428
DOC.
420 DECEMBER
1917
fractions
signify
second differential
quotients.
If
in
the last
term
saß
.(...),
the
[]
are
replaced by
[
]s,
Buv,ar
changes by
-1/4a/3
•
[favoifTpp
-
fTvafauß).
The
term
resulting
after subtraction
of this
tensor
differs from
the
fully symmetric
one
in
the
terms:
1
dßT
^
ducr
dur
"•fia
2
V7
ßT VT
One
can see
that
although
the fundamental
tensor is
asymmetric,
Buv,ar
becomes
symmetric
if the
4
equations
are
satisfied:
(2)
fuva
=
0.
I
would
like
to treat
equations
(1)
and
(2) as
Maxwellian,
but there
is
a
hitch in
the
energy compo-
nents.
e)
Out
of
the
expansion
of
the fundamental
tensor,
three
scalars
(for
Hamilton
principles,
or
the
like)
can
be formed:
sp°
•
s"7
•
suß
or
Aßv,
a
'
Aaß,
-y
• . .
svß
.
s°7
or
.
spa s"7
•
saß
.
2)
Gravitational
fields
derived from
a
“potential”
a)
Assumption:
C
.
guv =
gua
+
gaudAa/dxu
+
Aadguv/dxa.
If
special
coordinates
are
introduced,
so
that
A1
=
A2
=
A3
=
0;
A4
=
1,
then
results C
.
guv
=
guv/dx4;
guv
=
eCx(4)
.
huv(x1,
x2, x3).
For C
=
0,
the
guv’s
would thus
depend
on
only
three variables. Such
an
approach appeals
to
me
very much,
because
obviously,
in
reality
also, only
the
16
•
oo3
initial
values
can
be
given
for
the
guv’s,
out
of
which
the
subsequent
ones
develop by
natural
processes.
The
integral
curves
of
the
equations
dx1
:
dx2
:
dx3
:
dx4
=
A1
:
A2
:
A3
:
A4
must
mean something
like
the individual
local time.
(World
lines? Or
are
they
connected
to
a
kind of four
potential?)
At
any
rate,
my conceptions
of
this
are
still
very vague.
b)
Assumption.
A
vector
Ap
exists such
that
dAp/ax
-Ar.sat
[pv]
=
0.
Through
a
special
choice
of
coordinates
Ap
=
sp4
can
be obtained.
Then
re-
sults
(1)
dapv/d4
=
0;
(2)
dsp4/dv
=
dsv4/dp,
Ap
=
gradp;
rotUp
=
0;
Ap
thus
cannot
be
the four
potential.
3)
A
formal series for which
I
have not
yet
found
any proper
meaning: