Such sets of simultaneous measurements form the common ma terial of investigations in both pure and applied science. For in stance, 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 measured simultaneously, and an investigator can conveniently represent the result of the whole series of his measurements in graphical form ; a single read ing is represented by a point whose distances from two fixed per pendicular lines represent the quantities measured, and the curve obtained by joining these single points will give all the informa tion 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 ob tained 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 co-ordinates. The practical importance of such graphical representation is nil, since it is impossible to con struct a four-dimensional graph, but its theoretical importance to the theory of relativity is immense. For if the hypothesis of rela tivity 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, z 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 illus trative example. Imagine a number of aeroplanes flying over Eng land, and, in order to eliminate one of the three directions in space —the vertical—let us limit them to fly always at the same height, say i ,000ft. 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 2h.o.m. G.M.T. a plate is taken and the position of each aero plane marked by a thick black dot. At i 2h. im. let a second plate be taken and similarly marked, and let this he done every minute for an hour. The 6o 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 co-ordinates represent motions in any two rectangular directions over England, say east and north, while the third co-ordinate—the vertical—represents time. The indi vidual black dots which represent the positions of any one aero plane will form a dotted curve, and this curve gives a graphical representation of the motion of the particular aeroplane. Our rec tangle of glass contains the history, for one hour, of all the aero planes in graphical 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 terminology, 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 continuous line in the con tinuum, 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 uni verse equally well for all observers, the influence of their motions being shown only through their choosing different axes in the con tinuum 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 equations (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 continuum, 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 ob servers. 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 in terval 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 may be equally right, al though 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 par ticular observer. To obtain a section which would represent what an observer in motion relative to this first observer could regard as simultaneous positions of the aeroplanes, we should have to cut the mass of glass on the slant. The continuum is more closely rep resented 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 ordi nary 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."