DOC.
41
HAMILTON'S PRINCIPLE 245
J-
($;
+
o
=
o.
(21)
dxv
Equation (21) expresses
the conservation
of
the momentum and the
energy.
We
[p.
116]
call
zvo
the
components
of
the
energy
of
matter,
tvo
the
components
of the
energy
of
the
gravitational
field.
From the field
equations
(7)
of
gravitation
follows
(after
multiplication by
gouv,
summation
over
u.
and
v,
and
on
account
of
(20))
+
1
rMv
aa»
_
0,
dxv
280
agMv
or,
taking
(19)
and
(21)
into
account,
dzvo/dxv+1/2gouvZuv
=
0, (22)
[12]
{8}
where
Zuv
denotes the
quantities
gvoZou.
These
are
four
equations
that the
energy
components
of
matter have to
satisfy.
It is to be
emphasized
that the
(generally covariant)
conservation theorems
(21)
and
(22)
have been
derived-using
also the
postulate
of
general
covariance
(relativity)-from
the field
equations
(7)
of
gravitation
alone,
without
use
of
the
field
equations
(8)
for material
processes. [13]
Additional
notes
by
translator
In his footnote
1),
just
prior
to
equations (4)
and
(5),
Einstein introduces into tensor
calculus,
in
a
formal
manner,
the rule
of
abbreviated
writing
of
summations,
which
is
now generally
known
as
the Einstein summation convention. It
was
introduced in
Doc.
30,
p.
296.
{1}
The
"q"
with
2
subscripts
and
2
superscripts
has been corrected here to
"g";
the
2nd derivative
of
"q"
inside the
following parenthesis
has
also been corrected
to
"g,"
both with indices
as
indicated. Editorial notes
[6]
and
[7]
relate to
similar
typesetting
errors.
{2}
"quvor"
has
been corrected here to
"guvor."
{3}
"(4a)"
has
been corrected to
"(5)."
{4}
"q"
has been corrected here
to "guv".
{5}
"(13)"
and
"(14)"
have been corrected here to
"(11)"
and
"(12)."