DOCS.
66,
67 MARCH 1915
79
applies,
i.e.,
g
has
a
tensor
character.
It
is
thus
quite superfluous
to
introduce
some
limit
to
replace
the
Auv’s;
the
derivation
is independent
of it.
Any
consideration in which
the
ôguv’s
are
sub
jected to
more
constraints than
is
necessary
for
the
nature of
the
problem
must
be
rejected
as an
unnecessary complication.
With
cordial
greetings, yours,
A.
Einstein.
67. From
Tullio LeviCivita
[Padova,]
28
March
1915
Dearest
Colleague,
Yesterday
I
received
your
postcard of
the
20th,[1]
and
it
is
with
much
pleasure
that
I
reply
to
it
immediately,
as
you requested.
In
your view, my
observation
[that
there
are
adapted
(angepasste)
coordinate
systems
for which tensor
Euv
does not
vanish,
while
it
is
zero
when
the
guv's
are constant][2]
is not
conclusive,
because
a generic
gravitational field
cannot be
obtained with the
help
of
a
coordinate
transformation
starting
from
a
Euclidean
ds2
(guv
constant).
This is indeed
the
case.
Therefore it
is
necessary
to
give a
concrete
example
in which
not
all
Euv's
vanish
as a
result
of
some
admissible
transformation
from
a
Euclidean
ds2
(contrary
to what covariance would
require).
This
is
how
we
go
about
it.
We
start from
a
coordinate
system
in which
ds2
has the canonical Euclidean form
ds2
=
dx21
+
dx22
+
dx23
+
dx24
(guv
=
6uv),
and
we
proceed to perform
an
infinitesimal transformation,
putting
x'u
=
xu +
yu, (1)
where
the
yu's
designate
(a priori
any)
infinitesimal functions
of
x.
Putting
^uv
+
huv
(2)
for
the
guv's
relative
to
the
new
variables
x', we
have
hßn

dVn
dyv\
dxu
dXf,,)
(3)