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Derivation of the Hubble Redshift

in a Static Universe

Thomas B. ANDREWS

Brooklyn, NY

January 1, 2007

Abstract

This paper is the second of two papers on determining the dynamical state and metric of

the universe. In the ﬁrst paper, a new light curve broadening eﬀect for Ia supernovae (and

other short duration events) was discovered which resulted in the falsiﬁcation of the expanding

universe model. The static universe model was then conﬁrmed by Tolman surface brightness

tests of the brightest cluster galaxies.

In this paper, a pure wave system is proposed as a basis for a new model of the universe. It is

hypothesized that the constructive interference peaks of the wave modes are the mass particles.

Furthermore, it is proved that the intensity of the wave modes varies as 1/r instead of the

expected 1/r2. Consequently, the interactions of distant mass particles via the wave modes are

the source of the energy of elementary particles and photons. The energy contributed by these

wave mode interactions and the classical gravitational potential energy are then identically the

same.

It is proposed that the “local” universe is simply a large sphere within a “background”

universe with a constant mass density. A simple eigenvalue equation is used to calculate the

eigenvalues of the wave modes in the local universe assuming a constant mass density. Results

show that the frequency and energy of the mass particles increases with the number of interacting

mass particles. Therefore,an equilibrium state exists when the gravitational potential energy

equals the eigenvalue energy of a mass particle.

The Hubble redshift process is described as follows: As a photon propagates in the local

universe, it interacts with fewer mass particles and, therefore, has a lower gravitational potential

energy. This loss of energy by the photon over cosmological distances is no diﬀerent, in principle,

than the loss of gravitational potential energy by a photon rising in the gravitational ﬁeld of

the earth. The energy loss is proportional to exp s/R given R= 2RL/3 where sis the distance

the photon moves and RLis the radius of the local universe.

1

Contents

1 Introduction 2

2 Theory of the Wave System 3

2.1 The General Force Law in a Wave System . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Derivation of Newton’s Law of Gravitation 5

3.1 Proof of 1/r Intensity Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Gravitation as an Energy Source for Particles . . . . . . . . . . . . . . . . . . . . . . 7

4 Newtonian Gravitation Theory 8

4.1 Gravitational Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Cosmological Models of the Universe 10

5.1 Static Flat Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6 Calculation of Parameters of the Universe 11

6.1 The Local Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Derivation of the Hubble Redshift 14

7.1 Hubble Redshift of a Photon and Mass Particle . . . . . . . . . . . . . . . . . . . . . 18

8 Maximum Hubble Redshift 21

9 Comments 23

1 Introduction

This is the second of two papers on determining the dynamical state (static or expanding) and the

metric of the universe. Solving both problems is essential since valid interpretations of cosmological

observations, especially at high redshifts, require a correct model of the universe.

In the ﬁrst paper [1], a new light curve broadening eﬀect for supernovae was discovered. The

theory of this eﬀect was presented in the ﬁrst paper. Due to this discovery, the expanding universe

model was falsiﬁed based on the following logical argument. Because the new light curve broadening

eﬀect is in addition to the time-dilation broadening predicted in the expanding universe model,

two light curve broadening eﬀects should occur if the expanding universe model is true. However,

Goldhaber [2] observed only a single light curve broadening eﬀect. Consequently, the expanding

universe model is falsiﬁed. Finally, Tolman surface brightness tests of the brightest cluster galaxies

presented in the ﬁrst paper prove observationally that the universe is static.

However, just proving the universe is static does not really resolve the basic problems of a

static universe that were discussed by Einstein [3] over 80 years ago. The major problem is the

stability of the universe against collapse due to the attractive force of gravitation. Another problem

is the gravitational potential energy in an inﬁnite, homogenous universe. The potential energy

diﬀerences in such a universe could be inﬁnite. But, Einstein pointed out that great diﬀerences in

the potential are contradicted by the facts. In particular, the potential diﬀerences must be small

since the magnitude of stellar (or galactic) velocities are low. Both of these problems are addressed

in this paper.

2

The Hubble redshift in a static universe is derived in this paper. This proved to be a very diﬃcult

problem as is apparent from the attempts of many theorists, over a long period of time, to explain

the Hubble redshift in a static universe. To my knowledge, all these attempts failed.

It is useful to list the requirements for the Hubble redshift process. They are:

1. All frequencies are aﬀected

2. The decrease in frequency is proportional to the frequency

3. For small distances, the Hubble redshift is proportional to the distance

4. The Hubble redshift is a reciprocal eﬀect between observers and distant objects

Gravitation is the only known process other than the doppler shift or the expansion of the universe

which satisﬁes the ﬁrst three requirements. Therefore, without introducing a new ﬁeld, gravitation

is the best candidate for the Hubble redshift process. However, the Hubble redshift can not be

directly derived from only the properties of gravitation. Therefore, a new viewpoint on the nature

of the universe appears required.

2 Theory of the Wave System

The introduction of a new viewpoint requires speculative assumptions concerning the nature of the

universe. Brieﬂy, I assume the universe is a pure wave system and that mass particles and photons

result from the interactions between the waves. This is not an outlandish assumption. Actually,

quite the contrary is true. Pierce in his book “All About Waves” [4] says that “The idea of waves

is one of the great unifying concepts of physics. Once we recognize that in certain phenomena we

are dealing with waves, we can assert and predict a great deal about the phenomena even though

we do not clearly understand the mechanism by which the waves are generated and transmitted.”

Consequently, I use wave theory to deduce the properties of the universe. In particular, I develop

strong evidence that elementary particles are the constructive interference peaks produced by the

wave modes of the system. Similarly, “space” is composed of the destructive interference regions of

the wave system.

The stable state of the wave system is governed by the following two principles:

1. The frequency of the wave system is reduced when the wave parameters, ie, the mass or energy

density and the gradient of the tension, increase at the constructive interference peaks of the

wave modes.

2. As the number of mass particles mutually interacting increases, the frequency of the wave

system increases.

As an example of the ﬁrst principle, consider a string vibrating in a speciﬁc frequency mode.

When lead weights are placed at the peaks of the vibration, the frequency of the string is reduced.

Similarly, when the gradient of the tension is positive and is larger at the peaks, the frequency is

reduced.

