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General Relativity and Gravitation, Vol. 35, No. 5, May 2003 (C ?2003)

Machian Inertia and the Isotropic Universe

Peter Graneau1and Neal Graneau2

Received July 19, 2002, revised version January 3, 2003

This paper addresses the origin of the forces of inertia. It proposes a Newton-Mach

particle interaction force between all pairs of particles that depends on their relative

acceleration and is proportional to the gravitational force between them. The motion of

all objects therefore becomes directly influenced by all of the matter in the universe,

as prescribed by Mach’s principle. The effect of the observed hierarchical structure

of the universe is considered and is used to ensure that the inertial force on an object

is finite and isotropic. The instantaneous matter interaction force is justified and both

Einstein’s and Mach’s objections to a Newtonian framework are discussed and shown

to be absorbed by the proposed universal law of inertia.

KEY WORDS: Newton-Mach Paradigm; Cosmology.

1. NEWTON-MACH PARADIGM

Any Machian theory of inertia depends on instantaneous action at a distance, or

as one might prefer to call it, mutual simultaneous far-actions. The reason for this

is the requirement of simultaneous universal momentum and energy conservation

whichiswellknownfromexperimentandistheheartofNewtonianmechanics.To

illustrate this point we consider the simple example of the falling apple to which

Figure 1 refers. This diagram complies with d’Alembert’s principle of Newtonian

mechanics according to which all forces on a finite body or particle are in dynamic

equilibrium at any instant. It is equivalent to saying their vector sum is zero. In

Figure 1, IMD stands for an isotropic mass distribution. With M being the mass

of the earth, m, the mass of the apple, G, Newton’s constant of gravitation and

ra,e, the distance between the centres of gravity of the two objects, then Newton’s

1205 Holden Wood Road, Concord, Massachussetts, 01742, USA

2Department of Engineering Science, Oxford University, Oxford, OX1 3PJ, United Kingdom; e-mail:

neal.graneau@ eng.ox.ac.uk

751

0001-7701/03/0500-0751/0C ?2003 Plenum Publishing Corporation

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752 Graneau and Graneau

Figure1. Thefallofanapple,demonstratinginstantaneousmomentumconservationinaccordance

with D’Alembert’s principle.

universal law of gravitation gives the mutual force of attraction between the apple

and the earth as

Fg(a,e)= −GMm

r2

a,e

.

(1)

The force is always negative, implying attraction. Further, assuming no ex-

ternal forces such as air resistance, at every instant, the downward velocity of

the apple, va, and the associated upward velocity, ve, of the earth must ensure

momentum conservation. Therefore even while both are accelerating toward each

other

Mve+ mva= Constant.

(2)

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Machian Inertia and the Isotropic Universe 753

The velocities cannot be referred to the frame of the earth because vewould then

be zero and momentum could not be conserved as the bodies accelerate toward

each other. Mach (1960, p. 286) insisted that the two velocities have to be assessed

relative to the fixed stars. In this paper it will be argued that Mach’s unique inertial

reference frame is more sensibly taken to be an isotropic distribution of matter

which for our purposes may be treated as being at rest with respect to our galaxy.

The potential energy, Pa,e, of Newton’s universal gravitation for the apple-

earth combination is the energy stored when the earth and apple centres of gravity

were moved apart from 0 to ra,eagainst the force of gravity,

Pa,e=

ra,e

?

0

−Fg(a,e)dr.

(3)

In Newtonian dynamics, the negative gradient of the gravitational potential func-

tion defines the mutual gravitational attraction, or

Fg(a,e)= −dPa,e

dr

.

(4)

Inordertomaintaininstantaneousenergyconservation,thelossinpotentialenergy

must at all times be equal to the gain in kinetic energy when the velocities are

expressed relative to the Machian frame of inertia.

Thereislittledoubtthatkineticenergymustresideinthemovingbodywhich

possesses it, however the location of the storage of potential energy is not so

obvious. In non-Newtonian field theories, the stored potential energy is a physical

commodity which resides in the field surrounding the mobile bodies. If this were

correct, then the conversion of potential to kinetic energy would take travel time

and it would be impossible to conserve energy instantaneously.

In strictly Newtonian physics, energy is always associated with matter. It is

then logical to assume that the potential energy of gravitation is simply a mathe-

matical representation of distant matter force interactions. As well, the principles

of momentum and energy conservation require the forces of attraction, Fg(a,e),

(Figure 1) to act simultaneously on the apple and the earth. Consequently, the

experimentally well established concepts of both momentum and energy conser-

vation provide compelling support for the concept of instantaneous action at a

distance.

Figure 1 also shows the forces of inertia,? Fi, which Newton defined as being

equal and opposite to the external force causing the observed acceleration, ? Fe,

that is

? Fi= −? Fe= −m? a,

(5)

where ? a is the acceleration of m relative to Newton’s proposed absolute space.

Mach, however, insisted that ? a is the acceleration relative to the fixed stars, which

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754 Graneau and Graneau

in the present analysis, is taken to be equal to the acceleration relative to the

isotropic mass distribution, (IMD).

We now adhere to the Newtonian view that all fundamental forces of nature

are attractions or repulsions between two entities of matter (Graneau 1999). This

becomesthemostgenerallyvalidformofNewton’sthirdlaw.Asaresult,onemust

discoverwhatparticlesareinteractingwithanacceleratingobjectinordertocreate

the inertia force. These interacting particles must form an isotropic distribution as

the magnitude of the forces of inertia are independent of the direction of the

externally applied force. It is therefore proposed that the cause of and the reaction

to the inertia forces is distributed over an IMD, scattered throughout the universe.

The inertia force? Fiand its equal and opposite reaction force on the IMD can

be treated as having a line of action, as shown in Figure 1, which is co-linear with

the force? Fgon the apple. Since the earth is accelerating upward, it will also be

subjecttoaforceofinertiaequalandoppositeto? Fg.Thisleadstoasecondreaction

force,? Fi,ontheIMD.Hencewehavetoconsiderthreeattractions:(1)apple-earth,

(2) apple-IMD and (3) earth-IMD.

MachcriticizedmuchofNewton’swordingofthelatter’stheoryofdynamics.

He reserved the strongest objection for Newton’s concepts of absolute space and

absolute time. In the preface to the seventh (German) edition (1912) of his book,

The Science of Mechanics, Mach (1960, p. xxviii), wrote (in English translation):

“With respect to the monstrous conception of absolute space and absolute time I can retract

nothing.HereIhaveonlyshownmoreclearlythanhithertothatNewtonindeedspokemuch

about these things, but throughout made no serious application of it.”

The mechanically expressed fundamental laws of Newtonian mechanics are

still correct and used daily, although most scientists have agreed with Mach re-

garding the unreality of absolute space and time. The implication is that the force

of inertia,? Fi, on the apple of Fig. 1 is not a local interaction with absolute space,

but is the consequence of a vast number of remote interactions with all of the

matter in the universe. The interactions that significantly determine the magnitude

and direction of the inertia force are those that involve the vast isotropic matter

distribution of the distant universe. This philosophical change has no effect on the

equations of Newtonian dynamics and the magnitude of the force of inertia is still

given by Newton’s second law of motion, Eq. (5). Mach (1960, p. 287) developed

an argument which concludes:

“...we see that even in the simplest case, in which apparently we deal with the mutual

action of only two masses [apple and earth], the neglecting of the rest of the world is

impossible.”

This last statement comes nearest to what is now generally referred to as

Mach’s Principle. Einstein (1920, p. 71) accepted that Mach had corrected one

of the two perceived fundamental flaws of Newtonian mechanics, and thus he

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Machian Inertia and the Isotropic Universe 755

soughttoincorporateMach’sprincipleintohisownrelativitytheory.Thecomplete

paradigm suggested by this principle however still requires a law of nature which

describes the inertia force interaction between a particle in the laboratory and

another particle in the distant universe. We will call this the Machian inertial

particle interaction law.

