Relativity - Part 24-26 Einstein

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Correction for this chapter in mathematical formulaeinstead of I here one. This is
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Cambridge, Massachusetts. Relativity The Specialand General Theory by Albert Einstein,
continuing Part two The General Theory ofRelativity, sections twenty four through twenty six.
Section twenty four Euclidean and non Euclideancontinuum. The surface of a marble

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table is spread out in front ofme. I can get from any one
point on this table to any otherpoint by passing continuously from one point to
a neighboring one, and repeating thisprocess a large number of times, or
in other words, by going frompoint to point without executing jumps. I
am sure the reader will appreciate withsufficient clearness what I mean here by neighboring

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and by jumps, if he isnot too pedantic. We express this property
of the surface by describing the latteras a continuum. Let us now imagine
that a large number of little rodsof equal length have been made, their
lengths being small compared with the dimensionsof the marble slab. When I say

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they are of equal length, Imean that one can be laid on any
other one without the ends overlapping.We next lay four of these little rods
on the marble slab, so thatthey constitute a quadrilateral figure a square,
the diagonals of which are equally long. To ensure the equality of the diagonals,
we make use of a little testingrod. To this square, we

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add similar ones, each of whichhas one rod in common with the first.
We proceed in like manner with eachof these squares, until finally the
whole marble slab is laid out withsquares. The arrangement is such that each
side of a square belongs to twosquares, and each corner to four squares.
It is a veritable wonder that wecan carry out this business without getting

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into the greatest difficulties. We onlyneed to think of the following. If
at any moment three squares need ata corner, then two sides of the
fourth square are already laid and asa consequence, the arrangement of the remaining
two sides of the square is alreadycompletely determined. But I am now no

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longer able to adjust the quadrilateral sothat its diagonals may be equal. If
they are equal of their own accord, then this is an especial favor of
the marble slab and of the littlerods, about which I can only be
thankfully surprised. We must needs experiencemany such surprises if the construction is to
be successful. If everything has reallygone smoothly, then I say that the

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points of the marble slab constitute aEuclidean continuum with respect to the little rod,
which has been used as a distanceline interval. By choosing one corner
of a square as origin, Ican characterize every other corner of a square
with reference to this origin by meansof two numbers. I only need state

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how many rods I must pass overwhen starting from the origin, I proceed
towards the right and then upwards inorder to arrive at the corner of the
square under consideration. These two numbersare then the Cartesian coordinates of this corner
with reference to the Cartesian coordinate system, which is determined by the arrangement of
little rods. By making use ofthe following modification of this abstract experiment,

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we recognize that there must also becases in which the experiment would be unsuccessful.
We shall suppose that the rods expandby an amount proportional to the increase
of temperature. We heat the centralpart of the marble slab, but not
the periphery, in which case twoof our little rods can still be brought
into coincidence at every position on thetable. But our construction of squares must

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necessarily come into disorder during the heating, because the little rods on the central
region of the table expand, whereasthose on the outer part do not.
With reference to our little rods definedas unit lengths, the marble slab is
no longer an Euclidean continuum, andwe are also no longer in the position
of defining Cartesian coordinates directly with theiraid. Since the above construction can no

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longer be carried out. But sincethere are other things which are not influenced
in a similar manner to the littlerods, or perhaps not at all by
the temperature of the table, itis possible quite naturally to maintain the point
of view that the marble slab isa Euclidean continuum. This can be done
in a satisfactory manner by making amore subtle stipulation about the measurement or the

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comparison of leaneth. But if rodsof every kind i e. Of every
material were to behave in the sameway as regards the influence of temperature when
they are on the variably heated marbleslab, and if we had no other
means of detecting the effect of temperaturethan the geometrical behavior of our rods in
experiments analogous to the one described above, then our best plan would be to

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assign the distance one to two pointson the slab, provided that the ends
of one of our rods could bemade to coincide with these two points.
For how else should we define thedistance without our proceeding being in the highest
measure grossly arbitrary. The method ofCartesian coordinates must then be discarded and replaced

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by another which does not assume thevalidity of Euclidean geometry for rigid bodies.
Begin footnote. Mathematicians have been confrontedwith our problem in the following form.
If we are given a surface eg. An ellipsoid in Euclidean three dimensional
space, then there exist for thissurface a two dimensional geometry just as much

