WEBVTT

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Section twenty four Euclidean and non Euclidean
continuum. The surface of a marble table

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is spread out in front of me. I can get from any one point

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on this table to any other point
by passing continuously from one point to a

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neighboring one, and repeating this process
a large number of times, or in

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other words, by going from point
to point without executing jumps. I am

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sure the reader will appreciate with sufficient
clearness what I mean here by neighboring and

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by jumps, if he is not
too pedantic. We express this property of

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the surface by describing the latter as
a continuum. Let us now imagine that

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a large number of little rods of
equal length have been made, their lengths

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being small compared with the dimensions of
the marble slab. When I say they

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are of equal length, I mean
that one can be laid on any other

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one without the ends overlapping. We
next lay four of these little rods on

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the marble slab, so that they
constitute a quadrilateral figure, a square,

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the diagonals of which are equally long. To ensure the equality of the diagonals,

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we make use of a little testing
rod to this square. We add

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similar ones, each of which has
one rod in common with the first.

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We proceed in like manner with each
of these squares, until finally the whole

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marble slab is laid out with squares. The arrangement is such that each side

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of a square belongs to two squares, in each corner to four squares.

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It is a veritable wonder that we
can carry out this business without getting into

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the greatest difficulties. We only need
to think of the following. If at

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any moment, three squares meet at
a corner, then two sides of the

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fourth square are already laid and as
a consequence, the arrangement of the remaining

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two sides of the square is already
completely determined. But I am now no

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longer able to adjust the quadrilateral so
that its diagonals may be equal. If

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they are equal of their own accord, then this is an especial favor of

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the marble slab and of the little
rods, about which I can only be

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thankfully surprised. We must needs experience
many such surprises if the construction is to

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be successful. If everything has really
gone smoothly, then I say that the

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points of the marble slab constitute a
Euclidean continuum with respect to the little rod,

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which has been used as a distance
line interval. By choosing one corner

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of a square as origin, I
can characterize every other corner of a square

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with reference to this origin by means
of two numbers. I only need state

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how many rods I must pass over
when starting from the origin, I proceed

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towards the right and then upwards in
order to arrive at the corner of the

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square under consideration. These two numbers
are then the Cartesian coordinates of this corner

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with reference to the Cartesian coordinate system, which is determined by the arrangement of

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little rods. By making use of
the following modification of this abstract experiment,

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we recognize that there must also be
cases in which the experiment would be unsuccessful.

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We shall suppose that the rods expand
by an amount proportional to the increase

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of temperature we heat the central part
of the marble slab, but not the

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periphery, in which case two of
our little rods can still be brought into

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coincidence at every position on the table. But our construction of squares must necessarily

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come into disorder during the heating.
Because the little rods on the central region

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of the table expand whereas those on
the outer part do not. With reference

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to our little rods defined as unit
lengths, the marble slab is no longer

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a Euclidean continuum, and we are
also no longer in the position of defining

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Cartesian coordinates directly with their aid,
since the above construction can no longer be

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carried out. But since there are
other things which are not influenced in a

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similar manner to the little rods,
or perhaps not at all by the temperature

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of the table, it is possible
quite naturally to maintain the point of view

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that the marble slad is a Euclidean
continuum. This can be done in a

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satisfactory manner by making a more subtle
stipulation about the measurement or the comparison of

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lengths. But if rods of every
kind i e. Of every material were

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to behave in the same way as
regards the influence of temperature when they are

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on the variably heated marble slab,
and if we had no other means of

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detecting the effect of temperature than the
geometrical behavior of our rods in experiments analogous

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to the one described above, Then
our best plan would be to assign the

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distance one to two points on the
slab, provided that the ends of one

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of our rods could be made to
coincide with these two points. For how

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else should we define the distance without
our proceeding being in the highest measure grossly

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arbitrary. The method of Cartesian co
ordinates must then be discarded and replace by

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another which does not assume the validity
of Euclidean geometry for rigid bodies. Begin

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footnote. Mathematicians have been confronted with
our problem in the following form. If

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we are given a surface e g. An ellipsoid in Euclidean three dimensional space,

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then there exists for this surface a
two dimensional geometry, just as much

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as for a plane surface. Gauss
undertook the task of treating this two dimensional

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geometry from first principles without making use
of the fact that the surface belongs to

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a Euclidean continuum of three dimensions.
If we imagine constructions to be made with

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rigid rods in the surface similar to
that above with the marble slab, we

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should find that different laws hold for
these from those resulting on the basis of

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Euclidean plane geometry. The surface is
not a Euclidean continuum with respect to the

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rods, and we cannot define Cartesian
coordinates in the surface. Gaus indicated the

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principles according to which we can treat
the geometrical relationships the surface, and thus

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pointed out the way to the method
of Ramon of treating multidimensional non Euclidean continua.

