Relativistic Time Contraction: a Wave TheoreticalReinterpretation of the Lorentz Transformation Equations.

Rick D. Ballan

rickballan@sydney.edu.au

November 16, 2010

AbstractA fundamental asymmetry currently exists between the Einstein-

Minkowski definition of a single unified spacetime and the separationof space and time variables required by wave theory. The source ofthis asymmetry is traced back to Einstein’s definition of time dilationwhich, by following the ‘world line’ of a discrete material particle, notonly contravenes Heisenberg’s Uncertainty Principle but also the moregeneral condition that time-frequency measurement must be carriedout at a single rest point in space relative to each inertial system. Re-defining space-time accordingly, the space and time axes for “moving”systems, expressed in the coordinates of the “stationary” system, be-come identical to the phase and group velocities of spherical standingwaves. By recognising that Einstein’s “array” of synchronised clocks,and the Michelson-Morely experiment upon which it was based, hasall of the salient features of equal and opposite standing waves, theLorentz Transformation Equations can then be directly deduced as thewave arguments of these standing wave motions when transformed toother systems of coordinates. Only under this definition can the princi-ple of relativity be upheld. Assigning the proper frame to the observerhas the effect of inverting time dilation while leaving the mathemat-ical structure essentially in tact; the standard time-like, space-likeand light-like interpretations of the Minkowski interval, for example,find their most natural expression depending upon whether we aremeasuring frequency, wavelength, or the wavefront of an electromag-netic wave. Inasmuch as energy-momentum depend upon frequency-wavelength, then the equivalence of mass and energy still holds under

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this interpretation. However, this is no mere trivial change in our‘point of view’. Time is now proportional to and transformed togetherwith length, which was always a fundamental condition of wave theory.Since the array of clocks have been synchronised via light waves of uni-versal speed c according to Einstein’s second principle, then this leadsto a satisfactory definition of time “in general”. Reviewing some of theexperimental evidence or arguments that are usually cited in defenceof time dilation - the Transverse Doppler Effect, the Twins Paradoxand acceleration - it is shown that they can and must be reinterpretedin terms of time contraction if the equations are to remain consistent.The empirical evidence obtained by Ives-Stillwel, Hafele-Keating et al.merely proved the existence of a Transverse Doppler Effect, which ishere identified with time contraction; the “stationary” clocks, now de-fined as objective wave motions, have been contracted with respect toclocks in relative motion and not dilated inversely. Since the empiricalpredictions are precisely the same as the conventional interpretation -it is the “stationary” clock’s that are running fast - then it becomes aclear case of affirming the consequent. Hence, all of this implies thatour sense of space and time, and perhaps even our biological apparatusused in sensory perception, evolved from our experience of observablewave phenomena and not post hoc from the rods and clocks of our ownmaking. As a final consideration, if we generalise this Minkowski met-ric for curved space-time, then the motion of quantum wave-particlesunder the influence of gravity should follow as a matter of course.

1 IntroductionWhen Einstein was first working out the mathematical details of On theElectrodynamics of Moving Bodies prior to its publication in 1905, it had notyet been discovered that the elementary particles of matter, in addition tolight and other electromagnetic radiation, displayed all of the salient charac-teristics of wave motion. Coming out of the tradition of classical mechanicsand inevitably being unaware of future developments in quantum theory(ironically, developments that Einstein himself would play a large part ininstigating), it was natural for Einstein to look for ways of generalising New-ton’s laws and to account for them in light of the relatively new findingsmade in optics and electrodynamics. Even brushing aside the many apoc-ryphal stories that have since been attached to his name and reputation, the

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Figure 1: Minkowski’s original spacetime graph. Here Minkowski gives a de-tailed account of how Einstein’s definition of length contraction can be inter-preted in terms of hyperbolic geometry. Moving from the point D to D′ givesD′ = D

√1− v2. However, the analogous temporal points which would produce

time contraction, here added as E to E′, conspicuously find no mention in thetext.

fact that the twenty six year old Einstein not only discovered but correctlyreinterpreted the Lorentz Transformation Equations from the contemporarymechanical perspective, and this in complete independence from all of theleading scientific figures of his day, remains truly remarkable. Neverthe-less, it has always been the mark of any great scientific theory that it willlead to predictions that were not foreseen by its original author. And whileEinstein’s theory does in many respects represent a radical departure fromclassical mechanics, it is important not to see the differences exclusively atthe expense of their similarities. For example, it is often said that Einstein’srelativistic theory “swept away” the absolute space and time of Newton. Yetat the same time it is also shown that Einstein’s equations reduce to thoseof Newton’s in the limit of velocities small in comparison to light which, onecould argue, is proof enough that they were intended to be generalisations ofthem. Both of these theories still uphold an essentially mechanical view ofthe physical universe, or in other words, are based on the assumption thatmatter is reducible to interactions between discrete particles (rigid bodies)which are always and in every circumstance well-defined in both space and

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time:

The theory to be developed is based - like all electrodynamics - onthe kinematics of the rigid body, since the assertions of any suchtheory have to do with the relationship between rigid bodies (sys-tems of coordinates), clocks, and electromagnetic processes. . . Ifwe wish to describe the motion of a material point, we give thevalues of its co-ordinates as functions of time. Now we must bearcarefully in mind that a mathematical description of this kindhas no physical meaning unless we are quite clear as to what weunderstand by “time”. We have to take into account that all ourjudgments in which time plays a part are always judgments ofsimultaneous events. If, for instance, I say, “That train arriveshere at 7 o’ clock,” I mean something like this: “The pointingof the small hand of my watch to 7 and the arrival of the trainare simultaneous events”*. . . The “time” of an event is that whichis given simultaneously with the event by a stationary clock lo-cated at the place of the event, this clock being synchronous, andindeed synchronous for all time determinations, with a specifiedstationary clock.*We shall not here discuss the inexactitude which lurks in theconcept of simultaneity of two events at approximately the sameplace, which can only be removed by an abstraction.[1]

With this condition for the measurement of time in place, Einstein thenarrives at the only conclusion possible. Following convention, take two in-ertial frames of reference, S and S ′, with the latter moving uniformly withvelocity v along the common x-x′ axis towards positive x. A particle at restat the origin of S ′ has the equation x = vt according to S. Substitution intothe Lorentz Transformation Equation for Time we have

t′ = (t− vx/c2)/√

1− v2/c2 = t√

1− v2/c2,

t = t′/√

1− v2/c2 (1)

That is, a time measured to be at rest with a material body will appear tobe dilated according to every other inertial reference frame. In other words,moving clocks run slow.

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A potential problem with this description of “simultaneous events”, to-gether with its subsequent definition of time dilation, is that it seems tocontravene Heisenberg’s Uncertainty Principle (1928), which states that it isimpossible to determine both the position and momentum of such a materialpoint simultaneously. For instance, if we attempt to synchronise clocks in themanner prescribed by Einstein using a photon, then according to Heisenbergthere must be an uncertainty as to the times of both emission and arrival ofthis photon in the order of h/2. The smallness of this number might assureus that any uncertainty surrounding the time “7 o’ clock” is, for everydayintents and purposes, negligible. Yet in the realm of micro-physics, a trainand a photon are by no means of the same order. And when the position inspace-time of a particle is defined, then it has no time or space magnitude.How then are we to say that the time of such an event can be dilated? EvenEinstein himself supplies us with an uncanny premonition to this problem asseen in the footnote. In other words, this uncertainty cannot be “removed byan abstraction” because it cannot be removed at all, not even in principle.Consequently, if the precise point in time of this particle is now dubious, thenhow can we continue to legitimately uphold Einstein’s interpretation of theproper time of an event, the view of a single clock at rest with respect to awell-defined moving particle?