Assuming the mass density and tension are proportional to the energy density, any process which

increases the energy density at the peaks of the constructive interference lowers the frequency and

energy of the wave system. Furthermore, since an eigenvalue system generally tends to a minimum

energy system, this implies that there is complete constructive interference in the wave system.

3

By the second principle, there is a lower bound on the frequency, at least for mass particles,

determined by the number of particles which mutually interact in the universe. This frequency is

given exactly by a unique eigenvalue equation derived by Chen [5]

fp=1

2π√Npk/m (1)

where N is the number of wave modes and pk/m is the interaction constant. Assuming that

the interactions between particles have the same interaction constant, this eigenvalue system is

remarkable in that it has only two discrete frequencies, a lower frequency equal to fo= 1/(2π)pk/m

and a higher (degenerate) frequency fp=√Nfowith N−1 modes.

Except for the lower frequency mode, the wave system vibrates at the degenerate single frequency,

fp. The general solution [6] for each of the degenerate frequencies, fp, is

Ai(t) = Ci(0) cos(fpt) + D∗

i(0)/fpsin(fpt) (2)

where Ci(t) is the amplitude, D∗

i(t) is the velocity. The initial values of these have the argument

t= 0. The phase angle is determined by the relative magnitudes of Ci(0) and D∗

i(0)/fp. There are

on the order of 1080 such equations.

It is diﬃcult to interpret the above in terms of the wave system because there is constructive

interference. However, provisionally, the Ci(t) and the D∗

i(t)/fpare the amplitudes of the proton and

electron for the ith eigenvalue. Both particles vibrate at a single frequency but have diﬀerent energy

because their amplitudes are diﬀerent. Interactions between the particles are both gravitational

and electrostatic. The fact that the proton and electron are the only stable mass particles tends to

conﬁrm the existence of the above equations.

Furthermore, because the system is a wave system, the energy of the system is conserved and

the system is Lorentz invariant. These invariants constitute the most important relations in physics.

Since it is diﬃcult to imagine another physical system with these exact invariants, a strong case can

be made that the universe is a pure wave system.

Therefore, an important question is “Does the wave system physically exist?” A symmetry

argument due to Giorgi [7], when applied to the universe, proves, almost certainly, that a wave system

underlies all physical processes in the universe. The symmetry argument is as follows: Assume a

system which is inﬁnite and linear and where the laws of physics are space and time translation

invariant. Giorgi then argues that the modes of oscillation of the system are determined by the

representations of the space and time translation groups. Since the solution of each representation

is a complex exponential in space and time respectively, given by exp i(ωt ±kx), the system is

equivalent to a system of waves. If an underlying wave system does not exist, the laws of physics

would not be space and time translation invariant. Therefore, the existence of an underlying wave

system is proved.

2.1 The General Force Law in a Wave System

In order to determine the forces which occur in the wave system, the classical wave system equation

with variable parameters is solved for small changes in the variable parameters. These variable

parameters are the mass density and the tension. This solution assumes that all forces are due to

changes in the parameters of the wave system.

Since the equilibrium solution of the wave system equation is needed, the space dependent eigen-

value equation is used. This is derived from the one-dimensional classical wave equation by separation

of variables, resulting in

d

dx T(x)dY

dx +σ(x) (2πf )2Y= 0 (3)

4

where (2πf )2is the separation constant. T(x) and σ(x) are the variable tension and mass density

parameters respectively.

For small spatial variations in the parameters, the perturbational solution [8] of Eq. 3 is given

by

f2=f2

o 1−2

LσoZL

0

(σ(x)−σo) sin2(kx)dx

−2

kLToZL

o

∂T (x)

∂x cos (kx) sin (kx)dx!(4)

where fo=k2

o/(2π)2(To/σo) is the unperturbed frequency and Lis the size of the system. This

equation shows that fdecreases when σ(x) and ∂T (x)/∂x are larger at the constructive interference

peaks.

Eq. 4 may be simpliﬁed as follows: First, by noting that the two terms on the right are equal.

Second, by setting sin2(kx) and sin (kx) cos (kx) = 1/2 and taking the square root. Then, the

perturbed frequency, f, is approximately

f=fo 1−1

LσoZL

0

(σ(x)−σo)dx!.(5)

Now assume there is only a single particle in the system and the particle (or constructive inter-

ference peak) has a constant linear size, lo. Then, σ(x) = m/loand σo=m/L where mis the mass

of the particle. Integrating Eq. 5,

f=fo2−loσ(x)

Lσo.(6)

Note: σ(x)dx integrates to loσ(x) since σ(x) is a constructive interference peak which only exists over

the distance lowithin the much larger integration distance, L. Without the constructive interference

peak, f= 2fo. With the constructive interference peak, f=fo, the stable frequency of the wave

system.

Next, multiplying both sides of Eq. 6 by hand setting E=hf,mc2=hfoand m=Lσo, we

have

E=mc22−loσ(x)

m= 2mc2−loσE(x) (7)

where the energy density σE(x) = c2σ(x). Then the force required to move a mass particle “up” in

a gravitational ﬁeld is given by

F=dE

dx =−lo

d

dxσE(x).(8)

Eq. 8 is hereafter referred to as the “General Force Law.” The type of force depends upon the nature

of the physical process which changes the energy density at the particle.

3 Derivation of Newton’s Law of Gravitation

In the following, I derive Newton’s law of gravitation including the value of the gravitational constant,

G, from a physical process. Note that Newton’s law is entirely a mathematical statement of the law

and has not been derived previously from a physical process (see Feynman [9]).

5

To derive Newton’s law, I assume two protons, labeled m1and m2, are separated by a distance

r. The physical model for gravitation is as follows: Wave modes originating at m1interact with m2.

The gravitational force is then derived from the General Force Law in Eq. 8.

