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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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758Graneau and Graneau
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
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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
Page 16
766 Graneau and Graneau
in the shaded sphere can now be expressed from Eq. (30) as
Msphere= Z RD.
(32)
Therefore, in analogy with Eq. (28), the gravitational force on m0, in the arbitrary
direction ˆ r0,P, in a fractal mass distribution can be written
? Fg,ˆ r0,P= Gm0Msphere
R2
ˆ r0,P= Gm0Z RD
R2
ˆ r0,P.
(33)
ItcanbeseenfromEq.(33)thatif(D ≤ 2),thentheilldefinedforceofgravity
in any arbitrary direction and acting on every particle, m0, due to an isotropic
universe goes to zero in an infinite isotropic distribution. In this situation, the well
defineddirectionandmagnitudeoftheobservablegravitationalforceiscompletely
caused by the AMD as described by Eq. (23). The observation that (D ? 2) is
consistent with the requirement that (D ≤ 2) in order to ensure the resolution of
the Gravitational Paradox. Hoyle (1953) and Mandelbrot (1983, p. 92) have both
speculated on this intimate connection between the hierarchical structure of the
cosmos and Newtonian gravitation, and have suggested that it may be the force of
Newtonian gravity that creates the fractal structure that we observe.
In our treatment of Mach’s principle we measure local accelerations against
a background of point like galaxies, apparently fixed relative to each other over
the short timescales of human experience. This is similar to Roscoe (2002) who
has developed a model universe consisting, initially, of a stationary (but not static)
ensemble of identical particles existing in a formless continuum, without precon-
ceived notions of clocks and measuring rods. He concludes that, on very large
scales, all motion can be considered to be inertial, and the distribution of mass is
necessarily fractal with dimension (D = 2).
Einstein and his colleagues were apparently unaware that a fractal mass dis-
tribution such as described by Eqs. (29)–(32) was possible, in a manner that does
not pre-select any unique position as the centre of the universe. Consequently, he
felt forced to propose a geometry of curved space in which the mass contained
in the universe was finite (to avoid the gravitational paradox) but yet the universe
was unbounded, in that space was curved so that all of space could be filled with
a homogeneous but finite matter distribution (Einstein 1920, p. 108). While the
mathematics behind fractal geometry was slowly emerging in the late 19th cen-
tury, its application to the study of nature was only first attempted in the 1970’s
by Mandelbrot (1983). Thus Einstein was unaware of the power of such a matter
distribution for the purpose of resolving the gravitational paradox. With the even
more recent experimental confirmation of a fractal (D ? 2) mass distribution of
the galaxies up to the limits of our measuring equipment, there seems to be no
longer a conceptual requirement to abandon Newtonian dynamics or Euclidian
space.
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Machian Inertia and the Isotropic Universe 767
The discovery that D is less than or equal to 2 may resolve the gravitational
forceparadoxforaninfiniteuniverseundertheinfluenceofNewtoniangravitation,
butitisnotsufficienttopreventinfinitegravitationalpotentials.However,sincewe
never directly measure potentials, but only accelerations which are proportional
to forces, then infinite potentials with finite gradients are not a physical problem.
It is often claimed that the universe consists of up to 99% unobservable dark
matter. By definition, we know nothing directly about the nature or distribution of
this material. Dark matter distributions are conventionally invoked when a gravi-
tational theory is unable to explain the behaviour of observable bright matter. For
the purposes of this paper, the observed (D ? 2) distribution of bright matter and
its natural relationship with Newtonian gravity, offers no reason to suspect that if
there is dark matter that it should be distributed differently.
Returning to the force of inertia, in order to ensure that it is always finite, we
must confirm that Bnas defined by Eq. (25) remains finite. Using the relationship
for isotropic density, ρi(r), in a fractal distribution given in Eq. (31), we can write
Bn=D Z
4π
∞
?
t
rD−n−1dr.
(34)
For Bnto remain finite, n > D.
In the same manner that Hoyle (1953) and Mandelbrot (1983, p. 92) claimed
a connection between the inverse square law of gravitation and the hierarchical
(D ? 2) structure of the universe, we propose that this fractal mass distribution
also implies that a mass interaction law of inertia will also be an inverse square
interaction (n = 2). If this were so, then (n = D = 2) is the limiting case and
in order to maintain a finite value of inertia, D must actually be less than 2.
It is plausible that the universe is constantly trying to achieve a homogeneous
distribution (D = 3), but that as it approaches D ? 2 it cannot get beyond there
becauseatthatpoint,theinertialforcewouldbecomeinfiniteandallmotionwould
cease. Since D ? 2 is the observed universal fractal dimension, we feel justified
to assume that the inertial force is an inverse square law. This assumption seems
quite natural when it is noted that all the Newtonian matter interaction force laws
discovered to date are built on the inverse square relation.
Several important pieces of information can now be assimilated in order to
arriveataplausibleexpressionfortheproposedMachianinertialmatterinteraction
law.
1) The exponent of the distance of separation in Newton’s law of universal
gravitation has been proved to very high accuracies even down to length
scalesof200µm.Nodeviationfromtheinversesquarelawofgravitational
matter interaction has been detected (Hoyle et al. 2001).
Page 18
768 Graneau and Graneau
2) Themostrecentsurveysofthecosmosimplythatmatteronalargeenough
scale is distributed in an isotropic, but inhomogeneous manner with a
fractal dimension, D ? 2.
