78
DOC.
66
MARCH
1915
First of
all,
once
again
to
the
objection
relating
to
the
special
case
that with
an
appropriate choice of coordinates
the
guv's
are
constant.[3]
You state
that
in
this
case
the
Euv’s
do not vanish
if
the
adapted
system
is selected
in
such
a
way
that the
guv's
are
not constant. But
you
did
not support
this
statement,
and
I
am
considering
it
incorrect
as
long
as
you
have not
provided
an
example
or a
general proof
for this.
Now
to
the
second
objection. Although you
concede
that
T
_(tßu A^
+
£
=
invariant,
(A^
=
ƒ
SgV'y/^dr),
where
e
is
an
infinitesimal amount of
higher
order,
you
contest,
however,
that
from
this
it
can
be concluded
that
Euv/-g
is
a
tensor.
Well,
I must
admit
that
the
basis of this
objection
is not clear
to
me.
For,
since
I
have
proven
that the
Auv's
transform
contravariantly,[4]
in
my opinion
it
is quite
irrelevant to
the
proof
that
these
Auv’s
are
infinitesimal
quantities.
Nevertheless,
an
addition
must
be made
to
the
proof
that instead
of
an
infinitesimal
tensor
Auv,
a
finite
one
is
introduced
through
the
formation of
a
limit.
I
shall not
go
into
this,
however,
but
am
convinced
that
you
will
acknowledge
the
following
derivation
as
sound:
Up
to
a
relatively
oo
small
quantity,
E
• • •
(1)
for
any justified
substitution,
as
you yourself
concede.
Up
to
a
relatively
oo
small quantity-as
you
likewise
concede-and
as
I
have
proven
in
my
last
letter,[5]
dx1 dx1Ud/T
j^l(JT
_
UJy(T
J^LV ^
dxn
dxv
(2)
From
(1)
and
(2)
it
follows,
however,
that
up
to
a
relatively
oo
small
quantity,
the
equation
V""'
&CTT
dxa
dxT
V7?
dx"
dxu
thus
also
applies.
Since
the
Auv’s
can
be chosen
independently
of
one
another,
hence
_
y dx'a
dx'T
è'aT
V-9
y
dxn
dxv