A constructive approach to the special theory of relativity

A constructive approach to the special theory of relativity D. J. Millera兲 Centre for Time, University of Sydney, Sydney, New South Wales 2006, Australia and School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia

共Received 24 August 2009; accepted 6 January 2010兲 Simple physical models of a measuring rod and of a clock are used to demonstrate the contraction of objects and clock retardation in special relativity. It is argued that models can help student understanding of special relativity and distinguishing between dynamical and purely perspectival effects. © 2010 American Association of Physics Teachers. 关DOI: 10.1119/1.3298908兴 I. INTRODUCTION Several factors make special relativity initially difficult to understand. One factor is the strongly counterintuitive nature of some of the important results, especially the relativity of simultaneity, time dilation, and length contraction. Another factor is that special relativity is usually taught, both in the classroom and textbooks, differently from other areas of physics in the undergraduate curriculum except for classical thermodynamics. That is, its results are taught as a “principle” theory and obtained by deduction from the relativity postulate and the light postulate 共either in the forms originally proposed or versions of them兲.1 Teaching special relativity in this way is especially problematic because unlike classical thermodynamics, which is also taught as a principle theory, the two postulates or principles for the special theory of relativity are strongly counterintuitive when taken together because the natural models of light as either an electromagnetic wave propagating in a medium 共the ether兲 or a photon emitted by its source are both apparently ruled out by the relativity and the light postulates. Therefore it is unsatisfactory that it is the peculiar property of the speed of light 共often as it bounces about in stationary or moving railway carriages兲, which is relied on in many introductory treatments of special relativity to introduce students to time dilation, the relativity of simultaneity, and length contraction. In other areas of undergraduate physics, results are deduced on the basis of dynamical laws involving specific physical forces or fields. That is, the other areas are taught as “constructive” theories. As a consequence, part of the difficulty of understanding special relativity is unanswered questions such as the following: Is the contraction of the measuring rod the result of a force? and What changes inside the clock make it tick slower? Opinions differ on whether special relativity is best regarded as a principle or a constructive theory and, if the latter, as to the nature of the constructive theory. Brown2 argued in favor of a dynamical constructive approach, and Janssen3 argued in favor of a kinematical constructive theory based on the geometry of Minkowski spacetime. Norton4 argued that a dynamical constructive approach must fail 共unless an extreme form of operationalism is adopted兲 because it requires a realist conception of spacetime from the outset. 共In the following, a reference to a constructive approach means a dynamical constructive approach unless otherwise indicated.兲 Even if it is believed that it is best to approach special relativity as a principle theory and derive the results from the principles by kinematical arguments or as a kinematical constructive theory, there is common ground in most of the lit633

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erature that a constructive account of the phenomena of special relativity based on dynamical arguments 共see Ref. 5兲. This point is not made clear in many textbooks, and even in some of the very best textbooks, there are comments that might inadvertently imply the contrary to some readers. Generally the constructive approach, based as it is on the dynamics of the problem, provides a causal explanation and better physical insight into the problem.2,6 These are arguments in favor of introducing the concepts of special relativity via a dynamical constructive approach even if one is going to argue in hindsight that the concepts are best seen to flow directly from the principles or from the geometry of Minkowski spacetime. However, it is probably not suitable to replace the usual principle-based approach with a constructive approach in most courses on special relativity because of the need to rely on a fairly sophisticated understanding of electromagnetism and the need to develop the relativistic version of Newton’s second law before being able to derive time dilation constructively. A better option from a pedagogical point of view may be to refer to the constructive approach in qualitative terms as an alternative to the principle-based approach and motivate students to follow up the details if they are interested. The understanding of a subject is usually enhanced if it is presented from different points of view. In a well-known article, Bell6 advocated the constructive approach to teaching relativity and analyzed a physical model that could be used to illustrate the method. No subsequent textbook has adopted Bell’s approach.7 A problem with Bell’s model is that it cannot be solved analytically, and so the impact of the result is obscured by the need for a numerical solution. Another limitation is that the relativistic form of Newton’s second law needs to be anticipated at the outset. The aim of this work is to derive special relativity constructively in a way that avoids those limitations. In Sec. II A a model involving point charges is used to derive length contraction by relying only on Maxwell’s equations. This derivation both motivates and enables a derivation of the Lorentz transformation in Sec. II B. Given the Lorentz transformation, the relativistic form of Newton’s second law readily follows in Sec. II B, and with that result, the expression for time dilation is derived constructively from a modification of the system of point charges in Sec. II C. In the discussion of these results in Secs. III A and III B, it is argued that a significant advantage of at least referring to the constructive approach is that it brings home the difference between boosting an observer and boosting an object from one frame to another. © 2010 American Association of Physics Teachers

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q

q

+Q

2L

+Q

2L

+Q

x0 q

(a)

q

(b)

Fig. 1. 共a兲 The distance between the +Q charges in their equilibrium positions represents the measuring rod. The −q charges are held in position by an insulating bar of length 2L. When Q = 2冑2q and the charges are at rest with respect to the observer, the separation of the +Q charges is also 2L. If the whole arrangement is set in motion in the horizontal direction, the distance between the +Q charges is contracted. 共b兲 The periodic motion of the +Q charge after it is slightly displaced from the center of the rod represents a clock. When the whole arrangement is set in motion in the horizontal direction, the period of the motion becomes greater, and thus the clock becomes retarded.

II. A CONSTRUCTIVE APPROACH The development of most areas of physics begins with the discovery or proposal of the relevant laws of physics and proceeds by the elaboration of their consequences. A constructive approach to special relativity should proceed in the same way, that is, with a physical law that can be used to investigate the behavior of measuring rods and clocks. The obvious candidate now, as it was at the time of Fitzgerald, Lorentz, and others who took a constructive approach to some of the problems considered by special relativity, is Maxwell’s theory of electromagnetism. One of the reasons that Einstein preferred a principle approach to special relativity was that he was concerned that Maxwell’s theory might need to be modified in the light of contemporary developments.8 More than a century later, we need not share those concerns. To make the constructive derivation of length contraction and clock retardation as transparent as possible, it is desirable to construct a measuring rod and a clock from a very simple system such as a system of point charges in equilibrium. However, there is no stable configuration of charges that can be maintained by the interaction of the charges with the electric and magnetic fields they produce. Diamagnetic materials are an exception to this rule,9 which is an extension of Earnshaw’s theorem,10 but invoking diamagnetism is undesirable because of the need for simplicity. As emphasized by Swann,11 material objects are in equilibrium due to quantum mechanical effects, but considering those is also ruled out by the need for simplicity. Fortunately there are simple qualitative arguments12 that show that lengths of material objects transverse to the direction of their motion are independent of their speed, that is, transverse length is the same for all inertial observers. We can rely on this property to stabilize a system of charges, which can then be used to derive length contraction and clock retardation. Our simple “measuring rod” is shown in Fig. 1共a兲. It is the distance between the +Q charges, which are repelled by each other and attracted by two charges −q attached to the ends of a nonconducting bar of length 2L with relative permittivity of 1. We consider motion only along the x-axis, which is perpendicular to the length 2L. All observers agree on the 634

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separation 2L 共and do so independently of the definition of simultaneity, which is therefore not required at this stage兲. The two charges +Q are in stable equilibrium for displacements along the x-axis but in unstable equilibrium for small displacements perpendicular to the x-axis, and thus it is best to imagine that the +Q charges are inside a transparent and frictionless tube lying along the x-axis. To write Maxwell’s equations and perform the calculations, we need a system of coordinates, which requires rods and synchronized clocks to set up. We therefore must first assume that rods and clocks exist and that the clocks can be synchronized, say, by slow clock transport, and then show that the existence of such rods and clocks is consistent with the theory.