By applying Heisenberg’s principle we have discovered a small openingthrough which to manoeuvre. However, this problem becomes increasinglymore pronounced once it is seen that this definition of “the time of an event”appears to be fundamentally incompatible with the determination of timefor any waveform whatsoever. We could not, for example, use Einstein’sdefinition of “simultaneous events” above to explain the particular event ofmeasuring the frequency of the light waves emanating from the train andclock without falling into infinite regress. Since quantum wave-particles relyon Fourier analysis for their very existence - the particle aspect also beingdescribed by a wave packet- then this problem becomes one in a whole classof problems. When Einstein adopts the conditions for the measurement oflength, it is clear that moving lengths must be measured at the same instantin time relative to each inertial observer:

Let there be given a stationary rigid rod; and let its length bel as measured by a measuring-rod which is also stationary. Wenow imagine the axis of the rod lying along the axis of x of thestationary system of co-ordinates, and that a uniform motion of

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parallel translation with velocity v along the axis of x in thedirection of increasing x is then imparted to the rod. We nowinquire as to the length of the moving rod, and imagine its lengthto be ascertained by the following two operations:-

(a) The observer moves together with the given measuring-rodand the rod to be measured, and measures the length of therod directly by superposing the measuring-rod, in just thesame way as if all three were at rest.

(b) By means of stationary clocks set up in the stationary sys-tem and synchronizing in accordance with §1, the observerascertains at what points of the stationary system the twoends of the rod to be measured are located at a definitetime. The distance between these two points, measured bythe measuring-rod already employed, which in this case is atrest, is also a length which may be designated “the lengthof the rod.”[1]

Of course, adopting this condition shows that moving lengths are contractedrelative to the stationary observer. With t1 = t2 for our instant in S, theLorentz Space Transformation gives

(x′2 − x′1) = ((x2 − x1)− v(t2 − t1))/√

1− v2/c2 = (x2 − x1)/√

1− v2/c2,

(x2 − x1) = (x′2 − x′1)√

1− v2/c2 (2)Thus, a length at rest in S ′ appears contracted in S. And it is not difficultto show that this condition agrees with the measurement of wavelengths.1

Yet when it comes to the definition of time, Einstein does not adopt thecomplimentary condition that frequency is to be measured at a single restpoint in space according to each inertial observer. By employing this condi-tion it is seen that time is no longer dilated from S to S ′ but is contractedinversely. That is, with x1 = x2 as our stationary ‘measuring’ point in S, wehave:

(t′2 − t′1) = ((t2 − t1)− v(x2 − x1)/c2)/√

1− v2/c2 = (t2 − t1)/√

1− v2/c2,

1One way this can be obtained is by taking u = v where u is the velocity of thewavefront (u 6= c). It is then found that all wavelength are contracted from the properframe. But as we shall see, another example of length contraction can be achieved bytransforming the modulated wavelength of a standing wave.

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(t2 − t1) = (t′2 − t′1)√

1− v2/c2 (3)

Thus, in direct contrast to (1), time is now contracted from S ′ to S andis proportional to the transformation for length, which was always a fun-damental condition of wave theory. Looking back to Fig.1 where the timecontracted points are given in red, it is seen that not only does this still agreewith the mathematics of hyperbolic space-time but also helps to elucidatea problem that has been systematically overlooked for over a century [3].Similarly, Fig.2 represents a standard picture which has often been used toexplain time dilation. Yet the simple observation that frequency must bemeasured at a fixed location in space alters it’s whole meaning.

As the following theory will show, this time contracted period occurs nat-urally when the period of an equal and opposite standing wave is transformedalong the common line of motion to other systems of coordinates. A littlethought will reveal that this requires us to confine our attention to the singledimension of space between the two observers. Having become accustomedto the plane wave (or ’direction-cosine’) model in four-dimensions, this ini-tially seems insufficient. However, we must bear carefully in mind that theinitial choice of the x-x′ axis was always arbitrary. At any point in our in-quiry we could substitute x for the vector of magnitude r =

√x2 + y2 + z2

which would then also represent the line between the two observers. In otherwords, generality demands that we deal with a spherical rather than a planewave model. The condition that frequency measurement must be made ata stationary point along this line of motion assures that time will alwaysbe contracted together with length. Furthermore, it will be shown that Ein-stein’s synchronisation of clocks using light rays has all of the salient featuresof one of these standing waves and may in fact be defined accordingly. Byadopting this procedure, time is no longer measured by taking the readingof a single synchronised clock at rest in the “moving” frame but insteadtakes the reading of a continuum of synchronised clocks in the “stationary”frame as they pass this point of rest. In other words, the proper frame isnow the frame that is carrying out the measurement while the event beingmeasured - the rate of clocks/standing waves - is now located in the otherframe. This has the effect of inverting time transformation while leaving themathematical structure essentially in tact.

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′x

t1′′S

v

′x

t2 ′

′S

v

Sx1 x2

t2

t1

Monday, 17 November 14

Figure 2: 1. According to the standard mechanical interpretation, a clock is at restwith a material particle located at the point x′ in S′. The kinematic event beingmeasured by an observer at rest in S appears to be dilated; the rate of the S′ clockruns slow. 2. According to the wave theoretical interpretation, the rest point x′in S′ represents an observer who is carrying out a frequency measurement and thewave event being measured is now located in S. The continuum of S clocks havebeen synchronized by equal-and-opposite standing waves of light and the rate ofthese clocks is nothing more than the average frequency of these objective waves.The rate of S clocks as they pass this fixed point x′ appear to run fast. Hence,the period has been contracted.

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2 Physical Interpretation as Spherical Stand-ing Waves

It is evident that Einstein’s method of synchronising clocks requires the con-dition of symmetry i.e. the synchronisation takes place in both directions.Furthermore, since the choice of 1 second is arbitrary, then we may definethe rate of synchronised clocks in the S frame to correspond to the angularfrequency ω. We could imagine, for instance, that the hands on each clockface rotates once per period of T seconds according to an observer at rest inS. Let us now consider one possible wave used in this synchronisation:2

ψ(x, t) = A cos(ωt+ kx) + A cos(ωt− kx) = 2A cos kx cosωt. (4)

It is seen that this function is composed of two waves of equal amplitude,frequency and wavenumber moving in opposite directions. Alternatively, wecould consider this to be an averaged harmonic wave in time multiplied by amodulated amplitude in space. To see this, let us set ω1 = ω2 = ω, k1 = −kand k2 = k such that ω1/k1 = −u and ω2/k2 = u, where, contrary to theusual view, u can be any velocity whatsoever (the only condition being thatthe speed of light c remains invariant for all coordinate systems in the eventit is chosen. By definition, the time required for the wave to move from A toB is equal to the time required for it to move from B to A). We then have:

ωa = (1/2)(ω1 + ω2) = ω,

ωm = (1/2)(ω1 − ω2) = 0,ka = (1/2)(k1 + k2) = 0,km = (1/2)(k1 − k2) = −k. (5)

and (4) can be written as2Of course we could always replace (4) with Euler exponentials. And as mentioned,

we could at any time also substitute x in the following with a vector of magnituder =

√x2 + y2 + z2 giving spherical standing waves in S. This is necessary since time

measurement must now be carried out by observers at the place of measurement, andspace is isotropic. Thus, the Lorentz Space Transformation becomes r′ = (r− vt)β. Yet Iwill continue to use x due to custom and familiarity. Alternatively, we could create a planewave model simply by introducing y = y′ and z = z′ which do not effect the final results.Nevertheless, a third possibility arises that we may need to take separate time measure-ments for each spatial dimension which would require six-dimensions (x, y, z, tx, ty, tz).

