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Special relativity:interpretation and

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Citation: RAHNEJAT, H., 2005. Special relativity: interpretation and impli-cations for space-time geometry. Proceedings of the Institution of MechanicalEngineers, Part K: Journal of Multi-body Dynamics, 219 (2), pp. 133-146

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Special relativity: interpretation and implicationsfor space–time geometryH Rahnejat

Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, Loughborough,

Leicestershire LE11 3TU, UK

The manuscript was received on 20 January 2005 and was accepted after revision for publication on 6 April 2005.

DOI: 10.1243/146441905X10122

Abstract: The paper commemorates the centenary of the special theory of relativity, whicheffectively sets the limit for the structure of space–time to that of the stationary system. Thelong lasting debate for definition of concepts of instantaneity and simultaneity was thusresolved by the declaration of constancy of speed of light in vacuo as a law of physics. Allmotions were thus bounded by the light cone and described by the properties of differential geo-metry, firmly anchored in the calculus of variations. The key contribution underpinning thetheory was the resolution of the contradiction imposed by the Galilean transformation throughphysical explanation and the adoption of the Lorentzian transformation. This highlighted therelative nature of both space and time and the linkage of these to preserve the sanctity of thelight cone. The resulting space–time geometry was then founded on the traditional calculusof variation with the addition of this transformation. This retains the time as an independentcoordinate and its linkage to space in an explicit form. One implication of this approach hasbeen the retention of the concept of infinitum for some physical quantities as a drawback foruse of the Lorentzian transformation. The paper shows that this singular behaviour need notarise if the explicit linkage in space–time is abandoned in favour of the implicit inclusion oftime as a link between the curved structure of space and the speed of light, thus restating thecalculus of variation in line with special relativity. This points to a closed loop space–matterfield, which may belie the fabric of the continuum. One implication of this interpretation isthat a small variation in speed of light within the field would be required to dispense with theaforementioned singular nature of the Lorentzian boost, while still remaining within the spiritof special relativity.

Keywords: special theory of relativity, principle of incompleteness, closed loop space–matterfield

1 INTRODUCTION

With the acceptance of the Copernican heliocentricsystem, it became apparent to Galileo [1] that allbodies were subject to motion, thus measurementof kinematic quantities cannot be achieved in anabsolute sense, but relative to an observer, also insome state of motion. He declared this to be theprinciple of relativity. With this realization, theconcept of relativism was born. Galileo then statedthat no observer could distinguish between thestates of absolute rest and absolute motion.

Measurement of motion of an entity required atransformation to the frame of reference attachedto the observer. This transformation was providedby Galileo [1] for observers in relative uniformmotion, known as the Galilean transformation inshear, because the world-lines of motion of entitiesappear as straight lines. At the turn of the 20thcentury, Galilean transformation was used in anexperiment to measure the velocity of the Earth rela-tive to aether (the so-called aether wind), which wasassumed to occupy the empty space (see anystandard physics text on the Mickelson–Morley

SPECIAL ISSUE PAPER 133

JMBD9 # IMechE 2005 Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics

experiment and section 3). Furthermore, it was notedthat Maxwell’s equations for electromagnetic wavesdo not transform properly under Galilean transform-ation. Lorentz [2] had already established that atvelocities nearing the speed of light, the measureddistances in the direction of motion appear to signifi-cantly contract (referred to as the Fitzgerald’sContraction, who first described this phenomenon).Lorentz provided a new form of transformation,which took this anomaly into account and enabledproper transformation of Maxwell’s equation [3].Lorentzian transformation, referred to as the boostfunction, provides a relationship for time of anobserver as a mixture of time and spatial coordinatesof another. Thus, not knowing the physical impli-cations of this relationship, he rather uncomfortablycalled the time of an observer the local time. Animportant feature of the Lorentzian transformationis that it limits motion of observable matter withinthe light-cone. While much of the physics’ worldat the turn of the 20th century was concerned withthe seeming incompatibility of Maxwell’s equationswith the classical Newtonian physics, with a strokeof ingenuity, Einstein unified kinematics by declar-ing the constancy of velocity of light in vacuo as alaw of physics, and brushed aside the concepts ofinstantaneity and simultaneity by replacing themwith the time of the stationary system (the Galileanabsolute rest or absolute motion, see the Editorialto this issue), and declared the Galilean principle ofrelativity also as a law of physics. Later, he namedthe theory underpinning these declared laws ofphysics as the special theory of relativity, recogni-zing its limited validity to uniform motion ofmatter [4].

2 ASPECTS OF CALCULUS OF VARIATIONS

The foundation of physics of motion, as shown vari-ously here, is based upon differential geometry orsimply properties of curves. Additionally, the motionof matter on such curves can be described withregard to infinitesimal changes or evanescent quan-tities. In other words, small increments of timeonly are admissible. The study of vanishing changeswithin differential and integral geometry is knownas the calculus of variation. Newton placed thebasis of his theories of motion on this approach,using a series of lemmas and corollaries in the styleof the early Greek mathematicians such as Euclidand Pythagoras. This approach avoided the rathertedious and rigorous method later suggested byCauchy, but is equally effective and far moreaccessible.

Perhaps, the most important and underlyingproperties of limits in differential geometry are the

curvature properties of a path, giving physicalmeaning to kinematic quantities: velocity and accel-eration. For this purpose, Newton introducedLemmas VI–VII in the Principia [5], using a geo-metrical proof as depicted in Fig. 1. Note that theangle of contact made at point A should vanish forthe curve to continue beyond it. Newton put it as aLemma in the following form (refer to Fig. 1): ‘Ifany arc ACB, given in position, is subtended byits chord AB, and in any point A, in the middle ofcontinued curvature, is touched by a right line AD,produced both ways; then if the points A and Bapproach one another and meet, I say, the angleBAD, contained between the chord and the tangent,will be diminished in infinitum, and ultimatelywill vanish.’

The proof provided by Newton makes use of refut-ing the contrary: that the continuity of the curvatureis contradicted if the lemma is not upheld. Theextension of this lemma was made by a subsequentone, where Newton stated: ‘The same things beingsupposed, I say that the ultimate ratio of the arc,the chord, and tangent, any one to any other, isthe ratio of equality [5].’ Then, using elementary

Fig. 1 Limits in the Euclidean geometry

134 H Rahnejat

Proc. IMechE Vol. 219 Part K: J. Multi-body Dynamics JMBD9 # IMechE 2005

Euclidean geometry, one can see that (see Fig. 1)

arc ACB ¼ RDu

chord AB ¼ 2R sinDu

2

tangent AD ¼ 2R tanDu

2

(1)

It is clear that as Du! 0, sin Du � tan Du � Du and theratio of all these evanescent quantities tend to unity. Ifone is to assume that point B converges upon A in auniform manner with a velocity, v, and a correspond-ingly short time interval Dt! 0, then RDu ¼ vDt.This can be represented as v ¼ RDu/Dt ¼ Dx/Dt,where in Fig. 1, AD ¼ x. Thus, in limit with the dimin-ishing quantities

v ¼ limDt!0Dx

Dt

� �(2)

In differential geometry, one can represent this in thelimit as v ¼ dx/dt.