The ﬁrst step in the derivation is to calculate the energy density of the wave modes originating

from m2at m1as a function of r. Assume that m2is at an average distance Rgiven by Ro/2 where

Rois the radius of the universe. m2is one of n protons at the distance Rand m1interacts equally

with the n protons. Therefore, m2contributes to the energy density at m1only 1/n of the total

energy density of m1. Then, the energy density at m1originating from m2at a distance Ris given

by

σE= (m1c2)/(lon) (9)

Assuming σE(r) decreases as 1/r2, the energy density at m1from m2at the distance Ris

σE(r) = (m1c2)/(lonr2)R2.(10)

Then, applying the General Force Law, the force is found to vary as

F=−lo

dσE(r)

dr =−lo

d

dr m1c2

nlor2R2=m1c2

nr3∗R2.(11)

This is obviously an incorrect result since the gravitational force is observed to vary as 1/r2, not as

1/r3. But why is this incorrect result obtained?

Assuming the derivation of the general force law is valid (which does not specify the variation of

the energy density with distance), I was forced to assume that the energy density of the wave modes

decreases as 1/r, not as 1/r2. Then, the energy density at any distance is

σE(r) = (m1c2)/(lonr)R. (12)

Although the decrease in the energy density as 1/r appears physically improbable for wave propaga-

tion, I found that the assumption works perfectly. Applying the General Force Law, the gravitational

force is given by

F=−lo

dσE(r)

dr =−lo

d

dr m1c2

lonr R=m1c2

nr2R. (13)

Then, to obtain the exact form of Newton’s law, set R= 1.5 1028 and n= 1080 and multiply the

numerator and denominator of Eq. 13 by m2/m2obtaining

F=10−7m1m2

r2(14)

The coeﬃcient 10−7is close to G= 6 10−8, so that obviously Eq. 14 becomes

F=Gm1m2

r2.(15)

Here, the value of Gis approximate.

3.1 Proof of 1/r Intensity Relation

However, it is still diﬃcult intuitively to make sense of the concept that the energy density of the wave

modes decreases as 1/r in 3-dimensional space. Fortunately, I was able to prove this mathematically

by determining that eigenvectors producing this eﬀect exist in a spherical wave system under certain

6

conditions. The proof depends on the general properties of spherical wave systems and, speciﬁcally,

on the assumed 1/r dependence of the energy density. The wave modes are actually the normal

modes of the universe and theoretically completely deﬁne the universe. They are a new concept in

cosmology and, in fact, their physical existence, to my knowledge, has not even been conjectured

previously.

The proof is as follows: Begin with the classical wave equation in spherical coordinates and then

consider the separated radial wave equation, R(r), given by

d2R(r)

dr2+2

r

dR(r)

dr +k2−l(l+ 1)

r2R(r) = 0 (16)

where k2= (2πf )2/c2.

The standard method [10] of solving for R(r) is to separate Eq. 16 into two parts

R(r) = r−1/2B(r) (17)

where B(r) is Bessel’s equation of half-integral order. The solutions of B(r) are non-periodic except

when the angular momentum eigenvector l= 0. Since periodicity is essential for constructive

interference, this is the only solution consistent with the wave system. For this unique solution,

B(r) = s2

(πr)[sin (r) + cos (r)] .(18)

But, since cos (r) equals 1 at r= 0, p(2/(πr) cos (r) goes to inﬁnity at r= 0. Consequently, only

the sin (r) part of the solution can be used. However, this is still not the required solution to the

problem. Since the energy density σis proportional to R2(r) in Eq. 17, σis proportional to 1/r2.

Fortunately, the problem can be completely solved by noting that the above formulation of the

problem assumes the tension and mass density are constant for each wave mode. This is not true in

the wave system since the tension and mass density parameters are proportional to the local energy

density of each wave mode. Thus, if the local energy density at rfor a wave mode is proportional

to 1/r, we must have T=To/r and σ=σo/r.

For these parameter variations in the spherical wave equation, the amplitude B(r) becomes [11]

B0(r) = 1

(T σ)1/4B(r) = r1/2

(Toσo)1/4B(r) (19)

since (T σ)1/4=r−1/2(Toσo)1/4. Then, the radial amplitude function becomes

R0(r) = r−1/2B0(r)∝B(r).(20)

Consequently, using Eq. 18, the energy density of R2(r) is proportional to 1/r, as required. The

modes represented by the modiﬁed radial amplitude function, R0(r), in Eq. 20 are very important

physically since they are the normal wave modes of the universe.

3.2 Gravitation as an Energy Source for Particles

In current particle physics, the source for the energy of mass particles and photons is unknown.

However, this problem is solved in the wave system theory by considering the interactions between

the normal wave modes and mass particles.

7

The number of normal modes is an important parameter in the wave system theory because the

properties of the wave system depend on constructive interference. Since each particle is equivalent

to a local oscillator, the number of modes is equal to the three times the number of particles. Then,

for constructive interference, the intensity, I, at each particle is proportional to the square of the

sum of the amplitudes of each mode given by

I∝X3(A1+A2... AN))2= (3NA)2= 9N2A2(21)

where N is the number of particles and Aiis the amplitude of the mode at the ith particle.

If it is assumed that the energy of a particle is proportional to the frequency, the energy of a

particle should be proportional to the square root of the number of modes for gravitation. This

follows from the frequency dependence on the number of modes given in equation 1. Similarly, the

(positive or negative) electrostatic energy at a particle should be proportional to the square root

of the number of interactions. (Note: The interactions generally cancel at the particles.) Since the

number of particles is approximately 1080, the energy of particles is proportional to 1040 and the

electrostatic energy is proportional to 1080. Thus, the electrostatic force is approximately 1040 times

stronger than the gravitational force.

Furthermore, since the intensity of the normal wave modes vary as 1/r, it may be shown quan-

titatively that the energy of a mass particle or photon is entirely due to the interactions with all

the other particles in the universe via the gravitational terms. In the next section, this interaction

is identiﬁed as the negative of the gravitational potential energy.

4 Newtonian Gravitation Theory

First, a strictly Newtonian gravitation derivation of the attraction of the universe on a test particle

of mass, m, will be presented. Although Newtonian gravitation by itself does not explain the Hubble

redshift, some necessary concepts will be introduced. Fig. 1 serves to illustrate these concepts.

First, consider a thin spherical shell with a constant surface mass density. Given the inverse

square law, the net attractive force, F, of the shell on a mass particle located anywhere within the

shell is given by

F= 0.(22)

A proof is given by Ramsy [12]. The same proof was also given by Newton in the Principia.

Next, consider the case where a particle with mass, m, is located on the surface of the same thin

spherical mass shell with a radius, r. Then, the attractive force, F, towards the center of the shell

is given by Newton’s law

F=−GMSm

r2.(23)

where the surface mass of the shell is MS. The point made here is that, for the inverse square law,

the total mass of the shell can be considered located at the center of the shell. Note: The minus

sign indicates an attractive force.

Equation 23 is very important since it also applies if the single spherical shell is replaced by a

constant mass density sphere with a radius, RL. For a particle located within the sphere at a radius

r, by Eq. 22, the net force from mass particles outside the radius ris zero. This is due to an exact

cancelation of gravitational forces originating from mass particles outside the radius r. Then, by

Eq. 23, the attractive gravitational force on the particle depends only on the mass, M, within the

radius, r, given by

F=−GMm

r2.(24)

8

!"#$%'&

(

)*+# #,$.-0/12#034-.5",7689

(;:

<=>?2+3?@AB*C>2)

DCE8FG+

EH

Figure 1: A Sphere with a Constant Mass (or Particle) Density is Proposed as a Model of the Local

Universe. RLis the radius of the Local Universe. PSis a Particle on the Surface of the Sphere and

Pis a Particle in the Interior rdistant from the Center of the Sphere. Assuming the Inverse Square

Law, the Gravitational Attraction is directed towards the Center of the Sphere at the point, Po.

9

4.1 Gravitational Potential Energy

The gravitational potential energy, V, is deﬁned as

V=GMm

r.(25)

Since Vis set equal to zero at inﬁnity, Vis positive when deﬁned as the work done by gravity on

the particle as it falls. Consequently, the gravitational potential energy of a mass particle is the

maximum at the center of a sphere. In the derivation of Newton’s law, the wave modes interact with

particles and, most important, the intensity of the wave modes varies as 1/r. Since the gravitational

potential energy also varies as 1/r, the intensity of the wave modes is identiﬁed with the gravitational

potential energy.

Consequently, I propose that the gravitational potential energy deﬁned by Eq. 25 is the actual

particle energy. It should be viewed as the primary gravitational relation. Then, Newton’s law is

a derived relation. Thus, the gravitational potential energy is a real wave ﬁeld rather than simply

an auxiliary concept in Newton’s theory of gravitation. This revised concept of the gravitational

potential energy will be essential later to the derivation of the Hubble redshift process.

For the calculation of the gravitational potential energy, the mass particles within a constant

density sphere can be considered concentrated at the center of a sphere as a single mass, M. This

is explicitly proved in Ramsey [12].

The gravitational potential energy, V, at a point within a sphere is given by

V=2π

3Gσ(3R2

S−r2)m(26)

where RSis the radius of the sphere and ris the distance of a particle from the center of the sphere.

The maximum potential occurs at the center of the sphere. From Eq. 26, the attractive force on a

particle at a distance, r from the center of the sphere is

F=dV

dr =−4π

3Gσrm. (27)

Consequently, there is an attractive force proportional to ron every particle either within (or on the

surface) of the sphere towards the center of the sphere. This is the source of the instability problem

for a static universe as cited by Einstein.

5 Cosmological Models of the Universe

The standard expanding universe model is based on the expansion of the universe. In this section,

a new static model of the universe is developed. This model will then be used to derive the Hubble

redshift.

5.1 Static Flat Metric

Cosmology is considerably simpliﬁed by assuming the universe is homogenous and isotropic. This is

not true within the local supercluster, but does appear true on a larger scale. And, since most of the

mass of the visible universe occurs on the large scale, the assumption the universe is homogenous

and isotropic should be valid in practice. The universe, therefore, should have a constant mass (or

particle) density.

10

These two assumptions lead to the three-space Robertson-Walker metric by geometrical argu-

ments only [13]. There are then only three possibilities for the metric: elliptical, ﬂat and hyperbolic.

The three-space ﬂat metric [14] is given by

ds2=dt2−R2

c2[dr2+r(dθ2+ sin2θdφ2)] (28)

where Ris a constant and ris the coordinate distance in the radial direction.

Of the three metrics, only the ﬂat metric is consistent with the wave system model of the universe.

Actually, the ﬂat metric should be referred to as the“Minkowski” metric which is Lorentz invariant. A

periodic solution is essential in the wave system theory because constructive interference of the wave

modes cannot occur without exact periodicity of the waves. And without constructive interference,

the minimum energy of the wave system would be much greater. As was shown in the previous

section, a single periodic solution for the radial wave function exists in the Minkowski metric.

On the other hand, the corresponding radial wave function in the hyperbolic metric is known as

the hyperspherical Bessel equation [15] which does not have a periodic solution.

6 Calculation of Parameters of the Universe

As discussed in the section on the “Theory of the Wave System”, as more mass particles mutually

interact in the universe, the frequency (the eigenvalues) of the wave modes increase. This increase

in frequency is based on Eq. 1 in the Section on the “Wave System Theory”.

It cannot be overemphasized that this equation represents an enormous simpliﬁcation of the

physics of the wave system. Since the vibration modes associated with each mass particle are each

represented by an diﬀerential equation, on the order of 1080 equations are required to describe the

system. Additionally, each equation must include interactions with all the mass particles in the

universe. This number of equations each with the same number of interaction terms cannot be

solved by a brute force calculation.

However, assuming the interaction constants are all equal, the eigenvalues are easily calculated.

Chen proves that one mode vibrates at a low frequency given by 1/(2π)pk/m. The other N−1

modes all vibrate at a higher frequency √Ntimes larger than the low frequency mode. This is a

unique vibrating system since all the modes, except for the one mode at low frequency, vibrate at

the same frequency. The same frequency for each mode is essential for constructive interference.

The major problem in applying Chen’s equation to the vibration modes of the particles in the

sphere is that the distances between particles vary and, consequently, the interaction constants are

diﬀerent. But, from the Newtonian theory of the gravitational potential energy, all the particles

located within the sphere can be considered concentrated at the center of the sphere. Since the

distances between a particle on the surface of the sphere and the particles at the center are then

the same, all the interaction constants are the same. Therefore, Chen’s eigenvalue equation applies

exactly to the interactions between a surface particle and all other particles within the constant

density sphere.

To apply Chen’s equation to the sphere, the minimum frequency in the local universe, given

that the radius is RL, is fo=c/(2RL) where cis the velocity of light. This minimum frequency

corresponds to a full wavelength between the surface particle and particles at the furthest distance

within the sphere since the mass particles must be in-phase. However, the minimum frequency for

interaction between the surface particle and the particles comprising the mass, M, at the center of

the sphere is larger, given by c/RL. In addition, since Vvaries as 1/r, the minimum energy of the

interaction constant associated with this minimum frequency, f1, is E= 2hf1= 2hc/RL.

11

6.1 The Local Universe

Using this simple model of the universe, the equilibrium state of the local universe can be calculated.

There are two important features of this model. First, the mass particles at Poare stable since they

are subject to equal and opposite gravitational forces from the other mass particles. Second, the

interacting particles are limited to the particles within the local universe. Mass particles outside the

local universe do not increase the gravitational potential energy of mass particles inside.

This second feature results from the greater distance of mass particles outside the local universe.

Then, the interaction from these particles is too small to eﬀect a change in the energy level of a

mass particles at the center of the local universe and, thus, the outside particles are “invisible” to

the mass particles inside.

Next, from Newtonian gravitational theory and Chen’s eigenvalue equation, the gravitational

potential energy and the minimum mass energy of a proton located at Pocan be exactly calculated.

Speciﬁcally, the maximum gravitational potential energy is 2πGσR2

Lmfrom Eq. 26. The minimum

mass energy is given by √3n2hc/RLwhere nis the number of mass particles (protons + electrons).

The factor 3 multiplying nassumes three modes of vibration per particle based on the vibrations of

the particle perpendicular to the three planes in 3-dimensional space. Finally, the calculation of the

energy should apply, in principle, to two particles with equal frequencies. But, since the electrostatic

interaction results in the proton and electron, the calculated energy is doubled to represent the much

larger energy of the proton.

The above calculations are shown in Table 1. A stable equilibrium state of the local universe

exists when

V=E(29)

where Vis the gravitational potential energy and Eis the minimum energy of a proton calculated

using Chen’s eigenvalue equation. The variables are the mass density of the universe, σ, and the

radius of the local universe, RL. These are varied until V=E.

As shown in Table 1, the variational solution for the local universe does not exist. Fig. 2

illustrates why no equilibrium solution is obtained. In the next section, a possible solution to the

problem is proposed.

In the conﬁguration universe, the local universes ﬁll a larger sphere with radius RC= 2RL.

The mass particles within the local universes may be considered concentrated at the centers of the

spheres by Newtonian gravitational theory. As previously discussed, this was proved for the inverse

square gravitational force law. It is also true for the gravitational potential energy which varies as

1/r. Therefore, the mass particles in the local universes can be considered concentrated at their

centers. Then, for a continuous orientation of the local universes around the mass particle, PS, the

mass particles form a spherical shell RLdistant from the single surface particle at PS, the center of

the conﬁguration universe.

Because the conﬁguration universe has eight times the mass of the local universe, an equilibrium

state can be found. This equilibrium state also applies to the local universes. Also note that each

mass particle in the local universes can be considered, for a diﬀerent conﬁguration universe and a

diﬀerent set of local universes, to be at the center of a conﬁguration universe. Then, each mass

particle in the local universes has the same invariant gravitational potential energy.

Next, in Table 1, the equilibrium state of the conﬁguration universe is calculated. The only

change from the calculations in Table 1 is in the calculation of the frequency where the lowest

frequency is c/(RC/2) instead of c/RL.

TABLE 1

Calculation of the Equilibrium State

12

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Figure 2: Shows a Conﬁguration of Local Universes Intersecting at a Single Particle located at PS.

All Other Interacting Particles are Distributed Throughout the Local Universes. The Points Labeled

“O” are at the Centers of the local Universes. Assuming the Inverse Square Law, All the Particles

May be Considered Concentrated at the Center of Each Local Universe. In Actuality, the Local

Universes are Continuously Oriented Around PSso that the Local Universes are Contained within

an Outer Circumference of Radius RC= 2RL.

13

of the Conﬁguration Universe

Norm Dist Radius Grav Pot Grav Pot Mass Particles

R/RCRCPer Part Energy MCProt + Elect

cm ergs g

0.20 5.43E+027 0.040 6.01E-005 7.31E+053 8.74E+077

0.30 8.15E+027 0.090 1.35E-004 2.47E+054 2.95E+078

0.40 1.09E+028 0.160 2.40E-004 5.85E+054 6.99E+078

0.50 1.36E+028 0.250 3.76E-004 1.14E+055 1.37E+079

0.60 1.63E+028 0.360 5.41E-004 1.97E+055 2.36E+079

0.70 1.90E+028 0.490 7.36E-004 3.13E+055 3.75E+079

0.80 2.17E+028 0.640 9.62E-004 4.68E+055 5.59E+079

0.90 2.44E+028 0.810 1.22E-003 6.66E+055 7.97E+079

1.00 2.72E+028 1.000 1.50E-003 9.14E+055 1.09E+080

1.10 2.99E+028 1.209 1.82E-003 1.22E+056 1.45E+080

1.20 3.26E+028 1.439 2.16E-003 1.58E+056 1.89E+080

1.30 3.53E+028 1.689 2.54E-003 2.01E+056 2.40E+080

1.40 3.80E+028 1.959 2.94E-003 2.51E+056 3.00E+080

1.50 4.07E+028 2.249 3.38E-003 3.08E+056 3.69E+080

(Continued below)

TABLE 1 (Continued)

Calculation of the Equilibrium State

of the Conﬁguration Universe

Norm Dist Vibration Frequency Proton

R/RCModes Hertz Energy

3 per Part ergs

0.20 5.25E+078 5.06E+022 6.70E-004

0.30 1.77E+079 6.19E+022 8.21E-004

0.40 4.20E+079 7.15E+022 9.48E-004

0.50 8.20E+079 8.00E+022 1.06E-003

0.60 1.42E+080 8.76E+022 1.16E-003

0.70 2.25E+080 9.46E+022 1.25E-003

0.80 3.36E+080 1.01E+023 1.34E-003

0.90 4.78E+080 1.07E+023 1.42E-003

1.00 6.56E+080 1.13E+023 1.50E-003

1.10 8.73E+080 1.19E+023 1.57E-003

1.20 1.13E+081 1.24E+023 1.64E-003

1.30 1.44E+081 1.29E+023 1.71E-003

1.40 1.80E+081 1.34E+023 1.77E-003

1.50 2.21E+081 1.39E+023 1.84E-003

Two graphs illustrate the variational solution for the conﬁguration universe. Fig. 3 shows the

changes in Vand Eversus the radius of the conﬁguration universe with the mass density ﬁxed at

the equilibrium value. Fig. 4 similarly shows the changes in Vand Eversus the mass density with

the radius ﬁxed at the equilibrium value.