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ψ(x, t) = Acos(ω1t− k1x) + Acos(ω2t− k2x) =

2Acos(ωmt− kmx)cos(ωat− kax). (6)

Transforming this set of solutions to S ′ gives:

ω′1 = (ω1 − vk1)β = (ω + vk)β,

ω′2 = (ω2 − vk2)β = (ω − vk)β,

k′1 = (k1 − vω1/c2)β = −(k + vω/c2)β,

k′2 = (k2 − vω2/c2)β = (k − vω/c2)β, (7)

where β = 1/√

1− v2/c2. Dividing each of the two frequencies by theircorresponding wavenumbers it is apparent that these values agree with theusual R’c velocity transformations. To obtain the average and modulatedvalues according to S ′, we must assume that the same form applies in bothreference-frames. We then have:

ω′a = (1/2)(ω′1 + ω′2) = ωβ, (8)

ω′m = (1/2)(ω′1 − ω′2) = vkβ, (9)

k′a = (1/2)(k′1 + k′2) = −(vω/c2)β, (10)

k′m = (1/2)(k′1 − k′2) = −kβ. (11)

The phase and group velocities are obtained in the usual manner:

ω′a/k′a = −c2/v = u′p (12)

ω′m/k′m = −v = u′g (13)

It is to be noted that this interpretation agrees with both the standard R’cand wave theoretical definitions. Their product, for example, equals thespeed of light squared. Differentiating the R’c dispersion gives dω′a/dk′a = u′gwhich is the standard interpretation. And in accordance with orthodox RT,(12) and (13) represent the space and time axes, respectively, of the S-framerelative to S ′. Finally, if we take the respective average and modulated valuesfor the system S ′, we obtain,

(ω′at′ − k′ax′) = ω(t′ + vx′/c2)β = ωt, (14)

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(ω′mt′ − k′mx′) = k(x′ + vt′)β = kx. (15)which, we see, gives the Lorentz space and time transformation equationsfrom S to S ′. So, in a manner of speaking, our deduction has come full circle.We began by assuming that the space-time continuum was mathematicallyidentifiable with the frequency-wavelengths of observable wave phenomenawithout invoking the use of the Lorentz equations, and we have come someway in proving it. Of course, we could have simply substituted the Lorentzequations in (4) to begin with. But what is of importance here is that wehave gained an understanding of each term in the Lorentz equations. Itis a straightforward matter to prove that the average angular frequency ωβcorresponds to the rate of S-clocks as they pass a fixed point in S ′. In contrastto time dilation, the period of these clocks has clearly been contracted:

T ′a = T√

1− v2/c2. (16)

This is the time compliment of length contraction corresponding to modu-lated wavelength, both of which are now proportional. Furthermore, earlierit was said that here we would discover a new definition of “time in gen-eral” that was both legitimate and believable. Since all clocks in S aresynchronous, then an observer at the origin of S can measure the time ofany event whatsoever, which will also be the reading on all clocks in thatsystem. Therefore, to an observer at rest in S ′, the reading on these clocks,and consequently the “time of the event”, will appear to be contracted.

It is also important to note that Eq’s (9) and (10) represent ‘emerging’values of frequency and wavelength that come from the fact that the lengthsand times which appear constant in S - that is, they appear as ‘pure’ lengthsor times - no longer do so when viewed from S ′. The emergent frequencycan be deduced along the following lines: at each fixed location x in S, thewave θw = (ωt − kx) will oscillate at the same rate as the hands on theclock face θt = ωt but will lag by a constant angle θx = kx. However, astime increases in S ′, the difference between the time and wave readings is nolonger constant but grows progressively larger. Since frequency is measuredaccording to x = vt in S ′, we have

θx = −kx = −kvt. (17)

Using t′ = t√

1− v2/c2 and substituting for t gives (9).3 Similarly, a wave-length appears from the pure time θt of S when viewed from S ′. At any

3Or we can look at it this way: at t = t′ = 0 and x = 0, kx = 0. But at t = λ/v,

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instant in time for S, all clocks are synchronous so that θt = ωt always hasthe same reading at each location. However, when these clocks are viewed atan instant in time in S ′, not only do the clocks appear to be non-synchronous,they also have a definite wavelength. Since wavelength is measured accordingto x = c2t/v in S ′, we have

θt = −wt = wvx/c2t. (18)

Using x′ = x√

1− v2/c2 and substituting for x gives (10).4 Again, whatappears as a pure temporal quantity in S is a mixture of both space andtime in S ′. Of course, this non-synchronicity of clocks is well known instandard relativity, but it is only half the picture.

But perhaps most importantly, the complete wave values Eq’s (14) and(15) reveal the fact that the Lorentz Equations may be thought to representthe wave arguments of standing waves; that is their meaning. That this isat all possible should really come as no surprise. Given a simple sine waveof the form ψ(x, t) = A sin 2π(νt ± σx) = sin 2π(n ± m), where n and mrepresent the number of cycles per second and length, it follows that t = nTand x = mλ. Using elementary calculus we have

∂ψ/∂t = (∂ψ/∂n)(∂n/∂t) = (∂ψ/∂n)ν (19)

∂ψ/∂x = (∂ψ/∂n)σ (20)

and, by repeated applications to obtain higher order derivatives, the WaveEquation, Maxwell’s Equations, the Shrodinger and Dirac Equations - inshort, any DE giving sine wave solutions - can be readily expressed in thesenew variables.5 Thus, space and time are not “independent variables” at allbut are quantities defined in proportion to observable period and wavelength,

the origin of S′ has moved a distance x = λ according to S and kx = 2π. Substituting(x, t) = (λ, λ/v) into the Lorentz time transformation, taking the inverse and multiplying2π gives (9).

4Taking the times from t = 0 to t = T , then the clock in which one cycle has beenturned must be at the location x = c2T/v. Substituting the coordinates (x, t) = (c2T/v, T )into the Lorentz space transformation and taking 2π times the inverse gives (10).

5There is no a priori reason to suppose that these would not transfer to the differ-ential operators where the N ′th derivative is ∂N/∂tN = (∂N/∂nN )νN and ∂N/∂tN =(∂N/∂mN )σN . I am reminded of the quote by John Wheeler: “Should we be preparedto see some day a new structure for the foundations of physics that does away with time. . . yes because ‘time’ is in trouble”.

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the factor of proportionality being the number of cycles. Since dimensionlessnumbers do not transform, then we can only conclude that the Lorentz andDoppler equations are in fact the same equations in varying form. Further-more, it is also implied here that something of the so-called “solutions” tothese DE’s might have been known, or at least suspected, in advance. Thisserves as a reminder that no one has or could ever observe a DE; it is onlyvia wave “solutions” that a DE can be tested against physical reality. So re-turning to relativity, Einstein’s initial condition of finding an invariant formfor Maxwell’s Equation’s assumed that this invariance would automatically“filter down” into the wave solutions. It did not take into sufficient accountthat the waves too needed to meet this requirement.

