DOC.
71
PRINCETON LECTURES 321
THE GENERAL
THEORY
force of
Coriolis).
We therefore arrive
at
the result:
the
gravitational
field influences
and
even
determines the
metrical
laws of
the
space-time
continuum. If the
laws
of
configuration
of
ideal
rigid
bodies
are
to
be
expressed
geometrically,
then in the
presence
of
a
gravitational
field
the
geometry is not
Euclidean.
The
case
that
we
have been
considering is analogous
to
that which
is
presented
in
the two-dimensional
treat-
ment
of surfaces.
It
is
impossible
in
the latter
case
also,
to
introduce co-ordinates
on a
surface
(e.g.
the surface
of
an ellipsoid)
which have
a
simple
metrical
significance,
while
on a
plane
the Cartesian
co-ordinates,
x1, x2,
signify
directly lengths
measured
by a
unit
measuring
rod.
Gauss
overcame
this
difficulty,
in his
theory
of
surfaces,
by
intro-
ducing
curvilinear co-ordinates
which, apart
from
satisfying
conditions of
continuity,
were
wholly arbitrary,
and
only
afterwards these
co-ordinates
were
related
to
the
metrical
properties
of
the
surface.
In
an
analogous way
we
shall
introduce
in
the
general
theory
of
relativity arbitrary
co-ordinates,
x1, x2, x3, x4,
which
shall
number
uniquely
the
space-time points,
so
that
neighbouring events
are
associ-
ated with
neighbouring
values of
the
co-ordinates;
other-
wise,
the
choice of
co-ordinates
is
arbitrary.
We
shall be
true to
the
principle
of
relativity
in its
broadest
sense
if
we [82]
give
such
a
form
to
the
laws
that
they
are
valid in
every
such
four-dimensional
system
of co-ordinates,
that
is,
if
the
equations
expressing
the laws
are
co-variant with
respect to arbitrary
transformations.
The
most
important point
of
contact
between
Gauss’s
theory
of
surfaces
and the
general theory
of
relativity
lies in
the metrical
properties upon
which the
concepts
of
both
theories,
in
the
main,
are
based.
In the
case
of
the
theory
of
surfaces,
Gauss’s argument
is
as
follows.
[61]