X
Einstein reexamined the concept of simultaneity.
Accepting it as intuitively clear for events occurring
at the same place, he asked what meaning could be
given to it when the events were at different places.
Realizing that this must be a matter of convention,
he proposed a definition that we shall now illustrate.
Imagine the spaceships S, S′ equipped with identi-
cally constructed clocks fore and aft, as shown (Figure
4). Pretend that c is small, or that the spaceships are
of enormous length, so that we can use convenient
numbers in what follows. When clock C1 reads noon,
E sends a light signal from C1 to C2 where it is reflected
back to C1. Suppose that the light reached clock C2
when C2 read 1 second after noon, and returned to
C1 when C1 read 3 seconds after noon. Then Einstein
would have E say that his clocks C1 and C2 were not
synchronized. To synchronize the clocks Einstein
would have E advance C2 by half a second so that
according to the readings of C1 and C2 the light would
take equal times for the outward and return journeys.
With C1 and C2 thus synchronized, if events occurred
at C1 and C2 when these clocks read the same time,
the events would be deemed simultaneous.
In 1898 Poincaré had already questioned the concept
of simultaneity at different locations, and in 1900 he
had considered the use of light signals for adjusting
clocks in a manner strikingly similar to that used by
Einstein in 1905. However, Poincaré was concerned
with adjusting rather than synchronizing clocks. More-
over, he did not build on two kinematical postulates
but worked in terms of the Maxwell equations; nor
did he take the following step, and it is things such
as these that set Einstein's work sharply apart.
Consider E and E′ synchronizing the clocks in their
respective spaceships S and S′. E arranges C1 and C2
so that they indicate equal durations for the forward
and return light journeys, and declares them synchro-
nized. But E′, watching him from S′, sees S moving
backwards with speed v relative to him. Therefore,
according to S′, the light signals sent by E did not travel
equal distances there and back (Figure 5), but unequal
distances (Figure 6). And so, according to E′, the very
fact that C1 and C2 indicated equal durations for the
forward and backward light journeys showed that C1
and C2 were not synchronized.
However, E′ has synchronized his own clocks, C′1,
C′2, according to Einstein's recipe, and E says they are
not synchronized, since relative to S the light signals
used by E′ travel unequal distances there and back.
With E and E′ disagreeing about synchronization
we naturally ask which of them is correct. But the
principle of relativity, as Einstein rather than Poincaré
viewed it, forbids our favoring one over the other. E
and E′ are on an equal footing, and we have to regard
both as correct. Since events simultaneous according
to E are not simultaneous according to E′, and vice
versa, Einstein concluded that simultaneity is relative,
being dependent on the reference frame. By so doing
he gave up Newtonian universal absolute time.
Previously we spoke of measurements of the one-way
speed of light as not being feasible, deliberately sug-
gesting by the wording that this might be because of
practical difficulties. The lack of feasibility can now
be seen to have a deeper significance. To measure the
one-way speed of light over a path AB in a given
reference frame we need synchronized clocks at A and
B by which to time the journey of the light. With the
synchronization itself performed by means of light, the
measurement of the one-way speed of light becomes,
in principle, a tautology. The mode of synchronization
is a convention permitting the convenient spreading
of a time coordinate over the reference frame one uses.
What can be said to transcend convention (with apolo-
gies to conventionalists) is the rejection of Newtonian
absolute time, with its absolute simultaneity.
Once the concept of time is changed, havoc spreads
throughout science and philosophy. Speed, for exam-
ple, is altered, and acceleration too, and with it force,
and work, and energy, and mass, so that we wonder
if anything can remain unaffected.
Not even distance remains unscathed, as is easily
seen. Imagine the spaceships S and S′ marked off in
yard lengths by E and E′ respectively. E measures a
yard length of S′ by noting where the ends of the yard
are at some particular time. Since E and E′ disagree
about simultaneity, E′ accuses E of noting the positions
of the yard marks non-simultaneously, thus obtaining
an incorrect value for the length. When the roles are
reversed, E similarly accuses E′. Because of the princi-
ple of relativity, both are adjudged correct. Thus once
simultaneity is relative, so too is length. Indeed, the
disagreement as to lengths corresponds in magnitude
to the FitzGerald-Lorentz contraction, but here it is
a purely kinematical effect of relative motion and not
an absolute effect arising from motion relative to a fixed
aether. While E says that the yards of E′ are con-
tracted, E′ says the same about those of E.
Let x, y, z, and t denote the coordinates and syn-
chronized clock times used by E in his spaceship refer-
ence frame S; and let x′, y′, z′, and t′ denote the corre-
sponding quantities used by E′ in S′. Einstein derived
directly from his two postulates a mathematical rela-
tion between these quantities, and it turned out to be
the Lorentz transformation (2). The interpretation,
though, was different, because E and E′ were now on
an equal footing: t′, for example, was just as good a
time as t.
Because of the principle of relativity, the rela-
tionships between E and E′ are reciprocal. We have
already discussed this in connection with lengths. It
is instructive to consider it in relation to time: when
E and E′ compare clock rates each says the other's
clocks go the more slowly. This, like the reciprocal
contractions of lengths, is immediately derivable from
the Lorentz transformation in its new interpretation.
It can be understood more vividly by giving E and
E′ “clocks,” each of which consists of a framework
holding facing parallel mirrors with light reflected
tick-tock between them. Each experimenter regards his
own clocks as using light paths as indicated (Figure 8).
But because of the relative motion of E and E′, we
have the situation shown (Figure 9). Since these longer
light paths are also traversed with speed c, each ex-
perimenter finds the other's clocks ticking more slowly
than his own. Indeed, a simple, yet subtle, application
of Pythagoras' theorem to the above diagrams yields
the mutual time dilatation factor √ 1 - v
/c2. More-
over, since E′ and E, in addition to agreeing to dis-
agree, agree about the speed of light, each says that
the other's relative lengths are contracted in the
direction of the relative motion by this same factor
√1 - v/c2, for otherwise the ratio distance/time for
light would not be c for both.
A further kinematical consequence is easily deduced
directly from Einstein's postulates. We begin by noting
that no matter how fast E′ moves relative to E, light
waves recede from him with speed c: he cannot over-
take them. But these light waves also move with speed
c relative to E. Therefore E′ cannot move relative to
E with a speed greater than c. Nor can any material
object relative to any other: c is the speed limit. (It
has been proposed that particles exist that move faster
than light. They have been named tachyons. According
to the Lorentz transformation, tachyons can never
move slower than light; their speed exceeds c in all
reference frames.)
