Section twenty four Euclidean and non Euclidean continuum. The surface of a marble table
is spread out in front of me. I can get from any one point
on this table to any other point by passing continuously from one point to a
neighboring one, and repeating this process a large number of times, or in
other words, by going from point to point without executing jumps. I am
sure the reader will appreciate with sufficient clearness what I mean here by neighboring and
by jumps, if he is not too pedantic. We express this property of
the surface by describing the latter as a continuum. Let us now imagine that
a large number of little rods of equal length have been made, their lengths
being small compared with the dimensions of the marble slab. When I say they
are of equal length, I mean 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 that they 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 testing rod to this square. We add
similar ones, each of which has one rod in common with the first.
We proceed in like manner with each of these squares, until finally the whole
marble slab is laid out with squares. The arrangement is such that each side
of a square belongs to two squares, in each corner to four squares.
It is a veritable wonder that we can carry out this business without getting into
the greatest difficulties. We only need to think of the following. If at
any moment, three squares meet at a corner, then two sides of the
fourth square are already laid and as a consequence, the arrangement of the remaining
two sides of the square is already completely determined. But I am now no
longer able to adjust the quadrilateral so that 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 little rods, about which I can only be
thankfully surprised. We must needs experience many such surprises if the construction is to
be successful. If everything has really gone smoothly, then I say that the
points of the marble slab constitute a Euclidean continuum with respect to the little rod,
which has been used as a distance line interval. By choosing one corner
of a square as origin, I can characterize every other corner of a square
with reference to this origin by means of two numbers. I only need state
how many rods I must pass over when starting from the origin, I proceed
towards the right and then upwards in order to arrive at the corner of the
square under consideration. These two numbers are 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 of the following modification of this abstract experiment,
we recognize that there must also be cases in which the experiment would be unsuccessful.
We shall suppose that the rods expand by an amount proportional to the increase
of temperature we heat the central part of the marble slab, but not the
periphery, in which case two of our little rods can still be brought into
coincidence at every position on the table. But our construction of squares must necessarily
come into disorder during the heating. Because the little rods on the central region
of the table expand whereas those on the outer part do not. With reference
to our little rods defined as unit lengths, the marble slab is no longer
a Euclidean continuum, and we are also no longer in the position of defining
Cartesian coordinates directly with their aid, since the above construction can no longer be
carried out. But since there are other things which are not influenced in a
similar manner to the little rods, or perhaps not at all by the temperature
of the table, it is possible quite naturally to maintain the point of view
that the marble slad is a Euclidean continuum. This can be done in a
satisfactory manner by making a more subtle stipulation about the measurement or the comparison of
lengths. But if rods of every kind i e. Of every material were
to behave in the same way as regards the influence of temperature when they are
on the variably heated marble slab, and if we had no other means of
detecting the effect of temperature than the geometrical behavior of our rods in experiments analogous
to the one described above, Then our best plan would be to assign the
distance one to two points on the slab, provided that the ends of one
of our rods could be made to coincide with these two points. For how
else should we define the distance without our proceeding being in the highest measure grossly
arbitrary. The method of Cartesian co ordinates must then be discarded and replace by
another which does not assume the validity of Euclidean geometry for rigid bodies. Begin
footnote. Mathematicians have been confronted with our problem in the following form. If
we are given a surface e g. An ellipsoid in Euclidean three dimensional space,
then there exists for this surface a two dimensional geometry, just as much
as for a plane surface. Gauss undertook the task of treating this two dimensional
geometry from first principles without making use of 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 similar to that above with the marble slab, we
should find that different laws hold for these from those resulting on the basis of
Euclidean plane geometry. The surface is not a Euclidean continuum with respect to the
rods, and we cannot define Cartesian coordinates in the surface. Gaus indicated the
principles according to which we can treat the geometrical relationships the surface, and thus
pointed out the way to the method of Ramon of treating multidimensional non Euclidean continua.
Thus it is that mathematicians long ago solved the formal problems to which we
are led by the general postulate of relativity end footnote. The reader will notice
that the situation depicted here corresponds to the one brought about by the general postulate
of relativity Section twenty three end of section twenty four, Section twenty five Gaussian
coordinates. According to Gauss, this combined analytical and geometrical mode of handling the
problem can be arrived at in the following way. We imagine a system of
arbitrary curves see Figure four, drawn on the surface of the table. These
we designate as U curves, and we indicate each of them by means of
a number. The curves U equals one, U equals two, and U
equals three are drawn 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 which correspond to the real numbers lying between one and two. We
have then a system of U curves, and this infinitely dense system covers the
whole surface of the table. These U curves must not intersect each other,
and through each 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 of the marble slab. In like manner, we imagine a system of
V curves drawn on the surface. These satisfy the same conditions as the U
curves. They are provided with numbers in a corresponding manner, and they may
likewise be of arbitrary shape. It follows that a value of U and a
value of V belong to every point on the surface of the table. We
call these two numbers the coordinates of the surface of the table Gaussian coordinates.
