that the set of quantitiesgrs'will in such an arbitrarydistortion
differ from the correspondingquantitiessay grs, at the creors-ponding
point. Likewise their differential coefficients with
respect to the co-ordinates will take new values.
The problem then
essentiallya positivequantity if the two points do not coincide
(cf.what has been said above as to coincidences in physical
measurements). The simplest type of expression satisfying
this criterion for two near pointshe takes to be ds,the positive
Now justas Riemann found a sufficiemneatns of ipnregscriba
measure for the interval between adjacent events by
means of the quantitiesg^v and the quadraticform, we are
able to devise a sufficient means for our new purpose by ilangydown
the rate at which ^
We recall the identification of a straightline as the shortest
distance between two points. The lengthof a line in a fourdimensional
pictureis to be defined thus. In the absence of
a gravitationaflield the interval between two adjacentpoints
has been defi
by putting a = 2ym
and P = R2(i + 47^/R)
or b = R(i + 2ym/R.).
The total deflexion of the ray is the angle between the
asymptotes, viz :-"
which to the first order is
THE VERIFICATION OF EINSTEIN'S EQUATIONS 121
On drawing the correspond
The second means that the expression
must be built up from quantitieswhich have no specialrelation
to any particulardirection through the centre of the sun.
If we use sphericalpolar co-ordinates the only such
quantityin addition to r, the distance from th
electrical field and the measure system is so remote, that the
fundamental place accorded to electricityin modern thought
seems to be denied. In Weyl's theory it is completely
restored, and gravitation and electricityrank together as two
or, since F = i and therefore " = o
Also F?_ = - r'sintf
Now, at any specifiedmoment the particleis moving in a
certain planethrough the origin. Let the axis from which 0
is measured be taken normal to this plane. Then at thi
we should have liked to be able to predict the cveonr-gence.
And why ? Because if we had known
how to predict it in the one case, we should know
how to predict it in another. We have been fsucl,cessit
is true, but that is little in our eyes if we have
space correspondsto a planetat any moment of his conceptual
time; he can express the space-co-ordinatneumbers of the
planet in terms of the time co-ordinate number. When this
is done it is found that,except in the long standingdiscrepancy
in the motion of
Each measure-system may also have associated with it
certain quantitieswhose values are unaltered under co-ordinate
changes and guage-changes which leave any given interval
unaltered. For instance the integral
Illf*l/l*Jg ^ *d* *
where /* = ^ - ^, and ./"
so that there is an invariant element ,Jg d\".
87. HYPOTHETICAL EXPRESSION FOR ACTION.
We thus arrive at an expressionwhich for any given region
of the four dimensional world is an invariant,viz :"
The subjectof integrationhere is simply a function of the
is known with respect to which the laws of motion are
satisfied then any other frame which is moving relative to
that known frame with a constant velocityas judged by the
standards of dynamics,is equallysuitable as a basis for the
descriptionof the motion
In the chapters devoted to them, I am obliged to
treat of somewhat more abstract subjects,and, to begin
with, I have to speak of the notion of space. Every one
knows that space is relative,or rather every one says
so, but how many people think still as if
of view, to think of them piecingtogetherinto a sthiorenae-ldimenregion
which is,so to speak,a curved regionin a fourdimensional
In such a region a system of geometry can be built up
usingthe given expressionfor the element of distance,just as
them, by the greater number of our contemporaries.
He cares but little for the industrial applications of
science, for the marvels of electricity or of automobilism,
which he regards rather as hindrances to
For him the useful is exclusivel
gave to his writingsa tone of doubt which was hailed
with joy by scepticsand pragmatists. But he was in
truth no sceptic: however conscious of the difficulty
of attainingknowledge,he never admitted its isimbpiols-ity.
" It is a mistake to believe,"he said
interest: it may have been noted many times without
renderingany great service to science ; it only acquires
a value when some more careful thinker perceivesthe
connexion it brings out, and symbolizes it by a term.
The physicistsalso proceed in exactly th
portionof this will have advanced a distance CT, being now part
of the line X'N. In the same time the reflector has advanced a distance
AA' = vr ; so that,drawing A'B' parallel
to AB to meet X'N in C, C is now the
point of incidence of the wave-front on
each of these little atoms. Similarlythe biologisthas
been led instinctiveltyo regardthe cell as more ingterestthan
the whole animal, and the event has proved
him right,since cells belonging to the most diverse
organisms have greater resemblances, for tho
1 Actuallythis is given up in the new theory.
2 This is,in fact,the case in the generalPrincipleof Relativity.
THE RELATIVITY OF SPACE AND TIME 33
the choice of the variables which are used to describe the
motions is in any way limited. The Newtonian form
VI. Minkowski's Four-Dimension Vectors 72
THE GENERAL PRINCIPLE OF RELATIVITY.
VII. The General Theory .86
VIII. Verification of Einstein's Theory 113
IX. Further Generalization, Weyl's Theory of Electricity
. . 125
NOTE ON VECTOR
of ' simultaneous events at different places'
came to be
considered as one about which there was no lack of dteifoin.niIf we examine the development of the idea,however, we
see that it arose graduallyin some such way as this. In
the first instance,events
consistent with such a position.
2. The limited scope of the relativiatryisingout of dmiycna-l
theoryhas always been rather unsatisfyingto the mind,
and so it was almost with a sense of relief that the rise of the
electro-magnetic theoryof light,bringingw
number of ways of choosingtwo out of the four co-ordinates
of a point.
Now Minkowski was able to show that the transformations
(B, p. 48), affecting(exteyy ez(ihx,ihy"ih^ are exactly of
the same form as those which affect the six components of a
foresee whether the solution of these problems will
be simple,it shows us at least whether the calculation
is worth undertaking.
What I have justsaid is sufficient to show how vain
it would be to attempt to replacethe mathematician's
free initiative by a
Maxwell's equations or the law of gravitation. It is a general
1 " Raum und Zeit," 1908. Reprinted in " Das Relativitats prinzip," Leipzig^
MINKOWSKPS FOUR-DIMENSION VECTORS 73
mode of thought. We are not concerned with any particular
Eotvos thus describes his experiment :"
" I attached separate bodies of about 30 grms. weight to
111 Mathematische und Naturwissenschaft liche Berichte aus Ungarn," Bd. 8
THE GENERAL THEORY OF RELATIVITY 89
the ends of a balance beam about 2
the idea of a luminiferous aether came into prominence.
The question was at once asked, " What is the velocity of the
earth relative to this medium ? "
Arago, sometime before iSiS,1 devised an experiment
which he thought would answer that question. On the