neither measure of time is more accurate or more logical than the other. There are as many ways of measuring time as there are observers, and all are right.
The investigator who is trying to discover laws of nature will, in general, require to measure either directly or indirectly both time and space. If, to take a simple case, he is studying the motion of a single particle, he will measure out the position of the particle at definite instants as determined by his clock. He may specify the position of the particle at any instant by three measurements in space for instance, he may say that two seconds after his particle started it was 6 ft. to the E. of the point from which it started, 9 ft. to the N. and 12 ft. vertically upward. The mathematician would express this by taking axes x, y, z to the E., to the N. and vertically upwards, and saying that at time t=2 the particle had coordinates * = 6, y = 9, 2=12. Or he might, putting his time coordinate / on the same footing as the space coordinates x, y, z, simply say that x=6, y = 9, 2=12, t=2 represented one position of the particle. A complete set of readings of this type, each consisting of values of four coordinates, would give the complete history of the motion of the particle.
Such sets of simultaneous measurements form the common material of investigations in both pure and applied science. For instance, the engineer may measure the extension of a sample of steel corresponding to different loads; the electrician may measure the amount -of light given by an electric filament corresponding to different amounts of current passed through it. In each of these cases there are only two quantities to be meas- ured simultaneously, and an investigator can conveniently represent the result of the whole series of his measurements in graphical form; a single reading is represented by a point whose distances from two fixed perpendicular lines represent the quanti- ties measured, and the curve obtained by joining these single points will give all the information contained in the whole set of readings.
We have seen that, in studying the motion of a particle in space, four sets of quantities must be measured, so that the results obtained cannot be plotted graphically on a piece of paper. Their proper representation demands a four-dimensional space, in which x, y, z and t are taken as coordinates. The practical importance of such graphical representation is nil, since it is impossible to construct a four-dimensional graph, but its theoreti- cal importance to the theory of relativity is immense. For if the hypothesis of relativity is true, then the four-dimensional graphs of any natural event constructed by all observers, no matter what their relative motions, will be identical. The influence of their motion will be shown only in that the axes of x, y, 2 and t will be different for different observers, and the relations between these sets of axes will be those given by the foregoing equations (B).
The importance of this conception can hardly be overestimated, and it may be well to consider it further with the help of an illustrative example. Imagine a number of aeroplanes flying over England, and, in order to eliminate one of the three direc- tions in space -the vertical let us limit them to fly always at the same height, say 1,000 ft. above sea-level. Imagine a number of similar plates of glass prepared, each marked faintly with an outline map of England and with lines of latitude and longitude. Suppose that at 12 h. o m. G.M.T. a plate is taken and the position of each aeroplane marked by a thick black dot. At 12 h. i m. let a second plate be taken and similarly marked, and let this be done every minute for an hour. The 60 plates so marked will constitute a record of the motion of each aeroplane within this hour. If, now, we place the plates in order, one above the other, on a horizontal table, the mass of glass so formed will present a graphical representation, in three dimensions, of the motions of all the aeroplanes. In this graph the two horizontal coordinates represent motions in any two rectangular directions over England, say E. and N., while the third coordinate the vertical represents time. The individual black dots which represent the positions of any one aeroplane will form a dotted curve, and this curve gives a graphical representation of the
motion of the particular aeroplane. Our rectangle of glass contains the history, for one hour, of all the aeroplanes in graphi- cal form.
To represent the motion of particles in the whole world of space a four-dimensional graph is required. The four-dimensional space in which it is constructed may, following the usual termi- nology, be spoken of as a four-dimensional continuum. The history of any particle in the universe just as that of any aeroplane flying over England will be represented by a con- tinuous line in the continuum, and this is called the " world line " of the particle. If the hypothesis of relativity is true the same continuum and the same world lines will represent the history of the particles of the universe equally well for all observers, the influence of their motions being shown only through their choosing different axes in the continuum for their axes of space and time. Thus the continuum must be thought of as something real and objective, but the choice of axes is subjective and will vary with the observer, the relation between different choices being expressed mathematically by our equa- tions (B), the equations of the Lorentz transformation. An inspection of these equations shows that the sets of axes chosen by different observers have different orientations in the con- tinuum, so that what one observer describes as a pure space interval will appear to another to be a mixture of time and space.
The instant of time and point in space at which any event occurs can be fixed by a single point in the continuum, so that the interval between two events will be represented by a finite line. The events and the interval between them are absolute, but the interval will be split up into time and space in different ways by different observers. The interval between any two events, such as the great fire of London and the outBurst on the star Nova Persei, may be measured by one set of observers as so many years and so many millions of miles, but another set of observers may divide the interval quite differently. For instance a terrestrial astronomer may reckon that the outburst on Nova Persei occurred a century before the great fire of London, but an astronomer on the Nova may reckon with equal accuracy that the great fire occurred a century before the outburst on the Nova. A third astronomer may insist that the events were simultaneous. All will be equally right, although none will be right in an absolute sense. At this stage we may notice one respect in which our pile of glass plates failed to represent the true continuum. The mass of glass was stratified into different plates which represent different times for one particular observer. To obtain a section which would represent what an observer in motion relative to this first observer could regard as simul- taneous positions of the aeroplanes, we should have to cut the mass of glass on the slant. The continuum is more closely represented by our plates of glass if they are annealed into a solid mass from which all trace of the original stratification is made to disappear. All observers, no matter what their motion, are then equally free to cut a section to represent their individual ideas of simultaneity.
Thus space and time fade into subjective conceptions, just as subjective as right hand or left hand, front and behind, are in ordinary life. The continuum alone is objective and may be thought of as containing an objective record of the motion of every particle of the universe. The curve in which this record is embodied is spoken of as the world line of the particle in question. To use the words of Minkowski: " Space in itself and time in itself sink to mere shadows, and only a kind of union of the two retains an independent existence."
Gravitation and Relativity. Since all the phenomena of light and of electromagnetism are believed, on almost incontrovertible evidence, to be in accordance with the hypothesis of relativity, it is necessarily impossible to determine absolute velocity by optical or gravitational means. On the other hand, as we have already mentioned, the Newtonian law of gravitation is readily seen to be inconsistent with the hypothesis of relativity. Three alternatives are open:
(i.) The Newtonian law may be true, in which case it must be possible to determine absolute velocity by gravitational means.