It is remarkable that the mass density of the conﬁguration universe and the radius of the local

universes (RL=RC/2) are consistent approximately with the current observational estimates.

7 Derivation of the Hubble Redshift

There are currently two classical explanations of the redshift of light in a static gravitational ﬁeld,

for example, the gravitational ﬁeld of the Earth. In the ﬁrst, the frequency and energy of the photon

14

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Figure 3: Illustrates the Variational Solution to the Radius of the Conﬁguration Universe. The

Solid Line Represents the Gravitational Potential Energy of the Proton and the Dashed Line the

Eigenvalue Energy of the Proton. The Mass Density and Radius of the Conﬁguration Universe were

Varied until the Curves Crossed at the Invariant Energy of the Proton. Values of the parameters of

the Local Universe Close to Currently Observed Values were Obtained.

15

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Figure 4: Illustrates the Variational Solution to the Mass Density of the Conﬁguration Universe.

The Solid Line Represents the Gravitational Potential Energy of the Proton and the Dashed Line the

Eigenvalue Energy of the Proton. The Mass Density and Radius of the Conﬁguration Universe were

Varied until the Curves Crossed at the Invariant Energy of the Proton. Values of the parameters of

the Local Universe Close to Currently Observed Values of the were Obtained.

16

do not change as it moves higher in the ﬁeld. Then, the redshift is explained through the behavior of

clocks which run faster at the higher level. In eﬀect, the energy diﬀerence between atomic levels in

atoms is larger at the higher level. Then, the frequency of the light or the photon appears redshifted

when it is observed at a higher level.

In the second, the photon looses energy as it rises because of the gravitational attraction of

the earth. This interpretation requires the photon to have a mass, E/c2, in order for the gravi-

tational force to do work on the photon. Both interpretations result in the correct redshift, given

approximately by ∆f

fo

=−gs

c2(30)

where fois the initial frequency, gis the acceleration of gravity on the earth and sis the diﬀerence

in height between the emitting and absorbing atoms.

Okun [17] believes the ﬁrst explanation is “correct” because it is consistent with observations

showing that atomic clocks run faster when carried for many hours at a high altitude by airplanes. If

the second explanation were also correct, then, according to Okun, a doubling of the redshift would

be expected due to the sum of the gravitational eﬀects on the clock and the gravitational attraction

of the photon. Given the inputs to the arguments, the ﬁrst explanation, I believe, is preferred

over the second explanation. However, another explanation is based on the wave system theory of

gravitation. Because this explanation is based on a speciﬁc physical process for gravitation, it is

more likely to be correct.

In the wave system theory, the energy density of a mass particle varies as 1/r in a gravitational

ﬁeld. In addition, the size, lof the particle increases as r. Consequently, both the mass particle and

the photon maintain their initial energy when moved to a higher point. However, a mass particle

has an invariant minimum energy at each level in the gravitational ﬁeld. Consequently, it is not

possible to move the particle to a higher level unless additional energy is provided to maintain the

invariant energy of the mass particle. This additional energy is provided by the work done on the

particle to raise the particle against the force of gravity.

On the other hand, the photon has no minimum energy. Since no work is done on the photon, the

energy and frequency of the photon stays the same as the photon moves to a higher level. However,

since the energy level of a particle is greater at the higher level, the energy and frequency of the

photon are measured as less by ∆V. An advantage of this explanation is that the mass particle

remains in-phase with the underlying wave modes and has the same energy as it had at the lower

point in the ﬁeld. Also, note that when the mass particle or a photon emitted by the atom at the

higher point fall, the increase in the gravitational potential energy ﬁeld increases the energy density

of the mass particle and the photon.

Finally, since the mass particle at the higher point is in a ﬁeld with a lower energy density,

distances and the time rate are each increased by a factor ∆V/V and the velocity of light is increased

by a factor 2∆V/V . Then, clocks at the higher point run faster than clocks at the lower point.

In the gravitational ﬁeld of the earth, the change in the gravitational potential energy of a photon

rising a distance sis

∆V=hfo

c2GM

R+s−GM

R≈ −hfo

gs

c2(31)

where fois the initial frequency, gthe gravitational acceleration, Mthe mass of the earth and R

the radius of the earth. Then,

f=fo(1 −gs

c2).(32)

17

7.1 Hubble Redshift of a Photon and Mass Particle

The cosmological redshift is analogous to the gravitational redshift. One diﬀerence, however, is that

the distant atom which absorbs the photon has the same invariant gravitational potential energy

as the atom emitting the photon. Thus, the cosmological redshift of the photon is due only to the

decrease in the gravitational potential energy of the photon.

For the derivation of the Hubble redshift, a simpliﬁed diagram of the local universe is shown in

Fig. 6. As shown in the diagram, the photon originates at the point Poand propagates to the point

P1where it is observed. For example, Pocould be a distant galaxy and P1our local position on

the Earth. Or, the diagram applies equally to Poour local position on the earth and P1a distant

galaxy. Thus, the Hubble redshift eﬀect is reciprocal between the same points in the local universe

in this model.

The regions A and B include all the particles which interact initially with the photon at Po. At

P1, only the particles in region B interact with the photon. Then, the decrease in the gravitational

potential energy occurs as a result of two eﬀects. First, at P1, the photon does not interact with

the particles in region A. This is due to the non-interaction of the wave modes from mass particles

beyond the distance, RLwith particles inside the local universe. Second, a decrease in the volume

in region B to the right of P1. This decrease in volume is equal to the volume of region C. This

translates into an additional decrease in the gravitational potential energy equal to region A. There

is also an increase in the distance of a photon at P1from region A and a decrease in the distance

to region C. However, these changes in distance only have a second order eﬀect on the gravitational

potential energy and can be ignored.

