10
DOC.
2
COVARIANCE PROPERTIES
where
(Va)
H
=
12/(-g
E
dg dyq
aßtq
P
and the
yuv
are
varied
independently
of each other such that the variation
on
the
boundary
of the four-dimensional domain of
integration
vanishes.
Utilizing
for the calculation of
SH
the
easily
understood formulas
s(/-g)
=
-12 E/-gSuv,
Hv
d(dg)
=
-
Ed(gtugsyuv),
d(fe)
=
flV
s(dyt/dxB)
=
d(sytq),
and
considering
the fact that variations
of surface
integrals
vanish,
one
finds
SEI,-J2(-i
(y=5r",fe)
+
/-gy aytq
dgut
dgvq
fAvaptq
[16]
+
iy-g.
dgtq dgtq
-
1/4guvy
dgtq
dgtq)syuv.dt.
Utilizing
the definitions
(14)
and
(16)
of
the
"Outline,"
our
condition
(V)
takes
[p.
220]
the form
,J)
+
+
T
})9y
-y^gdx
=-0.
fiv
As the
Syuv
are
supposed
to be
mutually independent,
the
equations
(21)
of
the
"Outline," i.e.,
our
gravitational equations
in covariant
form,
now
become
a
consequence
of
this condition.
[17]
§4.
Proof
of
a
Lemma.
Adapted
Coordinate
Systems
Our next task is the
investigation
of the covariance
properties
of
equation (V).
For
this
purpose we
look first for the transformational
properties
of
the
integrals
J
=
f
Hdt
=
f/-g
EyaB
dgtq
dytq.dt.
a Bt
Q
Let there be
an
arbitrary
four-dimensional manifold
M,
referred to
a
coordinate
system
K
of
the
xv.
Furthermore,
we
refer the
same
manifold M
to
a
second
coordinate
system
K'
of
the
x'v
such that