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
Am. J. Phys. 78 共6兲, June 2010
http://aapt.org/ajp
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
633
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
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Am. J. Phys., Vol. 78, No. 6, June 2010
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 = vx̂,15
E共r,t兲 =
1 − ␤2
q r̂
,
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
634
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 S̄ the rod is at rest and so
x̄ = 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
x̄ = 2L in S̄, which leads to
v
+Q
2L
+Q
x = x1 + x2
q
x1 = vt
x̄ = ␥共x − vt兲.
x2 = 2 1 L
2
If we repeat a similar argument from the point of view of
reference frame S̄, we find17
x = 2L
Fig. 2. The measuring rod apparatus is at rest in frame S̄, 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 S̄. At
time t in S which is shown, the leading edge 共right +Q charge兲 is at x in S
and at x̄ in S̄.
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 S̄ moves at speed v with respect to S in
the x-direction and the measuring rod is at rest in frame S̄
with the trailing end at the origin at t = 0 in S when the
origins of S and S̄ coincide. As shown in Fig. 2, at time t in
S, the other end of the rod is located at x in S and x̄ in S̄. This
description requires clock synchronization in only a single
frame, which can be carried out by a variety of methods, for
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共6兲
Am. J. Phys., Vol. 78, No. 6, June 2010
x = ␥共x̄ + vt̄兲.
共7兲
Substitution of x̄ from Eq. 共6兲 into Eq. 共7兲 leads directly to
冉 冊
t̄ = ␥ 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
635
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 S̄,
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 S̄ 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 S̄ is ␥. Inertial observers disagree on the magnitudes of various quantities: The velocities of S and S̄, the
acceleration, the duration of the acceleration, and the lengths
separating the charged and the uncharged spheres in S and S̄.
They also disagree on the timing of the beginning of the
acceleration of the various parts of the apparatus. For ex636
Am. J. Phys., Vol. 78, No. 6, June 2010
ample, from the point of view of S̄, 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 S̄, they are separated by 2␥L according
to S̄. When the +Q charges are replaced, S̄ 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 S̄ 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 S̄, the receiving frame. The
uncharged spheres reach S̄ with their initial separation observed by S̄, namely, 2冑1 − ␤2L. On recharging in S̄, 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
636
the ions in a rod in the real world兲 reach S̄ 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
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Am. J. Phys., Vol. 78, No. 6, June 2010
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 S̄, 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
637
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.
638
Am. J. Phys., Vol. 78, No. 6, June 2010
a兲
Electronic mail: [email protected]
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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
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John D. Norton, “Why constructive relativity fails,” Br. J. Philos. Sci. 59
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John S. Bell, “How to teach special relativity,” reprinted in Speakable
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Hans C. Ohanion, Einstein’s Mistakes. The Human Failings of Genius
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Reference 2, pp. 69–71.
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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.
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Valery P. Dmitriyev, “The easiest way to the Heaviside ellipsoid,” Am. J.
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Ben Yu-Kuang Hu, “Comment on ‘The easiest way to the Heaviside
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David J. Griffiths, Introduction to Electrodynamics, 2nd ed. 共PrenticeHall, Englewood Cliffs, NJ, 1989兲, pp. 425–426.
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Reference 15, pp. 460–463.
17
In deriving Eq. 共7兲, it has been assumed that the speed of S with respect
to S̄ 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
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William C. Davidon, “Kinematics and dynamics of elastic rods,” Am. J.
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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.
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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.
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We consider only inertial observers whose relative velocity lies along the
x-axis.
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E. L. FeŽnberg, “Can the relativistic change in the scales of length and
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Wolfgang Rindler, Relativity. Special, General, and Cosmological, 2nd
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Reference 12, pp. 119–120.
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Alberto A. Martinez, “There’s no pain in the Fitzgerald contraction, is
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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.
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Savas Dimopoulos, Stuart A. Raby, and Frank Wilczek, “Unification of
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Frank Wilczek, “Anticipating a new golden age,” Eur. Phys. J. C 59 共2兲,
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1
D. J. Miller
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