SUMMARY
There is a general agreement that Einstein's investigations have one
fundamental merit irrespective of any criticisms which we may feel
inclined to pass on them. They have made us think. But when we have
admitted so far, we are most of us faced with a distressing perplexity.
What is it that we ought to think about? The purport of my lecture this
afternoon will be to meet this difficulty and, so far as I am able, to
set in a clear light the changes in the background of our scientific
thought which are necessitated by any acceptance, however qualified, of
Einstein's main positions. I remember that I am lecturing to the members
of a chemical society who are not for the most part versed in advanced
mathematics. The first point that I would urge upon you is that what
immediately concerns you is not so much the detailed deductions of the
new theory as this general change in the background of scientific
conceptions which will follow from its acceptance. Of course, the
detailed deductions are important, because unless our colleagues the
astronomers and the physicists find these predictions to be verified we
can neglect the theory altogether. But we may now take it as granted
that in many striking particulars these deductions have been found to be
in agreement with observation. Accordingly the theory has to be taken
seriously and we are anxious to know what will be the consequences of
its final acceptance. Furthermore during the last few weeks the
scientific journals and the lay press have been filled with articles as
to the nature of the crucial experiments which have been made and as to
some of the more striking expressions of the outcome of the new theory.
'Space caught bending' appeared on the news-sheet of a well-known
evening paper. This rendering is a terse but not inapt translation of
Einstein's own way of interpreting his results. I should say at once
that I am a heretic as to this explanation and that I shall expound to
you another explanation based upon some work of my own, an explanation
which seems to me to be more in accordance with our scientific ideas and
with the whole body of facts which have to be explained. We have to
remember that a new theory must take account of the old well-attested
facts of science just as much as of the very latest experimental results
which have led to its production.
To put ourselves in the position to assimilate and to criticise any
change in ultimate scientific conceptions we must begin at the
beginning. So you must bear with me if I commence by making some simple
and obvious reflections. Let us consider three statements, (i)
'Yesterday a man was run over on the Chelsea Embankment,' (ii)
'Cleopatra's Needle is on the Charing Cross Embankment,' and (iii)
'There are dark lines in the Solar Spectrum.' The first statement about
the accident to the man is about what we may term an 'occurrence,' a
'happening,' or an 'event.' I will use the term 'event' because it is
the shortest. In order to specify an observed event, the place, the
time, and character of the event are necessary. In specifying the place
and the time you are really stating the relation of the assigned event
to the general structure of other observed events. For example, the man
was run over between your tea and your dinner and adjacently to a
passing barge in the river and the traffic in the Strand. The point
which I want to make is this: Nature is known to us in our experience as
a complex of passing events. In this complex we discern definite mutual
relations between component events, which we may call their relative
positions, and these positions we express partly in terms of space and
partly in terms of time. Also in addition to its mere relative position
to other events, each particular event has its own peculiar character.
In other words, nature is a structure of events and each event has its
position in this structure and its own peculiar character or quality.
Let us now examine the other two statements in the light of this general
principle as to the meaning of nature. Take the second statement,
'Cleopatra's Needle is on the Charing Cross Embankment.' At first sight
we should hardly call this an event. It seems to lack the element of
time or transitoriness. But does it? If an angel had made the remark
some hundreds of millions of years ago, the earth was not in existence,
twenty millions of years ago there was no Thames, eighty years ago there
was no Thames Embankment, and when I was a small boy Cleopatra's Needle
was not there. And now that it is there, we none of us expect it to be
eternal. The static timeless element in the relation of Cleopatra's
Needle to the Embankment is a pure illusion generated by the fact that
for purposes of daily intercourse its emphasis is needless. What it
comes to is this: Amidst the structure of events which form the medium
within which the daily life of Londoners is passed we know how to
identify a certain stream of events which maintain permanence of
character, namely the character of being the situations of Cleopatra's
Needle. Day by day and hour by hour we can find a certain chunk in the
transitory life of nature and of that chunk we say, 'There is
Cleopatra's Needle.' If we define the Needle in a sufficiently abstract
manner we can say that it never changes. But a physicist who looks on
that part of the life of nature as a dance of electrons, will tell you
that daily it has lost some molecules and gained others, and even the
plain man can see that it gets dirtier and is occasionally washed. Thus
the question of change in the Needle is a mere matter of definition. The
more abstract your definition, the more permanent the Needle. But
whether your Needle change or be permanent, all you mean by stating that
it is situated on the Charing Cross Embankment, is that amid the
structure of events you know of a certain continuous limited stream of
events, such that any chunk of that stream, during any hour, or any day,
or any second, has the character of being the situation of Cleopatra's
Needle.
Finally, we come to the third statement, 'There are dark lines in the
Solar Spectrum.' This is a law of nature. But what does that mean? It
means merely this. If any event has the character of being an exhibition
of the solar spectrum under certain assigned circumstances, it will also
have the character of exhibiting dark lines in that spectrum.
This long discussion brings us to the final conclusion that the concrete
facts of nature are events exhibiting a certain structure in their
mutual relations and certain characters of their own. The aim of science
is to express the relations between their characters in terms of the
mutual structural relations between the events thus characterised. The
mutual structural relations between events are both spatial and
temporal. If you think of them as merely spatial you are omitting the
temporal element, and if you think of them as merely temporal you are
omitting the spatial element. Thus when you think of space alone, or of
time alone, you are dealing in abstractions, namely, you are leaving out
an essential element in the life of nature as known to you in the
experience of your senses. Furthermore there are different ways of
making these abstractions which we think of as space and as time; and
under some circumstances we adopt one way and under other circumstances
we adopt another way. Thus there is no paradox in holding that what we
mean by space under one set of circumstances is not what we mean by
space under another set of circumstances. And equally what we mean by
time under one set of circumstances is not what we mean by time under
another set of circumstances. By saying that space and time are
abstractions, I do not mean that they do not express for us real facts
about nature. What I mean is that there are no spatial facts or temporal
facts apart from physical nature, namely that space and time are merely
ways of expressing certain truths about the relations between events.
Also that under different circumstances there are different sets of
truths about the universe which are naturally presented to us as
statements about space. In such a case what a being under the one set of
circumstances means by space will be different from that meant by a
being under the other set of circumstances. Accordingly when we are
comparing two observations made under different circumstances we have to
ask 'Do the two observers mean the same thing by space and the same
thing by time?' The modern theory of relativity has arisen because
certain perplexities as to the concordance of certain delicate
observations such as the motion of the earth through the ether, the
perihelion of mercury, and the positions of the stars in the
neighbourhood of the sun, have been solved by reference to this purely
relative significance of space and time.
I want now to recall your attention to Cleopatra's Needle, which I have
not yet done with. As you are walking along the Embankment you suddenly
look up and say, 'Hullo, there's the Needle.' In other words, you
recognise it. You cannot recognise an event; because when it is gone, it
is gone. You may observe another event of analogous character, but the
actual chunk of the life of nature is inseparable from its unique
occurrence. But a character of an event can be recognised. We all know
that if we go to the Embankment near Charing Cross we shall observe an
event having the character which we recognise as Cleopatra's Needle.
Things which we thus recognise I call objects. An object is situated in
those events or in that stream of events of which it expresses the
character. There are many sorts of objects. For example, the colour
green is an object according to the above definition. It is the purpose
of science to trace the laws which govern the appearance of objects in
the various events in which they are found to be situated. For this
purpose we can mainly concentrate on two types of objects, which I will
call material physical objects and scientific objects. A material
physical object is an ordinary bit of matter, Cleopatra's Needle for
example. This is a much more complicated type of object than a mere
colour, such as the colour of the Needle. I call these simple objects,
such as colours or sounds, sense-objects. An artist will train himself
to attend more particularly to sense-objects where the ordinary person
attends normally to material objects. Thus if you were walking with an
artist, when you said 'There's Cleopatra's Needle,' perhaps he
simultaneously exclaimed 'There's a nice bit of colour.' Yet you were
both expressing your recognition of different component characters of
the same event. But in science we have found out that when we know all
about the adventures amid events of material physical objects and of
scientific objects we have most of the relevant information which will
enable us to predict the conditions under which we shall perceive
sense-objects in specific situations. For example, when we know that
there is a blazing fire (_i.e._ material and scientific objects
undergoing various exciting adventures amid events) and opposite to it a
mirror (which is another material object) and the positions of a man's
face and eyes gazing into the mirror, we know that he can perceive the
redness of the flame situated in an event behind the mirror--thus, to a
large extent, the appearance of sense-objects is conditioned by the
adventures of material objects. The analysis of these adventures makes
us aware of another character of events, namely their characters as
fields of activity which determine the subsequent events to which they
will pass on the objects situated in them. We express these fields of
activity in terms of gravitational, electromagnetic, or chemical forces
and attractions. But the exact expression of the nature of these fields
of activity forces us intellectually to acknowledge a less obvious type
of objects as situated in events. I mean molecules and electrons. These
objects are not recognised in isolation. We cannot well miss Cleopatra's
Needle, if we are in its neighbourhood; but no one has seen a single
molecule or a single electron, yet the characters of events are only
explicable to us by expressing them in terms of these scientific
objects. Undoubtedly molecules and electrons are abstractions. But then
so is Cleopatra's Needle. The concrete facts are the events
themselves--I have already explained to you that to be an abstraction
does not mean that an entity is nothing. It merely means that its
existence is only one factor of a more concrete element of nature. So an
electron is abstract because you cannot wipe out the whole structure of
events and yet retain the electron in existence. In the same way the
grin on the cat is abstract; and the molecule is really in the event in
the same sense as the grin is really on the cat's face. Now the more
ultimate sciences such as Chemistry or Physics cannot express their
ultimate laws in terms of such vague objects as the sun, the earth,
Cleopatra's Needle, or a human body. Such objects more properly belong
to Astronomy, to Geology, to Engineering, to Archaeology, or to Biology.
