350
DOC. 467
AUGUST
1913
seems
to be
the
skeleton,
so
to
speak,
on
which
everything hangs. But,
unfortunately,
the
gravitation equations
themselves do
not
possess
the
property
of
general
covariance.
Only
their covariance
with
respect
to linear
transformations
is
certain. But
all
of
our
confidence in
the
theory rests
on
the
conviction
that
an
acceleration of the reference
system
is
equivalent
to
a
gravitational
field.
Hence,
if not all
of
the
equation
systems
of
the
theory,
and thus
also
equations
(18),[13]
permit
other than linear
transformations,
then
the
theory
refutes
its
own
starting
point;
then
it has
no
foundation whatsoever.
Thus
far, however, we
have not
been
able to
specify any
nonlinear substitutions
with
respect to
which
equations
(18)
would be covariant.
There
are
two
possibilities
of
a
fundamentally
different
kind:
(1)
Transformations that
are
independent
of
the
guv-field
present
and that Ehrenfest
designated
as
"independent transformations"; to
the best of
my
knowledge,
group theory
has
so
far
been concerned
only
with such
transformations.
(2)
Transformations,
the
p
of
which would first have
to
be determined
by
differential
equations
for the
guv-field
under
consideration,[14]
that
is
to
say,
which
must
be
adapted
to
the
gßV-field
present.
As
far
as
I
know,
such
transformations
have
not
yet
been
investigated
in
a systematic
fashion.
("non-independent transformations")
The
existence
of
"independent"
nonlinear transformations
is
the
simpler
possibility;
but
this does
not
seem
to
me
to obtain,
even
though I
would
not
know how
to
prove
it.
But
the
existence
of
"non-independent"
nonlinear
transformations
already
suffices for
there
not
to
occur a
conflict with
the
equivalence
hypothesis.
In
principle,
the
matter is
simple.
One
asks:
What conditions
must
the
pik
of
a
transformation
satisfy
so
that,
under the
transformation,
rMv
= Auv(T)
- kUuv
transforms
as a
tensor?[15]
In this
way
one
obtains
partial
differential
equations
for the
pik.
The
question
is,
do
the latter
have solutions that
are
compatible
with
the
integrability
conditions?- But when
I
try
to
carry
out
the
calculation,
I
fail
because the
equations
are
very
complicated.- Should it
turn out
that nonlinear transformations
do
not
exist
at all,
then
the
theory
would lose all
credibility.
On
the other
hand,
it
is
very
interesting
that the
equations
yield
the
relativity
of
inertial
mass.
That
is to
say,
the
following things
emerge:[16]
1.
The
existence
of
an
inertial
spherical
shell at
rest
increases the inertia of
a mass m
that
it encloses.
2.
An acceleration of
K
induces
an
accelerating
force
acting on
m
in
the
same
direction.
3.
If K
rotates,
then
a
Coriolis field arises inside
K,
such
that
a
pendulum
set
up
inside
K
is
influenced
in such
a way
that
its
plane
of
oscillation
is
carried
along.