DOC. 2
COVARIANCE
PROPERTIES
9
Therefore,
if
all substitutions would be
permitted,
the
same
system
of
®uv
would
have
more
than
one
system
of the
yuv
belonging
to
it,
and this is
a
contradiction
to
assumption a).4
Once it
is
understood
that
an acceptable theory
of
gravitation implies
necessarily a specialization
of
the coordinate
system,
it is also
easily seen
that
the
gravitational equations, given by us,
are
based
upon
a
special
coordinate
system.
An
Xv-differentiation
of
equations (II)
and summation
over
v,
under simultaneous
consideration
of
equations
(III),
yields
the
relations
(IV)
a[i/uv
=
0,
and these
are
four differential conditions for the
quantities
guv.
We want
to
write
(IV)
in the abbreviated form
Bo
=
0.
These four
quantities
Ba
do not form
a generally-covariant
vector,
as
will be
shown
[p.
219]
in
§5.
From this
one can
conclude that the
equations
Ba
=
0
represent
a
true
condition for the choice of the coordinate
system.5
§3.
The
Hamiltonian Form
of the
Gravitational
Equations
In the
following proof
of covariance
of
the
gravitational equations
we
will
use
the
fact that these
equations can
be
brought
into the form of
a
variational
principle.6
The
gravitational equations (II) can
be shown to be
equivalent to
the statement
(V)
(SH-2XgTuvSyuv)dt
=
0, [15]
fiv
4This train
of
thought
is
already among
the notes in the
appendix
of
the
reprint
of
the
"Outline" in volume 62
of
the Zeitschr.
f.
Math.
u. Phys.
The claim
appended
there about
[12]
the restriction
on
the coordinate
systems,
however,
does
not
apply;
the restriction to linear
substitutions follows from
(III) only
if
the
quantities tav
/ -g have tensorial
character,
and
this turned out to be not
justified. [13]
5The
equations
Ba
=
0
can
also
be
obtained
by
imposing
the
divergence operator upon
the
gravitational equations
in the
manner
of
the absolute differential
calculus,
thereby using
the conservation law
of
matter.
6We
owe
thanks to Mr. Paul
Bernays
in Zurich for
suggesting
the idea
of
simplifying
[14]
the
proof
by
such
a procedure.
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