An important notion in curved space is the connection. In standard tensor language it
makes an appearance through the Clan.!!.!1~ symbols. The equivalents in NP language are
called the spin coefficients and are defined, in general, by
Y(i)(j)(k) = e i) j)v a"(k)b.
(2-4)
It follows from the definition that
(2-5)
There is a total of twelve spin complex coefficients, individually named as follows
is= lamb ab
o- = mamb a 6,
p = mamb a b,
7 = namb a by
S--Ra b anb,
S= --ma b anb,
I- = -ma b anbb,
_T --a b anb,
(2-6)
and
/7 = ~(manb alb me b amb>,
a~ = ~(mamb amb me b anb>,
e = ~(l"Rb alb la b a b>
y = ~(namb amb na b a~b>.
Our e") naturally define four independent, non-commuting directional derivatives
e(i) -- e(i) 8xa
(2-7)
which are also given individual names:
8 m
a = n
8x" '
8
(2-8)