1. During recent years mathematicians and philosophers have been much occupied in analysing the fundamental conceptions on which the different sciences are based, with the result that many things which were formerly regarded as quite simple and axiomatic can no longer be regarded as such. The tendency has, of course, been to make definitions as precise as possible, and to make descriptions of phenomena approximate to reality as we know it, and not to a preconceived idea of what the description ought to be.
Many difficulties arise, however, in a careful examination of the fundamental concepts of any science, and this is soon found to be the case when we commence to examine the ideas of space and time which are fundamental in all physical and metaphysical enquiries.
In the case of time, for instance, it is found that we have to examine the connection between time as it is known to us by the mind's experience, i.e., psychological time, and time as it is measured by the course of physical phenomena, i.e., physical time.
With regard to psychological time, it has been contested that it is purely qualitative, in other words that we are quite unable to decide intuitively whether two intervals of time are equal or not.^{[1]} This means that there is no fixed method by which two sequences of events may be compared in the mind. The comparisons which actually occur give a qualitative description of events, inasmuch as the sequence of processes is generally unaltered in direct perception and in memory, but the lack of a standard set of units invariably connected with the method of comparison, prevents the description from being a true quantitative one.
This being the case, we are met with a fundamental difficulty when we try to analyse the idea of simultaneity as presented to us by the mind.
If we could represent an event by a point on a line, the idea of simultaneity would be quite simple, for two events could be regarded as simultaneous when their representative points were coincident. In reality such a representation is not valid, there is no sensation of such a simple nature that it can be represented by a point on a line. If we adopt a representation by means of an interval on a line, we obtain what is probably a truer representation of an event as regards its duration; but if we suppose that two events are simultaneous when their representative intervals have a common part, it is clear that two events which are simultaneous with the same event would not necessarily be simultaneous with one another.
It will be realised after a little thought that we can only obtain a satisfactory definition of simultaneity by introducing the idea of the measurement of time; we are thus obliged to consider the physical aspect of time in order to understand the idea of simultaneity.
An observer provided with an instrument for measuring time, such as a clock or a pendulum, can attach a definite number to each event that occurs. In some cases he may find it difficult to decide as to which of two consecutive numbers should be attached to an event; but we shall suppose that he has a consistent method of avoiding the difficulty, as, for example, by always choosing the larger number of the two.
A satisfactory definition of simultaneity for two events which happen in the immediate vicinity of the observer can be given as soon as events are numbered, for we can say that two events are simultaneous when their corresponding numbers are equal. The actual enumeration may depend upon the personal equation of the observer, but discrepancies may be eliminated as soon as a method of comparing the observations of different observers has been adopted.
We now require a method of comparison by means of which we can decide whether the observations of time made by two different observers are equivalent or not. The criterion of equivalence must be such that if the observations of A are equivalent to those of B, and also to those of C, then the observations of B and C are equivalent to one another.
It is clear that if two observers are situated at different points of space a comparison of observations can only be made by means of something which travels from one to the other, and for the sake of simplicity it is convenient to choose something which can be supposed to travel in a straight line with constant velocity. It should be remarked, however, that these terms have no meaning until time and distance have been defined.
It is by no means obvious that a universal method of comparing observations can be found which will lead to consistent results, for this presupposes the existence of a universal time, an entity which has sometimes been regarded as the psychological time of an infinite mind governing the whole of the universe. The latter point of view is really not sound, because the universal time we are endeavouring to define is essentially quantitative in character. The best way of establishing the existence of a consistent method of comparison is to give an example of one, and so we shall consider Galileo's method of light signals,^{[2]} which was used in a first but unsuccessful attempt to measure the velocity of light.
The way in which this method is applied is as follows. An observer situated at a point A observes at time t an event which has taken place at another point B. If τ is the time which light takes to travel from B to A, the universal time to be associated with the event at B according to A's measurements is . As soon as A has observed the event he makes a signal, and it is clear that by a series of signals the two measures of an interval of time may be compared.
