DOC.
41
HAMILTON'S PRINCIPLE 241
depend only upon
the
guv.
The variational
principle (1) can
then
be
replaced by one
more
convenient for
us.
With suitable
partial
integration
one
gets
Jbdr
=
Jbdr
+
F,
(2)
where F is
an
integral
extended
over
the boundaries
of
the domain under
consider-
ation,
while the
quantity
b*
depends only upon
the
guv,
gouv,
q(p), q(p)a
but
no
longer upon
guvor.
For the variation
of
interest
to
us one gets
from
(2)
s{fbdr}
= d{fbd*r},
(3)
whereupon
we
can replace
the variational
principle
(1)
with the
more
convenient
one
d{fbdr}
=
0.
(1a)
By executing
the variation after the
guv
and the
q(p)
one
obtains for the field
equations
of
gravitation
and matter the
equations3
a
db
db*
dx
dgauv dguv
db*
db*
dxa
dq(p)
=
0
(4)
=
0.
(5)
§2.
Separate
Existence
of the Gravitational Field
The
energy components
cannot be
split
into two
separate parts
such that
one
belongs
to the
gravitational
field and the other to
matter,
unless
one
makes
special assump-
tions in which
manner
b
should
depend upon
the
guv,
goruv, q(p),
q(p)a.
In
order
to
bring
about this
property
of
the
theory
we
assume
b
=
R
+
M,
where
R
depends only upon
guv,
gouv, gorv
and
M
only upon guv,
q(p),
q(p)a.
Equations (4), (5)
then take the form
(6)
[p. 1113]
{3}
3As
an
abbreviation,
the summation
signs are
omitted in the formulas. A summation
has to
be
carried out
over
the indices that
occur
twice in
a
term.
For
example,
in
(4)
d
dx
db
denotes
the
term
Z
d
a
dxa
db*
dguva
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