The standard of length in the theory of relativity and Ehrenfest paradox

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  • IL NUOV0 CIMENT0 VOL. 74 B, N. 2 l l Aprile 1983

    The Standard of Length in the Theory of Relativity and Ehrenfest Paradox (*).

    W. A. RODRIGVES jr.

    Instituto de Matemdtica, Estatistica e Cidncia da Computafdo IMECC-UNICAMP, Caixa Postal 6155 - 13100 Campinas, SP, Brazil

    (ricevuto il 28 Giugno 1982; manoscritto revisionato ricevuto il 24 Gennaio 1983)

    Summary . - We investigate the physical systems which can be associated with the standard clocks and standard rulers of the theory of relativity. We show that, once the standard clock has been identified, the standard ruler is uniquely determined as being a ((light ruler )>. We then investi- gate under what conditions material rods can be used as standard rulers. This shows the existence of two distinct contractions in the theory that are often confused: the Einstein and the Lorentz contractions. We argue that the Lorentz contraction is a real phenomenon which results as a consequence of the interaction of material bodies with the ground-state vacuum of the Universe. These results permit us to give a definitive answer to the question (( do metric standards contract? * and also to solve Ehrenfest's paradox in an almost trivial way.

    PACS. 03.30. - Special relativity.

    1. - In t roduct ion .

    The l i terature on the Ehrenfest paradox, publ ished in 1909 (1), is very ex-

    tensive, and, as is well known, there is no general agreement on the solution.

    As clearly stated by CAVALLERI (2)~ NEWBURGH (a) and also PHIPPS ('), there

    (*) To speed up publication, the author of this paper has agreed to not receive the proofs for correction. (1) P. ERRENFEST: Phys. Z., 10, 918 (1909). (~) G. CAVALLERI: Nq~ovo Cimento B, 53, 415 (1968). (a) R. G. NEWSURGH: NUOVO Cimento B, 23, 365 (1974). (4) T. E. PHIPPS jr.: l~ound. Phys., 10, 289 (1980).

    199

  • 200 W.A . RODRIGUES jr.

    are indeed three distinct questions connected with the paradox that are often confused. The first question, posed by EttRENFEST, concerns a disk, originally at rest in an inertial frame S, being set into rotatory motion. How does the geometry of the disk change for an inertial observer in S?

    The second question is related to the first and involves an observer at rest with respect to the disk. Originally the disk and the observer are at rest in an inertial frame. When both are set rotating, how does the disk geometry change for the noninertial, co-moving observer?

    The third question is the following (~): A uniform disk of radius R and cireunferenee 2zR as determined by an inertial observer S is postulated to exist. I tow does the geometry seen by a noninertial observer co-moving with the disk differ from that seen by the inertial observer?

    I t is quite clear at least to us that an answer to the above questions requires an understanding of the following points:

    i) What is the physical meaning of the Lorentz contraction?

    if) Do all material rods Lorentz-contract when set in motion?

    iii) Which are the physical systems which realize the standard ruler and the standard clock in the general theory of relat ivity?

    In the next sections we study these points and show how a proper under- standing of the operational foundations of the general theory of relat ivity leads to a reasonable solution to the question ~ do metric standard contract? ~> and the Ehrenfest paradox.

    2. - Standard clocks and standard rulers in the theory of relativity.

    Assuming the well-known geometrical f ramework of the theory of rela- t iv ity (6-8), we give in what follows the definitions of the standard rulers and standard clocks of the theory and identify them with physical devices.

    To fix the notation, we denote the events of the world manifold ]74 by El, E~, ... and label them in each local chart valid for U c V~ by four real numbers x ~ = (x ~ x 1, x ~, xa), x ~ being the timelike co-ordinate and x ~ (i = 1, 2, 3 ) the spacelike co-ordinates. The 3-dimensional space for a given obser- ver is defined as a spaeelike hypersurfaee Va in V4 (see ref. (7,s) for details). We use in what follows units such that c ~ 1.