By means of this rule the clocks belonging to a number of observers can be regulated in a consistent manner, provided light always takes the same time to travel from an observer to an observer .
Let us suppose that a large number of observers find that their observations of one another's experiences give a consistent universal time as far as they are concerned; they can then regard themselves as being at constant distances from one another, the distance between two observers , being defined as , where is the time light takes to travel from to , and C is a constant called the velocity of light.
These observers may then form a standard system for the measurement of time and distances at other points of space. The measurements of four standard observers should suffice to determine the position and time of any occurrence, if space is of three dimensions. If there are more than four observers in the standard system the relations between all the different observations of an event will depend upon the nature of space, and will take a comparatively simple form if the space is Euclidean.
If the position and time associated with an object B is always determined from measurements by a number of standard observers , so that a consistent universal time exists for each point of space, the following conclusion may be deduced by elementary geometry for Euclidean space:
If two observers B and C are at rest or in motion relative to the standard system, and their velocities are less than that of light, there is only one instant^{[3]} at which B is able to observe an instantaneous event experienced by C, but if one of the observers is moving with a velocity greater than that of light this is not necessarily the case; in fact it may happen that B sees two or more pictures of the same event.^{[4]}
In order that Galileo's method of comparing times at different points of space may be suitable for a sober world, it seems necessary to suppose that a body cannot move with a velocity greater than that of light, and it may be of interest to remark that this view is supported by modern electrical theories.
Now let us suppose that a second system of observers, , find that their observations are in agreement, and so can regard themselves as a standard system. It may happen that according to their measurements the first system of observers are in motion, and then it is easy to see that the specifications of position and time as made by the A's and the B's will not agree. For instance, if two observers pass an observer B at different times, their distance apart as measured by B is zero, while measured from A's point of view it is not.
The relation between the two sets of measurements may be obtained by taking into account the fact that the analytical conditions that an observer P should be able at time to observe an event which happened at another point Q at time , ought to be of the same form in the two systems of coordinates.
If are the coordinates of P and Q, we have, in the first place, the necessary conditions
These conditions, combined with the kinematical character of the motion of the B's relative to the A's, are practically sufficient to determine the relation between the two systems of measurement in certain cases, as, for instance, when the B's are moving with a uniform velocity relative to the A's.^{[5]}
It is found that in the case of uniform relative motion the units of length and time in the two systems are different, and that two events occurring at different points of space may appear to be simultaneous according to measurements made by the A's, and not appear to be simultaneous according to measurements made by the B's. Also, the shape of a body is theoretically different according to the two series of measurements, but the difference is so very slight as to be unnoticeable.
The fact that the electromagnetic equations have the same form in the two systems of coordinates, indicates, that as far as our observations of electromagnetic phenomena are concerned, a uniform motion of a system of observers would remain undetected. This, of course, is in accordance with the view that position and motion are purely relative, and that the term absolute motion is meaningless.
A. Einstein^{[6]} has developed this theory of relativity starting with the fundamental idea of the constancy of the velocity of light, and has thus been able to present us with a new kinematics which is apparently more consistent with the modern theories of electrodynamics than the approximate kinematics to which we are accustomed. Some of the most interesting results of the theory are that the resultant of two velocities, both of which are less than that of light, is always a velocity less than that of light; the resultant of two velocities one of which is equal to that of light is a velocity equal to that of light; the resultant of two velocities equal to that of light but of opposite directions is indeterminate, and may have any value less than or equal to that of light.
This theory of relativity is not based simply on theoretical considerations; it has received considerable support from some very delicate experiments. It was first put forward in an approximate form by Lorentz^{[7]} and Larmor^{[8]}, following a suggestion made by FitzGerald^{[9]}, to explain the negative results of the MichelsonMorley experiment. It was then found that it provided an ample explanation of a number of other negative results concerning the effect of the motion of the Earth on double refraction^{[10]}, the rotation of the plane of polarisation^{[11]}, the resistance of a piece of metal^{[12]}, and other physical phenomena. Also, the theoretical formula deduced for the transverse mass of a moving electron^{[13]} is in fairly good agreement with the results of Kaufmann's experiments, and has been verified very closely in a recent experiment made by Bucherer.
