426 DOC.
420 DECEMBER
1917
principle
in
gravitation
theory.[2]
The
reason
I did
not
do
so
long ago
is
that
I
have been
busily making
calculations
using
the
theory,
in search of
something
new.
Some
little
things
have
come
out of
it, yet
it
seems
that
I
am
not
making any
more
progress
for
the
time
being
and would
like to
take
the
liberty
of
reporting
to
you
about
my
efforts.
Although
it
seems
probable to
me
that
you
have
long
been
acquainted
with these
sidetracks, among
the
lot connected
to
your theory,
and
have
come
farther
along
them
than
I,
it could nevertheless be
possible,
and
it would
please
me greatly,
if
you
were
to benefit from
one or
the other
in
some
way.
First
of
all,
I
would
like
to
correct
an error
in
my
earlier letter.[3]
I gave
an
example
there
of
a
space
of
nonvanishing
curvature in which
the
energy
tensor of
matter
was supposed
to vanish.
Subsequently
I
noticed
nonvanishing
components
in
a
hidden
corner
of
the
tensor, though.
1)
My
main efforts since
receipt
of
your
letter
have been
devoted toward
unify-
ing
the
electromagnetic
field
with
the
gravitational field.[4] I
have
not
yet
found
a
solution.
I
performed larger
calculations in
one
vein
by proceeding
from
an
asym-
metric fundamental
tensor
guv.
I would
like to
present
the
following approaches:
a)
Reduction into
a
symmetric
and
an
antisymmetric
part
guv
= suv
+
auv.
Deter-
minant:
g, or
subdeterminant
s,
divided
by
g
(or s)
itself:
guv, or
suv.
Yguagva
=
Evu
=
0
or
1,
likewise
gaugav
=
Evu,
by
contrast
gaugva
=
avu(
#
1);
guagav
=
ivu;
hence
guv
=
aavgau
=
iaugva; guv
=
aaugva
=
ivagau;
iva
.
aau
=
iau
.
avu
=
eva.
dguv
=
-gavgußdgaß;
dguv =
-gavgußdgaß;
logg/dt
=
guvdguv/dt
=
-guv
(:t
one
parameter:).
b)
If
Av is
a
vector,
then
ft»
=
+ A
'
~dx^
is a
tensor,
likewise
dAv
dA^ d(fu
dxa dxa dxa
Others
in
addition
to
this
through
multiplication by
guo, or
gov,
etc.
c)
From
the
64
derivatives of
guv
the
40
second derivatives of
the
xo,s
can
plainly
be
calculated
if 24
relations exist between
the
guv’s
and
g'u'v's.
These
relations
imply
that the
following
system
of quantities is
a
tensor: