A Simple Cosmology in G4v - symmetry-15-01309.pdf

Citation:

Mead, C. A Simple

Cosmology in G4v.

Symmetry

2023

, 1309. https://doi.org/

10.3390/sym15071309

Academic Editors: Xin Wu, Wenbiao

Han and Kazuharu Bamba

Received: 20 May 2023

Revised: 9 June 2023

Accepted: 20 June 2023

Published: 25 June 2023

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

symmetry

Article

A Simple Cosmology in G4v

Carver Mead

California Institute of Technology, Pasadena, CA 91125, USA; carver@caltech.edu

Abstract:

There has been a great deal of debate as to the role, if any, of Mach’s Principle in cosmology.

We propose that these questions have meaning only in the context of a cosmic gravitational potential.

G4v operates with such a four-potential, which imposes a different symmetry on its treatment of

light propagation than that employed by GR. That combination enables a simple cosmic solution

that is Lorentz-invariant in free-fall frames of reference. The solution is compared with supernovae

data, which, together with an internal self-consistency condition, fixes the parameters of the solution.

The resulting Hubble trajectory is, in broad brush, consistent with many cosmic observations.

Keywords:

cosmology; gravitational potential; G4v; Hubble; red shift; energy density; light cone

1. Introduction

Any cosmology is built on a few central tenets. The central tenet of G4v is Mach’s

Principle, which states that matter derives its inertia through interaction with other matter

in the universe. Because virtually all modern “metric” formulations of gravitation are

differential in nature, they cannot directly address integral relations such as Mach’s.

To directly address Mach’s Principle we must work with the gravitational potential rather

than merely its derivatives, commonly referred to as the “curvature of spacetime”. Einstein

himself often referred to the

μν

in his field equation as the “tensor gravitational potential”.

In the very last section of his wonderful little book

The Meaning of Relativity

[

], he outlines

a four-vector approach to Mach’s Principle, but does not develop it. We view G4v as the

development of Einstein’s Last Section, and show that it has many surprising merits.

2. Mach’s Principle

The (2004) successful launch of the historic Gravity Probe B (GPB) satellite has been

widely heralded as a test of General Relativity (GR). In a larger context, it can be viewed

as an investigation into the origin of inertia. In Newtonian physics, inertia was viewed

as an intrinsic local property of a massive body, moving with respect to absolute space.

Gravitation was a quite separate phenomenon, evidenced by the attraction of nearby

massive bodies. Elegant experiments by Eötvös showed that the inertial mass and the

gravitational mass were equal, to extremely high precision. Ernst Mach [

] was an

outspoken critic of Newton’s notion of “absolute space”. He believed that the motion

of an object had no meaning except with reference to other objects in the universe. In

his view, a mass would have no inertia if the rest of the mass in the universe were not

interacting with it. The idea that matter derives its inertia through interaction with other

matter in the universe is known as “Mach’s Principle”.

A number of workers have attempted to realize a mathematically precise formulation

of Mach’s Principle: an excellent account with many references can be found in Barbour and

Pfister [

]. A wonderfully well-reasoned and historically important exposition was given

by Sciama [

]. In a fully Machian universe, Gravitation becomes the universal interaction

of all matter. There is no longer such a thing as an isolated experiment. Local physics

cannot be separated from cosmology, and the identity of inertial and gravitational mass

is automatic.

Symmetry

2023

, 1309. https://doi.org/10.3390/sym15071309

https://www.mdpi.com/journal/symmetry

Symmetry

2023

, 1309

2 of 15

If gravitation is to serve as the mechanism for establishing a frame of reference, the

gravitational interaction of an element of matter with all other matter in the visible universe

must be responsible for the inertia of that element of matter. Inertia is the property that

allows matter moving with a velocity

to exhibit a momentum

. For gravitation to be the

origin of inertia, it must have a vector (momentum) effect as well as the usual scalar (energy)

effect associated with the gravitational potential energy of nearby massive bodies. In the

same way that magnetism is the vector aspect of electrical interaction, “gravitomagnetism”

must be responsible for inertia.

In the title of his 1912 paper, Einstein [5] asks the question:

“Is There a Gravitational Effect which is Analogous to Electrodynamic Induction?”

This paper contains the first concrete proposal for how distant matter in the universe

could serve as an inertial frame of reference:

This suggests that the

entire

inertia of a point mass is an effect of the presence of

all other masses, which is based on a kind of interaction with the latter.

