Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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Dictionary of the History of Ideas | ||

*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

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

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*
/*c*2. 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*/*c*2, 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.)

Dictionary of the History of Ideas | ||