arXiv:0704.0363v2  [physics.gen-ph]  10 Feb 2009
Time and motion in physics: the Reciprocity Principle,
relativistic invariance of the lengths of rulers and time
dilatation
J.H.Field
Département de Physique Nucléaire et Corpusculaire Université de Genève . 24, quai
Ernest-Ansermet CH-1211 Genève 4.
E-mail: john.field@cern.ch
Abstract
Ponderable objects moving in free space according to Newton’s First Law con-
stitute both rulers and clocks when one such object is viewed from the rest frame
of another. Together with the Reciprocity Principle this is used to demonstrate, in
both Galilean and special relativity, the invariance of the measured length of a ruler
in motion. The different times: ‘proper’, ‘improper’ and ‘apparent’ appearing in
different formulations of the relativistic time dilatation relation are discussed and
exemplified by experimental applications. A non-intuitive ‘length expansion’ effect
predicted by the Reciprocity Principle as a necessary consequence of time dilatation
is pointed out.
PACS 03.30.+p
http://arxiv.org/abs/0704.0363v2
1 Introduction
The standard text-book presentation of special relativity follows closely that of Ein-
stein’s seminal paper of 1905 [1] in basing the theory on the Special Relativity Principle,
classical electromagnetism and the postulate of constant light speed. However an alterna-
tive and conceptually simpler approach to the physics of space and time, in the absence
of gravitational fields, is possible in which it is not necessary to consider light signals,
classical electromagnetism, or indeed, any dynamical theory whatsoever. The Lorentz
transformation (LT) was first derived in this way by Ignatowsky [2] in 1910. Purely
mathematical considerations imply, in such a derivation of the LT, the existence of a
maximum relative velocity, V , of two inertial frames. Use of relativistic kinematics then
shows that V is equal to the speed of light, c, when light in identified as a manifestation
of the propgation in space-time of massless particles –photons [3]. In this way Einstein’s
mysterious second postulate is derived from first principles. The fundamental axiom un-
derlying such an approach is the Reciprocity Principle (RP) [4, 3], discussed in Section
3 below, relating the the relative velocities of two inertial frames. Derivations of the LT
and the parallel velocity addition formula based on the RP and other simple axioms are
given in Ref. [3].
In the present paper the space-time properties of ponderable1 physical bodies in free
space, as described by Newton’s First Law of mechanics, are used together with the RP,
to demonstrate the invariance of the measured length of a ruler in uniform motion. The
proof given is valid in both Galilean and special relativity, since Newton’s First Law and
the RP hold in both theories.
The analysis presented is based on a careful definition of physical time concepts. In
particular, the ‘frame time’ or ‘proper times’ that appear in in the RP, are distinguished
from the ‘improper time’ or ‘apparent time’ (of a moving clock) that appear in the Time
Dilatation (TD) relation of special relativity.
The paper is organised as follows: The following section contains an elementary discus-
sion of the concepts of ‘space’, ‘time’ and ‘motion’ in physics, in connection with Newton’s
First Law. In Section 3, the RP is introduced and discussed in relation to Newton’s First
Law. It is pointed out that, because of the RP, ‘rulers are clocks’ and ‘clocks are rulers’
when the motion of ponderable bodies in free space is considered. In Section 4 the RP is
used to demonstrate the invariance of the measured length of a uniformly moving ruler.
In Section 5 the operational meanings of the time symbols appearing in the TD formula
of special relativity are discussed. This may be done in a ‘clock oriented’ manner in terms
of ’proper’ and ’improper’ times of the observed clock, or in an ‘observer oriented’ manner
in terms of the proper time of the observer’s local clock and the ‘apparent time’, as seen
by the observer, of the moving clock. Two specific experiments are described to exemplify
the operational meanings of the time symbols in the TD formula. A non-intuitive ‘length
expansion’ effect is found to relate similarly defined spatial intervals corresponding to the
observation of an event either in the rest frame of the clock, or in a frame in which it is
in uniform motion.
1That is bodies, with a non-vanishing Newtonian mass, which may be associated with an inertial
frame in which the body is at rest. No such frame may be associated with a massless object.
The results of the present paper show that the ‘length contraction’ effect and the
correlated ‘relativity of simultaneity’ effect of conventional special relativity do not exist.
A detailed discussion of the reason for the spurious nature of these effects of conventional
special relativity theory may be found in Refs. [5, 6, 7, 8, 9, 10] .
However, a genuine ‘relativistic length contraction’ effect does occur when distances be-
tween spatial coincidences of moving objects are observed from different inertial frames [11].
Also a genuine ‘relativity of simultaneity’ effect occurs when clocks at rest in two different
inertial frames are viewed from a third one [12, 13]. An alternative derivation, directly
from the RP, of the invariance of the measured spatial separation of two objects at rest
in the same inertial frame as well as the absence of the conventional ‘relativity of simul-
taneity’ effect is given in Ref. [9].
2 Physical time and Newton’s First Law of Mechan-
In physics the concepts of ‘time’ and ‘motion’ are inseparable. In a world in which
motion did not exist the physical concept of time would be meaningless. Similarly the
physical concepts of ‘space’ and ‘motion’ are inseperable. Without the concept of space,
no operational definition of motion is possible. The concept of historical time –the time
of the everyday world of human existence– requires the introduction of the further, and
equivalent, concepts of ‘uniform motion’ and ‘cyclic motion with constant period’. For
example, the unit of time the ‘year’ is identified with the (assumed constant) period of
rotation of the Earth about the Sun.
The idea of uniform motion entered into physics in a quantitative way with the for-
mulation of Newton’s First Law [14]
Every body continues in its state of rest, or uniform motion in a right
line unless it is compelled to change that state by forces impressed upon
This law gives an operational meaning to the physical concept of ‘uniform motion’.
It is defined by observations of the position of any ponderable object in ‘free space’ i.e.
in the absence of any mechanical interaction of the object with other objects. There is
a one-to-one correspondence between such a ponderable object and an ‘inertial frame’
of relativity theory. As will be discussed in the following section, one such ponderable
object, O, constitutes both a ruler and a clock for an observer in the rest frame of another
such object, O’, and vice versa.
When time is measured by using a cyclic physical phenomenon, e.g. an analogue clock,
time measurement reduces to recording the result of a spatial (or angular) measurement.
There is a one-to-one correspondence between the spatial coincidence of a stationary
‘mark’ on the face of the clock and a moving ‘pointer’, constituted by the hand of the clock,
and the time measurement [6]. A ‘time interval’ is measured by the angular separation of
two such ‘pointer-mark coincidences’. The implicit assumption is that the motion of the
pointer is ‘uniform’. There is an evident logical circularity here since ‘equal’ time intervals
measured by such an analogue clock assume that the angular velocity of the hand is
constant, whereas constant angular velocity is established by observation of equal angular
increments for equal time time intervals (i.e. also equal angular increments) recorded by
a second clock of supposedly known uniform rate. In practice, this conundrum is resolved
by an appeal to physics. For example, an undamped pendulum in a uniform gravitational
field is predicted, by the laws of mechanics, to have a constant period of oscillation.
Quantum mechanics predicts the same transition frequency and mean lifetimes for two
identical atoms in the same excited state, in the same physical environment, etc.
Measurements of ‘time’ are then ultimately observations of spatial phenomena, e.g.
the time measurement corresponding to observation of the number displayed by a digital
clock is a spatial perception. This will also be the case for time measurements related to
observation of two ponderable objects O and O’ in motion in free space that will now be
discussed.
3 The Reciprocity Principle: rulers are clocks, and
clocks are rulers
Consider two non-interacting ponderable objects O and O’, with arbitary motions in
free space. They are placed at the origins of inertial coordinate systems S and S’ with
axes orientated so that the x and x′ axes are parallel to the relative velocity vector of O
and O’. Without any loss of generality for the following discussion, it may be assumed
that O and O’ lie on the common x-x′ axis.
The Reciprocity Principle (RP) [4, 3, 9] is defined by the equation:
v = vO′O =
∂xO′O
= −vOO′ (3.1)
where xO′O ≡ xO′−xO and x
, or in words: ‘If the velocity of O’ relative to O
is ~v, the velocity of O relative to O’ is - ~v’. In many discussions of special relativity, the RP
is taken as ‘obvious’ and is often not even declared as a separate axiom. This is the case,
for example, in Einstein’s 1905 special relativity paper [1]. However, as first demonstrated
by Ignatowsky in 1910 [2], it is sufficient, together with some other weaker axioms such as
the homogeneity of space or single-valuedness of the transfomation equations, to derive [3]
the space-time Lorentz transformation and hence the whole of special relativity theory.