However, we have also uncovered another possible fallacy in the standardinterpretation, the belief that dispersive waves represent some type of specialseparate category of waves which do not obey the principle of superposition.Assuming that the wave in question is a matter wave, we set ω = m0c

2/~giving

ω′ = ω′a = (m0c2/~)β, (21)

k′ = k′a = −(vω/c2)β = −(m0v/~)β, (22)

satisfying the dispersion relation

ω′a =√ω2 + c2k′2a . (23)

Clearly, matter waves correspond to the averaged values which were neverexpected to obey the superposition principle. In other words, the ‘problemof the square root’ has been circumvented. Once again this is not completelyunexpected. Given the ratio of two frequencies we have ν2/ν1 = ν2T1. But aswe have just seen, the product of frequency and time equals the number ofcycles.6 Since a) the formation of Fourier series depend upon the calculationof ratios and b) cycles/ratios do not transform, then relativity demands thatthe principle of superposition remains invariant for all wave-types in everysystem of coordinates. It was always absurd that a periodic wave in onereference frame might become aperiodic in another. The false premise seemsto be that the term frequency applies to that of a single sine wave and notto the GCD between sine wave components.7

6Of course the same applies for σ2/σ1 = σ2λ1.7The invariance of the number of cycles/ratio should really be exalted to the level of a

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Looking back at Einstein’s original article Does the Inertia of a BodyDepend Upon its Energy-Content? [2], we have:

Let there be a stationary body in the system (x, y, z), and let itsenergy - referred to the system (x, y, z) - be E0. Let the energy ofthe system (ξ, η, ζ), moving as above with the velocity v be H0.Let this body send out, in a direction making an angle φ with theaxis of x, plane waves of light, of energy 1

2L measured relativelyto (x, y, z), and simultaneously an equal quantity of light in theopposite direction. . . From this equation it directly follows that :-If a body gives off the energy L in the form of radiation, its massis diminished by L/c2.. . . The mass of a body is a measure of itsenergy-content.

Seeing that Einstein invokes the use of equal-opposite waves to derive mass-energy equivalence, it is now clear why E ′ = m0c

2β corresponds to the aver-aged value of frequency.

3 Reinterpretation of the Minkowski IntervalIt is known from conventional relativity theory that to each and every space-time event there corresponds an invariant space-time four vector, or MinkowskiInterval. Given

s2 = c2t2 − x2 = c2t′2 − x′2 (24)the interval s2 remains invariant under a general Lorentz transformation.For convenience, let us consider S ′ as the proper frame-of-reference. Thenaccording to the customary view, there are three possible cases:

1. s2 represents a time-like interval: If c2t2 > x2, then s2 = c2t′2, x′ = 0and t′ is known as the proper time.

2. s2 represents an interval which is space-like: If c2t2 < x2, then s2 =−x′2, t′ = 0 and x′ represents the proper length.

3. s2 signifies a light-like interval: c2t2 = x2 and s2 = 0.

first principle and credit given to Pythagoras (b. 570 BC). For it was he and not Euclid(b. 330 BC) who was the first to exclaim that “Everything is logos”, where in this contextlogos is the ancient Greek word for the Latin Ratio.

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Two events which are causally connected must be associated by a time orlight-like interval. However, this four-vector has a very specific interpreta-tion in wave theory. If S ′ is the measuring frame, then the “events” to beconsidered - and they are the only events relevant for wave theory - are themeasurements of time and space in S relative to S ′:

1. The time-like interval always represents the measurement of the S-clocks relative to a fixed location in the S ′-frame. If we consider themeasurement to be taken at the origin of S ′, then the coordinates ofthis event must be: (x = vt, t) and (x′ = 0, t′ = t

√1− v2/c2) and the

interval becomes:c2t2 − x2 = c2t′2. (25)

Note that the time transformation represents contraction from S to S ′and also that c2t2 > x2 is satisfied for these coordinates.

2. Similarly, the space-like interval always signifies the general transfor-mation of length from S to S ′. Taking this measurement at zerotime in S ′, the coordinates of this event must be (x, t = vx/c2) and(x′ = x

√1− v2/c2, t′ = 0) giving:

c2t2 − x2 = −x′2. (26)Evidently c2t2 < x2 is true for these coordinates.

Of course, both of these come together on the cusp of the light cone for elec-tromagnetic waves, giving light-like intervals.

Corollary: A useful way to keep track of these equations is to rememberthat the space and time variables of the ‘other’ frame now represent cycles.For example, taking x = 0 (or x′ = −vt′ in the inverse transformation)indicates that it is time that is being measured in S and cycles that are beingmeasured in S ′. Therefore, if we substitute t′ by n′, then t′ = tβ becomesn′ = tβsec−1 and n′/t = βsec−1. Similarly, taking t = 0 (or x′ = −c2t′/v inthe inverse) shows the condition for spatial measurement. x′ = xβ becomesm′ = xβm−1 and m′/x = βm−1.

4 The Transverse Doppler EffectNowhere can we find a clearer and more typical illustration of the allegedrelationship between time dilation and the transverse Doppler effect than

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in Resnick and Halliday’s Basic Concepts in Relativity and Early QuantumTheory [4]. Chapter 2.7 entitled Aberration and Doppler Effect in Relativitybegins by placing a source of plane monochromatic light waves at the originof the primed or S ′-frame of reference. After deriving the transformationequations for frequency and wavelength, the chapter continues on to even-tually explain how this effect can be interpreted by time dilation, therebyconfirming once again the predictions of Special Relativity. The transforma-tion equations are given as:

ν = ν ′(1 + v cos θ′/c)√1− v2/c2

, (27)

ν ′ = ν(1− v cos θ/c)√1− v2/c2

. (28)

What is interesting in the following passage is the extra trouble that theauthors have gone to, no doubt unwittingly, to interpret these equations interms of time dilation:

In the procedure that we have adopted here, we start with a lightwave in S ′ for which we know λ′, ν ′ and θ′ and we wish to findwhat the corresponding quantities λ, ν and θ are in the S -frame. . .More striking, however, is the fact that the relativistic formulapredicts a transverse Doppler effect, an effect that is purely rel-ativistic, for there is no transverse Doppler effect in classicalphysics at all. This prediction follows from (28), ν = ν ′

√1− v2/c2/

(1− v cos θ/c), when we set θ = 900, obtaining

ν = ν ′√

1− v2/c2. (29)

If our line of sight is θ = 900 to the relative motion, then we shouldobserve a frequency ν which is lower than the proper frequencyν ′ of the source which is sweeping by us. Ives and Stilwell [9] in1938 and 1941, and Otting [12] in 1939 confirmed the existenceof this transverse Doppler effect, and more recently Kundig [13]obtained excellent quantitative data confirming the relativisticformula to within the experimental error of 1.1 percent.

16

It is instructive to note that the transverse Doppler effect hasa simple time-dilation interpretation. The moving source is re-ally a moving clock, beating out electromagnetic oscillations. Wehave seen that moving clocks appear to run slow. Hence, we seea given number of oscillations in a time that is longer than theproper time. Or, equivalently, we see a smaller number of oscilla-tions in our unit time than is seen in the unit time of the properframe. Therefore, we observe a lower frequency than the properfrequency.

