474
DOC.
463
FEBRUARY
1918
only one
unknown each of
the 4th
order.). Unfortunately,
they
then
also
contain
some
nasty
terms of
the
2nd
order
that
are
quadratic
to
the other
unknowns.
On
this
occasion,
I also discovered
some
differential relations between
the
Buv's, or
Bao,
or
BaB,ro.
Those between
the
Bao’s
lead to
your energy-momentum
law
and
read
(without
the
constraining
conditions
\/-g
= 1):[9]
dB/dx-1/2dB/dxa=Bao.TBaB-BaBTBao.
These
relations
are
the direct
consequence
of
more
general,
apparently
still
entirely
unknown
relations:[10]
dBik,lm/dXa
+
dBik,mn/l
+
dBik,nl/m
=
TaklBai,mn
+
TakmBai,nl
+
TaknBai,lm- (3
analogous
terms
with
i and
k
exchanged.).
I
am
probably
going
to take
the
opportunity
to
publish
this,
after
all.
My
calculations showed
me now
that the
field
equations
are
inadequate
for
determining
the
guv’s,
that
is,
not
even
to
the
extent
of
allowing general
arbi-
trariness for
the
coordinates. Yet
the
space-time
geometry
in the
vicinity
of
a
point
should be determined
entirely by
the
energy
state-etc.-prevailing
within
it. Hence conditions must
necessarily be
added
to
your
field
equations.
For
this,
it
seems
to
me, my
condition
of potential is
very suitable,
which states
that
a
four
vector
Au must
exist such
that
gua-dAo/v
+
gua-dAo/u
+
Aadguv/a
=
0 (:or,
as
far
as
I
am
concerned, equal
to
another
2nd-order
tensor:).[11]
For it
seems
to
me
that,
on
the
basis of
this
assumption,
the
guv's
are
definitely
determined
essentially
by
the
energy
tensor and
the
electromagnet,
field (:if
the
electromagnet,
field is
derived from
Au
through
the
formulas
pa
= guo Au;
Fpa
=
dpp/o-dpo/p,
as
I
have
done
in
my
last
letter:).
All these reflections
are
terribly
drawn
out
and
tricky,
and
I
am
almost inclined
to
give
up hope
of
arriving
at
a
positive
result here with
the
current state of
mathematics,
with
my
lack
of
access
to
the literature
and
with
my
limited time.
I
cannot
spend
my 17-day
vacation
on
a
thorough study of
mathematics
either,
disregarding
that
nothing
would
come
out of
it
in such
a
short
time.
I
therefore
would like to
ask
you
to
direct
your
attention
to these
problems yourself or,
if
your
time
also is
taxed
too much
otherwise,
to refer
some
mathematicians
to
your
equations. My
esteemed teacher
Hilbert[12] will
certainly
find
some
results here.
The author
of
the
Encyclopedia
article cited
above,
Mr. Ed.
von
Weber,[13]
might
possibly
be in
the
position
to
shake
some
theorems
effortlessly
out of
his
rich store
of
special
mathematical
skills,
which
might
illuminate
the
applied
boundary
value
problem.