Space Time and Gravitation/Chapter 3
THE WORLD OF FOUR DIMENSIONS
The distinction between horizontal and vertical is not an illusion; and the man who thinks it can be disregarded is likely to come to an untimely end. Yet we cannot arrive at a comprehensive view of nature unless we combine horizontal and vertical dimensions into a three-dimensional space. By doing this we obtain a better idea of what the distinction of horizontal and vertical really is in those cases where it is relevant, e.g. the phenomena of motion of a projectile. We recognise also that vertical is not a universally differentiated direction in space, as the flat-earth philosophers might have imagined.
Similarly by combining the time-ordering and space-ordering of the events of nature into a single order of four dimensions, we shall not only obtain greater simplicity for the phenomena in which the separation of time and space is irrelevant, but we shall understand better the nature of the differentiation when it is relevant.
A point in this space-time, that is to say a given instant at a given place, is called an "event." An event in its customary meaning would be the physical happening which occurs at and identifies a particular place and time. However, we shall use the word in both senses, because it is scarcely possible to think of a point in space-time without imagining some identifying occurrence.
In the ordinary geometry of two or three dimensions, the distance between two points is something which can be measured, usually with a rigid scale; it is supposed to be the same for all observers, and there is no need to specify horizontal and vertical directions or a particular system of coordinates. In four-dimensional space-time there is likewise a certain extension or generalised distance between two events, of which the distance in space and the separation in time are particular components. This extension in space and time combined is called the "interval" between the two events; it is the same for all observers, however they resolve it into space and time separately. We may think of the interval as something intrinsic in external nature—an absolute relation of the two events, which postulates no particular observer. Its practical measurement is suggested by analogy with the distance of two points in space.
In two dimensions on a plane, two points , (Fig. 2) can be specified by their rectangular coordinates and , when arbitrary axes have been selected. In the figure, , , etc. We have so that if is the distance between and .
The extension to three dimensions is, as we should expect, . Introducing the times of the events , , we should naturally expect that the interval in the four-dimensional world would be given by .
An important point arises here. It was, of course, assumed that the same scale was used for measuring and and . But how are we to use the same scale for measuring ? We cannot use a scale at all; some kind of clock is needed. The most natural connection between the measure of time and length is given by the fact that light travels 300,000 kilometres in 1 second. For the four-dimensional world we shall accordingly regard 1 second as the equivalent of 300,000 kilometres, and measure lengths and times in seconds or kilometres indiscriminately; in other words we make the velocity of light the unit of velocity. It is not essential to do this, but it greatly simplifies the discussion.
Secondly, the formulae here given for are the characteristic formulae of Euclidean geometry. So far as three-dimensional space is concerned the applicability of Euclidean geometry is very closely confirmed by experiment. But space-time is not Euclidean; it does, however, conform (at least approximately) to a very simple modification of Euclidean geometry indicated by the corrected formula . There is only a sign altered; but that minus sign is the secret of the differences of the manifestations of time and space in nature.
This change of sign is often found puzzling at the start. We could not define by the expression originally proposed (with the positive sign), because the expression does not define anything objective. Using the space and time of one observer, one value is obtained; for another observer, another value is obtained. But if is defined by the expression now given, it is found that the same result is obtained by all observers. The quantity is thus something which concerns solely the two events chosen; we give it a name—the interval between the two events. In ordinary space the distance between two points is the corresponding property, which concerns only the two points and not the extraneous coordinate system of location which is used. Hence interval, as here defined, is the analogue of distance; and the analogy is strengthened by the evident resemblance of the formula for in both cases. Moreover, when the difference of time vanishes, the interval reduces to the distance. But the discrepancy of sign introduces certain important differences. These differences are summed up in the statement that the geometry of space is Euclidean, but the geometry of space-time is semi-Euclidean or "hyperbolic." The association of a geometry with any continuum always implies the existence of some uniquely measurable quantity like interval or distance; in ordinary space, geometry without the idea of distance would be meaningless.
For the moment the difficulty of thinking in terms of an unfamiliar geometry may be evaded by a dodge. Instead of real time , consider imaginary time ; that is to say, let . Then , so that . Everything is now symmetrical and there is no distinction between and the other variables. The continuum formed of space and imaginary time is completely isotropic for all measurements; no direction can be picked out in it as fundamentally distinct from any other.