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Thus it is that mathematicians long ago
solved the formal problems to which we

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are led by the general postulate of
relativity end footnote. The reader will notice

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that the situation depicted here corresponds to
the one brought about by the general postulate

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of relativity Section twenty three end of
section twenty four, Section twenty five Gaussian

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coordinates. According to Gauss, this
combined analytical and geometrical mode of handling the

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problem can be arrived at in the
following way. We imagine a system of

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arbitrary curves see Figure four, drawn
on the surface of the table. These

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we designate as U curves, and
we indicate each of them by means of

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a number. The curves U equals
one, U equals two, and U

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equals three are drawn in the diagram. Between the curves U equals one and

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U equals two. We must imagine
an infinitely large number to be drawn,

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all of which correspond to the real
numbers lying between one and two. We

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have then a system of U curves, and this infinitely dense system covers the

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whole surface of the table. These
U curves must not intersect each other,

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and through each point of the surface, one and only one curve must pass.

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Thus, a perfectly definite value of
U belongs to every point on the

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surface of the marble slab. In
like manner, we imagine a system of

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V curves drawn on the surface.
These satisfy the same conditions as the U

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curves. They are provided with numbers
in a corresponding manner, and they may

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likewise be of arbitrary shape. It
follows that a value of U and a

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value of V belong to every point
on the surface of the table. We

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call these two numbers the coordinates of
the surface of the table Gaussian coordinates.

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For example, the point capital P
in the diagram has the Gaussian coordinates U

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equals three, V equals one.
Two neighboring points capital P and capital P

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prime on the surface then correspond to
the coordinates capital P colon U comma v,

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capital p prime colon U plus d
u comma v plus d v,

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where du and dv signify very small
numbers. In a similar manner, we

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may indicate the distance line interval between
capital P and capital p prime has measured

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with a little rod by means of
the very small number d s. Then,

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according to Gaus, we have ds
squared equals g sub one one,

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d U squared plus two g sub
one two, d u d v plus

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g sub two two d V squared, Where g sub one one, g

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sub one two, g sub two
two are magnitudes which depend in a perfectly

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definite way on U and v.
The magnitudes g sub one one, g

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sub one two, and g sub
two two determine the behavior of the rods

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relative to the U curves and V
curves, and thus also relative to the

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surface of the table. For the
case in which the points of the surface

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considered form a Euclidean continuum with reference
to the measuring rods. But only in

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this case, it is possible to
draw the U curves and V curves and

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to attach numbers to them in such
a manner that we simply have ds squared

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equals DU squared plus dv squared.
Under these conditions, the U curves and

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V curves are straight lines in the
sense of Euclidean geometry, and they are

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perpendicular to each other. Here,
the Gaussian coordinates are simply Cartesian ones.

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It is clear that gas coordinates are
nothing more than an association of two sets

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of numbers with the points of the
surface considered of such a nature that numerical

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values differing slightly from each other are
associated with neighboring points in space. So

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far, these considerations hold for a
continuum of two dimensions, but the Gassian

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method can be applied also to a
continuum of three, four, or more

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dimensions. If, for instance,
a continuum of four dimensions be supposed available,

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we may represent it in the following
way. With every point of the

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continuum, we associate arbitrarily four numbers
x sub one x sub two, x

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sub three x sub four, which
are known as coordinates. Adjacent points correspond

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to adjacent values of the coordinates.
If a distance DS is associated with the

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adjacent points capital p and capital p
prime, this distance being measurable and well

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defined from a physical point of view. Then the following formula holds d s

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squared equals g sub one one,
d x sub one squared plus two g

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sub one two, d x sub
one, d x sub two dot dot

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plus g sub four four d x
sub four squared. Where the magnitudes g

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sub one, one, et cetera
have values which vary with the position in

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the continuum. Only when the continuum
is a Euclidean one is it possible to

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associate the coordinates x sub one to
x sub four with the points of the

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continuum, so that we have simply
d s squared equals d x sub one

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squared plus d x sub two squared
plus d x sub three squared plus d

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x sub four squared. In this
case, relations hold in the four dimensional

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continuum which are analogous to those holding
in our three dimensional measurements. However,

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the Gauss treatment for ds squared which
we have given above is not always possible.