Eqn(3.1) looks very similar to the equation defining the relative velocity of two objects
A and B as observed in a single inertial reference frame (say S):
vAB ≡ vA − vB =
d(xA − xB)
= −vBA (3.2)
The crucial difference is the appearence in the RP, (3.1), of two different times t and t′.
The time t is the ‘frame time’ of S. i.e. the time registered by a synchronised clock at rest,
at any position in S, according to an observer also a rest in S. The frame time t′ is similarly
defined by an array of synchronised clocks at rest in S’. Eqn(3.1) (and its integral) gives a
relation between the times t and t′ Both t and t′ correspond to ‘proper times’ of clocks at
rest, whereas, as explained in Section 4 below, the Lorentz transformation relates instead
a proper time to an ‘improper time’ –the observed time of a clock in uniform motion.
Suppose now that O and O’ are equipped with local clocks that are observed to run at
exactly the same rate when they are both at rest in the same inertial frame. The direction
of the relative velocity vector ~v of O’ relative to O is such that they are approaching each
other at the frame times t and t′. The spatial separations of O and O’ in S and S’ are ℓ(t)
and ℓ′(t′) respectively, at times t and t′. Using the RP, a spatial coincidence of O and O’
will be observed at the time
tOO′ = t+
(3.3)
in S, and
= t′ +
ℓ′(t′)
(3.4)
in S’. The OO’ coincidence event will be mutually simultaneous in the frames S and S’.
Note that the OO’ spatial coincidence that is mutually simultaneous in S and S’
constitutes a pair of reciprocal pointer mark coincidences. In S the mark is at the position
of O and the moving pointer at the position of O’, whereas in S’ the position of O’
constitutes the mark and the position of O the pointer. A corollary is that all such pairs
of reciprocal pointer mark coincidences are mutually simultaneous. This is the basis of the
‘system external synchronistation’ [15] as introduced in Einstein’s first special relativity
paper [1] to synchronise clocks at rest in different inertial frames when they are in spatial
coincidence.
The observation of the OO’ coincidence event in both frames can be used to give a
condition that any other pair of events, one observed in S, the other observed in S’ are
mutually simultaneous. If the time of an event in S is t̃ and another event in S’ is t̃′ they
will be ‘mutually simultaneous’ providing that:
t̃′ − t̃ = t′
− tOO′ (3.5)
Combining (3.3)-(3.5) gives:
t̃′ − t̃ = t′
− tOO′ = t
′ − t+
ℓ′(t′)− ℓ(t)
(3.6)
If now events occuring at times t in S and t′ in S’ are mutually simultaneous, it follows
from (3.5) and (3.6) that ℓ(t) = ℓ′(t′), so that events which occur when O and O’ have
the same spatial separation in S and S’ are mutually simultaneous. A special case occurs
if the clock arrays in S and S’ are mutually synchronised so that ℓ(t) = ℓ′(t′ = t). There
is then a direct correlation between either t or t′ and the spatial separation of O and O’:
When mutually synchronised clocks in the frames S and S’ have the same reading, O and
O’ have the equal spatial separations in S and S’, and conversely, When O and O’ have
equal spatial separations in the frames S and S’, mutually synchronised clocks in S and S’
have the same reading.
The dependence of ℓ on t in Eqn(3.3) and ℓ′ on t′ in Eqn(3.4) means that each of
the objects may be considered to be an ‘inertial clock’ by an observer in the rest frame
of the other one. That is, t is measured by the spatial separation of O’ from O in S
and t′ is measured by the spatial separation of O from O’ in S’. Conversely, after mutual
synchronisation of the clock arrays in S and S’ at the instant when O and O’ are in spatial
coincidence, t measures the spatial separation of O’ and O in S (and so is effectively a
ruler in this frame) while t′ measures the spatial separation of O’ and O in S’, constituting
a ruler in this frame. Matching of these measurements of the separation of O and O’ with
the lengths of physical rulers at rest in S and S’ is now used to demonstrate the invariance
of the measured length of the length of a ruler in uniform motion –that is, the absence of
any relativistic length contraction effect– in this case.
4 Invariance of the measured length of a ruler in uni-
form motion
Figure 1: Rulers attached to objects O and O’ are viewed from the frame S (left) and S’
(right). The equality of the separations of O and O’ in S and S’ at time t = t′ = L/v,
predicted by the RP, is used to establish the invariance of the measured length of the
moving ruler R’ in S, or of the moving ruler R in S’ (see text).
Suppose that O and O’ are equipped with rulers R and R’, parallel to the x-x′ axis
as shown in Fig.1. O coincides with the mark MR(0) of the ruler R and O’ with the
mark MR′(10) of the ruler R’. A t = t′ = 0 (Fig.1a) O and O’ are in spatial coincidence.
The clock arrays in S and S’ are mutually synchronised at this time. The length of each
ruler in its rest frame is L. The object O’ now moves along the ruler R, being in spatial
coincidence with different marks of the ruler at different times. The object O moves in a
similar manner along the ruler R’. At any given time t the separation of O and O’ in S is
given by the corresponding ‘Pointer Mark Coincidence’ (PMC):
PMC(O′, t) ≡ O′(t)@MR(J) (4.1)
where the symbol before the ampersand denotes the moving ‘pointer’, and the symbol
after it the stationary ‘mark’ with which it is spatial coincidence2. Since
PMC(O, t) ≡ O(t)@MR(0) for all t (4.2)
and x[MR(0)] = 0 it follows that the separation of O and O’ in the frame S at time t is
given by:
dO′O(t) = x[MR(J)]− x[MR(0)] = x[MR(J)] (4.3)
where
x[MR(J)] =
and where, in Fig.1, Jmax = 10, is the ordinal number of the mark at the end of the ruler.
Thus the x-coordinate origin is at MR(0). Defining in a similar manner a PMC in the
frame S’:
PMC(O, t′) ≡ O(t′)@MR′(K) (4.4)
and since
PMC(O′, t′) ≡ O′(t′)@MR′(10) for all t′ (4.5)
the separation of O and O’ in S’ at the time t′ is
(t′) = x′[MR′(10)]− x′[MR′(K)] (4.6)
where
x′[MR′(K)] =
and where, in Fig.1, Kmax = 10. The spatial configurations in S and S’ at the times
t = t′ = L/v are shown in Fig.1b. The corresponding PMC are:
PMC(O′, L/v) ≡ O′(L/v)@MR(10) (4.7)
PMC(O, L/v) ≡ O(L/v)@MR′(0) (4.8)
It follows from (4.3) and (4.6) that
dO′O(L/v) = x[MR(10)]−x[MR(0)] = L = x
′[MR′(10)]−x′[MR′(0)] = d′
(L/v) (4.9)
Since O’ coincides with MR′(10) at all times it follows that, at t = L/v
x[MR′(10)] = x[O′] = x[MR(10)] (4.10)
Also, since O is in spatial coincidence with MR′(0) at t = t′ = L/v it follows that at
t = L/v,
x[MR′(0)] = x[O] = x[MR(0)] = 0 (4.11)
2This notation was introduced in Ref. [6]. Note the similarity with an e-mail address
Eqns(4.9)-(4.11) then give at t = L/v:
x[MR′(10)]− x[MR′(0)] = x[MR(10)]− x[MR(0)] = L (4.12)
That is, the measured length of the moving ruler R’ in the frame S, at t = L/v, is the
same as the length of the same ruler at rest –there is no ‘length contraction’ effect. A
similar calculation for the length of the ruler R as measured in the frame S’ gives, at
t′ = L/v:
x′[MR(10)]− x′[MR(0)] = x′[MR′(10)]− x′[MR′(0)] = L (4.13)
The length of the moving ruler R as measured in S’, at t′ = L/v, is the same as the length
of the same ruler at rest. The above calculations have used the equality of the spatial
separations of O and O’ in S and S’ at equal times of mutually synchronised clocks in these
frames, that follows from the RP, to establish, via corresponding PMCs, the equality of
the measured lengths of a ruler at rest, or in motion. Note that nowhere in any of the
calculations was the Lorentz transformation invoked. In fact the calculations are the same
in Galilean and special relativity, since the RP is equally valid for both.
