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Physics Essays Volume 16, Number 1, 2003

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Einstein and Space-Time Units

Minoru Harada

Abstract

A grave conflicting aspect is uncovered for the conventional wisdom in relativitytheory regarding space-time units. No account has ever been taken of the undeni-able fact that these units are contingent on nature, resulting in the present con-vention in which the same space-time units are used in all reference frames. Thisconvention runs counter to Einsteins denunciation of an absolute space and anabsolute time propounded by Newton. This conflict can be resolved only by ourrecent proposal [Phys. Essays 11, 521 (1998); 12, 368 (1999)] for introducingseparate space-time units for each reference frame.

Key words:space-time units, Einstein, relativity theory, paradoxes

The most conspicuous consequence of Einsteinstheory of relativity is that length and time aredependent on reference frames, which denies thenotion of an absolute space and an absolute time, aswas expounded by Newton. The familiar phenomenaof time dilation and the Lorentz contraction areassociated with this feature. To be more specific, thespatial extension of an object and the temporalduration of a phenomenon take different numericalvalues depending on reference frames. As a matter ofcourse, these values are expressed in the same units in

all reference frames. Moreover it is taken for grantedthat the alteration in these numerical values representsreal physical changes, as evidenced by the apparentdilation of the traveling muons lifetime.

This interpretation further leads to the asymmetricaging of twins who move relative to each other,which is believed to resolve the twin paradox.

It is noted, however, that the well-known flying-clock experiment by Hafele and Keating(1) does notprovide any empirical verification of this asymmetricaging because it was recently revealed by Kelly(2) thatthe results of this experiment were fabricated.

It should also be noted that there is an indisputableinconsistency in the traditional interpretation of timedilation. According to this interpretation, the above-mentioned asymmetric aging is solely due to theacceleration experienced by the traveling twin, withthe relative motion at constant speed playing no role.In the problem of the muons lifetime, however, noacceleration is involved and the dilated lifetime isalleged to be a consequence of constant relativemotion, contrary to the case of the twin paradox.

In the present paper we take issue with the conven-tion for the space-time units as usually adopted inrelativity theory with an eye to eliminating the inconsis-tency just mentioned. In the conventional wisdom theseunits are regarded as theoretical constructs just likemathematical definitions. Thus no account is taken hereof the fact that space-time units are contingent onnature. For example, time units were once defined byusing the rotation of Earth, and nowadays they aremore reliably defined by utilizing the cesium atomSimilarly, length units were based on the meridian of

Earth, although the present-day definition rests onlight in addition to the cesium atom.These circumstances clearly indicate that space-

time units are physical quantities, implying that theuse of the same length and time units in all referenceframes is tantamount to the introduction of hypotheti-cal entities that are not subject to relativity theoryThis is because space-time units can only be definedon the basis of physical entities, which are governedby relativity theory.

In fact, Einstein once said,(3) The idea of themeasuring rod and the idea of the clock coordinated

with it in the theory of relativity do not find theirexact correspondence in the real world. In otherwords, Einstein maintained the relativity of space andtime measures, though he used the same name fortheir units. For example, 3 seconds in the expres-sion of a law of nature in a stationary reference framemight be called 2 seconds in the expression of thesame law in a frame that is in motion relative to thestationary one. The units here go by the same name ofseconds in both cases.

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This type of absolutism, however, is essentially thesame as Newtons idea of an absolute space and anabsolute time in that space-time units entail physicalentities defining these units.

On second thought, Einstein later said,(4) ... strictlyspeaking, measuring rods and clocks would have tobe represented as solutions of the basic equations(objects consisting of moving atomic configurations),not, as it were, as theoretically self-sufficient entities.

Apparently, Einstein was on the horns of a dilemmabecause the absolutism of space-time units collideswith the nature-contingency of the measuring rod andthe clock as implied by the latter statement.

Strangely enough, however, Einstein did not pursuethis problem and left it as an open question: If onedid not wish to forego a physical interpretation of thecoordinates in general, ... it was better to permit suchinconsistency with the obligation, however, of

eliminating it at a later stage of the theory.

(4)

In our view, what underlies the problem perceivedby Einstein is the unproved assumption that space-time units are independent of reference frames. Thisobservation thus points to the need to introduceseparate units of length and time for each referenceframe, as we have done in Refs. 5 and 6.

Specifically, we have introduced two types of meterand second as defined by

12 2

1 m-at-rest 1 m-in-motion,v

c

=

(1)

12 2

1 s-at-rest 1 s-in-motion.v

c

=

(2)

In (1) m-at-rest refers to the meter to be used forobjects in a stationary reference frame and m-in-motion refers to the meter to be applied to movingobjects, when viewed from the standpoint of the

stationary observer. v is the speed of the movingreference frame and c is the speed of light. Equation (2)has a similar meaning, with s representing second.

As shown in Refs. 5 and 6, these formulas werederived by using the Lorentz transformations,implementing our observation that space and timeunits are contingent on nature.

The salient feature of the new convention for space-time units is that it signifies that no physical changesare involved in time dilation or the Lorentz contrac-

tion, thus eradicating all paradoxes in relativitytheory, as elucidated in Refs. 5 and 6.

This result is only natural from our viewpointinitiated by Sachs,

(7,8)that the Lorentz transformations

simply induce the scale change of spatial or temporalmeasure and have nothing to do with physicalchanges. This viewpoint is no more than the embodi-ment of the principle of covariance, which is the basicpremise of relativity theory.

This principle signifies that space-time transforma-tions are only descriptive, kinematic relationsprescribing the rules for translating the languageparameters from one reference frame to another so asto preserve the objectivity of the expressions of thelaws of nature.

(9)

It is noted that the separate units for each referenceframe do not imply the nonuniversality of the intrinsicproperties of physical entities. The altered units only

represent changes in the description of invariablephysical entities, depending on the reference frame.Equations (1) and (2) have much analogy to the

exchange rate relation in economics, such as 1 U.Sdollar = 1.5 Canadian dollars. No Americans thinkthat they gain money by visiting Canada thanks to therelation 1 < 1.5. The same applies to the case of themuons lifetime. The alleged dilated lifetime for thetraveling muon is a consequence of the neglect of thefact that the lifetime is expressed in different units inthe traveling and stationary systems; the secondappearing in the proper lifetime as quoted by the

stationary observer should be taken as the second-in-motion, and the second used for the observedlifetime in the stationary system should be taken asthe second-at-rest.

In summary, we have shown that the logical inconsis-tency hiding out in the present convention for space-time units in relativity theory can only be eliminated by

our recent proposal(5,6) to introduce separate units of

length and time for each reference frame.

AcknowledgmentsI wish to dedicate this paper to my friend and

colleague Mendel Sachs, who introduced me to hisunparalleled but authentic philosophy of relativitywhich lies at the core of the present discussion. I firstbecame aware of Einsteins statements as quoted hereby reading Sachss writings (Ref. 9). I am muchindebted to him for calling my attention to Ref. 9 andfor helping me to improve the paper. He alsotranslated the abstract into French for me.

Received 10 April 2001.

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Einstein and Space-Time Units

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RsumUn aspect grave et contradictoire est dcouvert concernant la sagesse conven-

tionnelle de la thorie de la relativit, relativement aux lments de lespace-

temps. Aucun compte na dj t rendu du fait indniable que ces lments sont

contingents de la nature, rsultant dans la convention actuelle do les mmes

units despace-temps sont utilises dans tous les cadres de rfrences. Cette

convention est oppose la dnonciation dEinstein dun espace absolu ainsi quede temps absolu tels que soutenue par Newton. Cette contradiction ne peut tre

rsoute que par notre proposition rcente [Phys. Essays 11, 521 (1998); 12, 368

(1999)] pour lintroduction dunits despace-temps spares pour chaque cadre

de rfrence.

References1. J.C. Hafele and R.E. Keating, Science 177, 166,

168 (1972).

2. A.G. Kelly, Phys. Essays 13, 616 (2000).3. A. Einstein, Sidelights of Relativity (Dover, New

York, 1983), p. 35.

4. P.A. Schilpp (editor), Albert Einstein Philoso-

pher-Scientist(Open Court, La Salle, IL, 1949).

5. M. Harada and M. Sachs, Phys. Essays 11, 521(1998).

6. M. Harada, Phys. Essays 12, 368 (1999).

7. M. Sachs, Einstein Versus Bohr(Open Court, LaSalle, IL, 1988), Chap. 9. Also see the Japanese

translation by M. Harada and K. Sugimoto (Ma-

ruzen, Tokyo, 1991).8. M. Sachs, Relativity in Our Time (Taylor &

Francis, London, 1993), Chaps. 10 and 11. Also

see the Japanese translation by M. Harada (Nip-pon-Hyoronsha, Tokyo, 1998).

9. M. Sachs, Found. Phys. 15, 977 (1985); Relativity in Our Time (Taylor & Francis, London1993), pp. 7778.

Minoru HaradaInstitute of Liberal ArtsOtaru University of Commerce, Otaru

Hokkaido 047-8501 Japan

e-mail: mharada@res.otaru-uc.ac.jp