3 SOME LIMITATIONS WITH DEFINITIONS

The problem with the earlier mentioned definition ofvelocity is that Dt! 0, which means that in theextreme case of it attaining the value of zero, the mag-nitude of velocity reaches infinity. This breaches thevelocity limitation set at the speed of the stationarysystem, in other words, the speed of light. It would beasked naturally, what is wrong with that? The simplestexplanation would be: because physics is the study ofnature as perceived by humans, and only verifiable assuch, and that such observations are bounded by thespeed of light, then concepts such as infinite velocityare meaningless. If this view is adopted, then someunnatural conclusions can be drawn from some ofthe most fundamental axioms in physics. To demon-strate this, the following example is used.

Newton’s three axioms form the cornerstone ofphysics. Using his second axiom, and the samecalculus of variation as mentioned earlier

F ¼ ma ¼ mdv

dt(3)

The velocity is

ðdv ¼

F

m

ðdt (4)

or

v ¼F

mt þ C (5)

and choosing the boundary conditions v ¼ 0jt¼0,C ¼ 0, thus v ¼ (F/m)t.

For any small ratio of F/m (corresponding to aninappreciable force applied to a very massivebody), as long as sufficient time is allowed, t! 1,v! 1, which is simply not sensible.

This simple example, among others, meant thatEinstein’s special theory of relativity should beupheld. The special theory of relativity simplystates that velocity is limited by the speed of propa-gation of light in vacuo [4]. It further confirmsthe Galilean principle of relativity, which draws nodistinction between the state of absolute rest andabsolute motion, both of which are representationsof the speed of light in vacuo.

To correct the previously mentioned anomalousproblem and at the same time retain the Newtonianaxioms, it should be noted that the mass of the testentity is related to its velocity. Therefore, the specialtheory of relativity implies that as the velocity of theentity progressively increases, so should its resist-ance to motion. This is also a restatement of theGalilean principle of inertia [1]. Thus, mass of anentity is not a constant as treated earlier, but a func-tion of the state of its motion (say velocity). One canstate this simply as m ¼ f(v). An important con-clusion is that mass is a motion-like property, andit should be possible to describe it in terms of differ-ential geometry, like the kinematic quantities. Howdoes this affect Newton’s second axiom? Because

F ¼ ma ¼d(mv)

dt¼

d

dt( f (v)v) (6)

which means that

F

ðdt ¼

ðd( f (v)v) (7)

This is simply the impulse–momentum equation inwhich for a given impulse, the momentum shouldbe conserved. This means that an observer wouldnote an increasing resistance to the motion of theentity, if its velocity were to progressively increase,as suggested earlier. This means that its mass alsoappears to increase, although such will not be notedby the entity itself. Thus, mass, like other motion-likequantities, has a relativistic nature. The key point isthat it is not merely the simple traditional interpret-ation put upon its definition, and as yet it is notfundamentally understood. Nevertheless, to carryout the integration in the earlier equation, onewould need to know the function f(v) apriori.

Without going into a detailed historical review, itcan be noted that the way an observer perceivesthe velocity of a moving object, v, is relative to thespeed of observation itself (i.e. the speed of light, c).

Special relativity 135


Thus, if a beam of light is directed at an objectmoving away from an observer, the velocity of thebeam relative to it will be (c 2 v). When the beamreflects from this object and returns to the observer,then it would be (cþ v). It is clear that from the aver-age speed, the results obtained would be c. Thus, anyvariations in the velocity of the object, v, would beinsensible. The problem is in the way the transform-ation is made from the frame of the moving object tothat of the observation itself. This transformation isreferred to as the Galilean transformation in shear,and was noted in an attempt to measure the velocityof the Earth relative to the so-called aether in theMichelson–Morley experiment (readers are referredto any standard text book on the physics of motion).

It is, therefore, obvious that with the Galileantransformation, the irrational conclusion would bethat v ¼ 0 always. Subsequent to the Michelson–Morley experiment, it became clear that a new trans-formation was needed between the space–timecoordinates of a moving object X iji¼0!3 ; (t, x, y, z)and those of an observer j jjj¼0!3 ; (t, h, 6, z ). TheLorentzian transformation [2] shows that when thedistance moved by the object is, for example z, thisappears to be foreshortened to z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (v2=c2)

prelative

to the observer [4]. Of course, when v� c, then theobserver will measure the same actual distance.Otherwise, if the object has moved by a distance,z ¼ vt, the observer measures the same end pointas z ¼ z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (v2=c2

p)þ vt. The same relationship

can be obtained for all other spatial coordinates, thus

j j ¼X i � vX 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (v2=c2

p)

, i ¼ j ¼ 1! 3, X 0 ¼ t (8)

The relationship in the temporal sense is given as

t ¼z � vtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� (v2=c2p

)(9)

Therefore, the Lorentzian transformation embodiesthe relative nature of both space and time. The trans-formation function, Bv ¼ 1=


p, repre-

sents the velocity boost as the velocity of an objectnears that of the speed of light [2]. Using this trans-formation, it is obvious that like space and time,mass is also a relative quantity, and it transforms inthe same manner as them (using the principle ofpreservation of four- momenta: one in time andthree in space, see standard texts), thus

m ¼ f (v) ¼mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� (v2=c2)p (10)

where m is the rest mass of the entity. Mass of amoving object is perceived to increase dramatically

by an observer as it moves close to the speed oflight. This relationship suggests a relativistic massfunction of the form shown in Fig. 2.

This so-called definition for mass has certainappeal, in that it affords it a motion-like property,in line with properties of differential geometry.However, it seemingly provides it with no fundamen-tal physical meaning (i.e. the underlying reason forit). Using the definition in equation (10), it is clearthat the energy of matter is given as [6]

E ¼ mc2 ¼mc2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� (v2=c2)p (11)

which leads to the interchangeability of energy andmass, where clearly the quantity mc 2 is the restenergy of matter, implying an intrinsic property, forwhich no physical explanation can be made. Theconcept is not appealing to all because it removesfrom mass, the motion-like property that it affordedby the virtue of equation (10). It also demands theacceptance of point-like (atomistic) nature of massat the limit v ¼ 0 or rest Dt ¼ 0, which is beyondthe attainable increments of successive observationsor Dt ! 1=c. This last point makes the concept of restmass as an intrinsic quantity philosophically impon-derable, something that has concerned both schoolsof thought, positivist atomism (as in the CartesianSchool, see D’Alembert [7]) and rational empiricismof the field theorists (relativism) [8]. The problemis that although relativity affords mass a motion-like property, it does not tie it down explicitly tospace, time, or space–time geometry. This wouldbe the main reason for representatives of both theD’Alembert and Mach schools not accepting thetheory of relativity in its entirety, observing itshybrid nature with respect to the retention of massas an intrinsic property. To retain most of the

Fig. 2 Relativistic mass function

136 H Rahnejat


foregone discussion within the theory of relativity,there must be a linking between the matter andspace–time in the form of a closed field. The calculusof variation needs to be revisited to set the limits inline with the observable (i.e. banishment of conceptssuch as v! 1 and Dt! 0).