5 The time dilatation effect; proper, improper and
apparent time intervals
All the times considered above were ‘frame times’ i.e. t and t′ are the times recorded
by a synchronised clock at rest at any position in S and S’ as viewed by an observer at rest
in these respective frames. In order to discuss the time dilatation effect it will be found
convenient to use the notation t(S), t′(S ′) for the frame times where the arguments S, S’
specify the reference frame of the observer of the clock. Such times are proper times of
such a clock. The Lorentz transformation relates the space-time coordinates (x′,t′(S ′)) of
an event specified in the frame S’ to those of the same event, (x,t′(S)) as observed in S, or
vice versa. The times t(S ′)[ t′(S)] which are those of clocks at rest in S[S’], as viewed from
S’[S] are called improper times. The space-time LT gives the following invariant interval
relation between corresponding space and time intervals in the frames S and S’:
c2(∆τ ′)2 = c2(∆t′(S))2 − (∆x)2 = c2(∆t′(S ′))2 − (∆x′)2 (5.1)
where ∆x ≡ x2 − x1 etc, while the inverse LT gives:
c2(∆τ)2 = c2(∆t(S ′))2 − (∆x′)2 = c2(∆t(S))2 − (∆x)2 (5.2)
In order to use the general interval relation (5.1) to derive the time dilatation effect it is
necessary to identify the time interval ∆t′(S ′) with the proper time interval of a clock at
rest in S’ (∆x′ = 0), and with equation of motion in S: ∆x = v∆t′(S). Using the latter
equation to eliminate ∆x from (5.1) and setting ∆x′ = 0 yields the time dilatation (TD)
relation:
∆t′(S) = γ∆t′(S ′) (5.3)
Figure 2: An experiment to illustrate the TD effcet viewed from S (left) and S’ (right). a)
The pulsed lamp PL at rest in S flashes at time t(S) = L/v and PL’ at rest in S’ flashes
at time t′(S ′) = L/v. b) The light signal from PL is observed at time t(S ′) = γL/v in the
frame S’, that from PL’ at time t′(S) = γL/v in the frame S. The PMCs corresponding to
the positions of observation of the signals in the different frames are indicated. See text
for discussion.
Figure 3: Spatial configurations in the frame S (left) and the frame S’ (right) are viewed
at different times. a) t(S) = t′(S ′) = 0; the Λ is created and moves to the right in the
plane of the figure with speed v =
3c/2. b) t(S) = t′(S ′) = T ′; the Λ is observed to decay
in the frame S’. The decay products move in the plane of the figure perpendicular to the
direction of motion of the Λ. c) t(S) = t′(S ′) = γT ′; the Λ is observed to decay in the
frame S. See text for discussion. The momentum vectors of the p and π− are drawn to
scale in the different reference frames. The spatial position of each particle is at the tail
of the corresponding momentum vector.
where γ ≡ 1/
1− (v/c)2, relating the improper to the proper time of a clock at rest in
S’. In a similar manner the interval relation (5.2) gives the TD relation for a clock at rest
in S and observed from S’:
∆t(S ′) = γ∆t(S) (5.4)
It is important to note the existence of four different time symbols, with different opera-
tional meanings in Eqns(5.3) and (5.4). The proper times t(S) and t′(S ′) (corresponding
to the ‘frame times’ t and t′ of the previous sections) and the improper times t(S ′) and
t(S ′). The notation for these times just introduced may be called ‘clock oriented’ since
only the readings of a single clock (observed either at rest, or in motion) appear in the TD
relations. In any actual experiment where the TD effect in measured, two clocks are nec-
essary, the observed moving clock, and another one at rest to measure the corresponding
time interval in the observer’s proper frame. If a clock at rest in S’ is observed from S as
in Eqn(5.3), the time interval ∆t′(S) is actually that, ∆τ , recorded by a similar clock, at
rest in S while ∆t′(S ′) is the corresponding time interval recorded by the (slowed-down)
moving clock. Since the observed rate of the moving clock depends on its motion, ∆t′(S ′)
is not a proper time interval for the observer in S. From the view-point of the latter this
is an ‘apparent’ (velocity-dependent) time interval that may be denoted simply as ∆t′,
to distinguish it from the observer’s proper time interval ∆τ . This gives an alternative
‘observer oriented’ time notation for the TD relations (5.3) and (5.4) above:
∆τ = γ∆t′ (5.5)
∆τ ′ = γ∆t (5.6)
This alternative notation has beeen employed in several previous papers by the present
author [6, 8, 11, 12, 13, 16].
In order to apply the TD relations (5.3) and (5.4), or (5.5) and (5.6), to any actual
or imagined experiment an operational definition must be given to the improper time
intervals of Eqns(5.3) and (5.4) or the apparent time intervals of (5.5) and (5.6). Two
examples of such definitions will be given, the first in a thought experiment to illustrate
the physical meaning of the TD effect, the second in an actual experiment typical of
many performed in particle physics, where the TD effect is used to measure the proper
decay time of an unstable particle. However as will be seen, the thought experiment and
actually realisable (and many times realised) one are similar in all essential features.
What notation is most convenient depends on the experiment considered. In the
observation of the TD effect in the last CERN muon g-2 experiment [17] where the time
interval ∆τ was directly measured by clocks in the laboratory frame, and ∆t′ was the
known muon rest-frame lifetime, it was natural to use Eqn(5.5). For the second of the
two experiments considered below where ∆τ is not directly measured but inferred from
spatial measurements in the frame S, the relation (5.3) relating connecting a proper time
in the frame S’ to an improper time in the frame S, is used.
In the thought experiment it is imagined that the objects O, O’ are each equipped
with local pulsed lamps PL, PL’. The objects O, O’ are in spatial coincidence at times
t(S) = t′(S ′) = 0 and are attached to rulers of length 2L in similar spatial configurations to
that shown in Fig.1a. The objects move apart with relative velocity v =
3c/2. As shown
in Fig.2a, at the times t(S) = t′(S ′) = L/v, PL and PL’ both flash, producing an isotropic
pulse of photons. The observation times in S of the photon signal produced by PL’, and
in S’ of the photon signal produced by PL, are given by Eqns(5.3) and (5.4) respectively.
Since γ = 2, these observations occur at the times t(S) = t′(S ′) = γL/v = 2L/v. The
corresponding spatial configurations of O and O’ at these times shown in Fig.2b. It can
be seen that the observation times of the light flashes in S and S’ correspond to different
PMCs of the objects O and O’ and to different spatial separations of the objects:
In S PL : PMC(MR′(10), L/v) ≡ MR′(10)@0 = MR′(10)@MR(0) (5.7)
PL′ : PMC(O′, γL/v) ≡ O′@MR(20) = MR′(20)@MR(20) (5.8)
In S′ PL′ : PMC(MR(10), L/v) ≡ MR(10)@O′ = MR(10)@MR′(20) (5.9)
PL : PMC(O, γL/v) ≡ O@MR′(0) = MR(0)@MR′(0) (5.10)
ℓ(γL/v)
ℓ(L/v)
ℓ′(γL/v)
ℓ′(L/v)
vγL/v
vL/v)
= γ (5.11)
The relations in (5.11) follow directly from the RP, while the PMC in (5.7)-(5.10) are
obtained from the geometry of Fig.2 and the invariance of the lengths of the moving
rulers derived in Section 3 above.
The different PMC corresponding to observations of the light flashes emitted by PL
and PL’ in different frames in (5.7)-(5.10) is deeply perplexing for common-sense con-
cepts of space and time. For example the photon bunches emitted by PL’ correspond
to MR(10)@MR′(20) in S’ and to MR′(20)@MR(20) in S. In some discussions of time
dilatation this apparent paradox is avoided by invoking a hypothetical contraction of a
moving ruler by a factor 1/γ [18]. This has the effect of shortening the moving ruler
R by a factor 1/2 in the right hand figure in Fig.2a, so that the PMC corresponding
to the flashing of PL’ becomes MR(20)@MR′(20), the same as in S with inversion of
pointer and mark. However, as demonstrated above, there is no such length contraction
effect, which, as pointed out elsewhere [5, 6, 7, 8, 9] is a spurious consequence of misin-
terpreting the space-time Lorentz transformation. Indeed the possibility of such a length
contraction effect is already excluded by inspection of Fig.2a. In the right hand figure, the
PMC correponding to the moving object O considered as a pointer is MR(0)@MR′(10).
