[ Pobierz całość w formacie PDF ]
ds2 = g11dx12 + 2g12dx1dx2 . . . . g44dx42,
where the magnitudes g11, etc., have values which vary with the position in the continuum. Only
when the continuum is a Euclidean one is it possible to associate the co-ordinates x1 . . x4. with the
points of the continuum so that we have simply
ds2 = dx12 + dx22 + dx32 + dx42.
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 ds2 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 temperature. The temperature is practically constant for a small part of the
slab, and thus the geometrical behaviour 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 neighbouring 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
generalisation of the Cartesian co-ordinate system. It is also applicable to non-Euclidean continua,
but only when, 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.
Next: The Space-Time Continuum of the Speical Theory of Relativity Considered as a Euclidean
Continuum
Relativity: The Special and General Theory
54
Relativity: The Special and General Theory
Albert Einstein: Relativity
Part II: The General Theory of Relativity
The Space-Time Continuum of the Speical 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 17. In accordance with the special theory of relativity, certain co-ordinate
systems are given preference for the description of the four-dimensional, space-time continuum.
We called these " Galileian co-ordinate systems." For these systems, the four co-ordinates 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 Galileian 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 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 Galileian
systems of reference.
Minkowski found that the Lorentz transformations satisfy the following simple conditions. Let us
consider two neighbouring events, the relative position of which in the four-dimensional continuum
is given with respect to a Galileian reference-body K by the space co-ordinate differences dx, dy,
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 dx1, dy1, dz1, dt1. Then these magnitudes
always fulfil the condition 1)
dx2 + dy2 + dz2 - c2dt2 = dx1 2 + dy1 2 + dz1 2 - c2dt1 2.
The validity of the Lorentz transformation follows from this condition. We can express this as
follows: The magnitude
ds2 = dx2 + dy2 + dz2 - c2dt2,
which belongs to two adjacent points of the four-dimensional space-time continuum, has the same
value for all selected (Galileian) reference-bodies. If we replace x, y, z, , by x1, x2, x3, x4, we
also obtaill the result that
ds2 = dx12 + dx22 + dx32 + dx42.
is independent of the choice of the body of reference. We call the magnitude ds the " distance "
apart of the two events or four-dimensional points.
Thus, if we choose as time-variable the imaginary variable instead of the real quantity t, we
can regard the space-time contintium accordance with the special theory of relativity as a ",
Euclidean " four-dimensional continuum, a result which follows from the considerations of the
preceding section.
Next: The Space-Time Continuum of the General Theory of Realtiivty is Not a Eculidean Continuum
55
Relativity: The Special and General Theory
Footnotes
1)
Cf. Appendixes I and 2. The relations which are derived there for the co-ordlnates themselves
[ Pobierz całość w formacie PDF ]