    (5) H. ARZELI~S: Relativistic Kinematics (London, 1966). (s) A. EINSTEIN: The Meaning o] Relativity, fiftieth edition (Princeton, N.J., 1974). (7) R. K. SACHS and H. ~Vu: General Relativity ]or Mathematicians (New York, N.Y., 1977). (s) J. L. ANDERSON: Principles o] Relativity Physics (New York, N.Y., 1967).

  • THE STANDARD OF LENGTH IN THE THEORY OF RELAT IV ITY :ETC. 201

    The standard ruler (standard clock) of the theory is defined as a physical device that realizes the interval ds 2 ~-- - - 1 (ds 2 ---- 3- 1) in each point of V.~ (,,~o). When ds 2 > 0, dr = ds/c is called the proper time of the clock. We accept that atomic clocks provide the devices for measuring r (~,1~) (but see also the discussion of ref. (~s)). The theory also postulates that light follows null geodesics in V4, i.e. realizes the interval ds 2 = 0.

    The identification of the standard ruler of the theory as being a ((light ruler ~>, i.e. a device consisting of an atomic clock (that measures proper time), a light cannon and a reflecting system (mirror) follows at once if we remember how the temporal order of distinct events is defined (~4).

    Indeed, considering the event at the world-lines that pass through P . ,o x o and ~0 are the and P q -dPe V3, we see immediately that, if x~, x~, ~, ~

    timelike co-ordinates of the events E, E~, E2 and E' 1 in fig. 1 (at E1 the observer

    !

    i

    E.

    p P+ctP

    Fig. 1. - Diagram for the definition of simultaneity.

    at P sends a light signal to P Jr dP, where it arrives at the event E ' 1 and is then immedie~tely reflected back to P, where it arrives at the event E2), then

    (1) x~ 'x ~ x ~ 0

  • 202 w.A . RODRIGUES jr.

    line passing through P -~ dP, which has timelike co-ordinate x ~ -~ dx ~ with dx ~ : - - (g~o/goo)dx ~, where x ~ -t- dx ~ are the spatial co-ordinates of the point P ~- dP.

    The spatial distance between P and P -~ dP is then defined (~5) as dr/2, where dr is the proper time as measured at P for light to go from P to P ~- dP and come back. ~Te have

    (2) ( d~ ~ dl "~ : 7ij(x) dx i dx j ~]= 7i~(x) ---- - - giJ ~- goigoJ

    goo

    x= (x ~ 2,x 3), i, j = 1, 2, 3,

    The interval between the event E = (x ~ P) and the simultaneous event at P ~-dP will give exactly ds 2 ------1, for an appropriate value of dr, completing the identification of the standard ruler. ~u say that the light ruler is stress free.

    Now, in which conditions a material rod can be used as standard ruler? We define that an infinitesimal material rod is a standard ruler if, when its length is measured by the light ruler in the system where the rod is at rest, it is an invariant at all points of Vd. We say that in this case the material rod is Born rigid and call it the (( rigid ruler ~> for short.

    I t is important to observe that, if there does not exist in Nature a signal faster than light, then no rigid material in the classical sense can exist. The maximum rigidity compatible with the theory of relativity has been discussed by CLAI~K (16).

    We define when an infinitesimal material rod is a standard ruler, but it is not clear yet in what physical conditions a material rod is Born rigid. We study now this point.

    3. - The condit ions for a mater ia l rod to be a standard ruler.

    I t is an empirical fact that reference frames exist in Nature where g,v(x) ~-- ~l~,. = diag (~ 1 , - 1 , - 1 , - 1) over large regions of the world mani- fold Vd. These are the inertial frames. Thus, in an inertial frame, proper time is a direct measure of the timelike co-ordinate along the world-line of a given observer at rest in the inertial frame.

    Suppose we have a material rod at rest in the inertial frame S. I t is an

    (15) L. D. LANDAU and E. M. LIFSHITZ: The Classical Theory o] -Fields, fourth revised English edition (London, 1975). (16) G. L. CLARK: Proc. t~. Soc. Edinburg, Ser. A, 62, 434 (1947-48); also Proc. Cam- bridge Philos. Soc., 45, 405 (1949).

