298 DOC.
28
NORDSTRÖM'S THEORY OF GRAVITATION
ds2
=
&\Jdx2
+
dy2
+
dz2,
i.e., a measuring
rod of natural
length
ds
has the coordinate
length
ds/q.
For
a
timelike
connecting vector,
the
spatial components
vanish
upon
an
appropriate
choice
of
the reference
system,
and
one
obtains
ds
=
)J-dx42,
or
ds/i =E
dx4.
ds/i
is
nothing
other than the
temporal
duration measured with
a
clock
with
a
specific
constitution.
Thus,
ds/Qi
is the time difference in the coordinate
measure.
Hence,
1/O
is the factor
by
which the
naturally
measured times and
lengths
must
be
multiplied
in
order
to
obtain coordinate times
or
coordinate
lengths.
It follows from the form of the line element
ds2
= t2(dx2
+
dy2
+
dz2
-
c2dt2)
that the
equations
of Nordström's
theory
are
covariant
not
only
with
respect
to
the
Lorentz
transformations,
but also with
respect
to
similarity
transformations.
The
momentum
and
energy equations (3)
for
matter
take the form
(3a)
Y
^-v=
o*v dxa
It is
noteworthy that,
according
to
this
equation, only
the scalar
(1//-g)EEtt
T
determines the
influence
of
the
gravitational
field
on
a
system.
This conforms with
the
argument we
have
given
in connection with the derivation
of
equation
(5a).
The differential
equation
of
the
gravitational
field
(5a)
takes the form
(5b)
1 a2&
l)3
dxi
•
a2$
i
y#
a2o
0JC22
dx$ 3*4 f4
(where
k denotes
a new
constant), or
£ &
=
$"•
T
Since the relation between the
natural and the coordinate
lengths
at
one
location
can
be chosen
arbitrarily,
one can
choose the
constant
k
arbitrarily.
One
can,
for
example,
follow Nordström's
procedure
and
set
k
=
1.
We
see
that the derived
equations agree completely
with those
given by
Nordström.
§4.
Concluding
Remarks
We
were
able
to
show in the
foregoing
that if
one
bases oneself
on
the
principle
of
the
constancy
of
the
velocity
of
light, one can
arrive at
Nordström's
theory
by purely