DOC.
619
SEPTEMBER
1918 643
by a
complete
penetration
into the mathematical
foundations of
this
theory.
I
am
publishing
an
article
about
it entitled “Reine
Infinitesimalgeometrie” [“Pure
Infinitesimal
Geometry”]
in
the
Mathem.
Zeitschrift,
which
is appearing soon.[2]
There the natural
development
of
geometry
in
the three
stages
of
topology [reinen
“analysis situs"] (1),
affine
(2),
and metric
geometry
(3)
is followed through. I
say, now:
God
could have stopped
at
any
of
these
3
stages, namely,
created either
an
empty world,
or a
world
just
with
a
gravitational
field,[3]
or
the
one
familiar
to
us;
but
having
once
decided
on
the
metric
one,
the
most
richly
endowed,
he
had
to
adopt
mine,
the
pure one.[4]
I
can
perhaps
best
refute
the
objection
you
raised[5]
with the
following
remark: if
R
is
some
invariant of
weight -1,[6]
e.g.,
the
scalar
of
the
curvature,[7]
then the
quadratic
form
R(gikdxidxk)
is
an
absolute
invariant;[8] by multiplying by
this
factor,
an
absolute
normalization,
so
to
speak,
of
the unit
of
length
thus
is achieved.-[9]
When
I
sent
you my
note in
the
spring[10]
I
was already
convinced
that
your
gravitational
equations
follow
in
the
1st
approximation
from
my
theory
for static
fields,
that
is,
with
the
cosmological
A
term; yet
in such
a
way
that
from
the
laws
of nature it
only
follows
that
A
is
a
constant,
without
any specific
value
being prescribed
for this constant
(this
value
rather
appears
to
be
determined
by
the
world’s
total
mass).-[11]
In
my
case
also,
a
conservation
law for
the
total
energy
and momentum
is
valid,
dxk
=
0,
which
results
much
more
naturally
than
in
your
theory.[12]
For
the
matter in
the
more
restricted
sense,
tik,
the
equation[13]
dtf
-*_
_
pr
ts +
F•«*Sk
g
ZS r
+
=
0
(s
current)
applies;
whereas for
the
current,
the
postulate
si
=
pui
(u
four
velocity
dxi/ds,
p
charge
density)
is
possible,
tki
=
uuiuk
contradicts the
required
invariance
properties.[14]
In
consequence
of
this,
the
mechanical
equations
are
still
not
clear
to
me;
and
as long as
this
is not
settled,
it
is difficult
to
draw
any consequences
to
be
tested
against
experience
(the
influence of
the
electromagnetic
potential
on a
uncharged point-mass,
which
you
infer from
the
equation
for
the
geodesic
line,[15]
is
up
in
the
air
as
long
as
the
significance
the
geodesic
line has in mechanics
can-
not
be
ascertained).
This
will
be
the
next
task,
for
me
to arrive at
some
clarity
here.[16]
In
addition, I
have formulated the strict
equations
for
a
static,
rotation-
ally
symmetric
field;
they
are so complicated, though,
that
I
do not know
what
to do
with them
at
present;
the constitution
of
the
electron
and
the
possibility
of
its existence
ought
to result from
them.[17]
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