  • THE STA:NDAI~D OF LENGTH IN TI-YE THEORY OF RELATIVITY ETC. 203

    empirical fact (Michelson-Morley experiment) that the transit t ime of light from one end to the other and back is a constant independent of the position of the rod and its orientation in the inertial frame. Then according to our definitions the rod is Born rigid and can be used as a standard ruler.

    Let T be the period of the standard clocks at rest in S and let {L} be a standard material rod with length

    (3) L = 89

    as determined in S, by the light ruler method. Let S be an inertial frame which is moving with velocity v ---- v.~ (the stand-

    ard configuration). The standard material ruler {l} at rest in S is defined as the one that satisfies

    (4) 1 : 1T (in the S-frame),

    where T is the period of the standard clocks at rest in S. I f these clocks are of construction analogous to the clocks in S, then, according to the definition of the standard clock, T ---- T, and we have

    (5) t = /~.

    Now we have

    T ~ T~ (1 - - v2) -~ or - - = (1 - - v~) -~ , (6) T : T

    where T ~ is the period of the standard clock at rest in S as determined by S. Suppose now that the observer at rest in S (in S) wants to determine the

    length of the ((rigid ruler >> at rest in S (in S), i .e. he wants to determine 1 ~ (Ls). This determination needs a definition and the following is used: 1 ~ (L s) is the projection in S (in S) of the end points of the rod {1} (rod {L}) at a given instant of t ime T in S (T in S). i f all clocks in S (in S) have been synchro- nized (( el la Einstein >> (i.e. e ----- 89 in eq. (1)) we Ban immediately obtain the value of l ~ (L s) using the light ruler method. We obtain

    (7) ~(0 ,~)_ LV I - -v~ , v 2 sin e 0

    (8) Ls(O, T ) -= ~1 ~ v ~ sin e 0 -- V ' i - - v" sin e 0 '

    where {J (0) is the angle the rod {/} (rod {L}) makes with the X(X)-axes of the S(S)-frame. The results are independent of T or T.

  • 204 w.A . RODRIGUES jr.

    Now take one of the standard (( rigid rulers ~> {Z} at rest in S and accelerate it until it has velocity v, i.e. becomes at rest relative to the S-system. What is the proper length now, i.e. L, the length determined in the S-system, where the rod is now at rest? What is the length L s, i.e. the length of the rod {L} at rest in S as determined by S?

    \u see at once that, in order that the proper length of {L} be an invariant, i.e. L = l, it is necessary that

    (9) z~(O, f) =/~ ~/i - - v" sin" 0 '

    which me,~0ns th,~ot the rod s nmst in the direction of motion. This contraction must then be a real phenomenon, i t is called the Lorentz contraction.

    The contractions defined by eqs. (7) and (8), which, as is well known, can be derived from the Lorentz transformations, are a kinematical mctrogenie phenomena, which REmHES]~ACH (~4) calls the Einstein contraction. See ref. (~5) for an explicit example in which the Lorentz' and Einstein's contractions have different values due to a nonstandard synchronization procedure (i.e. s =/: 89 in eq. (1)).

    But does a rod suffer the Lorentz contraction independently of its accel- eration program? The answer is no, as is well illustrated by ~he famous examples of Dewan and Beran (~7) and of Oavalleri and Spinelli (~s).

    From the quoted examples it becomes clear that, if all points of a material rod suffer the same acceleration program relative to S, i.e. all points are simultaneously accelerated relative to S, the rod will not suffer the Lorentz contraction. From this it is clear that such a rod will not be Born rigid, during the acceleration program, since eq. (9) shows that the Lorentz con- traction is necessary in order that the proper length of the rod be an invariant as determined with the light ruler. For materials occupying a finite extension, there are some motions that are not possible to achieve if we insist on Born rigidity. We shall discuss explicitly the ease of the rotating disk in the next section.