Since O is in motion and R’ at rest no hypothetical length contraction effect operates
here. In the left hand figure the mutually simultaneous PMC in S is MR′(10)@MR(0)
so that at t(S) = t′(S ′) = L/v observers in S and S’ see reciprocal PMCs, i.e. ones
related by exchange of the pointer and mark symbols. If however the length contrac-
tion effect exists, the observer in S will see instead that the PMC corresponding to O is
MR′(0)@MR(0) at time t(S) = L/v. But from the RP this PMC must correspond to the
times t(S) = t′(S ′) = 2L/v (see Fig.2b) contrary to the assumption that t(S) = L/v. The
length contraction hypothesis therefore contradicts the corollory of the RP that states
that mutually simultaneous events in two frames have reciprocal PMCs, since it implies
that the reciprocal PMCs MR′(0)@MR(0) and MR(0)@MR′(0) are not mutually simul-
taneous.
The second example of a TD experiment illustrates a typical application of the effect
in particle physics (see Fig.3). A π− meson interacts with a proton in a thin plastic
target T to produce a Λ hyperon via the reaction3 π−p → ΛK0 The hyperon moves with
3The results of an actual such experiment constructed to test the ∆S = ∆Q rule in semileptonic
neutral kaon decays are described in Ref. [19].
velocity v =
3c/2 perpendicular to the plane of the target in the laboratory frame S.
After the time t′(S ′) = T ′ in its rest frame S’, it decays to a proton and a negative pion:
Λ → pπ−. These decay products are observed in the laboratory system. The experiment
is in every way similar to that shown in Fig.2. The object O is replaced by the target T,
the object O’ by the undecayed Λ or the kinematical system constructed from its decay
products. The photon pulse emitted by PL’ is replaced by the decay products of the
Λ. By reconstructing the trajectories of the decay p and π− in a particle detector the
position of the decay event and hence the decay length lD –the distance between the point
of production and decay of the Λ– in the frame S can be measured. Identification of the
p and π− and measurement of their momenta (typically by measurement of the curvature
of their trajectories in a known magnetic field ) enables the momentum P and the energy
E of the Λ to be determined. Since v = Pc2/E and γ = E/(mΛc
2) where mΛ is the mass
of the Λ, the proper decay time of the Λ is given by Eqn(5.3) as:
T ′ = ∆t′(S ′) =
∆t′(S)
(5.12)
The spatial configurations of T and the Λ at different times in the frames S and S’ are
shown in Fig.3. The spatial separations of T and the Λ at the observed instant of decay
in S and S’ obey the relation (5.11). This implies that this separation, in changing the
frame of observation from the rest frame of the Λ to the laboratory system in which it is
motion, undergoes a ‘length expansion’ by the factor γ. In accordance with Eqn(5.11), it
can be seen that this is a necessary consequence of the RP, given the existence of the TD
effect.
The mutally simultaneous events in S and S’ shown in Fig.3c, correspond, as they
must, to equal spatial separations of T and the physical object constituted by the decay
products, p and π−, of the Λ. However, in the frame S, these particles have just been
created and have vanishing spatial separation, whereas in S’ they are spatially separated by
a distance corresponding to a time-of-flight (γ− 1)T ′. This also seems highly paradoxical
when interpreted by commonsense classical concepts of space and time.
Acknowledgement
I thank the referee of the journal that rejected Ref. [11] for publication for correspon-
dence that was important for the clarificatiion of the ideas expressed in both the latest
version of Ref. [11] and the present paper.
Added Note
The calculations presented in the present paper are flawed by a major conceptual
misunderstanding which is rectified in later papers [20, 21] treating similar subjects.
At the time of writing the present paper, the author had correctly understood the
spurious nature of the ‘relativity of simultaneity’and ‘length contraction’ effects of con-
ventional special relativity theory [5, 7, 8, 10] but had not yet drawn the simple conclusion
that the existence of the genuine and experimentally-confirmed time dilatation effect then
necessarily implies that the Reciprocity Principle, as generally understood, also breaks
down in special relativity. This point is easily understood by considering the first member
of Eqn(3.1), written in a simplified notation as:
dxO′O
Transforming into the frame S’, the invariance of length intervals implies that
dxO′O = −dx
Since the time dilatation relation gives dt = γdt′, the Reciprocity Principle of (3.1) is
replaced by:
dxO′O
so that
= −γv
to be compared with v′ = −v given by (3.1).
The detailed calculations presented in Section 4 are correct and logically coherent
given the initial assumptions, but the configurations shown in the frame S’ in Fig.1 do not
correspond to observations in this frame of the coincidence events specified in the frame
S in the same space-time experiment. If this were the case, in the S’ frame configurations
in Fig.1 v should be replaced by γv and t and t′ should be related by time dilatation
relation t = γt′. In fact, what are shown in Fig.1 and considered in Section 4 are the
configurations in S of a primary experiment and in S’ of the corresponding but physically
independent reciprocal experiment [20, 22].
Nevertheless, the invariance of corresponding length intervals can be derived [21] by
considering the configurations in S and S’ in Fig.1b in the case that they are corresponding
ones, at the same epoch, in the same space-time experiment. In this case, as explained
above, the speed of O in S’ should be γv, not v. Consider, however, an object Õ with the
same x′ coordinate as O that does have the velocity v. The separation L′ of O and O’
in S’ is then equal to that between O’ and Õ. at the epoch of Fig.1b. Compare now the
configuration of O and O’ in S, with separation L with the corresponding one of Õ. and
O’ in S’ with separation L′. From the symmetry of the configurations it can be seen that
both L and L′ can depend only on v: L = L(v), L′ = L′(v). The reciprocity of the two
configurations is now invoked to give the condition, as stated by Pauli [23]:
The contraction of length at rest in S’ and observed from S is equal to
the length at rest in S as observed from S’.
The ‘length at rest in S’ ’ is L′ which ‘as observed from S’ is L, whereas the ‘length
at rest in S ’ is L which ‘as observed fron S’ ’ is L′. Denoting the contraction factor by
α(v), the above condition states that
L = α(v)L′, L′ = α(v)L
which implies that L = α(v)2L or α(v)2 = 1 so that L = L′ and the spatial separation
between O and O’ is the same in S and S’ at corresponding epochs. The same conclusion
is more simply reached by noting the symmetry of the configurations of O,O’ in S and
Õ,O’ in S’. and applying Leibnitz’ Principle of Sufficient Reason [21].
If, therefore, in the primary experiment, shown in S in Fig.2b and S’ in Fig.2a, the
configuration in S’ in Fig.2a is to correctly represent that corresponding to the configura-
tion in S in Fig.2b, the velocity v in S’ should be replaced by γv, so that when PL’ flashes
O’ is aligned with MR(20) in both S and S’. In the reciprocal experiment, shown in S in
Fig.2a and S’ in Fig.2b, v in S in Fig.2a should be replaced by γv so that O is aligned
with MR’(0) in both S and S’ when PL flashes.
Similarly, in the thought experiment of Fig.5, if the S’ frame configurations on the right
side of the figure are to represent observations in this frame of events shown in S by the
configurations on the left side, instead of what are actually shown which are configurations
of the physically independent reciprocal experiment, v should be replaced by γv in all the
S’ frame configurations. In this case, there is no mismatch between the spatial position of
the decay event in the two frames and the claimed ‘length expansion’ effect does not occur.
Indeed the claimed ‘... different PMC corresponding to observations of the light flashes
emitted by PL and PL’ in different frames in (5.7)-(5.10)’ is not only ‘...deeply perplex-
ing for common-sense concepts of space and time.’ it is the absurd (self-contradictory)
consequence of assuming, at the same time, that length intervals are invariant, time di-
latation occurs and the conventional interpretation of the Reciprocity Principle holds. In
conventional special relativity theory time dilatation and the Reciprocity Principle are
reconciled by invoking the spurious ‘length contraction’ effect dxO′O = −γx
[18]. so
that v′ = −v. The correct physical interpretation of the Reciprocity Principle is actually
the definition of the configuration in S’ of the physically-independent experiment that is
reciprocal to the primary one specified by the standard configuration of the frames S and
S’ [20, 22].