Chemistry and Physics only deal with them as exhibiting statistical
complexes of the effects of their more intimate laws. In a certain
sense, they only enter into Physics and Chemistry as technological
applications. The reason is that they are too vague. Where does
Cleopatra's Needle begin and where does it end? Is the soot part of it?
Is it a different object when it sheds a molecule or when its surface
enters into chemical combination with the acid of a London fog? The
definiteness and permanence of the Needle is nothing to the possible
permanent definiteness of a molecule as conceived by science, and the
permanent definiteness of a molecule in its turn yields to that of an
electron. Thus science in its most ultimate formulation of law seeks
objects with the most permanent definite simplicity of character and
expresses its final laws in terms of them.
Again when we seek definitely to express the relations of events which
arise from their spatio-temporal structure, we approximate to simplicity
by progressively diminishing the extent (both temporal and spatial) of
the events considered. For example, the event which is the life of the
chunk of nature which is the Needle during one minute has to the life of
nature within a passing barge during the same minute a very complex
spatio-temporal relation. But suppose we progressively diminish the time
considered to a second, to a hundredth of a second, to a thousandth of a
second, and so on. As we pass along such a series we approximate to an
ideal simplicity of structural relations of the pairs of events
successively considered, which ideal we call the spatial relations of
the Needle to the barge at some instant. Even these relations are too
complicated for us, and we consider smaller and smaller bits of the
Needle and of the barge. Thus we finally reach the ideal of an event so
restricted in its extension as to be without extension in space or
extension in time. Such an event is a mere spatial point-flash of
instantaneous duration. I call such an ideal event an 'event-particle.'
You must not think of the world as ultimately built up of
event-particles. That is to put the cart before the horse. The world we
know is a continuous stream of occurrence which we can discriminate into
finite events forming by their overlappings and containings of each
other and separations a spatio-temporal structure. We can express the
properties of this structure in terms of the ideal limits to routes of
approximation, which I have termed event-particles. Accordingly
event-particles are abstractions in their relations to the more concrete
events. But then by this time you will have comprehended that you cannot
analyse concrete nature without abstracting. Also I repeat, the
abstractions of science are entities which are truly in nature, though
they have no meaning in isolation from nature.
The character of the spatio-temporal structure of events can be fully
expressed in terms of relations between these more abstract
event-particles. The advantage of dealing with event-particles is that
though they are abstract and complex in respect to the finite events
which we directly observe, they are simpler than finite events in
respect to their mutual relations. Accordingly they express for us the
demands of an ideal accuracy, and of an ideal simplicity in the
exposition of relations. These event-particles are the ultimate elements
of the four-dimensional space-time manifold which the theory of
relativity presupposes. You will have observed that each event-particle
is as much an instant of time as it is a point of space. I have called
it an instantaneous point-flash. Thus in the structure of this
space-time manifold space is not finally discriminated from time, and
the possibility remains open for diverse modes of discrimination
according to the diverse circumstances of observers. It is this
possibility which makes the fundamental distinction between the new way
of conceiving the universe and the old way. The secret of understanding
relativity is to understand this. It is of no use rushing in with
picturesque paradoxes, such as 'Space caught bending,' if you have not
mastered this fundamental conception which underlies the whole theory.
When I say that it underlies the whole theory, I mean that in my opinion
it ought to underlie it, though I may confess some doubts as to how far
all expositions of the theory have really understood its implications
and its premises.
Our measurements when they are expressed in terms of an ideal accuracy
are measurements which express properties of the space-time manifold.
Now there are measurements of different sorts. You can measure lengths,
or angles, or areas, or volumes, or times. There are also other sorts of
measures such as measurements of intensity of illumination, but I will
disregard these for the moment and will confine attention to those
measurements which particularly interest us as being measurements of
space or of time. It is easy to see that four such measurements of the
proper characters are necessary to determine the position of an
event-particle in the space-time manifold in its relation to the rest of
the manifold. For example, in a rectangular field you start from one
corner at a given time, you measure a definite distance along one side,
you then strike out into the field at right angles, and then measure a
definite distance parallel to the other pair of sides, you then rise
vertically a definite height and take the time. At the point and at the
time which you thus reach there is occurring a definite instantaneous
point-flash of nature. In other words, your four measurements have
determined a definite event-particle belonging to the four-dimension
space-time manifold. These measurements have appeared to be very simple
to the land-surveyor and raise in his mind no philosophic difficulties.
