostriker, *Science, *** 284**,
1481 (1999).

8 A. Kamenshchick, U. Moschella and V. Pasquier, “An alternative to quintessence,” *Phys *Lett*. *B
**511**, 265 (2001).

9. N. Bilic, G.B. Tupper and R.D.Viollier, “Unification of Dark Matter and Dark Energy:
The Inhomogeneous Chaplygin Gas,” *Astrophysics*, astro-ph/0111325 ( 2002).

__Appendix B__

__A
New Explanation For the Anomalous Weak
Tensile Strengths of Liquids and Solids__

**B**ernard A. Power

December, 2006

The observed tensile strengths of liquids and solids are
orders of magnitude lower than the theoretical values. The discrepancy is
explained by heterogeneous nucleation of
the ruptures. However, the purely spherical expansion of rupture voids or bubbles may involve an
adiabatic expansion of pv^{7} =
const., giving much lower theoretical
tensile strengths which are in agreement
with the observations. The concept should be of interest to materials science.

__Introduction__

Theoretical estimates of the
tensile strength of solids and liquids give values of around 3 x 10^{9} to 3 x 10^{10} N/m^{2. } ( 3 x 10^{4 } to 3 x 10^{5} atm.). However, for solids the experimental values
are around 100 times smaller, while for
liquids, such as water, the
observed values are even smaller at 50
to 200 atmospheres [1,2].

The simple Frenkel derivation
[2,3] of the theoretical tensile
strength of solids or liquids gives a
fractional volumetric expansion
ratio ∆V/Vo needed to cause
rupture and form the bubble, and this
then is equated to an average numerical
value of about 1/3. Then, since liquids
and solids have compressibility moduli K which are about 10^{10}
to 10^{11 } kg/m^{2} ( i.e. 10^{5} to 10^{6}
atmospheres ), we have a rupture
pressure p_{max} = −K(∆V/Vo). Taking the average 1/3 value for ∆V/Vo, the heoretical rupture pressure p_{mac } then is between 3 x 10^{4 } and 3 x
10^{5} atmospheres

Curiously, however, the high theoretical values of tensile strength do match the observed critical temperature for rupture by boiling. Why then does the theory fail in the case of rupture by tension where the observed values are so low?

For solids, the discrepancy in tensile strength is usually ascribed to heterogeneous nucleation of rupture at defects such as cracks or dislocations in the lattice [1]. In the case of liquids, the even larger discrepancy is explained by invoking the presence of irremovable tiny gas or solid nuclei within the liquid which act to lower the pressures and tensions needed for mechanical rupture [3]. Still, there remain discrepancies, and the foreign nuclei explanation, or heterogeneous nucleation process, acting alone, have appeared somewhat artificial, especially since the thermal rupture (boiling) values do agree with the theory.

Recently it has been realized that a mechanism exists in the cavitation process which may act to greatly lower the tensile strength predictions of theory and to reconcile them with the observations.

The basic mechanical equilibrium equation for the production of a spherical void, or vapour- filled bubble, in a liquid by rupture is

p_{B}
− p_{L } =_{ }∆p_{max} = 2 σ /R_{C
}(1)

which gives the relationship between the vapour pressure in the bubble p_{B}, the pressure in the surrounding bulk liquid p_{L},
the interfacial surface tension σ and the radius of the spherical bubble
at the rupture radius R_{C} .

The formation of a bubble
by rupture requires a negative pressure exceeding the
tensile strength 2 σ/R in
order to create the spherical void, R being typically the bond length in
liquids, say around 2 x 10^{10} m. At this rupture radius, with σ
having the value for water of around 10^{-2} N/m ( 100 dynes/cm), ∆p_{max } has the value of 10^{8 } kg/m^{2} ( 10^{4} atm.). _{ }

Any incipient void in the liquid may fill with vapour molecules. In expansion this has little effect; in bubble collapse however, it is significant.

