200 DOC. 123 THEORY OF AFFINE FIELD
448
endoparasitic organisms,
will
determine
largely
the
extent to
which
he
can use
and
develop
the natural
resources
of the rich
tropical and subtropical zone
of
the earth.
Other
applications
of
zoology
to human wellbeing
cannot be
dealt with
here,
but mention
should
be made
of twothe researches
on
seafisheries
problems
which
[SEPTEMBER
22,
1923]
have formed
an important
branch of the
zoological
work of Great Britain for forty
years,
and
the
studies
on
genetics
which
made
possible
an explanation
of the
mode of
inheritance of
a particular bloodgroup,
and
of
some
of
the
defects
(e.g.
colourblindness and
haemophilia)
and malformations which
appear
in
the
human
race.
NATURE
[1]
[2]
[3]
[4]
The
Theory
of the Affine Field.1
By
Prof.
ALBERT
EINSTEIN,
For.
Mem. R.S.
THE
theory
of the
connexion between
gravitationgravitational
and
electromagnetism
outlined
below is
foundedconsideration
on
Eddington’s
idea, published
during
recent
years,
of
basing
“field
physics” mathematically on
the
theory
of
the
affine
relation.
We shall
first
briefly considerin
the
entire
development
of
ideas associated
with the
names LeviCivita, Weyl,
and
Eddington.
The
general
theory of relativity rests formally
on
the
geometry
of
Riemann,
which
bases
all its
concep
tions
on
that
of the
interval is
between
points
in
definitely near together,
in
accordance
with the
formula2
is*=g^ix,
...
(1)
These
magnitudes
g^ determine
the behaviour
of
measuringrods
and clocks
with
reference to
the
co
ordinate
system,
as
well
as
the
gravitational
field.
Thus
far
we
are
able
to
say
that,
from its
foundations,
the
general
theory
of
relativity
explains
the
gravita
tional
field.
In
contrast to this,
the
conceptual
founda
tions
of
the
theory
have
no
relations with the
electro
magnetic
field.
These facts
suggest
the
following
question.
Is
it
not
possible
to
generalise
the mathematical foundations
of
the
theory
in
such
a way
that
we can
derive
from them
not
only
the
properties
of
the
gravitational field,
but
also those of the
electromagnetic
field
?
The
possibility
of
a
generalisation
of the mathe
matical foundations
resulted
from the fact that
Levi
Civita
pointed
out
an
element
in the
geometry
of
Riemann that could be made
independent
of this
geometry,
to
wit,
the
“affine
relation”;
for
according
to Riemann’s
geometry every
indefinitely
small
part
of
the manifold
can
be
represented
approximately by amanner.
Euclidean
one.
Thus in this
elemental
region
there
exists
the
idea of
parallelism.
If
we subject a con
travariant vector
A"
at
the
point xy
to
a parallel
displacement
to the
indefinitely adjacent point *„+&»„,
then the
resulting
vector
A'’+8A”
is determined by an
expression
of the form
8A*=
T’^Sxy
.
.
(2)
The
magnitudes
F
are
symmetrical
in the lower
indices,
and
are
expressed
in
accordance
with
Riemann
geometry
by
the
ghv
and
their first derivatives
(Christoffel symbols
of
the second
kind).
We
obtain
these
expressions
by formulating
the condition that
the
length
of
a
contravariant
vector formed
in
accordance
with
(1)
does not
change
as a
result of
thewith
that the Riemann
tensor
of
curvature,
which is
fundamental for the
theory
of the
parallel
displacement
LeviCivita
has sh
1
Translated
by
Dr. R. W.
Lawson.
2
In
accordance
with
custom,
the
signs
of summation
are
omitted.
NO.
2812, VOL.
112]
field,
can
be obtained
from
a
geometrical
based
solely
on
the
law of the
affine
relation
given
by
(2)
above. The
manner
in which the
are
expressible
in
terms
of the
ghy
plays
no part
this
consideration. The
behaviour in the
case
of
differential
operations
of the
absolute
differential
calculus is
analogous.
These results
naturally
lead to
a
generalisation
of
Riemann’s
geometry.
Instead of
starting
off from the
metrical relation
(1)
and
deriving
from this the
co
efficients
T of the
affine
relation
characterised
by
(2),
we proceed
from
a general
affine
relation of the
type
(2)
without
postulating
(1).
The search
for the
mathematical
laws
which
shall
correspond
to
the
laws
of Nature then
resolves
itself into
the
solution of the
question
:
What
are
the formally most
natural
con
ditions
that
can
be
imposed
upon an affine
relation
?
The first
step
in this direction
was
taken
by
H.
Weyl.
His
theory is connected
with the fact that
light
rays
are simpler
structures
from the
physical
viewpoint
than
measuringrods
and
clocks,
and
that
only
the
ratios of the
g^v
are
determined
by
the law of
pro
pagation
of
light. Accordingly
he
ascribes objective
significance
not to
the
magnitude
is
in
(1),
i.e. to
the
length
of
a
vector,
but
only to
the ratio of the
lengths
of
two
vectors (thus
also to the
angles).
Those affine
relations
are
permissible
in which the
parallel
displace
ment is
angularly
accurate.
In this
way a theory was
arrived
at,
in
which,
along
with the
determinate
(except
for
a factor)
g^
other four
magnitudes
f,.
occurred,
which
Weyl
identified with
electromagnetic
potentials.
Eddington
attacked
the
problem
in
a
more
radical
He
proceeded
from
an
affine
relation of the
type
(2)
and
sought
to characterise
this without intro
ducing
into the
basis
of the
theory anything derived
from
(1),
i.e.
from the metric. The metric
was to
appear
as a
deduction from the
theory.
The
tensor
ôFa 9p
«

3*r+r¿
r^+isfr¿
(3)
is symmetrical
in the
special case
of
Riemann’s
geometry.
In the
general
case
Rßy
is
split up
into
a
symmetrical and
an
“
antisymmetrical ”
part :
R;»..=»+£,......(4)
One is
confronted with the
possibility
of
identifying
the
symmetrical
tensor
of
the metrical
or
gravitational field,
and
ƒ,,„
with the
antisymmetrical
tensor
of the
electromagnetic
field. This
was
the
course
taken
by Eddington.
But
his
theory remained
incomplete,
because at
first
no course
possessed
of
the
advantages
of
simplicity
and naturalness
presented