106
DOC.
3
STATICS OF
GRAVITATIONAL
FIELD
v
dx
+
dt
dt'
=
N
1
-
hold. dx' and
dt'
must
be total differentials. The
following equations
should
therefore
hold
1
Ö
dx,
-v
N
i
-
5
*
V
a
C2
a
l
&
1-
^
dx.
l-
v2
c2
N
c2
Let the
gravitational
field
in
the
unprimed system
be static. In that
case c
is
an
arbitrarily given
function of
x,
but
is
independent
of
t.
If
the
primed system
is
to
be
in "uniform"
motion,
then
v
must
be
independent
of
t
for fixed
x.
The left-hand sides
of the
equations,
and hence also the
right-hand sides,
must
therefore vanish. But the
latter is
impossible,
because
if
c
is
an
arbitrary
function of
x,
then the
two
right-hand
sides
cannot
both be
made to vanish
by choosing
v
appropriately
as a
function of
x.
This
proves, therefore,
that the Lorentz transformation
also
cannot
be
considered
valid
for
small
spacetime regions
as soon as one
gives up
the universal
constancy
of
c.
It
seems
to
me
that the
spacetime problem
is
as
follows. If
one
limits oneself
to
a
region
of
constant
gravitational potential,
then the laws of
nature
take
on an
extremely simple
and invariant form if
one
refers them
to
one
of the manifold
spacetime systems
that
are
connected with
one
another
by
Lorentz transformations
with
a
constant
c.
If
one
does
not
limit oneself
to
regions
of
constant
c,
then the
manifold
of
equivalent systems
as
well
as
the manifold of transformations that leave
the laws
of
nature
unchanged
will become
a
larger
one,
but for these the laws will
become
more
complicated.
Prague, February
1912.
(Received
on
26
February 1912)