Now, observe that instead of taking θ′ = 900 and using (27) directly,which would be the case if S ′ was the source as it was originally intended,the inverse Eq (28) is used, which is then inverted. What was to be theangle the wave makes in S ′ now becomes the “line of sight” in S. In the caseof the longitudinal Doppler effect which preceded this section, the resultsare identical and so the oversight may be understood. But as we shall seepresently, for the the transverse case not only are they no longer identicalbut they also give inverse results. Furthermore, when it comes to the time-dilation interpretation, why would we assume that a “moving clock, beatingout electromagnetic oscillations” is somehow more fundamental than the fre-quency of each of those electromagnetic waves that our hypothetical clockis supposedly beating out? First the time is transformed back-to-front andthen the number of oscillations is counted after the fact. This same problemarises with the so-called twins paradox where it is tacitly assumed that thelight signals used to communicate the “time” of the other twin, blue shiftedin one direction and red shifted in the other, do not carry their own time. Itdoes not take into consideration the possibility that the coordinate systemsthemselves, without which the velocity of the twins could not be measuredin the first place, are identical to the phase and group velocities of thosesignals which, since they occur in both directions, once again have the formof standing waves. This will be covered in the next chapter.

Returning to the problem at hand, reinterpreting the equations in termsof the original plan, let us take S ′ as our source which is emitting electro-magnetic waves at an angle θ′ = 900 relative to the line between source andreceiver. What is the measured frequency for an observer at rest with respectto S? Using Eq (27) directly, we obtain:

ν = ν ′√1− v2/c2

. (30)

17

That is, the direct inverse of (29). Expressed in terms of period, this becomes:

T = T ′√

1− v2/c2. (31)

Thus, the time is S ′ appears to be contracted when viewed by an observer atrest in S. Of course we could have achieved the same result in inverted formby taking θ = 900 in (28), in which case time would appear contracted in S ′.Needless to say, SR makes other predictions other than time dilation whichis only one way of interpreting the Lorentz equations. The fact that Ivesand Stilwell, Otting and Kundig confirmed the existence of this transverseDoppler effect does not necessarily mean that they proved the existence oftime dilation rather than time contraction.

Figure 3:

Corollary: Figure 3 shows another standard method used to demon-strate time dilation based on Pythagoras’ Theorem. Since the speed oflight is constant in all frames then the hypotenuse of the triangle must be√

(12vt′)2 + L2 = 1

2ct′ and an easy calculation gives t′ = (2L/c)/

√1− v2/c2.

Since 2L/c is the total time for the journey in the original frame, then it isclearly seen that time is dilated. However, given the frequency transforma-tion ν ′ = ν(1−v cos θ/c)/

√1− v2/c2, since the light is travelling wholly along

18

the y-axis we have θ = π/2 for the first half of the journey and θ = (3/2)π forthe second. Either way we obtain ν ′ = ν/

√1− v2/c2 and T ′ = T

√1− v2/c2.

Time is contracted.

5 The Clock or Twins Paradox

Figure 4:

Figure 4 represents a standard Minkowski spacetime graph used to resolvethe so-called twins paradox of Special Relativity. That is, the ‘naive’ viewthat each twin should measure the other to have aged by the same ratesimultaneously, wherein lies the paradox. The apparent solution is that thetravelling twin experiences a jump discontinuity at the turning point, inwhich case the times are no longer symmetrical. The essence of the paradoxis contained in section 4 of Einstein’s electrodynamics[1]:

If there are two synchronous clocks at A, and one of them ismoved along a closed curve with constant velocity v until it hasreturned to A, which takes, say t seconds, then this clock will lagon its arrival at A by 1

2t(v/c)2 seconds behind the clock that has

not been moved.

19

However, the form that is now well-known was first proposed by Paul Langevinin 1911. In this thought experiment a traveller is sent on a round trip awayfrom the earth at a speed comparable to that of light and the two observerskeep track of one another by the sending of radio signals:

As they move away from each other with a velocity close to thatof light, each seems to flee before the electromagnetic or opticalsignals that were sent to the other, so that there will be a verylong time to receive signals that were emitted during a given time.The calculation shows that each of them will see the other livetwo hundred times slower than usual. During the year of thisdistant motion, the explorer will receive the news from Earth ofthe first two days after his departure, and during this year he willsee the Earth make the motions of two days. Moreover, for thesame reason, the radiation he receives from the Earth during thistime has wavelengths that are two hundred times greater due tothe Doppler principle. [8]

Once again we see the tacit assumption that the frequency and wavelengthof these radio signals do not qualify as ‘real’ time or space which, apparently,have been reserved solely for the province of mechanics. Furthermore, byusing the “simultaneity planes” in this figure, which are the space-axes of themoving twin expressed in the coordinates of the stationary one, we see anexplicit use of time contraction that was spoken of in the original Minkowskidiagram (1). Let us calculate these planes in the usual manner. The lineof the outward journey is given by x = vt (black line).8 Consider the firstplane (blue line) at the time t1 in the “stationary” twin frame, which wewill continue to call S. Since the slope of this plane is c2/v, we have x =vt1 = c2t1/v +Q and the equation for this plane is x = c2t/v + (v − c2/v)t1.The time to be transformed occurs at the point where this line interceptsthe t-axis: x = 0 = c2t/v + (v − c2/v)t1 and solving for t produces t =t1(1 − v2/c2). Substitution of this point (0, t1(1 − v2/c2))9 into the LorentzTime Transformation we obtain:

t′1 = t1(1− v2/c2)/√

1− v2/c2 = t1√

1− v2/c2. (32)8For convenience, I will adopt the other convention of taking the time axis simply as t.9Note that this point is not the event being transformed but is a projection into the

coordinates of S. The point being transformed is (vt1, t1).

20

Mathematically speaking, there is nothing here that contradicts the standardtime dilation interpretation. The essential difference is that in the usualinterpretation the condition x′ = 0 is taken to be the proper frame whereaswave theory requires that this is now interpreted as the frame that is carryingout the measurement. The “event” so to speak is located in S ′ but the onusof the event lies with the reading of the S clocks. Therefore, if the traveller inS ′ measures the time in S by taking the rate of Einstein’s “lattice” of clocks,he will deduce that the rate is higher than his own and the time has beencontracted.

For the return leg of the journey, take S ′′ as the moving frame. If τ isthe total time for the journey relative to S, then the equation of this lineis easily calculated to be x = −vt + vτ/2 and that of the plane of motion(red lines) taken at the time t2 is x = −c2t/v + (c2/v − v)t2 + vτ/2. Takingx = 0 and solving for t gives t = t2(1 − v2/c2) + v2τ/2c2. Substitution intothe Lorentz equation yields:

t′′2 = t2√

1− v2/c2 + v2τ/2c2√

1− v2/c2. (33)

If we take the period between two times, then it is evident that the extraterms will be subtracted out. Therefore, the rate of S clocks as measured inS ′′ is still time contracted. This was to be expected since we are here dealingwith the transverse Doppler effect and not the longitudinal.

