DOC. 28 NORDSTRÖM'S THEORY OF GRAVITATION 297
Further,
it is evident from the
general theory
of
covariants that the
only
scalar
belonging
to
the
2ov
is
1/v/-gET 2TT
(or
a
function of this
quantity).
From
this it follows that the
equation
we
seek must have the form
(5a)
£
Y,mYki(jk,
bn)
=
$tt.
iklm \g
T
To be
sure,
it is assumed here that the second derivatives of the
guv
and the
2ov
enter
linearly
the
equation
we are seeking.
[13]
Equation (5a),
which
we
have
just
set
up,
and
equations
(3)
contain Nordström's
theory
of
gravitation
with
respect
to
arbitrary space-time
coordinates in its
entirety
if
one
attaches
the conditions which
the
guv
must fulfill
in
order for the
principle
of
the
constancy
of
the
velocity
of
light
to
be satisfied for
an
appropriately
chosen
reference
system.
§3.
The
Fundamental
Equations
of
Nordström's
Theory
with
Respect
to
Reference
Systems
That
Are
Adapted
to
the
Principle
of the
Constancy
of the
Velocity
of
Light
Let
us now
consider those reference
systems
with
respect
to which the
principle
of
the
constancy
of
the
velocity
of
light
is
satisfied
as privileged systems.
The
components guv
of
the fundamental
tensor
are
given, then, by
the values written
down in
(4).
The
corresponding
guv are
to
be found in the table
[14]
+-
0 0 0
J2
0
+-
0 0
(4a)
4,2
0 0
+-
0
0 0 0
-
-
J)2
In this
case
one
obtains
ds2
=
Oy'dx21
+
dx22
+
dx23
-
dx24.
As
already
mentioned,
ds
is
the
"naturally
measured" interval between
two
space-time points.
Now
one
can
distinguish
between the
cases
where the
connecting
vector
is
spacelike
or
timelike. In the first
case,
the
vector
can
be made into
a
purely spatial
one
by an
appropriate
choice
of
the reference
system;
for the connection between the
lengths
measured
"naturally"
and those measured in coordinate
measure,
one
then obtains