118
DOC. 25 FIELD
EQUATIONS
OF
GRAVITATION
[p.
845]
The
ten
generally-covariant equations
of the
gravitational
field in
spaces
where
"matter" is absent
are
obtained
by setting
Gim
=
0.
(2)
These
equations can
be
simplified by choosing
the
system
of
reference such that
v/-g
=
1.
Sim
then vanishes because
of
(16),
and
one
gets
instead
of
(2)
(3)
I
axi
p/
yFg
=
1.
(3a)
We
have
set
here
r'
am
(4)
which
quantities
we
call the
"components"
of
the
gravitational
field.
When there
is
"matter" in the
space
under
consideration,
its
energy
tensor
occurs
on
the
right-hand
sides of
(2)
and
(3),
respectively.
We set
Gim
=
Tim
1
(2a)
where
= =
(5)
pa a
T is the scalar
of
the
energy
tensor
of
"matter,"
and the
right-hand
side
of
(2a)
is
a
tensor.
If
we
specialize
the coordinate
system again
in the familiar
manner,
we get
in
place
of
(2a)
the
equivalent equations
Rim
=
E
^
+
IXC
=
-
ifc.r)
(6)
=
1.
(3a)
We
assume,
as
usual,
that the
divergence
of the
energy
tensor
of
matter
vanishes
when taken in the
sense
of
the
general
differential calculus
(energy-momentum
theorem). Specializing
the choice
of
coordinates
according
to
(3a),
this
means
basically
that the
Tim
should
satisfy
the conditions
(7)
i
«1
2^
oxa
"