7Chapter 1

A BRIEF ACCOUNT OF TIMENow entertain conjecture of a TIME,when creeping murmur and the poringdark fills the wide vessel of the Universe SHAKESPEARE(King Henry the Fifth)1.0 IntroductionTime has a central role in mathematics and all mathematical sciences, aswell as in history, religion etc. Yet it is one of the most baffling and mysteriousconcepts, on which the scientists have contradictory views: (i) existence ofabsolute time (ii) existence of local times only (iii) existence of both proper timeand local time (iv) truth of (i) to (iii). Newton put forward a theory of howbodies move in space and time. Philosophers argue that time is a basic propertyof the universe while some argue that it is an illusion or a property of the humanmind. Relativity and quantum mechanics have shown conflicting complexities inthe interpretation and meaning of time. In this chapter, we examine the varioustime-concepts such as absolute time, local time, retarded/advanced time andproper time, usually found in books on classical dynamics and relativity. As thefirst proposition, we prove that the concept of retarded/advanced time correctlyleads to the linear LT, and the concept of proper time τ. This can be used as thesubstitute for the absolute time, in defining velocity, acceleration etc. It is shownthat the Lorentz’ version of the retarded/advanced time is flawless whereas theweaker form in which it is restricted to -axis, along which two observers are inrelative motion, needs the additional law of the Fitzgerald-Larmour-Lorentzcontraction, which we prove as proposition 2. As the third proposition, we showthat the postulates of STR do not lead to the linear LT, but to a non-linear

8transformation. We examine some of the logical fallacies, such as the wrong useof retarded time/advanced time in the Einstein’s 1905 paper “On the Electro-Dynamics of Moving Bodies”. Next, we examine the relativistic time-concept, andfinally we review the literature on the SRT. In our discussions, is the maximumsignal velocity. In the case of electromagnetism, it is the speed ofelectromagnetic waves/photons, and in the case of gravito-dynamics, it is thespeed of gravitational waves/gravitons. Since signal-velocity depends on ,for gravitation and , for electro-magnetism, need not be a universalconstant. The main contents of this chapter are adaptations from the publishedpapers [*1] and [*2] of the researcher.1.1 Classical Concept of Time

A system of reference is a coordinate system, serving to indicate theposition and time of a particle in space-time, by means of measuring rods andidentical clocks showing the same pointer-positions, distributed throughout thespace. A frame is said to be inertial, if moving bodies, not acted upon by externalforces or acceleration, move with constant velocity. When we consider themotion of a free particle in inertial frame, it will be assumed that the space of theinertial frame is homogeneous and isotropic and the time-coordinate isassumed homogeneous. Let the space-time event in a system S with origin O bedenoted by O( , , , ) and let Sbe a system O′( , , , ′) in which O′ movesalong the positive -axis from O with constant velocity , the ′ and ′ axesbeing always parallel to the and respectively.

9The two systems of coordinates are related by the Galileantransformation = + , = , = , ′ = . In classical dynamics, time isconsidered absolute, which elapses uniformly, independent of other influences.If clocks of identical structure showing the same pointer-time, running at thesame rate, is distributed throughout a reference frame, some measuring deviceattached to each such point can determine the time of occurrence of any event ata point P. Since is a constant, both S and S′ are inertial relative to each otherbut if they are accelerated relative to a third frame, then they are not inertialwith respect to the third one. Thus the ‘times’ of an event are equal in all inertialframes and hence = ′ holds. The classical Galilean principle of relativitystates that all the laws of classical dynamics are invariant with respect toGalilean transformations and if, the Newton’s second law of motion F = holdsin an inertial frame, then it will hold in all inertial frames. Here P is themomentum of a moving body and F the external force acting on it. Further, inclassical dynamics, the force of inertia between two inter-acting bodies dependsonly on the spatial distance between the bodies, but not on the time. This meansthat interactions between bodies take place instantaneously and hence Newton’sLaw of Gravitation is a static law. It has to be modified to a dynamic form asLorentz did in the case of electro-dynamics.Roamer’s remarkable studies in 1675 in the eclipses of Jupiter and Saturnmade it clear that light takes time to travel, to be propagated from one point inspace to another. This fact has been accepted by pre-relativistic physics, longbefore the relativistic ideas were presented, in the form of retardedtime/advanced time. From this principle, the length contraction and time

10dilation can be proved. These are the basics of the pre-relativistic period. Themain contribution of the 19th century is the Maxwell-Lorentz equations ofelectro-dynamics. According to the theory, the motion of electron differs fromthe classical mass-particle. By the experiments in electro-magnetism, it isapparently concluded that the electro-magnetic signals/waves propagate in alldirections with the same velocity in empty space irrespective of the motion ofthe source. This assertion is in disagreement with the classical Galilean principleof relativity and the time relation = . In other words, the electro-dynamical equations are not co-variant with respect to the Galileantransformation. Maxwell, Lorentz and their contemporaries assumed that thereis one privileged frame, the ether frame, in which the above experimental truthholds. According to the classical Galilean principle of relativity, the addition ruleof velocity is = + . By letting = we have = − . Based on thisequation it is physically possible to determine the velocity of the earth in itsmotion through the ether under correct conditions. In 1887, the Michelson-Morley experiment [15] was conducted to ascertain the velocity of ether-windagainst the motion of the earth through the ether. The experiment had thesevere defect that it was conducted on the surface of the earth, by ignoringentrainment of ether. The experiment could not conclude anything in favour oragainst the existence of ether. However, the experiment led many scientists tosuspect the addition rule of velocity and the Galilean transformation.1.2 Michelson-Morley Experiment

A ray of light from the source L meets a plate of glass GG at an angle ofat the point O. The ray is partly reflected to the mirror M and partly refracted

11towards mirror M , 1M and M being equidistant from O at a distance . Afterreflection at 1M and 2M the rays return to O. The returned rays are partlydeflected to L and partly to the eyepiece E. On their way from O to E, the rayswill interfere with each other and the observer at E will observe an interferenceband in the tube of the eyepiece.