    Now, what are the possible accelerated motions which leave a solid Born rigid? The answer (~s.~o) is that, for all pairs of neighbouring particles, the two particles have to be accelerated simultaneously with the same acceleration as observed in their instantaneous inertial rest frame S. This point is quite obvious from the above discussion. This implies also that the material points

    (17) E. D]~wAN and M. B~,RA~: A~n. J . Phys., 27, 517 (1959). (is) G. CAVALZ~RI and G. SPI~]~zzI: Nuovo Cimento B, 56, 11 (1970). (19) O. GRON: Eound. Phys., 10, 499 (1980). (20) O. GRO~: .Found. Phys., l l , 623 (1981).

  • THiE 8TANDAItD OF LENGTt I IN THE THEORY OF RELAT IV ITY :ETC. 20~

    must be simultaneously accelerated in the rest frame 27 where the solid is at rest. Indeed, let Xt, and X, ~- dX~ with dX ~ = 0 be the co-ordinates of two simultaneous events at the points P and P '= P ~-dP as determined by S, where P is by definition at rest and P ' differentially at rest in the S-frame (so that the segment . PP ' is differentially at rest relative to S). Let now a,, ~:~ q-dx~ '~ be the co-ordinates of the above events for the instantaneous observer at P in the 2-frame. I t follows that dx~ (g~o/goo)dx ~ and then, according to the discussion of sect. 2, these two events are simultaneous in the rest frame 2: of the material system.

    For completeness, we remark that the square of the proper length of the segment PP ' in the S-frame is (dX~) 2 q- (dX~) ~ q- (dXa) ~, which is easily shown to be equal to the right member of eq. (2), which is the square of the proper length of PP ' in the X-frame as determined by the light ruler.

    4. - The solut ion of the Ehrenfest paradox.

    We come now to the answers of the three questions asked in sect. 1 in relation to the Ehrenfest paradox.

    According to our discussion of the l~.~st section, we see that the solution of the third question asked in sect. 1 is very simple. Indeed, one disk rotating at uniform angular velocity w can be described according to the observer at rest in the inertial frame S (here the laboratory) by the equations

    (10) 2=xcoswt - -ys inwt , y=xs inwt + y cos wt , z = z , i - - - - t ,

    where x, y, z are the co-ordinates in the disk's rest frame. If (r, 0) are polar co-ordinates in the disk's rest frame, eq. (2) then gives for the spatial geometry of the disk

    r ~ dO ~- (11) d /2= dr -~ q-

    1- w2r "~ '

    which is a non-Euclidean geometry. This is the geometry as determined with the light ruler.

    The second question asked in sect. 1 can be answered, once we know the solution for the first question (which is the real problem), using the same method as in the solution of the third question.

    To give an answer to the first question, we observe first of all that it is kinematieally impossible to set the disk into rotation, while the cireunferenee satisfies Born's definition of rigidity. The argument due to G~o~- (19) goes as follows: Let us associate an instantaneous inertial rest ~.rame Sk with each element between two neighbouring points P~ and P~+l on the circumference. I f the accelerations of each set of neighbouring points are simultaneous, as

  • 20~ w.A . RODRIGUE8 jr.

    measured in the associate rest frame (so that the proper lengths of the elements remain invariant), then the circumference is Lorentz-contracted as observed in the inertial frame S. This, however, is kinematical ly impossible, i.e. it is kinematical ly self-contradictory to assume that all n points on the circum- ference suffer accelerations simultaneously as measured in the successive iner- tial frames S~. This is because it implies that a given point must be accel- erated at the same t ime before and after itself!

    GRON then concludes that, if a disk is set in motion in such a way that all its points are simultaneously accelerated according to the cloks in the S-frame, then its circumference will not contract, and the intrinsic geometry will be given by eq. (1).

    GRo~ claims then to have obtained a purely kinematical solution to the Ehrenfest paradox. However, this assertion is a (( nonsequitur )). Indeed it as- sumes that the elastic dilation overcomes the Lorentz contraction (even when the acceleration program ceases) and this is an intrinsically dynamical prob- lem. This problem has been solved by CAVALLEI~I (2). He showed that, even for a rotat ion disk having the max imum degree of r igidity in agreement with special relativity, the elastic dilation is greater than the Lorentz contraction, if we take a stress-strain relationship which leads to a sound velocity equal to the light velocity.