While liquid water has a value for
k = c_{p}/c_{v} of
around 1.1, it does not ordinarily have a well defined equation of state
pv^{k} = const. However, Courant and Friedrichs [4] discuss the
expansion and contraction of spherical blast waves in water and fit the
experimental data to a quasi-equation of state for water under a pressure of
around 3000 atm. which is pv^{7} = const or p =A
ρ^{7 } + B. They also derive
the same value of k = 7 as a
theoretically solution to their
non-linear flow equations for spherical
( i.e. radial) shock expansions in fluids.

If we now apply k = 7 to purely radial ruptures and void or bubble initiation in water, the Frenkel derivation of tensile strength becomes

p_{
max} = −K ((∆V /Vo)_{}^{7} = −K(1/3)^{7}
(2)

Using the rupture pressures above, we get the theoretical
tensile strength of liquids such as
water to be − 10^{4} x [1/3]^{7} = 45.7 atm.
In addietion, −10^{5}
x [1/3]^{7} = 457 atm, which is now in general agreement with the
experimental values.

This result may be expressed in a more general form by noting that Equation 1 is derived on the assumption that
the temperature of the liquid is
constant [2,3]. In this case k has the
value of unity and the relation between p and
v is the isothermal one pv+^{1} = constant. Equation 1 can then be rewritten as

∆p = const./ ∆V = 2 σ /R_{C
}(2a)

But if the isothermal assumption
is replaced by an adiabatic radial expansion with k = 7 we would have pv^{7} = const., and so

∆p =
const./ (∆V)^{7} =
2 σ /R_{C
}(2b)

Consequently for any given value
of R_{C} the radial adiabatic volume change ∆V now enters to the
seventh power, explaining the observed order of magnitude of rupture forces in
the general theoretical context of an equation of state for the rupture
process.

To sum up, we now have two cases, the current theoretical one for random kinetic volume expansions with pv = const. and the spherical ( i.e. radial) volume expansion, each with different initiating rupture pressures.

Case 1.

__Random kinetic motion expansion__:.

Equation of State is pv = const.

V = 4/3 π R^{3 } ; R ≈ (V)^{1/3} and
p_{max} = 2σ /R ≈
2σ/ (∆V)^{1/3 } , so that
p_{max} is inversely proportional to (∆V)^{1/3}

Case 2.

__Spherical ( radial) expansion:__

Equaton of Sate is: pv^{7}
= const.

R ≈ (V^{7})^{1/3}
≈ V^{2.333}
p_{max} = 2σ
/R ≈ 2σ/ (∆V^{7}
)^{1/3 } ≈ 2σ/ (∆V)^{2.333} and here
p_{max} is inversely proportional to ^{ . }(∆V)^{2.333} .

Clearly, bond breaking in
spherical expansion will be much more efficient than random kinetic or thermal
expansion, since the increase in R will be large for smaller changes in
pressure. This agrees with the experimental rupture pressure data above. ^{ }

For example, for a volume
fluctuation ratio +∆V of 5, Case 2
would give a radius increase of 5^{2.333}
= 42.5. Clearly, radial expansion
is a very efficient bond breaking
and rupture mechanism

The energy source of the purely radial or spherical expansion motions can be thermal waves, ultrasonic source waves , point source cosmic ray impact etc. etc.

For solids the rupture flow, because of structural and steric hindrance to radial orientation by an appropriate energy source, may be only quasi- radial, and a value of k between 4 and 6 might then be appropriate, giving tensile strengths higher than liquids but still well below the classical theoretical estimates.

The proposed model would still require simultaneous radial rupture over a sufficient number of adjacent bonds, and therefore the theory of rate analysis still applies. At the physical level the rupture can still, of course, be either homogeneous or heterogeneous, and all the various physical mechanism are still in play. In particular, the vibration cavitation process [3] is of interest, since it would supply an orderly negative pressure perturbation over a wide enough field to bring about effective simultaneous rupture under the radial flow assumption. It would appear that the new model should be of interest to materials science.

