296 DOC.
28
NORDSTRÖM'S THEORY OF GRAVITATION
must
go over
into the
equation
dx2
+
dy2
+
dz2
-c2dt2
=
0.
From
this it follows that with such
a
choice of the reference
system,
we
must
have
£
g^v
=
+
®2dx£
+
E2dx32
- 42dx42,
where
we
have
now
set
x1 =
x,
x2 =
y,
x3 =
z
and
x4 =
ct.
Thus,
the
system
of the
guv
degenerates
into
Z*
0 0
0
0
02
0
0
0 0
02
0
0
0 0
-02
In order
to determine the
one
quantity Q2,
we
need
a
single
differential
equation
that will have
a
scalar character like
Poisson's
equation.
Just
as we
have done with
the
previous equations,
we
will
set
up
this
equation, too,
in
a
generally
covariant
form, i.e.,
without
at
first
carrying
out
the
specialization
of the reference
system
that
is
suggested by
the
principle
of the
constancy
of
the
velocity
of
light.
The
equation
being sought
is
completely
determined
by
the
assumption
that
it
is
of
the second
order if
one
also
takes into
account
the fact that
it
must be
a
generalization
of
Poisson's
equation.
Obviously,
it
will
have the form
(5)
r
=
k£,
where T is
a
scalar formed from the
quantities guv
and their first and second
derivatives,
and
2
is
a
scalar determined
by
the material
process,
that
is, according
to
what has been said
before, by
the
2ov.
K
denotes
a
constant.
Mathematical
investigations
of the differential
tensors
of
a
multidimensional
manifold show that the
only expression
to
be considered for T is
a
function
of
£
Y^Y« (»'*,/«)•
iklm
Here
(ik, lm)
denotes the familiar fourth-rank Riemann-Christoffel
tensor
which is
associated with the
measure
of
curvature
in the
theory
of surfaces and is defined
by
the
equation
[11]
[12]
«'
*»
-
*
u-B
+
d29
ki
d'-gt
il
d2g
m
k
xkoxi
'
dxidx
dxkdxm
d
Xi
d
x,
m
im
kl il
k
m
ao
.
Q
a o
(T
where
im
denotes
dg
m p
dgim
P
dX-
dx
P
V