---- The Special Relativity of Einstein ----
NOTE: This is a personal and partial interpretation of the original text of Einstein. This page is made to give the own interpretations of Einstein, allowing to understand the scope and objectives of his theory. In my personal case, if I can avoid intermediaries, I try to avoid them. In the text, and with green color, I will add comments trying to help them who do not feel comfortable with technical explanations. Those of you who may want the whole text (in English), can find it in The Principle of Relativity (A collection of original memoirs on the special and general theory of relativity) of DOVER INC. Publications. The full paper is available at http://www.fourmilab.ch/etexts/einstein/specrel/www/
It is known that Maxwell's electrodynamics, as usually understood at the present time, when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example ...
(I want to establish from the beginning, my admiration for the genius of Einstein, which is evident until in the small details that seem non to be necessary. From my own point of view, the observation " as usually understood at the present time" indicates the total conscience of the young Einstein about the theories evolution concerning that things that today may be granted as true, can tomorrow not to be taken in the same way).
It follows an example of about 20 lines, on the mutual interaction between electric fields and magnets, that with time I can translate, but I believe that it is only for those with a solid physical background.
Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the "light medium" suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the "Principle of Relativity") to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies. The introduction of a "luminiferous ether" will prove to be superfluous inasmuch as the view here to be developed will not require an "absolutely stationary space" provided with special properties, nor assign a velocity vector to a point of the empty space in which electromagnetic processes take place.
The theory to be developed is based - like all electrodynamics - on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters.
§ 1 - Definition of Simultaneity
Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good. In order to render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the "stationary system".
If a material point is at rest relatively to this system of co-ordinates, its position can be defined relatively thereto by the employment of rigid standards of measurement and the methods of Euclidean geometry, and can be expressed in Cartesian co-ordinates.
If we wish to describe the motion of a material point, we give the values of its co-ordinates as functions of the time. Now we must bear carefully in mind that a mathematical description of this kind has no physical meaning unless we are quite clear as to what we understand by "time". We have to take into account that all our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, "That train arrives here at 7 o'clock", I mean something like this: "The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events".
It might appear possible to overcome all the difficulties attending the definition of "time" by substituting "the position of the small hand of my watch" for "time". And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located ; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or - what comes to the same thing - to evaluate the times of events occurring at places remote from the watch.
We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this co-ordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought.
If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If. there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an "A time" and a "B time''. We have not defined a common "time" for A and B, for the latter cannot be defined at all, unless we establish by definition that the " time" required by light to travel from A to B equals the "time" it requires to travel from B to A.
(This it is the crucial point in the development of Special Relativity. It is very important to clearly recognize what it means to accept by definition, that the time used sending the signal is equal to the time used during the return. Einstein was conscientious that it was necessary to define this point, when a naive vision seems to indicate that it is not necessary to add special postulates. Just measuring both times we would avoid the previous definition. But Einstein was conscientious that the efforts to make this measurement had been already exhausted and that the results indicated clearly that the answer was the one that he "postulated").
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" tA'
In accordance with definition the two clocks synchronize if
tB - tA = t'A -tB
We assume that this definition (DEFINITION) of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:
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
2 AB
----------------
= c
t'A - tA
to be a universal constant: the velocity of light in empty apace.
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".
§2 - On the Relativity of Lengths and Times
The following reflexions are based on the principle of relativity and on the principle of the constancy of the velocity of light. These two principles we define as follows:
Light path
Velocity = -----------------------------
Time interval
where time interval is to be taken in the sense of the definition in § 1
(It must be noted that as "Velocity" means the constant "c", this equation is transformed into the mechanism to measure distances defined like "Light Path").
Let there be given a stationary rigid rod; and let its length be "l" as measured by a measuring-rod which is also stationary. We now imagine the axis of the rod lying along the axis of x of the stationary system of co-ordinates, and that a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod. We now inquire as to the length of the moving rod, and imagine its length to be ascertained by the following two operations:
(a) The observer moves together with the given measuring-rod and the rod to be measured; and measures the length of the rod directly by superposing the measuring-rod, in just the same way as if all three were at rest.
