Equations of State

for

Cosmic Fields

Summary

The presently disparate cosmic fields of dark energy, dark matter, visible or ordinary matter, quantum electrodynamics, and gravitation are examined by compressible energy flow theory and verifiable explanatory compressibility patterns are found. For example, it is possible to fit thermodynamic equations of state for each field.

These cosmic equations of state also display a remarkable symmetry, except for one set of quantum linear states on the pv energy diagram which intersect at the origin point where there is a curious physical discontinuity.

Shock wave condensations are proposed for the origin of ordinary matter; the strong shock process for the baryons and hadrons, and the weak shock option for the leptons and electron. The mass ratios of the elementary particles of matter are found to fit theoretical compressibility predictions to within about 1%.

Cosmic state interactions and transformations are discussed.

Contents

1.0 Introduction

2.0 Outline of Compressible Fluid Flow

3.0 The Main Cosmic Fields and Their Possible Equations of State:

3.1 Visible Matter

3.2 Quantum Fields and Electromagnetism

3.3 Dark Matter

3,4 Dark Energy

3.5 Gravitational Field

4.0 Interactions and Transformations between States

5.0 The Transformation of Visible (Baryonic) Matter to Dark

Matter May Yield an Accelerated Cosmic Expansion

6.0 The Problem of the Discontinuity at the pv-graphical Point of Origin

7.0 Cosmology, Empirical Science and an Integrated World View

8.0 Summary

References

Appendix A. Maxwell’s Electromagnetic Waves and Compressible Flow

Appendix B. Summary of a Universal Physics

1.0 Introduction

At present, the cosmic fields – dark energy (68.3%), dark matter (26.8%), ordinary baryonic matter (4.9%), electromagnetic and quantum fields, and gravitation-- are partially understood, and they remain mostly separate theoretically or conceptually.

We offer an outline of the concept of compressible fluid flow ( compressible energy flow) as a potential unifying principle for physical cosmology.

Some previous work towards this goal [1,2,3] has shown that basic quantum concepts, such as Planck’s constant, the quantum wave function, the de Broglie wave/particle equation and so on, can be expressed in terms of the formalism of compressible fluid flow. In addition, the mass ratios of the elementary particles of matter can be derived from compressible flow shocks and their compression ratios – the strong shock for the hadrons and the weak shock for the leptons.

Here we shall attempt to incorporate the cosmic fields into compressible Equations of State. We shall approach this with a brief review of compressible flow fundamentals with emphasis on examples of how the theory fits various physical fields.

Naturally, such an attempted overview of intricate and developed fields must be tentative, and the present approach is offered as such.

2.0 Outline of Compressible Fluid Flow

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

2.1 Steady State Energy Equation

c² = c_o² − V²/n (1)

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

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

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

V_max = √ n c_o.

2.2 Unsteady State Energy Equation

c² = c_o² – V²/n – 2/n (dφ/dt) (2a)

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

c² = c_o² – V² /n − 2 (cV)/n (2b)

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

2.3 Lagrangian Energy Function L

L = (Kinetic energy) – (Potential energy) = ( c²+ V²/n) – c_o² , and so, from (2b)

L = − (2cV)/n (3)

2.4 Equation of State for Compressible, Ordinary Matter Systems

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

pv = RT ; p/ρ = RT = constant (4)

Equation 4 is seen to be isothermal (T= constant) . For adiabatic changes it becomes pv^k = constant, where the adiabatic constant k = c_p/c_v is the ratio of the specific heats at constant pressure and at constant volume, respectively.

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

Hyperbolic Equation of State (Ideal Gas Law)

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

2.5 Waves and Flow

2.5.1 The Classical Wave Equation

∆²ψ = 1/c² [∂²ψ/∂t²] (5)

where ∆² = ∂²../∂x² + ∂²..^./∂y² + ∂²../∂z²; ψ is the wave amplitude, that is, it is the amplitude of a thermodynamic state variable such as the pressure p or the density ρ. The local wave speed is c.

The general solution of (5) is

ψ = φ₁ (x – ct) + φ₂ ( x+ ct) (6)

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

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

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

Unsteady State ( Accelerating) Compressible Flow

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

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

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

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

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

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

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

2.5.2 The Exact Wave Equation

Ѳ ψ = 1/c² ∂²ψ/∂t² [ 1 + Ñψ ]^{(k + 1)} (7)

where k, the adiabatic exponent, is k = c_p/c_v = ( n + 2) /n; and ( k + 1 ) = 2( n + 1)/n = 2(c_o/c*)². Here, pressure is a function of density only. This wave is isentropic, non-linear, unstable, and grows to a non-isentropic discontinuity called a shock wave.

2.5.3 Shock Waves

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

2.5.3.1 Normal Shocks

V₁ > V₂ (8)

p₁, ρ₁, T₁ < p₂, ρ_2, T₂ (9)

Entropy Change Across Shock:

∆S = S₁ – S₂ = − ln(ρ₀₂/ρ₀₁) (10)

Maximum Condensation Ratio:

ρ₁/ρ₂ = [n+1]^1/2 = V_max/c* (11)

2.5.3.2 Oblique Shocks

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

Oblique Shock

V_1N > V_2N

p₁ρ₁T₁ < p₂ρ₂

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

2.5.3.3 Strong and Weak Oblique Shock Options: The Shock Polar

For each inlet Mach number M₁ ( = V_N1/c), and turning angle of the flow θ, there are two physical options:

1) the strong shock ( intersection S) with strong compression ratio and large flow velocity reduction (p₂ >> p₁; V₂ << V₁, or

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

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

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

2.6 Types of Compressible Flow:

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

b) Steady, supercritical flow ( e.g. supersonic, V₁ > c) governed by hyperbolic, nonlinear, partial differential equations.

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

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

2.7 Wave Speeds

c = [ c_o² – V²/n]^1/2 (steady flow) (12)

c = c_o² – V²/n – [(2/n)cV ]^1/2 (unsteady flow) (13)

c² = (dp/dρ)_s, where s is an isentropic state.

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

The shock speed U is always supercritical (U > c) with respect to the upstream or oncoming flow V₁.

2.8 Wave Speed Ratio c/c_o and The Isentropic Thermodynamic Ratios

c/c_o = [1 -1/n(V/c_o)²]^1/2 = (p/p_o)^1/(n+2) = (ρ/ρ_o)^1/n = (T/T_o)^1/2 (14)

All the basic thermodynamic parameters of a compressible isentropic flow are therefore specified by the wave speed ratio c/c_o. .

2.9 Relativity Effects in Compressible Flow: In compressible flows all velocities [V or u] are relative only, and, moreover, the wave speed c is a variable which is dependent on V and n; it decreases for larger velocities V. and it reaches its maximum value c_o at the static state i.e. at zero flow (V =0).

Interestingly, Equation (14)

c/c_o = [1 -1/n(V/c_o)²]^1/2 = (p/p_o)^1/(n+2) = (ρ/ρ_o)^1/n = (T/T_o)^1/2 (15)

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

The differences from special relativity are that the wave speed c is now a variable and a function of the flow velocity V, and that there is the energy partition constant n. Since the wave speeds are low ( c = 334 m/s for air at m.s.l.), the ‘Lorentz’ corrections for physical compressible systems such as gases are relatively large. Also, the flow speeds can exceed the wave speed ( supersonic flow), whereas in special relativity theory, the wave speed c is a constant ( 3 x 10⁸ m/s) which can never be exceeded.

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

2.10 Wave Stability

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

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

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

2.11 Elliptical Equations of State and Rarefaction Waves

II

	+p
-v

+v

-p

Elliptical Equation of State favours Rarefaction Waves and Shocks

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

1. If d²p/dv² is > 0 (i.e. the hyperbolic curve) compression waves form and steepen, while rarefaction waves flatten and die out. Only compression waves of infinitely small amplitude (“acoustic” or “sound” waves” ) are stable.

2. If d²p/dv² is < 0 , ( i.e. the elliptical curve) rarefaction waves form and steepen, while compression waves flatten and die out.

.

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

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

3.0 The Cosmic Fields and Proposed Compressible Equations of State

The main cosmic fields known to physical cosmology are:

3.1 Ordinary, Visible Matter (4.9% of the cosmos)

3.2 Quantum Fields of the elementary particles of matter, and the electromagnetic field

3.3 Dark Matter (26.8%) of the cosmos)

3.4 Dark Energy ( 68.3 % of the cosmos)

3.5 Gravitation

Of these, the theory of the quantum field (including E/M) and the hyperbolic field of ordinary matter are the most fully developed. In fact, of course, it is from ordinary fluid matter ( liquids and gases) that the concepts of compressible flow have been developed.

Here, we shall briefly look at each field in turn from the viewpoint of compressible flow features and put forward equations of state.

3.1 Ordinary Visible Matter

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

First, we examine the question of What is Matter? Here we shall show that compressible flow theory gives a direct answer to this crucial question. Clearly, matter is formed from elementary particles. But a deeper or more ultimate question is: How do the elementary particles themselves arise? Here we shall show that, if we start with energy as a compressible entity, then the elementary particles can arise naturally from compressible theory as energy shock wave condensations in a compressible energy flow.

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

We propose that : All elementary particles of matter (with the possible exception of the neutrino) are condensed energy forms produced from hyperbolic equations of state by compression shocks. .

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

A. Origin of Hadrons (Baryons and Heavy Mesons)

Maximum Compression Ratio

m_b/m_q = V_max/c* = [n+1]^1/2 (16)

m_b is the mass of any hadron particle, m_q is a quark mass, V_max = c_o n^1/2 is the escape speed to a vacuum; that is, it is the maximum possible relative flow velocity in an energy flow for a given value of n, the number of degrees of freedom of the energy form, This is a non-isentropic relationship which corresponds physically to the maximum possible strong shock. .

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

Table A) Hadrons (Baryons and Heavy Mesons)

--------------------------------------------------------------------------------------------

n n +1 [n+1]^1/2 Particle Mass (m_b) Ratio to

( MeV) quark mass

_____________________________________________

0 1 1 quark (ud) 310 MeV 1

quark (s) 505

1

2 3 1.73 eta (η) 548.8 1.73

3

4

5 6 2.45 rho (ρ) 776 2.45

6

7

8 9 3 proton (p) 938.28 3.03 (1)

neutron (n) 939.57 3.03

Λ (uds) 1115.6 2.97 (2)

Ξ^o (uss) 1314.19 2.99 (3)

9 10 3.16 Σ⁺ (uus) 1189.36 3.17 (2)

10 11 3.32 Ω^- (sss) 1672.2 3.31 (4)

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

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

B. Origin of Leptons, Pion and Kaon

m_L/m_e^- = k/α² = [(n+2)/n]/α² = {(n+2)/n] x 137 (17)

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

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

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

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

Table B) Leptons, Pion and Kaon

a

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

(MeV) to x 1/137

Electron

__________________________________________________________

1/3 7 Kaon K^± 493.67 966.32 7.05

2 2 Pion π^± 139.57 273.15 1.99

4 1.5 Muon μ 105.66 206.77 1.51

- - Electron 0.511 1

Clearly, column 2 values for k ≈ m_l/m_e (1/137) closely match column 6 for the mass ratio reduced by 1/137, thus verifying Equation 17 and the theory that the leptons are formed by weak shock condensation. .

Summary

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

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

Equation of State of Ordinary Liquids and Gases

The Equation of State of ordinary compressible cosmic matter ( gas and some liquids) is some form of the ideal Gas Law, which a hyperbolic curve on the pressure volume diagram:

pv = constant = RT

(a) For isothermal motions (T = constant) in a real gas, the equation of state therefore nis just the ideal gas law.

pv = RT or p/ρ = RT

Ideal Gas Law: A hyperbolic equation of state for Visible matter

(b) For real physical gases undergoing adiabatic motions ( i.e.(no heat flow, dQ = 0) the general equation of state is :

pv^k = constant (20)

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

Figure 1. Pressure-volume in Compressible Fluids

Quadrant 1: Real gases and Tsien/Tangent gas (exotic)

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

Equation of State of the Visible Cosmos: The Hyperbolic Ideal Gas Law

pv = const. = RT

The Hyperbolic Cosmic Equation of State of Visible Matter

Linear Exotic States: The Tsien/ Tangent Gas and the Chaplygin Gas

These linear gases were first proposed by Chaplygin [8] and then the tangent gas by Tsien [7]. In this report we apply them in a much broader sense to define the linear quantum waves. [ Then, in conjunction with their orthogonal counterpart, they form a set of transverse waves having the form of Maxwell’s electromagnetic wave equations, as we shall see below in Section 3.2.]

Linear Chaplygin gas and Linear Tsien/ tangent gas

3.2 Quantum Fields ( principally the Electromagnetic Field or the Field of Quantum Electrodynamics)

These fields describe the sub-atomic world that produces the elementary particles of matter, the hadrons and leptons whose assemblages form the atoms described by the Schrödinger equation. These matters are described by quantum theory which is highly successful, highly developed, and highly verified.

We shall not attempt any comprehensive application of the concept of compressibility of energy to all the various highly developed fields of quantum physics. We shall attempt only to show that, in select examples, compressible wave theory can successfully formulate some important aspects of quantum theory, implying that compressible action is physically involved in some cases and at some level. On this basis, for example, we can see the linear wave equation as a basic descriptor of quantum electrodynamic phenomena . First the select examples:

The basic quantum wave function Ψ Is the wave amplitude. Then, , if the basic quantum wave function Ψ is defined in a wave se as Ψ = c + v , then the interaction energy term 2cv from [c + v]² = c² + 2cv + v² appears in most of the fundamental quantum relationships:

A. The above ‘Extra Energy’or ‘Interaction energy’ term 2cv then yields the following fundamental quantum relationships:

A) Planck’s Constant, h

For n = 1, if cV = constant energy for each set of waves, then cV/υ = constant energy per cycle or pulse:

cV/υ = h

cV = hυ = hω/2π = ħω = ε_υ

And, for the complex case:

cV/υ = ħ/i = −iħ

B) De Broglie Wave/Particle Equation

cV/υ = h

But c/υ = λ; V(m) = p (momentum), so λp = h, or

p = h/λ

C ) Lagrangian Function, L

L = 2cV

D) Quantum Wave Function Operators

a) Hamiltonian Energy Operator

cV= − hυ = −ħω = ε

icV = − ∂../∂t, and so

cV = ħ/i ( ∂../∂t) = +iħ∂../∂t = H_op

which is the Hamiltonian energy operator. ( To ensure correct dimensions, it must be applied to the normalized quantum function ψ_N).

b) Momentum Operator

cv = hυ = +ħω = ε

v = (1/c))ħω, or (m)V = p = (m)(1/c)ħω

Multiplying by i, we have:

(m)iV = (m)(1/c) iħω= (m)(1/c) ħ ∂../∂t

So, we have

(m) V = p = -(m)(1/c) iħ ∂../∂t

But,

(1/c) ∂../∂t = ∂../∂x, and so

(m)V = p = -iħ∂../∂x = p_op

which is the quantum wave operator, ( to ensure correct dimensions, it must be applied to the normalized quantum function ψ_N).

E) Heisenberg Uncertainty Principle

cV = hυ; cV/υ = h

λV = h

But λ = Δx and V(m) = Δp, so

Δx . Δp ≥ (m) h

which is the Heisenberg uncertainty principle.

E) Origin of Elementary Particles of Matter as Shock Condensations in a Quantum Compressible Flow

This has been dealt with above in Section 2 on the origin of baryonic material particles and leptons to form Visible Matter and is repeated here for convenience:

We propose that : All elementary particles of matter (with the possible exception of the neutrino) are condensed energy forms produced from hyperbolic equations of state by compression shocks. .

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

A. Origin of Hadrons (Baryons and Heavy Mesons)

Maximum Compression Ratio

m_b/m_q = V_max/c* = [n+1]^1/2 (16)

m_b is the mass of any hadron particle, m_q is a quark mass, V_max = c_o n^1/2 is the escape speed to a vacuum; that is, it is the maximum possible relative flow velocity in an energy flow for a given value of n, the number of degrees of freedom of the energy form, This compression ratio is a non-isentropic relationship which corresponds physically to the maximum possible strong shock. .

Experimental verification for this hadron mass ratio formula is given in Table A Section 2. above.

B. Origin of Leptons, Pion and Kaon

m_L/m_e^- = k/α² = [(n+2)/n]/α² = {(n+2)/n] x 137 (17)

where α = 1/11.703 is the fine structure constant, and k is the adiabatic exponent or ratio of specific heats, k = c_p/c_v = [(n+2)/n].

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

The leptons are formed via the weak shock option

The experimental verification for the lepton mass ratio formula is given in Table B Section 2 above.

Summary

Equations 16 and 17 give, uniquely , an experimentally verified explanation for the origin of matter as being a shock condensation from a compressible energetic field., and for the observed ratios for the masses of the elementary particles of matter. The principle of the compressibility of energy flow, therefore, appear to underlie all material particles and the whole material universe.

.