For example, the point capital P in the diagram has the Gaussian coordinates U
equals three, V equals one. Two neighboring points capital P and capital P
prime on the surface then correspond to the coordinates capital P colon U comma v,
capital p prime colon U plus d u comma v plus d v,
where du and dv signify very small numbers. In a similar manner, we
may indicate the distance line interval between capital P and capital p prime has measured
with a little rod by means of the very small number d s. Then,
according to Gaus, we have ds squared equals g sub one one,
d U 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 two are magnitudes which depend in a perfectly
definite way on U and v. The magnitudes g sub one one, g
sub one two, and g sub two two 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 case in which the points of the surface
considered form a Euclidean continuum with reference to the measuring rods. But only in
this case, it is possible to draw the U curves and V curves and
to attach numbers to them in such a manner that we simply have ds squared
equals DU squared plus dv squared. Under these conditions, the U curves and
V curves are straight lines in the sense of Euclidean geometry, and they are
perpendicular to each other. Here, the Gaussian coordinates are simply Cartesian ones.
It is clear that gas coordinates are nothing more than an association of two sets
of numbers with the points of the surface considered of such a nature that numerical
values differing slightly from each other are associated with neighboring points in space. So
far, these considerations hold for a continuum of two dimensions, but the Gassian
method can be applied also to a continuum of three, four, or more
dimensions. If, for instance, a continuum of four dimensions be supposed available,
we may represent it in the following way. With every point of the
continuum, we associate arbitrarily four numbers x sub one x sub two, x
sub three x sub four, which are known as coordinates. Adjacent points correspond
to adjacent values of the coordinates. If a distance DS is associated with the
adjacent points capital p and capital p prime, this distance being measurable and well
defined from a physical point of view. Then the following formula holds d s
squared equals g sub one one, d x sub one squared plus two g
sub one two, d x sub one, d x sub two dot dot
plus g sub four four 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 is it possible to
associate the coordinates x sub one to x sub four with the points of the
continuum, so that we have simply d s squared equals d x sub one
squared plus d x sub two squared plus d x sub three squared plus d
x sub four squared. In this case, relations hold in the four dimensional
continuum which are analogous to those holding in our three dimensional measurements. However,
the Gauss treatment for ds squared which we have given above is not always possible.
It is only possible when sufficiently small regions of the continuum under consideration may
be regarded as Euclidean continua. For example, this obviously holds in the case
of the marble slab of the table and local variation of the temperature. The
temperature is practically constant for a small part of the slab, and thus the
geometrical behavior of the rods is almost as it ought to be according to the
rules of Euclidean geometry. Hence, the imperfections of the construction of squares in
the previous section do not show themselves clearly until this construction is extended over a
considerable portion of the surface of the table. We can sum this up as
follows. Gauss invented a method for the mathematical treatment of continua in general,
in which size relations distances between neighboring points are defined. To every point of
a continuum are assigned as many numbers Gaussian coordinates as the continuum has dimensions.
This is done in such a way that only one meaning can be attached to
the assignment, and that numbers Gaussian coordinates which differ by an indefinitely small amount
are assigned to adjacent points. The Gaussian coordinate system is a logical generalization of
the Cartesian coordinate system. It is also applicable to non Euclidean continua, but
only one with respect to the defined size or distance. Small parts of the
continuum under consideration behave more nearly like a Euclidean system, the smaller the part
of the continuum. Under our notice end of section twenty five Section twenty six
the space time continuum of the Special theory of relativity considered as a Euclidean continuum,
we are now in a position to formulate more exactly the idea of Minkowski,
which was only vaguely indicated in section seventeen. In accordance with the special
theory of relativity, certain coordinate systems are given preference for the description of the
Fordament space time continuum. We called these Galileyan coordinate systems. For these systems,
the four coordinates x, y, z, t, which determine an
event or in other words, a point of the four dimensional continuum, are
defined physically in a simple manner, 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 with reference to the first, the equations of the Lorenz
transformation are valid. These last form the basis for the derivation of deductions from
the special theory of relativity, and in themselves they are nothing more than the
expression of the universal validity of the law of transmission of light. For all
Galilean systems of reference. Minkowski found that the Lorenz transformations satisfy the following simple
conditions. Let us consider two neighboring events, the relative position of which in
the four dimensional continuum is given with respect to a Galiley in reference body capital
K by the space coordinate differences d x dy dz and the time difference dt
with reference to a second Galiley in system. We shall suppose that the corresponding
differences for these two events are dx prime, dy prime, dz prime,
dt prime. Then these magnitudes always fulfill the condition. Begin footnote c F
appendices one and two. The relations which are derived there for the coordinates themselves
are valid also for coordinate differences, and thus also for coordinate differentials indefinitely small
differences. End footnote d x squared plus dy squared plus DZ squared c squared
dt squared equals dx prime squared plus dy prime squared plus dz prime squared minus
c squared dt prime squared. The validity of the Lorentz transformation follows from this
condition. We can express this as follows. The magnitude ds squared equals d
x squared. Plus d y squared plus dz squared c squared dt squared,
which belongs to two adjacent points of the four dimensional space time continuum, has
the same value for all selected Galilean reference bodies. If we replace x y
z square root of quantity minus one and quantity ct by x sub one x
sub two, x sub three x sub four, we also obtain the result
that ds squared equals d x sub one squared plus d x sub two squared
plus d x sub three squared plus d x sub 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 or four dimensional points. Thus, if we
choose as time variable the imaginary variable square root quantity minus one end quantity ct
instead of the real quantity T, we can regard the space time continuum in
accordance with the special theory of relativity as a Euclidean four dimensional continuum, a
result which follows from the considerations of the preceding section. End of section twenty six