430
DOC.
420
DECEMBER
1917
theory
of
relativity:
(dp/dt)2
=
(dp/dx)2 (dp/dy)2 (dp/dz)2
:][10]
apparently
has
a
complete integral
which
contains,
apart
from
an
additive
and
a
multiplicative
in-
tegration constant,
2
other
essential
ones. Knowing
one
such
integral,
dp/dC1
=
C3;
dp/dC2
=
C4
must
be
the
equations
for
the total
“light rays,”
or
better, “gravitation
rays.”
This would be
the
characteristics of
the total
diff[erential] eq.
ds2
=
0.
If
x'i
=
p(x1, x2, x3, x4, ai,
bi)
is
chosen for four
arbitrary
value
systems
and
ai,
bi
as new
coordinates,
then
all
the
g'ii's disappear.
If
one
can,
which
is
probable,
also
choose
the
ai,
bi's
such
that
V(a1,
b1);
p(a2,
b2))
=
0
and
V(p(a3,
b3);
p(a4,
b4))
=
0,
then
g12,
g34
also
vanish;
g11, g22, g33, g44, g12,
g34;
/g
becomes
rational
=
g13g14
g23g24;
the expressions
for
Buv,ar
become
only minimally simpler,
though.
For
special problems, however, light-ray
coordinates
offer great
advantages.
(d) Example.
For
the
sake of
variety, cylindrical symmetry
now.
The
processes
allow
the
group:
x'
= x cos
p
+
y
sin
p
y'
=
-x
sin
p
+
y cos
p
z'
=
z
+
c
.
x
and
y
may
occur
only
in
the
re-
lations
x2
+
y2;
xdx +
ydy;
dx2
+
dy2;
z
only
in
the
form: dz.
From
this with
x2
+
y2
=
r2;
ds2
=
ƒ(r,
t)dt2
-
g.
(dr2
+
r2dp2)
-
h(xdx +
ydy)2
-
ldzdt
-
mdz2
-
n(xdx +
ydy)dz
-
kdrdt
or
expressed
somewhat
differently:
ds2
=
fdt2
-
hdr2
-
gdQ2
-
mdz2
-
kdrdt
-
ldzdt
-
ndrdz.
If here in
the
r,
t
plane
light-ray
coordinates
are
inserted,
then the
underlined
terms
are
dropped.
The tensor
becomes
0
-k
-n
0
-k
0
-l
0
-n
-l
-m
0
0 0 0 -g
The
equations
for
the
contravariant four
vector
introduced under
(2a)
above
are
simplified considerably
as a
result,
since
two of
the
coordinates
are
parameters of
the characteristics
of
the
occurring
partial diff.
eq.
In
any
case, I
did
not want to
omit
turning
your
attention
to these coordinates.
In the
pleasant hope
of
receiving
a
reply
from
you soon,
I
remain
very respect-
fully yours truly,
Dr. Rud.
Förster
Engineer.
P. S.
Incidentally,
I would also like to
note
that
my employment
contract
prohibits
me
from
any writing
activities.[11]
(The
lot
of
the industrial
slave.)