DOC.
13
GENERALIZED THEORY OF RELATIVITY
167
where
us2
=
ds
is
an
invariant with
respect
to
arbitrary
substitutions. The
equations
to be
sought,
which determine the
course
of
some physical process or
other,
must
be
so
constructed
that the invariance of ds will entail the covariance of the
equation system
in
question.
But in the
pursuit
of solutions
to
these
general problems,
we
at
first
encounter
a
fundamental
difficulty.
We do
not
know with
respect
to
which
group
of transforma-
tions the
equations
we
are seeking
must
be covariant. At first it
seems
most
natural
to demand
that
the
systems
of
equations
should be covariant
with
respect
to
arbitrary
transformations. But
opposed
to
this is the fact that the
equations
of the
gravitational
field that
we
have
set
up
do
not
possess
this
property.
For the
equations
of
gravitation
we
have
only
been able
to
prove
that
they
are
covariant with
respect
to
arbitrary
linear
transformations;
but
we
do
not
know whether
there exists
a general
group
of transformations with
respect
to
which the
equations
are
covariant. The
question
as
to
the existence of such
a
group
for the
system
of
equations
(18)
and
(21)
is
the
most
important question
connected with the considerations
presented
here.
At
any
rate,
given
the
present
state
of the
theory,
it is not
justifiable
for
us
to
demand
a
covariance of
physical equations
with
respect
to
arbitrary
substitutions.
But
on
the other hand
we
have
seen
that for material
processes
it
is
indeed
possible
to set
up
an
energy-momentum
balance
equation
that does
permit
arbitrary
transformations
(§4, equation 10).
Therefore it nevertheless
seems
natural
to
assume
that
all
systems
of
physical
equations,
with the
exception
of the
gravitational
equations,
should be formulated in such
a
way
that
they
are
covariant with
respect
to
arbitrary
substitutions.
This
exceptional position
that the
gravitational equations
occupy
in this
respect,
as
compared
with
all
of the other
systems,
has
to do, in
my
opinion,
with the fact that
only
the former
can
contain second derivatives of the
components
of the
fundamental tensor.
The construction of such
systems
of
equations requires
the
resources
of
generalized
vector
analysis
as
it is
presented
in
Part
II.
Here
we
confine ourselves
to
indicating
how
one
obtains the
electromagnetic
field
equations
for the
vacuum
in
this
way.16
We
start
from the
assumption
that the
electrical
charge
is to
be viewed
as
something unchangeable. Suppose
that
an
infinitesimally
small,
arbitrarily moving body
has the
charge
e
and the volume
dV0
16On
this
point,
cf.
also the article
by Kottler,
§3,
cited
on
p.
23
[36]
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