24. Euclidean and non-Euclidean Continuum25. Gaussian Co-ordinates26. The Space-Time Continuum of the Speical Theory of Relativity

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Episode Transcript

Speaker 1 (00:00):
Correction for this chapter in mathematical formulae instead of I
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Relativity The Special and General Theory by Albert Einstein, continuing
Part two The General Theory of Relativity, sections twenty four
through twenty six. Section twenty four Euclidean and non Euclidean continuum.

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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

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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

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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

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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

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belongs to two squares, and 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 need at a corner, then two sides of the
fourth square are already laid and as a consequence, the

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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.

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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

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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

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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.

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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

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

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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 slab 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 leaneth.
But if rods of every kind i e. Of every

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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

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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 coordinates must then
be discarded and replaced by another which does not assume
the validity of Euclidean geometry for rigid bodies. Begin footnote.

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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 exist 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

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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

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with respect to the rods, and we cannot define Cartesian
coordinates in the surface. Gauss indicated the principles according to
which we can treat the geometrical relationships in the surface,
and thus pointed out the way to the method of
Rimon of treating multidimensional non Euclidean continua. Thus it is
that mathematicians long ago solved the formal problems to which

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we are led by the general postulate of relativity. End footnote.
The reader will not 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

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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 Fig. Four drawn on the surface of the table.
These we designate as U curves, and we indicate each

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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

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a system of U curves, and this in knightly 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,

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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

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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

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c colon U comma v, capital p prime colon U plus
d u comma v plus d v, where d U
and d v signify very small numbers. In a similar manner,
we may indicate the distance line interval between capital P
and capital p prime, as measured with a little rod,

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by means of the very small number d s. Then,
according to gus, we have d s 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

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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

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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
d S squared equals DU squared plus d v squared.
Under these conditions, the U curves and V curves are

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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 Gauss 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

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associated with neighboring points in space. So far, these considerations
hold for a continue of two dimensions, but the Gaussian
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

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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 d s is associated with the adjacent points

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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 dot plus g sub four four d

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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

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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 gause treatment for d s
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 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

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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

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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

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assigned to adjacent points. The Gaussian coordinate system is a
logical generalization of the Cartesian coordinate system. It is also
applicable to non Euclidean continuum, 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.

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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

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vaguely 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 time continuum. We
called these Galilean coordinate systems. For these systems, the four
coordinates x, y, z T, which determine an event or

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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 Lorentz transformation are valid. These last form the basis

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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 Lorentz transformations satisfy the following simple conditions. Let us

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consider two neighboring events, the relative position of which in
the four dimensional continuum is given with respect to a
Galileian reference body capital K, by the space coordinate differences
d x, d y, dz and the time difference dt
with reference to a second Galileian system. We shall suppose
that the corresponding differences for these two events are dx prime,

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d y prime, d z prime, d t 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.

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D x squared plus d y squared plus DZ squared
minus c squared dt squared equals d x prime squared
plus d y 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.

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The magnitude d S squared equals d x squared plus
d y squared plus dz squared minus 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

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root of quantity minus one and quantity c t by
x sub one x sub two, x sub three x
sub four, we also obtain the result that 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. Is independent of the choice

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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 and quantity c t instead of
the real quantity t, we can regard the space time

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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