4 USE OF LIGHT CONE AND HYPERBOLICGEOMETRY

There is a certain elegance in the axiomatic nature andmathematical simplicity of Euclidean geometry. As aresult, when finally a link was established betweenthe kinematic quantities and thus force, momentum,and energy with geometry, it furnished them withthe same attributes. The axiomatic nature of Eucli-dean geometry is due to its affinity with human obser-vation, in other words, the Euclidean eyes. Whetherthis attribute is intrinsic to human nature or oneevolved through power of suggestion is beyond thescope of this discourse, but is nevertheless interesting.In any case, it does reveal its shortcomings, when deal-ing with the problems highlighted earlier (i.e. the per-ceived increasing mass of an object at very high speedsby an observer, and the problem with the Michelson–Morley experiment). Almost all every day phenomenaon the Earth deviate substantially from the speed oflight, and this makes the Euclidean geometry akin tohuman observations. When attempts are made toextend the same to those events (generally of electro-magnetic nature) using Euclidean geometry, its lim-ited nature provides rather surprising results, such asthose described earlier; the main reason being thebreach of the light cone.

Therefore, it became clear that a new form ofgeometry should be adopted to complement theLorentzian transformation. Note that the Lorentziantransformation was used to correct the Euclideanobservations to fit the facts. This led to the introduc-tion of hyperbolic geometry (a form of non-Euclidean geometry) by Minkowski, with the norm

[9], jjEjj ¼ j 02

� Si¼1!3ji2

� �1=2

, where an event E in

(1þ 3) space–time is defined as

f (E) ¼ c2j 02

�X

i¼1!3

j i2

E ; (j 0 ¼ t, j i, i ¼ 1! 3) (12)

This not only empowered the Lorentzian transform-ation to preserve the light cone c2j 02

� Si¼1!3ji2

¼ 0,but also defined the kinematics of the space–time asf(E) ¼ 0 (light-like), f(E).0 (time-like), and f(E ),0(space-like) events.

Thus, the hyperbolic geometry lays down a solidfoundation upon which the special theory of relati-vity resides, and with the embodied Lorentzian

transformation the conservation of four momentaand that of matter–energy is achieved.

However, one problem still remains: no explicitalteration is made to differential geometry. Eventhough hyperbolic geometry confines the time axis(as described earlier) within the light cone, Dt isallowed to diminish to zero in the definition oflimits in differential geometry. In other words, theinterval of time between two successive observationscan diminish beyond the value 1/c in the definitionof equation (2), and then be corrected later by theconstraints introduced by hyperbolic geometry.This is almost the same as using the Euclideangeometry, and then correcting the results of obser-vations to fit the physical facts via the Lorentziantransformation. Furthermore, the relative nature ofevents is ascertained with respect to given observersin hyperbolic geometry, thus losing the opportunityof using it as a unified framework. Such would notbe the case if all events were measured as deviationswith respect to a unified frame of observation, or thespeed of light, as this remains a constant for allobservers in relative motion. Differential geometrycan, therefore, be made affine to special relativity,and deviations from it can be described in line withthe general theory. This necessitates, first re-definingthe calculus of variation and second introducingappropriate space–matter fields.

5 CALCULUS OF VARIATION MADE AFFINETO SPECIAL RELATIVITY

In this way, proper use can be made of the profoundnature of hyperbolic geometry. The re-interpretationof differential geometry within the context of specialrelativity can simply be described as kinematics rela-tive to the speed of light rather than to any particularobserver.

Using hyperbolic geometry in (1þ 3) space–time,(j i), i ¼ 0! 3. For light-like behaviour [9]

(cj 0)2 ¼X3

i¼1

(j i)2 ¼ z 2 (13)

Note that j0 ¼ t. Thus, in the usual symbols,c 2t 2 ¼ z2. Hence, t ¼ (1/c)z, which corresponds tothe uniform motion of light in vacuo.

To seek a definition for velocity as a slope in differ-ential geometry and bounded by special relativity

v ¼@s

@t¼@cs

@z¼ c

@s

@zþ s

@c

@z¼ c

@s

@z(14)

for c considered to be constant. Note that @s=@z ¼ 1for light-like behaviour.



This definition is independent of time in an expli-cit sense, and where s is the arc length of motion. Thekey point to bear in mind is that time is nowembedded in an implicit manner within a scalarfield rcs, as described later.

This can be defined as

v ¼ limDz!1=cDs

Dz

� �(15)

which is clearly a valid definition for velocity as thatset by its instantaneous nature (slope in differentialgeometry) by Cauchy in 1821 [10]. It has, however,a much more profound meaning that has a closeraffinity to natural phenomenon by its axiomaticnature through universal observation. In otherwords, it defines the instantaneous velocity asbeing in harmony with the frame of observation(i.e. in simultaneity with speed of observation orthe speed of light). This underpins the notion thatsimultaneity is the same as the speed of the station-ary system, in other words, the speed of light. Thisdefinition, unlike the time-based definition, is under-pinned by the special theory of relativity in itsinception.

Now the local curvature provides the acceleration as

a ¼@

@t

@s

@t

� �¼@

@t

@cs

@z

� �¼@

@zc@cs

@z

� �

¼@c

@z

@cs

@zþ c

@2cs

@z2¼ v

@c

@zþ c

@

@z

@cs

@z

¼ v@c

@zþ c

@v

@z(16)

which, because the speed of light is assumed tobe independent of spatial location, z providesthe usual definition for acceleration as a ¼c(@v=@z) ¼ (@v=@t). However, it should be noted thatthis relationship is only true for light-like behaviour,and thus, the equality c(@v=@z ) ¼ (@v=@t) is notgenerally permitted. This point can be exploitedadvantageously, because for light and otherelectromagnetic radiation a ¼ v(@c=@z ), the secondterm on the right-hand side of equation (16)being c(@v=@z ) ¼ c(@=@z )(c(@s=@z )) ¼ c2(@2s=@z 2) ¼ 0,

as v! c, therefore, (@s=@z )! 1 (see equation (14)).This is defined as the slope of unity, or the light-likebehaviour, where a ¼ 0. The first term on theright-hand side of equation (16) will also be zero,unless the speed of light was to alter, for example, inthe presence of a strong gravitational field.

Away from light-like behaviour, time-like andspace-like characteristics are noted by the propertiesof the Minkowski norm. In general, f(E) in equation(12) can have any real value, with a positivevalue corresponding to time-like behaviour. Thus,

c2j 02

� z 2 ¼ f (E). This relationship describes allmaterial points within the light cone in the frameof reference of an inertial observer (j 0,

z ; j i, i ¼ 1! 3). Admitting only small quantities,the world-curve of the motion can be stated asc2dj 02

� dz 2 ¼ dt 2, where dt is the proper time of amaterial point in motion in relativity. The properfour-velocity of this material point is given as(@t=@t, @s=@t), where ds is the arc length of motionin the Euclidean geometry, corresponding to theproper time, dt. The term @t=@t is the local time dila-tion factor. Then, the spatial proper three-velocitycomponents can be found from

u ¼@s

@t¼@s

@t

@t

@t¼

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (v2=c2)

p @s

@t

¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� (v2=c2)p @cs

@z(17)

Note that u ¼ @s=@t ¼ (1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (v2=c2)

p(@cs=@z) ¼

(v=c)=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (v2=c2)

p, which is in line with both the

special case given by equation (14) and the Lorent-zian transformation [2]. Two important relationsalso result, @s=@t ¼ @s=@tjv � c and @s=@z ¼ (1=c)(@s=@t)jv! c. Note that the slope of unity is achievedby the second of these relations, while in the generalcase @s=@z ¼ (1=c)


p(@s=@t), which yields

the same equality as the earlier one for v! c, anddeviates from unity inversely in proportion to thelocal time dilation factor as @t=@t:

Now, the acceleration of this material point can beobtained by differentiation of equation (17), alsoemploying the definition in equation (16) as

a ¼@u

@t¼@

@t

@s

@t

� �¼@

@t

@t

@t

@s

@t

� �¼@t

@t

@

@t

@cs

@z

� �

¼@t

@t

@

@t

@t

@t

@cs

@z

� �¼@t

@t

@t

@t

@v

@t¼

@t

@t

� �2@v

@t

¼@t

@t

� �2

v@c

@zþ c

@v

@z

� �(18)

6 KINEMATIC QUANTITIES ASINCOMPLETENESS FROM UNIFIEDFRAME OF OBSERVATION

Definition (16) is a special case of equation (18),where in the latter, the proper time is taken intoaccount. Furthermore, in this case the deviationfrom the slope of unity constitutes a = 0. In otherwords, an accelerated motion yields curved space.Therefore, with the exception of perfect straightline motion of slope of unity (i.e. the light-like beha-viour), all other motions are curvilinear, with their

138 H Rahnejat


slopes as deviations from the light-like behaviour inaccordance with the quantity @t=@t. Thecorresponding local curvature is obtained by assum-ing the constancy of the speed of light in equation(18) (i.e. @c=@z ¼ 0) as @2s=@z2 ¼ (a=c2) @t=@tð Þ

2. There-fore, the motion of all material points in any region ofspace, z, within a weak gravitational field may bedescribed with respect to the light-like behaviourby a pair of numbers: its deviation from a slope ofunity, and its corresponding local curvature:I ; (I1, I2) ; ((@t=@t), (a=c2)(@t=@t)2), where the vec-tor I is hereinafter defined as the incompletenessof the observed motion, defined previously in geo-metrical terms in reference [11].

The term cs in equation (14) assumes the meaningof kinematic viscosity in the traditional sense, as ithas the units of m2/s (s being the displacement). Ofcourse, in the current definition here, all quantitiesare of geometric attributes. The definition, therefore,relates to the extension in a given spatial directionor the stretching of a curve (or matter striving todescribe a path). If, as earlier, it can be stretched toits limit of unity, it represents a straight line. In otherwords, the state of motion of matter to traverse sucha path endows to such matter a light-like property.

Now referring to equation (18), note that the curva-ture is defined as the combination of interactionsbetween the moving matter and the fabric of spaceand the action of light itself. (i.e. the manner inwhich light propagates in space). This becomes thedefinition for acceleration, which sits very naturallywith fundamentals of observation itself. The finalterm on the right-hand side of equation (18) is theusual observation model, indicating curvatureof motion as viewed by the speed of observa-tion (i.e. the speed of light). If put in theform c2(@t=@t)2(@2s=@z 2) ¼ (1=(1� (v2=c2)))(@2s=@z 2),it shows the deviation from light-like behaviouras observed by any observer in uniform motion.However, one can use the light-like slope of unity asa frame of measurement and use the defined incom-pleteness to note that with the constancy of thespeed of light I2¼ (a=c2)(@t=@t)2¼ (a=c2)(1� (v2=c2)),which means the slower the material point, thegreater the incompleteness, making it easier todiscriminate. This is simply axiomatic. Although thevalue of c in vacuo and on a geometric slope ofunity remains constant, in line with special relativity,there is no reason for it not to adhere to other curva-tures to facilitate observation of other phenomena,indicating its wave-like motion.

7 CLOSED LOOP SPACE –MATTER FIELD

The earlier mentioned relationships are kinematic innature. In other words, they are concerned with

observation of phenomena by an observer. Thismeans that the inseparability of space and matter isnot strictly embodied in them, which accounts forthe underlying cause of accelerated motion. By thepreceding arguments, a curvature in space is createdby the action of matter upon it. Because the unitarynature of matter must be refuted as contrary to evol-utionary dynamics, then its multiplicity can bedescribed, in its simplest representation, by the pre-sence of motion of a conglomerate of matter, creat-ing the curvatures of space. Now, using equation(18), a ¼ (dt=dt)2(@2c2s=@z2), which for motionsv � c! dt=dt � 1. The term c 2s can be interpretedas the resistance of space (note that cs has the con-ventional unit of kinematic viscosity as previouslydescribed) to the propagation of matter at thespeed of light (i.e. the electromagnetic waves). There-fore, it represents a closed loop space–matter fieldinterpretation. It is a closed loop field, because thepresence of all matter in such a space is implicitwithin the quantity c 2s, as opposed to an atomisticdefinition, which is based on the presence of discretemasses such as a gravitational source and targetbodies. The quantity c 2s, in conventional terms,has the unit of m3/s2. In the Newtonian atomisticview (i.e. the discrete action of bodies in the law ofuniversal gravitation), the product GM provides thebasis of interpretation and has the same unit.Einstein’s weak gravitational field theory is a halfway house, where an open loop field is assumed, asthe underlying source for the curved space–time.

In the conventional interpretation of gravitationalaction

a ¼@w

@z¼ rw (19)

where the simplest form of gravitational potential is agradient form, inversely proportional to the distancefrom the source, M, as w ¼ �GM=r. The negative signis due to the work done by the gravitational field. Asthe field varies from position to position in space (i.e.z ; j i, i ¼ 1! 3), equation (19) can be written in itsgeneral form as a ¼ �rw. Note that the vector field,a, does not depend on any target body of mass, m,but only on the source mass, M. This preciselydemonstrates the open loop nature of the conven-tional interpretation, explained earlier. Similarly, inthe closed loop interpretation

a ¼dt

dt

� �2@2c2s

@z2¼

dt

dt

� �2 @

@z

@c2s

@z

� �

¼dt

dt

� �2@w

@z¼

dt

dt

� �2

rw (20)



where rw ¼ (@w=@ji), i ¼ 1! 3, and in this casew ¼ @c2s=@z ¼ c2rs for c remaining constant inaccord with special relativity.

8 UNIVERSAL GRAVITATIONAL CONSTANTAND EINSTEIN’S RIPPLES

In the strictest terms, the results of the earlier twoalternative interpretations cannot be equated, asone is atomistic in nature, while the other relies ona continuous closed loop field. However, in caseswhere the mass of a source is the dominant factorfor the curved space–time as in the case of thesolar system (i.e. the Sun accounts for nearly 99 percent of the mass of the solar system), this equival-ence may be assumed. Thus, the gravitational poten-tial may be regarded as equivalent to that of thespace–matter field, described earlier

c2rs ;GM

r(21)

Now, this can lead to a physical description for theuniversal gravitational constant, noting that G ¼(c2=M)rrs, where rs is the slope at the location z ;j i, i ¼ 1! 3 at a distance r from the source M. Thismay be surmised as a wave-like or ripple motion,proposed by Einstein as it varies with distance r.Now the earlier mentioned relationship can berepresented as follows

G ¼c2r

Mrs ¼

E

Mmrrs (22)

This relationship provides a physical description forG: as the ripple created by a mass M remotely fromany locality (at a distance r), given by rs, wherematter of total energy E and mass m can reside.This description ties the curved space with the resid-ing matter and the gravitational action of a nearbymassive body. For G to be regarded as a constant(which is in line with all measurements of orbitalmotions in the solar system), such systems shouldpossess a massive body and rs should be almostinsensible, which, as evidence suggests, is the case.Using equation (22), the ripple amplitude causedon the space proximate to the gravitational field ofthe Earth may be surmised. For example, at a dis-tance of 1000 km from the surface of the Earth, withG ¼ 6.673 � 10211 m3/kg s2, mass of Earth ¼ 5.976�1024 kg, its radius ¼ 6371 km and assumingc ¼ 300 000 km/s, the amplitude of ripple is sur-mised to be 6 nm. However, it should be noted thatthe ripple in any spatial location is determined bythe action of a many-body system. In the case of alocation described earlier, the gravitational potential

of the Sun cannot be ignored. If the same relation-ship was to be used in the case of the Sun’s gravita-tional potential alone, with its mass and distance as1.989 � 1030 kg and 1.5 � 108 km, respectively, thenthe ripple amplitude obtained is 10 nm. The import-ant point to note is that any object creates ripples inspace as it moves and at any location in a system, theripple amplitude is an amalgam of these constitu-ents, and not necessarily in an additive manner.

