66 DOC. 9 FORMAL FOUNDATION OF RELATIVITY
occur.
The
physical process
in
E
is
completely
determined when the
quantities
guv
are
given
as
functions
of
the
xv
relative to the coordinate
system
K which is used for
the
description.
The
totality
of
these functions is
symbolically
denoted
by G(x).
Let
us
introduce
a new
coordinate
system
K' which coincides with K
outside of
E,
but deviates from K inside of
E,
such, however,
that the
g'uv
relative to K'
as
well
as
the
guv (including
their
derivatives)
are everywhere
continuous. The
totality
of
the
g'ßV
is
symbolically
denoted
by
G'(x').
G'(x')
and
G(x)
describe the
same
gravitational
field. When
we replace
the coordinates
x'v
by
the coordinates
xv
in the
functions
g'uv
,
i.e.,
when
we
form
G'(x),
then
this
G'(x)
also
represents
a
gravitational
field relative
to
K,
which
however,
is
not the
same
field
as
the factual
(that is,
the
originally given) gravitational
field.
If
we
assume
the differential
equations
of
the
gravitational
field to
be
everywhere
covariant,
then
they
are
satisfied for G'
(x')
relative to K' whenever
they are
satisfied
for
G(x)
relative to
K. Therefore,
they
are
also satisfied for G'
(x)
relative to
K.
There
are
then
two
different solutions
G(x)
and
G'
(x) relative to
K, even though
the
solutions coincide
on
the
boundary
of the
domain
E.
In other
words,
the
course
of
events
in this domain
cannot be
determined
uniquely
by
general-covariant differential
equations.
Consequently,
when
we
demand that the
course
of
events
in
the
gravitational
field be
a completely
determined
one,
we are
forced to restrict the choice
of
the
coordinate
system
such
as
to make the introduction
of
a
new
coordinate
system
K'
(as
characterized
above) impossible
without
a
violation
of
these restrictions. The
continuation
of
the coordinate
system
into the interior
of
the domain
E
cannot
remain
arbitrary.
§13.
Covariance toward Linear
Transformations. Adapted
Coordinate
Systems
Since
we
have
seen
that the coordinate
system
has to be
subject
to conditions,
we
must
focus
upon
several kinds of
specializations
in the choice
of
coordinates. A
very
far-reaching specialization
is obtained
by
admitting only
linear transformations. Our
theory
would be
deprived
of
its main
support
if
we
demand that the
equations
of
[p.
1068] physics
be covariant
merely
toward
linear
transformations. A transformation to
an
accelerated
or
rotating
system
would
no longer
be
an
admissible
transformation,
and
the
physical equivalence
of
a
"centrifugal
field" and
a
gravitational
field-emphasized
in §1-would not be
interpreted by
the
theory
as
to
be,
in
essence,
of
like nature. On
the other
hand,
it is
advantageous
to demand that linear transformations
are
also
among
the admissible transformations
(as
will be shown
later).
We
have, therefore,
to
speak briefly
about the modifications
of
the
theory
of
covariants,
set out
in
section