242
DOC.
41
HAMILTON'S PRINCIPLE
a
(dR) dR
dm
dx
druva
duv
dguv
a
d
m
dm
dx
dq(p)
dq(p)
=
0.
(7)
(8)
R
is
here
in
the
same
relation
to
R
as
b
is
to b.
It
must
be noted that
equations (8)
or (5) respectively
would
have
to
be
replaced
by
others
if
we
would
assume
that
m
or b
resp.
would
depend upon higher
than the
first
derivatives
of
q(p)
Similarly,
one
could
imagine
the
q(p)
not
as
mutually
independent
but rather
as
connected
to
each other
by further,
conditional
equations.
All this is irrelevant for the
following development,
since it is
solely
based
upon
[7] {4}
equation
(7),
which is obtained
by varying our integral
after the
guv.
§3.
Properties
of
the Field
Equations
of Gravitation
Based
on
the
Theory
of Invariants
We
now
introduce the
assumption
that
ds2
=
guv dxudxv
(9)
is
an
invariant. This fixes the transformational character of the
guv.
We make
no
presuppositions
about the transformational character
of
the
q(p)
which describe
matter.
However,
the functions H
=
b/-g
and
G
= R/-g
and
M
= m/-g
shall
be
invariants under
arbitrary
substitutions
of
the
space-time
coordinates.
From these
suppositions
follows the
general
covariance of
equations
(7)
and
(8),
which have been
derived from
(1).
It follows furthermore that
(up
to
a
constant
factor)
G
is
equal
to
the scalar of the
Riemann
tensor
of
curvature,
because
there
is
no
other invariant
with the
properties
demanded for
G.4
With
this,
R,
and
hence
the
left-hand side
of
the field
equation (7)
is
completely
determined.5
[p. 1114]
The
postulate
of
general relativity
entails certain
properties
of
the function
R*
4This is the
reason why
the
requirements
of
general relativity
led to
a quite
distinct
theory
of
gravitation.
5Execution
of
the
partial
integration yields
R
=
-gg
uv
ua
ub
ab
b
a a
B.