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as for a plane surface. Gaussundertook the task of treating this two dimensional
geometry from first principles without making useof the fact that the surface belongs to
a Euclidean continuum of three dimensions.If we imagine constructions to be made with
rigid rods in the surface, similarto that above with the marble slab,

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we should find that different laws holdfor these from those resulting on the basis
of Euclidean plane geometry. The surfaceis not a Euclidean continuum with respect to
the rods, and we cannot defineCartesian coordinates in the surface. Gauss indicated
the principles according to which we cantreat the geometrical relationships in the surface,
and thus pointed out the way tothe method of Rimon of treating multidimensional non

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Euclidean continua. Thus it is thatmathematicians long ago solved the formal problems to
which we are led by the generalpostulate of relativity. End footnote. The
reader will not that the situation depictedhere corresponds to the one brought about by
the general postulate of relativity. Sectiontwenty three. End of section twenty four

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Section twenty five Gaussian coordinates. Accordingto Gauss, this combined analytical and geometrical
mode of handling the problem can bearrived at in the following way. We
imagine a system of arbitrary curves seeFig. Four drawn on the surface of
the table. These we designate asU curves, and we indicate each of

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them by means of a number.The curves U equals one, U equals
two, and U equals three aredrawn in the diagram. Between the curves
U equals one and U equals two. We must imagine an infinitely large number
to be drawn, all of whichcorrespond to the real numbers lying between one
and two. We have then asystem of U curves, and this in

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nately dense system covers the whole surfaceof the table. These U curves must
not intersect each other, and througheach point of the surface, one and
only one curve must pass. Thus, a perfectly definite value of U belongs
to every point on the surface ofthe marble slab. In like manner,
we imagine a system of V curvesdrawn on the surface. These satisfy the

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same conditions as the U curves.They are provided with numbers in a corresponding
manner, and they may likewise beof arbitrary shape. It follows that a
value of U and a value ofV belong to every point on the surface
of the table. We call thesetwo numbers the coordinates of the surface of
the table Gaussian coordinates. For example, the point capital P in the diagram

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has the Gaussian coordinates u equals three, v equals one two. Neighboring points
capital P and capital p prime onthe surface then correspond to the coordinates capital
p, u s colon U commav, capital p prime colon U plus
d u comma v plus d v, where d u and d v signify

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very small numbers. In a similarmanner, we may indicate the distance line
interval between capital P and capital pprime, as measured with a little rod,
by means of the very small numberd s. Then, according to
Guss, we have d S squaredequals g sub one one, d U

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squared plus two g sub one two, d u, d v plus g
sub two two d V squared.Where g sub one one, g sub
one two, g sub two twoare magnitudes which depend in a perfectly definite
way on U and v. Themagnitudes G sub one one, G sub

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one two, and G sub twotwo determine the behavior of the rods relative
to the U curves and V curves, and thus also relative to the surface
of the table. For the casein which the points of the surface considered
form a Euclidean continuum with reference tothe measuring rods. But only in this
case, it is possible to drawthe U curves and V curves and to

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attach numbers to them in such amanner that we simply have d S squared
equals DU squared plus d v squared. Under these conditions, the U curves
and V curves are straight lines inthe sense of Euclidean geometry, and they
are perpendicular to each other. Herethe Gaussian coordinates are simply Cartesian ones.

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It is clear that Gauss coordinates arenothing more than an association of two sets
of numbers, with the points ofthe surface considered of such a nature that
numerical values differing slightly from each otherare associated with neighboring points in space.
So far, these considerations hold fora continue of two dimensions, but the
Gaussian method can be applied also toa continuum of three four or more dimensions.

