Equations of State

for

Cosmic Fields

 

 

         

 

 

 

 

 

 

 

 

 

 

 

 

90

 

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 = co2    V2/n                                                                              (1)

 

where c is the local wave speed, co 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 = cp/cv   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/co) is also  a quantum state variable. The maximum flow velocity Vmax (when c = 0)  is the escape speed to a vacuu:

Vmax = √ n co.

 

 

2.2   Unsteady State Energy Equation

 

c2 = co2 – 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 = co2 – 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) – co2 , 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 pv­k = constant, where the adiabatic constant  k = cp/cv 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

 

Ñ2 ψ  = 1/c22ψ/∂t2 [ 1 + Ñψ ](k + 1)                                                            (7)

 

where k, the adiabatic exponent, is k = cp/cv = ( n + 2) /n; and ( k + 1 ) = 2( n + 1)/n =  2(co/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, ρ2,  T2                                                            (9)    

 

 

 

Entropy Change Across Shock: 

 

∆S = S1 – S2 = − ln(ρ0201)                                                      (10)

 

 

Maximum Condensation Ratio:

 

                                  ρ12 = [n+1]1/2 = Vmax/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 

                                                V1N  >  V2N

                                             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 ( = VN1/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 = [ co2 – V2/n]1/2     (steady flow)                                        (12)

 

c = co2 – 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 = co 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/co  and The  Isentropic Thermodynamic Ratios

 

c/co = [1 -1/n(V/co)2]1/2  = (p/po)1/(n+2) = (ρ/ρo)1/n = (T/To)1/2                                              (14)

 

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

 

 

 

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 co at the static state i.e. at  zero flow (V =0).

 Interestingly, Equation (14)

 

c/co = [1 -1/n(V/co)2]1/2  = (p/po)1/(n+2) = (ρ/ρo)1/n = (T/To)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

 

-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

 

mb/mq = Vmax/c* = [n+1]1/2                                                                                                                       (16)

 

mb is the mass of any hadron  particle, mq is a quark mass, Vmax = co n1/2 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 (mb)        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)  

                                  Ξo (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

 

mL/me-  =  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 = cp/cv = [(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 ≈ ml/me (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 :

 

pvk = 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 = ευ

 

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 = Hop

 

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

 

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 = pop

 

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

 

 

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

 

mb/mq = Vmax/c* = [n+1]1/2                                                                                                                       (16)

 

mb is the mass of any hadron  particle, mq is a quark mass, Vmax = co n1/2 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

 

mL/me-  =  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 = cp/cv = [(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 = mγ cV, where mγ 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 pvk = const. where k is the adiabatic constant.

 

Proposed 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.

 

 

 

 

 

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: 

 

 

(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.

 

 

 

 


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

                                  Fro m 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

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.

 

 

 

 

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

 

−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    6th 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 po. 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

 

dEy/dx  = (1/c) dB/dt and dBy/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  = ∂Ey/∂x = −(1/c) ∂B/∂t

 

Curl B = ∂By/∂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 = ∂Ay/∂x = − (1/c) ∂I/∂t

 

Curl I = ∂Iy/∂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 – ρo ) / ρo =  +∆ρ/ ρo). 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:

 

Ñ2 ψ  = 1/c22ψ/∂t2 [ 1 + Ñψ ](k + 1)                                           

 

where k, the adiabatic exponent is cp/cv = ( n + 2) /n; and ( k + 1 ) = 2( n + 1)/n =  2(co/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/co and V/co.

 

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/co and V/co, 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, λc

 

                                    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.  λdeB

 

                        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 = co2  -  V2/n                                                                               (1)                                                                           

 

where c is the local, compressive wave speed, co 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/co) is a quantum state variable ( Sects. 2.10b; 3.8). The maximum flow velocity Vmax (when c = 0) is the escape speed to a vacuum; Vmax = √ n co.

 

 

2.2   UNSTEADY-STATE ENERGY EQUATION

 

c2 = co2 – 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* = co* - 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) – co2 , 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

 

Ñ2ψ = 1/c22ψ/∂t2 [ 1 + Ñψ ](k + 1)                                                                                                                         (7)

 

where k, the adiabatic exponent is cp/cv = ( n + 2) /n; and ( k + 1 ) = 2( n + 1)/n =  2(co/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  >  V2>

                       

                              P1, ρ1, T1   <  p2, ρ2,  T2    

 

 

ENERGY;     

                                   V12/2 + h1 = V22/2  + h2 = ho                                                                                                                                                (8)          

 

where ho – 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 = ρ21                                                                                                                                     (13)

 

 

ENTROPY CHANGE ACROSS SHOCK:

 

∆S = S1 – S2 = -ln(ρ0201)                                                                     (14)

 

MAXIMUM CONDENSATION RATIO:

 

ρ12 = [n+1]1/2 = Vmax/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 

                                                V1N  >  V2N

                                             p1ρ1T1  <  p2ρ2T2

 

 

ENERGY:

 

[V1N2 – V1N2] /2  = (n+2)/n [p22 – p11]

 

or,

1/2[ V2N2 – V1N2]  =  cp [ T1 –T2 ]                                                             (16)

 

 

CONTINUITY:

 

ρ1V1N = ρ2V2N                                                                                                                          ( 17)

 

 

MOMENTUM:

 

p1 - p2 = ρ2V2N2 – ρ1V1N2                                                                                                             (18)

 

 

RANKINE-HUGONIOT EQUATION:

 

Ρ2/ρ1 = [(n+1)(p2/p1) +1] / [(n+1) + (p2/p1)]                                            (19)

 

 

PRANDTL EQUATION:

 

VN1VN2  = c*2 – 1/(n+1) Vmax2                                                                                         (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 ( = VN1/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 = co2 – V2 /n.

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

Ñ2ψ = 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 – φx2) φxx - 2φxφt – φtt = 0                                                                    (24)

where φ is a velocity potential, i.e. ∂φ/∂x = u, etc., and c2 = co2 –V2/n – (2n) dφ/dt.

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

 

Ñ2φ = 1/c22φ/∂t2                                                                                                                                  (25)

 

2.7   COMPRESIBLE FLOW RATIOS AND PARAMETER K

 

Vmax/co = n1/2;   Vmax/c* = (n+1)1/2                                                                                                                (26)

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

n = 2/(k-1);   k = cp/cv = (n+2)/n                                                               (28)

 

2.8  WAVE SPEEDS

c = [ co2 – V2/n]1/2     (steady flow)                                                                   (1)

c = [ co2 – 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 = co 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/co = [1 -1/n(V/co)2]1/2  = (p/po)1/(n+2) = (ρ/ρo)1/n = (T/To)1/2                   (0)

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

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/co). It yields all the thermodynamic variables of state from the isentropic ratio relationships via the wave speed ratio c/co  ( 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).

Ñ2ψ = 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 = co in the ‘at rest’ coordinate system chosen. (cf. Sects. 2.8; 5.2). For an energy flow, co 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/co)2 = 1 – 1/n (V/co)2,  (Eq. 11, Sect.2.1), we have

  ΨN = c/co + i V/co;        ψ*N = c/co – i V/co                                                                                              (5) 

For small V ,this reduces to  ψN ≈ c/co.

The same wave speed ratio c/co 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

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

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

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

(c + iV)(c – iV) = c2 + V2 = [Eψ]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 = co2 ; b) Unsteady : -2cV = co2

 

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 = ευ                                                                                                                (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 = Hop

which is the Hamiltonian energy operator.

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

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 = pop                                                                                                                (15)

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

 

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 = me cV, where me 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

2/2m Ñ2ψ = iħ Ñψ/∂t                                                                        (17)

 

3.7.2   DIRAC EQUATION

Eψ +  c(α . p )ψ + moco2βψ = 0

Or,

iħ∂ψ/∂t = iħcα . Ñψ + moc2βψ = 0                                                             (18)

where α12 = α22 = α32 ; β = 1.

 

3.7.3   KLEIN-GORDON EQUATION

For a relativistic, but spinless particle, we have

Ñ2ψ = 1/c22ψ/∂t2  [moc/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/co and V/co. 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/co, 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

mb/mq = Vmax/c* = [n+1]1/2                                                                                                                       (21)

 

mb is the mass of any baryon particle, mq is a quark mass, Vmax = co n1/2 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

mL/me-  =  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 = cp/cv = [(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 (mb)        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)  

                                    Ξo (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 ≈ ml/me (1/137), verifying Equation (22).

 

The lepton/meson mas

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

 

mL/me- = cp/cv 1/α2                                                                                                                             (23)                            

 

α2 = ml/me- cp/cv [ ΔT/ΔT] = (ΔQL+)p/(ΔQe) = 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 = 91/2 = 3 = mb/mq

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:

Sy – Sx = ΔS = ln(1+P) – k ln(1 +((k + 1)/2k))P) + k ln(21+ ((k-1)/2k))P)

Where k is cp/cv and P is the shock strength expressed in terms of the pressure rise across the shock py/px  and is given by

P = py/px – 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 = [Sy – Sx] = 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

ρ21 = [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

mb/mq = [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

ML/me- = k/α2 =  α2 cp/cv = (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 - ∂ho/∂N ]                                                                      (2)

ho 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- = ge).  The square of the charge constant is equal to the square of the fine structure constant of the atom:

 

e-2 = ge2 = 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*/rmin = [n+1]1/2                                                                                                                               (5b)

Vmax = n1/2 co                                                                                                                                     (5c)

4.1.5.1   COMPTON WAVE LENGTH  λc

In the compressible potentiall votex (CPV), we have

λ c = n1/2 rmin ;  rmin = λc/n1/2                                                                                                                      (6)

r* = λc [ (n+1)/n]1/2                                                                                                                              (7)

rBohr = λ? (1/α2)                                                                              (8)

 

4.1.6     ELECTRON VORTEX PRESSURE

 

c/co = (p/po)1/(n+2) = [ 1 – 1/n{V/co}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 =  Vmax  = n1/2 co.

 

B)     THE COMPRESSIBLE POTENTIAL VORTEX (CPV)

 

VT r = const.; rmin = λc/ n1/2

(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, λde B

λde B = 2πλc[c/V]                                                                                (15)?

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

Λc = [λde 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 - Ñ2A

 

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 = co, 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 = co2 – V2 , and so

Vmax = n1/2 co =  co

The photon speed Vγ is, therefore, the escape velocity to a vacuum Vmax  which equals co (= 3 x 108  m/s).  In free space the photon speed corresponds to Vi, 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 Aμ

Aμ = 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 = co2 – 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/co = [ 1 – (V/co)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 = co2 –V2/n

from which we have the following relationships

c = [co2 – V2/n]1/2             c/co = [1 – 1/n (V/co)2 ]1/2

 

V = [nco2 – nc2]1/2                  V/co = [1 – (c/co)2]1/2

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

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

Finally, we put

N≡ Vo/Vi

Where Vo is the speed of photons in vacuum, and Vi 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 co, 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 Vo/Vi has the identical form of the left hand side of the Prandtl equation for the flow velocity transformation across a normal shock front, namely

Vo/Vi = c*2 / Vo2

 

And, if Vo = co, then also

Vo/Vi = c*2/co2 = (n+1)/n

But we have set Vo/Vi equal to the refractive index N and so we also have

N = Vo/Vi = (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 Vo relative to a downstream flow velocity Vi 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 Vo in space ( = 3 x 1010 m/s) is a flow speed and not co, the free –space wave speed; the speed of light Vi in the medium is again the flow speed and not the wave speed ci .  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 Vm 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 = [co/N +  Vm] / [ 1 + (coVm/N)/co2]

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

co/N + Vm[ 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/co)2 = 1 – 1/n (V/co)2

If n = 1 (1-dimensional flow) then (V/co)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)/λoco)                                                                               (28)

Where A is the area enclosed by the light path, Ώ is the angular rate of rotation, λo and co 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

upvg = co2

This derivation requires that the phase velocity up should exceed co when vg 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 = co2 – V2/n – 2cV/n,  or

cV = n[co2/2 – c2/2 –V2/2]

instead of upvg = co2.

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 co, 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):                                               λde B =  h/m/λ

(Compton wavelength):                                                   λc = h/mc

so that                                                                         λcde  B = V/c

It is interesting that, if we now set

N λde B = 2π re                                                                                                                                           (29)

We have

λde B = (2π/N) re                                                                                                                                   (29a)

where N is a quantum number equal to 1,2,3….2π, and re 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

        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 re    λde 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 = co2 – V2/n                                                                                    (1)

we see that V can only equal co if n = 1 and c = 0; this is the condition where Vγ = Vmax 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 co (numerically equal to the speed of light Vγ 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 = co2 – (1/n) V2                                                                                                                                     (1)

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

c2 = co2 – V2, and

c/c0 = [1 – (V/co)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/co, as follows:

c/co = l′/l = dt′/dt = dτ/dt = x′/(x – vt) = t′/(t –(vx)/co2) = W/Wo = Eo/E = mo/m = Fo/F =

= λ/λ′ = ν/ν′ = p/po = T/To                                                                                                                                                                                         (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/co 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/co and V/co (Sect. 3.8).

 

 

 

 

In (b) above the variables cot 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) co is constant and the equation is elliptical in c and V. Multiplying through by  t2, we have

c2t2 =  co2t2 – v2t2

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

co2t2 –V2t2 = c2t2 = s2 = const,

5.2.3     THE RELATIVISTIC HAMILTONIAN

The standard form: H2 = co2p2 + mo2co4 is not quite correct, although the error is very small in most cases.  Instead,

H2 = c2p2 + mo2co2c2                                                                                                                                (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 co iss 3 x 108 m/s as usual.

Setting p = moV[1 – (V/co)2 ]1/2 , we have H = (mo)co2, 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 = co ( = 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/co (Sect. 5.2).

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

The compressibility ratio c/co 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 Vγ , (which is, however, numerically equal to the free-space wave speed co, since, for n = 1, we have  Vmax = n1/2 co = co = 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 =mo(c/co). The so=called ‘longitudinal mass’

M = mo] 1 – (V/co)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 = co 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 = ∑gijdxidxj                                                                                                                              (10)

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

δ∫ ds = 0                                                                                (11)

or, expressed as a tensor,

d2xμ/ds2 + Г (dxμds)(dxν/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 = mo/[1 – (V/c­o)2]1/2dV/dt + moV d[1 – (V/co)2]2/dt

 

=  mo (co/c) [a (1 + m2)]                                                                                                  (1)

 

where M = V/c is the Mach number of the flow, p is the momentum, mo is the rest mass, a is the acceleration and c/co 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 = mo[1- (V/co)]1/2                                                                                                                            (2)

 

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

 

m = mo(c/co)                                                                                 (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):                                                   Fp = 1/3 (c2/p) Ñp:           p = 1/3 ρc2

 

For n = 2 (k = 2)´                                                  Fp = 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 = co2 –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 εh at a point in space ( i.r. for a particle), we saw that

 

Εh = cV = hν                                                                                     (6)

 

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

 

Εg = 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

 

Εgh = 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 = co2 – V2/n                                                                                    (5)

where we hav