At this point, Einstein inserts a footnote

This is exactly the same point of view that E. Mach advanced in his astute

investigations on the subject. (E. Mach,

The Evolution of the Principles of Dynamics

Chapter 2. Newton’sViews on Time, Space and Motion).

Einstein strongly believed in the relativity of inertia: In 1918 he stated, as being on

an equal footing, three principles which a satisfactory theory of gravitation should be

based upon:

The principle of relativity as expressed by general covariance.

The principle of equivalence.

Mach’s principle: That the

μν

are completely determined by the mass of bodies,

more generally by

μν

Einstein noted in 1922 that others were content to proceed without Mach’s criterion,

and added: “This contentedness will appear incomprehensible to a later generation..”. [6]

In many places, particularly in his little book

The Meaning of Relativity

[

] Einstein

refers to

μν

as the Gravitational Tensor Potential. That interpretation of the Field Equation

has not been widely accepted, principally because of a symmetry choice adopted by GR,

which is different from a symmetry adopted by G4v.

3. Introduction to G4v

G4v is specifically formulated to be accessible to the broad range of individuals with

modest technical training. It uses ordinary trigonometry and calculus in ordinary “flat”

(Minkowski) spacetime, in which the value of units of measure do not depend upon the

contents. In this description I refrain from using the jargon of specialized areas theoretical

physics, since it is particularly opaque to anyone not trained in the specialty. A full

exposition of G4v is given in

Engineering View of Gravitation

[7].

Section 4 gives a simplified description of how G4v is related to GR, which necessarily

involves some tensors. A mathematical theorem, Sylvester ’s Law of Inertia, shows that we

can always find a free-fall frame of reference, and the Principle of Relativity requires that

Special Relativity must hold in all free-fall frames of reference.

In addition, there is a symmetry degree of freedom: a theory must specify, in a

self-consistent manner, how the speed of light in one free-fall frame of reference is related

to that in another free-fall frame of reference at a different gravitational scalar potential.

GR choses full symmetry, in which the speed of light will always be the “Universal

Speed of Light in Vacuum”

∼

m/s, independent of gravitational scalar potential.

This choice imposes a different symmetry on spacetime: the unit of length and the unit

of time at any point in a free-fall frame of reference become proportional to the gravitational

scalar potential at that point in that free-fall frame of reference.

Symmetry

2023

, 1309

3 of 15

G4v choses full symmetry with respect to spacetime, in which the unit of length

and the unit of time in any free-fall frame of reference remain constant, which imposes a

different symmetry on the propagation of light: The speed of light

in a free-fall frame of

reference becomes proportional to the gravitation scalar potential at every point in that

free-fall frame of reference. Special Relativity remains valid locally, in any free-fall frame of

reference, provided the local value of

in that free-fall frame of reference is used.

For nearly all physical problems, in weak gravity, results derived in the two systems

are identical; the relationships are represented by the well-known transformations between

the two systems. There are certain simplifications attendant to the G4v system, in that it

can be identical to the frame of reference in which the measurements are made. With that

simplification, the speed of clocks and the time of flight of Radar pulses become physical

interactions, rather than properties of spacetime, and must be shown to accord with physical

law. For cosmological purposes, the G4v system has the great advantage that a cosmic

gravitational four-potential and a universal cosmic time are defined quantities.

G4v works in frames of reference where lengths are independent of Gravitational

Scalar Potential, and the speed of light is, as directly observed in the Shapiro Delay [

proportional to the Gravitational Scalar Potential. To work properly with the quantum

(wave) nature of matter, G4v is formulated in wave units, where energies have the units of

frequency

, and momenta have the units of a propagation vector

, i.e., inverse length.

Together these quantities form a proper four-vector

[

]

(1)

The relativistic dot-product of two proper four-vectors defines a Lorentz invariant

scalar. In the case of Equation (1), we can dot

with itself:

−

(2)

The Lorentz invariant quantity

is called the

Compton wave number

. The behavior of

each element of matter is governed by this Lorentz invariant

dispersion relation

(3)

•

In G4v the speed of light c itself is used as the local gravitational scalar potential.

•

The quantity of matter is not the mass m but the Compton wave number k

•

The entire rest energy of matter

is due to its gravitational scalar potential.

Thus, for example, the mass

of the electron is not a fixed number of kilograms,

but the electron’s Compton wave number

is always 2.589605

−

Note about units:

Because G4v is formulated in wave units, energies have the units

of frequency

, and momenta have the units of a propagation vector

, i.e., inverse length.

