7 8 D O C . 7 5 N O B E L L E C T U R E
Implementing this idea requires an even more profound modification of the geo-
metrical-kinematical foundations than the special theory of relativity. The Lorentz
contraction, which is derived from the latter theory, leads to the following result.
Take an inertial frame K free of a gravitational field and a frame K′ which is in ar-
bitrary motion relative to K. Then the laws of Euclidean geometry governing the
possible positions of rigid bodies (at rest relative to K′) do not apply with respect
to K′. Consequently the Cartesian coordinate system also loses its meaning with re-
spect to the reality postulate. Analogous reasoning applies to time; with respect to
K′, time cannot be meaningfully defined by the readings of identically constructed
clocks at rest relative to K′ or by the law governing the propagation of light. Gen-
eralizing, one arrives at the result that the gravitational field and the metric are just
different kinds of manifestation of the same physical field.
We obtain the formal description of this field by the following consideration. For
the infinitesimal neighborhood of every point in an arbitrary gravitational field a
local coordinate frame can be given for a state of motion such that relative to this
local frame no gravitational field exists (local inertial frame). With respect to this
inertial frame we may view the results of the special theory of relativity as accurate
in a first approximation for this infinitesimally small region. There are an infinite
number of such local inertial frames at every point in space-time; they are linked
by Lorentz transformations. These are characterized in that they leave invariant the
“distance” ds between two infinitely adjacent point-events—defined by the equa-
tion
Such a distance can be measured by means of measuring rods and clocks. For x, y,
z and t denote coordinates and time, measured with reference to a local inertial
frame.
For the description of space-time regions of finite extension, arbitrary point-co-
ordinates in the four-dimensional manifold are needed which serve no other pur-
pose than to provide an unambiguous designation of the space-time points by 4
numbers each, and . These account for the continuity of this 4-di-
mensional manifold (Gaussian coordinates). Then the mathematical expression of
the general relativity principle is that the systems of equations expressing the gen-
eral laws of Nature are the same for all such coordinate systems.
Since the coordinate differentials of the local inertial frame are expressed linear-
ly by the differentials of a Gaussian system of coordinates, upon applying the
latter to the distance ds between two events one obtains an expression of the form
The ’s, which are continuous functions of the ’s, determine the metric in the
four-dimensional manifold, in that ds is defined as an (absolute) quantity measur-
[p. 6]
ds2 c2dt2 dx2 – dy2 – dz2 – =
x1, x2, x3, x4
dxν
ds2 ¦gμνdxμdxν
= gμν gνμ) = (
gμν xν