A. Contraction of a moving measuring rod If the charges are at rest in an inertial frame S, only electrostatic forces are involved. In equilibrium, the +Q charges occupy positions where the electric field is zero, and by symmetry, these are equidistant from the bar at ⫾x0 along the x-axis, which bisects the bar. Also by symmetry, only the x-component of the electric field on the x-axis Ex共x0兲 can be nonzero. Thus x0 can be found from the condition Ex共x0兲 =

冋

册

1 2qx0 Q = 0, 2 − 2 4␲⑀0 4x0 共x0 + L2兲3/2

共1兲

where ⑀0 is the permittivity of free space. If Q = 2冑2q, Eq. 共1兲 is satisfied when x0 = L, and thus the equilibrium separation of the +Q charges is 2L in the rest frame of the charges. To consider what happens when the charge assembly in Fig. 1共a兲 moves with speed v in the x-direction with respect to S, we require the electric and magnetic fields due to charges in uniform motion. Although some students in a course on special relativity might not have seen the required expressions, there are simple derivations of them.13,14 Of course, we must not assume at this stage any of the results of special relativity, and thus it is essential to the present approach that the required expressions be obtained directly from Maxwell’s equations. Alternatively students can be asked to accept the expression for the electric and magnetic fields of a moving charge as given. When all four charges move with speed v in the x-direction, we need the electric and magnetic fields at a distance from the current 共not retarded兲 position at time t of a charge q moving with velocity v = vxˆ,15 E共r,t兲 =

1 − ␤2 q rˆ , 2 4␲⑀0 r 共1 − ␤2 sin2 ␪兲3/2

共2兲

B共r,t兲 =

1 共v ⫻ E共r,t兲兲, c2

共3兲

where ␤ = v / c and ␪ is the angle between r and the x-axis. The equilibrium separation can be determined from the condition that the fields at each of the +Q charges are zero. There is no net magnetic field at the +Q charges. The separate conditions for equilibrium of each of the +Q charges show that they are still equidistant from the rod by a new distance we call xv. The equilibrium separation 2xv of the moving +Q charges can be found from the condition D. J. Miller

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S

q

S

example, slow clock transport, and which does not anticipate the relativistic results involving time that we are deriving. In terms of the units of length used in S, the rod is contracted to 2冑1 − ␤2L using Eq. 共5兲, while in ¯S the rod is at rest and so ¯x = 2L from Eq. 共1兲. From inspection of Fig. 2, the position of the +Q charge on the right is x = vt + 2冑1 − ␤2L in S and ¯x = 2L in ¯S, which leads to

v

+Q

2L

+Q

x = x1 + x2

q x1 = vt

¯x = ␥共x − vt兲.

x2 = 2 1 L 2

If we repeat a similar argument from the point of view of reference frame ¯S, we find17

x = 2L Fig. 2. The measuring rod apparatus is at rest in frame ¯S, which moves horizontally to the right with speed v with respect to frame S. At time t0 in S, the trailing edge of the rod 共left +Q charge兲 is at the origin of S and ¯S. At time t in S which is shown, the leading edge 共right +Q charge兲 is at x in S and at ¯x in ¯S.

Ex共xv兲 =

冤

共1 − ␤2兲 Q − 4␲⑀0 4x2v

冉

2qxv

共x2v + L2兲3/2 1 − ␤2

L2 x2v + L2

冊

3/2

冥

共4兲

= 0, which is satisfied when xv = 冑1 − ␤2L = L/␥ ,

共5兲

where ␥ = 1 / 冑1 − ␤ . Hence, when the charge assembly moves with constant speed v, the moving measuring rod represented by the equilibrium separation of the +Q charges is reduced to 2冑1 − ␤2L from their separation, the length 2L of the measuring rod when at rest. 2

