CHAPTER IV
CONGRUENCE
11. Simultaneity. 11.1 Einstein analysed the ideas of time-order and of simultaneity. Primarily (according to his analysis) time-order only refers to the succession of events at a given place. Accordingly each given place has its own time-order. But these time-orders are not independent in the system of nature, and their correlation is known to us by means of physical measurements. Now ultimately all physical measurement depends upon coincidence in time and place. If and be two places, the time-orders and which belong to and are correlated by observations of coincidences at and at respectively.
Thus, confining ourselves to the two places and , there are two distinct processes of correlating the time-order of events throughout the universe, namely by a series of observations of coincidences at based on time-order and by a series of observations of coincidences at based on time-order . These two processes are distinct and will only agree by some accident of special circumstance.
11.2 What are the observations at which will assign to an event at a position in the time-order ? Suppose some message—a wave disturbance, for example—starts from when event happens at , reaches when event happens at and is immediately reflected so as to return to when event ″ happens at . Now according to the method of time-measurement for , there is an event ′ which happens at mid-time between and ″. Then, when certain conditions have been fulfilled, the event at is defined as simultaneous with the event ′ at according to the method of correlation appropriate to place . In this way a time-order of events at is derived solely from observation of coincidence at and is based solely on the fundamental time-order at . Thus the time-order at is extended as a time-order for all events at all places.
11.3 There are questions which require elucidation before this definition can be understood. What is a place? We have chosen a vague term on purpose, so as to postpone its consideration until now. A place can only be marked by phenomena capable of recognition, for example the continued appearance of a material body. Thus we must construe and to be the names of material bodies, or of persistent sets of circumstances which will serve the same purpose. In general and will be in relative motion with respect to each other.
What of the message which passes from to and back to ? Its transmission must be uniform. Suppose the message travels with velocity , that is, with the velocity of light in vacuo. Then, assuming the electromagnetic formulae for relativity, this velocity relative to is independent (so far as its magnitude is concerned) of the velocity which we ascribe to through space.
11.4 Thus our recording body can be any body at rest in some consentient set of the Newtonian group, and we reckon motion as relative to the space of this set. We send our message with the velocity of light in vacuo. Then, according to the local time-order at , the event at is simultaneous with the event ′ at . This definition of simultaneity in the local time-order at is independent of any assumption of absolute rest for provided that the electromagnetic formulae for relativity are adopted. The local time-order at is also in complete agreement with the local time-order at any body which is rigidly connected with , i.e. which belongs to the same consentient set.
11.5 The reason why the velocity of light has been adopted as the standard velocity in the definition of simultaneity is because the negative results of the experiments to determine the earth's motion require that this velocity, which is the '' of Maxwell's equations, should have this property. Also light signals are after all our only way of detecting distant events.
Certainly, once granting the idea of time-order being a local affair connected with a specific body , the acceptance of the electromagnetic formula connecting and is a slight affair. There is no presumption against it, once granting the conception of diverse time-orders which had not hitherto been thought of.
11.6 But there are certain objections to the acceptance of Einstein's definition of simultaneity, the 'signal-theory' as we will call it. In the first place light signals are very important elements in our lives, but still we cannot but feel that the signal-theory somewhat exaggerates their position. The very meaning of simultaneity is made to depend on them. There are blind people and dark cloudy nights, and neither blind people nor people in the dark are deficient in a sense of simultaneity. They know quite well what it means to bark both their shins at the same instant. In fact the determination of simultaneity in this way is never made, and if it could be made would not be accurate; for we live in air and not in vacuo.
Also there are other physical messages from place to place; there is the transmission of material bodies, the transmission of sound, the transmission of waves and ripples on the surface of water, the transmission of nerve excitation through the body, and innumerable other forms which enter into habitual experience. The transmission of light is only one form among many.
Furthermore local time does not concern one material particle only. The same definition of simultaneity holds throughout the whole space of a consentient set in the Newtonian group. The message theory does not account for the consentience in time-reckoning which characterises a consentient set, nor does it account for the fundamental position of the Newtonian group.
12. Congruence and Recognition. 12.1 Again the theory that measurement is essentially coincidence requires severe qualification. For if it were true only coincident things, coincident both in time and space, could be equal, yet measurement can only be of the slightest importance in so far as some other element not coincidence enters into it.
Let us take a simple example. Two footrules are placed together and are found to coincide. Then at the moment of coincidence they are equal in length. But what is the use of that information? We want to use one rule to-morrow in London and the other rule a week hence in Manchester, and to know that the stuffs which they measure are of equal length. Now we know that, provided they are made of certain sorts of material (luckily, materials easy to procure) and treated with certain precautions (luckily, precautions easy to observe), the footrules will not have altered their lengths to any extent which can be detected. But that means a direct judgment of constancy. Without such a judgment in some form or other, measurement becomes trivial.
12.2 It may be objected that whenever the footrules are brought together, or when stuffs measured by them are brought together, the coincidences will be observed; and that this is all we need for the importance of measurement.
But the coincidences will not be observed unless the circumstances of the various experiments are sufficiently uniform. The stuffs must be under the same tension or at the same temperatures as on previous occasions. Sooner or later and somehow or other a judgment of constancy, that is, of the preservation of property, is required. Ultimately this judgment reposes upon direct common sense; namely, obviously the footrule is of good stiff material and has not perceptibly changed amid slight differences of circumstance. The coincidences which can easily be obtained between lengths of elastic thread inspire no such beliefs, because evidently the thread has been stretched.
12.3 Again, in Einstein's own example, there is the direct judgment of the uniformity of conditions for the uniform transmission of light. Thus any ordinary event among the fixed stars does not affect this uniformity for the transmission from the sun to the earth. Apart from such presuppositions, so obvious that they do not enter into consciousness, the whole theory collapses.
12.4 These judgments of constancy are based on an immediate comparison of circumstances at different times and at different places. Such judgments are not infallible and are capable of being tested under certain circumstances. For example it may be judged that two footrules would coincide if they were brought together; and this experiment can be made, and the judgment tested.
The rejection of an immediate judgment of constancy is no paradox. There are differences between any distinct sets of circumstances, and it is always possible that these differences cut deeper than we have perceived so as to produce unsuspected divergences of properties.
But a judgment of constancy is recognition, and recognition is the source of all our natural knowledge. Accordingly though isolated judgments may be rejected, it is essential that a rational consideration of nature should assume the truth of the greater part of such judgments and should issue in theories which embody them.
12.5 This recognition of congruity between distinct circumstances has no especial connection with coincidence and extends far beyond the mere judgments of time and space. Thus judgments of the matching of colours can be made without coincidence by most people to some slight extent, and by some people with surprising accuracy. It may be urged that only in the case of judgments of spatial and temporal coincidence can great accuracy be obtained. This may be true; but complete accuracy is never obtained, and the ideal of accuracy shows that the meaning is not derived from the measurement. Our recognitions are the ultimate facts of nature for science, and the whole scientific theory is nothing else than an attempt to systematise our knowledge of the circumstances in which such recognitions will occur. The theory of congruence is one branch of the more general theory of recognitions. Another branch is the theory of objects which is considered in the next part of this enquiry.