32
DOC.
9
FORMAL FOUNDATION OF RELATIVITY
field
gains justification by
all
means.
This
interpretation
is
reminiscent of the
one
in
the
original
(more special) theory
of
relativity
where the
ponderomotively acting
force,
upon
an
electrically charged mass
which
moves
in
a
magnetic
field,
is the
action
of
that electric field which is found at the location of the
mass
as
seen by
a
reference
system
at rest
with the
moving
mass.
[5]
From what has been
said,
one already sees
that
gravitation
must
play
a
fundamental
role in
any theory
of
relativity
that is extended
along
the lines
we
have
indicated above.
After
all,
if
one
changes by a
mere
transformation from
one system
of
reference
K
to another
one
of
K',
then there
exists
a gravitational
field relative
to
K'
which relative to
K
need not
be
there at all.
There arises
the
natural
question:
which
systems
of
reference
and which
transformations should be considered
as
"admissible" in
a
generalized theory
of
relativity?
This
question
can
only
be answered much later
(section D).
For the time
being
we
take the
position
to
admit all coordinate
systems
and
transformations that
are compatible
with the conditions
of
continuity, as
is
always
demanded for theories
in
physics.
It will
turn
out that the
theory
of
relativity
allows for
a
rather
far-reaching
generalization
that is free of almost
any
arbitrariness.
§2.
The
Gravitational Field
According
to the
original
theory
of
relativity,
a
material
point
that is free of
gravitational
and other forces
moves
in
a
straight
line and
uniformly according
to
the
formula
8{fds) =
0
(1)
[p.
1033]
where
we
set
=
-
£
dxl
(2)
V
[6]
We also
set
x1 =
x,
x2
=
y,
x3 =
z,
x4
=
it.
In
this,
ds is the
"eigen-time"-
differential, i.e.,
this
quantity gives
the
amount
by
how much the clock-time
of
a
clock
(which
is associated with
the
moving
material
point) progresses along
the
path-
element
(dx,
dy,
dz).
The
variation in
(1)
has
to be
formed such that the
coordinates
xv
at
the end
points
of
integration
remain unvaried.
Equation
(1)
remains valid after
any
coordinate
transformation,
while
(2)
is
replaced by
the
more general
form
=
(2a)
MV
The ten
quantities
guv are
now
functions
of
those
xv
which
are
determined
by
the