The units of both

and

are inverse length. The product of

and the gravitational

potential

is the

rest frequency

. In mechanical units, the rest momentum

, which,

when multiplied by the gravitational potential

is equal to the rest energy

Since lengths in G4v do not change when the gravitational potential

changes, we need to

express the gravitational constant

as a length. Planck [

] noted that a natural system

of units would have a length unit

involving the Newton gravitational potential G.

By definition the Planck length

√

⇒

(4)

In G4v we use

as our gravitational coupling constant rather than

because, as we

will see, it turns out to be independent of the gravitational potential.

Symmetry

2023

, 1309

4 of 15

4. Relation of G4v to GR

The Einstein Field Equations of GR can be written as:

μν













(5)

where

μν

is the stress-energy tensor and

is Newton’s gravitational constant. Element

is the energy per unit volume divided by

, and

are the momenta per unit

volume in, for example, the

directions. Thus the units of all elements are momentum

per unit volume. We write the element of volume as

vol.

It is well known [

] that, for weak gravity, Equation (5) reduces to a vastly simpler

linear equation of the form:

μν

[

∇

−

∂

]

μν

≈

−

μν

−

μν

(6)

where the last form is from Equation (4),

μν

is the Tensor Gravitational Potential

and the

μν

are constructed differently than the common practice for Equation (5), as discussed below.

We may express Equation (6) in integral form, which follows from Green’s theorem,

as Einstein indicated in

The Meaning of Relativity

, Equation (101), and the discussion

that follows.

μν

≈

−

∫

μν

(

)

(

)

vol where

∫

path

(

)

(7)

As written, Equation (7) can utilize all components of

μν

. When we only include the

first column, or, equivalently, the top row, we arrive at a four-vector version.

For the overall space-time evolution of the idealized universe considered in this paper,

we deal only with the energy elements

and

. Common practice in GR theories is

to attribute energy to the

and

fields of electromagnetic radiation. Quantum theories

that treat the vacuum radiation field as an array of harmonic oscillators end up with an

energy density that acts as a Cosmological Constant

that is a factor of

∼

120

too large

to accord with observation. Much hacking has reduced this factor to “only”

∼

. This is,

by far, the largest disconnect between theory and experiment in the history of physics.

As discussed at length by Cramer and Mead [

] Section 7, G4v’s companion theory

Collective Electrodynamics (CE) [

] works with the electromagnetic four-potential, and

electromagnetic energy-momentum four vector is the scalar product of the four-potential

and the charge-current-density four potential. Thus the electromagnetic field itself has no

energy of its own, and exists only in the presence of charged matter.

Like CE, G4v works directly with the gravitational four-potential, and the energy

in Equation (3) only exists where the gravitational scalar potential

interacts with matter.

So neither electromagnetic nor gravitational potential fields have energy of their own, and,

in both cases, the

originates in matter. This formulation thus avoids the Cosmological

Constant disaster in a manner consistent with both electromagnetism and gravitation.

The top row and left column of the stress-energy tensor contain the components

of the energy-momentum four-vector, which are attributes of any fundamental element

of matter. The diagonal elements

αα

represent the moments of inertia of a binary, and

are the largest source of the Gravitational Waves that directly alter the spacing of the

LIGO mirrors. Using these elements in Equation (7) gives the same gravitational wave

coupling as predicted by GR. For gravitational arrangements where elements of matter

such as electrons and neutrons, and massive objects such as stars, planets, etc., can be

approximated as discrete elements, we can ignore the

αβ

elements for either

Symmetry

2023

, 1309

5 of 15

and a straightforward four-vector formulation becomes, as long as the curvature is small,

an excellent approximation.

G4v works strictly with a wave description of elements of matter, for which we employ

the original three-dimensional Schrödinger complex-valued wave function

Ψ

(

)

. The

amount of any property such as, i.e., the energy

(

)

, of the element of matter

apportioned to a unit volume

vol is given by

Ψ

∗

Ψ

, so the total energy of that element

is given by

∫

Ψ

∗

Ψ

vol by the normalization of the wave-function

Ψ

. The quantities in

Equation (7), with

= [

]

and four-vector quantities in

bold

are thus:

∗

(8)

Using

[

]

[

]

∗

So, when working in wave units, the

cancels out, the coupling constant is represented

, and Equation (7) becomes:

≈

−

∗

−

∗

(9)

From Green’s Theorem, Equation (9) has the equivalent form for the gravitational

four-potential

at distance

along a light-path

≈−

∫

∗

(

)

(

)

vol where

∫

path

(

)

(10)

When formulated in this way, the gravitational coupling constant becomes the square

of the Planck length, the quantity of matter becomes the Compton wavenumber (inverse

length), and all purely gravitational calculations require only units of time and length.

