78
DOC.
9
FORMAL FOUNDATION OF RELATIVITY
time domain in several
seemingly mutually independent representations.
This
runs
counter
to
my physical imagination
in
the
most
vivid
manner.
However,
I
am
not
able
to
demonstrate that the
occurrence
of such
curves can
be excluded from the
theory
that has been
developed
here.
[48]
Since
I cannot
help
but
see,
after these
confessions,
a
pitying
smile
creep over
the face
of
the
reader,
I
cannot
suppress
the
following
remark about the
current
opinion
on
the foundation of
physics.
Before
Maxwell,
the laws of nature with
respect
to
their
space dependence were
in
principle integral
laws;
this is
to
say
that
in
elementary
laws the distances between
finitely
distinct
points
did
occur.
Euclidean
geometry
is the basis for this
description
of
nature.
This
geometry
means originally
only
the
essence
of
conclusions from
geometric axioms;
in this
regard
it has
no
physical
content. But
geometry
becomes
a
physical
science
by adding
the statement
that two
points
of
a
"rigid" body
shall have
a
distinct distance from each other that
is
independent
of
the
position
of
the
body.
After this
amendment,
the theorems of
this
amended
geometry are (in
a
physical sense)
either
factually
true
or
not
true. It is
geometry
in this extended
sense
which forms the basis
of
physics.
Seen from this
aspect,
the theorems of
geometry
are
to
be looked
at
as
integral
laws of
physics
insofar
as
they
deal with distance
of
points
at
a
finite range.
Since
Maxwell,
and
by
his
work,
physics
has
undergone
a
fundamental revision
insofar
as
the demand
gradually prevailed
that distances of
points
at
a
finite
range
[p. 1080]
should
not
occur
in the
elementary
laws;
i.e.,
theories
of
"action
at
a
distance"
are
now replaced
by
theories of "local action." One
forgot
in this
process
that the
Euclidean
geometry
too-as it is used in
physics-consists
of
physical
theorems
that,
from
a physical aspect, are on an equal footing
with the
integral
laws of
Newtonian mechanics of
points.
In
my opinion,
this is
an
inconsistent attitude
of
which
we
should free ourselves.
An
attempt
to free ourselves leads
again,
first, to
the
use
of
arbitrary parameters
for the
description
of
the four-dimensional continuum
around us-instead of
coordinates.
Again, we
arrive
at
the
same
considerations
we
have
given
in sections
B and
C
of this paper-with the sole difference that
a
correlation of the
guv
with the
gravitational
field
is
not
postulated.
But
if
we
want to
adhere to the demands
of
Euclidean
geometry
(in
the
sense
stated
above) we
would have
to
replace
the
equations given
in this section
by
others that derive from the
following supposition:
the coordinates
xv
can
be chosen such that the
guv
are independent
of
the
xv.
In this
manner we are
led to the demand that the
components
of
the
Riemann-Christoffel
tensor-as
developed
in §9-shall vanish.
By
this
method,
the theorems of
Euclidean
geometry
would be reduced
to
differential laws. But
by phrasing
the
situation in this
manner,
one
realizes that
a
rigorous implementation
of
a
"local
action"
theory
is neither the most
simple nor
the next closest
possibility
that
comes
to mind.