(b) By means of stationary clocks set up in the stationary system and synchronizing in accordance with § 1, the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring?rod already employed, which in this case is at rest, is also a length which may be designated " the length of the rod".
In accordance with the principle of relativity the length to be discovered by the operation (a) - we will call it "the length of the rod in the moving system " - must be equal to the length l of the stationary rod.
The length to be discovered by the operation (b) we will call "the length of the (moving) rod in the stationary system". This we shall determine on the basis of our two principles, and we shall find that it differs from l.
Current kinematics tacitly assumes that the lengths determined by these two operations are precisely equal, or in other words, that a moving rigid body at the epoch t may in geo metrical respects be perfectly represented by the same body at rest in a definite position.
We imagine further that at the two ends A and B of the rod, clocks are placed which synchronize with the clocks of the stationary system, that is to say that their indications correspond at any instant to the "time of the stationary system" at the places where they happen to be. These clocks are therefore "synchronous in the stationary system."
We imagine further that with each clock there is a moving observer, and that these observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks. Let a ray of light depart from A at the time tA, let it be reflected at B at the time tB, and reach A again at the time t'A. Taking into consideration the principle of the constancy of the velocity of light we find that:
rAB
t'A-tB = -------------
c + v
where rAB denotes the length of the moving rod - measured in the stationary system. Obbservers moving with the moving rod would thus find that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous.
(Remarkable Observation: In this deduction, like in other ones related to Special Relativity, Einstein uses, without special considerations, the addition of speeds. He adds and extract when necessary, the speed "v" of the moving system from "c", If he did not make this addition of speeds, and "really" he assumed that "c" is the same one in all the systems, tB-tA should be identical to t'A-tB. This way of work is in concordance with the initial affirmation: The introduction of a "luminiferous ether" will prove to be superfluous. Einstein did not eliminate the ether, since he uses it for his deductions. He simply show the impossibility of its detection. At this point I repeat a personal observation that I believe fundamental: The work of Einstein shown that the ether is undetectable in "to and fro" trips of the signals. It does not make demonstrations with respect to the "True" speed in one way routes. Their mathematical developments, as in the previous case, start assuming that the speed of the light is different in routes from Going or Return).
So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.
(Although this observation can seem trivial, is important to clarify that Einstein uses clocks and rods for his demonstrations with intention to clarify all the developments. But this does not imply that the results are only applicable to the clocks as we know them. Einstein assumes that all particle and physical phenomenon of interaction behave as a clock or a rod of measurement and for that reason is possible to generalize the results.)
§ 3 - Theory of the Transformation of Co-ordinates and Times from a Stationary System to another System in Uniform Motion of Translation Relatively to the Former.
Let us in "stationary" space take two systems of coordinates, i.e. two systems, each of three rigid material lines, perpendicular to one another, and issuing from a point. Let the axes of X of the two systems coincide, and their axes of Y and Z respectively be parallel. Let each system be provided with a rigid measuring-rod and a number of clocks, and let the two measuring-rods, and likewise all the clocks of the two systems, be in all respects alike.
Now to the origin of one of the two systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other stationary system (K), and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring-rod, and the clocks. To any time of the stationary system K there then will correspond a definite position of the axes of the moving system, and from reasons of symmetry we are entitled to assume that the motion of k may be such that the axes of the moving system are at the time t (this "t" always denotes a time of the stationary system) parallel to the axes of the stationary system.
We now imagine space to be measured from the stationary system K by means of the stationary measuring-rod, and also from the moving system k by means of the measuring-rod moving with it ; and that we thus obtain the co-ordinates X, Y, Z and x, y, z respectively. Further, let the time t of the stationary system be determined for all points thereof at which there are clocks by means of light signals in the manner indicated in § 1. Similarly let the time ...
......
---- The Special Relativity of Einstein ----