DOC. 9 FORMAL FOUNDATION OF RELATIVITY
33
substitution that is
applied.
In the
physical
sense,
the
guv
determine the
gravitational
field which exists relative to the
new
coordinate
system,
as
is also
seen
from the
discussion in the
previous paragraph. Equations
(1)
and
(2a)
determine the motion
of
a
material
point
in
a
gravitational
field which
can
be made to vanish with
a
suitable
choice
of
the reference
system.
However,
we
want to
assume
in
a generalizing
manner
that the movements of the
material
point
in
a gravitational
field
are
determined
by
these
equations.
The
quantities
guv
have
a
further,
second
significance.
Because
we
can always
set
fc2
=
E
Smv
=
~Y^dxZ
(2b)
/IV
where, however,
the
dXv
are
not
complete
differentials. But in the
infinitesimally
small,
the
quantities
dXv
can
still be used
as
coordinates.
Therefore,
it
is
plausible
to
assume
that the
theory
of
relativity
is valid in the
infinitesimally
small. The
dXv
are
then the
directly
measurable coordinates in
an infinitesimally
small
domain,
and
they
are
measured with
unit-measuring
rods and
a
suitably
chosen unit-clock. In this
sense,
the
quantity
ds2
can
be
called
the
naturally
measured distance between
two
space-time points.
In
contrast,
the
dxv
cannot
be
obtained
in
the
same manner by
measurements with clocks and
rigid
bodies.
They
are
rather connected
with
the
naturally
measured distance ds via
(2b)
in
a manner
that is determined
by
the
guv.
After what has been
said,
ds
can
be defined
independent
of
the choice
of
the
[p.
1034]
coordinate
system;
it is thus
a
scalar. In the
general theory
of
relativity,
ds takes the
same
role the element
of
a
worldline had in the
original theory
of
relativity.
In
the
following,
we
want to derive the most
important
theorems
of
absolute
differential
calculus,
since
they
take in
our theory
the
role of the theorems
of
ordinary
vector
and
tensor
theory
with its three- and four-dimensional
vector
calculus
(with
respect
to
the Euclidean element
ds).
The laws of
general relativity theory
which
correspond
to
known laws of the
original theory
of
relativity
can
easily
be derived
with the
help
of those
theorems.
B.
From the
Theory
of Covariants
§3.
Four-Vectors
The
covariant
four-vector.
Four functions Av
of
coordinates that
are
defined for
any
coordinate
system
are
called
a
covariant four-vector
or a
covariant
tensor
of rank
one
if for
any
arbitrarily
chosen line element with the
components
dxv
the
sum
=
*
(3)
V