B. Lorentz transformation As shown in Fig. 1共b兲, this arrangement can be modified to produce a clock by removing one of the +Q charges and moving the other +Q charge between the −q charges 共the vertical bar needs to have a suitable gap drilled through it to accommodate the +Q charge兲. If the center charge is given a horizontal displacement from the point midway between the −q charges, it will experience a linear restoring force and execute periodic motion, which can be used to measure time. To calculate the period, it is necessary to have the correct form of Newton’s second law, that is, the relativistic form of the second law. We can obtain this result in a logical manner in the present approach by first noting that length contraction is inconsistent with the Galilean transformation of coordinates between inertial frames. That motivates a reconsideration of the transformation of coordinates leading to the correct 共Lorentz兲 transformation. The correct 共relativistic兲 form of the laws of mechanics then follows as in any course on special relativity. A standard textbook argument16 can be adapted to the present case. Frame ¯S moves at speed v with respect to S in the x-direction and the measuring rod is at rest in frame ¯S with the trailing end at the origin at t = 0 in S when the origins of S and ¯S coincide. As shown in Fig. 2, at time t in S, the other end of the rod is located at x in S and ¯x in ¯S. This description requires clock synchronization in only a single frame, which can be carried out by a variety of methods, for 635

共6兲


¯ + v¯t兲. x = ␥共x

共7兲

Substitution of ¯x from Eq. 共6兲 into Eq. 共7兲 leads directly to

冉冊

¯t = ␥ t −

v x . c2

共8兲

Together with the general argument about the coordinates transverse to the motion remaining unchanged, Eqs. 共6兲 and 共8兲 constitute the familiar Lorentz transformation. Once the Lorentz transformation has been obtained, the rest of special relativity follows, including the relativistic form of Newton’s second law in the direction parallel to the motion, F储共x兲 =

⳵ p储 d 2x = ␥ 3m 2 . dt ⳵t

共9兲

C. Slowing of a moving clock Although clock retardation follows from the Lorentz transformation alone, the constructive approach requires us to show that a material clock can be constructed within the theory and that it behaves in the expected manner. When the middle charge +Q in Fig. 1共b兲 is displaced from its equilibrium position by xv, the relativistic restoring force F储共xv兲, with the charges moving at speed v with respect to S, is F储共xv兲 = −

Q共1 − ␤2兲 4␲⑀0

冉

2qxv

共x2v + L2兲3/2 1 − ␤2

L2 x2v + L2

冊

3/2

共10a兲 ⬇− ␥

qQ x 2 ␲ ⑀ 0L 3 v

for xv Ⰶ L.

共10b兲

If we use the relativistic form of Newton’s second law in Eq. 共9兲, we find that the center charge executes simple harmonic motion with a period Tv = 2␲␥

冑

2 ␲ ⑀ 0L 3m = ␥T0 , qQ

共11兲

where T0 = 2␲冑2␲⑀0L3m / qQ is the period when the clock is at rest. This motion constitutes a simple clock. It runs more slowly by the factor 冑1 − ␤2 when it is moving in S than when it is at rest. It is reasonable to suggest that more elaborate moving clocks will run slowly for similar reasons. D. J. Miller

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III. DISCUSSION An advantage of the arrangement of charges in Fig. 1 is that they can have macroscopic dimensions and imagined to be on a laboratory bench top 共the arrangement in Fig. 1 is reminiscent of the twin pith balls, which serve as an elementary demonstration of charge repulsion兲. Another significant advantage is that they don’t involve explicitly the invariance of the speed of light for all inertial observers, which is one of the counterintuitive consequences of special relativity18 and therefore should be avoided in demonstrating other counterintuitive consequences of special relativity. The main advantage is that the length contraction of an object is derived within a single frame and does not involve the comparison of the synchronization of clocks in two different inertial frames. This advantage eliminates the notion that the length contraction that occurs when a physical object changes frames somehow involves the relativity of simultaneity. The rod and clock models are reasonable because they are analogous to the way one imagines that ions are held in position in real materials. Of course, the treatment of real materials considered as an assembly of ions and electrons requires a quantum mechanical solution. However, the Hamiltonian involves only the electromagnetic field used in the proposed models. An alternative approach is to consider the measuring rod as an elastic solid. Davidon19 treated the kinematics and dynamics of moving and accelerated rods from that point of view and derived many interesting results, including length contraction, by considering the dynamics in a single frame. Unfortunately, the analysis is not sufficiently simple to be suitable for an introduction to special relativity. A. Dewan–Beran thought experiment It is interesting to apply the thought experiment originally proposed by Dewan and Beran20 共also referred to in Ref. 6兲 to the configuration of charges in Fig. 1共a兲. First, remove the +Q charges from the spheres when they are in their equilibrium configuration at rest in S. The spheres remain in position. As measured in S, let all the components of the apparatus be simultaneously subject to a uniform acceleration21 in the +x-direction for the same time interval so that their velocity changes from zero to v. When they reach rest in ¯S, both the spheres remain separated by 2L as measured in S.20 Now put back the +Q charges. The recharged spheres will move to their new equilibrium positions separated by 2冑1 − ␤2L as measured by S. All inertial observers22 agree on some aspects of this sequence of events and disagree on other aspects. All the inertial observers agree that all parts of the apparatus started at rest in S and eventually ended at rest in ¯S and that all parts of the apparatus were given the same acceleration for the same time. All inertial observers agree that the pair of spheres move closer together when recharged and that the ratio of the initial 共uncharged兲 to the final 共charged兲 separations of the spheres in ¯S is ␥. Inertial observers disagree on the magnitudes of various quantities: The velocities of S and ¯S, the acceleration, the duration of the acceleration, and the lengths separating the charged and the uncharged spheres in S and ¯S. They also disagree on the timing of the beginning of the acceleration of the various parts of the apparatus. For ex636