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It is only possible when sufficiently small
regions of the continuum under consideration may

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be regarded as Euclidean continua. For
example, this obviously holds in the case

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of the marble slab of the table
and local variation of the temperature. The

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temperature is practically constant for a small
part of the slab, and thus the

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geometrical behavior of the rods is almost
as it ought to be according to the

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rules of Euclidean geometry. Hence,
the imperfections of the construction of squares in

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the previous section do not show themselves
clearly until this construction is extended over a

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considerable portion of the surface of the
table. We can sum this up as

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follows. Gauss invented a method for
the mathematical treatment of continua in general,

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in which size relations distances between neighboring
points are defined. To every point of

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a continuum are assigned as many numbers
Gaussian coordinates as the continuum has dimensions.

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This is done in such a way
that only one meaning can be attached to

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the assignment, and that numbers Gaussian
coordinates which differ by an indefinitely small amount

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are assigned to adjacent points. The
Gaussian coordinate system is a logical generalization of

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the Cartesian coordinate system. It is
also applicable to non Euclidean continua, but

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only one with respect to the defined
size or distance. Small parts of the

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continuum under consideration behave more nearly like
a Euclidean system, the smaller the part

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of the continuum. Under our notice
end of section twenty five Section twenty six

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the space time continuum of the Special
theory of relativity considered as a Euclidean continuum,

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we are now in a position to
formulate more exactly the idea of Minkowski,

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which was only vaguely indicated in section
seventeen. In accordance with the special

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theory of relativity, certain coordinate systems
are given preference for the description of the

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Fordament space time continuum. We called
these Galileyan coordinate systems. For these systems,

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the four coordinates x, y,
z, t, which determine an

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event or in other words, a
point of the four dimensional continuum, are

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defined physically in a simple manner,
as set forth in detail in the first

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part of this book. For the
transition from one Galilean system to another,

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which is moving uniformly with reference to
the first, the equations of the Lorenz

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transformation are valid. These last form
the basis for the derivation of deductions from

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the special theory of relativity, and
in themselves they are nothing more than the

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expression of the universal validity of the
law of transmission of light. For all

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Galilean systems of reference. Minkowski found
that the Lorenz transformations satisfy the following simple

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conditions. Let us consider two neighboring
events, the relative position of which in

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the four dimensional continuum is given with
respect to a Galiley in reference body capital

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K by the space coordinate differences d
x dy dz and the time difference dt

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with reference to a second Galiley in
system. We shall suppose that the corresponding

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differences for these two events are dx
prime, dy prime, dz prime,

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dt prime. Then these magnitudes always
fulfill the condition. Begin footnote c F

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appendices one and two. The relations
which are derived there for the coordinates themselves

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are valid also for coordinate differences,
and thus also for coordinate differentials indefinitely small

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differences. End footnote d x squared
plus dy squared plus DZ squared c squared

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dt squared equals dx prime squared plus
dy prime squared plus dz prime squared minus

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c squared dt prime squared. The
validity of the Lorentz transformation follows from this

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condition. We can express this as
follows. The magnitude ds squared equals d

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x squared. Plus d y squared
plus dz squared c squared dt squared,

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which belongs to two adjacent points of
the four dimensional space time continuum, has

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the same value for all selected Galilean
reference bodies. If we replace x y

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z square root of quantity minus one
and quantity ct by x sub one x

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sub two, x sub three x
sub four, we also obtain the result

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that ds squared equals d x sub
one squared plus d x sub two squared

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plus d x sub three squared plus
d x sub four squared is independent of

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the choice of the body of reference. We call the magnitude d s the

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distance apart of the two events or
four dimensional points. Thus, if we

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choose as time variable the imaginary variable
square root quantity minus one end quantity ct

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instead of the real quantity T,
we can regard the space time continuum in

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accordance with the special theory of relativity
as a Euclidean four dimensional continuum, a

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result which follows from the considerations of
the preceding section. End of section twenty six