Finally, let us calculate the time difference at the point where all threelines (blue, red and black) coincide half-way through the journey. Sub-stituting t1 and t2 in (32) and (33) by τ/2 and subtracting gives ∆τ ′ =v2τ/2c2

√1− v2/c2. Multiplying throughout by unit frequency shows that

there will be a jump in the number of cycles read on the S clocks as S ′moves to S ′′. However, this has no bearing on the final result. At the timeτ for the total journey according to S, the moving twin does not come torest at all but merely passes the stationary twin at the origin of coordi-nates. Therefore, the time as read on the S clocks according to S ′′ for thereturn trip is ∆t′′ = (1/2)τ

√1− v2/c2 and the total time for the journey is

τ ′ = τ√

1− v2/c2. Apparently, the moving twin now deduces that the siblingis younger. However, if we repeat this process from the point of view of theS ′-S ′′ twin, keeping in mind the fact that we are here dealing with a purelyidealistic “inertial” universe which does not recognise a change in direction,the resultant graph is merely a reflection of Fig .(4) around the t-axis witha change of coordinates to t′ and −x′. Therefore, it is very easy to prove

21

that the rate of S ′ and S ′′ clocks relative to S have also been contracted byprecisely the same amount.

There is perhaps nothing paradoxical in this result since all frames arereckoning their measurements against Einstein’s “lattice” of synchronisedclocks, now identified as standing waves. In this world, the waves map aclass of congruent modulo T ’s onto the time line so that, in a manner ofspeaking, there is no “past” or “future” to speak of. The reading of “7 o’clock”, to run with Einstein’s example, just means that the wave is 7/12th’sof the way through one of its many cycles and does not point to anythingoutside of this fact. By contrast, it does seem a stretch to believe that thereexists some type of causal connection between the one twin reading a jump inthe number of cycles due to a reversal in direction and the biological ageingprocesses of the other. In other words, perhaps modern mechanics merelyexpected too much from its numerical representation of “time”?

6 Time and Relativistic AccelerationIt is beyond the scope of this article to give a full account of how to deducethe correct relativistic Doppler equations between inertial and acceleratedsystems of coordinates. These will be better introduced in an article solelydevoted to them. Nevertheless, since the use of accelerated frames is oftenused as a final bastion in defence of time dilation and against the so-called“common sense view”, a broad outline will be given together with the finalresult. Once again it is seen that the standard time dilation interpretationcan quickly be inverted to favour the time contraction more appropriate forwave mechanics.

Let us consider the transformation of a monochromatic light wave froman inertial system S to an accelerated system S ′. Following the methodologyfound for inertial systems , it is necessary to substitute the general trans-formation equations for space and time into the wave argument. By defini-tion, the coefficients of space and time will correspond to the wavenumber(inverse wavelength) and frequency, respectively. The space-time transfor-mation equations from primed to unprimed coordinates are given as:

x = (c2

g+ x′) cosh(gt′/c), (34)

t = ( cg

+ x′

c) sinh(gt′/c). (35)

22

where g is the intrinsic acceleration.10 Substitution of x and t into the waveargument yields:

(ωt− kx) = (ω( cg

+ x′

c) sinh(gt′/c)− k(c

2

g+ x′) cosh(gt′/c)). (36)

Using ω/k = c, then the coefficient of x′ is derived as:

k′ = k(sinh(gt′/c)− cosh(gt′/c)). (37)

It is evident that the remaining terms ωc/g((sinh(gt′/c)− cosh(gt′/c)) mustcorrespond to frequency. However, since t′ is not explicitly present (i.e. lieoutside of the hyperbolic functions), then the frequency must be obtainedby differentiation with respect to t′ i.e. the remaining terms here equals thenumber of cycles per t′ which, by definition, is the integrand of frequencywith respect to time. Hence,

d

dt′(ω cg

(sinh(gt′/c)− cosh(gt′/c)) = ω(cosh(gt′/c)− sinh(gt′/c)) = ω′. (38)

Using the identities cosh(gt′/c) = 1/√

1− u2/c2 and sinh(gt′/c) = (u/c)/√

1− u2/c2

the equations take on a more familiar form as:

k′ = (k − uω/c2)√1− u2/c2

, (39)

ω′ = (ω − uk)√1− u2/c2

. (40)

Surprisingly, these are identical in form to the standard relativistic Dopplerequations except that now the velocity between reference frames is no longerconstant. If u is increasing, then it follows that ω′ will be decreasing withincreasing time, which would imply that its period is increasing. However,since these are instantaneous values without magnitude, then the deductionof length and time contraction is not such a straightforward matter. Evenso, the basic outline for time contraction can be given.

10Taking the acceleration and velocity in S as a and u, respectively, then g =a√

1− u2/c2−(3/2).

23

Using (35), lett2 = ( c

g+ x′2

c) sinh(gt′2/c),

t1 = ( cg

+ x′1c

) sinh(gt′1/c),

such that

t2 − t1 = T = ( cg

+ x′2c

) sinh(gt′2/c)− ( cg

+ x′1c

) sinh(gt′1/c) (41)

where T is the time required for the clocks in S to pass through one cycle.However, since time-frequency must be measured at a fixed location in space,we have x′2 = x′1. Taking this point to be the origin, we obtain

T = c

g(sinh(gt′2/c)− sinh(gt′1/c) (42)

Solving for t′2 in terms of t′1 produces

t′2 = c

gsinh−1(sinh(gt′1/c) + gT/c). (43)

But this is just the inverse transformation equation

t′2 = c

gsinh−1(g

c(t1 + T )). (44)

Thus, the observable frequency becomes

f ′ = 1(t′2 − t′1) (45)

where it is recognised that we have a continuum of overlapping frequencies,each existing for the time of its own period T ′ = t′2 − t′1. This represents theobservable rate of synchronized clocks in S measured by an observer in theaccelerating system S ′. It is seen that this frequency becomes increasinglylarger as time progresses in S ′, which means that the period of the S clocksappears to be continuously contracting. Assuming that deceleration occurson the return leg of the round trip, the contracted times in S according toS ′ once again becomes increasingly larger. This would seem to imply thatthe “stationary” system S appears to age at a faster rate according to theobserver S ′ in the accelerating frame. Interestingly, in contrast to the resultobtained in the previous section, this seems to affirm the possibility that the“twins” may age at different rates.

24

7 Deducing the Doppler Equations from Ein-stein’s Principles

If the Lorentz and Doppler equations are truly equivalent, as t = nT andx = mλ would suggest, then it stands to reason that we should be able todeduce the Doppler equations directly from Einstein’s principles. Followinga mathematical procedure similar to Einstein’s original deduction for theLorentz Equations, we will use the principle of relativity and the principle ofthe universal constancy of the speed of light, together with our previously ac-quired principles, to first develop the transformation equations for frequencyand wavelength. The Lorentz Equations are then easily obtained by sub-stituting these equations into the wave argument and collecting coefficientsof frequency and wavelength. The fact that this is at all possible, that wecan derive the Doppler equations directly from Einstein’s principles withoutthe intermediary of the Lorentz equations, proves that they must enter intophysics on at least an equal footing.

We measure the (angular) frequency ω and wavenumber k in one inertialsystem S and wish to find the equations of transformation relating the fre-quency ω′ and wavenumber k′ for another inertial system S ′.