If the whole setup is at rest in a system, where the light rays arepresumed to propagate uniformly and with constant speed in all directions, thatis at rest in the ether, both rays will travel the same distance from the glass plateto the mirrors M and M and back. If we then turn the whole apparatus andenclosures through , the distance the rays will have to travel will be alike andthere would be no change in the interference fringes.If on the other hand the setup moves in relation to the ether in thedirection EM and we then turn the setup , the rays will travel differentdistances on their ways O‒M₁‒O, O‒M₂‒O, and this will result in a displacementof the interference fringes, which will be observed in E. [15] The time that the rayswould take to travel these distances under the path conditions can be calculated.

Figure 1.1L

E O GM1

M245°

G

12

If we let the velocity in the ether equal to and the velocity of earth along EGM₁equal to υ, the forward ray takes the time to go from G to M₁ and thebackward ray takes the time to go from M₁ to G. Therefore, total time for toand fro movement along GM₁G is

= − + + = − = + + ⋯ ( . . )For the ray going from G to M₂ and back will be similarly found to be

= √ − = + + ⋯ ( . . )In addition, the time difference will be

− ≈ ( . . )Since ≈ 30 km per second, the expectation was that, by turning the setupthrough 2, there would be a considerable change in the interference fringes,which could be observed. However, appreciable change had not been observedin many of the earlier repetitions of the experiments conducted on the surface of

G G

Figure 1.2E O O

2M1M 1M '

13earth. H. A. Lorentz explained the result of the experiment by the contractionhypothesis that length of a moving object contracts in the motion through ether[14]. Thus, Lorentz replaced the in equation (1.2.1) by 1 − so that ₁ of(1.2.1) and the ₂ of (1.2.2) will be equal and there will be no interference fringeand the right hand side of (1.2.3) will become zero. The basis of this replacementis proved in the proposition 2 of Section 1.5. Later Einstein proposed to do anexperiment for the verification of the reality of the Fitzgerald-Larmour-Lorentzcontraction but the proposal did not materialise. [1] In 1887, Voigt derived atransformation, relative to which the formula of Doppler-shift is invariant. Thiscoincides with the equation set of the Lorentz transformation, which Lorentzdiscovered as a result of his study of electromagnetism.

The apparent null result of the 1887 Michelson-Morley Experiment ledEinstein to postulate the ‘constancy of speed of light’ as an axiom in his paper onspecial relativity. The postulate on constancy of speed relative to two realobservers in relative motion was severely criticized by later writers. Scientists ofEuropean Centre for Nuclear Research (CERL) experimentally found thatneutrino particles travel faster than light. The scientists at CERL and the GrandSasso Laboratory in Italy scrutinised their results for nearly six months, beforemaking this announcement.[Website: http://arxiv.org/abs/1109.4897]. Evenbefore the paper of Einstein, H. A. Lorentz and H. Poincare had discussed theconcepts and theories, inherent in the 1905 paper of Einstein.[21] Lorentz’ EtherTheory came to include all of Einstein’s basic findings and is equivalent to thespecial relativity. Einstein negated the existence of ether in his paper;[12] but he

14returned to it after his Leiden Conference of 1920.[2] There, he said “according tothe general theory of relativity, space without ether is unthinkable”.[2] Thisreturn by Einstein to the ether has been well documented by Kostro.[2] In 1926Dayton Miller, using what was thought to be the most sensitive interferometeryet built, won a prize from the American Association for the Advancement ofScience (AAAS) for finding an ether-drift of 11 km per second. In 2005 Prof.Eugene I Shtyrkov of Kazan Physical-Technical Institute, Russia, in a paper [3]describes that an ether drift of 29.45 km, which is very close to the true averageannual velocity of 29.765 km/sec, has been discovered in the process of trackingof a geo-stationary satellite. It can be inferred that entrainment of ether resultedin the failure of Michelson-Morley Experiment and similar experimentsconducted on the surface of the earth.

Lorentz developed his concepts and theories now discussed in books onspecial relativity, from his study of electromagnetism from 1889 to 2004.[14,21]Poincare spoke from 1899, of what he called ‘the principle of relativity’. He saidthat we cannot determine absolute rest or uniform motion by conductingexperiments on the surface of earth. Poincare realized with prophetic precisionthat Newton’s theory has to be drastically modified.[7,21] In June 1905, Poincarewrote two papers entitled ‘On the Dynamics of Electron’ which containsessentially most of the contents of the 1905 paper of Einstein. Lorentz’formulation endorsed the absolute space even beyond Newton. Lorentz,Poincare, Minkowski, Eddington etc. based their discussions, without negatingthe existence of ether.

151.3. Temporal Illusion and Distortion of Images of Moving Objects

A temporal illusion is a distortion in the sensory perceptions orobservations made by equipments, when spatially separated events and their‘time’ of occurrence are measured, that is temporal intervals appear to beaffected by spatial distances or vice-versa. It is encountered because of finitespeed of signals/photons/gravitons /neutrinos, and spatial separation. A fastmoving vertical stick with its midpoint moving along -axis, and its lengthparallel to -axis appears to be bent

at the middle and have the appearance of a distorted > or < symbols, accordingas the stick is receding or approaching the observer at O. The bent occursbecause rays of light emanating from the midpoint M reaches O simultaneouslywith the rays starting from the endpoints A and B at an earlier/later time. As aresult of this phenomenon, a moving ′-axis will also be slightly bent at themiddle. Similarly the appearance of a moving sphere will not be spherical. Thisphenomenon can be compared with the Heisenberg’s uncertainty principle inquantum mechanics.1.3.1. About a thought experiment of STR

Let O be a real observer situated at the origin of a stationary system Susing O ( , , , ); O′ is his hypothetical position when he suspects that he is in

StationaryStick

RecedingStick

ApproachingStick

B

A

MMB

A

M

AO

B

Figure 1.3

16motion relative to the stationary system S. For example, O can be the earth-observer with the assumption that earth is not moving and O′ is the sameobserver when the earth is supposed to be moving in its orbit. The sameobserver designates him O in the stationary frame S and O′ in the moving frameS′. Thus the same observer uses O′ ( ′, ′, ′, ′) to account for his motion, if any.On the other hand, if possible, assume that O′ is also the origin of a real,independent moving co-ordinate system, such that the velocity of O′ relative to Ois ; this is the assumption in SRT.