ample, from the point of view of ¯S, who initially observes the apparatus receding at speed −v, the trailing uncharged sphere begins accelerating 共in the opposite direction to the velocity兲 before the leading uncharged sphere. As a consequence the separation between the charges increases so that when both charges reach rest in ¯S, they are separated by 2␥L according to ¯S. When the +Q charges are replaced, ¯S observes them contracting to a separation of 2L. This thought experiment makes explicit the two dynamical processes that are in operation when a physical object is transferred between inertial frames: The external force producing the required acceleration and the internal forces that maintain the object in equilibrium. Feinberg considered this question more generally.23 For a normal object in which the charges are left in place, the internal forces act continuously so as to maintain, or attempt to maintain, the charges in their instantaneous equilibrium positions. The effect of the internal forces depends on the mode of acceleration 共for example, the timing of the start of the acceleration and whether the acceleration is constant兲 involved in changing the velocity of the object. The direction of the acceleration in relation to the direction of the velocity makes a qualitative difference. We have just considered the case of the acceleration in the opposite direction to the velocity, which is always what is required from the point of view of the receiving frame. Some observers will say that the assembly with uncharged spheres has to speed up to reach ¯S from S, that is, the acceleration is in the same direction as the velocity of S. For those observers the trailing uncharged sphere begins to accelerate first 共we continue to consider the case when all parts of the apparatus begin to accelerate at the same rate in S兲, but now it is catching up with the other sphere so that the separation of the uncharged spheres decreases compared to its original value in S. Despite their disagreement on the magnitude of the separation of the uncharged spheres and whether this separation increased or decreased as the result of the change of frames, all observers agree that the spheres move together when the charges are replaced and that they move together by the same factor compared with their separation when uncharged. Thus the fractional change in length with the change in velocity caused by the forces that maintain the equilibrium configuration is real in the sense that all inertial observers agree on it. For completeness, suppose the equal acceleration of all parts of the apparatus originally at rest in S begins simultaneously from the point of view of ¯S, the receiving frame. The uncharged spheres reach ¯S with their initial separation observed by ¯S, namely, 2冑1 − ␤2L. On recharging in ¯S, the +Q charges move further apart by the factor ␥ to reach the points of equilibrium separated by 2L. All inertial observers agree on this movement 共while disagreeing, as before, about the magnitudes of the separations involved兲. Thus if a rod is transferred between frames, the movement of the charges in the rod from their old equilibrium separations in the original frame to their new equilibrium separations in the final frame, under the influence of the dynamical forces within the rod, will produce an expansion or a contraction depending on the manner in which the rod is accelerated between frames. It is possible to apply a position-dependent acceleration in such a way that the uncharged spheres in our example 共and D. J. Miller

