The field equations are obtained from the splitting of the Riemann tensor into a
trace-free part and its traces according to
1 1
Rabcd Cabcd + @ac bd + bd ac gbc ad gad bc) acgbd gbcgad) R. (2-9)
2 2
where Cabcd, abcd, ab and R denote the Weyl tensor, Riemann tensor, Ricci tensor and
Ricci scalar, respectively. Since both the Ricci tensor and the Ricci scalar vanish in the
absence of sources, the Weyl and Riemann tensors are identical in source-free spacetimes.
In that sense the Weyl tensor represents the purely gravitational degrees of freedom.
The Riemann tensor is then expressed purely in terms of the spin coefficients and their
derivatives by contracting all four vector indices with e )'s and making use of the Ricci
identity,
(Ve~b Vb a = Rabcd~d = abcd~d, (2-10)
where vd is an arbitrary vector. In four dimensions the Riemann tensor has twenty
independent components and the Ricci tensor has ten, leaving the Weyl tensor with ten
independent components. In the NP formalism, this translates into five complex scalars:
n = Cabcd a blc d
I = -Cabcd a blc d
',_=-Cabcdla blc d + a b c d), (2-11)
= Cabcdanb cnd
~4 -abcd Ra b c d