Let a ray of light be emitted along OP making an angle with the direction ofrelative motion containing O and O′, when O′ coincided with O. Also let time bemeasured from the instant of coincidence. The equations of the ray, when it is atP are given by = ≠ = ′ and ≠ ′ according to figure 1.4.1 in which O′is hypothetical. According to the second figure 1.4.2, O and O′ are two realdistinct observers who observe a ray of light making a constant inclination to thedirection of relative motion, the ray being emitted when O and O′ coincided. Thismeans = ′ or sin = sin ′. This implies = ; but = ′ so thatOP = OP′. Therefore = ′ or = ′. (But ≠ ′ when O′ is hypothetical).Hence, Galilean time invariance holds according to the relativistic thoughtexperiment. Now we have the following inferences:

Figure 1.4.1

PO O′ ′ O′

P′P′O Figure 1.4.2

17(i)A single real observer O emitting a ray of light along OP will estimate (seefig. 1.4.1) O′P as the hypothetical position of OP, when he is supposed to bemoving and will have ≠ ′ and ≠(ii) In the relativistic thought experiment two real observers O and O′ areobserving a ray of light at a constant inclination with OO′, emitted when Oand O′ coincided will have = ′ and = ′.

In our discussion, we retain the first version and figure 1.4.1 in which O′ isthe apparent position when O is supposedly moving since the possibility of twoindependent observers emitting a ray at a constant inclination with OO′ cannotbe depicted by figure 1.4.1. Therefore, of the two positions O and O′, only one isreal and the other is the estimated position in a mutually exclusive situation so asto account for the motion of the single real observer. Hence the relativisticassertion that O and O′ are equally valid real observers in all discussionsinvolving LT, is not an experimental truth, but a choice of claim only. In the restof our discussion we shall develop all deductions according to the realisticconcepts of Lorentz and Poincare contained in the above discussion. Since ≠ ′the two observers (one real and the other hypothetical) in relative motion willapparently have different duration of time. Lorentz calls them ‘local time’(orzeit). On the other hand we see the following description of the thoughtexperiment in special relativity. Suppose a ray of light is emitted at the instant oftime = ′ = 0 when O and O′ coincided during the motion of O′ relative to O;then the expression = is transformed into ′ = ′ ′ . This argument isillogical; transformation between two systems must be relations betweencoordinates of a point-event. But here an infinite set is assumed to be

18transformed into another such set of points. If a ray of light is emitted at someinclination with the common -axis of the two systems, then the ray willcorrespond to two straight lines in the two frames, and the transformation mustbe related to two straight lines and not two spheres/two cones.1.4 Retardation/Advancement of Time

We continue the discussion of the previous paragraph. Suppose that O ( ,, , ) as the origin and coordinates of an event in the S frame and O′ ( ′, ′, ′,′) as the origin and coordinates of the same event in the hypothetical S′ frameattached to a moving object at O′. Let P be any point on the -axis

Let O′P = ′. Now we calculate OP = as follows:OP = OO + O′ P

i.e. = + + ′that is 1 + = − (1.4.1)Since the motion is relative, the roles of ( , ) and ( , ′) can be interchangedafter changing into − . Thus

1 − = + (1.4.2)From the last two equations we have

O PnewPoldO′newO′old Figure 1.5

191 + = +1 − = − and − ≡ − (1.4.3)

That is, time is retarded in the -direction and the retarded times in the -direction are equal. From the above equations we also have+ ≡ + (1.4.4)

and − ≡ ( ′ − ′ )where stands for . The above argument holds if < 1. When > 1, 1 −will be replaced by |1 − | and 1 − will be replaced by |1 − | in all ourdiscussions.If we further take

= 1 +1 − ′ and= 1 +1 − ′

then it follows that− ≡ 1 +1 − ( ′ − ′ ) (1.4.5)

In other words, the true speed and the apparent speed of light is the same ;otherwise a single observer will have two different values for the speed of light,

20one for the actual frame and another for the hypothetical frame. As in theprevious section, we assert that, of the two O and O′, only one of them is real andthe remaining one is his hypothetical location. Further, none has ever conductedany experiment in which two different real observers O and O′ in relative motion,observed the same ray of light emitted by the same source of light and found thatthe velocity of the ray of light is the same . On the other hand, according to therealistic Lorentz-Poincare concepts, there is no need for two separate observersin the S-frame and S′ frame. A single observer uses O ( , , , ) when he isassumed to be stationary and uses O′ ( ′, ′, ′, ′) when he is assumed to be inmotion, relative to some other stationary frame O ( , , , ). Thus only one setof space-time coordinates is real and the remaining set is a correction to accountfor his relative motion.In the above analysis, we got − ≡ − . Here the retarded time/advancedtime, does not depend on the relative velocity of the frames of references. Sowe, again start with equation (1.4.3) and (1.4.4) and modify them by writing

− = − (1.4.6)+ = + ′ (1.4.7)

where depends on , so that there is symmetry of inverse transformation. Byletting =(i) = ℓ − and = ℓ + we have ℓ = 1 + and

21= ℓ + (1.4.8)

= ℓ + (1.4.9)Also, we get = ℓ − (1.4.10)

= − (1.4.11)By appending = , = we have − ≡ ′ − ′ ,where ℓ = √1 + (1.4.12)Here we see that there is no restriction on the velocity . i.e. we can let → ∞.(ii) = and =we may takeℓ = + (1.4.13)ℓ = − (1.4.14)where ℓ = |1 − |. By appending = , = ′, we get the complete Lorentz’Transformation. Also, we getℓ = + (1.4.15)ℓ = − (1.4.16)and lastly − ≡ ′ − ′

22A question arises at this stage. Why did we restrict retardation in the -direction, which is equivalent to assuming no contraction along and axes?This is proved in proposition 2. Suppose we try to include and in theequations for retardation/advancement of time, in the form.