The volume of region A is equal to the decrease in the “lens” volume [16] given by

lens vol =π

12(4RL+s)(2RL−s)2.(33)

For small s, the change in the lens volume is

∆lens vol ≈ −πR2

Ls. (34)

This reduction in the lens volume equals the volume of region A. The same volume reduction, equal

to region C, occurs in region B when the photon has moved to P1.

Then, the decrease in the gravitational potential energy of the photon is given by

∆V=−πGσV /c2R2

Ls

RL+s+R2

Ls

RL−s≈ −2πGσV /c2RLs. (35)

where Vis the energy of the photon.

Dividing Eq. 35 by the initial gravitational potential energy of the photon given by

V= 2πGσV /c2R2

L,(36)

we have the simple expression for the proportional energy loss of a photon moving a distance ds

dV

V=−ds

RL

.(37)

This is a diﬀerential equation for determining the energy loss of a photon moving in the gravitational

ﬁeld of the local universe. Note that Eq. 36 contains Von both sides of the equation. This is not a

mistake since V=mc2and the factor 2πGσR2

L/c2= 1.

18

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Figure 5: Local Universes are Represented by a Sphere Centered on the Point Po. The Second

Sphere is Centered on the Point P1.PoRepresents a Galaxy at a Distance sand P1an Observer

on the Earth. A Photon Originates at Poand is Observed at P1. The Regions A and B Include the

Mass Particles which Interact with the Photon at Po.

19

Integrating Eq. 37 and determining the integration constant, the following equation is obtained

ln(V /Vo) = −s

RL

(38)

where Vois the initial energy of a photon and Vis the observed energy after the photon moves a

distance s. Finally, we obtain

V=Voexp (−s/RL).(39)

This equation has an exponential form because the energy, V, of the photon is continually decreasing

proportional to the distance, s, the photon has moved.

Next, from the deﬁnition of the Hubble redshift, z= (λ−λo)/λowhere λis the observed

wavelength, we have

ln (f/fo) = ln (V/Vo) = −ln (1 + z).(40)

Then, from Eqs. 38 and 40, we ﬁnd that

r=s

RL

= ln (1 + z) (41)

where ris the normalized distance the photon has moved.

Finally, for zsmall, ln (1 + z) is approximately equal to z. Then, since z=v/c where vis the

recession velocity in an expanding universe model, we have approximately

v=c

RL

s=Hs (42)

where His the Hubble redshift constant. This equation is the famous Hubble Redshift Law.

In summary, the normalized distance, r, of a galaxy or supernovae can be determined from the

Hubble redshift, z, using Eq. 41. Then, the observed energy of a photon after a redshift of zis

calculated in Eq. 39.

The theory of the Hubble redshift developed for the photon also applies to mass particles. The

kinetic energy of a mass particle is analogous to the energy of a photon. Therefore, the change in

the gravitational potential energy of a mass particle as it moves a small distance, ds, is given by

dV =−Vds

RL

.(43)

If the mass particle is stationary with respect to the local universe, moving the mass particle from

its equilibrium position requires an applied force as if the particle was going up in a gravitational

ﬁeld of a massive body. However, if the mass particle is moving with respect to the local universe,

the mass particle due to special relativity has a kinetic energy. It is only the kinetic energy which is

reduced due to the Hubble redshift since the rest mass energy of a mass particle is invariant. For a

moving particle, the Hubble redshift eﬀect results in a reduction in the velocity of the particle, that

is, a deceleration of the particle until the particle velocity becomes zero.

This mass particle “Hubble redshift” eﬀect explains the observed, very small, anomalous accel-

eration toward the sun of the Pioneer 10 and 11 spacecraft [18]. The acceleration can be derived

from Eq. 43 by setting F=dV/ds =ma and using the relations V=mc2and RL=c/H. Then,

the acceleration due to the Hubble redshift of a mass particle is

a=−cH. (44)

The acceleration is directed opposite to the velocity of the spacecraft.

For H= 1.945 10−18/sec (H= 60 Km/sec/Mpc), the predicted acceleration from Eq. 44 is 5.83

(0.5) 10−8cm/sec2. This compares with the recent measurement [18] of 8.74 (1.25) 10−8cm/sec2

directed towards the sun.

20

8 Maximum Hubble Redshift

The two spheres shown in Fig. 6 do not intersect for s > 2RL. Consequently, the photon gravitational

potential energy becomes zero for s= 2RLand the Hubble redshift eﬀect only occurs for s < 2RL.

Then, the maximum possible redshift is 6.3891 obtained from the equation

z= exp (s/RL)−1 (45)

with s/RL= 2. Coincidentally, the highest spectroscopic redshift observed (as of September 4,

2005) is 6.29 from a powerful gamma-ray burst [19].

To test whether higher redshifts do occur, a sample of 23 long duration gamma-ray bursts (GRB)

with spectroscopic redshifts from Jakobsson [20] were analyzed. The 23 GRB in the sample are part

of a select sample of GRB detected by the Burst Alert Telescope on the Swift satellite over the

period March 1, 2005 to March 29, 2006. The sample includes the very high redshift GRB 050904

with z= 6.29 detected on September 4, 2005. Additional GRB in the select sample have estimated

redshifts greater than 6.39. However, since these estimated redshifts are upper limits to the actual

redshifts of the GRB, they are ignored in this analysis.

Assuming a static universe with a Minkowski metric, four bins were used, each with a width

∆r= 0.5. The average distance, r= ln (1 + z), of each bin and the number of observed GRB in

each bin are shown in Table 3 and plotted in Fig. 7.

TABLE 3

Sample of 23 Gamma-Ray Bursts with Spectroscopic Redshifts

r z r # GRB

Bin Bin Avg Bin with

Boundaries Boundaries Distance Redshift

0.0 0.00

0.5 0.65 0.25 2

1.0 1.72 0.75 5

1.5 3.48 1.25 8

2.0 6.39 1.75 8

Assuming the number of GRB in each bin is proportional to the volume of each bin, the number

of observed GRB should scale as r2. Consequently, for a log-log plot, the plot should be linear with

a slope of 2.