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If, for instance, a continuumof four dimensions be supposed available,
we may represent it in the followingway. With every point of the continuum,
we associate arbitrarily four numbers x subone, x sub two, x
sub three, x sub four,which are known as coordinates. Adjacent points

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correspond to adjacent values of the coordinates. If a distance d s is associated
with the adjacent points capital p andcapital p prime, this distance being measurable
and well defined from a physical pointof view, then the following formula holds
d s squared equals g sub oneone, d x sub one squared plus

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two g sub one two, dx sub one, d x sub two
dot dot dot plus g sub fourfour d x sub four squared, Where
the magnitudes g sub one, one, et cetera, have values which vary
with the position in the continuum.Only when the continuum is a Euclidean one

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is it possible to associate the coordinatesx sub one to x sub four with
the points of the continuum, sothat we have simply d s squared equals
d x sub one squared plus dx sub two squared plus d x sub
three squared plus d x sub foursquared. In this case, relations hold
in the four dimensional continuum which areanalogous to those holding in our three dimensional

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measurements. However, the gause treatmentfor d s squared which we have given
above is not always possible. Itis only possible when sufficiently small regions of
the continuum under consideration may be regardedas Euclidean continua. For example, this
obviously holds in the case of themarble slab of the table and local variation

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of the temperature. The temperature ispractically constant for a small part of the
slab, and thus the geometrical behaviorof the rods is almost as it ought
to be according to the rules ofEuclidean geometry. Hence, the imperfections of
the construction of squares in the previoussection do not show themselves clearly until this
construction is extended over a considerable portionof the surface of the table. We

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can sum this up as follows.Gauss invented a method for the mathematical treatment
of continua in general, in whichsize relations distances between neighboring points are defined
to every point of a continuum areassigned as many numbers Gaussian coordinates as the
continuum has dimensions. This is donein such a way that only one meaning

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can be attached to the assignment,and that numbers Gaussian coordinates which differ by
an indefinitely small amount are assigned toadjacent points. The Gaussian coordinate system is
a logical generalization of the Cartesian coordinatesystem. It is also applicable to non
Euclidean continuum, but only one withrespect to the defined size or distance.

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Small parts of the continuum under considerationbehave more nearly like a Euclidean system.
The smaller the part of the continuumunder our notice end of section twenty five
Section twenty six the space time continuumof the special theory of relativity considered as
a Euclidean continuum. We are nowin a position to formulate more exactly the

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idea of Minkowski, which was onlyvaguely indicated in section seventeen. In accordance
with the special theory of relativity,certain coordinate systems are given preference for the
description of the four dimensional space timecontinuum. We called these Galilean coordinate systems.
For these systems, the four coordinatesx, y, z T,

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which determine an event or In otherwords, a point of the four dimensional
continuum are defined physically in a simplemanner, as set forth in detail in
the first part of this book.For the transition from one Galilean system to
another, which is moving uniformly withreference to the first, the equations of
the Lorentz transformation are valid. Theselast form the basis for the derivation of

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deductions from the special theory of relativity, and in themselves they are nothing more
than the expression of the universal validityof the law of transmission of light for
all Galilean systems of reference. Minkowskifound that the Lorentz transformations satisfy the following
simple conditions. Let us consider twoneighboring events, the relative position of which

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in the four dimensional continuum is givenwith respect to a Galileian reference body capital
K, by the space coordinate differencesd x, d y, dz and
the time difference dt with reference toa second Galileian system. We shall suppose
that the corresponding differences for these twoevents are dx prime, d y prime,

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d z prime, d t prime. Then these magnitudes always fulfill the
condition begin footnote c F Appendices oneand two. The relations which are derived
there for the coordinates themselves are validalso for coordinate differences, and thus also
for coordinate differentials indefinitely small differences.End footnote. D x squared plus d

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y squared plus DZ squared minus csquared dt squared equals d x prime squared
plus d y prime squared plus dzprime squared minus c squared dt prime squared.
The validity of the Lorentz transformation followsfrom this condition. We can express
this as follows. The magnitude dS squared equals d x squared plus d

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y squared plus dz squared minus csquared dt squared, which belongs to two
adjacent points of the four dimensional spacetime continuum, has the same value for
all selected Galilean reference bodies. Ifwe replace x y z square root of
quantity minus one and quantity c tby x sub one x sub two,

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x sub three x sub four,we also obtain the result that d S
squared equals d x sub one squaredplus d x sub two squared plus d
x sub three squared plus d xsub four squared. Is independent of the
choice of the body of reference.We call the magnitude d s. The
distance apart of the two events orfour dimensional points. Thus, if we

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choose as time variable the imaginary variablesquare root quantity minus one and quantity c
t instead of the real quantity t, we can regard the space time continuum
in accordance with the special theory ofrelativity as a Euclidean four dimensional continuum,
a result which follows from the considerationsof the preceding section. End of section twenty six