__References:__

1. Kittell, Charles. (1968) *Introduction
to **Solid** **State** Physics*. , 6^{th}. ed. John Wiley & Sons Inc.,

2. Brennen, Christopher E., *Cavitation and Bubble Dynamics*.

3. Frenkel, J. (1955). *Kinetic Theory of Liquids*.

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

** **

__Appendix C__

__Physical Derivation of the Schrödinger
Equation of Quantum Mechanics from the Concept of a Universal Wave Field__

Bernard A. Power^{a)}

July 2006

**1. Introduction**

**2. Schrödinger’s Derivation**

**3. UF Derivation**

**4. New Insights**

**5. Analogies to classical mechanics**

**6. Conclusions**

__1. Introduction__

The Schrödinger equation for
the hydrogen atom is based on the classical wave equation in the form of the
Helmholtz oscillator. But the classical wave equation holds exactly only for
the special case of the theoretical, compressible field or fluid called the
Chaplygin/Tangent gas.

The
theoretical or exotic fluid known as the Chaplygin/Tangent gas has an
isentropic/adiabatic equation of state which has been widely studied and
applied in aerodynamics and fluid
dynamics for many years [3-7]. Recently,
because of negative pressure attribute of the Chaplygin gas it has been
proposed by cosmologists as a possible
solution to the observed increased expansion of the cosmos [1,2]. Such a potential role as a universal cosmic
fluid, however, would now seem to merit an examination of this field in
relation to the quantum mechanics, starting with the basic Schrödinger equation
for the hydrogen atom..

__2. The Schrödinger Equation for the
Hydrogen atom__

Schrödinger based his derivation on the assumption
of the classical wave or exact equation
which, for compressive waves and expressed in terms
of the wave function ψ for amplitude, is [7],

Ñ^{2} ψ^{ } =^{ }1/c^{2}
∂^{2}ψ/∂t^{2}/ [ 1 + Ñψ ]^{(k
}^{+ 1)
}(1)

or, for one
dimensional motion in the x-direction

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

This exact
equation means that for material gases ( k > 0) compressive waves are always
unstable and grow with time . For very small amplitude waves however, the term
in the denominator involving ∂ ψ /∂x approximates to unity, and the equation
simplifies to becomes **the classical wave
equation **

Ñ^{2 }ψ = (1/c^{2 })∂^{2}ψ/∂t^{2}
(3)

^{ }

which has the general solution

ψ = ψ_{1(}x – ct) + ψ_{2} (
x – ct)
(4)

In the case of the
UF, however we see that, since k = – 1, the exponent ( k + 1) in the denominator of Eqs. 1 and 2 becomes
zero, thereby automatically
reducing the equation to the
simple classical wave equation, but without any of the approximation needed for
real gases such as air.

The UF is
therefore truly unique in that it automatically becomes exact
for waves of any wave amplitude, large or small
and is no longer limited to infinitesimal waves as is the case with real gases .The UF is therefore unique
among gases it is the only field in which the
Classical Wave Equation is strictly valid and which therefore can transmit
stable waves, of either condensation or rarefaction, of any amplitude.

For periodic motions the classical wave equation assumes the form

Ñ^{2 }ψ + [4π^{2}ν^{2}/c^{2}]
ψ = 0
(5)

called the
Helmholtz equation which applies to periodic motions such as for
the simple harmonic oscillator or for sinusoidal oscillations.

Since c/ ν =
λ we can express it also as

Ñ^{2 }ψ + [4π^{2}ν^{2}/
λ^{ 2}] ψ = 0
(6)

So far this is
classical mechanics, but at this point Schrödinger introduced the de Broglie
relationship

λ = h/mv
(7)

which gives us
the equation

Ñ^{2 }ψ + [4π^{2}m^{2
}v ^{2}/h^{2}]ψ = 0
(8)

Next he expressed
the kinetic energy ½ mv^{2 } as
[E – V] to get

Ñ^{2 }ψ + [8π^{2}m^{2
}/h^{2}] [ E –mV]ψ = 0
(9)

which is the
time-independent form of his famous equation.

We see that
Schrödinger’s derivation is straightforward classical wave theory up to the point of his introduction of the de
Broglie relationship λ = h/mv.