Summary of Compressible Flow and Quantum State Effects: We have been able to formally relate fundamental quantum relationships to a single compressible energy pulse term, 2cV.

We have, in effect, quantized the various energy ‘fields’ represented by 2cV/n for various values of n, by equating them to the ‘time-like’ condition set by the frequency υ in the quantum equation hυ = 2cV/n.

(Note that these equations, as is usual in compressible flow theory, are for ‘specific’ energy, that is, for unit mass flow. For a definite particle, the numerical value of the mass is to be inserted; the dimensions of the equations being not thereby changed, since in our system, mass (m) is a dimensionless ratio. Thus for the photon, we have hυ = m_γ cV, where m_γ is the relativistic mass of the photon. In terms of the momentum, p we have

hυ = cp,

which is the de Broglie equation].

Considering the above relationships, it seems reasonable to conclude that there is something fluidic and compressible involved in quantum physics. We suggest that it is the compressible flow of wave energy (c +V)² which underlies this apparent unification of the various quantum relationships listed above.

Quantum Equation of State : The Equation of State concept has not at present been applied to all types of quantum fields.

The quantum reactions produce baryonic matter composed of elementary particles organized into atoms and molecules. We have shown above that these elementary particles are associated with compressible flow shock condensations. The appropriate equations of state for these particles would seen to be the quantum wave-particle equations such as the Schrödinger Equation, the Dirac Equation, the Klein-Gordon wave equation, the Weyl Equation , the de Broglie wave particle equation.

However, for the cosmic quantum electrodynamic field i.e. for electromagnetic radiation, we can propose as its Quantum Equation of State the following linear wave equation corresponding to the Tsien/ tangent gas equation of state and the Chaplygin gas, but now extended from Quadrant I into all four Quadrants:

p = ± Av ±B

which, for intercept values A = 1 and B = 0, becomes just

p = ± v

.

− v

Linear Equation of State p = ±Av ±B Linear Quantum Fields Equation of State

p = ± v

Note

The linear radiation cosmic state equations p = ± v ,which pass through the pv-diagram’s Origin Point, raise a problem of thermodynamic continuity at the point where the wave or flow is postulated to move from one Quadrant to another. A barrier arises at this origin point where the specific volume v values and pressure p must go to zero. Thus, the thermodynamic energy pv equals zero and the thermodynamic temperature goes to absolute zero.

We must now ask How can a physical wave of either rarefaction or compression pass such a physically discontinuous zero energy state/. Physical i.e. thermodynamic existence seems impossible in this case.

These and other questions are troubling because the linear wave equation of the quantum state is the dynamic link with the other four cosmic states. In some cases it is the only communication within and between those states, for example, of the hyperbolic state with its isolated versions in all four quadrants. The cosmic symmetry which would unify the various cosmic states via linear radiation state interaction is broken.

To repeat, this is clear from the fact that a state of absolute zero tempersture is required at the origin point [since from pv = RT = 0] . This thermodynamic barrier apparently prevents any passage or transmission of a wave or signal through the origin from one quadrant to another. The geometrical or mathematical symmetry remains, but the physical or thermodynamic symmetry is broken.

[We should perhaps note that the discontinuity at the origin in the case of the p = ± v wave equation refers only to the physically real equation. In the case of a purely mathematical or geometrical equation y = ± x there would be no such discontinuity problem since zero values for the variables are permitted and the equations are then properly seen as mathematically continuous through the origin point.]

Compressibility and Quantum Electromagnetic Waves : Evidence for Transverse Waves in a Tenuous Fluid

We have indicated that some quantum waves are linear and are supported by linear equation of state. Since there are two such linear states one of which is adiabatic and the other which is orthogonal to it and is isothermal, we then have the interesting possibility that this orthogonal set can support transverse waves in form identical to Maxwells electromagnetic waves.

There is a detailed discussion of this in: Appendix A: Maxwell’s Electromagnetic Waves and Compressible Flow

.

3.3 Dark Matter

Some known characteristics of the dark matter are as follows :

1. It interacts with ordinary matter very weakly, and apparently only with the weak force. 2. It causes gravitational lensing of electromagnetic radiation. 3. It clumps to form denser entities more readily then ordinary matter. 4. It appears to be denser than ordinary matter. 5. It makes up about 26.8% of total cosmic substance. There have been no generally accepted equations of state for dark matter. The reactions with neutrinos are important but not considered here.

In the case of ordinary visible matter, we have stated above that its state as a cosmic expanding fluid is approximately fitted by the ordinary hyperbolic gas law pv = RT and that its adiabatic form is pv^k = const. where k is the adiabatic constant.

Proposed Elliptical Equation of State

We propose an elliptical equation of state for the dark matter as follows :

P² /b² - v²/a² = a²b²⁼

	Visible Matter pv = const. = RT
+p

Dark Matter: Elliptical Equation of State

State Properties and Dark Matter Properties: In the elliptical state, rarefaction waves are demanded and rarefaction shocks are possible. This is seen to be the opposite or counterpart to visible matter’s hyperbolic behavior which requires compression waves and compression shocks.

Dark matter, if elliptical as theorized, may thus be a ‘form’ of rarefied matter, interacting with visible matter only via the weak interaction or possibly by neutrinos. Its rarefied forms would be the product of rarefied shocks, both the strong rarefaction shock and weak rarefaction shock, Its stable entities would thus be ‘rarefied forms’, both strong and weak.

In the graph of the elliptical equation of state for dark matter, we have shown the case where dark matter and visible matter contact one another at a tangent point in Quadrant This suggests a possible transformation from one system to another, for example a transformation from visible condensed matter to invisible rarefied dark matter. In Sections 4.0 and 5.0 below we deal with such an interaction in some detail.

Depiction of the Conjunction of Visible matter, Dark matter and Linear State matter

3.4 Dark Energy

This form of energy, filling all space, is currently calculated to constitute about 68.3% of the total observable cosmos.. Because of its property of having negative pressure, it is considered to act to accelerate the expansion of the cosmos, according to its equation of state:

p/ρ = − 1

How do we depict this equation of state? First, since the specific density ρ = 1/v, then we have

pv = − 1

While this negative value for pv energy cannot be depicted in Quadrants I and III, which have pv always positive, it does fit into Quadrants II and IV as shown.

Negative values for compressible

energy pv = -1 In Quadrants II and Quadrant IV

The Proposed Dark Energy Equation of State: To derive the corresponding equation of state of which the −1 value for pv is a point, we note that pv = const is an equilateral hyperbola. This corresponds to the ideal gas hyperbolic equation of state in Quadrant 1, but now we are in the negative [−pv] Quadrants II and IV, so that the Equation of State is

(-p)v = const.

It is depicted as an equilateral hyperbola in Quadrant IV.

Visible matter

pv = const. =RT

.

The dark energy’s most striking property is its negative pressure, which fills the need of physical cosmology to explain the observed acceleration in the expansion rate of the cosmos. The negative pressure property of Quadrant IV has already been proposed [9,10,11] via a linear extension of the Chaplygin/Tsien/ Tangent Gas from Quadrant I down into Quadrant IV for the same purpose of explaining the observed acceleration in the rate of expansion of the cosmos [See also Section 5 below].

[The theoretical possibility of a dark energy state with negative energy but positive pressure also existing in Quadrant II exists, but its properties and possibilities for physical cosmology remain unexplored.]

The dark energy’s wave forms, being hyperbolic, would be compressive and compression shocks would be permissible. Low amplitude, acoustic type compression waves of dark energy would be allowable. Rarefaction waves of dark energy would be suppressed.

3.5 The Gravitational Field

Characteristics of Gravitation: The principal characteristics of gravitational force to be properly accounted for in a new theory are: (1) Its universal action on all mass entities, (2) its exclusively attractive nature for mass, (3 )its weakness relative to the electromagnetic force, and (4) its 1/r² decline in strength with distance.

The nature of gravitational force:

a) Newtonian and General Relativistic Theories: Newton very successfully described gravitation as a force acting on all point masses which fell off in strength as the inverse square of the distance separating the mass points.

In his General Relativity, Albert Einstein expressed gravitational force as a consequence of motion in a tensorial curved space time continuum. It was successful in correctly calculating certain small aberrations from Newton’s theory, such as the magnitude of the advance in the perihelion of the planet Mercury. It also correctly describes other astronomical problems so that General Relativity is the core of the standard theory of physical cosmology today.

Uniform compressible flow theory and special relativity have some formal similarities. Both, for example, yield the Lorentz relationship. [see Sect. 2.10 above ]. General relativity can be seen as a limiting case of compressible flow when the energy distribution parameter or degrees of freedom n approaches infinity so that virtual incompressibility sets in. In this incompressible limit the field acquires a tensor character which can be related to non-uniform motion and their forces.

b) Gravity and compressible flow theory:

We propose a circular equation of state for the gravitational field ( i.e. in all four Quadrants):

Visible matter

Hyperbolic Eqn. of State pv = const. = RT

= const. = RT

+p

P² + v² = r²

			+v
−v

(1) This equation is universal with respect to the other fields in that, since it is circular, it affects all four Quadrants equally,

(2) Because of its negative curvature [ d²p/dv² = −ve] on the pv diagram, the gravity waves it carries will be rarefaction or dilatational ; as such they will exert an attractive force see (Section 2 above, under Wave Stability ) on any baryon mass object they encounter.

In the above depiction, the radius of the circular equation of state for gravity has, for symmetry depiction, arbitrarily been set at the dimension to give tangency with the hyperbolic equations of state for visible matter and dark energy in Quadrants I and IV.

While the simplicity of the circle as an equation of state for gravity is appealing, still, the conceptual schemes and procedures for applying it to actual masses and the other states are still obscure. Undoubtedly, the concept of ‘centre of mass point’ from Newtonian theory , i.e. the point at which all the mass of an extended body is concentrated for calculation of gravitational force, will be involved. The definition of ‘interaction’ and of an allowable range of interaction intervals, for example at a tangent point between states as compared to an intersection point, will need thought.

These and other implications remain to be explored, and the uncertainty will remain until observational data are available to support the proposed circular equation of state.

4.0 Interactions and Transformations Between Cosmic States

The possibility would seem to exist that, at a tangent point or at an intersection, where two states share identical values of pressure and specific volume, one state may transform into the other.

Such interaction concepts should be straightforward for visible matter and quantum fields where so much work has already been done. Dark matter and dark energy are more problematical, and gravitation, as has been pointed out in Section 3.5 above, may be more difficult still.

Elliptical (circular) and linear states can (theoretically) exist in all four Quadrants and assume different or exotic forms as the numerical values of the state variables change sign. The possibility of these transformation might be open to experimental verification on the local scale.

Sharp pressure fluctuations or pulses would seem necessary for the pv energy point to cross from one Quadrant to another as, for example , with linear and elliptical or circular states. In such changes, one state variable must assume the zero value for the pulse to cross from one Quadrant to another. Pressure pulses of large magnitude would probably be of special interest in triggering state transformation of this type.

One example of a cosmic transformation between quadrants would be the proposed Chaplygin gas transformation from Quadrant 1, with its positive cosmic pressure, to a Quadrant IV state where the pressure is negative state. This possibility has been used to explain the observed acceleration in the rate of expansion of the visible universe [9,10,11].

Hyperbolic states are confined to one quadrant only. Our world of baryonic matter can therefore exist in Quadrant I only.

A second class of transformations would be those from one state to another state within a Quadrant, e.g. from hyperbolic to a tangent or intersecting linear or elliptical state. Possibly this transformation type may be open to experimental verification on the local scale. At tangent points other interactions may occur. For example, in Quadrant I quantum state interactions with baryonic matter interactions must certainly occur. We explore the possibility of a transformation of visible matter ( hyperbolic) into dark matter ( elliptical) in Section 5 below.

Cosmic Equations of State

−v

Note: In the above graph, State No.2 – Elliptical – has been graphed as being circular simply for reasons of graphical economy, the difference between elliptical and circular being simply one of a difference in the magnitude of the axial intercepts a and b for the elliptical state , and being equal for the case of the circle.

Interactions: Between gravity (circular) and all other States.

Between quantum radiation states ( linear) and all other States in the same Quadrant

Transformations: From hyperbolic to elliptical in all quadrants, and vice versa

From elliptical in one quadrant to elliptical in all other quadrants

We see that hyperbolic forms are fixed to a single quadrant. This is the case with our world of visible, baryonic condensed energy type matter. On the other hand, rarefied dark matter, if elliptical, as assigned here, might exist in four Quadrant forms..

5.0 The Transformation of Visible (Baryonic) Matter to Dark Matter May Yield an Observed Accelerated Cosmic Expansion

Let us first assume that a visible to dark matter transformation can occur, and that it yields a change of state but no change in total rest mass. This leaves us with a need to export any kinetic flow energy in the transformation system. We can estimate this exportable energy as follows:

K.E._visible = c² + V² +2ncV Assume n = 1

K.E. _{dark matter} = c² + V² + 2ncV Assume n = − 1, then we have

the numerical difference, or net exportable kinetic flow energy, in the said transformation:

Δ K.E._transf. = 0 + 0 +4cV= 4cV

Our question now is: Where does the exported energy 4cV go?

Since the proposed physical change is one from the compressed energy matter of the visible hyperbolic world to a rarefied energy of the elliptical dark matter the physical change must consist of a pressure drop or rarefaction pulse. And, reflection will show that, while such a large pressure drop could take either or both matter states to the zero pressure line , still their energy could pass into the negative pressure of Quadrant IV only for a pulse in which there is no rest mass.

Now, in our equation of state proposed system, the state with no rest mass is the Linear Wave State, shown in the Figure as the tangent line from the interaction point I running down into Quadrant IV where the 4cV energy is now negative. This is because p and pv are negative) in Quadrant IV). This linear wave in Quadrant I is called the the Chaplygin gas or the extended Tsien tangent gas system [7 ,8 ]. In our system it is also the quantum wave system with equation of state p = ± Av +B].)

Such a transformed flow into negative pressure in Quadrant IV has already been proposed by a number of researchers [ 9, 10, 11] to explain the observed acceleration in the expansion of the cosmos.

It seems supportive of our proposed cosmic equations of state and state transformation hypothesis that it should yield a straightforward physical basis for the negative pressure hypothesis which had little physical basis when it was first put forward [9,10,11].

We have thus supplied a physical basis for a possible export of transformed excess kinetic energy from Quadrant I into Quadrant IV via the linear wave state, where it supplies negative cosmic pressure and merges with the dark energy pool of the cosmos.

6.0 The Problem of the Discontinuity at the pv-Graphical Point of Origin

The simplest linear cosmic quantum radiation equations of state p = ± v which pass through the pv-diagram’s Origin Point raise a problem of thermodynamic or physical continuity. A barrier arises at this origin point where the specific volume v and pressure p both go to zero. Thus the thermodynamic energy pv = 0 and the thermodynamic temperature T = 0. .

We must now ask: Can a physical wave, of either rarefaction or compression, pass continuously through such a physical discontinuous zero energy state? It would seem clear that that they cannot.

These questions are troubling because the linear wave equation of the quantum state is the dynamic link with all the four Quadrants of physical states. . In some cases this would seem to be the only physical communication within and between those states, for example, of the hyperbolic state in Quadrant I with its isolated versions in the other three. The cosmic symmetry which would unify the various cosmic states via interaction with linear radiation states is broken.

To repeat, zero tempersture is required at the origin point [since pv = RT = 0] . This thermodynamic barrier prevents any passage or transmission of a wave or signal through the origin from one quadrant to another. The geometrical symmetry is physically and thermodynamically broken.

We should perhaps note here that the discontinuity at the origin in the case of the p = ± v wave equation refers only to the physically real equation. In the case of a purely mathematical or geometrical equation y = ± x there would be no such discontinuity problem since zero values for the variables are permitted and the equations are then properly seen as mathematically continuous through the origin point.

Summary of Cosmic States

−v

Note: In the above graph, State No.2 – Elliptical, has been graphed as being circular simply for reasons of graphical economy, the difference between elliptical and circular being simply one of a difference in the magnitude of the axial intercepts a and b for the elliptical state , and being equal for the case of the circle.

The understanding that we have hopefully achieved is that cosmological structures and processes involve principles of compressible energy flows and their interactions. The cosmic thermodynamic state equations that we have proposed exhibit a strong symmetry, especially strong for the linear radiation state which for the case of p = ± v pass symmetrically through the pv-energy diagram’s origin. And yet, this latter appealing symmetry is broken by the problem of a continuous wave or radiation entity being unable to pass through the thermodynamic barrier of pv = 0 at the origin.

This raises an unexpected , speculative, but rationally positive possible solution as follows:

1. Cosmic thermodynamic symmetry is strongly suggested by the compressible equations of state; but this is broken by the discontinuity at the Origin Point for the case of the linear wave state p = ± v.

2. A solution, preserving complete cosmic symmetry, would require the presence at the Point of Origin of some intrinsically unquantified dynamic entity.

3. But, an ‘intrinsically unquantified dynamic entity’ is, probably what philosophy would term a requirement for spirit.

4. Therefore, the logical argument emerges that “if complete thermodynamic cosmic symmetry exists, then spirit exists.”

This rather astonishing but intriguing result is one for Philosophy and Natural Theology. It rests logically, of course, on the appropriateness and validity of the proposed scheme of cosmic equations of state. This scheme has been proposed to explain facts of physical cosmology. If the requirement for complete thermodynamic cosmic symmetry is discarded then the above theoretical argument is also to be discarded.

The argument, of course, also cannot be compelling, since experimental verification is barred, and it is therefore based on postulates and logic, and, any deletion or defect in these eliminates the argument. Still, it seems somehow persistent and calling for an answer. The assessment must be rational with input data from the three disciplines of science ( cosmology), philosophy and natural theology.

Conclusion

To repeat, if complete thermodynamic symmetry is desired for the cosmic equations of state, then the key quantum radiation equation of state [ p = ±v] lacks continuity at and across the Origin.

Therefore, if this thermodynamic cosmic symmetry is made a requisite, it may require the introduction, at the origin point discontinuity, of an intrinsically unquantified dynamism, -which is philosophically a spiritual, dynamic entity, -- in order to supply dynamic continuity through the origin from one quadrant to another and so to complete the desired cosmic symmetry.

+p

	ХХХ
		+v
-v

A Perfectly Symmetrical Thermodynamic Cosmos, (Made Possible by the Presence of ‘Intrinsically Unquantified Dynamism’ i.e. of Spirit Х ?)

-p

F]’

7.0 Cosmology, Empirical Science and A More Integrated World View

In the previous Section 6.0 on the physical discontinuity at the origin, we came to a tentative, very unusual conclusion which invoked a philosophical definition. This may reasonably seem unusual in a report on physical cosmology. Therefore some additional remarks on this point may be in order.

The understanding that we have hopefully achieved is that cosmological structure, process and cosmic states involve principles of compressible energy flows and their interactions, and that they therefore show a common structure and nature. Since compressibility is a well established field of physical knowledge, we should thereby have achieved some of the same scientific unification for cosmology. But have we?

For there is an important difference here. John Polkinghorne [1below] has pointed out this difference between cosmology and evolutionary biology from the rest of science. Both of these scientific fields, he points out, employ all the methods of scientific inquiry except for one, namely experimental verification, which is denied to them. The cosmos cannot be experimented on and the evolutionary past is not experimentally accessible either.

Still, cosmology is a true science and reaches valid insights and valid and fundamental knowledge. Polkinghorne also points out that in this respect cosmologists and evolutionary biology methodologically have much in common with philosophy and natural theology, which also reach their understandings [2 below] in the same rational intellectual manner, and yet are likewise denied the satisfaction of experimental verification.

Thus, in a deep sense, when the results of cosmology are examined by philosophers and theologians and discussed by them with scientists, all are then employing the same intellectual methods and the same tools of logic and reason and all three stand on the common ground of reason. This necessary communality merits some thought, since in it we may have the seeds of a wider mutual acceptance and understanding concerning various views of ultimate reality,

It may be useful at this point, then to re-think the situation from the viewpoint that therein may lie the beginnings or the elements of A Re-integrated World View , one acknowledging the distinctions and the similarities of Science, Philosophy and Natural Theology.

The fields of human theoretical inquiry are commonly taken to be Science, Philosophy, Theology.

1. Science ( excluding Cosmology and Evolutionary Biology) :

Its data and scope concern the physical world.

Its method is rational, insightful [2 below], mathematical and empirical.

Its verification or validation is rational and experimental (empirical).

2. Philosophy: Its data and scope concerns the entire word of reality i.e .the world of being.

Its method is rational, insightful and logical.

Its verification or validation is rational, critical and judgmental

3. Natural Theology: Its data and scope are the data and conclusions from Science and Philosophy

Its method is rational, insightful and logical i.e. as with philosophy.

Its verification or validation is rational, critical and judgmental i.e. as with

philosophy.

4.Theology :

See Lonergan’s “Method in Theology” [2 below].

5. Cosmology and Evolutionary Biology

Their data are observations from the physical world.

Their methods are rational, insightful and mathematical.

Their verification or validation are those of Philosophy and Natural Theology, namely

rational, critical and judgmental. Scientific Experimentation is ruled out by their historical and cosmic nature.

Thus we seem to have the possibility of a merging of science at its cosmic margins with philosophy and natural theology, and perhaps therefore the outlines of a Reintegrated World View. The latter would eventually involve a general understanding and acknowledgement of the separate aims, methods and rational validation standards for each main field of human intellectual endeavor.

The influence of such a rational and valid Integral World Vew on the multifarious areas of human civilized activity --artistic, social, economic, governmental and political--would appear to be undoubtedly beneficial.

Section 7 References:

1. Polkinghorne, John, C., Science and Creation SPCK, London , 1988

2. Lonergan, S.J., Bernard. Insight: A Study of Human Understanding. Philosophical Library Inc., New York, N.Y, 1957. World views surveyed by Lonergan are: Aristotelian, Galilean, Darwinian, Quantum Indeterminate.

………………, Method in Theology. Herder and Herder, New York, 1972

8.0 Summary

We have shown how a postulated compressible energy field and its flows can fit and unite the main cosmic fields known to physical cosmology via symmetrical equations of state. Clearly also, there are many known facts and aspects not considered here. The proposed equations of state will have to undergo much critical examination.

In particular, we would emphasize the need for a thorough thermodynamic analysis for each equation of state and field, and in all four pv- quadrants in each case. The entropy behaviour in each instance is also of great interest.

A caveat seems appropriate here. Namely, that the above treatment using equations of state, although it touches on exotic and alternative states and possibilities, is nevertheless always scientific and physical. These alternate states in physical cosmology should not be an invitation to uncritical extrapolation or imaginative speculation. They do touch on natural theology and philosophy and such matters have been mentioned in Section 6 above.

[There is one persistent problem that remains. This is the matter of the broken cosmic symmetry with the simplest linear wave equation of state equations p = ± v, that is to say with the set of linear state equations that pass through the coordinate origin where the pv energy is zero. This problem can raises extra-scientific questions which are dealt with in Section 6 ].

−v

+v

−v

−p

+p

+p

References

1. Power, Bernard A., Unification of Forces and Particle Production at an Oblique Radiation Shock Front. Contr. Paper N0. 462. American Association for the Advancement of Science, Annual Meeting, Washington, D.C., Jan. 1982.

2. .---------------, Baryon Mass-ratios and Degrees of Freedom in a Compressible Radiation Flow. Contr. Paper No. 505. American Association for the Advancement of Science, Annual Meeting, Detroit, May 1983.

3. .---------------, Summary of a Universal Physics. Monograph (Private distribution) pp 92. Tempress, Dorval, Quebec, 1992. ( Appendix B: Summary of a Universal Physics”)

4. . Shapiro, A. H. The Dynamics and Thermodynamics of Compressible Fluid Flow. 2 vols. John Wiley and Sons, New York, 1953

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

6. Lamb, Horace., Hydrodynamics 6^th ed. Dover, New York, 1932.

7. Chaplygin, S., Sci. Mem. Moscow Univ. Math. Phys. 21 (1904).

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

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

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

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

Copyright, Bernard A. Power, September 2017

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Appendix A

Transverse Waves in a Tenuous Field:

Maxwell’s Electromagnetic Waves and Compressible Flow

,

Compressibility and Electromagnetic Waves : Evidence for transverse waves in a tenuous fluid

Here we shall show that compressible flow theory and the two proposed orthogonal linear equations of state p = ± v can produce transverse waves in a shear free compressible fluid, so as to fit with the established transverse nature of electromagnetic waves.

( The following insert is from UF pages) needs editing to fit in here,,

Material 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. Its 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 Linear Wave Field with its mutually orthogonal adiabatics and isotherms?

We consider a simple pressure pulse ( ±∆p) in the orthogonal wave field:

A pressure pulse ( ±∆p) in the Orthogonal Wave Field

The initial or static state is designated as p_o. When the pressure pulse ( +∆p) is imposed from outside in some way, the wave field 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 constant pressure disturbance ( +∆p) must propagate in the direction of the initial impulse ,but, since the there are two orthogonal components of the pulse are the only way 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. Vectorially, this requires an axial wave vector V in the direction of propagation ( say z) 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.

E

Electromagnetic Poynting energy /vector

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

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 formally established in outline a two component wave system in theLinear Wave Field with (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 Linear Wave Field, having the above described thermodynamic properties for adiabatic and isothermal motions initiated in the wave field and initiated by pressure pulses ( presumably by accelerated motions of electric charges.) The compressibility of the wave state now accounts on physical grounds for the finite 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

∂²E/∂x² = (1/c²) ∂²E/∂t²

∂²B/∂x² = (1/c²) ∂²B/∂t²

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

∂²A/∂x² = (1/c²) ∂²A/∂t²

∂²I/∂x² = (1/c²) ∂²I/∂t²

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 again striking.

Instead of taking our initial external perturbation as a pressure pulse ( +∆p) we should 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.

S

The physical ambiguity which results from a pressure/density perturbation in the Orthogonal UF

An oscillating density perturbation ( ±∆ρ) then 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 E and magnetic field intensity B.

We have thus established a case for the compressible linear wave field being a cosmic entity which transmits transverse electromagnetic waves through space. A necessary next step will be to examine the field or state 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 the resulting magnetic fields, etc. etc. Preliminary work has indicated that this additional reconciliation will be successful.

Nite: The appropriate wave equation for the compressible flow field, from which the quantum shock compressions that generate the elementary particles of matter are produced, would seem to be the exact Classical Wave Equation:

Ѳ ψ = 1/c² ∂²ψ/∂t² [ 1 + Ñψ ]^{(k + 1)}

where k, the adiabatic exponent is c_p/c_v = ( n + 2) /n; and ( k + 1 ) = 2( n + 1)/n = 2(c_o/c*)². Here, pressure is a function of density only. This wave is isentropic, non-linear, unstable, and grows to a non-isentropic discontinuity called a shock wave. It is at these shock discontinuities that the elementary particles can form – the hadrons at the strong shock and the leptons at the weak shock option.

In many quantum actions stable waves are involved, such as the electromagnetic waves. For these we propose the linearised classical wave equation, as follows

∆²ψ = 1/c² ∂²ψ/∂t²

where ∆² = ∂../∂x² + ∂^2../∂y² + ∂²../∂z²; ψ is the wave amplitude, that is, it is the amplitude of a thermodynamic state variable such as the pressure p or the density ρ. The local wave speed is c.

The general solution is

ψ = φ₁x – ct) + φ₂( x = ct)

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

Summary

We have presented examples of a close conection of compressible flow theory and quantum mechanics fundamental relationships. We have related the formation of the elementary prticles of matter to energy condensation occurring in compression shocks in a compressible flow.

We have assigned a Linear Equation of State to the quantum fields of electromagnetic radiation. This equation has two forms, one being adiabatic and the other being isothermal. In the case wheer these two ewuations are orthogonal. the resultant wave would appesr to be transverse to the direction of motion.. Then, the transverse wave equations are shown to formally match Maxwell’s electromagnetic equations.

Copyright, Bernard A. Power, September 2017

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Appendix B

Summary of a Universal Physics

Prepublication Copy

May 1992

SUMMARY OF A UNIVERSAL PHYSICS

Bernard A. Power

Consulting Meteorologist (ret.)

TEMPRESS

Dorval, Quebec, Canada

© 1992

PREFACE

The central concept of the new unified theory of physics which is the subject of this book, is that energy flows and transformations are compressible, and that this single concept makes possible the unification of the presently separate fields of physics – classical mechanics, quantum mechanics, nuclear physics, relativity and gravitation.

We may roughly characterize this compressibility as : (1) the capacity of energy to exist in the form of various elementary particles having varying concentrations or densities of energy, and (2) transformations from one particle form to another take place via energy flows which involve energy compressibility.

The existence of compressibility has very important physical consequences. For example, it permits certain supercritical motions to exist which now provide a long-sought-for physical explanation for the existence of the elementary particles of matter in their various mass-ratios, for whole ranges of quantum and nucleqar phenomena, and for their unification with other branches of physics.

Of course, a new, central scientific concept, to be valid, must relate in a fundamental way to an enormous range of scientific topics and experimental data. The complete exposition of such a new theory in all its details would obviously be quite impossible to accomplish in a single book, or even in several; and, given the current high level of development and complexity of physics, it would not only be far beyond the ability of any one individual, but would undoubtedly tax the capabilities of a whole team of specialists. Still, any fundamental, new concept must have a beginning, and beginnings are usually small – hence the present small book with such an extended scope.

The evidence presented for the general correctness of the new theory is both theoretical and experimental.

Theoretically, Section 3 presents a new physical basis for quantum physics which is compatible with the standard model in most respects, by making use of the concept of energy waves, and compressible flows and transformations. The quantum state variables are the wave speed c and the relative flow velocity V; the normalized quantum wave functions are c/c_o and V/c_o.

The energy of the wave function contains a new energy term, 2cV, which in turn then grounds all the quantum relationships on a basis of extreme simplicity – Planck’s constant h, the de Broglie equation, the Lagrangian function, the quantum operators for momentum and position, and the Heisenberg uncertainty principle. The entropy is derived from a basic, compressible energy equation, and this in turn yields the fine structure constant of the atom α in an extremely simple manner.

Experimentally, the problem of the mass-ratios of the elementary particles of all matter is explained by the compressible flow expression [n+1]^1/2 to within about 1% of experimental values. This grounds the creation of matter in either the strong compressible shock for the baryons, or the weak shock for the electron and the leptons.

In this book, mass becomes a dimensionless quantity – a measure of the degree of energy condensation of the various elementary particles of matter.

In section 4, the physical basis for electric charge is related to compressible vortex motion ; charge is then related to the entropy and the fine structure constant. The results of the optical experiments of Michelson and Morley are shown to be consequences of compressibility. The refractive index in optical media is derived from the compressibility equation, as well as Fermat’s least-time principle for optical rays. The experimental discrepancies in some versions of the Fizeau experiment are explainable as shock phenomena.

In Section 5, the basic relativity relationships, such as the Fitzgerald contraction factor and the Lorentz transformations, are shown to be simple consequences of the compressibility ratios c/c_o and V/c_o, which, as mentioned above, are also the normalized quantum state variables.

In Section 6, the 2cV compressibility perturbation term is shown to lead to a quantum theory of gravitation; the nature and physical generation of the quantum of gravity are shown to be related directly to the generation of the elementary particles of matter by the strong and weak shock condensations of energy. Mach’s Principle becomes expressed on the physical basis of compressibility.

The quantum nature of gravity leads to a deeply-rooted cosmology in a universe evolving through Unitary → Binary → Unitary transformations; this process provides a solution to the current mystery of the so-called ‘missing mass’ or “ dark matter” of the universe.

Experimentally, Section 8 resolves the conflict over the cosmological constant; that is, it reconciles quantum predictions of an enormous vacuum energy with the conclusions of astronomy of a zero energy vacuum and a nearly flat structure for space. The observed early abundance of quasars in space is explained by the new theory in an experimentally testable prediction. The A → B transformation of matter yields an energy-change equation from which the prediction might be tested against energy requirements for astronomical processes.

Since a new theory cannot be exhaustive, the present summary aims to present to specialists only a brief overview of applications and some preliminary results, so as to encourage and assist in a more detailed examination.

Those already familiar with compressible flow theory e.g. aerodynamicists, astronomers, meteorologists ( such as the author) and so on will probably find Section 2 merely a review. Those unfamiliar with it will hopefully find the Section to be a concise presentation of compressible flow topics which will provide a key to the application of the new theory to their various specialties in the later sections.

Dorval, Quebec, Canada

May 1992

CONTENTS

Section 1 Introduction

Section 2 Compressible Fluid Dynamics

2.1 Steady state energy equation

2.2 Unsteady state energy equation

2.3 Lagrangian energy function L

2.4 Equation of state for compressible systems

2.5 Waves

2.5.1 Classical wave equation

2.5.2 Exact wave equation

2.5.3 Shock waves

2.5.3.1 Normal shocks

2.5.3.2 Oblique shocks

2.5.3.3 Strong and weak shock options: The shock polar

2.6 Types of compressible flow

2.7 Compressible flow ratios and parameter k

2.8 Wave speeds

2.9 Wave speed ratios and isentropic ratios

2.10 Graphical depiction of waves and shocks

2.11 Wave stability

Section 3 Quantum Physics

3.0 General

3.1 Compressibility and quantum mechanics

3.2 Basic quantum wave

3.3 Interacting ( non-parallel) quantum waves

3.4 Curved characteristics

3.5 Quantum wave function

3.5.1 Normalized quantum wave function

3.5.2 The specific energy of the wave function

3.5.3 The ‘extra energy’ term 2cV

3.6 The ‘extra energy’ term 2cV yields the following fundamental quantum relationships

A) Planck’s constant h

B) de Broglie wave/particle equation

C) Lagrangian function L

D) Quantum wave function operators

E) Heisenberg uncertainty principle

3.7 Quantum wave/particle equations

3.7.1 Schrodinger equation

3.7.2 Dirac equation

3.7.3 Klein-Gordon wave equation

3.7.4 Weyl equation (neutrino)

3.8 Quantum state functions

3.9 Origin of matter and energy compressibility

A) Baryons and heavy mesons

B) Leptons, pion and kaon

3.9.1 Experimental verification of mass-ratio equations

3.9.2 Summary

3.10 Entropy and the mass ratios

3.11 Entropy and the fine structure constant

3.12 Quantum interactions and elementary particle interactions; Graphical depiction

3.13 Radioactive Decay

3.14 Spin, reflection, particle confinement and spinors

3.15 Quantum creation and annihilation operators

3.16 The ‘collapse of the wave function’ problem

3.17 Nature of the quantum wave function

3.18 Feynman diagrams and virtual particles

3.19 Dirac delta function

Section 4 Electromagnetism

4.1 The physical basis of electrical charge

4.1.1 Charge as vortex motion

4.1.2 Vorticity at a contact discontinuity

4.1.3 Entropy and electric charge

4.1.4 Entropy and pressure ratio

4.1..5 Compressible potential vortex

4.1.5.1 Compton wave length, λ_c

4.1.6 Electron vortex pressure

4.1.7 Model of the hydrogen atom:

A) The rotational vortex

B) Compressible vortex

C) The hydrogen atom: nucleus + electron

4.1.8 de Broglie wavelength. λ_deB

4.2 Electrostatic field

4.3 Magnetism: The compressed virtual photon field of a moving charge

4.4 Electromagnetic radiation

4.5 The photon structure:

4.5.1 Cusp vector field

4.5.2 Photon energy

4.5.3 Photon speed

4.5.4 Relativistic form

4.5.5 Photon spin

4.6 The Michelson-Morley experiment

4.7 Some Optical Effects:

4.7.1 The refractive index N

4.7.2 The Fizeau Effect

4.7.3 Fermat’s Least Time principle for optical rays

4.8 The Sagnac Effect

4.9 The de Broglie equation

4.10 The Periodic Table

Section 5 Relativity: A Compressibility Effect

5.1 Galilean relativity

5.2 Special Relativity:

5.2.1 The Fitzgerald contraction factor

5.2.2 The Lorentz transformations

5.2.3 The relativistic Hamiltonian

5.2.4 The Einstein formulation of special relativity

5.2.5 Compressibility formulation of special relativity

5..2.6 Transverse and longitudinal mass

5..3 General relativity

Section 6 Gravitation and Cosmology

6.0 General

6.1 Force in compressibility terms

6.1.1 ‘Longitudinal mass’

6.1.2 Force in pressure gradient terms

6.1.3 Wave speed c

6.2 Derivation and dimensions of the gravitational constant G

6.2.1 Gravitational wave speed

6.3 The gravitational quantum or gravitino

6.4 Force of gravity in compressibility terms

6.5 Exclusively attractive nature of gravity

6.6 A new force law

6.7 Equivalence of gravitational and inertial force: Mach’s Principle

6.8 The problem of the cosmological constant

6.9 The ‘flatness’ problem: Ώ equals unity

6.10 Action-at-a-distance : The experiments of Aspect et al.

6.11 The hidden mass, or “dark matter”, of the universe:

6.11.1 The equation of state has two forms

6.11.2 Rarefied or celeston matter

6.11.3 Local pressure lowering in vortices

6.11.4 Quasars in early cosmic times [ p >>p*]

6.11.5 Spiral and elliptical galaxies

6.12 Experimental text of the proposed A → B transformation: Quasars vs. galaxies

Selected Bibliography

SUMMARY

OF

A UNIVERSAL PHYSICS

1. INTRODUCTION

1.1 The whole of physics – classical mechanics and thermodynamics, quantum mechanics, electromagnetism, relativity, gravitation and cosmology – can be unified on the basis of a single, theoretical principle – the compressibility of energy flows or energy transformations. This principle thus introduces a universal physics.

1.2 It is postulated that all energy transformations and processes involve compressible motions which are governed by the laws of compressible fluid flow.

1.3 Energy always exists in a definite physical configuration or ‘form’, such as a massive elementary particle, a wave pulse, a photon, etc., but never as ‘energy per se’.

1.4 Section 2 is an outline of some essentials of standard, compressible fluid flow theory. Sections 3 to 6 then apply these to some fundamental topics in quantum physics, electromagnetism, relativity and cosmology, to show the universality of the new theory.

2. COMPRESSIBLE FLUID DYNAMICS

2.1 STEADY STATE ENERGY EQUATION

c² = c_o² - V²/n (1)

where c is the local, compressive wave speed, c_o is the static wave speed or maximum wav e speed, V is the relative flow speed, n is the variety of the energy configuration ( the number of degrees of freedom) of the system.

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

The case where V = c = c* is called the critical state. The ratio (V/c) is the Mach number M of the flow. The ratio (V/c_o) is a quantum state variable ( Sects. 2.10b; 3.8). The maximum flow velocity V_max (when c = 0) is the escape speed to a vacuum; V_max = √ n c_o.

2.2 UNSTEADY-STATE ENERGY EQUATION

c² = c_o² – V²/n – 2/n dφ/dt (2)

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

dφ/dt = V (dx/dt) = Vc

and

c* = c_o* - V*/n - 2/n cV (2b)

(See also sects. 3.6; 6.2; 6.9; and 8.10 for far-reaching implications of the cV term).

2.3 LAGRANGIAN ENERGY FUNCTION L

L = (Kinetic energy) – (Potential energy)

= ( c² = V²/n) – c_o² , and so, from (2b)

L = - 2cV/n (3)

2.4 EQUATION OF STATE FOR COMPRESSIBLE SYSTEMS

pv = RT ; p/ρ = RT (4)

p = pressure, v = volume, 1/v = ρ, the density; R = gas constant; t = temperature.

Equation of State

2.5 WAVES

2.5.1 CLASSICAL WAVE EQUATION

∆²ψ = 1/c² ∂²ψ/∂t² (5)

∆² = ∂../∂x² + ∂^2../∂y² + ∂²../∂z²; ψ is the wave amplitude, that is, it is the amplitude of a thermodynamic state variable such as the pressure p or the density ρ.

The general solution of (5) is

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

ψ = φ₁x – ct) + φ₂( x = ct) (6)

of compressible flows and their waves is on the (x,t), or space-time diagram.

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

2.5.2 EXACT WAVE EQUATION

Ñ^{2ψ =} 1/c² ∂²ψ/∂t² [ 1 + Ñψ ]^{(k + 1)} (7)

where k, the adiabatic exponent is c_p/c_v = ( n + 2) /n; and ( k + 1 ) = 2( n + 1)/n = 2(c_o/c*)². Here, pressure is a function of density only. This wave is isentropic, non-linear, unstable, and grows to a non-isentropic discontinuity called a shock wave.

2.5.3 SHOCK WAVES

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

2.5.3.1 NORMAL SHOCKS

V₁ > V_2>

P₁, ρ₁, T₁ < p₂, ρ_2, T₂

ENERGY;

V₁²/2 + h₁ = V₂²/2 + h₂ = h_o (8)

where h_o – stagnation enthalpy.

CONTINUITY:

ρ₁V₁ = ρ₂V₂ (9)

MOMENTUM:

p₁ + ρ₁V₁² = p₂ + ρ₂V₂² (10)

EQUATION OF STATE:

h = h_(s,p); S = S_(p,ρ) (11)

PRANDTL’S EQUATION:

V₁V₂ = c*² (12)

V₁V₂ = ρ₂/ρ₁ (13)

ENTROPY CHANGE ACROSS SHOCK:

∆S = S₁ – S₂ = -ln(ρ₀₂/ρ₀₁) (14)

MAXIMUM CONDENSATION RATIO:

ρ₁/ρ₂ = [n+1]^1/2 = V_max/c* (15)

2.5.3.2 OBLIQUE SHOCKS

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

Oblique Shock

V_1N > V_2N

p₁ρ₁T₁ < p₂ρ₂T₂

ENERGY:

[V_1N² – V_1N²] /2 = (n+2)/n [p₂/ρ₂ – p₁/ρ₁]

or,

1/2[ V_2N² – V_1N²] = c_p [ T₁ –T₂ ] (16)

CONTINUITY:

ρ₁V_1N = ρ₂V_2N ( 17)

MOMENTUM:

p₁ - p₂ = ρ₂V_2N² – ρ₁V_1N² (18)

RANKINE-HUGONIOT EQUATION:

Ρ_2/ρ₁ = [(n+1)(p₂/p₁) +1] / [(n+1) + (p₂/p₁)] (19)

PRANDTL EQUATION:

V_N1V_N2 = c*² – 1/(n+1) V_max² (20a)

V₁/c* + c*/V₂ = V₂/c* + c*/V₂ (20b)

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

2.5.3.3 STRONG AND WEAK OBLIQUE SHOCK OPTIONS: THE SHOCK POLAR

For each inlet Mach number M₁ ( = V_N1/c), and turning angle of the flow θ, there are two physical options:

1) the strong shock ( intersection S) with strong compression ratio and large flow velocity reduction (p₂ >> p₁; V₂ << V₁, or

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

Which of the two options occurs depends on the boundary conditions: low back, or downstream, pressure favours the weak

shock occurrence; high diownsgream pressure favours the stroing shock.

When the turning angle θ of the oncoming flow is zero the strong shock becomes the normal or maximum strong shock, and the weak shock becomes an infinitesimal, low-amplitude, acoustic wave ( described by Eq. 5).

2.6 TYPES OF COMPRESSIBLE FLOW

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

b) Steady, supercritical ( e.g. supersonic, V₁ > c) governeddd by hyperbolic, nonlinear, partial differential equations.

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

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

TWO-DIMENSIONAL, STEADY, SUBCRITICAL FLOW

(1 – M²) ∂u/∂x + ∂v/∂y = 0 (21)

where M = V/c and c² = c_o² – V² /n.

Thus equation is elliptical, and reduces by a simple transformation to the Laplace equation

Ѳψ = 0 (22)

TWO-DIMENSIONAL, STEADY, SUPERCRITICAL FLOW

(M² – 1) ∂u/∂x - ∂v/∂y = 0 (23)

where M and c are as given in (21).

This equation is hyperbolic and reduces to the classical wave equation (5) by a simple transformation.

UNSTEADY, ONE-DIMENSIONAL MOTION

(c² – φ_x²) φ_xx - 2φ_xφ_t – φ_tt = 0 (24)

where φ is a velocity potential, i.e. ∂φ/∂x = u, etc., and c² = c_o² –V²/n – (2n) dφ/dt.

In linearised form, we have the classical wave equation in terms of the velocity potential φ:

Ѳφ = 1/c² ∂²φ/∂t² (25)

2.7 COMPRESIBLE FLOW RATIOS AND PARAMETER K

V_max/c_o = n^1/2; V_max/c* = (n+1)^1/2 (26)

(c_o/c*)² = n/(n+1); c/c_o = [1 – 1/n)(V/c_o)² (27)

n = 2/(k-1); k = c_p/c_v = (n+2)/n (28)

2.8 WAVE SPEEDS

c = [ c_o² – V²/n]^1/2 (steady flow) (1)

c = [ c_o² – V²/n – (2/n)cV ]^1/2 (unsteady flow) (2)

c² = kp/ρ = (dp/dρ)_s, where s is an isentropic state (29)

Since V is relative, it may be arbitrarily set to zero to give a stationary or “local” coordinate system moving with the flow; this automatically puts c = c_o and transforms the variable wave speed to any other relatively moving coordinate system. (See also Sect. 5.2).

The shock speed U is always supercritical (U > c) with respect to the upstream or oncoming flow V₁.

2.9 WAVE SPEED RATIOS AND ISENTROPIC RATIOS

c/c_o = [1 -1/n(V/c_o)²]^1/2 = (p/p_o)^1/(n+2) = (ρ/ρ_o)^1/n = (T/T_o)^1/2 (0)

All the basic physical parameters of a compressible isentropic flow are therefore specified by the wave speed ratio c/c_o.

2.10 GRAPHICAL DEPICTION OF WAVES AND SHOCKS

a) Waves, shocks and their various interactions are depicted in the space-time or x,t-coordinate system, which is also called the physical plane.

b) State plane representation

On the state (c,V) plane the characteristics Γ^±_1,2 show as two sets of lines inclined at slopes of +(1/n) and –(1/n). The normalised state plane has state coordinates (c/_o) and (V/c_o). It yields all the thermodynamic variables of state from the isentropic ratio relationships via the wave speed ratio c/c_o ( Eq. 30, Sect. 2.9).

2.11 WAVE STABILITY

Compression waves are the rule in the physical world. Density waves are always compressive in nature, and all compression waves of finite amplitude grow towards shocks. Only acoustic compression waves ( i.e. infinitely low-amplitude compressions) are stable.

Finite rarefaction waves and rarefaction shocks are impossible in material gases; only infinitely low-amplitude rarefacttion waves can persist. ( however, see Sect. 6.11).

3. QUANTUM PHYSICS

3.0 The standard model of quantum physics is one of the most remarkable achievements of science. It explains an enormous range of nuclear an atomic phenomena to a very high degree of accuracy, and is logically quite consistent. Yet, for all its success, it has some serious deficiencies.

For example, while it is very successful in describing many aspects of particle creations, annihilations and interactions, it still has no predictive power to specify the values of the observed masses, and the mass-ratios of the elementary particles – the experimentally determined values of these masses must still be put into the theory by hand. Again, it requires the arbitrary introduction of various fundamental constants, such as the fine structure constant of the atom α, which are essential for its calculations, but for whose existence or numerical value it has no explanation whatsoever. Again, quantum physics is still essentially unrelated to classical mechanics, and although it can be related to electromagnetic theory through quantum electrodynamics, this is so only for the cases of the electron – the phenomena of the nucleus of the atom are as yet still essentially unrelated to the rest of physical science in the standard model of quantum theory. Yet again, quantum physics has little relationship to relativity, gravitation and cosmology.

3.1 COMPRESSIBILITY AND QUANTUM MECHANICS

All quantum phenomena are basically compressible energy flow manifestations. The basic energy form ( variously to be called a characteristic, enform, enray, ray, wave pulse, wavelet, etc.) is the quantum wave function ψ which refers to a single, compressible, enray pulse or wavelet.

More complex waves ( elementary particles, etc. ) are built up from the linear enray ψ by superposition.

3.2 BASIC QUANTUM WAVE

The enray ψ is the linear, ‘characteristic’, or ‘ray’ solution, of the hyperbolic, linearised, approximate differential equation called the classical wave equation ( Sect. 2.5.1; 2.6).

Ѳψ = 1/c² ∂²ψ/∂t² ; ∂²ψ/∂x² = 1/c² ∂²ψ/∂t²

Therefore, the basic formula for the characteristic or enray is as follows:

Ψ = c ± V (1,2)

3.3 INTERACTING (NON-PARALLEL) QUANTUM WAVES

..

3.4 CURVED CHARACTERISTICS

In the general case, characteristics may be curved in space-time, ( i.e. on the physical plane), indicating the presence of force and acceleration.

3.5 QUANTUM WAVE FUNCTION

ψ = c ± V (1,2)

Here, V is the relative velocity and may be set to zero, making c = c_o in the ‘at rest’ coordinate system chosen. (cf. Sects. 2.8; 5.2). For an energy flow, c_o is 3 x 10⁸m/s.

In general, ψ is complex, and we then have

Ψ = c ± iV (3)

3.5.1 NORMALISED QUANTUM WAVE FUNCTION

From (c/c_o)² = 1 – 1/n (V/c_o)², (Eq. 11, Sect.2.1), we have

Ψ_N = c/c_o + i V/c_o; ψ*_N = c/c_o – i V/c_o (5)

For small V ,this reduces to ψ_N ≈ c/c_o.

The same wave speed ratio c/c_o gives all the isentropic thermodynamic variable ratios (Sect. 2.9). It also gives the Fitzgerald contraction factor of special relativity ( Sect. 5.2.1).

3.5.2 THE (SPECIFIC) ENERGY OF THE WAVE FUNCTION

E_ψ = (c – iV)² = (c + iV)² – 2icV (6)

E_ψ* = (c + iV)² = (c² –V²) + 2icV (7)

For ‘mixed’ energies we would have no 2cV term:

(c + iV)(c – iV) = c² + V² = [E_ψ]² (8)

(c+ V)(c – V) = c² - V²

3.5.3 THE ‘EXTRA ENERGY’ TERM, 2CV

The term (c² + V²) in (8) is the familiar, elliptical ,steady state energy from the basic energy equation (

Sect. 2.1). The ‘extra energy’ term 2cV in (8) can be interpreted as being a transformation of the coordinate axes c and V , by rotation through an angle θ.