The source of the electrical four-potential is the charge, current-density four-vector, and

that for the gravitational four potential is the energy-momentum four-vector. Both quantities

are defined for the wave function of the source matter, and appear directly as terms in the

affected matter wave function. The concept of force is not necessary.

For a macroscopic “classical” object composed of the incoherent arrangement of

many localized elementary quantum elements, for each element

∫

∗

vol

1, so

∫

∗

vol

, the total source value is just the sum of each individual

th source,

and Equation (10) becomes:

≈−

∑

(

)

(

)

vol where

∫

path

(

)

(11)

Thus, for such a classical object, the energy is just the sum of the energies of all the

constituent elements and the momentum is just the vector sum of the momenta of the

constituent elements, as common sense would dictate.

We have thus found that, in weak gravity, where only the energy–momentum elements

of the stress–energy tensor play a significant role, we have a four-vector version of Einstein’s

field equation that is parallel to, and integrated with CE four-vector, quantum-based

electromagnetism. We call this formulation G4v.

5. Basic Cosmological Assumptions

It is not obvious at the outset that G4v is compatible with a universe having the

properties we observe. Here we analyze the simplest and most basic properties of a

G4v universe. In accord with our goal of obtaining the maximum physical insight with

the minimum of mathematical complexity, we introduce what we see as the minimal

assumptions which could underlie any form of cosmology whatsoever.

Symmetry

2023

, 1309

6 of 15

The assumptions upon which this formulation is based are:

The universe is homogeneous, isotropic, and expanding;

Energy density

is the source of gravitational scalar potential,

which acts along both past and future light cones;

The gravitational scalar potential

is the speed of light c;

Energy density satisfies a continuity equation.

6. Gravitational Potential

In G4v, the gravitational vector potential takes the same place in the matter wave

function as does the electromagnetic vector potential, each with their individual coupling

constant: The source of the gravitational four-potential is the energy-momentum four-vector

. The coupling constant

is equal to the square of the Planck length, and the scalar

potential

is equal to the local speed of light

Whatever the makeup of matter in the universe, it will have some average energy

density

, which we abbreviate

, and that energy density is the source for gravitational

scalar potential

, governed by the Abraham’s version of Equation (9). (Abraham, in his

1913 paper [

] (p. 356), saw that this form of the equation must be used when the speed

of light

is changing with time. Unfortunately he applied it to a theory in which the

variable was

√

instead of

. Nonetheless we have chosen to refer to this form of the

equation, when used for gravitational potential calculations, as Abraham’s):

∇

−

∂

(

∂

)

πχρ

(12)

where

is the coupling constant. Because the potential is equal to the speed of

light

, the equation is highly non-linear due to the

terms, so it has no scalable

solutions. We are accustomed to solving it in the limit of nearly-constant gravitational

potential, in which case it becomes nearly linear, and ordinary methods apply. In the case

of cosmology, we must look for a particular solution.

Assuming a uniform solution throughout space, consistent with the cosmological

principle, both

and

are functions of cosmic time

alone and Equation (12) becomes

−

∂

(

∂

)

πχρ

(13)

which has solution

∞

tanh

(

)

cosh

(

)

cosh

(

)

sinh

(

)

πχ

∞

(14)

where

is a parameter of the theory that is determined by the self-consistency condition

described in Section 12. So the speed of light is uniquely determined by the Hubble

Constant,

, and the energy density.

7. Continuity Condition

A necessary constraint on the solution is the continuity of energy density

. We express

the expansion in a spherical coordinate system, in which the expansion velocity

is in

the

direction. For a universe whose density is uniform at any given cosmic time,

is a

function of

alone:

−

∂

(

)

∂

∇·

(

) =

∇·

−

∂ρ

∂

(15)

Symmetry

2023

, 1309

7 of 15

We assume a separable solution for the velocity

, where

is a function of

alone:

∂ρ

∂

−

(16)

So the full solution for the expansion velocity is

(

3 coth

(

)

tanh

(

)

−

2 tanh

(

)

(17)

Henceforth we shall express the relations of Equations (14) and (17) in terms of

dimensionless variables as follows:

(

) =

∞

tanh

(

)

(

) =

cosh

(

)

cosh

(

)

sinh

(

)

(

) =

∞

(

3 coth

(

)

tanh

(

)

−

2 tanh

(

)

where

∞

(18)

This model has two free parameters: (1) the present cosmic time of our observations,

and (2) the constant

. These will be determined by: (a) internal consistency of the model

with Mach’s Principle, described in Section 12, and (b) fitting the model to SN1a Hubble

observations, as described in Section 11.3. The evolution of these quantities with cosmic

time is shown in Figure 1.

