Newman–Penrose formalism

Definition

Let $(M, g)$ be a four-dimensional Lorentzian manifold with signature $(+,-,-,-)$, and let $\nabla$ be its Levi-Civita connection. A null tetrad on an open set $U \subseteq M$ is a frame field $(l, n, m, \bar m)$ such that:

1. $l$ and $n$ are real vector fields, and $m$ is a complex vector field with complex conjugate $\bar m$.
2. All four vectors are null: $g(l,l) = g(n,n) = g(m,m) = 0$.
3. $g(l, n) = 1$ and $g(m, \bar m) = -1$, and all other products of two of them vanish.

Then $g_{ab} = l_a n_b + n_a l_b - m_a \bar m_b - \bar m_a m_b$. The Newman–Penrose formalism describes the geometry on $U$ by the following components in a null tetrad.

• The directional derivatives $D = l^a \nabla_a$, $\Delta = n^a \nabla_a$, $\delta = m^a \nabla_a$ and $\bar\delta = \bar m^a \nabla_a$.
• The twelve complex spin coefficients $\kappa, \sigma, \rho, \tau, \nu, \lambda, \mu, \pi, \epsilon, \gamma, \alpha, \beta$, which are the components of $\nabla$ in the tetrad.
• The five complex Weyl scalars built from the Weyl tensor $C_{abcd}$: $$\Psi_0 = C_{abcd} l^a m^b l^c m^d, \quad \Psi_1 = C_{abcd} l^a n^b l^c m^d, \quad \Psi_2 = C_{abcd} l^a m^b \bar m^c n^d,$$ $$\Psi_3 = C_{abcd} l^a n^b \bar m^c n^d, \quad \Psi_4 = C_{abcd} n^a \bar m^b n^c \bar m^d.$$
• The components $\Phi_{ij}$, $0 \le i, j \le 2$, of the trace-free Ricci tensor, and $\Lambda = R/24$, where $R$ is the scalar curvature.

The Einstein equations and the Bianchi identities become first-order equations for these quantities, the Newman–Penrose equations.