− = − ′and − ≡ ′ − ′These imply that the apparent speed of light is the same in the primed and un-primed co-ordinate systems; but this choice is not desirable, since , , , ′ donot occur explicitly. Next we consider a light-emission method of deriving theLT:Again consider

ℓ = + ′ℓ = −

By letting = = for a ray of light proceeding towards thepositive -axis in these equations, and then multiply the two equations, we get= − when < 1

or = − or = √ −Now by imposing = , = ′ we get the complete set of LT. This method isalso illogical, by considering the following reasons:

Figure 1.6O O′ P

23(i) when we assume = , = ′ we are indirectly borrowing theFitzgerald contraction hypothesis.(ii) suppose we apply this last procedure for a ray of light proceedingalong the -axis/ ′-axis

In this case there are two possible diagrams. The relations among thespace-time coordinates must be expressible in the formℓ = ′ + ′ℓ ′ = −By imposing = and = ′We get ℓ = 1 − or ℓ = √1 − ≠ √1 −This value of ℓ indicates that this method of proof is invalid.

Another variation of the light-emission method is to assume that theequation = changes into ′ = ′ , but this must be proved in advance,as proved by Lorentz based on the law of retarded/advance time. Further, weknow that it is meaningful to find transformation between ‘points’ or betweenemanating ‘rays’ and not between two spheres/cones.

O O′, v t′

′′

Figure 1.7

241.5 Lorentz’ Concept of Time and LT

Lorentz developed the Maxwellian electromagnetic theory, applicable toan electron in motion[11,14], in the 1890s. It is widely recognized within thephysics community that the Lorentzian theory of electrodynamics (ether-basedunderlying preferred frame) is indeed in accord with all that has been observed.Lorentz theory is based on the concept of retarded/advanced time [7] whereasSRT is based on two postulates plus the ‘retarded time’ concept, at a later stagewhen motion of electrons in an electromagnetic field, is considered. This showsthat the relativistic theory and the postulates of SRT are superfluous and,depending on a single concept, Lorentz’ theory is superior.We have earlier noticed that Newtonian dynamics is based on theassumption of instantaneous propagation of interaction and Roamer’s studiesshowed that the velocity of light is finite. Later it was found to be = 2.998 ×10 mps . The potential due to a charge Q at a distant point P ( , , ) from it canbe taken as = in suitable units. In classical physics, it is considered that theinfluence of the source charge Q at the field point P is transmittedinstantaneously. However, from experiments, instantaneous interaction isimpossible[10]; the maximum possible velocity of interaction is, at the speed oflight/graviton/neutrino. The solution of Poisson equation ∇ ∅ = −4πρ is∅ = ∭ ( ⁄ ) V whereas that of ∇ ∅ − c = −4 is given by ∅ =∭ ([ρ]⁄ ) V where [ρ] is evaluated at the retarded time − and is thedistance between the source and the field point. Lorentz [7] observed that, if anychange takes place in one of the interacting charges, it will influence the other

25charges, only after a lapse of time . If P ( , , ) is a fixed point of S-frame, then

− = . Hence the local t ime of the S -frame and the retardedtime interval are equal. But if P ( , , ) is a variable point of S then the retardedtime interval and the local time interval are not equal. Hence the local timeintervals of two frames, in apparent relative motion, also will not be equal.Hence two observers in relative motion, cannot have a common local time.Hence, Lorentz introduced two local times- , ′As discussed earlier, let an observer be situated at O of S-frame and aparticle of mass be situated at the moving origin O′ of the S′-frame. Let ( , , ,) be coordinates of an event at P according to S-frame and ( ′, ′, ′, ′ ′) thecoordinates of the event, relative to the hypothetical observer at the origin O′ ofthe moving mass.

It is further assumed that = 0 = ′ when the origins O′ and O coincidedduring the motion of the mass particle . The coordinates ( , , , ) will betrue for an observer situated in the immediate vicinity of P but these will becorrected to , , , ( − ) for an observer situated at O. Hence if ( , , , )are the coordinates of an event at P then ( , , , − ) will be the coordinates ofthe same event relative to O. This shows that observers, who are spatially

Figure 1.8O O′P( , , , )( ′, ′, ′, ′ ′)

26separated, even if in the same frame of reference, do not observe a given eventsimultaneously.

After duration of time , according to the real observer at O and ′according to the hypothetical observer placed at the moving mass at O′, thedistances of the mass in motion will be determined as follows. The projecteddistance of or O′ is , the retarded distance is and the current/presentdistance of and the hypothetical observer O′ at is ′. Lorentz assumedthat the retarded time is given by [7 ]= − | − | ( . . )

and then derived the formula∅ = [( − ) (1 − )⁄ ] + + (1.5.2)for the potential at P ( , , ) due to the moving charge, where = . Let= ( − ), = , =

= 1 √1 −⁄ . By changing ( , ) into (x , t′) and changing to − we furtherhave= ( + ′)Here is the beginning of LT in electro-magnetism [7]In this context, we shall modify (1.5.1) by the equation= − 1 (| − |) ∝ − ′

27i.e. the retarded time in the S′-frame is proportional to the estimate of in the S-frame.∴ = − | − | = − (1.5.3)where λ may depend on v and λ → 1 as → 0.1.5.1 Proposition I

The concept of retarded time represented by (1.5.3) correctly leads to thefull equation set of the Lorentz transformation, and the concept of proper time,whereas the relativistic assumption does not. Let R = | − |, = | − |and = | ′| = ′ + + ′ and = + + , = . From (1.5.3)we have( − ) = + − 2( ∙ )∴ − ∙ ⁄ + ⁄ = 0 where = ⁄∴ = ( − ∙ ⁄ )1 − − ( − ∙ ⁄ )(1 − ) − ( − ⁄ )1 − (1.5.4)omitting plus sign since we discard ‘advanced time’ in favour of retarded time.Comparing RHS of (1.5.3) with (1.5.4), we have

= − ( ∙ ⁄ )√1 − (1.5.5)= 1√1 − and = − ( − ⁄ )

28Here out of several possible values of λ, we chose the one that satisfiessymmetry of inverse transformation. The latter gives

′ − ′ = − (1.5.6)√1 − = − | | where = ( , − )and

= 1√1 − | − | − sin= 1√1 − ( − ) + + − ( + )= ( − )(1 − ) + + (1.5.7)But ∙ = ∙ ( − ) = ∙ − and R = − ,∴ R − ( ∙ )⁄ = − − ∙ + ⁄⁄= − ( ∙ ⁄ ) − (1 − )

( ∙ ⁄ )√ = − √1 − = by (1.5.3)i.e. = ( ∙ ⁄ )√ = √ R(1 − cos ) (1.5.8)where = ( , ). Also, R = | − | = | − ( − R⁄ )|∴ R = + ( − ⁄ ) − 2( ∙ )( − ⁄ )

29= + − 2 R + − 2( ∙ ) + 2( ∙ )i.e. − + 2 ⁄ − 2( ∙ ⁄ )⁄= + − 2( ∙ ). . (1 − ) − 2 R ( − ) = ( − ) + +

∴ 1 − = ( − )√1 − ± ( − )(1 − ) + ( − ) + +∴ = 1√1 − ( − )√1 − + ( − )(1 − ) + +omitting minus sign, since R is positive. By using (1.5.7), this can be re-writtenasR = (1 + cos )√1 − where = , (1.5.9)Multiplying (1.5.8) and (1.5.9) we get(1 − cos )(1 + cos ) = 1 − (1.5.10)Equations (1.5.8), (1.5.9), (1.5.10) give the formulae for Doppler shift andaberration, in the general form.From (1.5.3) and (1.5.9), we have

= − ′ (1 + cos ′)√1 − = − ′⁄√1 −

30√1 − − ( ⁄ ) cos ′ = ′

∴ √1 − = + ( ⁄ ) cos∴ = ( + cos ⁄ )√1 − (1.5.11)(1.5.5) and (1.5.11) give√1 − ≡ + ′, √1 − ≡ −These give

− = ’ − . . But − = − by (1.5.6).Hence + = ′ + ′ . This must hold when = = 0. Thus we get = ′and hence z= ′. Thus the postulate of retarded time, correctly leads to theLorentz transformation and the formulae for aberration and Doppler shift. Fromthe above analysis we have the invertible sets of equations.

(i) R = ′ ( )√ = √ ,= √ , tan = √1 −= R (1 − ) √1 − = ( )√= ( )√ , tan ′ = √( )

(ii) = ′√1 − cos ′, tan = √= √ √ , tan ′ = √1 − tan

31(iii) = R 1 + − 2 cos , tan = ( )

R = √ , tan = √These formulae give the space coordinates relative to (i) projected position ofthe origin O′ (ii) retarded position of the origin O′ and (iii) the current positionof the origin O′. It is clear that < ′ < ; hence O′ precedes O′ , whichprecedes O′ . Hence if is the speed of the origin O′, then its projectedposition is ahead of current position ′ and the current position ′ isahead of the retarded position .

Since − ≡ ′ − ′ = τ⁄⁄ , say, we consider τ as the proper time.More generally τ = − ⁄ = ′ − ′ ⁄ is defined as proper timeinterval associated with the two frames of reference. Minkowski definedτ = ∫ − as proper time. [12] Accordingly the proper velocityvector is defined by τ⁄ and acceleration is defined as τ⁄ . The properLagrangian and time-rates can be defined, replacing ‘time’ by ‘proper time’

R ′P ( , , )

′O′O O′O′Figure 1.9

321.5.2 Proposition 2 (Fitzgerald – Larmor - Lorentz contraction)

From the postulate of retarded/advanced time, we have proved that+ + − ≡ + + − ( 1.5.6 ) which reduces to+ − ≡ ′ + ′ − ( 1.5.6′ )when = ′ =0

Let the relative velocity = + . The matrix of lineartransformation carrying ( , , ) into ( , , ′) can be expressed in the formL = ℓ ℓ ℓ ℓ ℓℓ + ℓ+ (1.5.12 )Since a 3 × 3 symmetric matrix must have ( ) = 6 components, we took theseunknown constants as ℓ, , , ℓ , ℓ , and ℓ and the combinations + ′ and+ ′ should occur. Thus we take

= ℓ + ℓ ℓ + ℓ ℓ (1.5.13)= ℓ( + ′) + ( + ) (1.5.14)= ( + ′) + ( + ′) (1.5.15)

along with the symmetric conditionsℓ ℓ = ℓ + and ℓ ℓ = + .Thus we get the above matrix of linear transformation (1.5.12).By imposing the conditions (1.5.13), (1.5.14) and (1.5.15) into equation (1.5.6′)we have the six conditions

33ℓ − ( ℓ + ) − ( + ) = 1 (1.5.16)ℓ + − ℓ ℓ = 1 (1.5.17)

+ − ℓ ℓ = 1 (1.5.18)ℓ( ℓ + ) + ( + ) = ℓ ℓ (1.5.19)

( ℓ + ) + ( + ) = ℓ ℓ (1.5.20)ℓ + = ℓ ℓ ℓ (1.5.21)From (1.5.14) and ( 1.5.15 ) we have(ℓ + ) + (ℓ + ) = ℓ ℓ (1.5.19 )( + ) + (ℓ + ) = ℓ ℓi.e. (1 + ℓ ℓ ) + ℓ ℓ ℓ = ℓ ℓ by using (1.5.17) and (1.5.21)and (1 + ℓ ℓ ) + ℓ ℓ ℓ = ℓ ℓ by using (1.5.18) and (1.5.21)∴ ℓ = ℓ ( ℓ ℓ ) = ℓ ( ℓ ℓ ) ⇒ ℓ = ℓ = , say (1.5.22)

∴ ℓ = ( ) where = +From (1.5.21), ℓ + = ℓ ℓ ℓ (1.5.23)and from(1.5.19)and (1.5.20)ℓ − + (ℓ + ) − = ℓ ℓ − ℓ = 0 ⟹ℓ − + − = 0

34∴ ℓ − = − (1.5.24)∴ 4ℓ = [(ℓ + ) + (ℓ − )] = ℓ ℓ ℓ + − (1.5.25)From ( 1.5.19′ ), ℓ + + ℓ ℓ ℓ = ℓ ℓ∴ ℓ + = ℓ − ℓ ℓ ℓ = ℓ (1 − ) (1.5.26)(1.5.25) ⇒ 4ℓ + 4= ℓ ℓ ℓ + − + 2 ℓ ℓ ℓ − + 4 (1.5.27)From (1.5.26) and (1.5.27) we have4ℓ (1 − ) = ℓ ℓ ℓ + + + 2ℓ ( − )i. e. 4 (1 − ) = ℓ ℓ ℓ + 2 ( − ) + ℓ. . ℓ ℓ ℓ + ℓ = 4 ∴ ℓ ℓ ℓ + ℓ = √4

