Newman–Penrose formalism

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 and n μ are just a pair of standard (real) null vectors such that l a n a = − 1 . For example, we can think in terms of spherical coordinates, and take l a to be the outgoing null vector, and 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

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 } and { ( − , + , + , + ) ; l a n a = − 1 , m a 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 } 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 } to { ( − , + , + , + ) , l a n a = − 1 , m a m ¯ a = 1 } , definitions of the spin coefficients, Weyl-NP scalars Ψ i and Ricci-NP scalars Φ i j 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 } and two complex null (co)vectors { m , m ¯ } . Being null (co)vectors, self-normalization of { ℓ , n } are naturally vanishes,

l a l a = n a n a = m a m a = m ¯ a 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 ,

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 .

Here the indices can be raised and lowered by the global metric g a b 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 .

Four directional derivatives

First of all, there are four directional covariant derivatives along with each tetrad vector,

D := ∇ ℓ = l a ∇ a , Δ := ∇ n = n a ∇ a , δ := ∇ m = m a ∇ a , δ ¯ := ∇ m ¯ = m ¯ a ∇ a ,

which are reduced to { D = l a ∂ a , Δ = n a ∂ a , δ = m a ∂ a , δ ¯ = m ¯ a ∂ 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 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 ;

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 ) ,
β := − 1 2 ( n a δ l a − m ¯ a δ m a ) = − 1 2 ( n a m b ∇ b l a − m ¯ a m b ∇ b m a ) ,
α := − 1 2 ( n a δ ¯ l a − m ¯ a δ ¯ m a ) = − 1 2 ( n a m ¯ b ∇ b l a − m ¯ a m ¯ b ∇ b m a ) .

Spin coefficients are the primary quantities in NP formalism, with which all other NP quantities (as defined below) could be calculated indirectly using the NP field equations. Thus, NP formalism is sometimes referred to as spin-coefficient formalism as well.

Transportation equations

Apply the directional derivative operators to tetrad vectors and one could obtain the transportation/propagation equations:

D l a = ( ε + ε ¯ ) l a − κ ¯ m a − κ m ¯ a ,
Δ l a = ( γ + γ ¯ ) l a − τ ¯ m a − τ m ¯ a ,
δ l a = ( α ¯ + β ) l a − ρ ¯ m a − σ m ¯ a ,
δ ¯ l a = ( α + β ¯ ) l a − σ ¯ m a − ρ m ¯ a ;

D n a = π m a + π ¯ m ¯ a − ( ε + ε ¯ ) n a ,
Δ n a = ν m a + ν ¯ m ¯ a − ( γ + γ ¯ ) n a ,
δ n a = μ m a + λ ¯ m ¯ a − ( α ¯ + β ) n a ,
δ ¯ n a = λ m a + μ ¯ m ¯ a − ( α + β ¯ ) n a ;

D m a = ( ε − ε ¯ ) m a + π ¯ l a − κ n a ,
Δ m a = ( γ − γ ¯ ) m a + ν ¯ l a − τ n a ,
δ m a = ( β − α ¯ ) m a + λ ¯ l a − σ n a ,
δ ¯ m a = ( α − β ¯ ) m a + μ ¯ l a − ρ n a ;

D m ¯ a = ( ε ¯ − ε ) m ¯ a + π l a − κ ¯ n a ,
Δ m ¯ a = ( γ ¯ − γ ) m ¯ a + ν l a − τ ¯ n a ,
δ m ¯ a = ( β − α ¯ ) m ¯ a + μ l a − ρ ¯ n a ,
δ ¯ m ¯ a = ( α − β ¯ ) m ¯ a + λ l a − σ ¯ n a .

