74 DOC.
9
FORMAL FOUNDATION OF RELATIVITY
[40]
and
gvo
=
1 or
0
depending upon
o
=
v or
o
#
v.
The
ten
equations
(74) can
be used
to
determine the
ten
functions
guv
if
the
ZOT
are
given.
Furthermore,
the
guv
must also
satisfy
the four
equations
(67)
because the
coordinate
system
is
to
be
an
adapted
one.
We
have, therefore, more equations
than
we
have functions
to
be found. This is
only possible
if the
equations
are
not all
mutually independent
of
each other. It
must
be demanded that
satisfying equations
(74) implies
that
equations
(67)
are
also satisfied.
A
glance
at
(76)
and
(76a)
shows
that
this is achieved
if
Sav
(a
quantity
which is
a
function
of
the
guv
and the
gouv
just
as
H
is)
vanishes
identically
for
every
combination of
indices. H
then has to be
chosen in
agreement
with
the conditions
[41]
{18}
Svo
=
0.
(77)
Without
being
able to state
a
formal
reason, I
demand furthermore that H is
an
integral homogeneous
function
of
the second
degree
in the
guvo.
In this
case
H
is
completely
determined
up
to
a
constant factor. Since
H
shall be
a
scalar under linear
transformations,
it
must8
(considering
what
we
just
postulated)
be
a
linear combina-
tion of the
following
five
quantities:
dg^dgZ.
y
goog
Mil-9
yg
dg°»
dg"v.
Zs
Ö/AV
-\ -\ 9
6
Ofiv
ö/xv'
iL-/
~v
"
fjLVor
uXa
ÜXT
ßVfi'v'aa' ®Xa
OXa aa'fxv ^X^
OXv
y
o a
.
Yf,
dgZdgfl
g's"e,,dgßV
a."
a*".'
x
a*,
&"
.
Conditions
(77), finally,
lead
us
to
equate
H-aside
from
a
constant
factor-to
[p. 1076]
the fourth
one
of
these
quantities.
We therefore
set9
under consideration
of
(35)
and
[42] making
free
use
of
the
constant,
(78)
4
aßrp OXa
ÖXß
We limit ourselves to show that this choice of H
actually
satisfies
(77). Utilizing
the relations
8The proof
is
simple
but
involved,
and for this
reason
I delete
it.
9Expressing
H
by
the
components
Tvot
of
the
gravitational
field
(see (46)), one
obtains
H
=
-
E
r"
/xprr'