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P233.
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  Ponderable objects moving in free space according to Newton's First Law
constitute both rulers and clocks when one such object is viewed from the rest
frame of another. Together with the Reciprocity Principle this is used to
demonstrate, in both Galilean and special relativity, the invariance of the
measured length of a ruler in motion. The different times: `proper', `improper'
and `apparent' appearing in different formulations of the relativistic time
dilatation relation are discussed and exemplified by experimental applications.
A non-intuitive `length expansion' effect predicted by the Reciprocity
Principle as a necessary consequence of time dilatation is pointed out

Introduction
The standard text-book presentation of special relativity follows closely that of Ein-
stein’s seminal paper of 1905 [1] in basing the theory on the Special Relativity Principle,
classical electromagnetism and the postulate of constant light speed. However an alterna-
tive and conceptually simpler approach to the physics of space and time, in the absence
of gravitational fields, is possible in which it is not necessary to consider light signals,
classical electromagnetism, or indeed, any dynamical theory whatsoever. The Lorentz
transformation (LT) was first derived in this way by Ignatowsky [2] in 1910. Purely
mathematical considerations imply, in such a derivation of the LT, the existence of a
maximum relative velocity, V , of two inertial frames. Use of relativistic kinematics then
shows that V is equal to the speed of light, c, when light in identified as a manifestation
of the propgation in space-time of massless particles –photons [3]. In this way Einstein’s
mysterious second postulate is derived from first principles. The fundamental axiom un-
derlying such an approach is the Reciprocity Principle (RP) [4, 3], discussed in Section
3 below, relating the the relative velocities of two inertial frames. Derivations of the LT
and the parallel velocity addition formula based on the RP and other simple axioms are
given in Ref. [3].
In the present paper the space-time properties of ponderable1 physical bodies in free
space, as described by Newton’s First Law of mechanics, are used together with the RP,
to demonstrate the invariance of the measured length of a ruler in uniform motion. The
proof given is valid in both Galilean and special relativity, since Newton’s First Law and
the RP hold in both theories.
The analysis presented is based on a careful definition of physical time concepts. In
particular, the ‘frame time’ or ‘proper times’ that appear in in the RP, are distinguished
from the ‘improper time’ or ‘apparent time’ (of a moving clock) that appear in the Time
Dilatation (TD) relation of special relativity.
The paper is organised as follows: The following section contains an elementary discus-
sion of the concepts of ‘space’, ‘time’ and ‘motion’ in physics, in connection with Newton’s
First Law. In Section 3, the RP is introduced and discussed in relation to Newton’s First
Law. It is pointed out that, because of the RP, ‘rulers are clocks’ and ‘clocks are rulers’
when the motion of ponderable bodies in free space is considered. In Section 4 the RP is
used to demonstrate the invariance of the measured length of a uniformly moving ruler.
In Section 5 the operational meanings of the time symbols appearing in the TD formula
of special relativity are discussed. This may be done in a ‘clock oriented’ manner in terms
of ’proper’ and ’improper’ times of the observed clock, or in an ‘observer oriented’ manner
in terms of the proper time of the observer’s local clock and the ‘apparent time’, as seen
by the observer, of the moving clock. Two specific experiments are described to exemplify
the operational meanings of the time symbols in the TD formula. A non-intuitive ‘length
expansion’ effect is found to relate similarly defined spatial intervals corresponding to the
observation of an event either in the rest frame of the clock, or in a frame in which it is
in uniform motion.
1That is bodies, with a non-vanishing Newtonian mass, which may be associated with an inertial
frame in which the body is at rest. No such frame may be associated with a massless object.
The results of the present paper show that the ‘length contraction’ effect and the
correlated ‘relativity of simultaneity’ effect of conventional special relativity do not exist.
A detailed discussion of the reason for the spurious nature of these effects of conventional
special relativity theory may be found in Refs. [5, 6, 7, 8, 9, 10] .
However, a genuine ‘relativistic length contraction’ effect does occur when distances be-
tween spatial coincidences of moving objects are observed from different inertial frames [11].
Also a genuine ‘relativity of simultaneity’ effect occurs when clocks at rest in two different
inertial frames are viewed from a third one [12, 13]. An alternative derivation, directly
from the RP, of the invariance of the measured spatial separation of two objects at rest
in the same inertial frame as well as the absence of the conventional ‘relativity of simul-
taneity’ effect is given in Ref. [9].
2 Physical time and Newton’s First Law of Mechan-
In physics the concepts of ‘time’ and ‘motion’ are inseparable. In a world in which
motion did not exist the physical concept of time would be meaningless. Similarly the
physical concepts of ‘space’ and ‘motion’ are inseperable. Without the concept of space,
no operational definition of motion is possible. The concept of historical time –the time
of the everyday world of human existence– requires the introduction of the further, and
equivalent, concepts of ‘uniform motion’ and ‘cyclic motion with constant period’. For
example, the unit of time the ‘year’ is identified with the (assumed constant) period of
rotation of the Earth about the Sun.
The idea of uniform motion entered into physics in a quantitative way with the for-
mulation of Newton’s First Law [14]
Every body continues in its state of rest, or uniform motion in a right
line unless it is compelled to change that state by forces impressed upon
This law gives an operational meaning to the physical concept of ‘uniform motion’.
It is defined by observations of the position of any ponderable object in ‘free space’ i.e.
in the absence of any mechanical interaction of the object with other objects. There is
a one-to-one correspondence between such a ponderable object and an ‘inertial frame’
of relativity theory. As will be discussed in the following section, one such ponderable
object, O, constitutes both a ruler and a clock for an observer in the rest frame of another
such object, O’, and vice versa.
When time is measured by using a cyclic physical phenomenon, e.g. an analogue clock,
time measurement reduces to recording the result of a spatial (or angular) measurement.
There is a one-to-one correspondence between the spatial coincidence of a stationary
‘mark’ on the face of the clock and a moving ‘pointer’, constituted by the hand of the clock,
and the time measurement [6]. A ‘time interval’ is measured by the angular separation of
two such ‘pointer-mark coincidences’. The implicit assumption is that the motion of the
pointer is ‘uniform’. There is an evident logical circularity here since ‘equal’ time intervals
measured by such an analogue clock assume that the angular velocity of the hand is
constant, whereas constant angular velocity is established by observation of equal angular
increments for equal time time intervals (i.e. also equal angular increments) recorded by
a second clock of supposedly known uniform rate. In practice, this conundrum is resolved
by an appeal to physics. For example, an undamped pendulum in a uniform gravitational
field is predicted, by the laws of mechanics, to have a constant period of oscillation.
Quantum mechanics predicts the same transition frequency and mean lifetimes for two
identical atoms in the same excited state, in the same physical environment, etc.
Measurements of ‘time’ are then ultimately observations of spatial phenomena, e.g.
the time measurement corresponding to observation of the number displayed by a digital
clock is a spatial perception. This will also be the case for time measurements related to
observation of two ponderable objects O and O’ in motion in free space that will now be
discussed.
3 The Reciprocity Principle: rulers are clocks, and
clocks are rulers
Consider two non-interacting ponderable objects O and O’, with arbitary motions in
free space. They are placed at the origins of inertial coordinate systems S and S’ with
axes orientated so that the x and x′ axes are parallel to the relative velocity vector of O
and O’. Without any loss of generality for the following discussion, it may be assumed
that O and O’ lie on the common x-x′ axis.
The Reciprocity Principle (RP) [4, 3, 9] is defined by the equation:
v = vO′O =
∂xO′O
= −vOO′ (3.1)
where xO′O ≡ xO′−xO and x
, or in words: ‘If the velocity of O’ relative to O
is ~v, the velocity of O relative to O’ is - ~v’. In many discussions of special relativity, the RP
is taken as ‘obvious’ and is often not even declared as a separate axiom. This is the case,
for example, in Einstein’s 1905 special relativity paper [1]. However, as first demonstrated
by Ignatowsky in 1910 [2], it is sufficient, together with some other weaker axioms such as
the homogeneity of space or single-valuedness of the transfomation equations, to derive [3]
the space-time Lorentz transformation and hence the whole of special relativity theory.