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the ions in a rod in the real world兲 reach ¯S in their equilibrium positions in that inertial frame. Then recharging the spheres would cause no change in their separation at all—an example of rigid motion of a body.24 The motion of an accelerating body can be described as “rigid” if each infinitesimal and finite part of the body has the same length and is under zero stress as a consequence of the acceleration in the frame in which it is momentarily at rest. If the leading end of the body has constant acceleration, then the other parts of the body must also have constant but different accelerations, increasing toward the rear, to achieve rigid motion of the body. It is noteworthy that for the case of constant acceleration imposed on one part of the body, the differing accelerations of the respective parts of the rod required for rigid motion remain constant despite the ever-increasing contraction of the rod.24 Taylor and Wheeler described a concrete method for achieving rigid-body acceleration of a measuring rod.25 B. Dynamical versus perspectival effects The treatment of the thought experiment in Sec. III A relies on the constructive or dynamical approach to special relativity. It clearly exposes several aspects of special relativity that remain implicit or hidden in a purely principle or kinematical approach. It distinguishes between perspectival and dynamical effects. The changes in the separations of the spheres in their destination inertial frame when they are uncharged and then recharged show that when a physical object in equilibrium changes frames, there are physical changes in it that can be calculated from a consideration of the forces which keep it in equilibrium, and it is these changes that lead to satisfying the Lorentz transformation conditions. The changes in an object when it changes frames will be referred to as dynamical effects. When an observer changes frames 共or when we compare the results of observers in different frames兲, there are no dynamical effects in the physical object being observed. The differences among the observations of the single observer in the various frames 共or the different observers兲 are due to the differences in their respective measuring instruments and will be referred to as perspectival effects. 共Of course, these perspectival effects ultimately have a dynamical origin because the properties of measuring instruments are determined by the forces that keep them in equilibrium in their respective frames.兲 Thus, in the previous thought experiment, the disagreements among the inertial observers about the magnitudes of quantities and the relative timings of the beginnings and ends of the acceleration period for the various parts of the apparatus are perspectival. The different inertial observers have no dynamical explanation in terms of a change in the object, nor do they require one. This point can be confusing because the factors involved in the perspectival and dynamical effects involve the same numerical values 共determined by ␥兲. When the measuring rods and clocks are moved between inertial observers, they suffer dynamical changes. When the observers use their dynamically altered rods and clocks to make measurements, it is not surprising that their results differ and that they differ by the same factors that are involved in the dynamical changes. Although the point might appear trivial and is obvious from the constructive approach to special relativity, failure to appreciate the differences between moving a physical object between frames and moving an observer between frames sometimes produces confusion. For example, it has been 637


Table I. The ratio of lengths involving two different inertial frames moving at relative speed v along the direction of the lengths with ␥ = 共1 − v2 / c2兲−1/2. Connected lengths refer to lengths of objects in equilibrium under internal forces, and unconnected lengths refer to distances between independent objects. The first row is a perspectival effect: The ratio of the length observed from one inertial frame to the length observed from the other inertial frame by two different observers. The second row is due to dynamical effects. The ratio of the unconnected lengths is variable because it depends on the mode of acceleration of the physical objects between the inertial frames. Ratio of lengths when Observer changes frames Objects change frames