Similarly ℓ0ℓ1ℓ2 − 2ℓ0 1 2 = 4 − 4 2 2By subtraction 2ℓ0 1 2 = − − 2 2∴ = √ − √ − ℓ

35∴ ∝ ∴ when = 0 or = 0, then = 0. Without loss of generality wemay take e = 0∴ = 0, ℓ = = 0, ℓ = = But ℓ ℓ = ℓ′Also ℓ − ℓ = 1, ℓ − ℓ ℓ = 1 ∴ ℓ = ℓ = 11 − ℓ = 1√ −

2 = 1 or = 1 and λ = 1. Thus the matrix of linear transformationbecomes (when e < 1) (1.5.12) becomesL =

⎣⎢⎢⎢⎢⎢⎢⎡ 1√1 − √1 − 0

√1 − 1√1 − 00 0 1 ⎦⎥⎥

⎥⎥⎥⎥⎤

Thus we get √1 − e = + t , y = y′∴ There is no contraction or enlargement along the -direction, but there is acontraction by the factor √1 − e in the -direction, the direction of relativemotion. Thus, the apparent length of a moving rod parallel to its length with avelocity will be ℓ = √1 − ℓ (1.5.28)where ℓ is the rest-length.When e >1 we take ℓ = |1 − | instead of √1 − e .This is the Fitzgerald-Larmor-Lorentz contraction hypothesis of the realistictheory of relativity. The determinant of L is unity ∴ we have

≡ ≡ ℓ (1.5.29)

36where ℓ is the proper displacement or proper length.1.6 Relativistic Concept of Time & LTWe shall begin the discussion by examining the contradictory opinions of aformer relativist H. Dingle and two adherents of STR.(i) Prof. Herbert Dingle in his 1972 book [4] ‘Science at the Cross-roads’wrote: ‘the question is left by the experimenters to the mathematicalspecialists, who either ignore it or shroud it in various obscurities’. Hecontinues that ‘obviously something must be logically and mathematicallywrong with either the LT or the principle of relativity or both, and that tobe logically and mathematically consistent, one would have to jettisoneither LT or the principle of relativity or both’.(ii) In his book, ‘Cranks, Quarks & the Cosmos’ (published by OxfordUniversity Press, 1997) the author Jeremy Bernstein points out that “Iwould insist that any proposal for a radically new theory in physics orscience, should contain a clear explanation of why the precedent scienceworked. Einstein did this as the first page of the paper, ‘on the electro-dynamics of moving bodies’ illustrates perfectly”.(iii) In the paper ‘Special Relativity Invalid?” by Vesselin Petkov,[http://groupkos.com/rnboyd/special-relativity-invalid1.html] the authorattempts to convince the readers with a slim hope that, “what I will writebelow might be helpful to some of the people who have reservationsabout relativity”. The author continues “As a rule those who criticize it,do not see the whole picture – they pick up only individual relativistic

37results. I know this even from personal experience – for twenty years, allletters/papers claiming to have ‘finished’ relativity, sent to twodepartments (where I have worked) have been regularly forwarded tome. Since the above opinions are conflicting with each other, it is veryessential to examine the time-concept of Special Relativity in details.The first paper [12] on relativity, viz. “On the Electrodynamics of MovingBodies”, published in 1905, contains a definition of simultaneity. In §1 of thepaper, the author considers the following thought experiment. Let there be twoobservers at A and B in a stationary system having clocks of identical timereading. Let a ray of light start from A at ‘A time’ towards B, reflected at B at ‘Btime’ back to A, arriving at A time ′ . The two clocks synchronize if

− = −The author further assumed that 2 AB ( ′ − ) =⁄ , as the velocity oflight in empty space. From the above two equations, it is clear that = −AB⁄ , i.e. the author has used the concept of retarded time to explain‘simultaneity’. After this, the paper continues, “It is essential to have timedefined by means of stationary clocks in the stationary system, and the time nowdefined being appropriate to the stationary system, we call it the time of“stationary system”. In §1 of the paper, the author has used the idea of retardedtime − ⁄ with in the denominator. But in §2 of the paper, dealing with the“relativity of lengths and time”, we come across the following postulates ofrelativity, viz.

38Postulate 1. The laws by which the states of physical systems undergochange are not affected, whether these changes of state are referred to one orthe other of the two systems of coordinates in uniform translatory motion.Postulate 2. Any ray of light moves in the stationary system of coordinateswith the determined velocity , whether the ray be emitted by a stationary orby a moving body. Hence, velocity = , where time interval is to betaken in the sense of definition in §1.Again the author considers the clock ray emission and reflection experimentwith a difference and then assumes

− = ( − )⁄ and ′ − = ( + )⁄where A and B are the ends of a moving rod of speed ν and is its length in thestationary system. The above equations give

= − ( − )⁄ and = ′ − ( + )⁄Again we see that the author uses the idea of retarded time with differentvelocities ( − ) and ( + ) in the denominator, whereas the correctdenominator is , from Lorentzian time-concept. It is clear that the authorwrongly uses the concept of retarded time. In §3, dealing with the derivation ofLorentz transformation the author again uses the concept of retarded timerestricted to -axis, where we read the equation

12 (0,0,0, ) + , , , + ′− + ′+ = , 0,0, + ′−

39in which τ is the time in the S′ frame (in the place of ′ of the Lorentzian theory)Here also, we see that + is a velocity greater than the velocity of light. This iscontrary to the content of the second postulate of relativity and contrary to the‘principle of retarded time/advanced time’.

