DOC.
71
PRINCETON
LECTURES 365
THE GENERAL THEORY
Inside the brackets
are
the
guv
of
the manifold
in
the
neighbourhood
of
the
origin.
Since
the
first
derivatives
of
the
guv,
and therefore
also
the
Tauv,
vanish
at
the
origin,
the calculation
of
the
Ruv
for
this manifold,
by
(88), is very
simple
at
the
origin.
We have
R
HV
2
_
2
a2ÔM-
a2
g*’- [139]
Since
the
relation
Ruv
=
-2/a2guv
is
generally
co-variant,
and
since
all
points
of
the
manifold
are
geometrically
equivalent,
this
relation
holds for
every system
of
co-
ordinates,
and
everywhere
in
the manifold. In order
to
avoid confusion
with
the four-dimensional
continuum,
we
shall,
in the
following,
designate
quantities
that
refer
to
the three-dimensional continuum
by
Greek
letters,
and
put
(120)
PM,
=
-
^
7,.-
We
now
proceed to apply
the
field
equations (96) to
our
special
case.
From
(119)
we
get
for
the four-dimen-
sional
manifold,
(121)
(R-
P"y
for the indices
1
to
3
R14
=
R24
=
R34
=
R44
=
0
For the
right-hand
side of
(96)
we
have
to
consider the
energy tensor
for
matter
distributed
like
a
cloud
of
dust.
According to
what
has
gone
before
we
must
therefore
put
TV
-
T
dx"
dx,
ds ds
specialized
for
the
case
of
rest.
But
in
addition,
we
shall
add
a
pressure
term
that
may
be
physically
established
as
[105]