2. PREVIOUS ATTEMPTS TO DISCOVER

THE MACHIAN INTERACTION LAW

Fiveseriousattemptshavebeenmadeinthesecondhalfofthe20thcenturyto

discover the Machian interaction law that could explain inertia. The first was due

to Sciama (1953). He argued that matter had inertia only in the presence of other

matter. In other words, inertia in a particle was induced by other remote particles.

Hecalleduponananalogywithelectromagneticinduction.Thisbecamethepattern

followed by all five previous investigators of the Machian particle interaction law.

Eighteen years after Sciama, French (1971, p. 542) derived an inertia induc-

tion law in his textbook, Newtonian Mechanics. He called it a speculation on the

originofinertia.ApparentlyunawareofSciama’sefforts,Frenchalsoreliedonthe

electromagneticanalogy.OnthebasisofMach’sprinciple,hearguedthatthelinear

inertia force (? Fiin Figure 1) and defined by Eq. (5) as −m? a, must be ascribable

to the acceleration of other bodies in the universe relative to a particle on earth.

This implied a mutual simultaneous interaction of widely separated particles and

bodies in a manner comparable to Newton’s universal theory of gravitation but in

a manner that also depended on relative acceleration.

Todiscovertheoriginofinertia,Frenchusedtheelectromagneticanalogyde-

pictedinFigure2.Twoelectriccharges,+q1and−q2,attracteachotheraccording

to Coulomb’s law by the force

Fc(1,2) =

1

4πε0

q1q2a

r2

,

(6)

wherer isthedistancebetweenthechargesand(1/4πε0)isadimensionalconstant.

Fc(1,2)is a ponderomotive (mechanical ) force and it obeys Newton’s third law.

French proposes that q2be given an acceleration, a, relative to q1caused by an

externalforce, Fe.Thereforeatanysubsequentinstant,q2ismovingwithavelocity,

v, relative to q1. The latter he assumes is stationary in the laboratory.

French then calculates the electrodynamic interaction of the two charges in

motionrelativetoeachother.Themagneticvectorpotentialofthecurrentelement,

q2v, at the position of q1is (q2v/r) in the direction of the relative acceleration, a.

In relativistic electromagnetism, the rate of change of the vector potential,

d

dt

q2v

r

=q2a

r

= E,

(7)

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756 Graneau and Graneau

Figure 2. French’s (1971) interaction of two electric charges.

results in an electric field strength, E, which then exerts an electromotive (not

mechanical)force, F?

c(1,2)onq1(seeFigure2).FromthisfollowsFrench’sequation

1

4πε0

c2r

The speed of light, c, has entered Eq. (8) as a consequence of the conversion from

electrostatic units of charge to electrodynamic units of charge. This was in fact the

contextinwhichtheconstant,c,wasfirstintroducedintophysicsbyWeber(1848)

in his attempts to unify the existing action at a distance laws of electrostatics and

electrodynamics.ConsequentlythechargesinEq.(8)arenolongertheelectrostatic

charges expressed in Coulomb’s law, Eq. (6).

It will be noted that from Eq. (6),

c(1,2)= Fc(1,2)r a

In French’s speculation about the origin of inertia, Coulomb’s law is taken as

an analogy of Newton’s law of universal gravitation. In order to achieve this, he

substitutestwomassesm and M forthetwochargesofEq.(6),andthedimensional

constantisreplacedbyNewton’sGravitationalconstant,G.Fromthishespeculates

that the same substitutions will also be valid in Eq. (8) yielding

F?

c(1,2)=

q1q2a

.

(8)

F?

c2.

(9)

F?

i(1,2)= GM ma

c2r

.

