KINEMATICAL PART

1. Definition of Simultaneity

Let us take a system of co-ordinates, as a “stationary system” in which the equations of Newtonian mechanics hold good.2

A material point is at rest relatively to this system of coordinates.

Its position can be defined relatively through:

• rigid standards of measurement and
• the methods of Euclidean geometry

This can be expressed in Cartesian co-ordinates.

We describe the motion of a material point as the values of its co-ordinates as functions of the time.

A mathematical description of this kind has no physical meaning unless we define “time.”

All our judgments concerning time are judgments of simultaneous events.

“That train arrives here at 7:00” means: “My watch showing 7:00 and the arrival of the train are simultaneous events”.

I replace my watch showing 7:00 with “time.”

This definition is satisfactory for time exclusively for the place where the watch is.

But it is not satisfactory when we have to:

• connect in time a series of events at different places.
• evaluate the times of events at places far from the watch.

An observer with the watch at the origin could coordinate the positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space.

But this coordination has the disadvantage that it is dependent on the observer with the watch.

(Simultaneity)

An observer at point A with a watch can determine the time values of events near A by finding the watch time which is simultaneous with those events.

Likewise, an observer at point B with the same watch can determine the time values of events near B by finding the watch time which is simultaneous with those events.

But Event A will take Time A, and Event B will take Time B.

(Relativity)

How can we define a common “time” for A and B?

This can be done by defining “time” as the time required by light to travel from A to B being equal to the “time” that it requires to travel from B to A.

Let a ray of light start at the “A time” tA from A towards B, let it at the “B time” tB be reflected at B in the direction of A, and arrive again at A at the “A time” t0A .

In accordance with definition the two clocks synchronize if tB − tA = t0A − tB .

We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:

1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.

2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.

Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary clocks located at different places, and have evidently obtained a definition of “simultaneous,” or “synchronous,” and of “time.”

The “time” of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event, this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock.

In agreement with experience we further assume the quantity

2AB = c, t0A − tA to be a universal constant—the velocity of light in empty space. It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it “the time of the stationary system.”