On the other hand, the energy term (c² – V²) in (7) is a theoretical oddity, namely a steady state, ‘hyperbolic energy’ with no aparentg physical meaning. ( But, see Sect. 6.2.1). We may also note in passing that, if the ‘extra enegy’ term 2cV is constant, then it, taken by itself, has a hyperbolic interpretation, namely the form of an equilateral hyperbola ( xy = const.)

Hyperbolic energy: a) Steady state: c² – V² = c_o² ; b) Unsteady : -2cV = c_o²

3.6 THE ‘EXTRA ENERGY’ TERM (2CV) YIELDS THE FOLLOWING FUNDAMENTAL QUANTUM RELATIONSHIPS:

A) PLANCK’S CONSTANT, h

For n = 1, if cV = constant energy for each set of waves, then cV/υ = constant energy per cycle or pulse:

cV/υ = h

cV = hυ = hω/2π = ħω = ε_υ (9)

For the complex case,

cV/υ = ħ/I = -iħ (10)

B) DE BROGLIE WAVE/PARTICLE EQUATION

cV/υ = h

But c/υ = λ; V(m) = p (momentum), so

λp = h, or

p = h/λ (11)

C) LAGANGIAN FUNCTION, L

L = 2cV (see Sect. 2.3)

See also Sect. 4.7.3. for application to Hamilton’s principle of Least Action for mechanical systems and to Fermat’s principle of Least time for optical rays.

D) QUANTUM WAVE FUNCTION OPERATORS:

a) HAMILTONIAN ENERGY

cV= hυ = -ħω = ε

icV = -iħω

But

iω = ∂../∂t, and so cV = h/I ∂../∂t = +iħ∂../∂t = H_op

which is the Hamiltonian energy operator.

( To ensure correct dimensions, it must be applied to the normalized quantum function ψ_N).

b) MOMENTUM

cv = hυ = +hω = ε

V = (1/c))ħω, or (m)V = p = (m)(1/c)ħω

Multiplying by i, we have:

(m)iV = (m)(1/c) iħω

= (m)(1/c) ħ ∂../∂t

So, we have

(m) V = p = -(m)(1/c) iħ ∂../∂t

But,

I1/c) ∂../∂t = ∂../∂x, and so

(m)V = p = -iħ∂../∂x = p_op (15)

which is the quantum wave operator, ( to ensure correct dimensions, it must be applied to the normalized quantum function ψ_N).

E) HEISENBERG UNCERTAINTY PRINCIPLE

cV = hυ; cv/υ = h

λV = h

But x = Δx and V(m) = Δp, so

Δx . Δp ≥ (m) h (16)

which is the Heisenberg uncertainty principle.

F) SUMMARY: We have, therefore, related all the above fundamental quantum relationships to a single energy term 2cV.

We have, in effect, quantized the various energy ‘fields’ represented by 2cV/n for various values of n, by equating them to the ‘time-like’ condition set by the frequency υ in the quantum equation hυ = 2cV/n.

(Note that these equations are for ‘specific’ energy, that is, for unit mass flow. For a definite particle, the numerical value of the mass is to be inserted; the dimensions of the equations are not thereby changed, since in our system, mass (m) is dimensionless. Thus for the photon, we have hυ = m_e cV, where m_e is the relativistic mass of the electron. In terms of the momentum, we have

hυ = cp,

which is the de Broglie equation (Eq.11).

3.7 QUANTUM WAVE/PARTICLE EQUATIONS

The quantum operators for momentum and Hamiltonian energy in Sect. 3.6 D then lead to the various standard quantum wave equations for the elementary particles:

3.7.1 SCHRODINGER EQUATION

For a non-relativistic, spinless particle we have

-ħ²/2m Ѳψ = iħ Ñψ/∂t (17)

3.7.2 DIRAC EQUATION

Eψ + c(α . p )ψ + m_oc_o²βψ = 0

Or,

iħ∂ψ/∂t = iħcα . Ñψ + m_oc²βψ = 0 (18)

where α₁² = α₂² = α₃² ; β = 1.

3.7.3 KLEIN-GORDON EQUATION

For a relativistic, but spinless particle, we have

Ѳψ = 1/c² ∂²ψ/∂t² [m_oc/h]² ψ 19)

3.7.4 WEYL EQUATION (NEUTRINO)

For a massless, spin one-half particle we have

Iħ ∂ψ/∂t = iħ cα . Ñψ

Or,

Iħ ∂ψ/ t = (σ . p) ψ (20)

3.8 QUANTUM STATE FUNCTIONS

The quantum state functions are the normalized form of the state variables, namely c/c_o and V/c_o. The state plane characteristics Γ^± (see Sect. 2.10 b) are:

Γ⁺ : dc/dV = +1/n; Γ^- : dc/dV = -1/n

The isentropic ratios can be obtained from c/c_o, just as in Sect. 2.9.

3.9 THE ORIGIN OF MATTER: ENERGY COMPRESSIBILITY

All elementary particles of matter (with the possible exception of the neutrino) are condensed energy forms. The forms are given in terms of a simple, integral number n ( n = degrees of freedom of the compressible energy flow):

A. BARYONS AND HEAVY MESONS

m_b/m_q = V_max/c* = [n+1]^1/2 (21)

m_b is the mass of any baryon particle, m_q is a quark mass, V_max = c_o n^1/2 is the escape speed to a vacuum; that is, it is the maximum possible relative flow velocity in an energy flow for a given value of n, the number of degrees of freedom of the energy form, (see Sect. 2.7 for compressible flow velocity ratios). This is a non-isentropic relationship, and it corresponds physically to the maximum possible strong shock. (See Sects. 2.5.3.3 and 3.13).

Experimental verification for thisw baryon mass ratio formula is given in Table 3.9.1A below.

B. LEPTONS, PION AND KAON

m_L/m_e^- = k/α² = [(n+2)/n]/α² = {(n+2)/n] x 137 (22)

where α = 1/11.703 is the fine structure constant, and k is the adiabatic exponent or ratio of specific heats, k = c_p/c_v = [(n+2)/n]. (See Sects. 3.10; 3.11). Because of the presence of k, this formula for the mass of the leptons is a thermodynamic and quasi-isentropic one.

The leptons are formed via the weak shock option 9 See Sects. 2.5.3.3 and 3.13).

The experimental verification for the lepton mass ratio formula is given in Table 3.9.1B below.

3.9.1 EXPERIMENTAL VERIFICATION OF MASS-RATIO EQUATIONS (21) AND (22)

A) BARYONS AND HEAVY MESONS

--------------------------------------------------------------------------------------------

n n +1 [n+1]^1/2 Particle Mass (m_b) Ratio to

(MeV) quark mass

0 1 1 quark (ud) 310 MeV 1

(s) 505

1

2 3 1.73 eta (η) 548.8 1.73

3

4

5 6 2.45 rho (ρ) 776 2.45

6

7

8 9 3 proton (p) 938.28 3.03 (1)

neutron (n) 939.57 3.03

Λ (uds) 1115.6 2.97 (2)

Ξ^o (uss) 1314.19 2.99 (3)

9 10 3.16 Σ⁺ (uus) 1189.36 3.17 (2)

10 11 3.32 Ω^- (sss) 1672.2 3.31 (4)

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

Therefore, Equation 21 is verified.

B) LEPTONS, PION AND KAON

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

(MeV) to x 1/137

Electron

1/3 7 Kaon K^± 493.67 966.32 7.05

2 2 Pion π^± 139.57 273.15 1.99

4 1.5 Muon μ 105.66 206.77 1.51

- - Electron 0.511 1

Clearly, k ≈ m_l/m_e (1/137), verifying Equation (22).

The lepton/meson mas

s formula (22) for a weak shock condensation also has a thermodynamic interpretation:

m_L/m_e^- = c_p/c_v 1/α² (23)

α² = m_l/m_e^- c_p/c_v [ ΔT/ΔT] = (ΔQ_L+)_p/(ΔQ_e) = 1/137 (24)

3.9.2 SUMMARY

Equations 21 and 22 give, for the first time, an experimentally verified explanation for the origin of matter, and for the observed ratios for the masses of the elementary particles of matter. The principle of the compressibility of energy flow, therefore, underlines all material particles and the whole material universe.

\The text of Summary of a Universal Physics has been revised to this point as of Sept. 29/03. The remaining sections which were composed in 1992 and previous years will be successively reviewed and revised as necessary in future Website Updates.

3.10 ENTROPY AND THE MASS RATIOS

If the entropy is taken to a binary logarithmic base, that is, if

S = log₂ n (25)

Where n =1,2,3,….., and if these values for S are then compared with the values of [n+1]^1/2, which give the baryon and heavy meson mass ratios from Eqn. (21), we see that there is only one value for n which gives the same value to S as computed and for [n+1+]^1/2. This is n = 8, the value for n for the proton (Sect. 3.9.1). We then have

[8+1]^1/2 = 9^1/2 = 3 = m_b/m_q

log₂ 8 = 3 = S_(n=8)

The proton is the most stable elementary particle of nature, whereas all other baryons have fleeting lifetimes. It appears that n = 8 is a unique value, since it is only for this value that the binary logarithmic formula for computing the entropy gives a value which equals tke necessary baryon mass condensation ratio [n+1]^1/2. This unique equality may underlie the observed stability of the proton.

aryon particles.

3.11 ENTROPY AND THE FINE STRUCTURE CONSTANT

The fine structure constant of the atom α = -1/11.706 is the fundamental electron/photon interaction, or coupling constant. Its square is approximately 1/137, and this number determines the probability of emission and absorption of photons in electromagnetic interactions.

We recall that entropy is given by S = ln n, dS = dn/n/ and ΔS = Δn/n, n being a large number. However, at the elementary particle level, n, the number of degrees of freedom of the compressible energy flow, is small; typically n equals 1 to 8 (see Sect. 3.9.1). This would suggest that entropy change here is given by

Δs = Δn/n

For example by ½ -1/3, 1/3 -1/4, and so on, the value of Δn being unity.

However, in compressible flow, the entropy change ΔS accompanying any non-isentropic process, such as a shock transition or a contact discontinuity, would be given by a formula such as the following one:

S_y – S_x = ΔS = ln(1+P) – k ln(1 +((k + 1)/2k))P) + k ln(21+ ((k-1)/2k))P)

Where k is c_p/c_v and P is the shock strength expressed in terms of the pressure rise across the shock p_y/p_x and is given by

P = p_y/p_x – 1

If the shock is very weak ( and this would correspond to electromagnetic interactions controlled by 1/137), then the following approximation applies:

Δs/R = [S_y – S_x] = 1/12 (k-1)/k² p³ – 1/8 (k+1)/k² p⁴ + …..

Now the mean value of the first term factor which multiplies p³ , computed over a range of values of k corresponding to n varying from 1 to7 is 1/11.6907; this is within about one part in a thousand of the accepted value for α, the fine structure constant.

It is even more interesting that, if we suppose that the formation of proton plus electron involves an ‘all-possible-paths’ entropy, and then compute this for n = 8 (proton) and n = 9 ( i.e. (n +1), we get

ΔS = [ln 8! + ln 9!)/2 = - 1/11.703215 ≈ α (28)

The experimental value for α is about -1/11.706244, so that this extraordinarily simple calculation gives a theoretical computed value for α to within 3 parts in 10⁴.

Again, various shock formulae also give values which are quite close to the experimental value for α when n in the range indicated. Further refinement of this new approach to a derivation of the fine structure constant α on physical compressibility grounds in the stanounds is left to specialists in quantum perturbation theory and charge renormalization.

It should also be noted here that the supposition of the involvement of shock transitions in particle structure also imposes a physical limitation to the close approach of electrons to one another, as is actually required on theoretical grounds in the standard model of quantum electrodynamics in order to avoid the problem of infinities in charge renormalization procedures.

On the state plane, we see that a change in entropy ΔS plots as a rotation of the state plane characteristic Γ through an angle θ as the value of n changes (Sect. 3.8).

3.12 QUANTUM INTERACTIONS AND ELEMENTARY PARTICLE GENERATION: GRAPHICAL DEPICTION

Two interacting enrays or quantum functions of the same family (ψ⁺_1,2) form a finite-amplitude, third wave which immediately grows to form a shock (δ⁺_s), ( see Sect. 2.10).

Two interacting shocks (δ⁺_1,2) produce a third shock (See Sect. 2.9.1).

If the interacting shocks are quarks (q⁺_1,2) the resulting elementary particle formed is a roton or neutron (p,n) .

These interactions are in order of increasing strength.

Continuity requirements and entropy considerations require that the proton generation process also produces a contact discontinuity (D) which is the physical basis for the electron (e^-) Sect. 4.1.1), as well as a balancing, simple rarefaction wave R which is the source of the (anti-) neutrino (ν) Sect. 2.10).

The u,d quarks combine two-by-two (q⁺_1,2) in the above model to form the proton. This means that the entropy changes involved in the process have a binary logarithmic basis. (Sect. 3.11.1).

The mass ratios for the generating particle and the generated particle are in the ratio of the shock condensation densities. For the baryons, this is

ρ₂/ρ₁ = [n+1]^1/2 (21)

The leptons and electron form via the weak shock option.

3.13 RADIOACTIVE DECAY

Baryon particles form via the strong shock condensation (Sect. 3.9;2.5.3.3), with the condensation ratio typically being specified by the parameter [n+1+]^1/2 in (21).

Baryon particles decay via the weak shock otin (Sect. 3.9; 2.5.3.3);

The weak shock typically occurs when the back, or downstream pressure is low. The continual formation, reflection, reformation process for particle stabilization, plus the two shock options underlie the observed probabilistic basic for radioactive isotopic decay of b

3.14 SPIN, REFLECTION, PARTICLE CONFINEMENT AND SPINORS

The interaction of two enrays produces the elementary baryon particles by shock condensation ( Sect. 3.9). Total internal reflection of the shock regenerates the enray pair, and the process is then endlessly repeated unless and until the weak shock option is chosen, whereupon radioactive decay occurs.

The particle stabilizes by confinement under total internal reflections of the 2-enray/shock ensemble at the refractive index interface between the particle and the surrounding vacuum, (see sect. 4.7.1).

Each flow reflection generates a ‘flip’ or spin of 180° (spin ½) for a coupled , 2-enray spinor, and a 360° rotation or spin 1, for a 3-enray set. (see Sectd. 4.5.5).

Each shock reformation (particle formation) entails the choice of either the strong shock option ( i.e. particle stabilization and continuity of existence), or the weak shock option (particle dissolution via radioactive decay). (cf. Sect 2.5.3.3).

The spin is related to the number of enrays interacting according to the usual formula

N = 2S + 1

The behavior of sets of coupled interacting enrays is described mathematically by the spinor calculus.

The details of confinement and spin must be left for the appropriate specialists to evaluate.

3.15 QUANTUM CREATION AND ANNIHILATION OPERATORS

In the standard model of quantum theory, these have the following form:

A* |n> = (n+1)^1/2|n+1>

The new theory equates the [n+1]^1/2 term to the maximum possible compressible flow condensation ratio (Sect. 3.9, Eq. (21)), which specifies the masses of the baryons and heavy mesons.

In our compressible system, we designate each separate energy ‘field’ which is to be subjected to the ‘second quantization’ so as to generate an elementary particle, by choosing values for n, the number of degrees of freedom of the energy transformation. Since the quantization also involves the expression hν, (Sect. 3.6), where ν is the frequency kit is a ‘time-like’ quantization. ( For a “space-like” quantization see Sect. 6.2).

3.16 THE COLLAPSE OF THE WAVE FUNCTION PROBLEM

The physical act of measurement or detection of the quantum wave pulse converts it to a shock wave, and this physical transformation is involved in the so-called ‘collapse of the wave function’ problem. (See also sects. 6.3 and 6.10).

The probabilities involved in the self-interference of the wave function ( for example in the two-slit interference experiment) are also associated with the extra energy term or wave perturbation term 2cV/n.

3.17 NATURE OF THE QUANTUM WAVE FUNCTION

The position of the new theory as to the nature of the quantum wave function is one of scientific realism. That is, the quantum wave function represents a basic physical entity of compressible energy flow, one which, however, is grasped, not imaginatively, nor symbolically, but rather in the intelligibility of its correlation with the scientific formalism of compressible flow theory.

3.18 FEYNMAN DIAGRAMS AND VIRTUAL PARTICLES

The Feynman diagram ( and space-time relativity effects see Chapter 5) can be interpreted as depictions of compressibility effects in the physical plane, which is the x,t or space-time diagram.

The intermittent aspect of the existence of the elementary particles (Sect. 3.14) is closely related to the virtual particle concept of standard quantum physics.

3.19 DIRAC DELTA FUNCTION

The mathematical delta function δ corresponds to a wave pulse discontinuity, such as a shock wave in the compressibility formalism.

4. ELECTROMAGNETISM

4.1 THE PHYSICAL BASIS OF ELECTRIC CHARGE

4.1.1 CHARGE AND VORTEX MOTION

As already seen in the new theory (Sect. 3.9), matter has the nature of a localized condensation in energy density, with its magnitude being given for the baryons and heavy mesons by the equation

m_b/m_q = [n+1]^1/2

and with the condensation taking place by the strong shock.

