The Newman–Penrose (NP) formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members often asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— Ψ 4 {\displaystyle \Psi _{4}} in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system. Newman and Penrose introduced the following functions as primary quantities using this tetrad:
Twelve complex spin coefficients (in three groups) which describe the change in the tetrad from point to point: κ , ρ , σ , τ ; λ , μ , ν , π ; ϵ , γ , β , α . {\displaystyle \kappa ,\rho ,\sigma ,\tau \,;\lambda ,\mu ,\nu ,\pi \,;\epsilon ,\gamma ,\beta ,\alpha .} . Five complex functions encoding Weyl tensors in the tetrad basis: Ψ 0 , … , Ψ 4 {\displaystyle \Psi _{0},\ldots ,\Psi _{4}} . Ten functions encoding Ricci tensors in the tetrad basis: Φ 00 , Φ 11 , Φ 22 , Λ {\displaystyle \Phi _{00},\Phi _{11},\Phi _{22},\Lambda } (real); Φ 01 , Φ 10 , Φ 02 , Φ 20 , Φ 12 , Φ 21 {\displaystyle \Phi _{01},\Phi _{10},\Phi _{02},\Phi _{20},\Phi _{12},\Phi _{21}} (complex). In many situations—especially algebraically special spacetimes or vacuum spacetimes—the Newman–Penrose formalism simplifies dramatically, as many of the functions go to zero. This simplification allows for various theorems to be proven more easily than using the standard form of Einstein's equations. In this article, we will only employ the tensorial rather than spinorial version of NP formalism, because the former is easier to understand and more popular in relevant papers. One can refer to ref. for a unified formulation of these two versions.
Null tetrad and sign convention The formalism is developed for four-dimensional spacetime, with a Lorentzian-signature metric. At each point, a tetrad (set of four vectors) is introduced. The first two vectors, ℓ μ {\displaystyle \ell ^{\mu }} and n μ {\displaystyle n^{\mu }} are just a pair of standard (real) null vectors such that ℓ a n a = − 1 {\displaystyle \ell ^{a}n_{a}=-1} . For example, we can think in terms of spherical coordinates, and take ℓ a {\displaystyle \ell ^{a}} to be the outgoing null vector, and n a {\displaystyle n^{a}} to be the ingoing null vector. A complex null vector is then constructed by combining a pair of real, orthogonal unit space-like vectors. In the case of spherical coordinates, the standard choice is
m μ = 1 2 ( θ ^ + i ϕ ^ ) μ . {\displaystyle m^{\mu }={\frac {1}{\sqrt {2}}}\left({\hat {\theta }}+i{\hat {\phi }}\right)^{\mu }\ .}