Eqn(3.1) looks very similar to the equation defining the relative velocity of two objects
A and B as observed in a single inertial reference frame (say S):
vAB ≡ vA − vB =
d(xA − xB)
= −vBA (3.2)
The crucial difference is the appearence in the RP, (3.1), of two different times t and t′.
The time t is the ‘frame time’ of S. i.e. the time registered by a synchronised clock at rest,
at any position in S, according to an observer also a rest in S. The frame time t′ is similarly
defined by an array of synchronised clocks at rest in S’. Eqn(3.1) (and its integral) gives a
relation between the times t and t′ Both t and t′ correspond to ‘proper times’ of clocks at
rest, whereas, as explained in Section 4 below, the Lorentz transformation relates instead
a proper time to an ‘improper time’ –the observed time of a clock in uniform motion.
Suppose now that O and O’ are equipped with local clocks that are observed to run at
exactly the same rate when they are both at rest in the same inertial frame. The direction
of the relative velocity vector ~v of O’ relative to O is such that they are approaching each
other at the frame times t and t′. The spatial separations of O and O’ in S and S’ are ℓ(t)
and ℓ′(t′) respectively, at times t and t′. Using the RP, a spatial coincidence of O and O’
will be observed at the time
tOO′ = t+
(3.3)
in S, and
= t′ +
ℓ′(t′)
(3.4)
in S’. The OO’ coincidence event will be mutually simultaneous in the frames S and S’.
Note that the OO’ spatial coincidence that is mutually simultaneous in S and S’
constitutes a pair of reciprocal pointer mark coincidences. In S the mark is at the position
of O and the moving pointer at the position of O’, whereas in S’ the position of O’
constitutes the mark and the position of O the pointer. A corollary is that all such pairs
of reciprocal pointer mark coincidences are mutually simultaneous. This is the basis of the
‘system external synchronistation’ [15] as introduced in Einstein’s first special relativity
paper [1] to synchronise clocks at rest in different inertial frames when they are in spatial
coincidence.
The observation of the OO’ coincidence event in both frames can be used to give a
condition that any other pair of events, one observed in S, the other observed in S’ are
mutually simultaneous. If the time of an event in S is t̃ and another event in S’ is t̃′ they
will be ‘mutually simultaneous’ providing that:
t̃′ − t̃ = t′
− tOO′ (3.5)
Combining (3.3)-(3.5) gives:
t̃′ − t̃ = t′
− tOO′ = t
′ − t+
ℓ′(t′)− ℓ(t)
(3.6)
If now events occuring at times t in S and t′ in S’ are mutually simultaneous, it follows
from (3.5) and (3.6) that ℓ(t) = ℓ′(t′), so that events which occur when O and O’ have
the same spatial separation in S and S’ are mutually simultaneous. A special case occurs
if the clock arrays in S and S’ are mutually synchronised so that ℓ(t) = ℓ′(t′ = t). There
is then a direct correlation between either t or t′ and the spatial separation of O and O’:
When mutually synchronised clocks in the frames S and S’ have the same reading, O and
O’ have the equal spatial separations in S and S’, and conversely, When O and O’ have
equal spatial separations in the frames S and S’, mutually synchronised clocks in S and S’
have the same reading.
The dependence of ℓ on t in Eqn(3.3) and ℓ′ on t′ in Eqn(3.4) means that each of
the objects may be considered to be an ‘inertial clock’ by an observer in the rest frame
of the other one. That is, t is measured by the spatial separation of O’ from O in S
and t′ is measured by the spatial separation of O from O’ in S’. Conversely, after mutual
synchronisation of the clock arrays in S and S’ at the instant when O and O’ are in spatial
coincidence, t measures the spatial separation of O’ and O in S (and so is effectively a
ruler in this frame) while t′ measures the spatial separation of O’ and O in S’, constituting
a ruler in this frame. Matching of these measurements of the separation of O and O’ with
the lengths of physical rulers at rest in S and S’ is now used to demonstrate the invariance
of the measured length of the length of a ruler in uniform motion –that is, the absence of
any relativistic length contraction effect– in this case.
4 Invariance of the measured length of a ruler in uni-
form motion
Figure 1: Rulers attached to objects O and O’ are viewed from the frame S (left) and S’
(right). The equality of the separations of O and O’ in S and S’ at time t = t′ = L/v,
predicted by the RP, is used to establish the invariance of the measured length of the
moving ruler R’ in S, or of the moving ruler R in S’ (see text).
Suppose that O and O’ are equipped with rulers R and R’, parallel to the x-x′ axis
as shown in Fig.1. O coincides with the mark MR(0) of the ruler R and O’ with the
mark MR′(10) of the ruler R’. A t = t′ = 0 (Fig.1a) O and O’ are in spatial coincidence.
The clock arrays in S and S’ are mutually synchronised at this time. The length of each
ruler in its rest frame is L. The object O’ now moves along the ruler R, being in spatial
coincidence with different marks of the ruler at different times. The object O moves in a
similar manner along the ruler R’. At any given time t the separation of O and O’ in S is
given by the corresponding ‘Pointer Mark Coincidence’ (PMC):
PMC(O′, t) ≡ O′(t)@MR(J) (4.1)
where the symbol before the ampersand denotes the moving ‘pointer’, and the symbol
after it the stationary ‘mark’ with which it is spatial coincidence2. Since
PMC(O, t) ≡ O(t)@MR(0) for all t (4.2)
and x[MR(0)] = 0 it follows that the separation of O and O’ in the frame S at time t is
given by:
dO′O(t) = x[MR(J)]− x[MR(0)] = x[MR(J)] (4.3)
where
x[MR(J)] =
and where, in Fig.1, Jmax = 10, is the ordinal number of the mark at the end of the ruler.
Thus the x-coordinate origin is at MR(0). Defining in a similar manner a PMC in the
frame S’:
PMC(O, t′) ≡ O(t′)@MR′(K) (4.4)
and since
PMC(O′, t′) ≡ O′(t′)@MR′(10) for all t′ (4.5)
the separation of O and O’ in S’ at the time t′ is
(t′) = x′[MR′(10)]− x′[MR′(K)] (4.6)
where
x′[MR′(K)] =
and where, in Fig.1, Kmax = 10. The spatial configurations in S and S’ at the times
t = t′ = L/v are shown in Fig.1b. The corresponding PMC are:
PMC(O′, L/v) ≡ O′(L/v)@MR(10) (4.7)
PMC(O, L/v) ≡ O(L/v)@MR′(0) (4.8)
It follows from (4.3) and (4.6) that
dO′O(L/v) = x[MR(10)]−x[MR(0)] = L = x
′[MR′(10)]−x′[MR′(0)] = d′
(L/v) (4.9)
Since O’ coincides with MR′(10) at all times it follows that, at t = L/v
x[MR′(10)] = x[O′] = x[MR(10)] (4.10)
Also, since O is in spatial coincidence with MR′(0) at t = t′ = L/v it follows that at
t = L/v,
x[MR′(0)] = x[O] = x[MR(0)] = 0 (4.11)
2This notation was introduced in Ref. [6]. Note the similarity with an e-mail address
Eqns(4.9)-(4.11) then give at t = L/v:
x[MR′(10)]− x[MR′(0)] = x[MR(10)]− x[MR(0)] = L (4.12)
That is, the measured length of the moving ruler R’ in the frame S, at t = L/v, is the
same as the length of the same ruler at rest –there is no ‘length contraction’ effect. A
similar calculation for the length of the ruler R as measured in the frame S’ gives, at
t′ = L/v:
x′[MR(10)]− x′[MR(0)] = x′[MR′(10)]− x′[MR′(0)] = L (4.13)
The length of the moving ruler R as measured in S’, at t′ = L/v, is the same as the length
of the same ruler at rest. The above calculations have used the equality of the spatial
separations of O and O’ in S and S’ at equal times of mutually synchronised clocks in these
frames, that follows from the RP, to establish, via corresponding PMCs, the equality of
the measured lengths of a ruler at rest, or in motion. Note that nowhere in any of the
calculations was the Lorentz transformation invoked. In fact the calculations are the same
in Galilean and special relativity, since the RP is equally valid for both.