Figure 1.

The three dimensionless functions of cosmic time in Equation (18) for the self-consistent

value

1.33

determined in Section 12. The green curve is the normalized speed of light

, equal to

the gravitational potential

. It is determined by the energy density and expansion rate, and is nearly

constant after about two Hubble times. The blue curve of the normalized expansion velocity can be

thought of as the effective Hubble constant, which also becomes constant after about two Hubble

times. The red curve is the normalized energy density.

8. Light Cone Distance

A light signal leaving a source at time

will reach an observer at distance

, where

∂

(

) =

tanh

(

)

⇒

∫

(

)

log

(

cosh

(

)

cosh

(

)

where

∞

(19)

We note that

is not the distance at one particular cosmic time, but

the distance along

the light cone

. When

is taken at the origin of our coordinates,

. For a

Symmetry

2023

, 1309

8 of 15

well-evolved universe, older than about two Hubble times, Equation (19) reduces to the

usual expression where the speed of light is constant. In dimensional units:

≈

(

−

) =

∞

(

−

)

for

(20)

9. Co-Moving Distance

We wish to follow the evolution of a small volume

as it moves with the Hubble flow

(called a co-moving volume in the cosmology literature). The dimensionless radius

reached by a co-moving volume at cosmic time

will be the solution to

∂

(

3 coth

(

)

tanh

(

)

−

2 tanh

(

)

(

) =

(

cosh

(

)

−

(

cosh

(

)

−

(

sinh

(

)

(

sinh

(

)

(

sinh

(

)

(21)

The co-moving radius has an arbitrary scale constant

which, when viewed as a

vector, is unique to each element in the universe. It identifies that element’s position,

relative to our chosen origin, at any given cosmic time.

10. Horizon

From Equation (19) we have a “lookback radius”—the distance from which light can

reach us. The maximum such distance occurs at the normalized

horizon radius

, where

in the chosen frame of reference. From Equation (18)

3 tanh

(

)

3 coth

(

)

tanh

(

)

−

2 tanh

(

)

(22)

The light-cone distance, co-moving distance, and horizon radius, normalized to the

Hubble radius, are shown in Figure 2.

Figure 2.

Cosmic distances for

1.33

, in units of the Hubble Radius

, relative to an arbitrarily

chosen origin, plotted as a function of cosmic time in units of the Hubble time. The Horizon Radius

is shown in black. The observer is at

0, on the

axis. The green, cyan, and blue curves

are co-moving distances

for objects with

0.1

0.2

and

0.4

. The right plot is an

enlargement of the lower-left corner of the left plot.

The red curves are the light paths emitted by the green, cyan, and blue sources at

earlier times

, and received by observers at

0.66, 1.5, 2.75, and 4.0. The

1.33

0.66

curve will turn out to be internally consistent, and to give the best fit to recently

observed SN1a Hubble data. The early universe is beyond the horizon in this model.

This plot would look the same, independent of which element in the visible universe

is chosen as the origin. We have chosen the frame of reference to be the co-moving frame of

the observer at

0. The expansion decelerates in the early stages of cosmic evolution,

and accelerates during the later stages. It is clear that, at late cosmic time, more and more

volume elements disappear beyond the horizon.

Symmetry

2023

, 1309

9 of 15

11. Comparison with Astrophysical Observations

11.1. Redshift

We compare the rest-frame frequency

of an atomic transition at

, where the

speed of light is

to the corresponding rest-frame frequency

of the same atomic

transition at the observer on the light come of the emitter, at

where the speed

of light = gravitational potential is

. The rest-frame frequencies will be proportional to

the corresponding gravitational potentials:

(23)

The frequency

obs

we observe at

where the speed of light is

, is further reduced

by the Doppler effect due to the relative motion of the source and observer. The commonly

used

relativistic redshift z

is therefore:

≡

obs

√

−

(24)

11.2. Magnitude

A certain number of excited atoms in the supernova we are observing radiate energy

at a frequency given by Equation (24). All time scales change inversely with gravitational

potential, and are affected the same way by the Doppler effect, so both the energy per

photon and the rate at which photons are received will be inversely proportional to the

factor given by Equation (24). It follows that the received energy per second

will be

inversely proportional to the square of that factor, and of the distance

∝

(

√

−

)

(25)

The Equation (25) relation is valid in the frame of reference of the source, where

is the radius of a sphere upon whose surface the energy flux

is constant. However, in

the frame of reference of the observer, the spheres of constant flux are lengthened in the

direction of source motion and flattened in the direction opposite to the source motion.