The observer's separation of this continuum into space and time consists in slicing it in some direction, viz. that perpendicular to the path along which he is himself travelling. The section gives three-dimensional space at some moment, and the perpendicular dimension is (imaginary) time. Clearly the slice may be taken in any direction; there is no question of a true separation and a fictitious separation. There is no conspiracy of the forces of nature to conceal our absolute motion—because, looked at from this broader point of view, there is nothing to conceal. The observer is at liberty to orient his rectangular axes of , , and arbitrarily, just as in three-dimensions he can orient his axes of , , arbitrarily.
It can be shown that the different space and time used by the aviator in Chapter i correspond to an orientation of the time-axis along his own course in the four-dimensional world, whereas the ordinary time and space are given when the time-axis is oriented along the course of a terrestrial observer. The FitzGerald contraction and the change of time-measurement are given exactly by the usual formulae for rotation of rectangular axes.
It is not very profitable to speculate on the implication of the mysterious factor , which seems to have the property of turning time into space. It can scarcely be regarded as more than an analytical device. To follow out the theory of the four-dimensional world in more detail, it is necessary to return to real time, and face the difficulties of a strange geometry.
Consider a particular observer, , and represent time according to his reckoning by distance up the page parallel to . One dimension of his space will be represented by horizontal distance parallel to ; another will stand out at right angles from the page; and the reader must imagine the third as best he can. Fortunately it will be sufficient for us to consider only the one dimension of space and deal with the phenomena of "line-land," i.e. we limit ourselves to motion to and fro in one straight line in space.
The two lines , , at 45° to the axes, represent the tracks of points which progress 1 unit horizontally (in space) for 1 unit vertically (in time); thus they represent points moving with unit velocity. We have chosen the velocity of light as unit velocity; hence , will be the tracks of pulses of light in opposite directions along the straight line.
Any event within the sector is indubitably after the event , whatever system of time-reckoning is adopted. For it would be possible for a material particle to travel from to , the necessary velocity being less than that of light; and no rational observer would venture to state that the particle had completed its journey before it had begun it. It would, in fact, be possible for an observer travelling along to receive a light-signal or wireless telegram announcing the event , just as he reached , since is the track of such a message; and then after the time he would have direct experience of the event . To have actual evidence of the occurrence of one event before experiencing the second is a clear proof of their absolute order in nature, which should convince not merely the observer concerned but any other observer with whom he can communicate.
Similarly events in the sector are indubitably before the event .
With regard to an event in the sector or we cannot assert that it is absolutely before or after . According to the time-reckoning of our chosen observer , is after , because it lies above the line ; but there is nothing absolute about this. The track corresponds to a velocity greater than that of light, so that we know of no particle or physical impulse which could follow the track. An observer experiencing the event could not get news of the event by any known means until after had happened. The order of the two events can therefore only be inferred by estimating the delay of the message and this estimate will depend on the observer's mode of reckoning space and time.
Space-time is thus divided into three zones with respect to the event . belongs to the indubitable past. is the indubitable future. and are (absolutely) neither past nor future, but simply "elsewhere." It may be remarked that, as we have no means of identifying points in space as "the same point," and as the events and might quite well happen to the same particle of matter, the events are not necessarily to be regarded as in different places, though the observer will judge them so; but the events and cannot happen to the same particle, and no observer could regard them as happening at the same place. The main interest of this analysis is that it shows that the arbitrariness of time-direction is not inconsistent with the existence of regions of absolute past and future.
Although there is an absolute past and future, there is between them an extended neutral zone; and simultaneity of events at different places has no absolute meaning. For our selected observer all events along are simultaneous with one another; for another observer the line of events simultaneous with would lie in a different direction. The denial of absolute simultaneity is a natural complement to the denial of absolute motion. The latter asserts that we cannot find out what is the same place at two different times; the former that we cannot find out what is the same time at two different places. It is curious that the philosophical denial of absolute motion is readily accepted, whilst the denial of absolute simultaneity appears to many people revolutionary.