5 The time dilatation effect; proper, improper and
apparent time intervals
All the times considered above were ‘frame times’ i.e. t and t′ are the times recorded
by a synchronised clock at rest at any position in S and S’ as viewed by an observer at rest
in these respective frames. In order to discuss the time dilatation effect it will be found
convenient to use the notation t(S), t′(S ′) for the frame times where the arguments S, S’
specify the reference frame of the observer of the clock. Such times are proper times of
such a clock. The Lorentz transformation relates the space-time coordinates (x′,t′(S ′)) of
an event specified in the frame S’ to those of the same event, (x,t′(S)) as observed in S, or
vice versa. The times t(S ′)[ t′(S)] which are those of clocks at rest in S[S’], as viewed from
S’[S] are called improper times. The space-time LT gives the following invariant interval
relation between corresponding space and time intervals in the frames S and S’:
c2(∆τ ′)2 = c2(∆t′(S))2 − (∆x)2 = c2(∆t′(S ′))2 − (∆x′)2 (5.1)
where ∆x ≡ x2 − x1 etc, while the inverse LT gives:
c2(∆τ)2 = c2(∆t(S ′))2 − (∆x′)2 = c2(∆t(S))2 − (∆x)2 (5.2)
In order to use the general interval relation (5.1) to derive the time dilatation effect it is
necessary to identify the time interval ∆t′(S ′) with the proper time interval of a clock at
rest in S’ (∆x′ = 0), and with equation of motion in S: ∆x = v∆t′(S). Using the latter
equation to eliminate ∆x from (5.1) and setting ∆x′ = 0 yields the time dilatation (TD)
relation:
∆t′(S) = γ∆t′(S ′) (5.3)
Figure 2: An experiment to illustrate the TD effcet viewed from S (left) and S’ (right). a)
The pulsed lamp PL at rest in S flashes at time t(S) = L/v and PL’ at rest in S’ flashes
at time t′(S ′) = L/v. b) The light signal from PL is observed at time t(S ′) = γL/v in the
frame S’, that from PL’ at time t′(S) = γL/v in the frame S. The PMCs corresponding to
the positions of observation of the signals in the different frames are indicated. See text
for discussion.
Figure 3: Spatial configurations in the frame S (left) and the frame S’ (right) are viewed
at different times. a) t(S) = t′(S ′) = 0; the Λ is created and moves to the right in the
plane of the figure with speed v =
3c/2. b) t(S) = t′(S ′) = T ′; the Λ is observed to decay
in the frame S’. The decay products move in the plane of the figure perpendicular to the
direction of motion of the Λ. c) t(S) = t′(S ′) = γT ′; the Λ is observed to decay in the
frame S. See text for discussion. The momentum vectors of the p and π− are drawn to
scale in the different reference frames. The spatial position of each particle is at the tail
of the corresponding momentum vector.
where γ ≡ 1/
1− (v/c)2, relating the improper to the proper time of a clock at rest in
S’. In a similar manner the interval relation (5.2) gives the TD relation for a clock at rest
in S and observed from S’:
∆t(S ′) = γ∆t(S) (5.4)
It is important to note the existence of four different time symbols, with different opera-
tional meanings in Eqns(5.3) and (5.4). The proper times t(S) and t′(S ′) (corresponding
to the ‘frame times’ t and t′ of the previous sections) and the improper times t(S ′) and
t(S ′). The notation for these times just introduced may be called ‘clock oriented’ since
only the readings of a single clock (observed either at rest, or in motion) appear in the TD
relations. In any actual experiment where the TD effect in measured, two clocks are nec-
essary, the observed moving clock, and another one at rest to measure the corresponding
time interval in the observer’s proper frame. If a clock at rest in S’ is observed from S as
in Eqn(5.3), the time interval ∆t′(S) is actually that, ∆τ , recorded by a similar clock, at
rest in S while ∆t′(S ′) is the corresponding time interval recorded by the (slowed-down)
moving clock. Since the observed rate of the moving clock depends on its motion, ∆t′(S ′)
is not a proper time interval for the observer in S. From the view-point of the latter this
is an ‘apparent’ (velocity-dependent) time interval that may be denoted simply as ∆t′,
to distinguish it from the observer’s proper time interval ∆τ . This gives an alternative
‘observer oriented’ time notation for the TD relations (5.3) and (5.4) above:
∆τ = γ∆t′ (5.5)
∆τ ′ = γ∆t (5.6)
This alternative notation has beeen employed in several previous papers by the present
author [6, 8, 11, 12, 13, 16].
In order to apply the TD relations (5.3) and (5.4), or (5.5) and (5.6), to any actual
or imagined experiment an operational definition must be given to the improper time
intervals of Eqns(5.3) and (5.4) or the apparent time intervals of (5.5) and (5.6). Two
examples of such definitions will be given, the first in a thought experiment to illustrate
the physical meaning of the TD effect, the second in an actual experiment typical of
many performed in particle physics, where the TD effect is used to measure the proper
decay time of an unstable particle. However as will be seen, the thought experiment and
actually realisable (and many times realised) one are similar in all essential features.
What notation is most convenient depends on the experiment considered. In the
observation of the TD effect in the last CERN muon g-2 experiment [17] where the time
interval ∆τ was directly measured by clocks in the laboratory frame, and ∆t′ was the
known muon rest-frame lifetime, it was natural to use Eqn(5.5). For the second of the
two experiments considered below where ∆τ is not directly measured but inferred from
spatial measurements in the frame S, the relation (5.3) relating connecting a proper time
in the frame S’ to an improper time in the frame S, is used.
In the thought experiment it is imagined that the objects O, O’ are each equipped
with local pulsed lamps PL, PL’. The objects O, O’ are in spatial coincidence at times
t(S) = t′(S ′) = 0 and are attached to rulers of length 2L in similar spatial configurations to
that shown in Fig.1a. The objects move apart with relative velocity v =
3c/2. As shown
in Fig.2a, at the times t(S) = t′(S ′) = L/v, PL and PL’ both flash, producing an isotropic
pulse of photons. The observation times in S of the photon signal produced by PL’, and
in S’ of the photon signal produced by PL, are given by Eqns(5.3) and (5.4) respectively.
Since γ = 2, these observations occur at the times t(S) = t′(S ′) = γL/v = 2L/v. The
corresponding spatial configurations of O and O’ at these times shown in Fig.2b. It can
be seen that the observation times of the light flashes in S and S’ correspond to different
PMCs of the objects O and O’ and to different spatial separations of the objects:
In S PL : PMC(MR′(10), L/v) ≡ MR′(10)@0 = MR′(10)@MR(0) (5.7)
PL′ : PMC(O′, γL/v) ≡ O′@MR(20) = MR′(20)@MR(20) (5.8)
In S′ PL′ : PMC(MR(10), L/v) ≡ MR(10)@O′ = MR(10)@MR′(20) (5.9)
PL : PMC(O, γL/v) ≡ O@MR′(0) = MR(0)@MR′(0) (5.10)
ℓ(γL/v)
ℓ(L/v)
ℓ′(γL/v)
ℓ′(L/v)
vγL/v
vL/v)
= γ (5.11)
The relations in (5.11) follow directly from the RP, while the PMC in (5.7)-(5.10) are
obtained from the geometry of Fig.2 and the invariance of the lengths of the moving
rulers derived in Section 3 above.
The different PMC corresponding to observations of the light flashes emitted by PL
and PL’ in different frames in (5.7)-(5.10) is deeply perplexing for common-sense con-
cepts of space and time. For example the photon bunches emitted by PL’ correspond
to MR(10)@MR′(20) in S’ and to MR′(20)@MR(20) in S. In some discussions of time
dilatation this apparent paradox is avoided by invoking a hypothetical contraction of a
moving ruler by a factor 1/γ [18]. This has the effect of shortening the moving ruler
R by a factor 1/2 in the right hand figure in Fig.2a, so that the PMC corresponding
to the flashing of PL’ becomes MR(20)@MR′(20), the same as in S with inversion of
pointer and mark. However, as demonstrated above, there is no such length contraction
effect, which, as pointed out elsewhere [5, 6, 7, 8, 9] is a spurious consequence of misin-
terpreting the space-time Lorentz transformation. Indeed the possibility of such a length
contraction effect is already excluded by inspection of Fig.2a. In the right hand figure, the
PMC correponding to the moving object O considered as a pointer is MR(0)@MR′(10).