For the leptons, pion and kaon, he condensation is given by

M_L/m_e- = k/α² = α² c_p/c_v = (n+2)/n (1/α²)

With the condensation process taking place by the weak shock.

Physically, the condensation takes place in either shock option at the intersection of two enrays in an energy flow (Mach lines, shocks, quarks, etc.). Furthermore, since the tgo interacting enrays must be of unequal strength ( i.e. having unequal slopes on the space-time diagram), then a third phenomena – a contact discontinuity line D -emerges.

In standard quantum mechanics, electric charge has some formal resemblance to angular momentum, which, however, is considered to be only a mathematical coincidence; in the new theory, electric charge has a physical origin in vortex motion.

4.1.2 VORTICITY AT A CONTACT DISCONINUITY

In the case of a compressible flow interaction, we have

2ω = -∂V/∂N -(1/ρV) ∂p/∂N (1)

where N is a space direction normal to the flow streamlines, ω is the rotation and p is the pressure.

In the case of two intersecting pressure pulses (waves) we have

2ω = -11/V [ T ∂S/∂N - ∂h_o/∂N ] (2)

h_o is the stagnation enthalpy ( h = U + pv), T is the temperature. The vorticity then depends on the rate of change of entropy and stagnation enthalpy normal to the flow directions.

4.1.3 ENTROPY AND CHARGE

In quantum theory, the charge e^-, has the nature of an electromagnetic force coupling constant (e^- = g_e). The square of the charge constant is equal to the square of the fine structure constant of the atom:

e^-2 = g_e² = e2/ћc = α² = [1/11.7]² = 1/137 (3)

But, (Sect. 3.11), the fine structure constant α is also an entropy, and we have , when n := 8 and (n + 1) = 9 :

ΔS = ( ln 8! + ln 9!)/2 = α ≈ -[ 1/11.7] 3a)

The entropy change across a contact discontinuity resulting from the interactin of tdwo unequal generating sshocks, or unequal characteristics, (ψ⁺_1,2), is also a measure of electgric charge e^-.

4.1.4 ENTROPY AND PRESSURE RATIO

ΔS = S₂ –S₁ = -R ln(p₀₂/p₀₁) (4)

4.1.5 COMPRESSIBLE POTENTIAL VORTEX (CPV)

Vr = const. (5a)

R*/r_min = [n+1]^1/2 (5b)

V_max = n^1/2 c_o (5c)

4.1.5.1 COMPTON WAVE LENGTH λ_c

In the compressible potentiall votex (CPV), we have

λ _c = n^1/2 r_min ; r_min = λ_c/n^1/2 (6)

r* = λ_c [ (n+1)/n]^1/2 (7)

r_Bohr = λ_? (1/α²) (8)

4.1.6 ELECTRON VORTEX PRESSURE

c/c_o = (p/p_o)^1/(n+2) = [ 1 – 1/n{V/c_o}²]^1/2 (9)

4.1.7 MODEL OF THE HYDROGEN ATOM

We have two vortex flows and their combinations:

A) THE ROTATIONAL VORTEX (RV)

V _T/r = const; V _T = rω = (Г/2π)r (14)?

At the rim of the vortex: V _T = V_max = n^1/2 c_o.

B) THE COMPRESSIBLE POTENTIAL VORTEX (CPV)

V_T r = const.; r_min = λ_c/ n^1/2

(see Sect. 4.1.5).

C) THE HYDROGEN ATOM: (NUCLEUS + ELECTRON)

Various combinations of A and B above may be considered for a model of the atom. We present here a double CPV model, with the nucleus being a CPV of dimensions approximately 10^-15 m , surrounding an electron CPV with a radius which is approximately the Bohr radius ( 10^-10 m).

4.1.8 de BROGLIE WAVELENGTH, λ_{de B}

λ_{de B} = 2πλ_c[c/V] (15)?

Wgere c/V is the invcerse Mach number (1/M) of the energiy flow.

Λ_c = [λ_{de B}/2π] V/c (16)?

4.2 ELECTROSTATIC FIELD

A stationary charge is surrounded by an electrostatic field, which, on the new theory, consists of a field of virtual photons emanating from the CPV of the electron. The virtual photons are Mach line characteristics of the flow, emitted symmetrically into space in the near field surrounding the electron. The characteristics terminate on a sink ( i.e. on another charge) confined to the near field, that is to within one wavelength. These curved characteristics are identified with the electromagnetic field lines. The curvature of the characteristics in space-time reveals the existence of force in a compressible flow (Sect. 3.4).

4.3 MAGNETISM: THE COMPRESSED VIRTUAL PHOTON FIELD OF A MOVING CHARGE

The electric field of a moving charge is compressed in the direction of the motion: a solenoidal magnetic field then emerges as a relatsivity ( i.e. compressibility ) effect (Sect. 5.2).

4.4 ELECTROMAGNETIC RADIATION

A stationary charge has a symmetrical, virtual photon field; a uniformly moving charge has a compressed virtual photon field which generates a solenoidal magnetic field; an accelerated charge represents a discontinuity ( or pulse) in virtual photon concentration. When the charge is accelerated, this concentrative gradient in the virtual photon field detaches itself from the charge or charges, (source or sink) and travels outwards as a set of photons which are no longer virtual but real. Maxwell’s classical equations for the electromagnetic field follow naturally (Sect. 4.5.1).

4.5 THE PHOTON STRUCTURE

When two characteristics interact ( sect. 2.19), for example in so-called corner flow, they encounter a structural flow ambiguity which is resolved by the formation of a shock.

When three characteristics interact, they for a cusp/envelope region in the flow and the three characteristics then meet at a point/ There is again an ambiguity in the flow.

The ambiguity at the crossing point of the three characteristics is resolved by a 360°rotation or spin, and the transverse, rotating vector constitutes the photon.

4.5.1 CUSP VECTOR FIELD (NON-CO-PLANAR)

Three, non-coplanar vectors combine to give

Ψ₁ x (ψ₂ + ψ₃) = (ψ₁ . ψ₃ψ₂) – (ψ₁ . ψ₂ψ₃)

Which is the curl of a vector field, B = curl A: so that

Curl B = curl curl A = Ñ( Ñ x A) = Ñ×ÑA - ÑÑA

= grad div A - ѲA

This connects the three characteristics of the compressible flow ( i.e. three non-coplanar vectors) in the new theory with Maxwell’s equations:

Ñ X E = 1/c ∂B/∂t div E = D (17)

Ñ X B = 1/c ∂E/∂t div B = 0 (17)

4.5.2 PHOTON ENERGY

ε = hν = cV (18)

h = (c/ν)V = λV

But, for the photon, V = c_o, and so

H = λ(m)V = λp

which is the de Broglie equation for the photon.

4.5.3 PHOTON SPEED

In the energy equation for 1-dimensionl motion, n equals unity, and we have

c² = c_o² – V² , and so

V_max = n^1/2 c_o = c_o

The photon speed V_γ is, therefore, the escape velocity to a vacuum V_max which equals c_o (= 3 x 10⁸ m/s). In free space the photon speed corresponds to V_i, the particle speed of the flow. In matter, or in interactions with matter, the photon speed corresponds to V, the local relative flow speed (Sect. 4.7.1).

4.5.4 RELATIVISTIC FORM

The above 3-vector description of the photon equation I classical Maxwellian form for a sokple cokmpressible flow an be cast in relativistic form by introducing the 4-vector potential A^μ

A^μ = A(x,y,z,φ)

Where φ is the velocity potential.

4.5.5 PHOTON SPIN

Spin involves an analysis of the streamline ambiguity which arises from some flow reflections, the relationship between the streamline directions and the characteristic direction requires that a 2-cmponent flow undergoes a 180° ‘flip’ or spin upon reflection of the flow (Sect. 3.14).

A 3-component flow, such as for the photon, requires a 360° rotation or spin upon reflection. This agrees with the usual spin equation, where N is the number of flow components:

2S + 1 = N

For example, if N = 3, we have S = 1, and this corresponds to the photon spin.

4.6 THE MICHELSON-MORLEY EXPERIMENT

In compressible flow theory, the flow velocity V in the energy equation

c² = c_o² – V²/n

is always purely relative, and this same equation also yields the Fitzgerald contraction factor (Sect. 5.2.1), ( provided that n in the energy equation is equal to unity, for example for 1-dimensional motions):

c/c_o = [ 1 – (V/c_o)²]^1/2

Thus, since only relative velocity V enters, there are no absolute motions involved which can effect the system; the observed null result of the Michelson-Morley experiment is therefore to he expected as a simple consequence of the existence of compressibility in the energy flow. Note also that in compressible flow theory, the M/M experiment is also seen as an isentropic process.

4.7 SOME OPTICAL EFFECTS AND THE FIZEAU EFFECT

4.7.1 THE REFRACTIVE INDEX N

The standard definition of the refractive index N is

So that N is the ratio of the speed of light in vacuum to the speed in some optical medium.

To express N physically as a compressibility phenomenon, we proceed as follows:

c² = c_o² –V²/n

from which we have the following relationships

c = [c_o² – V²/n]^1/2 c/c_o = [1 – 1/n (V/c_o)² ]^1/2

V = [nc_o² – nc²]^1/2 V/c_o = [1 – (c/c_o)²]^1/2

And, for V = c = c* , we have c* = V*

c*/c_o = [n/(n + 1)]^1/2

Finally, we put

N≡ V_o/V_i

Where V_o is the speed of photons in vacuum, and V_i is the speed of photons in the optical medium.. We therefore have expressed N in terms of two compressible flow velocities.

Note that the photon speed, or speed of light, is not identified with the wave speeds, c or c_o, but with the relative flow velocity V instead. However, the energy of the photon involves the wave speed c and the relative flow velocity V ( Sect. 3.6).

Now, the ratio of the two speeds of flow V_o/V_i has the identical form of the left hand side of the Prandtl equation for the flow velocity transformation across a normal shock front, namely

V_o/V_i = c*² / V_o²

And, if V_o = c_o, then also

V_o/V_i = c*²/c_o² = (n+1)/n

But we have set V_o/V_i equal to the refractive index N and so we also have

N = V_o/V_i = (n+1)/n

relating the refractive indeed N to the compressible flow parameter n.

Thus the reduction in the speed of light in an optical medium having refractive index N with respect to the speed of light in space is explainable physically as a reduction in an upstream compressible flow velocity V_o relative to a downstream flow velocity V_i across a normal shock front occurring in a compressible energy flow, these two flow velocities being identified with the photon velocities outside and inside the optical medium.

The relationship N = (n+1)/n Is shown in the following table for integral vales of n:

Degrees of Freedom n versus

Refractive Index N

n N

1 2

2 1.5

3 1.33

∞ 1

We note that the predicted range of the values for N, corresponding to simple vales of n ranging from 1-dimensionlal flow to 3-dimensional flow, covers the known values for N for over 99% of all optical media where N lies between 4/3 and 2. The convention of making N for free space equal to unity also corresponds to a value for n of infinity, and this is the value for a pure field. The electromagnetic shock hypothesis thus yields a prediction for the value of N corresponding to observed data. We conclude that N is in fact, physically based in a normal shock transition across the interface between an optical medium and free space.

We reiterate that the speed of light V_o in space ( = 3 x 10¹⁰ m/s) is a flow speed and not c_o, the free –space wave speed; the speed of light V_i in the medium is again the flow speed and not the wave speed c_i . The wave/particle nature of the photon is, however, not affected by this since the photon energy is given by the product of c the wave speed and V the particle speed ( ε = hν = 2cV) (Sect. 3.6)) .

Refractive indices intermediate in value between the values in the table should be explainable in terms of flows of mixed dimensionality. Media with N less than 1 or greater than 2 would involve either fractional degrees of flow freedom or flows of dimensionality higher than three.

In interpreting the mechanism further, it is worth noting tht the interface between an optical medium and space is, on the atomic scale, not ‘flat’, and so a transition interfacial state should be considered in any detailed or rigorous modeling.

4.7.2 THE FIZEAU EFFECT

The above derivation of N refers to stationary optical media. When the optical medium moves, (e.g. flowing water), then the experimentally observed fractional addition of velocity to the speed of light caused by the speed of the medium V_m is usually explained as a relativity effect which can be calculated from the Lorentz/Einstein velocity-addition formula for transformation of velocities between coordinate systems in uniform relative motion:

V = [c_o/N + V_m] / [ 1 + (c_oV_m/N)/c_o²]

Which, when expanded as a power series in V/c_o, gives

c_o/N + V_m[ 1 – 1/N²]

where the bracketed factor is the Fresnel drag coefficient.

This drag coefficient, however, is an isentropic flow factor derived from the compressible energy equation, since

(c/c_o)² = 1 – 1/n (V/c_o)²

If n = 1 (1-dimensional flow) then (V/c_o)² = 1/N², so that

Fresnel/Einstein drag coefficient.

This raises a logical problem, because, if the index of refraction N is to be explained by a shock mechanism for a stationary flow, then this is a non-isentropic process. Why then is the addition-of-velocity drag formula for a moving medium explained by an isentropic flow formula?

In the past, it has been proposed that there are observable discrepancies in the experimental data, which in fact point to a failure of the isentropic Fresnel/Einstein drag formula for moving media, and which argue instead for a shock transition formula for moving media as being the correct one. Another possible explanation lies in the well-known fact that, for weak shocks the differences between the non-isentropic or shock formula and the purely isentropic formulae show up only in terms higher than second order in the shock strength. Shocks involved in compressible theory for values of N between 4/3 and 2 would be relatively weak, and so the numerical differences in the predictions of the two mechanisms - isentropic and non-isentropic- would be small.

It is pointed out that the Fizeau effect, unlike the Michelson and Morley effect, is first order in (V/c) and is readily observable; it merits further analysis, and the experimental discrepancies should be carefully examined in the light of the various possible mechanisms involved.

4.7.3 FERMAT’S LEAST TIME PRIINCIPLE FOR OPTICAL RAYS

In classical mechanics, Hamilton’s principle of Least Action applies widely. It is expressed as

δò L dt = o

where L is the Lagrangian function.

But, in our new theory L = cV (Sect. 3.6C), and so

δò L dt = δò Vc dt = δò V ds =0

where ds is the invariant metric interval. In such systems, for example simple harmonic motions, L varies constantly or periodically and so, therefore, do c and V.

But, in a homogeneous optical medium, c and V are constant, and so we must put

δò L dt = δò cv dt = δò dt = 0

which is Fermat’s principle of a least time path for optical rays.

Both Hamilton’s principle and Fermat’s can be derived from the cV term in the unsteady flow compressible energy equation; if cV varies, Hamilton’s least action result, if cV is constant, Fermat’s least time applies.

4.8 THE SAGNAC EFFECT

In this interesting effect, an interferometer is rotated and a fringe shift δ is then observed which is given by

Δ = (4Ώ A)/λ_oc_o) (28)

Where A is the area enclosed by the light path, Ώ is the angular rate of rotation, λ_o and c_o are free space values.

4.9 DE BROGLIE EQUATION

In the de Broglie equation λ = h/p = h/mV, upon setting ε = hν = (m) c². we obtain the equation

u_pv_g = c_o²

This derivation requires that the phase velocity u_p should exceed c_o when v_g is small.

The new theory ( Sect. 3.6B) has, however, put

ε = cV

instead of ε = (m) c².

From the unsteady state energy equation we get _.

c² = c_o² – V²/n – 2cV/n, or

cV = n[c_o²/2 – c²/2 –V²/2]

instead of u_pv_{g =} c_o².

The supercritical phase speeds which emerge from the customary de Broglie derivation are avoided. Note that cV emerges as a :sum of kinetic energy terms in c_o, c and V.

4.10 THE PERIODIC TABLE

We have seen (Sects. 3.6B; 4.1.5.1; 4.1.8) that

( de Broglie wavelength): λ_{de B} = h/m/λ

(Compton wavelength): λ_c = h/mc

so that λ_c/λ_{de B} = V/c

It is interesting that, if we now set

N λ_{de B} = 2π r_e (29)

We have

λ_{de B} = (2π/N) r_e (29a)

where N is a quantum number equal to 1,2,3….2π, and r_e is an electron radius. E valuating (2π/N) we get the following table:

N (2π/N) Electron Shell

1 0.15915 K

2 0.31831 L

3 0.47746 M

4 0.63682 N

5 0.79577 O

6 0.95493 P

2π 1.00000 Q

We see that there are only seven possible configurations available from Equation 29, if N is a positive integral quantum number and r_e ≤ λ_{de B}.

This, and Eq. 29 are, of course, just the Bohr model of the atom, except that the model has now been related to a compressible (vortex) flow.

The full quantum theory of the atom requires the Schrodinger equation ( Sect. 3.7.1), which has its compressibility basis in the quantum operators for momentum and position as given in Sect. 3.6D.