    The acceleration program described by Gnow (~9) is not the only possible one. Other programs are possible, which result in different geometries. As an example see the recent papers by GlCU]~BAU~ and JANIS (21) and by Gno~x (:0). I t must be recalled here that also in these cases the above remark applies.

    We also mention the paper by IVES (22) where he studied the resulting geometry supposing that the Lorentz contraction overcomes the elastic dilation. CAVALLERI (2) argues that, if this happens, the disk will bend, and there results a violation of Einstein's equivalence principle. ~'Ioreover, it must be said that, if I res solution is correct, then there must be a fifty percent chance for the disk to curve up or down relative to the original plane, otherwise par i ty would be violated in this interaction of the matter with the physical vacuum. Sum- ming up, although Cavalleri's solution is the most reasonable ever presented, we cannot deny the possibil ity of the Lorentz contraction of the periphery when the acceleration program ceases. This is because we do not have a good theory for the interaction of material systems with the physical vacuum. We come again to this point in the next section.

    We remark here the important point that the intrinsic geometry of the rotat ing disk as determined with small material rods will give the same geo- met ry as the one determined with the light ruler, only if the rods have an acceleration program different from the acceleration program of the disk's

    (~1) A. GR~NBAUN and A. I. JANIS: LYound. Phys., 10, 495 (1980). (-~2) H. E. IVES: J. Opt. Soc. Am., 29, 472 (1939).

  • THE STANDARD OF LENGTH IN T I tE THEORY OF RELAT IV ITY ETC. 207

    points. In particular, they must be stress free at the end of the acceleration program (see sect. 5). I t is quite obvious from the above discussion that, if we fix both ends of small but finite material rods along the periphery of the disk when it is at rest, then, when the disk is set in motion, the rods will suffer the same acceleration program of the disk and for Gr in 's program the geo- metry will result Euclidean, but in this case they are not standard rulers.

    To end this section we analyse the following interesting problem: whut is the tendency of contraction if we accelerate the disk by a shaft that pusses through its centre?

    Suppose the disk is initially at rest in a given inertial frame; then the answer to the above question, accord.;ng to the theory of relativity, is that there will be no Lorentz contraction of the periphery. To understand this statement, it is necessary to observe that any pair of radial ly opposite points in the disk must be put in motion at the same instant of t ime according to Einstein's synchronization procedure of the inertial frame. Indeed, if this is not the case, we would find an internal synchronization procedure of clocks at rest in the inertial frame that does not agree with Einstein:s method, and breakdown of Lorentz invariance will result, for it is a theorem of the theory of relativity that all internal synchronization procedures in a given inertial f rame must agree with Einstein's method. For more details, see ref. (23,24).

    5. - The heart o f the ~ do metr ic standards cont rac t - controversy.

    We discuss now the heart of the (~ do metric standards contract ~) contro- versy ol Cantoni (25), Phipps (26.27) and Gron (2o). We think we explained in a clear way the difference between the Einstein contraction and the Lorentz contraction. The former is a kinematical metrogenie phenomenon (a term invented by REINCHESI:BACIt (14)) which has the same numerical value as the latter due to the co-ordinative definitions adopted. Indeed, if a nonstandard synchronization is used (i.e. z ~ 1), then the contractions are different (an ex- plicit example can be found in ref. (15)). I f we accept the standard mode of propagation of l ight and the normal t ime dilation phenomenon and eo-ordi- native definitions, then the sole explanation of Michelson's experiment is found in the Lorentz contraction, which is then a real phenomenon.

    Obviously, someone can have another view of the light propagation phenom-

    (23) W. A. RODRIGUES jr.: Proc. o] the 3rd Brasilian National Con]erence on the Physics o] Fields and Particles (Sao Paulo, 1981), p. 175. (0"4) W. A. ODRIGUE8 jr. and J. TIOMNO: preprint IMECC-UNICAMP 219 (1982) (submitted to Found. Phys.). (25) V. CANTONS: Found. Phys., 10, 809 (1980). (2~) T. E. PHIPPS jr.: Found. Phys., 10, 811 (1980).