This problem is known as

relativistic-beaming

, and is well known in particle-physics

circles. It has has been carefully worked out by K. T. McDonald [

]. The result is that, for

an isotropically radiating source moving away from the observer, the received energy flux

obs

is reduced from that of a stationary source

sta

by the factor:

obs

sta

(

−

)

(26)

so the total flux received at the observer, whose speed of light is

, becomes:

∝

(

√

−

)

(

−

)

−

2.5 log

(

) +

constant

(27)

11.3. Hubble Fits

The predictions of this model, plotted along with supernovae observations in Figure 3,

were arrived at by the following iterative trial-and-error process:

Choose a value of

and

, and

Calculate

at the source atom from Equation (18)

Calculate distance along light cone from Equation (19)

Symmetry

2023

, 1309

10 of 15

Calculate

and

from Equation (18)

Calculate

1 from Equation (24)

Calculate

√

from Equation (25)

Calculate

√

from Equation (27)

Make Hubble Plot such as Figure 3 and calculate residuals

Record slope and rms error of residuals, and adjust

Return to Step 1

Until an

is found that zeros slope and minimizes rms error for this

Experimentally, it was found that equally good Hubble plot fits could be obtained

for a wide range of the parameter

, each of which has a different value for the present

age of the universe

that gives the best fit. By carrying out a number of best fits with

different

values, the dependence of

was found as shown in Figure 4. This

“manual” method of finding self-consistent solutions might seem tedious, but it enables the

investigator to get an intuitive feeling for the solution space. With experience, values of

could be reproduced within

1% without undue effort.

Figure 3.

Hubble plot showing the magnitude-redshift relation predicted by the G4v cosmological

model (

red curve

). The blue points are the Union2.1 data set, augmented with the 2013

1.71

observation. The upturn of the curves at high redshift is predominantly due to the moving-source

factor (Equation (26)).

Figure 4.

Present age of the universe

giving the best Hubble plot fit, as a function of the

parameter

. The red line merely connects the points.

Symmetry

2023

, 1309

11 of 15

12. Internal Consistency

Thus far we have used only differential relations that follow from Equation (12), from

which

is determined in Equation (14). Our solution has a free parameter

, which

can only be determined by comparison with the observable Universe. We have found, by

fitting our model to observed Hubble plots, that equally good fits could be obtained as long

as the values of

for each

shown in Figure 4 were used. We should not be surprised at

this result—the cosmic gravitational potential is inherently an integral quantity and cannot

be determined by any differential theory alone.

It is a principle of G4v that the speed of light is equal to the gravitational potential,

and thus determined by Mach’s Principle i.e., by the gravitational interaction with every

other element of matter on both our future and past light cones within our horizon.

For each

pair, we can determine the value of every variable, including the

speed of light and density, at every point in space-time from Equation (18). Because we

have the density at every point on the light cone of our present point of observation, we can

carry out the Mach integral and thereby determine the gravitational potential, and hence

the speed of light. When our theory is internally consistent, these two values must agree.

Mach Integral

According to the G4v interpretation of Mach’s Principle, the gravitational potential,

and therefore the rest energy and hence the inertia of matter, is determined by the energy

density of matter, weighted inversely with distance, and corrected for the Doppler effect.

Because of the non-linear nature of the equations involved, we are not guaranteed that an

integral determination of the potential will yield the same result as that assumed in the

differential formulation. So we must carry out the integration to the point where the two

values can be compared to evaluate the internal consistency of the entire approach. From

the definitions of the dimensionless variables in Equations (17) and (18) we obtain

∞

−

∫

(

)

√

−

(

)

(

)

(

)

(

)

−

∞

πχ

∞

∫

(

)

√

−

(

)

(

)

(

)

(

)

r dr

πχρ

∞

∫

(

)

√

−

(

)

(

)

(

)

(

)

(

)

(

)

πχρ

∫

(

)

√

−

(

)

(

)

(

)

(

)

(

)

∞

(

)

(

)

πχρ

∞

∫

(

)

√

−

(

)

(

)

(

)

(

)

(

)

(

)

(

)

∫

(

)

√

−

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(28)

As with any Mach’s-Principle-based theory, the integration must be carried out over

both the past and future light cones. The integrand is shown as a function of

in Figure 5.