1. In the first place, it is clear that our equations must be linear. Oth-erwise, we would arrive at the absurd state of affairs that a periodicmotion in one inertial frame might be aperiodic in another, or vice-versa, a clear violation of the relativity principle. In other words, it isnecessary to conserve the principle of superposition under transforma-tion. Thus, we expect our equations to be of the form

ω′ = a11ω + a12k,

k′ = a21ω + a22k. (46)

2. We wish not only to obtain the transformations for light waves but alsothose of any velocity ω/k = u whatsoever, the only condition being thatthe speed of light is constant in the event that u = c is chosen. If wethen take the inertial frame at rest with respect to the wave such thatu = v, we would expect ω′ = 0. Further, in the limit where v � c, ourequation should reduce to the classical transformations ω′ = (ω − vk)and k′ = k. On both counts, given vk = ω , then it is easy to see that

25

ω′ = 0 would require a12 = −va11. Thus, our equations take on theform

ω′ = a11(ω − vk)

k′ = a21ω + a22k. (47)

3. In the event that u = c is chosen, then from the wave equation wehave ω2 = c2k2 and, by the principles of relativity and the universalconstancy of the speed of light, require ω′2 = c2k′2. From this lastequation we obtain

a211(ω − vk)2 = c2(a21ω + a22k)2. (48)

Rearranging terms according to ω2 = c2k2 produces

(a211 − c2a2

21)ω2 − 2(va211 + c2a21a22)ωk = (a2

22c2 − a2

11v2)k2. (49)

Thus, we have three equations in three unknowns

(a211 − c2a2

21) = 1,

(va211 + c2a21a22) = 0,

(a222c

2 − a211v

2) = c2, (50)

the solutions of which are

a11 = a22 = 1/√

1− v2/c2,

a21 = −(v/c2)/√

1− v2/c2. (51)

4. Finally, substitution into our original equations (47) produces the soughtafter relativistic transformation equations for Dopplers principle.

ω′ = (ω − vk)/√

1− v2/c2, (52)

k′ = (k − vω/c2)/√

1− v2/c2. (53)

26

It is evident that these are the standard equations for the Doppler Effect,the only difference being the manner in which they were here derived. Thisimplies that the initial principles were correct.

The Lorentz equations can now be seen to be derived from frequency andwavenumber. Substituting the Doppler equations into the wave argument(ωt′ − k′x′) = (ωt− kx), and collecting like-terms of ω and k gives

((ω − vk)t′ − (k − vω/c2)x′)/√

1− v2/c2 =

(ω(t′ + (vx′/c2))− k(x′ + vt′))/√

1− v2/c2. (54)

Time and space in S are just the coefficients of ω and k:

t = (t′ + vx′/c2)/√

1− v2/c2, (55)

x = (x′ + vt′)/√

1− v2/c2. (56)

Of course, we could just as well have started in S ′ to obtain the inverseequations.

8 Further DiscussionWhen the de Broglie matter wave was experimentally confirmed in 1927,perhaps it should have become apparent that wave theory was no longermerely a “branch” of theoretical physics but its raison d’etre. Since then, awave theoretical definition of relativistic space and time has been immanent.Seeing that the space and time axes, which have the equations x = c2t/vand x = vt in S, are now identified with the phase and group velocities forobservable waves, then this implies that our sense of space and time comesfrom our accumulated perceptions of wave phenomena and not from the mea-suring rods and clocks of our own making. This is the “stage” so to speakupon which nature plays her part. Since light propagates as a wave, thenthis remains true when our array of clocks has been synchronised by light ac-cording to Einstein’s formula. This might initially seem like a reinstatementof the “luminiferous aether” theory except for the fact that light propagatesin vacuo, while the introduction of matter waves makes the whole questionmute. When Einstein states that “all of our judgments in which time plays apart are always judgments of simultaneous events”, this is the condition for

27

the measurement of moving lengths, not time, which requires instead thatit is measured at a fixed location in space. Basing our theories on the ob-servable fact that the speed of light is a universal constant is a step in theright direction for a science which claims to be empirical. Nevertheless, it isuntenable to regard this speed as preceding the observable period when theformer requires the latter for its definition as wavelength per period, spaceper time, and all together must be regarded as given facts of nature. For-tunately the inconsistency seems to lie solely in the interpretation and notin the equations themselves. The particle view of the existence of a singleunified spacetime never did tally with the separation of variables used rou-tinely and without a second thought throughout all of wave theory. To insistthat the laws of mechanics must be invariant for all coordinate systems andyet overlook the most basic tenet of wave theory, the principle of superposi-tion, bears witness to a branch of science that has perhaps overstepped itsboundary. It is absurd to insist that a harmonic complex whose componentfrequencies are integer multiples of a fundamental F0, which is also the great-est common divisor between these components and the very definition of aperiodic motion, would somehow, miraculously become irrational and ape-riodic when transformed to all other systems of coordinates. The negativedesignation of “non-dispersive” waves overlooks the simple fact that disper-sion applies to the average and modulation values, the sum and differencebetween frequency components, and not to the ratio/number of cycles whichis axiomatic and must remain inviolate. The simple equations t = nT andx = mλ state quite explicitly that time and length are defined in terms ofperiod and wavelength. This forms the basis of our very representations ofsinusoidal waves. To deny this would be a contradictio in adjecto and theentire domain of Fourier analysis, including quantum mechanics, would nothave legs to stand on. Thus, the non-dispersive condition that “the sum oftransformations equal the transformation of their sum” is simply a statementof the principle of relativity.

From a philosophical standpoint, by continuing in Einstein’s footstepsand basing our theories on empirical evidence, it is no longer necessary toassume space and time to be metaphysical a priori “intuitive categories”that somehow exist in advance of, or at least in separation from, perceptualexperience [9]. The assumption that space and time must be linear andhomogenous, for example, is quite easily replaced by the mere existence ofperiodicities within nature; science is predictable because nature allows it tobe so. Arguably, Hume’s problem of induction is solved in absentia because

28

the variable “t” does not represent subjective qualities such as memory oranticipation but the class of all possible wave motions which may containperiodic phenomena (see footnote 13). Otherwise science is no longer basedin physical reality. The uncertainty principle, for instance, defines a featureof Fourier transformation as we navigate between the frequency and timedomains. A more precise definition of the “wave-particle” duality, therefore,would be a “wave- wave packet” duality because the uncertainty pertainsonly to the location-momentum of the particle aspect and not to the wavesthat describe them. Applying uncertainty to the waves themselves would beto compound uncertainties and ultimately fall into infinite regress. More-over, it was always dubious to speak of a “space” that does not distinguishbetween the presence or absence of an object, of a “time” that “flows at anequable rate” [10] and yet is somehow also a measure of constant change. Itwas always somewhat an appeal to convenience to point to the world andsay “look, time is changing” and then to a clock and say “look, time staysthe same”. This same contradiction applies when we measure the frequencyof the Caesium atom, call it “time” and then insist that the two conceptsare separate. It applies when we attempt to use the Doppler equations toconfirm time dilation then deny this connection for every other possible in-terpretation. It applies when we speak of particle accelerators rather thanwave-particle accelerators in order to - and it must be said - secure fundingby appealing to the “common sense” view of the general public. By the sametoken, we could also argue that it applies when quantum physicists criticiseEinstein for believing in “hidden variables” while their own expectation val-ues are still clothed in the metaphysical language of an independent “spaceand time”11. Isn’t it at least just as plausible to believe that the matterwaves are “out there” even though no observation has been made? After all,despite its stochastic nature, it still has faith that the wave function will bethere “tomorrow as it was yesterday” [11]. Hence, contrary to widespreadbelief, quantum mechanics is still based in inductive reasoning.12

In terms of temporal perception, we remember events like the “train ar-11Observe, for example, that the variable “t” that decides the quantum state at any given

time is still assumed as separate from the “t” that appears in the Euler wave representation.12The argument of course is that our logic evolved or was conditioned by the world of

everyday affairs and was not designed to cope with the strange phenomena occurring at themicro-level. This, however, is a very logical explanation. Unfortunately, the CopenhagenAgreement is sometimes used to justify untenable arguments and to let basic contradictionsslide.