__2. The Derivation of Schrödinger’s Equation from the theory of the Universal Field ,
including derivation of the de Broglie
Relationship __

If we now turn to
the concept of the Universal Wave field, we see that the classical wave
procedure obviously also fits right up to the same key point of the de
Broglie insertion. In fact, since this is the unique field supporting the wave
equation,in the UF it fits even better than with Schrödinger’s original
derivation.

In addition, we now show that the de Broglie
relationship, in UF theory, becomes a *physical * requirement and no longer an arbitrary step
as before.

To see this,
consider the kinetic energy equation for compressible fluid flow

c ^{2} = c_{o}^{2} – V^{2}/n −2cV/n
(10)

where c is the
actual wave speed, c_{o }is the static wave speed, V is the relative
flow velocity and n is the energy partition parameter related to the adiabatic
exponent k as k = (n+2)/n. As in all compressible flow theory the mass m is
taken as being unsteady and is not
explicitly shown. In the case of the UF we have n = − 1 .

The term 2cV/n is the unsteady energy of accelerated
and pulsed flows. It leads directly to the de Broglie relationship as follows

2/n cV= E = hν = (m) λν V, so per cycle of the pulse ( i.e. dividing
through by ν) we have

h = mν λ = p λ
(11)

From this new
viewpoint the de Broglie relationship is
just the energy per pulse cycle (
E/ν) of an accelerated or pulsed flow in the UF expressed in terms of momentum ( p = mV)
and the pulse wavelength λ. Therefore, its introduction into the
Helmholtz periodic or standing wave relationship becomes natural and necessary to complete the energy
balance. All previous arbitrariness in the derivation of the Schrödinger
equation thereby vanishes.

Again we can
understand the derivation of de Broglie relationship from the UF theory by
considering the velocity additions c + V and c – V. The corresponding energies
for unit mass m are the squares of the velocity sums, namely
m(c^{2} + V^{2} + 2cV)
and m(c^{2} –V^{2} – 2cV) respectively. If we understand mc^{2}
as the energy of the UF, and mV^{2} as twice the kinetic energy , then
mcV becomes the pulse energy .

Finally, we can
look at the velocity addition of a pulse and the corresponding interaction
energy as

(c + dc)(V + dV)
= cv + cdV + Vdc +dcdV

Clearly for small V relative to c all terms are
negligible except for the cV term which we then take as being the
non-relativistic interaction energy of the mass particle. We also point out that in the UF any flow speed V whatever results in a wave speed c greater than the
static or “no-flow” wave speed c_{o}.
On the other hand, in the UF, shock
waves can never form, since, while the Mach 1 condition can be approached, it can never be reached.

The
non-relativistic” character of the Schrödinger equation is also in accord with
this derivation since, for small flow velocity V, the energy term (m)V^{2} is negligible in comparison
with the 2cV term where c is always large.
Hence, for “non-relativistic” flows only the latter term ( expressed as
the de Broglie relationship ) is needed. Also, with small relative velocity V,
the departure of the local wave speed c
from the maximum or static value c_{o}
is very small and so c_{o} can be used.

We can see this
from the flow kinetic energy equation

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

The relativistic
correction centres around the departure of c from the stagnation wave speed c_{o}, that is to say, the
Schrödinger equation assumes implicitly that c = c_{o,} as do all special relativity theories. This will be small either for small V or for
larger n. Therefore, for more complex
assemblages or interactions of the UF with matter such as with the de Broglie pulse interaction, the
departure becomes progressively smaller and the non-relativistic approximation
becomes improved.

__References __

1] A. Kamenshchick, U. Moschella, and V.Pasquier, Phys. Lett. B 511
(2001) 265-268.

[2] N. A. Bachall, J.P. Ostriker,

[3] S. Chaplygin, Sci. Mem.

[4] H.-S. Tsien,
J. Aeron. Sci. 6 (1939) 399.