The determination of the self-consistent values of the parameters proceeds by using

the values of

pairs found in the previous section (plotted in Figure 4) to calculate

(

)

∞

from Equation (28). A value of

(

)

∞

for the same

is also given directly

in Equation (18). Then a plot can be made of both

(

)

∞

vs n

dependencies, as shown

in Figure 6. The best fit self-consistent values obtained are

1.33

0.66

, and

(

)

∞

0.706

. Plots such as ccvsn for each of the

pairs plotted in Figure 4

looked almost identical.

Symmetry

2023

, 1309

12 of 15

Figure 5.

Integrand of Equation (28) as a function of cosmic time at the radius on the light cone where

the source matter is located. The blue curve is the contribution of the past light cone and the red

curve is that of the future light cone. Both curves cut off where the light cone encounters the horizon.

The value of

0.66 used for this plot is that which gives a self-consistent solution.

This particular plot uses

1.33,

0.66

∞

≈

0.706

, which are the values

that are also consistent with the Mach integral, as shown in Figure 6. Out of the many

Hubble plots that were used to determined the self-consistent

pairs, the one that

is also consistent with the Mach integral is determined by the intersection at

≈

1.33

∞

≈

0.706

in Figure 6. It has

≈

0.66

from Figure 4, and is the best estimate of an

overall self-consistent solution. That particular Hubble plot is shown in Figure 3.

Figure 6.

Velocity of light determined from Equation (18) (

blue crosses

) and from the full

Equation (28) Mach integral (

red circles

), both plotted as functions of the parameter

13. Absolute Distance and Density

To establish the

1 absolute distance scale, we need a single measurement of

both

and distance. In 1999, the water masers in NGC4258 were observed with VLBI

parallax, and the distance to that galaxy has been followed and updated regularly. A recent

(2016) distance was reported by Humphreys et al. [

] as

7.6

0.23

Mpc (One megaparsec

(Mpc)= 3.09

m). This distance is too close to obtain a reliable Hubble-flow velocity,

so Riess et al. [

] used Cepheid variable star magnitudes in NGC4258 and more distant

galaxies to infer the magnitude of a hypothetical SN Ia in NGC4258 of

4258

10.25

. In

more recent studies Riess et al. [18] inferred a magnitude of

4258

10.18.

These two values of

4258

and the

Humphreys distance estimates give us four

values for the conversion factor from redshift to distance. For each combination we follow

our model Hubble plot back to

4258

and read off the redshift inferred for NGC4258 if it

were in the Hubble flow. The mean value so obtained is

4258

.00192

. Then, using the

Humphreys distance, we can determine the absolute conversion factor between the redshift

and the distance

z r

where

= (

3.96

0.16

)

Mpc

(29)

Symmetry

2023

, 1309

13 of 15

More recent values are more precise, but the papers [

–

] are much harder to follow.

13.1. The Hubble Constant

Although it is not an object of this analysis to derive an independent estimate of the

Hubble Constant

, the Equation (29) relation between the redshift z and the distance

equivalent to one. All theories, even the most naive Doppler interpretation, predict

for small redshift, where

is the recession velocity. The

.00192

redshift inferred for

NGC4258 (if it were in the Hubble flow) certainly meets the

1 criterion. The Hubble

constant

thus determined is

z c

z r

75.9

km/sec

Mpc

(30)

Of the

5 estimated uncertainty in the result,

3 is attributable to the uncertainty in

the distance to NGC4258 and the uncertainty in

4258

. The balance is directly attributable

to the value for

chosen as best fit, and the particular fit adopted to the data set.

To use this class of G4v models for a serious determination of

would require a

much more exacting analysis. In recent years, vast new data sets have become available,

with more appearing regularly. Elaborate evaluation technologies are evolving to integrate,

and get the most out of, the new data. The point of the present paper has been to develop

a simple conceptual way of thinking that can be understood by anyone with a modest

technical background. The data set used was updated in 2013 when the analysis was

carried out. It would seem rather remarkable if any of the results could relate to those of

the modern “gold rush”. Nevertheless, the value obtained here is near the value labled

“Best

” in Figure 7. Good 2022 overviews of the growing disagreement between values

derived from models fit to the Cosmic Microwave Background and those from more direct

distance measurements can be found in references [22,23].

