DOC. 9 FORMAL FOUNDATION OF RELATIVITY 45
is used
as an
abbreviation. From
this,
it follows that
Ko
=
0
(23)
[15]
is the
equation
of the
geodesic
line.
In the
original
theory
of
relativity,
those
geodesic
lines for which ds2
0
correspond
to
the motion of material
points,
those where ds
=
0
represent light rays.
[p.
1046]
This will also be the
case
in
a
generalized theory
of
relativity. Excluding
the latter
case
(ds
=
0)
from
our
consideration,
we can
choose the
"length
of arc"
s
along
a
geodesic
line
as our parameter A.
The
equation
of
a geodesic
line then transforms
into
y!
Saß
d2
ds
E
ßV
MV
a
^
=
0
ds ds
(23a)
where
we introduced,
after Christoffel, the abbreviation
/xv
1
o
2
dg,«,
dx..
dgyq
dx
dgßV
ß
dx
(24)
This
expression
is
symmetric
in the indices
u
and
v.
Finally,
(23a)
is
multiplied by
got
and summed
over
0.
Considering
(10)
and
using
the well-known
Christoffel
symbols
/xv
=
5
or
/xv
a
(24a)
one
gets
in
place
of
(23a)
d2x,
ds2
E
ßV
fxvldx^
d^
r
I
ds ds
(23b)
This is the
equation
of
the
geodesic
line in
its
most
comprehensive
form.
It
expresses
the second derivatives of the
xv
with
respect
to
s
by means
of
the first
derivatives.
Differentiating
(23b)
with
respect
to
s
would
yield equations
that would
also allow to reduce
higher
differential
quotients
of
coordinates with
respect
to
s
to
their first derivatives.
In this
manner one
would obtain the coordinates in
a
taylor
expansion
of
the variable
s. Equation (23b)
is
equivalent
to
the
equation
of
motion
of
a
material
point
in its
Minkowski
form where
s
denotes the
"eigen-time."
[16]
[17]
§8.
Forming
Tensors
by
Differentiation
The fundamental
significance
of
the
concept
of
a
tensor
rests,
as
is
well
known, upon