44 DOC.
9
FORMAL FOUNDATION OF RELATIVITY
§7.
The
Geodesic
Line and
Equations
of
Motions
of
Points
It has
already
been
explained
in
§2
that the
movement
of
a
material
point
in
a
gravitational
field is
governed by
O{Jds}
=
0.
(1)
From
a
mathematical
point
of
view,
the movement of
a
point, therefore, corresponds
[14]
to
a
geodesic
line in
a
four-dimensional manifold. We insert here the well-known
[p. 1045]
derivation
of
the
explicit equations
of
this line for the sake
of
completeness.
We
are dealing
here with
a
line between two
points
P(1)
and
P(2)
relative to
which all lines
infinitely
close
and
joining
both
points satisfy equation (1).
If
X
denotes
a
function
of
the coordinates
xv,
then the "surface" of constant
X
will cut
just one
point
from each
one
of
these
infinitely adjacent lines;
and the coordinates
of
this
point
can
be looked
at
as a
function of
X
alone,
provided
the
curve
is
given.
If
we put
w
2
_ =
E
8^'dxHx'
/XV
dxß
dxv
we
can
rewrite
(1)
in the form
JSwdX
=
0, (1a)
because the limits
of
integration
X1
and
X2
are
the
same
for all
curves
under
consideration.
If
Sxv
denotes the increases that have
to
be
given
to
the
xv
in order
to
go
from
a point
of the
desired
geodesic
line to
a point
of
the
same X on one
of
the
varied
lines,
then
one gets
8w
=
1
w
1
dgMV
dxß
dxv
+
2^
dxa
dX dX
°
^8
"
/XV
dx
^
~dX
dx.v
/
Substituting
this
expression
into
(1a), partially integrating
the last
term
and
considering
that the
8xv
vanish for
A
=
X1
and
X
=
X2
one
obtains
Ao
/
£
(Ka 8xa)
=
0,
1
where
Ka
=
E
/XV
1
dgflV
dx dx
/X
V
2w
dxa
dX dX