DOC. 30 FOUNDATION OF GENERAL
RELATIVITY
167
(the "complements"
of
covariant and contravariant tensors
respectively),
and
Buv
=
guvgabAaß.
We
call
Buv
the
reduced
tensor
associated
with
Auv.
Similarly,
Buv
=
guvgaßA.
It
may
be
noted that
guv
is
nothing more
than the
comple-
ment
of
guv,
since
g^g^gaß
=
g^B"a
=
gw.
§
9.
The
Equation of
the Geodetic Line. The Motion
of
a
Particle
As
the linear element ds is
defined
independently
of
the
system
of co-ordinates,
the
line
drawn between two
points
P
and
P'
of
the four-dimensional continuum in
such
a
way
that
Jds
is stationary-a
geodetic
line-has
a
meaning
which
also
is
independent
of the choice of co-ordinates.
Its
equation
is
8
ds
=
0
....
(20)
p
Carrying
out the variation
in
the
usual
way, we
obtain
from
this
equation
four differential
equations
which
define
the
geodetic
line;
this
operation
will be
inserted here
for
the
sake
of
completeness.
Let
A
be
a
function
of
the
co-ordinates
xv,
and let this
define
a
family
of surfaces
which intersect the
required
geodetic
line
as
well
as
all
the
lines
in immediate
proximity
to
it which
are
drawn
through
the
points
P
and P'.
Any
such
line
may
then
be
supposed
to be
given
by
expres-
sing
its co-ordinates
xv
as
functions
of
A.
Let the
symbol S
indicate the transition
from
a
point
of
the
required
geodetic
to the
point corresponding
to
the
same A on a
neighbouring
line.
Then
for
(20) we
may
substitute
fAaSweZX
=
0
JAj
n
dXn
d/Xy
I
...(20a)
But
since