The division into past and future (a feature of time-order which has no analogy in space-order) is closely associated with our ideas of causation and free-will. In a perfectly determinate scheme the past and future may be regarded as lying mapped out—as much available to present exploration as the distant parts of space. Events do not happen; they are just there, and we come across them. "The formality of taking place" is merely the indication that the observer has on his voyage of exploration passed into the absolute future of the event in question; and it has no important significance. We can be aware of an eclipse in the year 1999, very much as we are aware of an unseen companion to Algol. Our knowledge of things where we are not, and of things when we are not, is essentially the same—an inference (sometimes a mistaken inference) from brain impressions, including memory, here and now.
So, if events are determinate, there is nothing to prevent a person from being aware of an event before it happens; and an event may cause other events previous to it. Thus the eclipse of the Sun in May 1919 caused observers to embark in March. It may be said that it was not the eclipse, but the calculations of the eclipse, which caused the embarkation; but I do not think any such distinction is possible, having regard to the indirect character of our acquaintance with all events except those at the precise point of space where we stand. A detached observer contemplating our world would see some events apparently causing events in their future, others apparently causing events in their past— the truth being that all are linked by determinate laws, the so-called causal events being merely conspicuous foci from which the links radiate.The recognition of an absolute past and future seems to depend on the possibility of events which are not governed by a determinate scheme. If, say, the event is an ultimatum, and the person describing the path is a ruler of the country affected, then it may be manifest to all observers that it is his knowledge of the actual occurrence of the event which has caused him to create the event . must then be in the absolute , and, as we have seen, must lie in the sector . But the inference is only permissible, if the event could be determined by the event , and was not predetermined by causes anterior to both—if it was possible for it to happen or not, consistently with the laws of nature. Since physics does not attempt to cover indeterminate events of this kind, the distinction of absolute past and future is not directly important for physics; but it is of interest to show that the theory of four-dimensional space-time provides an absolute past and future, in accordance with common requirements, although this can usually be ignored in applications to physics.
Consider now all the events which are at an interval of one unit from , according to the definition of the interval . We have changed the sign of , because usually (though not always) the original would have come out negative. In Euclidean space points distant a unit interval lie on a circle; but, owing to the change in geometry due to the altered sign of , they now lie on a rectangular hyperbola with two branches , . Since the interval is an absolute quantity, all observers will agree that these points are at unit interval from .
Now make the following construction:—draw a straight line , to meet the hyperbola in ; draw the tangent at , meeting the light-line in ; complete the parallelogram ; produce to . We now assert that an observer who chooses for his time-direction will regard as his space direction and will consider and to be the units of time and space.
The two observers make their partitions of space and time in different ways, as illustrated in Figs. 5 and 6, where in each case the partitions are at unit distance (in space and time) according to the observers' own reckoning. The same diagram of events in the world will serve for both observers; merely removes 's partitions and overlays his own, locating the events in his space and time accordingly. It will be seen at once that the lines of unit velocity—progress of one unit of space for one unit of time—agree, so that the velocity of a pulse of light is unity for both observers. It can be shown from the properties of the hyperbola that the locus of points at any interval from , given by equation (1), viz. , is the same locus (a hyperbola) for both systems of reckoning and . The two observers will always agree on the measures of intervals, though they will disagree about lengths, durations, and the velocities of everything except light. This rather complex transformation is mathematically equivalent to the simple rotation of the axes required when imaginary time is used.
It must not be supposed that there is any natural distinction corresponding to the difference between the square-partitions of observer and the diamond-shaped partitions of observer . We might say that transplants the space-time world unchanged from Fig. 5 to Fig. 6, and then distorts it until the diamonds shown become squares; or we might equally well start with this distorted space-time, partitioned by into squares, and then 's partitions would be represented by diamonds. It cannot be said that either observer's space-time is distorted absolutely, but one is distorted relatively to the other. It is the relation of order which is intrinsic in nature, and is the same both for the squares and diamonds; shape is put into nature by the observer when he has chosen his partitions.
We can now deduce the FitzGerald contraction. Consider a rod of unit length at rest relatively to the observer . The two extremities are at rest in his space, and consequently remain on the same space-partitions; hence their tracks in four dimensions , (Fig. 7) are entirely in the time-direction. The real rod in nature is the four-dimensional object shown in section as . Overlay the same figure with 's space and time partitions, shown by the dotted lines. Taking a section at any one "time," the instantaneous rod is , viz. the section of by 's time-line. Although on paper is actually longer than , it is seen that it is a little shorter than one of 's space-partitions; and accordingly judges that it is less than one unit long—it has contracted on account of its motion relative to him.Similarly is a rod of unit length at rest relatively to . Overlaying 's partitions we see that it occupies at ; and this is less than one of 's partitions. Thus judges it to have contracted on account of its motion relative to him. (Fig. 8) to represent the duration of 's cigar (two units), we see that in 's reckoning it reaches over a little more than two time-partitions. Moreover it has not kept to one space-partition, i.e. it has moved. Similarly is the duration of 's cigar (two time-units for him); and it lasts a little beyond two unit-partitions in 's time-reckoning. (Note, in comparing the two diagrams, , , are the same points as , , .)
If in Fig. 4 we had taken the line very near to , our diamonds would have been very elongated, and the unit-divisions , very large. This kind of partition would be made by an observer whose course through the world is , and who is accordingly travelling with a velocity approaching that of light relative to . In the limit, when the velocity reaches that of light, both space-unit and time-unit become infinite, so that in the natural units for an observer travelling with the speed of light, all the events in the finite experience of take place "in no time" and the size of every object is zero. This applies, however, only to the two dimensions and ; the space-partitions parallel to the plane of the paper are not affected by this motion along . Consequently for an observer travelling with the speed of light all ordinary objects become two-dimensional, preserving their lateral dimensions, but infinitely thin longitudinally. The fact that events take place "in no time" is usually explained by saying that the inertia of any particle moving with the velocity of light becomes infinite so that all molecular processes in the observer must stop; many things may happen in 's world in a twinkling of an eye—of 's eye.
However successful the theory of a four-dimensional world may be, it is difficult to ignore a voice inside us which whispers "At the back of your mind, you know that a fourth dimension is all nonsense." I fancy that that voice must often have had a busy time in the past history of physics. What nonsense to say that this solid table on which I am writing is a collection of electrons moving with prodigious speeds in empty spaces, which relatively to electronic dimensions are as wide as the spaces between the planets in the solar system! What nonsense to say that the thin air is trying to crush my body with a load of 14 Ibs. to the square inch! What nonsense that the star-cluster, which I see through the telescope obviously there now, is a glimpse into a past age 50,000 years ago! Let us not be beguiled by this voice. It is discredited.
But the statement that time is a fourth dimension may suggest unnecessary difficulties which a more precise definition avoids. It is in the external world that the four dimensions are united—not in the relations of the external world to the individual which constitute his direct acquaintance with space and time. Just in that process of relation to an individual, the order falls apart into the distinct manifestations of space and time. An individual is a four-dimensional object of greatly elongated form; in ordinary language we say that he has considerable extension in time and insignificant extension in space. Practically he is represented by a line—his track through the world. When the world is related to such an individual, his own asymmetry is introduced into the relation; and that order of events which is parallel with his track, that is to say with himself, appears in his experience to be differentiated from all other orders of events.
Probably the best known exposition of the fourth dimension is that given in E. Abbott's popular book Flatland. It may be of interest to see how far the four-dimensional world of space-time conforms with his anticipations. He lays stress on three points.
(1) As a four-dimensional body moves, its section by the three-dimensional world may vary; thus a rigid body can alter size and shape.
(2) It should be possible for a body to enter a completely closed room, by travelling into it in the direction of the fourth dimension, just as we can bring our pencil down on to any point within a square without crossing its sides.
(3) It should be possible to see the inside of a solid, just as we can see the inside of a square by viewing it from a point outside its plane.
The first phenomenon is manifested by the FitzGerald contraction.
If quantity of matter is to be identified with its mass, the second phenomenon does not happen. It could easily be conceived of as happening, but it is provided against by a special law of nature—the conservation of mass. It could happen, but it does not happen.