The least squares regression line is given by

log (GRB) = 0.76 log (r)+0.77 (46)

where the slope has a standard deviation of 0.07. Since the slope is signiﬁcantly less than 2, there

must be selection eﬀects at higher redshifts. For example, if either the detection of GRB or the

measurement of spectroscopic redshifts at higher redshifts requires higher luminosity GRB, a slope

less than 2 would occur.

This analysis of GRB at high redshift was made to test the hypothesis that there is a largest

possible redshift. Referring to Fig. 7, the least squares regression predicts that 11 GRB should be

observed in the range z= 6.4 to 11.2. Then, the evidence is overwhelming that there is a maximum

redshift; otherwise, at least 8 GRB (2 standard deviations less than 11 GRB) with spectroscopic

redshifts greater than 6.4 should have been observed in the next bin.

The only other known explanation for the non-observance of GRB at z > 6.4 is the existence of

neutral hydrogen at redshifts beyond 6.4. Songaila [21] has measured the transmitted ﬂux of Lγα

21

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Figure 6: The Number of Long Duration Gamma-Ray Bursts with Spectroscopic Redshifts are

Plotted in Bins of Width ∆r= 0.5 versus the Average Distance, r, of the Bin. The Black Squares

are the Number of Observed GRB in Each Bin. The Blue Diamond is the Predicted Number of

GRB Between z= 6.4 and z= 11.2. The Line is the Least Squares Regression.

22

emission from quasars as a function of redshift and found that the transmission goes to zero for

z > 6. The onset of cosmic reionization of the intergalactic medium by the ﬁrst generation stars

and quasars is predicted by expanding universe models between redshifts of 20 and 6. However, if

the universe is static, the reionization explanation can not possibly be correct since it depends on

an evolving expanding universe model.

9 Comments

The derivation of the Hubble redshift in a static universe is the main result of this paper. Together

with the ﬁrst paper which showed the universe to be static, this paper completes the task of de-

termining the “true” cosmological model with the proof that the universe must have a Minkowski

metric.

Also important, a new model of the universe is proposed based on the concept that the universe is

a pure wave system. This model has, I believe, considerable potential for future theoretical studies.

Furthermore, this model of the universe has lead to important changes in the current interpretation

of physical quantities. The gravitational potential energy, generally considered in current physics as

an auxiliary concept, is found to be the primary gravitational concept. Most important, the energy

of mass particles and photons is shown equal to the gravitational potential energy.

Finally, little doubt should remain that the universe is static with a Minkowski metric and ﬁnite,

in the sense, that a ﬁnite number of mass particles interact to produce a local universe with a ﬁnite

radius.

References

[1] Andrews, T. B. (2006), Falsiﬁcation of the Expanding Universe Model, “1st Crisis in Cosmology

Conference, CCC-1”, AIP Conference Proceedings, 3, 822.

[2] Goldhaber, G. et al. (2001), Timescale Stretch Parameterization of Type Ia Supernova B-band

Light Curves, preprint (astro-ph/0104382).

[3] Einstein, A. (1923), Cosmological Considerations on the General Theory of Relativity, “The

Principle of Relativity,” Dover Publications, Inc., 175–188.

[4] Pierce, J.R. (1974), “All About Waves,” The MIT Press, 1–2.

[5] Chen, F.Y. (1970), Similarity Transformations and the Eigenvalue Problem of Certain Far-

Coupled Systems, AJP 38, 1036.

[6] Huelsman, L.P. (1963) “Circuits, Matrices, and Linear Vector Spaces”, McGraw-Hill Book

Company, Inc., Page 226.

[7] Giorgi, H. (1990), An Overview of Symmetry Groups in Physics, Talk presented at 1990 Fall

meeting of the New England Section of the American Physical Society at Yale University,

Harvard preprint number HUTP-90/A065. Also Giorgi, H. (1993), “The Physics of Waves,”

Prentice Hall, Inc., Englewood Cliﬀs, N.J.

[8] Slater, J.C. (1947), “Mechanics,” McGraw-Hill Book Co., 188–192.

[9] Feynman, R. (1965), “The Character of Physical Law,” The M.I.T. Press, 39.

23

[10] Croxton, C.A. (1974), “Introductory Eigenphysics, An Approach to the Theory of Fields,” John

Wiley & Sons, 142–145.

[11] Morse, P.M. & Feshbach, H. (1953), “Methods of Theoretical Physics,” McGraw-Hill Book

Company, Inc., 739.

[12] Ramsey, A.S. (1961), An Introduction to the Theory of Newtonian Attraction,” Cambridge at

the University Press, 45–50.

[13] Weinberg, S. (1972), Gravitation and Cosmology: “Principles and Applications of the General

Theory of Relativity,” John Wiley & Sons, New York, NY, Page 404.

[14] Longair, M.S. (1995), The Physics of Background Radiation, “The Deep Universe”, ed. Sandage,

A.R., Kron, R.G. Longair, M.S., Springer-Verlag, 363.

[15] e Costa, S.S., (2001), A Description of Several Coordinate Systems for Hyperbolic Spaces,

preprint (astro-ph/0112039).

[16] Intersection Volume of Two Equal Spheres, Internet at http://mathworld.wolfram.com/Sphere-

SphereIntersection.html.

[17] Okun, L.B. (2000), On the Interpretation of the Redshift in a Static Gravitational Field, AJP

68, 115.

[18] Anderson, J.D. et al. (2001), Un-Prosaic Exposition of a Prosaic Explanation, preprint (gr-

qc/0107022).

[19] Price, P.A. et al. (2005), GRB 050904: A Very High Redshift Gamma Ray Burst, preprint

(astro-ph/0509697).

[20] Jakobsson, P. et al. (2005), A Mean Redshift of 2.8 for Swift Gamma-Ray Bursts, preprint

(astro-ph/0509888 v2).

[21] Songaila, A. (2004), The Evolution of the Intergalactic Medium Transmission to Redshift Six,

preprint (astro-ph/0402347).

All preprints refer to the http://arxiv.org/ website

24