5. RELATIVITY: A COMPRESSIBILITY EFFECT

5.1 GALILEAN RELATIVITY

Communication signals are instantaneous (c = ∞ ). Transformations are as follows:

l = l′ y = y′ t = t′ V′ = V₁ + V₂

z = z ′

x = x′ + vt

5.2 SPECIAL RELATIVITY (Relativity of uniform motion)

Signals travel at a finite speed (c ‹ ∞ ), that is, at the speed of light (3 x 10⁸ m/s) in free space.

The present theory identifies a finite speed of light as a physically-based compressibility effect.

The photon speed is now explained as a flow velocity. The free space speed is Vγ = 3 x 10 m/s. From the basic energy equation

c² = c_o² – V²/n (1)

we see that V can only equal c_o if n = 1 and c = 0; this is the condition where V_γ = V_max the escape speed to vacuum.

The wave speed c is a local variable determined by V and vice versa. The wave/particle nature of the photon is evidenced in the new theory since the photon energy is

Ε = hν = 2cV

So that it depends on both a wave speed c and a particle of flow speed V. (Sect. 3.6).

If an arbitrary choice is then made to assume an inertial coordinate system, namely one ‘at rest’, then the wave speed in free space c_o (numerically equal to the speed of light V_γ in free space) becomes equal to the free space speed of light relative to the inertial system, and the variable wave speed c becomes transferred from the inertial system to any other system moving at relative speed V. The famous postulate of the ‘constancy of the speed of light in vacuum for all inertial observers’ has its roots in this essentially arbitrary choice in the compressible energy equation.

5.2.1 THE FITZGERALD CONTRACTION FACTOR

This follows from the steady state energy equation (1)

c² = c_o² – (1/n) V² (1)

where, for 1-dimensional flow where n = 1, we have

c² = c_o² – V², and

c/c₀ = [1 – (V/c_o)²]^1/2 (2)

which is the Fitzgerald contraction factor, and which is now seen being a compressibility ratio of wave speeds.

5.2.2 THE LORENTZ TRANSFORMATIONS

These now become a series of simple, kinematic, compressibility ratios, all equal to the wave speed ratio c/c_o, as follows:

c/c_o = l′/l = dt′/dt = dτ/dt = x′/(x – vt) = t′/(t –(vx)/c_o²) = W/W_o = E_o/E = m_o/m = F_o/F =

= λ/λ′ = ν/ν′ = p/p_o = T/T_o (3)

where τ is the proper time, W is work, E is energy, m is mass, F is force, λ is wavelength, ν is frequency, p is momentum, T is temperature.

Since c/c_o is also the fundamental isentropic ratio, giving all the thermodynamic ratio values for pressure, density and temperature , (see Sect. 2.9), the new theory now relates the Lorentz transformations to these thermodynamic ratios, as well as to quantum physics via the quantum state variables, c/c_o and V/c_o (Sect. 3.8).

In (b) above the variables c_ot and Vt are numerically identical to Minkowski’s c and x, but they are conceptually very different; they are now physically-based, compressibility quantities involving the quantum state variables.

The metric interval is

ds = c dt (4)

where c is now the local wave speed calculated from the energy equation (1), and dt is the local time differential.

We derive the metric interval ds from the energy equation as follows ( for n = 1): In the energy equation (1) c_o is constant and the equation is elliptical in c and V. Multiplying through by t², we have

c²t² = c_o²t² – v²t²

which is now hyperbolic in the variables c_ot and vt, with ct becoming the constant. Hence we have

c_o²t² –V²t² = c²t² = s² = const,

5.2.3 THE RELATIVISTIC HAMILTONIAN

The standard form: H² = c_o²p² + m_o²c_o⁴ is not quite correct, although the error is very small in most cases. Instead,

H² = c²p² + m_o²c_o²c² (5)

Or,

H² = c²p² + m²c⁴ (6)

Here, the c’s are the local wave speeds, and m is the relativistic mass. The c’s are as calculated from the energy equation (1), and c_o iss 3 x 10⁸ m/s as usual.

Setting p = m_oV[1 – (V/c_o)² ]^1/2 , we have H = (m_o)c_o², which is a static invariant energy. (Compare this with L = 2cV in Sect. 3.6C).

5.2.4 THE EINSTEIN FORMULATION OF SPECIAL RELATIVITY

Postulate 1: The laws of physics are the same for all inertial observers.

Postulate 21: The speed of light is constant (c = 3 x 10⁸ (Sect. 3.8). for all inertial observers, even those in uniform relative motion with respect to one another.

Postulate 2 has no physical basis, and the Lorentz transformations follow as a consequence of the ‘absolute’ nature of space-time.

5.2.5 THE COMPRESSIBILITY FORMULATION OF SPECIAL RELATIVITY

The wave speed c is a local variable given by the compressible energy equation (1). It may arbitrarily be set to c = c_o ( = 3x10⁸m/s) by any inertial observe r, thus transferring the variable local wave speed to any other observers in uniform relative motion.

The Lorentz transformations are simple, physically based, compressibility ratios equal to c/c_o (Sect. 5.2).

The Fitzgerald contraction factor is now this same compressibility ratio c/c_o (Sect. 5.2.1).

The compressibility ratio c/c_o is now a quantum state variable (Sect. 3.8).

Space and time, and space-time have no absolute character, nor any physical character apart from the energy forms ( particles). Space-time is simply the natural graphical expression of the compressibility relations which involve distance and time.

The compressibility formulation and the Einstein formulation of special relativity yield identical numerical results for most cases of uniform motion. Exceptions are the Hamiltonian energy (Sect. 5.2.3) and the Fizeau Effect (Sect. 4.7.2).

A subtlety of nature in this matter, which has been the hidden source of great perplexity, is that the speed of light in free space is not a wave speed at all, but is, instead, a particle or photon speed V_γ , (which is, however, numerically equal to the free-space wave speed c_o, since, for n = 1, we have V_max = n^1/2 c_o = c_o = Vγ ). Yet, upon its interception or measurement, the photon again regenerates its wave-like properties and a local or variable wave speed c reappears (See Sect. 4.7.1).

5.2.6 TRANSVERSE AND LONGITUDINAL MASS

There is only one relativistic mass, m =m_o(c/c_o). The so=called ‘longitudinal mass’

M = m_o] 1 – (V/c_o)²]^1/2 (7)

Is now a force correction tgerm (see Sect. 6.1.1).

5.3 GENERAL RELATIVITY

In compressible flow, forces, when present, introduce a curvature of the characteristics on the space-time diagram; space-time curvature thus indicates acceleration and the presence of force.

In compressible flow, the forces are both physically real and fully relativistic. They may be ‘transformed away’ if desired, by choosing a Lagrangian coordinate system which moves with the relative flow so as to make V = o and c = c_o in the basic energy equation (1).

The relationship of the Lagrangian coordinate system to the space-time coordinate system is one of distortion; that is to say, it is a tensorial relationship.

The transformation

h = x(h,t); h = h)x,t) (8)

between the two systems represents a distortion in the relativistic, space-time, graphical representation. Here, dh = c dt = ds.

The analogous (tensorial) distortion of 4-space (x,y,z,t) in general relativity – which is needed in order to obtain a force-free representation- is valid, but ‘curved space-time’ is a graphical concept only in compressibility terms, and has no physical foundation or physical reality. The particle path curvature in 3-space is real; the absence of curvature (force) in a Lagrangian system is a purely computational or representational option, and its value is to be judged solely on its utility or convenience, and not on any supposed insights it may offer into the nature of physical reality.

The ‘force-free’ condition is

∫ds = ∫dh = ∫c dt =0 (9)

In general relatatvity, ds , the geodesic arc element, is

ds² = ∑g_ij^dx_idx_j (10)

The fundamental equation of general relativity, needed to gve force free motions, is therefore

δ∫ ds = 0 (11)

or, expressed as a tensor,

d²x_μ/ds² + Г (dx_μds)(dx_ν/ds) = 0 (12)

General relativity is a continuous field theory, and as such, excludes discontinuities or singularities. Therefore, it appears to be fundamentally incompatible with quantum physics, where we have shown shock discontinuities to be the physical basis for the emergence of the particles of matter from compressible energy flows (Sect. 3.9).

General relativity is compatible with the restricted field of isentropic, classical mechanical motions, and of isentropic, optical ray curvatures ( bending of light rays near gravitating masses).

Space-time coordinates arise from the introduction of the relative velocity V in the compressible energy equation (1). So-called space-time ‘warping’ is a coordinate effect, tat is to say it is a purely graphical distortion in the depiction of an accelerated, compressible energy flow.

General relativity is a valid computational means for accelerated flows in a compressible field such as that of electromagnetic radiation which has a finite signal speed. But in the new quantum theory of gravitation ( see Sect. 6.3), which is essentially beyond the space-time ‘event horizon’ of relativity, it would appear to be inapplicable.

6. GRAVITATION AND COSMOLOGY

6.0 Since the new theory has been successful in reformulating quantum physics on a new basis and unifying it with relativity theory, it is natural to apply it to gravitation. We shall now see that it offers a new quantum theory of gravitation , and a new cosmology.

6.1 FORCE IN COMPRESSIBILITY TERMS

The basic compressibility force equation, expressed in terms of wave and flow velocities is

F = dp/dt = m_o/[1 – (V/c_o)²]^1/2dV/dt + m_oV d[1 – (V/c_o)²]²/dt

= m_o (c_o/c) [a (1 + m²)] (1)

where M = V/c is the Mach number of the flow, p is the momentum, m_o is the rest mass, a is the acceleration and c/c_o is the Fitzgerald contraction factor ( Sect. 5.2.1).

If we set p = ε/c, we get

F = d(ε/c)/dt

F = 1/c dε/dt – (ε/c²) dc/dt (1a)

6.1.1 ‘LONGITUDINAL MASS’

The long-standing puzzle of the so-called ‘longitudinal mass’ in relativity theory *Sect. 5.2.6), namely,

m = m_o[1- (V/c_o)]^1/2 (2)

no longer remains. There is only one (relativistic) mass, which is

m = m_o(c/c_o) (3)

Instead of mass, it is now force which has a compressibility correction term in the square of the Mach number M², and it can also be shown that it is this correction which introduces the 3/2 power factor in Eq.2 when the velocity and the acceleration are not co-linear.

6.1.2 FORCE IN PRESSURE GRADIENT TERMS

F = - 1/ρ dp/dx ( = 1/ρ Ñp, in general) (4)

For n = 1 (k = 3): F_p = 1/3 (c²/p) Ñp: p = 1/3 ρc²

For n = 2 (k = 2)´ F_p = 1/2 (c²/p) Ñp: p = 1/2 ρc²

6.1.3 WAVE SPEED, C

c² = k p/ρ = (n+2)/n p/ρ (5a)

where 1/ρ = v = c²/kp, and so

c² = c_o² –V²/n = k p/ρ (5)

6.2 DERIVATION AND DIMENSIONS OF THE GRAVITATIONAL CONSTANT G

From Sect. 3.5 and 3.6, where we introduced the electromagnetic vibrational energy ε_h at a point in space ( i.r. for a particle), we saw that

Ε_h = cV = hν (6)

Now, for gravitational energy ε_g attracting over the space intervening between masses, we can, by symmetry, write

Ε_g = cV = Gk (7)

where k = 1/λ is the wave number in tht space.

If cV is the same in both Eqs. (6) and (7), then

Ε_g/ε_h = cV/cV = 1 = Gk/hν

(G/h) (k/ν) = 1

G/(h λν) = G/hc = 1

G/h = c (8)

Where G has the dimensions of energy times length. This can be compared with Planck’s constant h ,which has dimensions of energy times time.

On the above derivation, Planck’s constant has a relationship to vibrational energy-density at a point in time via he frequency ν ( h = cV/ν); whereas the gravitational constant G relates to an energy-density at point in space between gravitating masses via the wave length λ (G = cV/λ).

6.2.1 GRAVITATIONAL WAVE SPEED

If we now insert the numerical values for h and G in (8), we get for the wave speed c

C = G/h = (6.67 x 10^-11)(6.63 x 10^-34 = 10²³ m/s) (9)

This raises the question: Can such an enormous wave speed be possible?

To answer, we plot the ordinary, compressible wave speed’s energy (c²) from the basic energy equation

c² = c_o² – V²/n (5)

where we have the familiar energy ellipse in quantum variables c and V, or the linear plot in energy variables c² and V²:

Here, c_o is 3 x 10¹⁰ m/s; the maximum flow speed is V_max = n^1/2 c_o. But, if in the energy equation (5), we now put n as being negative, then we have

c² = c_o² – V²/(-n) = c_o² + V²/(+n) (10)

which plots as

Here, with n being a negative quantity, c_o² becomes a minimum wave speed ( = 3 x 10⁸ m/s, and there is no upper limit to either c or V.

The enormous wave speed ( c = 10²³ m/s) indicated in Eq. (9) is therefore compatible with an energy equation where n is a negative quantity (Eq. (10). Furthermore, we see that , in variables c and V, this plots as an hyperbola, instead of the usual energy ellipse.

Further support for the new wave speed will be found in Sect. 6.8 in connection with the solution to the problem of the cosmological constant.

Next, we seek a physical basis for this extraordinary wave speed.

6.3 THE GRAVITATIONAL QUANTUM OR GRAVITINO

We have shown in Sect. 3.9 that ordinary matter is a condensed energy form related to the entropy-associated variable n through the factor [n+1)]^1/2, where the number of degrees of freedom in the basic, steady-sate, energy equation (5) is a positive integer: this energy equation is also elliptical in the quantum state variables c and V.

However, in our new formulation of quantum mechanics on the basis of compressible energy ( Sect. 3.6), we alsofound an ‘estra’ energyi term 2cV which afrises from either (a) the unsteady=stste energy equation

c² = c_o² – V²/n – 2cV/n

which is also elliptical when rearranged to

c² + V²/n + 2cV/n = c_o² (elliptical)

or, (b) from squaring the quantum wave function to obtain the energy

ψ² = (c + iV)² = c² – V²/n + 2icV/n

ψ*² = (c – iV/n)² = (c² + V²/n) – 2icV/n

Now, if n is taken to be a negative ineger,then the unsteady-state energy equation becomes

c² + V²/(-n) + 2cV/(-n) = c_o²

c² – V²/n – 2cV/n = c_o² (hyperbolic)

Since the cV ‘extra’ energy term can appear in either an elliptical energy relationship or in the hyperbolic form, therefore, our derivation of Eq. 3 ( and hence also Eq. 4) from the same cV term is justified. Also, recalling that the cV term generates all of the fundamental quantum relationships ( Sect. 3.60, it is not unreasonable that it should also generate the gravitational constant G, as we have suggested above – only now in a context of a hyperbolic energy equation rather than the standard elliptical one.

As to a physical basis for this extraordinary hyperbolic speed, we recall that the process of generation of the proton and electron ( Sect. 3.12) also yields a stable, infinitely low-amplitude, pressure balancing, rarefaction energy wave (R).

We propose that the rarefaction energy wave R must be governed by the hyperbolic energy equation (1), and that this rarefied particle is the gravitational quantum or gravitino, which we shall designate by ν_g. For its energy, we put

ε_g = G k = cV (7)

(analogously to hν = vV)) but with c now given by the hyperbolic energyi equation (10).

c² = c_o² - V²/(-n) (10)

c² = c_o² + V²/(+n)

This grounds the speed of the gravitino as being either the wave speed c ( = 10²³ m/s), or the flow speed V ( = 10²³ m/s0. Such an enormous speed for the rarefaction energy form that we have here called the gravitino ν_g. the carrier of the force of gravity, would mean that gravitational influences in the universe are quasi-instantaneous, since, at such a speed, the gravitational force would cross the entire cosmos in a mater of minutes.

Newton’s hypothesis of instantaneous gravitational interaction is thus supported by this conclusion. If this is so, then Galilean, and not Lorentzian or relativistic relationships, would govern gravitational signals.

To conclude: the energy of a quantum wave function has a 2cV term. In the case of ordinary elliptic energy, the emitted electromagnetic radiation ( photon) is a compressed energy wave form; in the case of hyperbolic energy, the 2cV term gives rise instead to a rarefied energy emission, or rarefied particle, the quantum of gravity or gravitino ν_g.

Gravitational force thus arises from the tield of rarefied energy gravitinos which are continuously generated and emitted by elementary particles of matter ( Sect. 3.14) at a speed of 10²³ m/s. The gravitational field now consists of the massless gravitinos setting up a gravitational pressure gradient which affects all elementary massive particles.

In effect, we have generated the gravitino by quantizing the energy field cV/(-n) in a ‘space-like’ manner by inserting the wave number or ‘inverse length’ in the formula Gk = cV; this arises from symmetry considerations of the frequency or ‘time-like’m quantization of the same field in the electromagnetic case by the Planck relationship for the photon ε = hν = cV.