From the above analysis, it is clear that there is no new concept of time, inthe SRT, which contains a faulty use of retarded/advanced time. In this contextwe prove the following.1.6.1 Proposition 3

The postulates of special relativity do not prove the truth of the linearLorentz transformation; it leads to a non-linear transformation with ≠ ′ and≠ ′.To derive the transformation between S frame and S′ frame referred inthe previous discussion, let us take = 0 = ′ for convenience. Let = + bethe complex variable. The second postulate leads us to the assumption that thecircle | | = is transformed into the circle | ′| = ′. The most general bilineartransformation is ′ = ( − ) ( − ̅ ⁄ )⁄⁄ where ̅ is the complexconjugate of . Hence we may take

L = − , L = − ̅ ⁄ (1.6.1)where L′ is constant depending on . Inverting these equations we have

(1 − ) L = +⁄ (1.6.2)(1 − ⁄ ) L = + ̅ ′⁄⁄

40which can be written in the formL = + (1.6.3), L = + ̅ ⁄ (1.6.4)where LL = 1 − ⁄ . Thus L and L′ are complex conjugates with magnitude1 − ⁄ . Hence we may take L = ∅ 1 − ⁄ and L′ = ∅ 1 − ⁄ .Clearlytan∅ = ′ ⁄+ ′⁄ = ⁄− ⁄ (1.6.5)

This transformation is non-linear and in general ≠ ′. Hence therelativistic claim ‘since there is no motion of S′-frame along and axis, itfollows that = ′, = ′ ’ is logically false. The fault lies in the assumption “asphere of light = centred at O is transformed into the sphere of light= ′ centred at O′”.It is meaningful to assume that a ray of light emitted by O′ has equations′ ℓ = ′ = ′ ′⁄⁄⁄ in S′ frame and ( − ) ℓ = = ⁄⁄⁄ in S frame.Similarly a ray of light emitted by O has equations L⁄ = M⁄ = N⁄ in S frameand ( + ′) L′⁄ = ′ M′⁄ = ′ N′⁄ . Now by applying the Lorentzian concepts ofretarded time, local time, present time as detailed in the proof of proposition 1,we get the values of ℓ, , , etc., thus obtaining the whole set of Lorentztransformations.The above analysis shows that postulates of SRT, does not lead to Lorentztransformation, whereas the concept of retarded time correctly leads to Lorentz

41transformation. From a study of classical dynamics, we see that there are threedifferent principles of relativity [15]. They are:(i) The laws of motion recognize no distinction between rest and uniformmotion, known as Galilean relativity (Newton, Fitzgerald and earlyPoincare).(ii) The laws of motion do recognize such a distinction but the real (notconceptual) effects of motion such as to make it impossible for anyone todetermine his own motion without reference to outside bodies-known asthe realistic version of relativity. Stated differently, it is impossible byperforming experiments on the surface of the earth to detect uniformmotion relative to ether (Lorentz and later Poincare).(iii) All inertial frames are equivalent, and the speed of light is an absoluteconstant, known as special theory of relativity. Stated differently, thereexists no natural standard of rest that will make it meaningful, so that anysingle body has motion rather than any other (Einstein). [6,12,15]

According to the Galilean principle of relativity, the velocity of motion of abody has different values for two observers in relative motion. This shows thatthe velocity of light cannot be constant in two real frames in relative motion. Inthe Lorentz-Poincare realistic version of the theory of relativity there is one realreference frame S and a second frame S’, which is a mutually exclusive substituteor replacement for the former, when the former has a hypothetical motionrelative to some other frame. Thus one of two frames S, S’ can be avoided oncewe have obtained the proper time interval: = − − − −= − − − . . one need only to retain either (i) O( , , ,

42) and O( , , , τ) or (ii) O′( ′, ′, ′, ′) and O′( ′, ′, ′, τ) but not both.Simultaneous use of both sets is irrelevant. Special relativity uses both the setssimultaneously. This is the difference between Lorentz realistic version and theSTR. But in GTR the primed coordinates are completely discarded, instead theproper time interval is the main object of study. Allowing τ in addition to and ′in STR, is equivalent to admit the existence of an absolute time τ and the ether-frame. By using both the primed and un-primed coordinates, as simultaneouslytrue, Einstein and Dingle arrived at contradictory views on time dilation; oneused the un-primed coordinates, the other used the primed ones. It is a futileattempt to conclude anything from < ′ or ′ < where the time-coordinates are both local. But we can write τ < and τ < ′. This showsthat misinterpretations are inherent in simultaneous use of the primed and un-primed coordinates. The paradox of ‘twin paradox’ is such a misconceived one.A.S. Eddington[15] accepts the concepts of ‘local time’ proper time and contractionhypothesis and the existence of ether. Once the contraction hypothesis isaccepted in STR, there will not be any contradiction. Thus there were only twomethods available for the explanation of the null result of Michelson-MorleyExperiment in 1905:

(i) By using the contraction hypothesis of Fitzgerald Larmor, Lorentz andothers explained by means of Galilean principles and a preferred frame ofreference.(ii) By using STR postulate of ‘constancy of speed of light, independent ofsource emitting the light’.

43The second method is based on the negation of ether, and this method islikely to fall flat, when the existence of ether is established finally. Even then theLorentz-Poincare version will survive. We have already shown that theconstancy of speed of light does not imply LT or any other linear transformation,but to a non-linear transformation. Still special relativists claim that they haveproved LT from the postulates, without admitting the fact that they are usingcontraction hypothesis in disguise.

1.7 Review of the literature on Special RelativityWe shall review two of the books dealing with special relativity. Whitrowtreats with such problems as universal time, individual time, mathematical time,and from these arrive at relativistic time. [15] The author presents seven axioms,and after most detailed and subtle discussions, arrives at Einstein’s results:= ( + ) and = ( − ). This is actually the weak form of the conceptof retarded/advanced time, since the above equations can be re-written as= − and = − . From earlier discussions, we know that these weakerforms, not containing the relative velocity ̅ , can explain only the clock-synchronization and simultaneity, but not the truth of the LT. Knowing this fact,Einstein attempted to use the retarded time/advanced time by means of theexpressions like − , + . This is wrong and contrary to the constancypostulate of STR as well as the postulate of retarded/advanced time. Thus thebook does not contain anything different from other books on STR.Now we shall examine Bergson [15] who made a serious study of STR, pinpointing the weak logical basis. In the book, the author explains the unexpected