Commutators

The metric-compatibility or torsion-freeness of the covariant derivative is recast into the commutators of the directional derivatives,

Δ D − D Δ = ( γ + γ ¯ ) D + ( ε + ε ¯ ) Δ − ( τ ¯ + π ) δ − ( τ + π ¯ ) δ ¯ ,
δ D − D δ = ( α ¯ + β − π ¯ ) D + κ Δ − ( ρ ¯ + ε − ε ¯ ) δ − σ δ ¯ ,
δ Δ − Δ δ = − ν ¯ D + ( τ − α ¯ − β ) Δ + ( μ − γ + γ ¯ ) δ + λ ¯ δ ¯ ,
δ ¯ δ − δ δ ¯ = ( μ ¯ − μ ) D + ( ρ ¯ − ρ ) Δ + ( α − β ¯ ) δ − ( α ¯ − β ) δ ¯ ,

which imply that

Δ l a − D n a = ( γ + γ ¯ ) l a + ( ε + ε ¯ ) n a − ( τ ¯ + π ) m a − ( τ + π ¯ ) m ¯ a ,
δ l a − D m a = ( α ¯ + β − π ¯ ) l a + κ n a − ( ρ ¯ + ε − ε ¯ ) m a − σ m ¯ a ,
δ n a − Δ m a = − ν ¯ l a + ( τ − α ¯ − β ) n a + ( μ − γ + γ ¯ ) m a + λ ¯ m ¯ a ,
δ ¯ m a − δ m ¯ a = ( μ ¯ − μ ) l a + ( ρ ¯ − ρ ) n a + ( α − β ¯ ) m a − ( α ¯ − β ) m ¯ a .

Note: (i) The above equations can be regarded either as implications of the commutators or combinations of the transportation equations; (ii) In these implied equations, the vectors { l a , n a , m a , m ¯ a } can be replaced by the covectors and the equations still hold.

Weyl–NP and Ricci–NP scalars

The 10 independent components of the Weyl tensor can be encoded into 5 complex Weyl-NP scalars,

Ψ 0 := C a b c d l a m b l c m d , Ψ 1 := C a b c d l a n b l c m d , Ψ 2 := C a b c d l a m b m ¯ c n d , Ψ 3 := C a b c d l a n b m ¯ c n d , Ψ 4 := C a b c d n a m ¯ b n c m ¯ d .

The 10 independent components of the Ricci tensor are encoded into 4 real scalars { Φ 00 , Φ 11 , Φ 22 , Λ } and 3 complex scalars { Φ 10 , Φ 20 , Φ 21 } (with their complex conjugates),

Φ 00 := 1 2 R a b l a l b , Φ 11 := 1 4 R a b ( l a n b + m a m ¯ b ) , Φ 22 := 1 2 R a b n a n b , Λ := R 24 ;

Φ 01 := 1 2 R a b l a m b , Φ 10 := 1 2 R a b l a m ¯ b = Φ 01 ¯ ,
Φ 02 := 1 2 R a b m a m b , Φ 20 := 1 2 R a b m ¯ a m ¯ b = Φ 02 ¯ ,
Φ 12 := 1 2 R a b m a n b , Φ 21 := 1 2 R a b m ¯ a n b = Φ 12 ¯ .

In these definitions, R a b could be replaced by its trace-free part Q a b = R a b − 1 4 g a b R or by the Einstein tensor G a b = R a b − 1 2 g a b R because of the normalization relations. Also, Φ 11 is reduced to Φ 11 = 1 2 R a b l a n b = 1 2 R a b m a m ¯ a for electrovacuum ( Λ = 0 ).

NP field equations

In a complex null tetrad, Ricci identities give rise to the following NP field equations connecting spin coefficients, Weyl-NP and Ricci-NP scalars (recall that in an orthogonal tetrad, Ricci rotation coefficients would respect Cartan's first and second structure equations),