At present the theory is being widely used as a working hypothesis, and is of great theoretical importance, as it enables us to pass from the known analytical specification of a phenomenon for a medium at rest to the corresponding case of a medium in motion. It has been used in this way by the late Hermann Minkowski to obtain a scheme of electromagnetic relations for ponderable bodies in uniform motion, and has been shown by Planck and von Mosengeil to provide a very useful method of studying the properties of radiation in a cavity in a moving body.
With regard to the ideas of time that have arisen in connection with the theory, it may be mentioned that two new terms have been introduced. The term local time is used by Lorentz to denote time as it is measured by a set of observers who are moving uniformly in a straight line relative to a standard set of observers. The relation between the local time and the standard time is a reciprocal one, the local time for one set of observers being the standard time for the other set, and viceversa.
A second term, used by Minkowski, is Eigenzeit or the "proper time." It is defined for each particle and may be regarded as the age of a particle. As a particle moves from one point to another, the increase of age depends upon the increase of the standard time, and also upon the velocity of the particle. If the particle is moving uniformly, the increase in age is equal to the increase in local time. The advantage of using the age of a particle in forming the equations of motion is that there is a gain in simplicity. The analytical methods based upon the use of the age of a particle may be compared with Lagrange's method of dealing with problems in hydrodynamics, while the methods based on the use of a standard time may be compared with the Eulerian method.
When a particle is moving in an arbitrary manner it is by no means certain that its age can be derived from a knowledge of an initial position and the position at a given time. It is to be expected, in fact, that the age of the particle will depend upon the path from one point to the other, and also upon the rates at which it describes different parts of the path.
There is, at present, considerable uncertainty with regard to the exact laws of the kinematics and dynamics of a body whose motion is not uniform. Systems of nonNewtonian mechanics and kinematics of a rigid electron have been based upon the theory of relativity for the case of uniform motion, but they can hardly be regarded as satisfactory, and difficulties arise as soon as a uniform motion of rotation about an axis is considered.^{[14]}
If mechanics is to be based on the science of electromagnetism, we must make a complete study of the transformations connected with the fundamental equations of electromagnetism.
Now the general problem of determining the transformations which leave the electrodynamical equations unaltered in form, may be partially solved by simply paying attention to the conditions which must be satisfied in order that an observer P who is at a point at time may be in a position to record the effect of a disturbance which issued from a point at time .
It should be remarked that the conditions given above with regard to the possibility of P seeing Q at the given times are necessary, but not sufficient. This accounts for the fact that the transformations for which the condition
is invariant are limited to a certain group.
It is clear that P is able to see Q if there is a flow of energy from P to Q. If denotes the direction in which energy is flowing from Q at time , and we regard the differences , etc., as small, so that
we may consider transformations such that the equations

are invariant. Since the flow of energy depends upon the state of the electromagnetic field, the formulæ of transformation will depend upon the character of the electromagnetic field, but it can be shown that if the above equations are invariant, the fundamental equations which describe the sequence of electromagnetic disturbances are also invariant.^{[15]} The description of any series of phenomena is thus qualitatively the same in the two series of coordinates.
It is interesting to compare the result just obtained with some ideas with regard to the way in which experience is interpreted by the mind.
If we suppose that some physical process taking place in the brain is the physical adjunct of a sensation in the mind, such as visual perception, we may regard the mind's interpretation of the sensation as a transformation of the actual physical process as it would be described by external observers if observed directly by them.
If now the exact nature of this transformation depends upon the physical process taking place as in the case of the transformations just considered, but at the same time leaves invariant the fundamental laws on which a description of the process depends, the description of the process by means of the transformation will be a correct qualitative description but will not be a true quantitative one, since the transformation varies for each independent event and so there are no fixed units of measurement. It is possible that when the mind is concentrated on a subject the type of transformation is practically constant, and so our interpretations of sensations become clearer as they become more of a quantitative nature, but it is dangerous to speculate and so I shall leave the subject at this point.