The complex conjugate of this vector then forms the fourth element of the tetrad. Two sets of signature and normalization conventions are in use for NP formalism: { ( + , − , − , − ) ; ℓ a n a = 1 , m a m ¯ a = − 1 } {\displaystyle \{(+,-,-,-);\ell ^{a}n_{a}=1\,,m^{a}{\bar {m}}_{a}=-1\}} and { ( − , + , + , + ) ; ℓ a n a = − 1 , m a m ¯ a = 1 } {\displaystyle \{(-,+,+,+);\ell ^{a}n_{a}=-1\,,m^{a}{\bar {m}}_{a}=1\}} . The former is the original one that was adopted when NP formalism was developed and has been widely used in black-hole physics, gravitational waves and various other areas in general relativity. However, it is the latter convention that is usually employed in contemporary study of black holes from quasilocal perspectives (such as isolated horizons and dynamical horizons). In this article, we will utilize { ( − , + , + , + ) ; ℓ a n a = − 1 , m a m ¯ a = 1 } {\displaystyle \{(-,+,+,+);\ell ^{a}n_{a}=-1\,,m^{a}{\bar {m}}_{a}=1\}} for a systematic review of the NP formalism (see also refs.). It's important to note that, when switching from { ( + , − , − , − ) , ℓ a n a = 1 , m a m ¯ a = − 1 } {\displaystyle \{(+,-,-,-)\,,\ell ^{a}n_{a}=1\,,m^{a}{\bar {m}}_{a}=-1\}} to { ( − , + , + , + ) , ℓ a n a = − 1 , m a m ¯ a = 1 } {\displaystyle \{(-,+,+,+)\,,\ell ^{a}n_{a}=-1\,,m^{a}{\bar {m}}_{a}=1\}} , definitions of the spin coefficients, Weyl-NP scalars Ψ i {\displaystyle \Psi _{i}} and Ricci-NP scalars Φ i j {\displaystyle \Phi _{ij}} need to change their signs; this way, the Einstein-Maxwell equations can be left unchanged. In NP formalism, the complex null tetrad contains two real null (co)vectors { ℓ , n } {\displaystyle \{\ell \,,n\}} and two complex null (co)vectors { m , m ¯ } {\displaystyle \{m\,,{\bar {m}}\}} . Being null (co)vectors, self-normalization of { ℓ , n } {\displaystyle \{\ell \,,n\}} naturally vanishes,
ℓ a ℓ a = n a n a = m a m a = m ¯ a m ¯ a = 0 , {\displaystyle \ell _{a}\ell ^{a}=n_{a}n^{a}=m_{a}m^{a}={\bar {m}}_{a}{\bar {m}}^{a}=0,}
so the following two pairs of cross-normalization are adopted
ℓ a n a = − 1 = ℓ a n a , m a m ¯ a = 1 = m a m ¯ a , {\displaystyle \ell _{a}n^{a}=-1=\ell ^{a}n_{a}\,,\quad m_{a}{\bar {m}}^{a}=1=m^{a}{\bar {m}}_{a}\,,}
while contractions between the two pairs are also vanishing,
ℓ a m a = ℓ a m ¯ a = n a m a = n a m ¯ a = 0. {\displaystyle \ell _{a}m^{a}=\ell _{a}{\bar {m}}^{a}=n_{a}m^{a}=n_{a}{\bar {m}}^{a}=0.}
Here the indices can be raised and lowered by the global metric g a b {\displaystyle g_{ab}} which in turn can be obtained via
g a b = − ℓ a n b − n a ℓ b + m a m ¯ b + m ¯ a m b , g a b = − ℓ a n b − n a ℓ b + m a m ¯ b + m ¯ a m b . {\displaystyle {\begin{aligned}g_{ab}&=-\ell _{a}n_{b}-n_{a}\ell _{b}+m_{a}{\bar {m}}_{b}+{\bar {m}}_{a}m_{b}\,,\\[1ex]g^{ab}&=-\ell ^{a}n^{b}-n^{a}\ell ^{b}+m^{a}{\bar {m}}^{b}+{\bar {m}}^{a}m^{b}\,.\end{aligned}}}
NP quantities and tetrad equations