44result of Michelson-Morley Experiment, by using the Lorentz-Fitzgerald theoryof contraction of length of a moving body in the direction of its motion, inrelation to ether and as compared with its length at rest in the ether. Hesupposes that the rest length ℓ in ether will have the apparent length ℓ 1 −when moving with velocity relative to ether and as judged from the ether. Hefurther declares that the time of the system dilates in the ratio 1: 1 − . Thisis equivalent to the statement = 1 − = 1 + . Fromthese bases, he deduces the LT and various other results found in STR. Sincethese concepts have no discordance with the postulate of retarded/advancedtime, the conclusions obtained by Bergson are valid. He gives a very elaboratecritical survey of some of the consequences drawn by STR. He raises thequestion as to, what extent the Einstein ‘times’ are real times. He proclaims thatwe cannot speak about a reality reigning without introducing consciousness. Hedeclares: when we want to know if we have to do with a real time or a fictitioustime, we only have to ask if the object presented could or could not be observed,become conscious. He elaborately discusses the relation between observationsof events and processes by observers as well in the moving system S′ as in thesystem S at rest, and comes to the conclusion that, by comparison of the ‘times’ ofobservers in different systems, there is only one ‘time’ that is ‘real’, the time thatis experienced by the real observer. The other times are fictions. Bergsonexpresses it thus: ‘we thus always come back to the same point. There is one realtime and the others are fictitious. What is a real time if not a time experienced,

45or which could be? What is an unreal, auxiliary, fictitious time, if not one whichcould not be effectively experienced by anything or anybody? ‘The twoobservers in S and S′ live exactly the same length of time and the two systemsthus have the same real time. This is more closely discussed in connection withthe train problem, which is the basis of Einstein’s definition of simultaneity. Hecompares the observations of the observer on the embankment in the midpointM between the points A and B where the hypothetical lightning strokes occurand the observer in the midpoint M′ on the moving train between the points A′and B′, where the lightning strokes occur with regard to the train. He comesto the result that one has to do with only one time. What is simultaneous withregard to the embankment is also simultaneous with regard to the train.

Bergson comes here in flagrant opposition to Einstein’s results. He explains it bypointing out that, we must suppose that the observations are really made by anobserver – ‘un-physician’ – in the system. Only what this observer measures isreal. But the observer can only be in one place. He is in M, and consequentlycannot also be in M′. Bergson concludes that nothing has been really observed inM′ because that would presuppose another observer in M′, which is not thesituation.In a later discussion of the observations made by the observers in the twosystems, one of which is resting on the earth, the other moving, he declares that

A BMB′M′A′

Figure 1.10

46the observer in the former system alone is real and the other observer aphantom. The exclusion of the privileged system of reference is the essence ofSTR. From the Lorentzian concept of time, we know that among the two localtimes , ′, only one is real, and the other is the correction or estimate toaccommodate the relative motion of the single observer and the finite signalvelocity of gravitational/electromagnetic signals. It is clear that Bergson’sarguments are in agreement with the realistic version of relativity, but not inagreement with STR.

Bergson, in his criticism of Einstein’s interpretation of the ‘simultaneity’met with the rejoinder from Charles Nordmann [15] who argued that if Einstein’stheory was really based on the demonstration of simultaneity, his theory wouldcollapse and remarked that the real foundation of STR is to be found in Einstein’s1905 paper ‘Elektrodynamik’. Failing to defend Bergson’s criticisms, he virtuallyadmitted the truth that the clock-synchronization, simultaneity etc. have no rolesin the foundation of the theory of relativity. On the other hand, the realisticversion of relativity based on the postulate of retarded/advanced time and theconcept of proper time, has the correct logical foundations. The ‘proper-time’can take the role of the classical time. In this manner, the classical Newton-Galilee principle of relativity will be intact. Further, since ≈ , the local timeis a good substitute for the proper time, in most practical purposes.1.8 Conclusions

In STR there are two simultaneously real coordinate systems S and S′ inconstant relative motion. In the realistic version of the principle of relativity, one

47real observer uses either of two coordinate systems: (i) ( , , , ) and ( , , ,τ) or (ii) ( ′, ′, ′, ′) and ( ′, ′, ′, τ) but not both. In this approach, there isno need for simultaneous use of primed and the un-primed system, as they arerelevant in mutually exclusive occasions. However, in Einstein’s SRT, there aretwo simultaneously real coordinate systems ( , , , ), ( ′, ′, ′, ′) andanother coordinate system with proper time τ. Thus there are two real spacecoordinate systems with three time coordinates. The relativistic postulates donot lead to LT or any linear transformation, but to a non-linear transformation.The double use of and ′ as real times, led A. Einstein and H. Dingle to havecontradictory views on time dilation. Einstein proved time dilation in 1905 byusing LT and Dingle proved that moving clocks run fast, by using the inverse LT.The postulate of retarded time/advance time is superior to the ‘postulate ofconstancy of speed of light’. The proper time τ can be used as a substitute forthe classical absolute time of Newtonian dynamics in infinitesimal region of aworld-point. Since the local time is also a good approximation for τ thedifferences among the three types of relativity discussed in section 1.6 becomenegligible. If Prof. Eugene I Shtyrkov’s[3] experimental evidence of ether-driftvelocity of 29.45 km/sec is finally accepted as a universal truth, then thepostulates of STR remain invalid and the Galilean transformation will prevail.Since the LT is not related to two separate real observers in relative motion, itwill be valid according to the realistic interpretations of Lorentz, Fitzgerald,Larmor, Poincare, Minkowski et al.

We know that a parabola is a one-dimensional curved space, but it can beembedded in a two-dimensional space-the plane of the parabola. Similarly, the

48regions of space-time become curved in the presence of gravitational and/orelectromagnetic fields. However, such regions can be embedded in a higherdimensional space[19]. Hence, the apparent universe may be considered as aunion of

(i) Euclidean sub-spaces ( ≈ and intensity of negligible)(ii)Pseudo-Euclidean sub-spaces ( ≠ and intensity of fieldsnegligible)(iii) Non-Euclidean sub-spaces (presence of fields)Hence, it is not appropriate to infer that absolute time/absolute spacedoes not exist.