The third phenomenon does not happen for two reasons. A natural body extends in time as well as in space, and is therefore four-dimensional; but for the analogy to hold, the object must have one dimension less than the world, like the square seen from the third dimension. If the solid suddenly went out of existence so as to present a plane section towards time, we should still fail to see the interior of it; because light-tracks in four-dimensions are restricted to certain lines like , in Fig. 3, whereas in three-dimensions light can traverse any straight line. This could be remedied by interposing some kind of dispersive medium, so that light of some wavelength could be found travelling with every velocity and following every track in space-time; then, looking at a solid which suddenly went out of existence, we should receive at the same moment light-impressions from every particle in its interior supposing them self-luminous). We actually should see the inside of it.
How our poor eyes are to disentangle this overwhelming experience is quite another question.The interval is a quantity so fundamental for us that we may consider its measurement in some detail. Suppose we have a scale AB divided into kilometres, say, and at each division is placed a clock also registering kilometres. (It will be remembered that time can be measured in seconds or kilometres indifferently.) . This is shown in Fig. 9 where the clock-readings are given as though they were being viewed from .
Now lay the scale in line with the two events; note the clock and scale-readings , , of the first event, and the corresponding readings , , of the second event. Then by the formula already given . But suppose we took a different standard of rest, and set the scale moving uniformly in the direction . Then the divisions would have advanced to meet the second event, and would be smaller. This is compensated, because also becomes altered. is now advancing to meet the light coming from any of the clocks along the rod; the light arrives too quickly, and in the initial adjustment described above the clock must be set back a little. The clock-reading of the event is thus smaller. There are other small corrections arising from the FitzGerald contraction, etc.; and the net result is that, it does not matter what uniform motion is given to the scale, the final result for is always the same.
In elementary mechanics we are taught that velocities can be compounded by adding. If 's velocity relative to (as observed by either of them) is 100 km. per sec., and 's velocity relative to is 100 km. per sec. in the same direction, then 's velocity relative to should be 200 km. per sec. This is not quite accurate; the true answer is 199.999978 km. per sec. The discrepancy is not difficult to explain. The two velocities and their resultant are not all reckoned with respect to the same partitions of space and time. When measures 's velocity relative to him he uses his own space and time, and it must be corrected to reduce to 's space and time units, before it can be added on to a velocity measured by .
If we continue the chain, introducing whose velocity relative to , and measured by , is 100 km. per sec., and so on ad infinitum, we never obtain an infinite velocity with respect to , but gradually approach the limiting velocity of 300,000 km. per sec., the speed of light. This speed has the remarkable property of being absolute, whereas every other speed is relative. If a speed of 100 km. per sec. or of 100,000 km. per sec. is mentioned, we have to ask—speed relative to what? But if a speed of 300,000 km. per sec. is mentioned, there is no need to ask the question; the answer is relative to any and every piece of matter. A β particle shot off from radium can move at more than 200,000 km. per sec.; but the speed of light relative to an observer travelling with it is still 300,000 km. per sec. It reminds us of the mathematicians transfinite number Aleph; you can subtract any number you like from it, and it still remains the same.
The velocity of light plays a conspicuous part in the relativity theory, and it is of importance to understand what is the property associated with it which makes it fundamental. The fact that the velocity of light is the same for all observers is a consequence rather than a cause of its pre-eminent character. Our first introduction of it, for the purpose of coordinating units of length and time, was merely conventional with a view to simplifying the algebraic expressions. Subsequently, considerable use has been made of the fact that nothing is known in physics which travels with greater speed, so that in practice our determinations of simultaneity depend on signals transmitted with this speed. If some new kind of ray with a higher speed were discovered, it would perhaps tend to displace light-signals and light-velocity in this part of the work, time-reckoning being modified to correspond; on the other hand, this would lead to greater complexity in the formulae, because the FitzGerald contraction which affects space-measurement depends on light-velocity. But the chief importance of the velocity of light is that no material body can exceed this velocity. This gives a general physical distinction between paths which are time-like and space-like, respectively—those which can be traversed by matter, and those which cannot. The material structure of the four-dimensional world is fibrous, with the threads all running along time-like tracks; it is a tangled warp without a woof. Hence, even if the discovery of a new ray led us to modify the reckoning of time and space, it would still be necessary in the study of material systems to preserve the present absolute distinction of time-like and space-like intervals, under a new name if necessary.
