So far in speaking of space we liave been talking of the timeless space of physical science, namely, of our concept of eternal space in which the world adventures. But the space which we see as we look about is instan- taneous space. Thus if our natural perceptions are adjustable to the /)-system of measurements we see instantaneously all the event-particles at some definite time p^, and observe a succession of such spaces as time moves on. The timeless space is achieved by stringing together all these instantaneous spaces. The points of an instantaneous space are event-particles, and the points of an eternal space are strings of event-particles occurring in succession. But the man on Mars will never perceive the same instantaneous spaces as the man on the earth. This system of instantaneous spaces will cut across the earth-man's system. For the earth- man there is one instantaneous space which is the instantaneous present, there are the past spaces and the future spaces. But the present space of the man on Mars cuts across the present space of the man on the earth. So that of the event-particles which the earth- man thinks of as happening now in the present, the man on Mars thinks that some are already past and are ancient history, that others are in the future, and others are in the immediate present. This break-down in the neat conception of a past, a present, and a future is a serious paradox. I call two event-particles which on some or other system of measurement are in the same instantaneous space 'co-present' event-particles. Then it is possible that A and B may be co-present, and that A and C may be co-present, but that B and C may not be co-present. For example, at some inconceivable distance from us there are events co-present with us
178 THE CONCEPT OF NAi UKii l^«.
now and also co-present with the birth of Queen
Victoria. If A and B are co-present there will be some
systems in which A precedes B and some in which B
precedes A. Also there can be no velocity quick enough
to carry a material particle from A to B or from B to A.
These different measure-systems with their divergences
of time-reckoning are puzzling, and to some extent
affront our common sense. It is not the usual way in
which we think of the Universe. We think of one
necessary time-system and one necessary space. Ac-
cording to the new theory, there are an indefinite
number of discordant time-series and an indefinite
number of distinct spaces. Any correlated pair, a
time-system and a space-system, will do in which to fit
our description of the Universe. We find that under
given conditions our measurements are necessarily made
in some one pair which together form our natural
measure-system. The difficulty as to discordant time-
systems is partly solved by distinguishing between what
I call the creative advance of nature, which is not
properly serial at all, and any one time series. We
habitually muddle together this creative advance, which
we experience and know as the perpetual transition of
nature into novelty, with the single-time series which
we naturally employ for measurement. The various
time-series each measure some aspect of the creative
advance, and the whole bundle of them express all the
properties of this advance which are measurable. The
reason why we have not previously noted this difference
of time-series is the very small difference of properties
between any two such series. Any observable pheno-
mena due to this cause depend on the square of the
ratio of any velocity entering into the observation to
vin] SUMMARY
the velocity of light. Now Ught takes about fifty minutes to get round the earth's orbit; and the earth takes rather more than 17,531 half-hours to do the same. Hence all the effects due to this motion are of the order of the ratio of one to the square of 10,000. Accordingly an earth-man and a sun-man have only neglected effects whose quantitative magnitudes all contain the factor i/io^. Evidently such effects can only be noted by means of the most refined observations. They have been observed however. Suppose we compare two observations on the velocity of light made with the same apparatus as we turn it through a right angle. The velocity of the earth relatively to the sun is in one direction, the velocity of light relatively to the ether should be the same in all directions. Hence if space when we take the ether as at rest means the same thing as space when we take the earth as at rest, we ought to find that the velocity of light relatively to the earth varies according to the direction from which it comes.
These observations on earth constitute the basic principle of the famous experiments designed to detect the motion of the earth through the ether. You all know that, quite unexpectedly, they gave a null result. This is completely explained by the fact that, the space- system and the time-system which we are using are in certain minute ways different from the space and the time relatively to the sun or relatively to any other body with respect to which it is moving.
All this discussion as to the nature of time and space
has lifted above our horizon a great difficulty which
affects the formulation of all the ultimate laws of physics
— ^for example, the laws of the electromagnetic field,
and the law of gravitation. Let us take the law of
i8o THE CONCEPT OF NATURE [ch.
gravitation as an example. Its formulation is as follows : Two material bodies attract each other with a force proportional to the product of their masses and uni- versely proportional to the square of their distances. In this statement the bodies are supposed to be small enough to be treated as material particles in relation to their distances; and we need not bother further about that minor point. The difficulty to which I want to draw your attention is this : In the formulation of the law one definite time and one definite space are pre- supposed. The two masses are assumed to be in simul- taneous positions.
But what is simultaneous in one time-system may not be simultaneous in another time-system. So according to our new views the law is in this respect not formulated so as to have any exact meaning. Furthermore an analogous difficulty arises over the question of distance. The distance between two instantaneous positions, i.e. between two event-particles, is different in different space-systems. What space is to be chosen ? Thus again the law lacks precise formulation, if relativity is accepted ./^ Our problem is to seek a fresh interpretation of the law of gravity in which these difficulties are evaded. In the first place we must avoid the abstractions of space and time in the formulation of our fundamental ideas and must recur to the ultimate facts of nature, namely to events. Also in order to find the ideal simplicity of expressions of the relations between events, we restrict ourselves to event-particles. Thus the life of a material particle is its adventure amid a track of event-particles strung out as a continuous series or path in the four- dimensional space-time manifold. These event-particles are the various situations of the material particle. We
viiij SUMMARY i8i
usually express this fact by adopting our natural space-
time system and by talking of the path in space of the
material particle as it exists at successive instants of time.
We have to ask ourselves what are the laws of nature
which lead the material particle to adopt just this path
among event-particles and no other. Think of the path
as a whole. What characteristic has that path got which
would not be shared by any other slightly varied path ?
We are asking for more than a law of gravity. We want
laws of motion and a general idea of the way to formulate
the effects of physical forces.
In order to answer our question we put the idea of the attracting masses in the background and concentrate attention on the field of activity of the events in the neighbourhood of the path. In so doing we are acting in conformity with the whole trend of scientific thought during the last hundred years, which has more and more concentrated attention on the field of force as the im- mediate agent in directing motion, to the exclusion of the consideration of the immediate mutual influence between two distant bodies. We have got to find the way of expressing the field of activity of events in the neighbourhood of some definite event-particle E of the four-dimensional manifold. I bring in a fundamental physical idea which I call the * impetus ' to express this physical field. The event-particle E is related to any neighbouring event-particle P by an element of impetus. The assemblage of all the elements of impetus relating E to the assemblage of event-particles in the neighbour- hood of E expresses the character of the field of activity in the neighbourhood of E. Where I differ from Einstein is that he conceives this quantity which I call the impetus as merely expressing the characters of the space and
i82 THE CONCEPT OF NATURE [cH.
time to be adopted and thus ends by talking of the
gravitational field expressing a curvature in the space-
time manifold. I cannot attach any clear conception to
his interpretation of space and time. My formulae
differ slightly from his, though they agree in those
instances where his results have been verified. I need
hardly say that in this particular of the formulation of the
law of gravitation I have drawn on the general method
of procedure which constitutes his great discovery.
Einstein showed how to express the characters of the assemblage of elements of impetus of the field sur- rounding an event-particle E in terms of ten quantities which I will call J^, J^^ (- J21), J^^, J23 (= Jzz)^ etc. It will be noted that there are four spatio-temporal measurements relating E to its neighbour P, and that there are ten pairs of such measurements if we are allowed to take any one measurement twice over to make one such pair. The ten J's depend merely on the position of E in the four-dimensional manifold, and the element of impetus between E and P can be expressed in terms of the ten J's and the ten pairs of the four spatio-temporal measurements relating E and P. The numerical values of the J's will depend on the system of measurement adopted, but are so adjusted to each particular system that the same value is obtained for the element of impetus between E and P, whatever be the system of measurement adopted. This fact is ex- pressed by saying that the ten J's form a ' tensor.' It is not going too far to say that the announcement that physicists would have in future to study the theory of tensors created a veritable panic among them when the verification of Einstein's predictions was first announced.
vin] SUMMARY 183
The ten J's at any event-particle E can be expressed in terms of two functions which I call the potential and the 'associate-potential' at E. The potential is practically what is meant by the ordinary gravitation potential, when we express ourselves in terms of the Euclidean space in reference to which the attracting mass is at rest. The associate-potential is defined by the modifi- cation of substituting the direct distance for the inverse distance in the definition of the potential, and its calcu- lation can easily be made to depend on that of the old- fashioned potential. Thus the calculation of the J's — the coefficients of impetus, as I will call them — does not involve anything very revolutionary in the mathematical knowledge of physicists. We now return to the path of the attracted particle. We add up all the elements of impetus in the whole path, and obtain thereby what I call the 'integral impetus.' The characteristic of the actual path as compared with neighbouring alternative paths is that in the actual paths the integral impetus would neither gain nor lose, if the particle wobbled out of it into a small extremely near alternative path. Mathe- maticians would express this by saying, that the integral impetus is stationary for an infinitesimal displacement. In this statement of the law of motion I have neglected the existence of other forces. But that would lead me too far afield.
The electromagnetic theory has to be modified to
allow for the presence of a gravitational field. Thus
Einstein's investigations lead to the first discovery of
any relation between gravity and other physical pheno-
mena. In the form in which I have put this modification,
we deduce Einstein's fundamental principle, as to tht
motion of light along its rays, as a first approximation