  • 20~ W.A . RODRIGUES jr.

    enon like the one recently exposed by PItIPPS (27), but we suspect that no NASA satellite would ever arrive at Jupiter if he is right, since all commands from Earth to the satellite have been sent under the usual hypothesis in connection with light propagation.

    PHIrPs thinks that the existence of azimuthal stresses along the disk's ring divorced from strain is nonphysical and that stresses have no place in kinematics.

    This conclusion arises because he does not recognize the difference between the Einstein and the Lorentz contraction. Jndeed stress is not divorced from strain, as we shall show below.

    We think that the fundamental way to understand this question is the realization that the motion of material bodies is not a simple kinematieM problem-- i t involves the interaction of the body with the (~ground-state vacuum ~) of our Universe. This interaction, in our opinion, is at the origin of the Lorentz contraction. The existence of this dynamical ground-state vacuum, or ether, for short, is not inconsistent with the theory of relativity, as was shown by E~SSTEI~ (2s) and DIm~C (29).

    Indeed, we said at the begining of sect. 3 that the existence of inertial frames is an empirical fact. But where are these frames?

    The answer is given by modern cosmology which is based on the cosmo- logical principle--the hypothesis that the Universe is spatially homogeneous and isotropie.

    As is well known (so), the cosmological principle can be formulated as a statement about the existence of equivalent co-ordinate systems. Let x, be a set of cosmic standard co-ordinates (so). A different set of space-time co- ordinates x'~ may be considered equivalent to the eosmic standard co-ordinates, if the whole history of the Universe appears the same in the x' , co-ordinate system as in the cosmic standard co-ordinate system. This requires that the metric field g'(x') and every other cosmic field, like T',~(x') and so on, must be the same function of the x ' , as the corresponding quantities g,~(x), T,.(x) and so on are of the standard co-ordinates x,. That is, at any co-ordinate point I"z, we must have

    (].o) g~,~(~-) = ' " T . , (~) g~. (~ ) , T f ,~(~ ~) = , ~ .

    This means that the transformations x~x ' must be a symmetry of these geometrical objects in the so-called intermediate point of view of the trans-

    (27) T. E. P~IIPPS jr.: ~ound. Phys., l l , 633 (1981). (2s) A. EINSTEIN: ~ber den .Ather, Verh. Schweiz. Natnr/ovsch. Ges., 105, Teil I I, 85 (1924); b) L'ether et la thdorie de la relativit~ (Paris, 1921). (29) p. A. M. DIRge: Nature (London), 168, 906 (1951). (so) S. WEINBERG: Gravitation and Cosmology (New York, N.Y., 1972).

  • THE STANDARD OF LENGTH IN TIIE THEORY OF RELATIVITY ETC. 20~

    format ions (23,31). We said that the ground-state vacuum spontaneously breaks the original symmetry of the theory (invari~nce under the manifold mapping group) and determines in this way a class of privileged frames. These frames seem to coincide with She ones ill which the cosmic black-body radiation is isotropic.

    ]n sect. 3 we assume that S is the privileged Lorentz f rame in which our Universe appears isotropic (according to optical phenomena). This frame hap- pens to coincide more or less with our own Galaxy.

    The set of privileged Lorentz frames (inertial frames) is not of infinite spatial extension and they are not moving uniformly with respect to each other. In fact, they are accelerating (if we use Lorentz co-ordinates) due to the tidal force field of the global gravitat ional field of our Universe.

    In (( resum6 )>, we think that it cannot be an accident that two measuring rods which have the same proper length at one place ~lways have the same proper length when brought to a defferent place along different paths under the same conditions. The same thing can be said with respect to the standard clocks. These facts must be explained as an adjustment to the field in which the measuring rods and clocks are embedded as test bodies. The final answer can, of course, be given only by a detailed theory of matter, which unfortu- nately does not yet exist. Nevertheless, the explanation for the Lorentz con- traction may happen to be something as simple as the one devised by Lo- ~E~'Tz (~).