To observe why G is regarded as a constant, therelativistic nature of the mass in equation (22) canbe replaced, using equation (10). Then

G ¼c2

Mrrs ¼

c2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (v2=c2)

pm

rrs

’ c2 1�1

2

v2

c2�

1

8

v4

c4� � � �

� �rrs

m(23)

It is clear that for massive bodies moving atv � c, G � (c2rrs=m), which is independent of thestate of their motion and that G is of electromagneticnature (thus a universal constant). It is also aproperty of the closed field, c2rs.

9 SPECIAL RELATIVITY AND SINGULARITYDUE TO THE LORENTZIAN BOOST

A problem still remaining, among others, with theapproach expounded is retaining the Lorentzianboost function. This can still lead to the attainmentof infinitum, for example in the evaluation of rsfrom equations (22) or (23), where for v! c,

rs! 1. The main reason for this is the acceptanceof the notion of mass in its traditional atomisticsense. Furthermore, c2rs cannot be claimed as thespace–matter field, if mass remains alongside it asa discrete entity. This would indeed be the sourceof objection by the proponents of both the atomisticschool of thought and the field theorists alike. Thecurrent exposition favours the latter, while sympathyalso exists with D’Alembert’s view of the imponder-able nature of mass as an entity [7, 8].

The most poignant manifestation of the singularityat the heart of the theory of relativity is the Lorent-zian boost (1=


p), which shows that as

v! c, Bv ! 1. Thus, many important relations,discussed earlier, including time dilation and relativemass tend to infinity. These are the result of the con-straint imposed by the constancy of speed of light invacuo, or in other words: special relativity. The grow-ing trend in disputing special relativity has comeabout as a result of a desire to remove this sourceof singularity, thus extending the theory of relativity,which is felt by many to be the fundamental basisupon which the observed nature resides. Some

140 H Rahnejat


have resorted to observations of accelerated motionof light in vacuo in certain circumstances as anoma-lous dispersion [12], although Einstein himself hadonly accepted the same within a gravitational field.If the notion of a closed loop space–matter field, asdescribed above, is accepted then it can be shownthat such concept as vacuum has no place sincespace itself cannot exist in such circumstances. Toshow this, tidal motion in vacuo must be considered.

In the traditional atomistic view, tidal accelerationcreated by a source may be defined by derivatives@a=@j i, i ¼ 1! 3. Using equation (19), this can berepresented in terms of the potential as @a=@j i ¼

@2w=@j i@j j, i, j ¼ 1! 3, yielding a 3�3 matrix forr2w. At the origin of the source, r2w ¼ 0 for i = j.Thus, only the diagonal terms have a finite value.Now, using the Newtonian gravitational potential, itis simple to show that r2w ¼ 0 (the Laplacian of thepotential) at any location in space z ¼ ji, i ¼ 1! 3,away from any source. This is, in fact, the vacuumfield equation. In the closed field view of space, theanalysis has to be extended to the case of a distri-bution of matter in order to remain in line with thespace–matter field. In this case, r2w ¼ r†rw ¼ r†a(using equation (21)). Note that the product r†r ¼

r2 is defined as the inner product of the vector differ-ential operator, thus, r† a is the divergence of a.Hence, empty space can be defined as the place,where the average tidal acceleration is given byr†a ¼ r†rw ¼ c2r†r2s ¼ c2r3s ¼ 0, when the con-stancy of speed of light is upheld. Because c = 0,then r3s ¼ 0, or the Laplacian r2s is finite. Thismeans that passage of light creates the space,which otherwise will have a null content (see sub-sequently). For the limiting case of the slope ofunity, this is regarded as the theoretical perfection(completeness, (I 1, I 2) ¼ (1, 0), see section 5, andalso reference [11]). In special relativity, this con-dition is designated the vacuo. In incompleteness,such a space is only created by the passage of light.

This simple analysis qualifies the concept of vacuo,where in this case, it is replaced by the notion of nullspace: one containing no matter (where the sanctityof inseparability of matter and space is thus upheld).When matter enters a place, space is created, andvice-versa, space must exist as a condition for thepresence of matter. Therefore, vacuum in itstraditional sense is not the same as the null space.There is a noteworthy subtle difference here. Inrelativity, the speed of light is constant in vacuo,but in incompleteness, slope of unity remainsmerely a theoretical perfection in a space where noother matter but light can reside. It may then be sur-mised that such defined null space is created by thepassage of light and ceases to exist without it. Toprovide some theoretical basis for this, the relationsin the preceding paragraph can be investigated

again for the general case of the space–matterfield, where for rc = 0, rw ¼ vrc þ crv (see equa-tions (18) and (20), and note that in general,w ¼ cv). Now

r†a ¼ r†rw ¼ r†(vrc þ crv)

¼ vr2c þ cr2v ¼ 0 (24)

The obvious solution for this empty space, other thanfor presence of light is v! c and r2v ¼ 2r2c, andnoting that jjr2cjj,0 indicates that rc must attainits maximum value to reach the slope of unity. Inincompleteness, a space is created by this action oflight, whereas in special relativity, this location rep-resents a form of vacuo.

The boost has now the form

Bv ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� (v2=c2)p ¼

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (1=c2){c(@s=@z)þ s(@s=@z)}2

p¼

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (1=c2){crsþ src}2

p (25)

The first term in the curly brackets is the restatementof the special theory, where in vacuo, and in line withthe incompleteness theorem: rs ¼ 1. If the secondterm in the curly bracket is ignored, then the singu-larity in dilation time and the relativistic mass func-tion would be reached. The second term, however,small (and this will always be the case, except forthe case of a very strong gravitational field nearby,such as a black hole) is finite, and close examinationshows that for proper determinism

rs , 1�s

crc, rc , 0, s = 0 (26)

The first condition states that the slope of unity ismerely theoretical, as no other matter than lightshould reside here. This is also qualified by the argu-ments in the preceding section, where the maximumvalue for

rc¼ 0: rs! 1 and w! c2 (v! c) (27)