Since O is in motion and R’ at rest no hypothetical length contraction effect operates
here. In the left hand figure the mutually simultaneous PMC in S is MR′(10)@MR(0)
so that at t(S) = t′(S ′) = L/v observers in S and S’ see reciprocal PMCs, i.e. ones
related by exchange of the pointer and mark symbols. If however the length contrac-
tion effect exists, the observer in S will see instead that the PMC corresponding to O is
MR′(0)@MR(0) at time t(S) = L/v. But from the RP this PMC must correspond to the
times t(S) = t′(S ′) = 2L/v (see Fig.2b) contrary to the assumption that t(S) = L/v. The
length contraction hypothesis therefore contradicts the corollory of the RP that states
that mutually simultaneous events in two frames have reciprocal PMCs, since it implies
that the reciprocal PMCs MR′(0)@MR(0) and MR(0)@MR′(0) are not mutually simul-
taneous.
The second example of a TD experiment illustrates a typical application of the effect
in particle physics (see Fig.3). A π− meson interacts with a proton in a thin plastic
target T to produce a Λ hyperon via the reaction3 π−p → ΛK0 The hyperon moves with
3The results of an actual such experiment constructed to test the ∆S = ∆Q rule in semileptonic
neutral kaon decays are described in Ref. [19].
velocity v =
3c/2 perpendicular to the plane of the target in the laboratory frame S.
After the time t′(S ′) = T ′ in its rest frame S’, it decays to a proton and a negative pion:
Λ → pπ−. These decay products are observed in the laboratory system. The experiment
is in every way similar to that shown in Fig.2. The object O is replaced by the target T,
the object O’ by the undecayed Λ or the kinematical system constructed from its decay
products. The photon pulse emitted by PL’ is replaced by the decay products of the
Λ. By reconstructing the trajectories of the decay p and π− in a particle detector the
position of the decay event and hence the decay length lD –the distance between the point
of production and decay of the Λ– in the frame S can be measured. Identification of the
p and π− and measurement of their momenta (typically by measurement of the curvature
of their trajectories in a known magnetic field ) enables the momentum P and the energy
E of the Λ to be determined. Since v = Pc2/E and γ = E/(mΛc
2) where mΛ is the mass
of the Λ, the proper decay time of the Λ is given by Eqn(5.3) as:
T ′ = ∆t′(S ′) =
∆t′(S)
(5.12)
The spatial configurations of T and the Λ at different times in the frames S and S’ are
shown in Fig.3. The spatial separations of T and the Λ at the observed instant of decay
in S and S’ obey the relation (5.11). This implies that this separation, in changing the
frame of observation from the rest frame of the Λ to the laboratory system in which it is
motion, undergoes a ‘length expansion’ by the factor γ. In accordance with Eqn(5.11), it
can be seen that this is a necessary consequence of the RP, given the existence of the TD
effect.
The mutally simultaneous events in S and S’ shown in Fig.3c, correspond, as they
must, to equal spatial separations of T and the physical object constituted by the decay
products, p and π−, of the Λ. However, in the frame S, these particles have just been
created and have vanishing spatial separation, whereas in S’ they are spatially separated by
a distance corresponding to a time-of-flight (γ− 1)T ′. This also seems highly paradoxical
when interpreted by commonsense classical concepts of space and time.
Acknowledgement
I thank the referee of the journal that rejected Ref. [11] for publication for correspon-
dence that was important for the clarificatiion of the ideas expressed in both the latest
version of Ref. [11] and the present paper.
Added Note
The calculations presented in the present paper are flawed by a major conceptual
misunderstanding which is rectified in later papers [20, 21] treating similar subjects.
At the time of writing the present paper, the author had correctly understood the
spurious nature of the ‘relativity of simultaneity’and ‘length contraction’ effects of con-
ventional special relativity theory [5, 7, 8, 10] but had not yet drawn the simple conclusion
that the existence of the genuine and experimentally-confirmed time dilatation effect then
necessarily implies that the Reciprocity Principle, as generally understood, also breaks
down in special relativity. This point is easily understood by considering the first member
of Eqn(3.1), written in a simplified notation as:
dxO′O
Transforming into the frame S’, the invariance of length intervals implies that
dxO′O = −dx
Since the time dilatation relation gives dt = γdt′, the Reciprocity Principle of (3.1) is
replaced by:
dxO′O
so that
= −γv
to be compared with v′ = −v given by (3.1).
The detailed calculations presented in Section 4 are correct and logically coherent
given the initial assumptions, but the configurations shown in the frame S’ in Fig.1 do not
correspond to observations in this frame of the coincidence events specified in the frame
S in the same space-time experiment. If this were the case, in the S’ frame configurations
in Fig.1 v should be replaced by γv and t and t′ should be related by time dilatation
relation t = γt′. In fact, what are shown in Fig.1 and considered in Section 4 are the
configurations in S of a primary experiment and in S’ of the corresponding but physically
independent reciprocal experiment [20, 22].
Nevertheless, the invariance of corresponding length intervals can be derived [21] by
considering the configurations in S and S’ in Fig.1b in the case that they are corresponding
ones, at the same epoch, in the same space-time experiment. In this case, as explained
above, the speed of O in S’ should be γv, not v. Consider, however, an object Õ with the
same x′ coordinate as O that does have the velocity v. The separation L′ of O and O’
in S’ is then equal to that between O’ and Õ. at the epoch of Fig.1b. Compare now the
configuration of O and O’ in S, with separation L with the corresponding one of Õ. and
O’ in S’ with separation L′. From the symmetry of the configurations it can be seen that
both L and L′ can depend only on v: L = L(v), L′ = L′(v). The reciprocity of the two
configurations is now invoked to give the condition, as stated by Pauli [23]:
The contraction of length at rest in S’ and observed from S is equal to
the length at rest in S as observed from S’.
The ‘length at rest in S’ ’ is L′ which ‘as observed from S’ is L, whereas the ‘length
at rest in S ’ is L which ‘as observed fron S’ ’ is L′. Denoting the contraction factor by
α(v), the above condition states that
L = α(v)L′, L′ = α(v)L
which implies that L = α(v)2L or α(v)2 = 1 so that L = L′ and the spatial separation
between O and O’ is the same in S and S’ at corresponding epochs. The same conclusion
is more simply reached by noting the symmetry of the configurations of O,O’ in S and
Õ,O’ in S’. and applying Leibnitz’ Principle of Sufficient Reason [21].
If, therefore, in the primary experiment, shown in S in Fig.2b and S’ in Fig.2a, the
configuration in S’ in Fig.2a is to correctly represent that corresponding to the configura-
tion in S in Fig.2b, the velocity v in S’ should be replaced by γv, so that when PL’ flashes
O’ is aligned with MR(20) in both S and S’. In the reciprocal experiment, shown in S in
Fig.2a and S’ in Fig.2b, v in S in Fig.2a should be replaced by γv so that O is aligned
with MR’(0) in both S and S’ when PL flashes.
Similarly, in the thought experiment of Fig.5, if the S’ frame configurations on the right
side of the figure are to represent observations in this frame of events shown in S by the
configurations on the left side, instead of what are actually shown which are configurations
of the physically independent reciprocal experiment, v should be replaced by γv in all the
S’ frame configurations. In this case, there is no mismatch between the spatial position of
the decay event in the two frames and the claimed ‘length expansion’ effect does not occur.
Indeed the claimed ‘... different PMC corresponding to observations of the light flashes
emitted by PL and PL’ in different frames in (5.7)-(5.10)’ is not only ‘...deeply perplex-
ing for common-sense concepts of space and time.’ it is the absurd (self-contradictory)
consequence of assuming, at the same time, that length intervals are invariant, time di-
latation occurs and the conventional interpretation of the Reciprocity Principle holds. In
conventional special relativity theory time dilatation and the Reciprocity Principle are
reconciled by invoking the spurious ‘length contraction’ effect dxO′O = −γx
[18]. so
that v′ = −v. The correct physical interpretation of the Reciprocity Principle is actually
the definition of the configuration in S’ of the physically-independent experiment that is
reciprocal to the primary one specified by the standard configuration of the frames S and
S’ [20, 22].
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