3) The force of gravity is measurable and well defined on all objects. The
resolutionoftheGravitationalParadoxbyusingafractalmassdistribution
implies that D ≤ 2, which is consistent with the previous two points. In
order for our proposed Machian inertial matter interaction law, Eq. (27),
to predict a finite force of inertia, it is essential that D < n and therefore
n ≥ 2.
4) The measured force of inertia is proportional to acceleration and acts
to oppose an external force applied to an object. It always has finite
magnitude.
Point(4)highlightsthefinitenatureoftheinertialforceandinordertoensure
this behaviour and also to absorb the suspected connection between the universal
fractal dimension and the proposed inertial force law, points (1–3) justify our use
of n = 2. Consequently, there are now two good reasons (well defined gravity
forces and finite inertial forces) to suspect that the observed hierarchical structure
of the universe is a consequence of our proposed Machian inertial force law which
is closely related to the Newtonian gravitational law.
6. CONCLUSIONS
Eq. (27), the elemental form of the proposed Machian inertial matter interac-
tion law, can now be justified as containing n = 2, and can then be summed over
all particles in the universe yielding the total inertial force on a particle, m0as
? Fi,ˆ a= −
1
π2Bm0? a
?
x
?
mx
r2
0,x
cos2θ
?
.
(35)
ItisalsoclaimedthatEq.(27)isafiniteinstantaneousactionatadistanceforce
betweenparticlesofmatterwhichcreatesaresistancetotheaccelerationcausedby
an external force acting on one of them. Our knowledge of the fractal distribution
of matter throughout the universe combined with the finite, inverse square nature
of the gravitational force allow us propose that n = 2 for both the gravitational
and inertial force laws and D is approximately equal to but slightly less than 2.
Therefore the elemental form of our proposed inertial force law, Eq. (27), can be
expressed as
?Fi(0,x)= −
1
π2B
d2r0,x
dt2
m0mx
r0,x2=
1
π2B Ga(0,x)?Fg(0,x).
(36)
Eq. (36) therefore represents an instantaneous Newtonian force of either attraction
or repulsion between mass particles that is proportional to their relative accelera-
Page 19
Machian Inertia and the Isotropic Universe769
tion, a(0,x), and is also proportional to the gravitational force between the objects,
?Fg(0,x). It is also inversely proportional to a constant B described by
B =D Z
4π
∞
?
t
rD−3dr.
(37)
where Z isauniversalfractalmassdensitydefinedbyEq.(30), D isthedimension
of the fractal distribution and t is the radius from m0of a spherical shell in which
themassdensitydistributionbecomesdominantlyisotropic.If D < 2,then B must
have a finite value, but we need a much more precise measure of D, Z and t in
order to put a magnitude to it.
Eq.(36)invitesabriefspeculationregardingtheverypreciselymeasured,but
nevertheless mysterious, Newtonian gravitational constant, G (6.67 × 10−11m3
kg s−2). If the infinite cosmos was expanding in such a way that every object was
accelerating from every other with an acceleration of (π2B G), then our proposed
force of inertia would become the cause of the gravitational force. With ever in-
creasing cosmological observations, it will eventually be within our powers to
estimate B (kg m−2) in Eq. (37), and thus our local laboratory determination of G
may be the measurement of a universal expansion acceleration. This unexpected
acceleration may be the mechanism by which the universe avoids becoming ho-
mogeneous and retains its hierarchical structure. However this pure speculation is
only built on the rather hopeful human desire to unify the known force laws and
cannot be justified by any existing experimental knowledge.
More importantly, Eq. (36) complements Newton’s universal law of gravita-
tion and thereby completes Newton’s theory of instantaneous action at a distance
mechanics in a manner which answers the cosmological doubts of both Mach (ab-
solute space) and Einstein (gravitational paradox) which were responsible for the
general relativistic revolution. Recent knowledge of the hierarchical structure of
the universe and the consequent finite nature of our proposed inertial force law
opens the door for a return to a simpler cosmological model, based on Newtonian
forces between pieces of matter, acting in a Euclidean geometry. It is important to
remember that Newtonian forces and Euclidean geometry have never been found
in error in any laboratory controlled experiment and are still used with complete
accuracy to predict the motion of all man-made objects in our solar system. A
famous apparent discrepancy is the anomalous precession of the perihelion of
mercury, but it represents an example of an uncontrolled experiment in which the
variables such as solar oblateness and mass distribution cannot be independently
manipulated and thus it lacks the rigor with which Newtonian theory has been
evaluated.
Tosummariseourproposedinteractionmechanism,theNewtonianelemental
inertial force, Eq. (36), always acts as an attraction or repulsion between the two
bodies, m0and mx, at the same time as an external applied force acts on one of
Page 20
770 Graneau and Graneau
them, m0. The inertial force always opposes the relative acceleration between m0
and every body, mx. In the spirit of Mach’s principle, summing over all objects
in the universe yields a finite value for the force of inertia on an accelerating
particle. Employing the now well confirmed fractal matter distribution consistent
with(D < 2),thefinitemagnitudeoftheforceofinertiaoccursdespitetheinfinite
number of non-cancelling instantaneous interactions. The mass related force of
inertia is therefore responsible for controlling the magnitude of the accelerations
that are caused by applied forces and is the mechanism that lies behind Newton’s
2nd law of motion.
ACKNOWLEDGMENTS
We are indebted to Andre Assis (1989) for first drawing our attention to
Machian inertial particle interactions. He was also responsible for clarifying the
concept of radial acceleration and with it the fact that mass interactions can be
repulsive as well as attractive.
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