One can surmise that in the absence of matter andany gravitational field, the value of s! 1. Astronom-ical observations do not support this. However, theyshow large areas of empty space, presumed to beoccupied by the so-called grey matter. Speculationshould not be favoured, but an equally valid postu-late to that of the presence of grey matter can beput forward here, as follows. The Sun’s gravitationalpotential in the vicinity of the Earth is ��8:85�108 m2=s2 and declines within the solar system to21.84�107 m2/s2 in the vicinity of Pluto at its furthest



orbital position (the negative signs indicate theattractive nature of the gravitational potential).Note that the potential in light propagation in slopeof unity, creating space is w¼ c2¼ 9�1016 m2=s2,which is much larger than the gravitational potentialto bring the matter together, even near a star thou-sands of times larger than our Sun. Therefore, itmay be surmised that wherever the speed of lightexceeds that governed by a nearby gravitationalfield, expansive (or dispersive) action will take placein the space. The creation of this space will pushthe galaxies apart, as indeed observed by the astron-omers. Whether other matter can enter this domainis a question of conjecture. It should be noted thatthe quantity cs, described in section 6, will assumea very large value, and breaching the required poten-tial to enter this domain appears to be foreboding.Inferring from Einstein’s analogy of the fabric ofspace–time as a warped sheet due to the action ofpotential fields, a field such as that proposed hereis likely to stretch it beyond endurance. Therefore,the alternative proposal to large areas of grey mattermay be tears in the fabric of space. As the light nearsany significant gravitational field rc , 0, and its dis-persive potential attenuates. This is the reason forthe second condition in equation (26), which alsofurther underpins the first stated condition.

Another extreme condition is when rs! 0 and(s2=c2)(rc)2 ! 1. Thus, rc=c! 1=s, which is thelimiting case in the first condition in equation (26).The ratio rc=c is the curvature of a ray of light, asshown in the next paragraph (note that it is a nega-tive quantity). Therefore, in this limiting case (usingequation (30)) ru! 1=s. Note from equation (1)that ru ¼ Du=Ds ¼ 1=R. Therefore, this limiting con-dition points to a place, where the curvature of aray of light adheres completely to the curvature ofspace, and with rs! 0, s ¼ R! 0 (and both beingnegative), which is presumably determined by avery strong gravitational field. Now using the generalcase of the space–matter field potential for this case

w¼ cv¼ c(crsþ src)¼ c2rsþ csrc

! csrc¼ c2src

c¼ c2s �

1

s

� �¼�c2jrs!0 (28)

It can be observed that the space–matter field poten-tial has an attractive nature under this condition.Equations (27) and (28) specify the bounds of thedescribed potential, �c2 ,w, c2. At the upperlimit, uncontrolled dispersion occurs in the con-verted null space created by the passage of lightalone. At the lower bound, the curve of spacediminishes. This is interpreted as a singular pointwith infinite matter. However, at the very limit ofpotential attaining its lower bound, space cannotexist as shown in equation (28). Thus,

electromagnetic radiation from such a locationmust inhibit this condition to arise. This suggeststhat blackholes should radiate energy, which con-forms to recent observations by Integral, ESA’sgamma ray observatory [13], which has shown thatblack holes at the centre of each galaxy, such as theSagittarius A-star at the centre of Milky Way act likecosmic vacuum cleaners, drawing in gas and mattertrapped due to their large gravitational pull. Thematter is then crushed and heated until outwardbursts of X-rays and gamma rays take place.

Using equation (28) to represent the boost in termsof the potential w, the second term in the radicalbecomes. (1=c2){crs þ src}2 ¼ (1=c2){w=c}2 ¼ {w=c2}2,and the bounded potential (justified by the earlier dis-cussion) prohibits the attainment of singularity ateither of the extreme cases described here. All otherphenomena fall in between the two extremes describedearlier, for all of which the term in the curly brackets inequation (25) conforms to the conditions in equation(26). Thus, the singularity is removed from theLorentzian boost function. All that now remains to beaddressed is to define mass within the closed loopspace–matter field interpretation.

Light propagates in space as a series of wavefronts.Owing to the wave-particle duality interpretation puton the propagating nature of light, its rays are con-sidered as paths described by photons of light, andremain orthogonal to the wavefronts at any givenlocation in the field. By virtue of the principle ofincompleteness everywhere in ordinary space thewavefronts are curved, and thus the speed of lightmust alter from point to point within the field, eventhough this may be by an inappreciable amount.This argument is in line with the general theory ofrelativity and also the Huygens’ principle [14].Furthermore, referring to the foregone arguments,it is clear that the speed of light will alter in all suchordinary spaces (spaces other than the defined nullspace to replace notions of vacuo or empty space).This is true, because the variation of speed of lightin gravitational fields and in passage through differ-ent media is now well established. Two very proxi-mate locations, z1 and z1 þ Dz on a currentwavefront can be considered. The new position ofthis front is considered at the short interval of Dt,as shown in Fig. 3. The speed of light is consideredto be different at these locations according to theHuygens’ principle and the corresponding rays oflight are shown in the figure. Therefore, the front isslanted (not parallel) with respect to its precedingposition. This being the case, there is a small anglebetween the rays of light from these very proximatelocations (i.e. Du). Thus

Du � sin Du ¼c(z1)Dt � c(z1 þ Dz )Dt

Dz(29)

142 H Rahnejat


The initial wavefront is taken as a straight line, corre-sponding to the theoretical limit of rs! 1, indi-cating that the rays of light are in transition to anordinary field. The overall spatial coordinate is z ¼

j i, i ¼ 1! 3 and as shown in section 2 in the limit-ing case Dt ! (1=c)Dz. It follows from equation (27)that in the limit, c(du=dz) ¼ �dc=dz, or

ru ¼ �rc

c(30)

Note that the motion of rays of light in all ordinaryspaces (other than the described null space) is dece-lerative in nature.

10 INTERPRETATION OF MASS IN THECLOSED SPACE–MATTER FIELD

With the singularity removed from the Lorentzianboost function for all matter, the relativistic natureof mass in equation (10) can be retained with thenew description of the boost given in the previoussection. However, two problems remain. First, adefinition is required for the rest mass and second,the atomistic nature of mass should also be aban-doned to sit comfortably within the closed loopspace–matter field concept.

Because no place in relativistic space–time andwithin incompleteness is regarded as static, it isclear that the concept of rest mass is merely retainedas a mathematical convenience and has no physicalmeaning or reason. Returning to equation (25), it isclear that for the boost to attain a value of unity

rs

s¼rc

c, rc , 0 (31)

The second condition mentioned earlier has beenjustified in the previous sections. The first condi-tion states that the curvature of motion at a givenlocation is the same as that of the bending of lightrays. Wherever this condition holds, the mass of theobject and that regarded as its rest mass are one

and the same. Clearly, the curvatures in equation(31) are in the same field, induced by the divergencer† a, where by the virtue of Mach’s principle theacceleration is due to the presence of all the matterin the field. For the condition given in equation(31), a balance is found between the light propa-gation field potential and the gravitational field ofmatter within the field. For this condition

w ¼ c2rs ¼ c2src

c

� �(32)

Noting that the Sun accounts for �99 per cent of allmatter in the solar system, this equality was used insection 8. Here, it can be noted that equation (23)with the use of the earlier mentioned relationshipyields

M ¼c2sr

G

rc

c

�� (33)

This relationship provides an atomistic estimate ofmass of a closed space–matter field system as deter-mined from a specific location within the field, wherethe balance described in equation (31) can beassumed. Here, Bv ¼ 1, M ¼ m. Because for mostfields Bv is close to unity, the concept of rest massassumes the same as the mass itself for them. Allparameters in equation (33) are measurable andthus the concentrated equivalent mass of the fieldcan be determined. As regards the field interpret-ation (or local mass within a given location in thefield), it is not necessary to be concerned at all withthis, as shown in the following paragraph.

