12
DOC. 2 COVARIANCE PROPERTIES
we
abbreviated
as
01
and
02,
respectively.
The factor of
Axu
in the third
integral
is
readily
recognized as
Bm
according
to
the notation
introduced
subsequent
to
equation
(V). Equation
(3)
in abbreviated notation is
now
(3a)
J'
-
0,
+
02
-
dr.
m
The
reasons
which led
to
a
preference
for coordinate
systems
in which the
quantities
Bm =
0 have been
explicated
in
§2.
We want to call these coordinate
systems "adapted"
to
the manifold.
It follows from
equation (3a)
that
adapted
coordinate
systems
are
selected such that under fixed
boundary
values of the
coordinates and their first derivatives
(considered
in
an
arbitrary
coordinate
system),
the
integral
J
becomes
an
extremum.
We
now
want to call
a
transformation between
appropriate
coordinate
systems
admissible.* When the transformation from K
to
K' is
admissible,
equation
(3a)
yields
J'
-
J
- O1
+
O2.
[p. 222] §5.
Proof
of
Covariance
of the
Gravitational
Equations
In
§4 we investigated a
manifold
M.
We shall
now
consider
a
second manifold
M
which differs
only infinitesimally
from the
former,
and for which the
quantities guv
and their first derivatives coincide
on
the
boundary
of
the
domain
L
with those
of
the
corresponding
manifold
M.
We
impose
the coordinate
systems
K and
K'
in the
following
manner:
a)
Both coordinate
systems
be
adapted ones
for the manifold M.
b)
On the
boundary
of
the
domain
L,
let
the
coordinates
xv
coincide with the
xv
and the
x'v
with the
x'v.
c)
The coincidence
of
the coordinate
systems
shall
not
only apply
on
the
boundary
of the
domain,
but also for
quantities
of
first order that
are
infinitesimally
close
to
the
boundary;
this condition
implies
that the
d(Axv)/dxa
coincide with the
d(Axv)/dxa.
Conditions
(b)
and
(c)
do not
contradict each
other,
as can
be
seen
in the
following
manner.
Since manifold M is referred
to
as an
adapted
coordinate
system,
§4
shows the choice of coordinate
system
K
is
such
as
to
make the
integral
J
an
*Translator's
note.
The word
"berechtigt"
in the German
original
is
today
mathemati-
cally
understood
as
"admissible
because of
justification by previously
stated
conditions";
modern German texts also favor
"zulässig" (=
admissible)
over
the
older
"berechtigt" (=
justified).