Connected lengths

Unconnected lengths

␥ ␥

␥ Variable

used recently as an argument against the constructive approach to special relativity.26 The difference between transferring physical objects between inertial frames and transferring observers between inertial frames is summarized in Table I where the length of a physical object in equilibrium is labeled a “connected” length to distinguish it from the distance between two objects that are independent of each other, which is an “unconnected” length. We see that for the perspectival changes resulting from a change of inertial frame by an observer 共or the comparison of observations by two observers兲, a distance appears to change by ␥ whether or not the distance is a connected or an unconnected length. However, when physical objects are transferred between inertial frames, the connected lengths change by the ratio ␥, but the unconnected lengths change by a factor that depends on the mode of acceleration involved in transferring the objects between inertial frames. The last point has been illustrated by the measuring rod apparatus in Sec. III A. For example, in the first case considered in Sec. III A, the separation of the uncharged spheres is an unconnected length, and on transfer from S to ¯S, the separation of the uncharged spheres remains at 2L. But when charged spheres are transferred in an identical manner and remain in equilibrium, their final separation is 2冑1 − ␤2L. The Lorentz transformation can be considered as an active or a passive transformation of coordinates. The passive sense, when the coordinate system is transformed to a different inertial frame, corresponds to a perspectival change. The active sense involves the physical system under consideration changing frames 共being given a “boost”兲. The previous discussion shows that it is an oversimplification in the context of a Lorentz transformation to consider an active boost as the system as an entity acquiring the velocity of the new frame. That is because connected and unconnected lengths would behave differently during the boost 共except when the change in velocity was especially contrived, but that restriction is not part of the concept of an active Lorentz transformation symmetry兲. Therefore an active Lorentz transformation should be pictured 共if the process is physically pictured at all兲 as a transfer and reassembly of the apparatus in the new frame. The connected lengths of the objects comprising the system can then act as a guide for the placement of the unconnected objects. C. The common origin inference argument Leaving aside pedagogical considerations, we consider one of the arguments3 that special relativity should not be D. J. Miller

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presented as a dynamical constructive theory. The argument is that the Lorentz invariance of the nongravitational laws of physics can be traced back to the structure of Minkowski spacetime, and so it is preferable and more logical to go directly from Minkowski spacetime to the results of special relativity 共kinematically兲 than via the physical fields 共dynamically兲. Put another way, if we proceed dynamically, the Lorentz invariance of all the physical fields is vital, but the latter has its origin in the Minkowski spacetime, and thus proceeding from the latter is preferable.27 The argument that Lorentz invariance of the physical fields has its origin in the Minkowski spacetime is justified by the principle of common origin inference.3 In brief, either we propose that the Lorentz invariance of the three nongravitational fields 共the electromagnetic, weak, and strong interactions兲 is a remarkable coincidence, or we seek a common origin and find it in Minkowski spacetime. As mentioned by Janssen,28 an alternative is that the three nongravitational fields might be different manifestations of a single field, and this property is the common origin inference for the Lorentz invariance of all the nongravitational physical fields. There are strong arguments that the differences between the strong, weak, and electromagnetic fields in the present era are due to symmetry breaking of a single field in an earlier era.29,30 If the unification of the physical fields turns out to be correct, then the common origin inference argument for the role of Minkowski spacetime in determining the Lorentz invariance of the physical fields involved in special relativity loses its force. IV. CONCLUSION Presenting special relativity from a purely kinematical point of view leaves students’ understanding of the subject deficient and might mean they are less able to deal confidently with non-text-book examples like the Dewan–Beran thought experiment. In contrast, including a constructive or dynamical approach enriches student understanding of the subject.6 The thought experiment involving arrays of charges capable of acting as a measuring rod and a simple clock described here facilitates the presentation of special relativity from the constructive point of view. On the more general question of whether special relativity is better interpreted as a principle theory or a dynamical 共or kinematical兲 constructive theory of physics, there does not seem to be any compelling reason that justifies the almost universal emphasis on special relativity as a principle theory. As for the question of whether we should take a dynamical or kinematical approach to the subject, we have argued that the changes in a connected object 共object in equilibrium under internal forces兲 when it is transferred between inertial frames involves dynamical effects. In contrast, the different observations of a single connected object by different inertial observers are a perspectival, not a dynamical, effect. ACKNOWLEDGMENTS The author thanks Michel Janssen and John Norton for correcting a 共hopefully former兲 misunderstanding of their work on the author’s part.