Figure 7.

Reported values of the Hubble Constant as of 2023. Figure 4 from arXiv:2304.06693

(https://arxiv.org/pdf/2304.06693.pdf, accessed on 1 May 2023). The rough value derived in the

present treatment is

≈

75.9 km/sec/Mpc.

Symmetry

2023

, 1309

14 of 15

13.2. Energy Density

Using the values quoted above, we can, with Equation (14), determine the value of

and thus the present energy density

in the universe (An

is needed to convert from our

standard G4v frequency units to ordinary energy units):

πχ

∞

cosh

(

)

πχ

(

)

cosh

(

)

sinh

(

)

≈

11.5 H atoms equivalent per

(31)

This value is about twice the often quoted values of

∼

6 equivalent protons/cubic

meter, arrived at with GR-based theory. Our approach has the advantage that it does not

place any restrictions on the particular form of energy making up the universe.

14. Conclusions

A simple G4v model based on Mach’s Principle, the wave representation of matter,

and a speed of light equal to the gravitational potential is formulated in “flat” Minkowski

space-time, and has a minimum of other assumptions. When fit to

.015

1.71

supernova data, it predicts a universe whose expansion decelerates in the early stages of

cosmic evolution and accelerates during the later stages, a present age of 0.66 Hubble times,

and a Hubble Constant

≈

75.9 km/sec/Mpc.

A substantial fraction of the observed redshift is due to lower gravitational potential

at earlier epochs. This approach naturally avoids the “Cosmological Constant problem”

and places no restrictions on the makeup of the universe. In this way it may also be a first

step toward resolving the “dark energy” puzzle that arises in GR cosmologies.

We have thus demonstrated that a simple, internally consistent G4v model cosmology

based on Mach’s Principle gives results that are within the bounds of reason. It can be

argued that using such an approximate approach as a “broad brush” introduction to cosmic

thinking has the advantage of conceptual clarity over present complex GR based models.

Funding:

This research received no external funding.

Data Availability Statement:

Union2.1 data are available at https://supernova.lbl.gov/Union/.

Acknowledgments:

The author is indebted to many colleagues for enlightening discussions down

through the years, and to Donna Fox for preparing the references. Critical comments and suggestions

from referees and editors greatly improved the manuscript.

Conflicts of Interest:

The author declares no conflict of interest.

References

Einstein, A.

The Meaning of Relativity: Four Lectures Delivered at Princeton University, May, 1921

; Methuen & Company, Ltd.:

London, UK, 1922.

Mach, E.

Die Mechanik in Ihrer Entwicklung

, 7th ed.; F. A. Brockhaus: Leipzig, Germany, 1883.

Barbour, J.; Pfister, H. (Eds.)

Mach’s Principle

; Birkhäuser: Basel, Switzerland, 1995; p. 536.

Sciama, D.W. On the Origin of Inertia.

Mon. Not. R. Astron. Soc.

1953

113

, 34–42. Available online: https://articles.adsabs.harvar

d.edu/full/1953MNRAS.113...34S (accessed on 19 June 2023). [CrossRef]

Einstein, A. Gibt es eine Gravitationswirkung, die der Electrodynamischen Induktionswirkung Analog ist?

Vier. Gericht. Med.

Sanit.

1912

, 37–40.

Pais, A.

The Science and the Life of Albert Einstein

; Oxford University Press Inc.: New York, NY, USA, 2005; pp. 287–288. ISBN

978-0-19-280672-7.

Mead, C. Engineering View of Gravitation. 2023. Available online: https://resolver.caltech.edu/CaltechAUTHORS:20230321-161151363

(accessed on 19 June 2023).

Shapiro, I.I. Fourth Test of General Relativity: Preliminary Results.

Phys. Rev. Lett.

1968

, 1265–1269. [CrossRef]

Shapiro, I.I.; Ash, M.E.; Ingalls, R.P.; Smith, W.B.; Campbell, D.B.; Dyce, R.B.; Jurgens, R.F.; Pettengill, G.H. Fourth test of general

relativity: New radar result.

Phys. Rev. Lett.

1971

, 1132. [CrossRef]

10.

Planck, M. The Theory of Heat Radiation.

Entropie

1900

144

, 164.