Four covariant derivative operators In keeping with the formalism's practice of using distinct unindexed symbols for each component of an object, the covariant derivative operator ∇ a {\displaystyle \nabla _{a}} is expressed using four separate symbols ( D , Δ , δ , δ ¯ {\displaystyle D,\Delta ,\delta ,{\bar {\delta }}} ) which name a directional covariant derivative operator for each tetrad direction. Given a linear combination of tetrad vectors, X a = a ℓ a + b n a + c m a + d m ¯ a {\displaystyle X^{a}=\mathrm {a} \ell ^{a}+\mathrm {b} n^{a}+\mathrm {c} m^{a}+\mathrm {d} {\bar {m}}^{a}} , the covariant derivative operator in the X a {\displaystyle X^{a}} direction is ∇ X = X a ∇ a = ( a D + b Δ + c δ + d δ ¯ ) {\displaystyle \nabla _{X}=X^{a}\nabla _{a}=(\mathrm {a} D+\mathrm {b} \Delta +\mathrm {c} \delta +\mathrm {d} {\bar {\delta }})} . The operators are defined as
D := ∇ ℓ = ℓ a ∇ a , Δ := ∇ n = n a ∇ a , δ := ∇ m = m a ∇ a , δ ¯ := ∇ m ¯ = m ¯ a ∇ a , {\displaystyle {\begin{aligned}D&:=\nabla _{\boldsymbol {\ell }}=\ell ^{a}\nabla _{a}\,,&\Delta &:=\nabla _{\boldsymbol {n}}=n^{a}\nabla _{a}\,,\\[1ex]\delta &:=\nabla _{\boldsymbol {m}}=m^{a}\nabla _{a}\,,&{\bar {\delta }}&:=\nabla _{\boldsymbol {\bar {m}}}={\bar {m}}^{a}\nabla _{a}\,,\end{aligned}}}
which reduce to D = ℓ a ∂ a , Δ = n a ∂ a , δ = m a ∂ a , δ ¯ = m ¯ a ∂ a {\displaystyle D=\ell ^{a}\partial _{a}\,,\Delta =n^{a}\partial _{a}\,,\delta =m^{a}\partial _{a}\,,{\bar {\delta }}={\bar {m}}^{a}\partial _{a}} when acting on scalar functions.
Twelve spin coefficients In NP formalism, instead of using index notations as in orthogonal tetrads, each Ricci rotation coefficient γ i j k {\displaystyle \gamma _{ijk}} in the null tetrad is assigned a lower-case Greek letter, which constitute the 12 complex spin coefficients (in three groups),
κ := − m a D ℓ a = − m a ℓ b ∇ b ℓ a , τ := − m a Δ ℓ a = − m a n b ∇ b ℓ a , σ := − m a δ ℓ a = − m a m b ∇ b ℓ a , ρ := − m a δ ¯ ℓ a = − m a m ¯ b ∇ b ℓ a ; π := m ¯ a D n a = m ¯ a ℓ b ∇ b n a , ν := m ¯ a Δ n a = m ¯ a n b ∇ b n a , μ := m ¯ a δ n a = m ¯ a m b ∇ b n a , λ := m ¯ a δ ¯ n a = m ¯ a m ¯ b ∇ b n a ; {\displaystyle {\begin{aligned}\kappa &:=-m^{a}D\ell _{a}=-m^{a}\ell ^{b}\nabla _{b}\ell _{a}\,,&\tau &:=-m^{a}\Delta \ell _{a}=-m^{a}n^{b}\nabla _{b}\ell _{a}\,,\\[1ex]\sigma &:=-m^{a}\delta \ell _{a}=-m^{a}m^{b}\nabla _{b}\ell _{a}\,,&\rho &:=-m^{a}{\bar {\delta }}\ell _{a}=-m^{a}{\bar {m}}^{b}\nabla _{b}\ell _{a}\,;\\[1ex]\pi &:={\bar {m}}^{a}Dn_{a}={\bar {m}}^{a}\ell ^{b}\nabla _{b}n_{a}\,,&\nu &:={\bar {m}}^{a}\Delta n_{a}={\bar {m}}^{a}n^{b}\nabla _{b}n_{a}\,,\\[1ex]\mu &:={\bar {m}}^{a}\delta n_{a}={\bar {m}}^{a}m^{b}\nabla _{b}n_{a}\,,&\lambda &:={\bar {m}}^{a}{\bar {\delta }}n_{a}={\bar {m}}^{a}{\bar {m}}^{b}\nabla _{b}n_{a}\,;\end{aligned}}}
ε := − 1 2 ( n a D ℓ a − m ¯ a D m a ) = − 1 2 ( n a ℓ b ∇ b ℓ a − m ¯ a ℓ b ∇ b m a ) , γ := − 1 2 ( n a Δ ℓ a − m ¯ a Δ m a ) = − 1 2 ( n a n b ∇ b ℓ a − m ¯ a n b ∇ b m a ) ,