[5] T.. von Karman, J. Aeron. Sci. 8 (1941) 337

[6]
A. H. Shapiro,* The Dynamics and
Thermodynamics of Compressible Fluid Flow.* 2 Vols. John Wiley & Sons,

[7]
Lamb, Horace, *Hydrodynamics*, ^{th} edition, 1932 .

__APPENDIX D__

__Philosophical Caveat__

Any
theory of cosmology inevitably brings up matters of interest to philosophy and
theology such as the origin and nature of the universe, creation, evolution,
and so on. It may be useful, therefore, to state as clearly as possible what
new scientific raw material this new cosmological theory may provide for these
two associated fields of human interest and study. The three intellectual disciplines of
science, philosophy and natural theology, while rationally related, are
autonomous and have different legitimate aims and methods. Most scientists are
not expert in philosophy or theology, nor do all philosophers and theologians
always properly understand science and scientific theories. In some cases, on the part of science, this
can be the unavoidable consequence of a technical language or of a compressed
presentation of theory. Consequently, we add a few words to try and avoid some
unnecessary potential misinterpretations which might arise in the present case.

First, the new
theory’s basic postulate is one of physical realism. The UF is experimentally
verifiable. It is not a purely theoretical or ideal construct.

Second, the
theory is rational. The Greek philosophical principle of non-contradiction
applies. That is to say, it must be self-consistent and open to experimental
falsification or verification according to ever emerging experimental data.
There can be no internal ambiguities or inherent contradictions here, such as
are held to be possible in some philosophies.
If the theory is shown to be self-contradictory or significantly at
variance with the data, it falls. Even if it should become widely accepted, it
will still be subject to adjustment, and revision, even radical revision,
according to the data.

Third, the basic
physical postulate is radically contingent. That is to say the initial
postulate of the existence of a universal primordial cosmic compressible fluid
entity does not contain within itself any compelling reason that explains its
own existence. It is not self-evident
that it must exist, or has always existed and so on. Its scientific justification lies in its
power to explain the experimental data.
In other words, we maintain that the UF is real, and that it can be
verified experimentally, but we have no knowledge of why it exists at all.

This can perhaps
be seen most succinctly in its equation of state pv^{k} = constant, where the constant
has the physical dimensions of an energy. It can theoretically have any value,
positive or negative; so we cannot on theoretical or on mathematical grounds
exclude any value, including zero and infinity.
This raises the philosophical question of why the constant has the
actual value that it does. Why is it
this value, and why not some other value, or even why is it not zero, nor why
can’t it eventually become zero and then reappear, or not, or even become
infinity? These questions, while very relevant, are not for science to
consider; science takes the best observed value or estimate and proceeds to
study how the compressible entity actually works or behaves. Therefore, this new scientific theory is
apparently a philosophical statement that its basic postulate is radically
contingent, with no self-contained reason for its existence. Its validity is
based on its remarkable ability to resolve many important and basic scientific
questions from a single scientific postulate of a compressible continuum, and
to do so in significant conformity with the experimental data.

This radical
contingency apparently raises a fundamental philosophical and theological
matter, since *radical contingency* according to philosophers brings up the
philosophical necessity of a creation or a beginning, and thus of a
Creator. One self-styled “modern pagan philosopher”,
Mortimer Adler, has drawn attention to this, and even labels the existence of
such radical contingency as being ‘proof
of a Creator’ at an__ __acceptance level of
‘beyond reasonable doubt’ [29].
On an even broader basis, the assertion of the physical reality of the
new cosmology would appear to meet Lonergan’s criterion [29] for what constitutes philosophical *reality*, namely something which
can be ‘ intellectually grasped and reasonably affirmed’, with the same *exnihilation *consequences for
philosophy. That is to say, the UF is
real if it obeys the rational laws of compressible fluid flow and if its
theoretical predictions are critically verifiable in the experimental data.