29

rived at 7’ o clock” because we built clocks, put them into the world andmade it so. In other words, they have insinuated themselves into our eventsafter the fact. Similarly, to speak of “empty space” now becomes as mean-ingless as trying to imagine a “non-wavelength”. Asked by a reporter of theNew York Times as to how mass warps space in Einstein’s general theory,Nicola Tesla stated:

It might be inferred that I am alluding to the curvature of spacesupposed to exist according to the teachings of relativity, butnothing could be further from my mind. I hold that space cannotbe curved, for the simple reason that it can have no properties.It might as well be said that God has properties. He has not, butonly attributes and these are of our own making. Of propertieswe can only speak when dealing with matter filling the space. Tosay that in the presence of large bodies space becomes curved, isequivalent to stating that something can act upon nothing. I, forone, refuse to subscribe to such a view.

[12]. The essential truth of the matter is that so-called empty space is literallyfilled with a wave-particle “zoo” of light and matter waves. We do not observethe motion of celestial bodies per se but only the light waves emanating fromthem. It is only by spectroscopy that we can deduce the chemical compositionof distant stars and planets. In Gravity and Light [13], Einstein is arguingfor the need to separate “absolute” time and frequency. After some initialcalculation, he states:

If we did not satisfy this condition, we should arrive at a definitionof time by the application of which time would merge explicitlyinto the laws of nature, and this would certainly be unnaturaland unpractical.

This not only implies that Einstein secretly believed time to be metaphys-ical, and therefore beyond the realm of observation, but also contains theuncharacteristically strange contradiction that if time were to merge withthe laws of nature it would somehow be unnatural. This doubt is furtherstrengthened by recalling Einstein’s now famous quote contained in a letterto his friend Michele Besso:

People like us, who believe in physics, know that the distinctionbetween past, present, and future is only a stubbornly persistentillusion.

30

However, this seemingly paradoxical situation might be circumvented whenit is identified that a class or universal is not identical to the empiricalelements of that class. It is not unreasonable to suggest that time, at leastin it’s enumerated and isotropic aspects, represents the class of all periodicphenomena, manmade or otherwise, and that this a priori class has formed aposteriori from these observable elements. Similarly, our perception of changealso finds an empirical explanation from the fact that aperiodic waves exist.In this way, time both merges with and remains outside the “laws of nature”.They are not mutually exclusive.

The currently held view that time and frequency are separate conceptsnot only contravenes the most basic empirical principle of theoretical physics- it is waves that are natural and observable, not measuring rods and clocks,imaginary or otherwise - but also violates the principle of relativity withrespect to the fundamental axioms of wave theory. The notion that the uni-versal class of waves are identified with space-time is at least no less general,and certainly more falsifiable13, than the idea that a material particle couldsomehow represent time as historical record. At any rate, since symmetry isnow perfectly attained, at least with regard to wave motion, then all of thebeauty and eloquence we have come to expect from Einstein’s equations isperhaps served better under this interpretation.

References[1] A. Einstein, On the Electrodynamics of Moving Bodies, Annalen der

Physik, 17, 1905.

[2] A. Einstein, Does the Inertia of a Body Depend Upon its Energy-Content?, Annalen der Physik, 17, 1905.

13A valid argument could be made that these definitions do not apply to Hume’s Problemof Induction or Popper’s fasifiability[14]. This follows from the fact that periodicity mapsa congruent modulo T onto the time-line so that these laws will be just as valid tomorrowbecause there is no tomorrow to speak of. In other words, this is not just a law thathappens to be T-invariant but is T-Invariance itself. This is perhaps more believable thana “time travel” which would have history, including that of scientific progress itself, subjectto alteration and thereby possibly undermining its own premise. On the other hand, theresults obtained in section 6 here do seem to imply that time itself may still be mutable.

31

[3] H. Minkowski, Space and Time, Address delivered at the 80th Assemblyof German Natural Scientists and Physicians, at Cologne, 21 September,1908.

[4] R. Resnick and D. Halliday, Basic Concepts in Relativity and Early Quan-tum Theory, Maxwell Macmillan International, 1992.

[5] G.F. Torres del Castillo and C.I. Perez Sanchez, Uniformly AcceleratedObservers in Special Relativity, Universidad Autonoma de Puebla, 2005.

[6] H. Weyl, Space Time Matter,, Dover Publications Inc., 1952.

[7] R. E. Turner, Relativity Physics, Routledge and Kegan Paul,1984.

[8] P. Langevin, L’evolution de l’espace et du temps, Scientia 10 31Ð54, 1911.

[9] I. Kant, Inaugural Dissertation, 1770.

[10] I. Newton, Philosophae Naturalis Principia Mathematica, 1687.

[11] D. Hume, An Enquiry Concerning Human Understanding, 1748.

[12] N. Tesla, Interview in The New York Times, July 10, 1932.

[13] A. Einstein, Gravity and Light, Annalen der Physik, 35, 1911.

[14] K. Popper, Logik der Forschung, 1934.

A Outline for General RelativityAs a preliminary sketch, perhaps all that is initially required to develop thesedefinitions into GR is to simply extend the separate Minkowski space andtime four-vectors just derived to include Einstein’s definition of space andtime curvature. In other words, given that Einstein’s metric for curvature gµυis a generalisation from the Minkowski metric ηµυ, we simply follow Einstein’sline of thought once again, except that now it is understood that separateinitial conditions must be applied for space and time. Furthermore, sincewe can no longer assume that Pythagoras’ theorem remains valid in curved

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space, we reinstate the three separate spatial dimensions. Thus, using theabove equations, the initial conditions would be given as:14

1. Invariant time interval:dτs

2 = dτc2t′2 = dτc

2t2 − dτx21 − dτx2

2 − dτx23 = ηµυdτx

ηdτxυ

is generalized todτs

2 = gµυdτxηdτx

υ.

2. Invariant space interval:dξs

2 = −dξx′2 = dξc2t2 − dξx2

1 − dξx22 − dξx2

3 = ηµυdξxηdξx

υ

becomes dξs2 = gµυdξxηdξx

υ.

Finally, seeing that many currently known solutions for the EFE’s are basedon the parameter of proper time, it may be necessary to reinterpret theseresults or else find yet another set of initial conditions and solutions for theinvariant space interval as well.

If we were to generalize this inertial theory by inverting the “reading” ofthe synchronised clocks within curved space-time and interpret it as a uni-versal class of standing electromagnetic and matter waves, then gravitationalmatter waves might be procured as a matter of course. This would automat-ically produce the motion of wave packets within curved space-time and allof the principles of quantum mechanics - uncertainty, duality and so forth -would still retain their meaning. There is no longer any need to invoke thead hoc theory of a hypothetical and undetectable graviton particle, whichwas never in agreement with Einstein’s initial explanation i.e. we don’t needto introduce some type of external agent to explain gravity, as if it were anobject. It is already written into the very fabric of GR as space-time curva-ture. The same can be said for all the recent flurry of activity surroundingthe search for gravitational waves. For even if some type of “ripples” in spaceand time were eventually detected, it would be utterly impossible to deter-mine whether or not this was the effect of gravity or gravity itself. Thatspace and time are already identified with observable waves makes the wholeenterprise an unnecessary complication.

14The forward subscripts τ and ξ have been added simply as a trace for time-like andspace-like elements. They have nothing to do with the indices µ and υ of the relevanttensors.

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