Like D’Alembert, Mach could not accept theatomistic concept of mass as an intrinsic propertyof matter [15]. In Mach’s interpretation, mass ofan object is the result of its many interactions withother surrounding matter. This is a relativisticinterpretation, except that, for one thing, it was con-ceived before the theory of relativity and for another,it requires a closed loop field interpretation, which isnot a necessary condition in relativity. However, it isprecisely in accord with the interpretation putforward here. With Mach’s interpretation, asexpounded by Einstein

�m ¼m1

m2¼ C �F ¼ C

F1

F2(34)

Now a location in space in such a closed field may beassumed, where a body resides and the condition inequation (31) holds true, and the same body inanother location within the same field, where thiscondition is not upheld, then clearly the ratio:�m = 1, and thus using Mach’s interpretation C / Bv

in equation (34). It is clear that it is not necessaryto be concerned about mass to describe theconditions locally within the field, which is

Fig. 3 Bending of light rays



completely defined by Bv and the field in equation(24), where

a ¼ rw

r2w ¼ vr2c þ cr2v

where

v ¼ crs þ src (35)

Nevertheless, the earlier arguments suggest thatm/ 1=I1, and for I1 !+1, �m ¼ 1. Thus, one caninterpret rest mass as the maximum incompleteness.At the other end of the scale, mass of matter tends tonothing, as indeed is the case for photons of light andnearly all elementary particles such as electrons,where I1 ! 0 (i.e. absolute motion). Note that theseare measured with respect to the frame of obser-vation (being universal in nature), and not withrespect to any particular observer. Mathematically

I1 ¼@t

@t¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

w

c2

� �2r

¼ 1�1

2

w

c2

� �2

�1

8

w

c2

� �4

� � � �

� �

w �! 0 I1 �! 1

w �!+c2 I1 �! 0 (36)

In this interpretation, incompleteness of motiongives rise to a sense of discerning matter by a quan-tity chosen to be called mass, which is an entirelymotion like property, described by the way matteris compelled to move in order to remain on curva-tures of space predefined by the presence of allother matter and the local electromagnetic potentialof light there. Therefore, elsewhere (other than forconditions in equation (36): rw = 0. The curvaturesof space are defined by the incompleteness, I2 (seesection 6), where in general

I2 ¼rc

cþrv

v

� �¼

a

cv(I1)2 ¼

rw

w(37)

The ratio rw=w is the curvature of the local potentialfield. For uniform motion of matter I2 ¼ 0, whichrelates to absolute rest or absolute motion only. Itshows that for the former condition in equation(36) krv=vk ¼ krc=ck, rc , 0, which means thatthe curvature of motion of matter at such locationsis the same as the curvature of light and electro-magnetic radiation. This suggests w! 0 fromequation (36) (an equilibrium between dispersiveelectromagnetic and local gravitational potentials).These are locations, which demarcate the boundarybetween electromagnetic radiation and gravitationalimplosion, such as near the surface of the Sun. Thiscondition is the general case of that described in sec-tion 9, where by virtue of equation (28), w!�c2,where absolute rest can only theoretically exist.

Thus, for w . 0, the curvature of the field is positiveand all objects are in a propagating field (increasingkinetic energy). Conversely, for w , 0 there is anincreasing stored potential, which is due to gravita-tion (an increasing mass).

When I2= 0, non-uniform motions (rw = 0) take

place on curvatures created by the presence of a con-glomerate of matter. If curvature of space with anincompleteness I2 ¼ 0 is thought of as a masslesstaut rope, then any ordinary curved space, given byequation (37), is as a result of a conglomerate ofmatter, which may be regarded as mass. Then, I2

= 0provides the curvature or sagging of the rope. Thisinterpretation indicates that

m/1

I1I2¼

1

I1

w

rw(38)

which indicates that mass is a form of local stretch-ing of the rope (note that 1=I2 is the radius of curva-ture of the potential field, and 1=I1 ¼ Bv describes thelocal kinematic conditions). Thus, this is a geometricextension of the space–matter field, a notion whichwas paradoxically first introduced by Descartes[16]. Now using Einstein’s famous equationE ¼ mc2, and noting that w!+c2 in the limitingcases described earlier, it is clear that mass can beinterpreted as

m ¼1

I1

1

rw¼

Bv

rw¼

1

rw1þ

1

2

w

c2

� �2

þ � � �

� �

¼1

rw1þ

1

2

v

c

� �2

þ � � �

� �(39)

This can become the definition for the local mass,which has the unit of s2=m: the rate of sagging ofthe curve, which may be expressed in kilogram.Using Einstein’s equation

EI1 rw

w¼ 1 (40)

With m given by equation (39), one would arrive atEinstein’s equation with w! c2. It should be notedthat, in general, w ¼ cv, unless all mass is convertedto pure energy. With net gravitational implosionw , 0, indicating that some of the energy is con-verted into mass. This would be very small unlesskvk ! c, v , 0, and in the extreme case, w!�c2

means that all energy is converted to mass. Thesetwo extreme conditions are unattainable asdescribed earlier. It is also clear that for energy–mass balance to hold

I2� ¼ kI2 jw,0 k¼ kI2 jw.0 k¼ I2þ

I2�� I2þ= 0:rw��rwþ= 0

I1 . 0 for v!+c (41)

144 H Rahnejat


Thus, the imploding matter towards a negativepotential (i.e. gravitational) should undergo fusion,and the energy released would propagate electro-magnetically with a net positive potential. Thus, theextreme case of w!�c2 cannot be reached, sincethe converted null space, described in section 9necessitates rs = 0. The value of rc¼ 0 is alsounattainable, and energy will convert to massimmediately that such a space is created (meaningthat the limit w! c2 is also theoretical). Therefore,black holes should radiate electromagnetic energyin planes orthogonal to the curvature of their eventhorizon, as indeed observed [13].

It is more natural in the analogy of the taut mass-less rope put forward above to use an equality inequation (38) to define mass, which would nowretain the form m ¼ (1=I1)(w=rw), which is purelygeometrical and has the conventional unit of metre,in which case E ¼ m.

It is clear that Newton’s second axiom can alsobe put in geometric form, using equations (20) and(39) as

(I1)3F ¼ 1 (42)

As in the case of relativity, force has no dimensions.One may represent it as F ¼ (1=I1rw)B2

vrw ¼

(a=I1rw) ¼ ma and with the earlier definition formass, it retains the usual unit of N.