But suppose there are beings on Mars sufficiently advanced in
scientific invention to be able to watch in detail the operations of
this survey on earth. Suppose that they construe the operations of the
English land-surveyors in reference to the space natural to a being on
Mars, namely a Martio-centric space in which that planet is fixed. The
earth is moving relatively to Mars and is rotating. To the beings on
Mars the operations, construed in this fashion, effect measurements of
the greatest complication. Furthermore, according to the relativistic
doctrine, the operation of time-measurement on earth will not correspond
quite exactly to any time-measurement on Mars.
I have discussed this example in order to make you realise that in
thinking of the possibilities of measurement in the space-time manifold,
we must not confine ourselves merely to those minor variations which
might seem natural to human beings on the earth. Let us make therefore
the general statement that four measurements, respectively of
independent types (such as measurements of lengths in three directions
and a time), can be found such that a definite event-particle is
determined by them in its relations to other parts of the manifold.
If (p_1, p_2, p_3, p_4) be a set of measurements of this system, then
the event-particle which is thus determined will be said to have p_1,
p_2, p_3, p_4 as its co-ordinates in this system of measurement. Suppose
that we name it the p-system of measurement. Then in the same p-system
by properly varying (p_1, p_2, p_3, p_4) every event-particle that has
been, or will be, or instantaneously is now, can be indicated.
Furthermore, according to any system of measurement that is natural to
us, three of the co-ordinates will be measurements of space and one will
be a measurement of time. Let us always take the last co-ordinate to
represent the time-measurement. Then we should naturally say that (p_1,
p_2, p_3) determined a point in space and that the event-particle
happened at that point at the time p_4. But we must not make the mistake
of thinking that there is a space in addition to the space-time
manifold. That manifold is all that there is for the determination of
the meaning of space and time. We have got to determine the meaning of a
space-point in terms of the event-particles of the four-dimensional
manifold. There is only one way to do this. Note that if we vary the
time and take times with the same three space co-ordinates, then the
event-particles, thus indicated, are all at the same point. But seeing
that there is nothing else except the event-particles, this can only
mean that the point (p_1, p_2, p_3) of the space in the p-system is
merely the collection of event-particles (p_1, p_2, p_3, [p_4]), where
p_4 is varied and (p_1, p_2, p_3) is kept fixed. It is rather
disconcerting to find that a point in space is not a simple entity; but
it is a conclusion which follows immediately from the relative theory of
space.
Furthermore the inhabitant of Mars determines event-particles by another
system of measurements. Call his system the q-system. According to him
(q_1, q_2, q_3, q_4) determines an event-particle, and (q_1, q_2, q_3)
determines a point and q_4 a time. But the collection of event-particles
which he thinks of as a point is entirely different from any such
collection which the man on earth thinks of as a point. Thus the q-space
for the man on Mars is quite different from the p-space for the
land-surveyor on earth.
So far in speaking of space we have 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
instantaneous space. Thus if our natural perceptions are adjustable to
the p-system of measurements we see instantaneously all the
event-particles at some definite time p_4, 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 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.
According 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
phenomena due to this cause depend on the square of the ratio of any
velocity entering into the observation to the velocity of light. Now
light 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
1/10^8. 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 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 inversely 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 presupposed. The two masses are
assumed to be in simultaneous 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 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 immediate 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 neighbourhood 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 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 surrounding an event-particle E in
terms of ten quantities which I will call J_{11}, J_{12}
(=J_{21}), J_{22}, J_{23}(=J_{32}), 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 expressed 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.
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 modification of substituting
the direct distance for the inverse distance in the definition of the
potential, and its calculation 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. Mathematicians
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
phenomena. In the form in which I have put this modification, we deduce
Einstein's fundamental principle, as to the motion of light along its
rays, as a first approximation which is absolutely true for infinitely
short waves. Einstein's principle, thus partially verified, stated in my
language is that a ray of light always follows a path such that the
integral impetus along it is zero. This involves that every element of
impetus along it is zero.
In conclusion, I must apologise. In the first place I have considerably
toned down the various exciting peculiarities of the original theory and
have reduced it to a greater conformity with the older physics. I do not
allow that physical phenomena are due to oddities of space. Also I have
added to the dullness of the lecture by my respect for the audience. You
would have enjoyed a more popular lecture with illustrations of
delightful paradoxes. But I know also that you are serious students who
are here because you really want to know how the new theories may affect
your scientific researches.