    We turn now the strain-stress issue in the rotating disk (accelerated with Gron's program). Let P and P ' : P + dP be two marks on the disk'sring. Let L be the proper distance PP' when the disk is at rest in the inertial frame S (the laboratory frame, here). According to sect. 4, the proper distance PP', when determined by the light ruler (which, by the way, is always stress free) at rest in the disk's frame, S, will be

    (13) L s=yL , y=(1--v~) -},

    when the disk is rotating. Thus, according to an observer always at rest at the point P, the element

    _PP' of the disk's ring is strained by

    (~4.) AX = (} , - -~) [ ,

    and is obviously stressed according to the usual connection between stress and strain.

    (31) E. RECAMI and W. A. RODRIGUE8 jr . : l~ound. Phys., 12, 709 (1982). (a2) H~. A. LORENTZ: in The Principle o] Relativity, ed i ted by A. SOMMERFELD (New York , N.Y . , 1952).

    14 - Il Nuovo Cimento B.

  • 210 w.A . RODRIGUES jr.

    NOW in the S-system (the inertial system) we are tempted to define the strain of the elements PP ' as

    (]5) AX = La--L,

    where /fi is the distance between the marks P and P ' at rest in S as deter- mined in the inertial f rame S (according to the general procedure explained earlier). This gives

    (16) L i = ~- 'L ~ = L -+ AX ~ 0

    and we would say that there is no strain and no stress. Definition (15) implies that~ if a rod suffers the Lorentz contraetion~ i.e.

    (17) L a = L 1Vic- v , ,

    then it is strained according to S. Indeed in this case

    (]8) ax = (? - ' - - ] )L

    and we would say that it is stressed. Now, who stressed a rod that suffers the Lorentz contraction? The answer is given above: it is the interaction of the rod with the physical vacuum.

    Another way to define the strain in S which is more appropriate to the orthodox interpretation of the special theory of relativity, which does not attr ibute physical propert ies to the vacuum (despite the contrary views of Einstein and Dirac), is to use Gren's definition (1,). Gren's definition is inter- esting since~ if we have strain-stress in one frame, we will have strain-stress in all other frames. But this is only a formalism~ no new physics is involved.

    6. - Conc lus ions .

    In this paper we think we have presented a sufficiently clear operational definition of the standard of length in the theory of relativity.

    We identify the light ruler with a realization of the standard of length and discuss under what conditions material rods can be used as standard rulers.

    I n this way, the three questions connected with the Ehrenfest paradox have reasonable answers, if we have in mind the distinction between the Einstein and the Lorentz contraction.

    To end we must say that we agree with CAVALLEm (=) that there is no simple kinematical solution of the Ehrenfest paradox. This point is clear from the above discussions.

  • T}IE STANDARD OF LENGTH IN THE THEORY OF R3ELATIVITY 3ETC. 211

    The ~uthor th~nks Prof. E. REcA~I for several useful comments ~nd dis-

    cussions ~nd ulso ,~ referee for po int ing to us some references znd for ~sking

    ~ very s t imu la t ing question. We ~re grateful Mso to CNPq (Br~sil) for

    research gr~nt.

    9 R IASSUNTO (*)

    Si studiano i sistemi fisici che possono essere associati agli orologi standard e ai re- goli standard della teoria di relativitY. Si mostra ehe, una volta che l'orologio stan- dard ~ stato identifieato, il regolo standard ~ determinato unicamente come (,regolo diluce)>. Si mostra quindi in quali eondizioni le aste materiali possono essere usate come regoli standard. Cib mostra l'esistenz~ di due distinte contrazioni nella teoria ehe sono spesso confuse: te contr~zioni di Einsteill e quelle di Lorentz. Si deduce ehe la contr~zione di Lorentz 5 un fenomeno reale che risult~ come conseguenz,~ deli'intera- zione di corpi materi~li con il vuoto dello stato fondamentale dell'Universo. Questi risultati ci pernlettono di dare una risposta definitiva alla domanda (,le metriche standard si contraggono? ,> e anche di risolvere il paradosso di Ehrenfest in modo abb~stanza banale.

    (*) Traduzione a cura della Redazione.

    Pe3ioMe i/e IIo.rlyqerlo.

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