Now referring back to Fig. 1 and noting that forconvenience the tangent AD may be regarded asthe slope of unity (the theoretical perfection), thecurved motions occur with respect to this slope. Thiscauses the sagging of the taut rope in the aforemen-tioned analogy (note that the curve of motion is unli-kely to be circular). BD is the subtense of the angle ofdeviation from the slope of unity and is the versedsine of the arc AB, shown as AC. The deviation BDtends to AC in the limit, where clearly according tosection 10, rc! 0. By virtue of corollary IV in thePrincipia [5], these displacements are caused by theimpulses acting at point B, and it is clear thatin the limit, here set as Dt ! 1=c, they will have acontinuous nature. These impulses are interpretedas the centripetal force in the Principia, being equiv-alent to the force F in equation (42). Because theincompleteness is measured from the slope ofunity, where the converted space (mentioned earlier)is void of resistance to electromagnetic motion, otherbodies are drawn away to undertake curvilinearpaths, described by I2. The ratio of motions of thesebodies to that on the slope of unity is the versedsine of the arc AB, as described earlier. This meansthat the ratio of intervals of local time to the timeof the stationary system should be given by(I1)3 ¼ {1� (w=c2)2}3=2, which is precisely that in

equation (42). This, in fact, is the general case of aparticle/body in curvilinear orbital motion about amassive body (relative to the particle itself ), subjectto a rectilinear motion due to the action of significantexternal forces, in which the particle describes areasalong the orbit in equal times (i.e. all the observedplanetary motions). An important point to note isthe unified nature of equation (42), irrespective ofthe form of potential w. It indicates that force is alocal interpretation of geometry of space as a devi-ation from uniformity, or in other words, as anincompleteness.

11 CONCLUDING REMARKS

The special theory of relativity declares the con-stancy of speed of light in vacuo and the Galileanprinciple of relativity both as laws of physics. Tolimit the motion of all matter within the light cone,and remain true to the principle of relativity, theunbounded Galilean transformation was replacedby the Lorentzian boost. The introduction of thespecial theory of relativity had profound effect inrationalization of many unnatural conclusions ofNewtonian physics at limiting conditions, anumber of which are referred to in section 3.However, the use of the Lorentzian boost function,together with the constancy of the speed of lightled to the retention of singular behaviour at limitsof motion. Einstein’s quest to extend the theory tothe case of accelerated motion in non-inertialframes of reference led him to abandon his atomisticview. Through the exposition of the special theory hehad unified the action of electromagnetic fields withthe atomistic nature of the classical physics. He wasstruck by the views of Mach, and interpreted them inthe form of Mach’s principle, described in the pre-vious section. This allowed him to develop a newtheory for gravity, finally leading to curved space–time and motion in non-inertial frames of reference(i.e. the general theory of relativity [17]). Einstein,as a field theorist, favoured the description ofspace–time in terms of action of weak gravitationalfields, and accepted the limitations of the law of con-stancy of the speed of light. However, the retention ofthe Lorentzian boost function kept the concept ofatomism at the heart of the special theory. Thisemanated from his desire to describe all forms ofpotential action by a unified theory for gravity. Theopen loop nature of such a field, and the lack of anexplanation for observed dispersive action have leftthe special theory open to some interpretations,such as anamolous dispersion of accelerated pulsesof light. For one thing, the aforementioned singularbehaviour had remained. For another, the atomisticnature of the special theory somewhat decouples it



from the general theory through retention of some(in the words of D’Alembert and interpretationsof Mach) imponderable quantities such as mass.Furthermore, some opportunities may not havebeen realized. For example, using the Galileanprinciple of relativity (noting that there is no distinc-tion between the states of absolute rest andabsolute motion) instantaneity was set as the timeof the stationary system, but this was not used toreplace the concept of time as the interaction oflight with space, which would necessitate its devi-ation from that in vacuo. In this manner, timewould be implicit in the fabric of space, and mani-fested by variations in speed of light at any position,as indeed is observed as rays of light bend in travel-ling through different media. This undertakingnecessitates the description of a closed loop space–matter field, in accord with Mach’s principle, whichrenders this principle the status of a law of physics.The aforementioned field will then be bounded inline with the light cone and behave as a dispersiveaction in what is defined as a converted null spaceto replace the concept of vacuo at one extreme, andstrongly attractive at the other extreme to accommo-date very strong gravitational fields. In between, thefield is an amalgam of gravitating action of matterand dispersive nature of electromagnetic potentialin the form of light. The behaviour of light in suchfields conforms to Huygens’ principle. Theseadditions, embodied in the principle of incomplete-ness result from an interpretation of the specialtheory and underpin the same to lead naturally tothe general theory as any theory for kinematicsshould extend to the general case of dynamics.

REFERENCES

1 Galilei, G. Discourses Concerning Two New Sciences,1638 (Elsevier, Leiden).

2 Lorentz, H. Versuch einer Theorie der elektrischen undoptischen Erscheinungenin bewegten Korpen, 1895(E. J. Brill, Leiden).

3 Maxwell, J. C. A Dynamical Theory of the Electro-magnetic Field, 1865 (Royal Society, London).

4 Einstein, A. Zur Elektrodynamik bewegten Korper.Ann. Phys., 1905, 17, 891–921.

5 Newton, I. Philosophiae Naturalis Principia Mathe-matica, 1687 (Royal Society, London).

6 Einstein, A. Ist die Traegheit eines Koerpers vonseinem Energieinhalt abhaengig. Ann. Phys., 1905, 18,639–641.

7 D’Alembert, J. Le R. Traite de dynamique, 1743(Academie de Sciences, Paris).

8 Rahnejat, H. Foreword – a tribute to Jean D’Alembertand Albert Einstein: geometrical interpretation ofmotion: an ironic legacy of apparently irreconcilable

atomistic and continuum philosophies. In Multi-bodyDynamics: Monitoring and Simulation Techniques – III,(Professional Engineering Publishing Ltd, Bury StEdmunds and London, UK).

9 Minkowski, H. Space and time. In The Principle ofRelativity, 1952 (Dover Publication, New York).

10 Cauchy, A. L. Cours d’analyse, 1821 (Ecole Polytech-nique, Paris).

11 Rahnejat, H. Relativity: 300 years from a principle toreality. Proc. Instn Mech. Engrs, Part K: J. Multi-bodyDynamics, 2001, 215, 1–6.

12 Wang, L. J., Kuzmich, A., and Dogariu, A. Gain-assistedsuperluminal light propagation. Nat. Lett., 2000, 406,277–279.

13 Revnivtsev, M. G., Churazov, E. M., Sazonov, S. Y.,Sunyaev, R. A., Lutovinov, A. A., Gilfanov, M. R.,Vikhlinin, A. A., Shtykovsky, P. E., and Pavlinsky,M. N. Hard X-ray view of the past activity of Sgr A� ina natural Compton mirror. Eur. J. Astron. Astrophys.,2004.

14 Huygens, C. H. R. Traite de la lumiere, 1690 (Van derAa, Leiden).

15 Mach, E. Die Mechanik in ihrer Entwicklung: histerisch-kritisch dagerstellt, 1883 (Brockhaus, Leipzig).

16 Descartes, R. Principia Philosophiae, 1644 (Elsevier,Amsterdam).

17 Einstein, A. Die Grundlagen der allgemeinen Relativi-tatstheorie. Ann. Phys., 1916, 49, 769–822.

APPENDIX

Notation

a accelerationBv Lorentzian boostc speed of lightE energyF forceG universal gravitational constantI 1 first incompleteness ¼ Bv

21

I 2 second incompletenessM source massm massR radiusr distance from source masss displacement or arc of motiont timev velocity

Dt increment of timew potentialm rest masst local timej i time coordinate, i ¼ 0z ¼ j i spatial coordinates, i ¼ 1! 3r gradientr† divergencer

2 Laplacian

146 H Rahnejat


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