DOC.
185
JANUARY
1916 185
Now I
write the
conservation law of matter
by
introducing
it
formally
without
assuming
its
validity[9]
[[Au’s
are]
unknown
space functions]
+'
-aaTT
=
V--
2¿V
£q
(3)
The
second
term
on
the
left-hand
side
can
be
converted
because of
(ß)
into
the
form
-1/2idgaT/aTaT.
The
{}
can
likewise
be
introduced,
which
I
do not need
here,
however.
With the
aid of
this,
the
right-hand
side
of
the last
equation can
be
substituted
by
dTf
-
2kAß,
oxa
and
you
obtain:
-
Í-
LÖß
-
9ß "aT
8L
'
+
t;
= Aß
...
dxa
\2k
dglT.
ta
(4)
If
the
Aß's were
to
vanish,
this would be
the
conservation
equation
both
for
matter
and
for
gravitation.
For
the
moment
we
shall work with
(4). Taking
into
account
(1)[10]
it
follows
from
the
definition of
the
taB's
í;
=
í
=
1/L...
(5)
3)
Mixed
form
of
the
gravitation
equations.
We
write the
latter
in
the
form
d{
}
Í
aß
1
Í ra
dxa
\
a
I
\
0
(2a)
and
multiply by
gTv.
The first term
on
the
left
yields
through partial diff.
refor-
mulation
d
ar rr I
dgTV
dxa a
+
a
ƒ
dxa
’
of
which
the
second term
can
be modified with the aid
of
the formula
(r),
deduced
here to
the
side,[11]
so
that
you
obtain
-d/dxa(gTv{oTa})-gTE{Eav}{Taa}-gvE{EaT}{oTa}.