(10)

Consequently, he proposes that the total inertial force could be calculated if

all of the objects in the universe acquire an acceleration, a, with respect to the

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Machian Inertia and the Isotropic Universe 757

mass, m. By summing over all masses except m, the inertial force on it can be

expressed as

Finertial= ma

?

all masses

G M

c2r.

(11)

In order to agree with Newton’s well known second law of motion,

?

universe

G M

c2r

= 1.

(12)

Relying on figures which have at times been quoted for the radius of a spheri-

cal cosmos and the total mass contained in it, French claimed that Eq. (12) was not

unreasonable.AcontroversialfeatureofFrench’stheory,howeveristhattheveloc-

ity of light, a fundamentally electrodynamic quantity, now enters the Newtonian

dynamics of forces of gravitation and inertia in which it has no obvious meaning.

Asmentionedearlier,itappearedbecauseFrenchusedtwodimensionallydiffering

types of charge in his electrical analogy whereas there is only one type of mass.

From this, it can be concluded that the electrodynamic analogy is an artefact and

French’s Eq. (10), which he considered to be the Machian particle interaction law,

is probably incompatible with Newtonian mechanics.

Three more attempts were made to discover the Machian particle interaction

law which must underlie Newton’s force of inertia, Eq. (5). These investigations

were carried out by Burniston Brown (1982, chap. 7), Assis (1989) and Ghosh

(2000, chap. 3). Although they all arrived at the same result as French, the latter

authors provided more qualitative discussion on the nature of the universe. They

agreed on the following premises:

rThe Machian particle interaction is based on an action at a distance mech-

rThere exists much isotropically distributed matter in the universe outside

inertia observed on earth. Burniston Brown includes in this all visible

matter while Assis and Ghosh speak of an isotropic matter distribution

superimposed on an anisotropic distribution.

anism

rThe observable universe is a sphere of finite radius with the Milky Way at

its centre.

our home galaxy. This matter is responsible for the isotropic forces of

rOn earth, we experience local gravitational attractions described correctly

ies of the solar system. As a consequence of the apparent isotropy of the

extra-galactic cosmos, its gravitational effect cannot be measured. The

observable Newtonian gravitational attractions involve so little matter that

their anisotropic contribution to inertia forces is negligible.

by Newton’s universal law of gravitation. This involves primarily the bod-

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Burniston Brown discussed retarded action at a distance, while Assis utilised

Weber’s instantaneous action at a distance. Ghosh mixed instantaneous with re-

tardedactionatadistance.Theyall,however,arrivedatFrench’sresultofEq.(10).

ThisisduetothefactthatBrown’sforcecalculationsignoretheretardationaspect,

presumably because it became unmanageable. French’s electrodynamic formula,

Eq.(8),wasderivedwiththehelpofrelativisticfieldtheory,whileBurnistonBrown

and Assis relied on Weberian electrodynamics which did not contain fields. This

is very surprising and suggests that special relativity, and field theory in general,

is to some extent contained in Weber’s electrodynamics. While Burniston Brown

and Assis argue that their forces of inertia are of Newtonian gravitational origin,

this cannot be true because Eq. (10) is not an inverse square law and it contains

the velocity of light. None of these five authors addressed the issue of how their

equation could lead to a finite and measurable force of inertia in a possibly infinite

universe.

3. PROPOSED MACHIAN PARTICLE INTERACTION LAW

Accepting the Newtonian principle of inertia, which states that the force of

inertia counteracts acceleration, we expect that a particle which accelerates in

the midst of an isotropic mass distribution (IMD), in any arbitrary direction, will

experience a repulsion from half the distribution in front of it and an attraction

from the other half behind it. These repulsions and attractions must combine to

create the measurable force of inertial resistance to acceleration as quantified

by Newton’s principle of inertia as expressed in Eq. (5). Further we never detect

a velocity dependent Newtonian force of attraction or repulsion as expressed

in Newton’s first law and the principle of Galilean invariance. Therefore we

only need to consider an interaction which is a function of relative position and

acceleration.

We will now hypothesize the Machian particle interaction with distant mat-

ter on the basis of Eq. (5) without calling upon an electrodynamic analogy. We

feel justified to utilise an instantaneous mass interaction law because it has been

revealedexperimentallythatthespeedofpropagationofacentralNewtoniangrav-

itational attraction is at least 2 × 1010c, (Van Flandern 1998) where c is the speed

of light. Such a velocity is experimentally indistinguishable from an instantaneous

interaction. Consider the diagram of Figure 3 in which a particle of mass, m0, in

the laboratory is being acted on by an upward external force,? Fe. If the particle

is free to move, it will accelerate with respect to the fixed stars (Machian inertial

system)inthedirectionof? Fe(θ = 0),perpendiculartotheplaneEE.Iftheinertial

force,? Fi, is proportional to the magnitude of the acceleration, ? a, and acts in the

opposite direction, then it will increase from zero as soon as the particle begins to

accelerate. The inertial force increases as the acceleration increases, ensuring that

the force of inertia is always equal and opposite to the applied external force. This

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Machian Inertia and the Isotropic Universe 759

Figure 3. Machian inertial force interactions between an observable particle, m0and

particles in the distant isotropic mass distribution (IDM), (mx& my).

dynamic equilibrium is stable and thus determines the value of the acceleration

that is caused by the application of a given external force. If the particle were to

accelerate faster, then? Fiwould increase and retard the extra acceleration, Simi-

larly,iftheparticleweretoslightlydecelerate,then? Fiwoulddecreasecausingthe

particle to feel a net increased force in the? Fedirection, thus resisting the deceler-

ation. This stability caused by a real force is the mechanism behind Newton’s 1st

law, ensuring that an object does not accelerate with respect to the distant universe

unless acted upon by another body.

If mxis the mass of another particle as shown in Figure 3, then a repulsive

Machian inertial interaction force, ?Fi,(0,x), will act between m0and mxwhich

opposestheirrelativeacceleration,a0,x.Themagnitudeoftherelativeacceleration

is quantified by

a0,x=d2r0,x

dt2,

(13)

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760Graneau and Graneau

where r0,x is the distance between m0and mx. We propose that the elemental

inertial force law is a force of mass interaction which takes the form

?Fi(0,x)= −K a(0,x)m0mx

rn

0,x

,

(14)

where K isadimensionalconstantwhosemeaningwillbediscussedlater.Eq.(14)

represents a mutual Newtonian force of attraction or repulsion between the two

particles. It is positive, representing repulsion when a0,xis negative as a result of

the two particles accelerating toward each other. Similarly, the force is negative,

representing attraction when the two particles accelerate away from each other. In

spite of the mathematical similarity with Newton’s law of gravitation, Eq. (1), the

Machian particle interaction, Eq. (14) is an additional force which vanishes when

the two particles are not accelerating with respect to each other, even though they

are still subject to mutual gravitational attraction.

A general expression for ? aˆ x, the acceleration vector of m0in the direction of

mx, defined by the unit vector, ˆ x, can be formulated from the total acceleration

vector, ? a, and θ, the angle between the two as shown in Figure 3.

? aˆ x= |?2a| cosθ ˆ x.

Resolving in the ˆ x direction, Eq. (14) can define the inertial force on m0due to

relative acceleration with respect to mxas

?? Fi,ˆ x= −K ? aˆ xm0mx

rn

0,x

(15)

= −K|? a| cosθm0mx

rn

0,x

ˆ x.

(16)

Due to the Newtonian nature of the force described in Eq. (14), the reaction force

on the particle, mx, naturally has the same magnitude but the opposite direction as

depicted in Figure 3.

If there is a particle, my, of the same mass as mxand symmetrically opposed

to mxabout m0, then the interaction between m0and mycan also be calculated.

Since the distance r0,yincreases as a result of the acceleration, aˆ y, there is an

attractive inertia force ?Fi,ˆ yof the same form as Eq. (16) which will also oppose

the acceleration toward mx. The two linear inertia forces on m0due to mxand my

therefore add together as shown in Figure 3, so that for the system of three masses

? Fi,(x+y)= ?? Fi,ˆ x+ ?? Fi,ˆ y= 2?? Fi,ˆ x.

(17)

ThisleadstotheimportantconclusionthatusingtheforcelawproposedinEq.(14),

anisotropicmassdistributionwillleadtoanon-zeroinertiaforceonanaccelerating

particle.

It can thus be seen that the linear force of inertia between m0and any mass,

mx, will result in a downward directed component of ?? Fi,ˆ x, perpendicular to EE

andopposing ? a.UsingEq.(16),itfollowsthatthesumofthecomponentsresolved

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Machian Inertia and the Isotropic Universe 761

in the direction of the acceleration, ˆ a, is

? Fi,ˆ a=

?

x

???? Fi,ˆ x

??cosθ ˆ a = −m0? a

?

K

?

x

mxcos2θ

rn

0,x

?

.

(18)

The summation is taken over all of the particles in the universe. It will be seen that

for an isotropic mass distribution, the inertial force components in the EE plane

will cancel by symmetry.

Wemustnowattempttodiscoverthepowerofthedenominator,n,andprovide

aninterpretationoftheconstant, K.Toachievethis,itishelpfultorewriteEq.(18)

in spherical coordinates centred on m0. It will be shown in the later discussion that

the mass density of the observable universe is a function of the distance, r, from

any observer. Therefore we can use the substitution,

?

x

mx

rn

0,x

=

2π

?

0

π

?

0

∞

?

t

ρ(r,θ,φ)r2dr dθ dφ

rn

.

(19)

t is a non-zero distance which ensures there is no force singularity due to self-

interaction. Its physical meaning will be discussed later. Eq. (18) can now be

rewritten as

? Fi,ˆ a= −m0? a K

2π

?

0

π

?

0

∞

?

t

ρ(r,θ,φ)r2−ncos2θ dr dθ dφ.

(20)

From experience, we know that the magnitude of the force of inertia is in-

dependent of the direction of the observed acceleration, ˆ a. In order for Eq. (20)

to satisfy this condition, ρ(r,θ,φ) must be invariant for all directions (θ,φ), and

thus approximate to an isotropic density function which is purely dependent on

distance, ρi(r). To achieve this, we can write

00

t

00

t

? Fi,ˆ a= −m0? a K

2π

?

2π

?

π

?

?

∞

?

?

ρi(r)r2−ncos2θ dr dθ dφ

+

π

∞

ρa(r,θ,φ)r2−ncos2θ dr dθ dφ

,

(21)

where ρa(r,θ,φ) describes the anisotropic density distribution defined by

ρa(r,θ,φ) = ρ(r,θ,φ) − ρi(r).

(22)

The direction invariance of? Fi,ˆ aimplies that the first integral in Eq. (21) rep-

resents the dominating contribution to the inertial force from a very large isotropic

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762Graneau and Graneau

massdistribution(IMD),whilethesecondintegraldescribesanegligiblecontribu-

tion to the inertial force as a result of interaction with a much smaller anisotropic

mass distribution (AMD).

The AMD however is well known to us for it causes the gravitational forces

that directly affect us, for instance those caused by the sun and moon and to a

lesser extent the other bodies in the solar system. We know that our galaxy has a

planar structure and thus must also be included in the (AMD). If n, the value of the

power of rn

distributiondefinedinEq.(22),Newton’suniversallawofgravitation,Eq.(1),can

be employed to describe the net gravitational force on m0in an arbitrary direction,

ˆ z, as

0,xin Eqs. (14)–(21), is taken to be 2, then using the anisotropic density

? Fg,ˆ z= −G m0ˆ z

2π

?

0

π

?

0

∞

?

t

ρa(r,θ,φ) cosθ dr dθ dφ,

(23)

where G is Newton’s gravitational constant and θ is the angle between dr and

ˆ z. Eq. (23) is valid because the contributions to the gravity force from the much

larger isotropic mass distribution will come to zero by symmetry.

InthecaseofEq.(23)andthesecondintegralofEq.(21),thevalueoft mustbe

takenasanydistanceoutsidethetestbody, m0,butlessthanthenearestinteracting

body. In the case of the first integral in Eq. (21), t must take on a value which

represents the distance at which the anisotropic distribution, ρa(r,θ,φ), becomes

insignificant in the determination of the local value of ρ(r). Observation indicates

that such a distance is much larger than our galaxy or in fact much larger than our

local cluster of galaxies. By inspection of recent maps of galaxies in the known

universe,ourbestestimateofthedistanceatwhichthisdistributionbecomesfairly

isotropic, is in the region of t = 70–100 Mpc (∼3 × 108light years).

The dominance of the first integral in Eq. (21), as a result of inertial isotropy,

allows us to neglect the second integral when we perform the integration, leaving

? Fi,ˆ a= −m0? a

π2K

∞

?

t

ρi(r)r2−ndr

.

(24)

In order to ensure that Eq. (24) remains equivalent to Eq. (5), Newton’s

empirical principle of inertia, we must ensure that the quantity in brackets is

dimensionless and equal to unity. The integral in Eq. (24) can be represented by a

constant whose value depends on the value of n. This implies that if

Bn=

∞

?

t

ρi(r)r2−ndr

(25)

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Machian Inertia and the Isotropic Universe 763

then

K =

1

π2Bn,

(26)

and therefore Eq. (14), the Machian particle interaction that predicts the force of

inertia can be rewritten as

?Fi(0,x)= −

1

π2Bn

d2r0,x

dt2

m0mx

rn

0,x

.

(27)

4. THE PARADOX OF A NEWTONIAN HOMOGENEOUS UNIVERSE

The major problem faced by this analysis so far is the possibility that Bnis

infinite, since Eq. (25) represents an integration to infinity of the mass distribution

in a possibly infinite universe. Since the time of Galileo, we have been aware that

we are not occupying a privileged position in the universe. Consequently, until

recently,ithasbeenassumedthattheuniverseisafairlyhomogeneousdistribution

of matter with a constant density. Newton was aware that he was caught between

two awkward universe scenarios, namely a) the apparently atheistic viewpoint

that the universe was infinite in extent or b) that it represented a finite amount

of matter in an infinite amount of space. The first, (a) was unsatisfactory from a

theological and mathematical point of view and the second, (b) would imply that

the universe should have collapsed as a consequence of his own law of universal

gravitation. The debate regarding the validity of these two systems has developed

furtherintheintervening300years(Jaki1990,chap.8)andultimatelyledtooneof

Einstein’s conjectures regarding a finite and curved space that led to the formation

of his theory of General Relativity. The issue has usually been debated under the

banner of the Gravitational Paradox and will now be investigated with regard to

the proposed Machian inertial mass interaction force.

Inaninfinitehomogeneousuniverseinwhichgravitationalmatterinteractions

are governed by the Newtonian inverse square law,? Fg,ˆ ris not in general a defined

value. This can be demonstrated by dividing such a universe into two regions as

shown in Figure 4. The surface of division is a spherical surface of radius, R,

whose centre is at P. The test particle, m0, lies on this surface. Newton (1962,

Book I, Section XII, Prop. LXX, Theorem XXX) showed that a constant density

spherical shell causes no net gravitational force on any particle inside the shell

as a consequence of the inverse square force law. Consequently, there is no net

gravitational force on m0due to matter outside the spherical dividing surface

since it is surrounded by concentric spherical shells of constant mass density. Still

assuming a homogeneous mass distribution, the gravitational force on m0due to

the mass inside the surface can be calculated by assuming that the entire mass of

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764Graneau and Graneau

Figure 4. Demonstraction of the gravitational force due to an isotropic density

function,ρi(r),asdescribedbyEq.(31).Theuniverseisdividedintotworegions,

(shaded and unshaded) centred on P, and containing concentric shells of constant

mass density.

theshadedsphereisactingatthecentreofmass,P,andthusthetotalforceonm0is

?4/3π R3ρh

where ρhis the density of the homogeneous mass distribution. Eq. (28) reveals

that the magnitude and direction of the gravitational force is dependent on the

arbitrary choice of the position of P which determines R. This demonstrates

that the force of gravity as predicted by Newton’s law of universal gravitation is

undefined in an infinite homogeneous mass distribution.

? Fg,ˆ r0,P= Gm0

?

R2

ˆ r0,P=4/3π G m0ρhR ˆ r0,P,

(28)

5. THE DISCOVERY OF COSMIC HIERARCHICAL STRUCTURE

Recent astronomical measurements have indicated that contrary to the as-

sumptionsoftheprevious300years,theuniverseisnothomogeneous,butappears

to have a hierarchical structure, meaning that galaxy clusters are highly irregular

Page 15

Machian Inertia and the Isotropic Universe 765

and yet self-similar, with a fractal structure which is asymptotically dominated by

voids. This isotropic structure has been described by Mandelbrot (1983, p. 87) as

fractal homogeneity. There is general agreement that galactic structures are fractal

up to a distance scale of 30–50 Mpc. Some have claimed that the data has revealed

fractalcorrelationswithdimension(D ? 2)uptothedeepestscalesprobedtodate

(1000Mpc)(SylosLabini,Montuori&Pietronero1998).Thesegalaxiesrepresent

the furthest matter for which we have redshift data. The fractal dimension, D, is

defined by

M ∝ rD,

(29)

whereMisthemassofthemattercontainedinasphereofradius,r,centredonany

observer.Clearly,inahomogeneousdistribution,(D = 3).Thereisalivelycontro-

versy regarding whether the mass distribution becomes homogenous at the largest

length scales, which is an important feature of the Friedmann-Robertson-Walker

(FRW) metric and the standard big-bang model of cosmology (Wu, Lahav &

Rees 1999). Measured values of D were initially reported as low as (D ∼ 1.3)

(de Vaucouleurs 1970), but as more redshift measurements have become avail-

able, it has become clear that out to a depth of ∼50 Mpc, the galaxies appear

to have a fractal distribution of dimension of (D = 2 ± 0.2) (Roscoe 2002). It

would appear that the analysis of the redshift data from the more distant galaxies

is shrouded in controversy over the statistical methods used to analyse the data.

However, there appears to be no observational basis preventing the conjecture that

the entire universe has a fractal dimension of (D ? 2).

Using Eq. (29), we can define a fractal mass density, Z, to describe the mass

contained in an arbitrarily positioned sphere of radius, R, such as the shaded one

in Figure 4, as

Z =Msphere(R)

RD

.

(30)

Thisimpliesthatthemassdensityisconstantinanygivensphericalshell,(r + dr),

but for (D < 3) the density of each shell decreases as r increases (Mandelbrot

1983, p. 88). Using Eq. (30), an isotropic density function in a fractal distribution

of dimension, D, can be defined as

ρi(r) =D Z

4πrD−3.

(31)

We can now calculate the Newtonian gravitational force on the particle, m0, in

the arbitrary direction ˆ r0,Pin an isotropic fractal universe again using Figure 4.

The mass outside the sphere is still in spherical shells of constant density and thus

causes no net gravitational force on m0. Even in a fractal distribution, the centre

of mass of the sphere remains at its centre, P. However, the total mass contained

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