(Note that the speed of rarefied particles in general, including neutrinos, need not be 10²³ m/s, since from Eqn. (10), the wave speed can vary from the minimum value of 3 x 10⁸ m/s on up to infinity).ρ

6.4 THE FORCE OF GRAVITY IN COMPRESSIBILITY TERMS

Newton’s graotational force equation

F = G m₁m₂/ r² (11)

can be derived in terrms of compressible energy parameters, namely the relative flow velocity V, and the pressure p, as follows:

F = - 1/ρ dp/dx (12)

= (m) a = (m) dV/dt

= - 1/ρ d(ρc²/k)/dx = - d(c²)/dx

provided that dρ/dx is small.

If c = U, the gravitational potential, then we have

F_g = -dU/dx = - - G(m₁m₂)/r² = -1/ρ dp/dx (13)

which in general gives F_g = -1/ρ Ñp.

This gives the gravitational force in terms of the pressure and density of a gravitino pressure field. Gravitational force, therefore, appears to be analogous to a pressure gradient force.

In terms of the usual Newtonian equation of motion for a gravitational field we have

Q^r = (m) f^r = F^μ = V/cv (1/ρn) Ñp = V/c (1/ρn) Ñp = V/c (1/ρn) dp/dx (14)

Here, V/v the Mach number of the energy flow, acts as a force coupling constant; the acceleration is denoted by f^r, and Q^r and F^μ are forces.

6.5 EXCLUSIVELY ATTRACTIVE NATURE OF GRAVITY

In the above force equation (12), the negative sign may be interpreted as a unidirectional force, that is a force of attraction.

Furthermore, on a physical basis, so long as the wave acceleration (dc/dt) inside the gravitating mass particle is positive, the entry of a rarefaction gravitino into the particle must decrease the energy density. From the wave speed equation

c² = k p/ρ (5)

we see that such an energy density decrease must cause an increase in the wave speed c: from Newton’s thir Law there will then beset up an equal and opposite force towards the source of the absorbed gravitino, that is to say, there will be a force of attraction.

6.6 A NEW FORCE LAW

In compressibility terms, we have

F^r = - dU/ds = -ÑU = - 1/(nρ) V/c Ñp

ѲU = Ñ× [1/(nρ) V/c Ñp ] = -4πG (m) (Poisson equation)

ѲU = 0 (Laplace equation)

In relativistic terms, for a continuous, perfect fluid, we have

T^μν….,ν + ρ_oF^μ = 0

Ρ_o is an invariant proper density, and F^μ a world force vector.

The field equations of general relativity

G^λσ = -8πρ^λσ

could also be employed , if desired, (but see Sect. 6.12 regarding the ‘dark matter’ of the cosmos).

6.7 EQUIVALENCE OF GRAVITATIONAL AND INERTIAL MASS: MACH’S PRINCIPLE

In the new theory, the equivalence principle becomes one of force equivalence, rather than the equivalence of gravitational and inertial mass as before. All force becomes gravitational in the sense that all force arises from acceleration with respect to the same background gravitino pressure field.

Because of the enormous gravitino speed ( c = 3 x 10²³ m/s), the local energy density is always in balance with, and can be related to, the total mass of the universe. Mach’s Principle is clearly complied with, although now on a new, and physical basis.

We can see this more clearliy from the energy equation

c² = c_o² – V²/n (5)

For constant energy (c²), the term V²/n must also be constant. If V changes ( i.e. if there is acceleration) then energy must be supplied to or released from the system. Thus necessary input or export of energy is the source of inertia in the new theory, and the rate of this energy lflow is the measure of force.

Mass is governed by – indeed, is generated by- the factor [n+1]^1/2 (Sect. 3.9), and so, for gravitational mass to differ from the inertial mass, n would have to change. Therefore, for a particle of mass m, with n being constant, we have

m_inertial = m_{gravitational}

The force causing the curvature of the water surface in Newton’s celebrated rotating water bucket experiment , thus arises from the energy flow or acceleration with respect to the universal background gravitino field; the latter now replaces Berkeley’s ‘fixed stars’ as a background reference; this also essentially agrees with Mach’s Principle, but now, for the first time, on a physical basis.

Note that, since V in the energy equation is purely relative, the new theory is fully relativistic. The theory, however, becomes quasi-Galilean for the case of gravitational radiation. Here, the relativistic corrections are now negligible because of the enormous gravitino speed which gives quasi-instantaneous signals cross the entire cosmos.

6.8 THE PROBLEM OF THE COSMOLOGICAL CONSANT

The cosmological constant Λ is given by Λ = (8π G ρ_v(/c_o²), where ρ_v is the vacuum energy density.

Friedmann’s calculations in general relativity theory led to the conclusion that Λ = 0. Quantum field theory concludes that the vacuum energy density must, however, be between 10²⁹ and 10⁴⁶ J/m³. But astronomical observations agree with the Friedmann conclusion that Λ must be zero. The conflict between quantum physics and astronomical observations lies in the enormous difference-ratio of 10⁴⁶. This conflict is now resolvable in the new theory through the new and enormous gravitino wave speed of 10²³ m/s, since

Ε = c_o² – [10²³]² = 10⁴⁶ J/kg

That is to say, the estimates of the vacuum energy density arising from standard quantum theory are in fact correct, but the energy is not our ordinary, measurable, elliptic energy. Instead, it is a rarefied or hyperbolic condensation of energy, and this is ordinarily undetected in our condensed-energy material world. The conflict in the estimates of the value of the cosmological constant is thus resolved by the new theory – the value of Λ is zero.

6.9 THE FLATNESS PROBLEM: Ω EQUALS UNITY

In the expanding universe (Big Bang model), the ratio of the initial gravitational energy to the initial kinetic energy of expansion, denoted by Ω, determine whether the universe is open ( Ω < 1), closed ( Ω > 1) or ‘flat’ (Ω = 1). In general relativity , these three criteria correspond to whether space-time is hyperbolic, elliptical, or Euclidean (flat). In Sect. 6.2 we have put

(radiated kinetic energy) ε_h = hν = cV (6)

(gravitational energy) ε_g = Gk = cV (7)

Both cV energies are generated in the same physical process, namely, the interaction of two quantum wave functions, characteristics, or enrays (Sects. 6.3; 4.1.1; 3.13). There is no theoretical reason why this same equality of kinetic and gravitational energies in Eqs. (6) and (7) should not also apply at the Initial Event of creation, and so the new theory states that the value of Ω is always unity. The universe is therefore ‘flat’ or Euclidean.

6.10 ACTION-AT-A-DISTANCE: THE ASPECT et al. EXPERIMENTS

The quantum wave function’s extra-energy term (2cV) which is involved with the fast moving gravitino, is generated and emitted in the same process which is also the source of the emission of the electromagnetic quantum or photon. The former, of course, must then contain the same phase and momentum information as the latter, that is, the gravitino emitted at 10²³ m/s contains the same quantum information as the photon emitted at 3 x 10⁸ m/s.

The experiments of Aspect et al. confirming the universal and instantaneous spread of wave function information are now explainable by the 2cV phase, momentum and polarization information carried by the gravitino at quasi-instantaneous speed to the detecting apparatus, well ahead of the same information carried by the slow-moving photon.

6.11 THE HIDDEN MASS,OR ‘DARK MATTER’ OF THE UNIVERSE: A NEW SOLUTION

Currently, the observed astronomical motions of certain structures of the universe, such as galaxies and clusters, can be explained by Newton’ laws of motion only with the assumption that there is present in and around the structures additional ‘hidden’ , unobservable, or ‘ dark matter’ amounting to from 90% to as much as 99% of the total mass of the universe; the optically and electromagnetically observed mass, therefore, constitutes only from 10% to as little as 1% of the total mass required to explain the motions. The existence and nature of this dark matter is a central problem of cosmology today.

Now in the new theory, we have assigned the force of gravity to a pressure field of massless, gravitinos, which are continuously emitted during the unceasing formation and reformation of the quantum wave interaction which generates the elementary particles of mater and stablizes their existence (Sect. 3.14).

Because the rarefied gravitino has such an enormous speed, it easily escapes the shock wave which encases and stabilizes all elementary particles, and it then speeds out across the universe. Since this gravitino does not itself emit gravitinos, it cannot be the source of the hidden mass, and so we must look further.

6.11.1 THE EQUATION OF STATE HAS TWO FORMS

The equation of state for compressible energy has not just one, but two possible forms:

A) pv = RT (isothermal); kpv = RT ( adiabatic) (15)

p²/a² –v²/b² = const. (hyperbolic)

B) p²/a² + v²/b² = const. (elliptical)

In World A: (d²p/dv²)_s > 0. Compression waves (condensed energy waves) steepen with time to form compression shocks, and these in turn form th elementary particles of matter. Rarefaction waves, on the other hand, die out, and only infinitely low-amplitude rarefaction waves, such as the gravitino ν_g can exist (Sect. 2.11).

In World B: (d²p/dv²)_s < 0). Rarefaction waves ( rarefied energy waves) steepen to form rarefaction shocks. Compression shocks would die out, and only infinitely low-amplitude compression waves would exist.

In all known material, compressible fluids only compression waves can exist, and n, the number of degrees of freedom parameter, is positive. Rarefaction states in a World B would, therefore, necessarily have n negative.

6.11.2 RAREFIED OR CELESTON MATTER

Rarefaction shocks in World B would form rarefied, elementary particles ( which we shall designate celestons).

These are the counterparts to the baryons, mesons and leptons of out World A of condensed energy forms . ( The electron-neutrino ν_e, muon-neutrino ν_μ and tau-neutrino ν_τ may possibly be already known examples of World B particles).

We now propose a transformation from World A matter to World B matter. This transformation A→ B takes place at a critical pressure p* ( see Figure page xx). Because of an entropy relationship

ΔS_A = Δn/(+n); and ΔS_B = Δn/(-n)

The transformation A → B, once begun, would be irreversible.

At an Initial or Creation Event ( e.g. a Big Bang) in a compressible expanding universe we would have p >>>p*, and so only condensed elementary particles of World A matter could emerge.

World B (rarefied or celeston matter) could only form when the critical pressure p* was reached. This point, however, might be reached locally (for example, in the centre of a vortex where the pressure is lower) before the expansion of the universe had reduced the overall cosmic pressure to the critical value p*.

The Initial Event would usher in a Unitary World A of condensed matter only ( p > p*) . Our present era (p ≥ p* ) is that of a Binary Universe ( A + B) where both compressed and rarefied energy forms of matter can exist.

A final cosmic state ( p < p*), which will begin when the overall cosmic pressure falls to the critical value, will again usher in a Unitary universe, but this time it will consist of World B, rarefied, or celeston matter only.

The gravitino ν_g would be the simplest possible rarefaction wave form, or celeston. How the graviton relates to the electron-neutrino, or to the muon-neutrino is a matter for study by quantum specialists, as is also the further theoretical investigation of the various possible other types of celeston particles which might be generated by a rarefaction shock process using negative values for the entropy parameter n. The scientific grounding of such a theory would necessarily rest on astronomical data ( See Sects. 6.11.4 and 6.11.5).

6.11.3 LOCAL PRESSURE LOWERING IN VORTICES

The pressure is given by

p = p_o - ò (ρw²/r) dr, or

p = p_o - ρò (w²/r) dr

The core pressure p_c is given by

p_c= p_o – (ρw²)/8π²r²

The compressible vortex ratios are:

V_max / r* = c*/r_min = [(k-1)/(k+1)]^1/2 = [ 1/(n+1)]^1/2

r/r* = 1/(V/c*)

The transformation of ordinary matter via the process A → B at r*. produces the necessary sink for continuous spiral inflow, because the matter flowing in to the vortex core is transformed from ordinary World a matter to the celeston or rarefied matte of World B.

(The ‘event horizon’ of general relativity’s black hole theory might correspond roughly to our r_min.).

6.11.4 QUASARS IN EARLY COSMIC TIMES (P >> P*)

Local pressure lowering in vortices in the early universe would not permit the pressure to drop sufficiently for p to reach p*. In this case, with no A → B effect, the sink causing the pressure lowering would have to be due to some process such as divergent flow.

Inflowing A-matter would be subjected to very large dynamic and thermodynamic effects near the quasar’s critical radius r*, and would therefore radiate strongly, but the A-matter would not transform to the B-state. Thus, quasars could form in the primitive universe, as is observed.

6.11.5 SPIRAL AND ELLIPTICAL GALALXIES (P_LOC < P*)

With an overall pressure lowering due to cosmic expansion, at some later time in cosmic history the pressure in the core of a vortex would drop to the critical pressure p*, and the transformation to B=matter could then take place locally.

In such cases, the central core would become a (transformation) sink. Quasars, having no such sink, could not persist; instead, spiral and elliptical galaxies with central sinks would emerge as the dominant cosmological structures.

Transformations of A-matter ( condensed, ordinary matter to World B matter ( rarefied, celeston matter) then would take place in the galactic cores. The ‘dark matter’ produced in the vortex core then would expand and spread throughout the galaxy producing a ‘halo’ of ‘dark matter’ as is observed. The universe is then in a binary state ( A + B).

The model can readily be extended to include the observed distinction between active and normal spirals.

6.12 EXPERIMENTAL TEST OF THE PROPOSED A → B TRANSFORMATION: QUASARS VS. GALAXIES

In the new theory, the change from quasar abundance in the early universe to their replacement by spiral and elliptical galaxies reflects and marks the critical pressure p* which controls the A → B transformation process just described.

Quasars, having no B or ‘dark matter’ should then have their experimentally observed motions explainable by Newton’s laws of motion using only their optically or electromagnetically observable mass; this prediction should, in principle, be open to experimental testing.

The energy change ΔE in the transformation A → B is given from the energy equations

ΔE = c_B² – c_A² = V²[ 1/n_A – 1/n_B]

Since n_A is positive and n_B is negative, energy must be evolved in the A → B transformations.

It might, therefore, be possible to calculate the energy evolved in galactic transformations of he type A → B from the above equation, that is to calculate the energy released the process of production of the proposed celeston matter from our ordinary condensed World A matter in the vortex core of a galaxy. Thus might be another experimental test of the A→ B transformation.

SELECTED BIBLIOGRAPHY

COMPRESSIBLE FLOW

Shapiro, A.H., The Dynamics and Thermodynamics of Compressible Fluid Flow.2 vols., J

John Wiley & Sons, New York, 1953.

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

Lamb, Horace, Hydrodynamics.8^th ed., Cambridge University Press, 1932.

Morse, P.M., Vibration and Sound. McGraw-Hill, New York, 1948.

Tietjens, O.G/, Fundamentals of Hydro- and Aeromechanics, (based on the lectures of L.Prandtl). McGraw Hill, 1969. Deve reprint, New York, 1957.

Rayleigh, J.W.S., The Theory of Sound. 2 vols. 1886. Dever reprint, New York 1945.

QUANTUM PHYSICS

Ziman, J.M., Elements of Advanced Quantum Theory. Cambridge University Press, 1969.

Sudbery, A., Quantum Mechanics and the Particle of Nature. Cambridge Universityi Press, 1986.

Van der Waerden, B.L., (ed.), Sources of Quantum Mechanics. Dovefr, New York,1949.

Heisenberg, W., The Physical Principles of the Quantum Theory. 1939. Dever reprint, New York, 1949.

Penrose, R. and Rindler, W., Spinors and Space-time. Vol .1, Two-spinor Calculus and Relativistic fields. Cambridge University Press, 1984.

Planck, Max, Theory of Heat Radiation, 2^nd ed. 1913. Dover reprint, New York, 1959.

RELATIVITY

Bergman, P.C., Introduction to the Theory of Relativity, Dover reprint, New York, 1976.

Pauli, W., Theory of Relativity (1921), Dover reprint, New York, 1981.

Einstein,A., Lorentz, H.A.,Weyl, H, and Minkowski, H.J., The Principle of Relativity ( (Collection of original memoirs on the Special and General Theory of Relativity with Notes by A. Sommerfeld). Dover reprint, New York.1952.

Morse, P.M., and Feshbach, H., Methods of Theoretical Physics.2 vols. McGraw Hill, New York, 1953.

MECHANICS, GRAVITATION AND COSMOLOGY

Lindsay, R.B., and Margenau, H., Fundamentals of Physics. Dover reprint, New York, 1957.

Menzel, D.H., Mathematical Physics, Dover reprint, New York, 1961.

Power, B.A., Shock Waves in a Photon Gas, Contributed Paper No.203, American Association for the Advancement of Science, Annual Meeting, Toronto, Jan.1981.

___________, Unification of Forces C. and Particle Production at an Oblique Radiation Shock, Contr. Paper No. 462, American Association for the Advancement of Science, Annual Meeting, Washington, D.C., Jan. 1982.

__________, 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, Detroit, May 1963.

___________, Gravitation: An Energy Compressibility Effect. Monograph, July 1983.

____________, Prediction of Elementary Particle Mass-ratios from Compressible FlowTheory. Monograph, May 1985.

Menzel, D.H., (ed) Fundamental Formulas of Physics. 2 vols., Dover, New York, 1960.

Copyright, Bernard A. Power, September 1992

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