It may be asked whether it is possible for anything to have a speed greater than the velocity of light. Certainly matter cannot attain a greater speed; but there might be other things in nature which could. "Mr Speaker," said Sir Boyle Roche, "not being a bird, I could not be in two places at the same time." Any entity with a speed greater than light would have the peculiarity of Sir Boyle Roche's bird. It can scarcely be said to be a self-contradictory property to be in two places at the same time any more than for an object to be at two times in the same place. The perplexities of the quantum theory of energy sometimes seem to suggest that the possibility ought not to be overlooked; but, on the whole, the evidence seems to be against the existence of anything moving with a speed beyond that of light.
The standpoint of relativity and the principle of relativity are quite independent of any views as to the constitution of matter or light. Hitherto our only reference to electrical theory has been in connection with Larmor and Lorentz's explanation of the FitzGerald contraction; but now from the discussion of the four-dimensional world, we have found a more general explanation of the change of length. The case for the electrical theory of matter is actually weakened, because many experimental effects formerly thought to depend on the peculiar properties of electrical forces are now found to be perfectly general consequences of the relativity of observational knowledge.
Whilst the evidence for the electrical theory of matter is not so conclusive, as at one time appeared, the theory may be accepted without serious misgivings. To postulate two entities, matter and electric charges, when one will suffice is an arbitrary hypothesis, unjustifiable in our present state of knowledge. The great contribution of the electrical theory to this subject is a precise explanation of the property of inertia. It was shown theoretically by J. J. Thompson that if a charged conductor is to be moved or stopped, additional effort will be necessary simply on account of the charge. The conductor has to carry its electric field with it, and force is needed to set the field moving. This property is called inertia, and it is measured by mass. If, keeping the charge constant, the size of the conductor is diminished, this inertia increases. Since the smallest separable particles of matter are found by experiment to be very minute and to carry charges, the suggestion arises that these charges may be responsible for the whole of the inertia detected in matter. The explanation is sufficient; and there seems no reason to doubt that all inertia is of this electrical kind.
When the calculations are extended to charges moving with high velocities, it is found that the electrical inertia is not strictly constant but depends on the speed; in all cases the variation is summed up in the statement that the inertia is simply proportional to the total energy of the electromagnetic field. We can say if we like that the mass of a charged particle at rest belongs to its electrostatic energy; when the charge is set in motion, kinetic energy is added, and this kinetic energy also has mass. Hence it appears that mass (inertia) and energy are essentially the same thing, or, at the most, two aspects of the same thing. It must be remembered that on this view the greater part of the mass of matter is due to concealed energy, which is not as yet releasable.
The question whether electrical energy not bound to electric charges has mass, is answered in the affirmative in the case of light. Light has mass. Presumably also gravitational energy has mass; or, if not, mass will be created when, as often happens, gravitational energy is converted into kinetic energy. The mass of the whole (negative) gravitational energy of the earth is of the order minus a billion tons.
The theoretical increase of the mass of an electron with speed has been confirmed experimentally, the agreement with calculation being perfect if the electron undergoes the FitzGerald contraction by its motion. This has been held to indicate that the electron cannot have any inertia other than that due to the electromagnetic field carried with it. But the conclusion (though probable enough) is not a fair inference; because these results, obtained by special calculation for electrical inertia, are found to be predicted by the theory of relativity for any kind of inertia. This will be shown in Chapter ix. The factor giving the increase of mass with speed is the same as that which affects length and time. Thus if a rod moves at such a speed that its length is halved, its mass will be doubled. Its density will be increased four-fold, since it is both heavier and less in volume.
We have thought it necessary to include this brief summary of the electrical theory of matter and mass, because, although it is not required by the relativity theory, it is so universally accepted in physics that we can scarcely ignore it. Later on we shall reach in a more general way the identification of mass with energy and the variation of mass with speed; but, since the experimental measurement of inertia involves the study of a body in non-uniform motion, it is not possible to enter on a satisfactory discussion of mass until the more general theory of relativity for non-uniform motion has been developed.
- Appendix, Note 2.
- Appendix, Note 3.