B1 Primo: The Relativity of Simultaneousness
Deep insights are often not at first sight perceptible as such …
First we want to convince ourselves that it is possible to synchronize several identical clocks which are at rest in an inertial frame at different places. Often the following method is suggested: two clocks are at points A and B respectively. A flash of light is released at the midpoint of AB and on arrival of the light each clock is set to 0000 and started.
The problem is: how does one synchronize a third clock C with clock A without losing the synchronization between A and B? And isn't finding the midpoint already a problem? This ‘standard method’ is actually unworkable.
It is however quite possible to synchronize as many clocks as desired with a given clock A: The ‘master’ clock A emits a flash of light at an arbitrary but well-known time t0. As soon as the light arrives at clock B, it is firstly reflected, secondly B’s clock is set to 0000 and thirdly it is started. Clock A records time t1, when the light reflected from B arrives again at A. One calculates the elapsed time (t1 - t0) / 2 for the light from A to B, records the value t0 + (t1 - t0) / 2 and sends it by snail mail to B. The (continuously running) clock B is then advanced by this value. One does not need the midpoint AB at all and in addition one obtains the distance between the two clocks.
Hans Reichenbach pointed out in different publications starting from 1920 that this definition implies a further assumption, i.e. the isotropy of space (a collection of Reichenbach's early writings on space, time and motion in English translation has been edited by Steven Gimbel and Anke Walz in ). In particular the speed of light should be equal in all directions. Measuring the one-way speed of light presupposes distant clocks which have already been synchronized. Therefore the synchronization of distant clocks and the measuring of the one-way speed of light have a circular relationship to each other. When we computed the elapsed time for the light from A to B as (t1 - t0) / 2 we tacitly assumed that the light needs equal time to travel in both directions! The postulate of isotropy was hidden in this assumption. The book “Concepts of Simultaneity” by Max Jammer [13-218] presents two simple axioms which a set of clocks must meet, in order to be synchronizeable. The formulation of the first axiom is ours:
- If a clock A sends out two light signals with ∆tA time difference, then each further clock B must receive the signals with ∆tB time difference,
where ∆tB = ∆tA.
- The time required for light to traverse a triangle is independent of the direction taken around the triangle.
The first axiom must surely be fulfilled if synchronized clocks are to remain synchronized. Obviously it can be fulfilled only by clocks which are at rest relative to each other! The second axiom (called the “round trip axiom") guarantees that the speed of light is independent of direction. Taken together the two axioms are necessary and sufficient so that a set of clocks can be synchronized.
Since we will need the postulate of isotropy of space in B3, we introduce it here to the STR. Its operational formulation concerning the speed of light is found in the “round trip” axiom.
Thus, in an inertial frame one can have clocks at arbitrary locations, which are all synchronized in the sense described above. Measuring the point of time of an event means to read the time from such a synchronized clock positioned at the location of the event. Thus we now have a conception in terms of a hardware view of an inertial frame, as represented in [14-37]:
What does one see in this scaffolding of clocks on the dial of one of the distant clocks?
For all physical quantities the following three aspects can never be completely disassociated: a) the definition of the quantity, b) the method of measurement of this quantity and c) the definition of its units of measurement. To define 'time' as a physical quantity means to construct some copies of an accurate clock and to define how to compare the time indicated by these clocks (consult the Wiki articles for the second and for the international atomic time !). Einstein's insightful contribution to the development of the STR was to show that this operational approach eliminates all difficulties.
Thus in an inertial frame we can get along with one single time. We are allowed to speak of the time t in an inertial frame. However different inertial frames usually have different chronologies for events. Epstein demonstrates this very beautifully in [15-33ff]. He considers an interstellar fleet of three spaceships, which travel in a row in space at a constant distance from each other:
There is an inertial frame, in which the three spaceships are at rest. A radio call from the flagship will reach the other two spaceships at the same time in this inertial frame (movie strip on the left, should be read from the bottom to the top). On the other hand, for a viewer in an inertial frame in which the fleet moves, the radio call reaches the leading spaceship later than it does the following ship (movie strip on the right)! This is a direct consequence of the fact that the radio signal in each inertial frame travels in all directions with the constant speed c.
Quantitatively however the strip on the right is not completely correctly drawn. Here c appears to be somewhat larger than for the one on the left, and above all the fleet does not move with uniform speed, which the reader can easily verify with a ruler.
Given the postulate that c has the same constant value in every coordinate system and that light or radio signals spread in each inertial frame with the same speed into all directions in space, it follows immediately that it makes sense only within an inertial frame to say that two events take place at the same time. Poincaré had already pointed out in 1902 :
"The English teach mechanics as an experimental science; on the Continent it is taught always more or less as a deductive and a priori science. The English are right, no doubt. How is it that the other method has been persisted in for so long; how is it that Continental scientists who have tried to escape from the practice of their predecessors have in most cases been unsuccessful? On the other hand, if the principles of mechanics are only of experimental origin, are they not merely approximate and provisory? May we not be some day compelled by new experiments to modify or even to abandon them? These are the questions which naturally arise, and the difficulty of solution is largely due to the fact that treatises on mechanics do not clearly distinguish between what is experiment, what is mathematical reasoning, what is convention, and what is hypothesis. This is not all.
1. There is no absolute space, and we only conceive of relative motion; and yet in most cases mechanical facts are enunciated as if there is an absolute space to which they can be referred.
2. There is no absolute time. When we say that two periods are equal, the statement has no meaning, and can only acquire a meaning by a convention.
3. Not only have we no direct intuition of the equality of two periods, but we have not even direct intuition of the simultaneity of two events
occurring in two different places. I have explained this in an article entitled "La Mesure du Temps".
4. Finally, is not our Euclidean geometry in itself only a kind of convention of language? Mechanical facts might be enunciated with reference to a non-Euclidean space which would be less convenient but quite as legitimate as our ordinary space; the enunciation would become more complicated, but it still would be possible.
Thus, absolute space, absolute time, and even geometry are not conditions which are imposed on mechanics. All these things no more existed before mechanics than the French language can be logically said to have existed before the truths which are expressed in French. We might endeavour to enunciate the fundamental law of mechanics in a language independent of all these conventions; and no doubt we should in this way get a clearer idea of those laws in themselves." [16-89ff]
Einstein and his friends Solovine and Habicht carefully studied Poincaré’s book at the “Akademie Olympia”. With his operational definitions Einstein analyzed the "conventions", which permit us to speak about simultaneousness within an inertial frame. Just as clearly he showed that clocks which are synchronized in one frame do not run synchronously when observed from another moving frame. Also the amount of de-synchronization was given an exact numerical value. We address this quantitative aspect in B6.