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a兲

Electronic mail: [email protected] Albert Einstein, “What is the theory of relativity?,” reprinted in Ideas and Opinions 共Crown, New York, 1982兲, pp. 227–232. 2 Harvey R. Brown, Physical Relativity. Space-Time Structure from a Dynamical Perspective 共Oxford U. P., Oxford, 2005兲. 3 Michel Janssen, “Drawing the line between kinematics and dynamics in special relativity,” Stud. Hist. Philos. Mod. Phys. 40 共1兲, 26–52 共2009兲. 4 John D. Norton, “Why constructive relativity fails,” Br. J. Philos. Sci. 59 共4兲, 821–834 共2008兲. 5 N. David Mermin, It’s About Time. Understanding Einstein’s Relativity 共Princeton U. P., Princeton, 2005兲, pp. 179–186. 6 John S. Bell, “How to teach special relativity,” reprinted in Speakable and Unspeakable in Quantum Mechanics, 2nd ed. 共Cambridge U. P., Cambridge, 2004兲, pp. 67–80. 7 Hans C. Ohanion, Einstein’s Mistakes. The Human Failings of Genius 共Norton, New York, 2008兲, p. 283. 8 Reference 2, pp. 69–71. 9 M. D. Simon, L. O. Heflinger, and A. K. Geim, “Diamagnetically stabilized magnet levitation,” Am. J. Phys. 69 共6兲, 702–713 共2001兲. 10 William T. Scott, “Who was Earnshaw?,” Am. J. Phys. 27 共6兲, 418–419 共1959兲. 11 W. F. G. Swann, “Relativity, the Fitzgerald–Lorentz contraction and quantum theory,” Rev. Mod. Phys. 13 共3兲, 197–202 共1941兲. 12 Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics, 2nd ed. 共Freeman, New York, 1992兲, pp. 65–67. 13 Valery P. Dmitriyev, “The easiest way to the Heaviside ellipsoid,” Am. J. Phys. 70 共7兲, 717–718 共2002兲. 14 Ben Yu-Kuang Hu, “Comment on ‘The easiest way to the Heaviside ellipsoid’ by Valery P. Dmitriyev,” Am. J. Phys. 71 共3兲, 281 共2003兲. 15 David J. Griffiths, Introduction to Electrodynamics, 2nd ed. 共PrenticeHall, Englewood Cliffs, NJ, 1989兲, pp. 425–426. 16 Reference 15, pp. 460–463. 17 In deriving Eq. 共7兲, it has been assumed that the speed of S with respect to ¯S is −v. This result follows from the method of clock synchronization we have assumed. It is less desirable to rely on the principle of reciprocity, which requires attributing some properties to space and time that are best to avoid in a constructive approach to special relativity. 18 Sometimes the invariance of the speed of light for all inertial observers is made one of the two postulates in a presentation of special theory of relativity, but that is not the minimalist form of the postulates as originally proposed by Einstein. 19 William C. Davidon, “Kinematics and dynamics of elastic rods,” Am. J. Phys. 43 共8兲, 705–713 共1975兲. 20 E. Dewan and M. Beran, “Note on stress effects due to relativistic contraction,” Am. J. Phys. 27 共7兲, 517–518 共1959兲. 21 The acceleration could be achieved by immersing the apparatus in a uniform gravitational field for the required period of time 共a gravitational field in which tidal effects are negligible over the dimensions of the apparatus兲. 22 We consider only inertial observers whose relative velocity lies along the x-axis. 23 E. L. Fenberg, “Can the relativistic change in the scales of length and time be considered the result of the action of certain forces?,” Sov. Phys. Usp. 18 共8兲, 624–635 共1975兲. 24 Wolfgang Rindler, Relativity. Special, General, and Cosmological, 2nd ed. 共Oxford U. P., Oxford, 2006兲, pp. 71–73. 25 Reference 12, pp. 119–120. 26 Alberto A. Martinez, “There’s no pain in the Fitzgerald contraction, is there?,” Stud. Hist. Philos. Mod. Phys. 38 共1兲, 209–215 共2007兲. 27 In the argument here it is tacitly assumed that if all the physical fields are Lorentz invariant, then all the laws of physics involving them must be Lorentz invariant. 28 Reference 3, p. 49. 29 Savas Dimopoulos, Stuart A. Raby, and Frank Wilczek, “Unification of couplings,” Phys. Today 44 共10兲, 25–33 共1991兲. 30 Frank Wilczek, “Anticipating a new golden age,” Eur. Phys. J. C 59 共2兲, 185–196 共2009兲. 1

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