 ↑ See for instance Poincaré, "La Valeur de la Science," ch. II.
 ↑ This method is used for the purpose of studying the properties of time by A. Einstein, Ann. der Physik, vol. 17 (1905).
 ↑ An analytical proof of this result is given by Prof. A. W. Conway. Proc. London Math. Soc., Ser. 2, vol. 1. (1903).
 ↑ If the times associated with B and A in two views of them are and respectively, B will be able to witness at time an event experienced by B at time if a sphere of radius having the point B as centre is touched internally by a sphere of radius having the point A as centre.
Now if B is moving with a velocity less than that of light, the spheres associated with consecutive positions of B surround one another in succession as in Fig. 1.
It is clear then that there is only one sphere of the series which is touched
internally by a given sphere associated with A, or if we make it is clear that there is only one sphere of the series which passes through a given point in space, provided the radii of the spheres associated with B are all positive; the equivalent supposition in the other case is that .If B is moving with a velocity greater than that of light, two of the spheres associated with it may intersect, and so two of them may be touched internally by the same sphere belonging to A, and then B is able to see more than one picture of the same event. See Figs. 2 and 3.
Again, if A is moving with a velocity less than that of light the spheres associated with A lie within one another or surround one another in succession, and it is clear that there is only one sphere of the series which touches internally a given sphere associated with B. Hence B cannot see two different states of A at the same time. If A is moving with a velocity greater than that of light two of its spheres may intersect, and then it is quite possible for B to see two or more different positions of A at a given time.
If B and A are moving with velocities less than that of light, and we establish a correspondence between the spheres connected with the different positions of B and A by associating together two spheres which touch internally, then since the two series of spheres are such that any sphere of a series surrounds all the smaller ones of the series, it is clear that as the radius of a B sphere increases the radius of the corresponding A sphere also increases. Fig. 4. This shows that B observes the events happening to A in the
correct order. This is not necessarily the case if one or both of the observers are moving with a velocity greater than that of light; in fact under certain conditions it would be possible for one observer to witness the other's experiences in the reverse order. For instance in figure 3 if the four crosses denote successive positions of B, it appears that the first and last positions are seen by A at one instant and the two intermediate positions at another instant. The two earlier positions of B are thus seen in the reverse order, and the two later positions in the correct order.  ↑ The transformation for the case of uniform velocity was given by Voigt, Larmor and Lorentz, it is
This transformation makes
a consequence of
It can be shown that any transformation of coordinates which leaves the last equation unaltered is such that the electrodynamical equations are unaltered in form. See papers by E. Cunningham and the author in the Proceedings of the London Mathematical Society (1910).
 ↑ Ann. der. Physik, vol. 17 (1905). Jahrb. der Radioaktivität (1907).
 ↑ "Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern." Leiden. (1895.)
 ↑ Aether and Matter. (1900.) Ch. x., xi., xiii.
 ↑ Public lectures in Trinity College, Dublin.
 ↑ D. C. Brace. Phil. Mag., (6), vol. 7, p. 317, 1904.
 ↑ Rayleigh. Phil. Mag., (6), vol. 4, p. 215. Brace. Phil. Mag., (6), vol. 10, p. 383, 1905. Ibid., p. 391.
 ↑ Trouton and Rankine. Phil. Trans. A. (1908), p. 420.
 ↑ This formula is where is the mass which the electron would have if it were at rest.
 ↑ This difficulty was pointed out by Ehrenfest, Physikal. Zeitschr., vol. 10, p. 918 (1909).
 ↑ These are not the only transformations which may be used to transform a particular electromagnetic field into another, there appear to be many different types of transformation which are suitable for this purpose.
This work is in the public domain in the United States because it was published before January 1, 1927.
The author died in 1946, so this work is also in the public domain in countries and areas where the copyright term is the author's life plus 75 years or less. This work may also be in the public domain in countries and areas with longer native copyright terms that apply the rule of the shorter term to foreign works.