As an example of the difficulties that can arise out of an apparently
minor alteration in a concept in one discipline, there is the definition of ** physical matter** in the present
theory. While the philosophical doctrine of materialism is no longer much in
fashion since the advent of quantum physics, clearly it will be important in
any new definition of matter, or in a new theory of its origin, to get things
straight. Here, we have defined matter
as a condensed energetic ‘form’ emerging from the UF, and originating in the
compression of the UF immediately preceding the Big Bang. This makes physical
matter a ‘form’ with compressed energy, and definitely a quantified or physical
entity. Also, the ‘dark’ matter as described in the theory is just as physical
as ordinary molecular matter, but it is of a different nature in that it
appears to be a physical form with a rarefaction of its energy density. Again,
the electromagnetic invisibility and tenuousness of the UF may thereby make it
prone to misinterpretation by those imaginatively inclined who may be tempted
to confuse it with “spirit”. Here, a
distinction between matter and spirit by Lonergan [30] may be helpful. He
maintains that the spiritual is what the material is not. For example, he
points out that the material is always

Clearly the fundamental concept of energy needs to be
treated with care in any passage from science to philosophy and vice versa.
Millennia ago Anaximenes proposed,
following Thales and Anaximander , that all creation is a manifestation
of one essential basic substance, but then in an extraordinary insight he added
that all the different physical manifestations of existence we observe are
various __condensations and rarefactions__
of this elemental substance. The element he chose was atmospheric air --
an excellent basis for meteorology but not for modern physics. It was however,
a remarkably close miss, being apparently the first introduction into cosmology
of the notion of material transformations resulting from compressibility.

And yet the
concept of the UF introduced here, while describing a compressible, energetic
or dynamic continuum or field, does not depict any imagined “sea of energy”. We
have no scientific basis for that concept. Energy as we know it does not exist *per se * but only in various definite quantified forms,
such as elementary particles, atoms, molecules, photon, quanta and so on. The
nature of the UF is that which is to be described by or grasped from its
physics of compressible fluid flow; it is not to be found in an imaginative
extension or reinterpretation. The
pitfalls attendant upon the introduction of imaginative constructs into science
are dealt with in Lonergan’s *Insight*
[30] where he calls them “extra- scientific categories”. Such “attempts to use images in the place of
concepts or to seek the absolute in the physical sphere” [31] have crept into
cosmology as far back as

Finally, any scientific
theory must be revisable as other more concise theories are formulated, and
especially as new experimental data become available. Consequently, any
extension of a theory to philosophy and natural theology will also be revisable.

To take one historical
example of what can happen, Albert Einstein had a very clear grasp of the
physical consequences of his theories.
A reading of the literature on relativity shows, however, that it was almost
immediately reinterpreted philosophically and ideologically in innumerable
ways, such as, for example, in the assertion that science now had established
the ‘relativity’ of everything, even of
truth and objective reality itself. Einstein never sanctioned this
nonsense, and his direct and vigorous approach to scientific truth and reality
is evident in one of his later remarks where he was discussing the possibility
that Prof. Dayton C. Miller’s exhaustive
optical repetitions of the Michelson Morley experiment at

The postulate of
the existence of the UF as a universal, physical, compressible continuum will
also probably bring up the question of whether it also constitutes a
quintessence or plenum [28]. These questions are at the interface between science
and philosophy and care needs to be taken to hold to the actual scientific
properties postulated for the UF and not to uncritically extend them during
philosophical examination of the theory.

28. Sciama,
Denis, W., Mach’s Principle 1979-1979.
Great Ideas 1978. 57-67, Encyl.
Britannica, Inc. 1978.

29. Adler,
Mortimer, *How to Think about God: A Guide
for the 20 ^{th-} Century Pagan. *

30. Lonergan,
S.J., Bernard, *Insight: A Study of Human
Understanding.* Philosophical Library Inc.,

31 Power, J.E., Henry More and Isaac Newton on
Absolute Space: An Extra-scientific Category. *J. History of Ideas*. pp
289-296.Vol. XXXI, No. 2, April-June 1970.

__Note: __**Part
II of this monograph entitled Origins of Matter, Dark Matter and Dark Energy
is under preparation and will be posted here when it**

**
becomes available. **

__Copyright Bernard A. Power, April 2007__