D ρ − δ ¯ κ = ( ρ 2 + σ σ ¯ ) + ( ε + ε ¯ ) ρ − κ ¯ τ − κ ( 3 α + β ¯ − π ) + Φ 00 ,
D σ − δ κ = ( ρ + ρ ¯ ) σ + ( 3 ε − ε ¯ ) σ − ( τ − π ¯ + α ¯ + 3 β ) κ + Ψ 0 ,
D τ − Δ κ = ( τ + π ¯ ) ρ + ( τ ¯ + π ) σ + ( ε − ε ¯ ) τ − ( 3 γ + γ ¯ ) κ + Ψ 1 + Φ 01 ,
D α − δ ¯ ε = ( ρ + ε ¯ − 2 ε ) α + β σ ¯ − β ¯ ε − κ λ − κ ¯ γ + ( ε + ρ ) π + Φ 10 ,
D β − δ ε = ( α + π ) σ + ( ρ ¯ − ε ¯ ) β − ( μ + γ ) κ − ( α ¯ − π ¯ ) ε + Ψ 1 ,
D γ − Δ ε = ( τ + π ¯ ) α + ( τ ¯ + π ) β − ( ε + ε ¯ ) γ − ( γ + γ ¯ ) ε + τ π − ν κ + Ψ 2 + Φ 11 − Λ ,
D λ − δ ¯ π = ( ρ λ + σ ¯ μ ) + π 2 + ( α − β ¯ ) π − ν κ ¯ − ( 3 ε − ε ¯ ) λ + Φ 20 ,
D μ − δ π = ( ρ ¯ μ + σ λ ) + π π ¯ − ( ε + ε ¯ ) μ − ( α ¯ − β ) π − ν κ + Ψ 2 + 2 Λ ,
D ν − Δ π = ( π + τ ¯ ) μ + ( π ¯ + τ ) λ + ( γ − γ ¯ ) π − ( 3 ε + ε ¯ ) ν + Ψ 3 + Φ 21 ,
Δ λ − δ ¯ ν = − ( μ + μ ¯ ) λ − ( 3 γ − γ ¯ ) λ + ( 3 α + β ¯ + π − τ ¯ ) ν − Ψ 4 ,
δ ρ − δ ¯ σ = ρ ( α ¯ + β ) − σ ( 3 α − β ¯ ) + ( ρ − ρ ¯ ) τ + ( μ − μ ¯ ) κ − Ψ 1 + Φ 01 ,
δ α − δ ¯ β = ( μ ρ − λ σ ) + α α ¯ + β β ¯ − 2 α β + γ ( ρ − ρ ¯ ) + ε ( μ − μ ¯ ) − Ψ 2 + Φ 11 + Λ ,
δ λ − δ ¯ μ = ( ρ − ρ ¯ ) ν + ( μ − μ ¯ ) π + ( α + β ¯ ) μ + ( α ¯ − 3 β ) λ − Ψ 3 + Φ 21 ,
δ ν − Δ μ = ( μ 2 + λ λ ¯ ) + ( γ + γ ¯ ) μ − ν ¯ π + ( τ − 3 β − α ¯ ) ν + Φ 22 ,
δ γ − Δ β = ( τ − α ¯ − β ) γ + μ τ − σ ν − ε ν ¯ − ( γ − γ ¯ − μ ) β + α λ ¯ + Φ 12 ,
δ τ − Δ σ = ( μ σ + λ ¯ ρ ) + ( τ + β − α ¯ ) τ − ( 3 γ − γ ¯ ) σ − κ ν ¯ + Φ 02 ,
Δ ρ − δ ¯ τ = − ( ρ μ ¯ + σ λ ) + ( β ¯ − α − τ ¯ ) τ + ( γ + γ ¯ ) ρ + ν κ − Ψ 2 − 2 Λ ,
Δ α − δ ¯ γ = ( ρ + ε ) ν − ( τ + β ) λ + ( γ ¯ − μ ¯ ) α + ( β ¯ − τ ¯ ) γ − Ψ 3 .

Also, the Weyl-NP scalars Ψ i and the Ricci-NP scalars Φ i j can be calculated indirectly from the above NP field equations after obtaining the spin coefficients rather than directly using their definitions.

Maxwell–NP scalars, Maxwell equations in NP formalism

The six independent components of the Faraday-Maxwell 2-form (i.e. the electromagnetic field strength tensor) F a b can be encoded into three complex Maxwell-NP scalars

ϕ 0 := F a b l a m b , ϕ 1 := 1 2 F a b ( l a n b + m ¯ a m b ) , ϕ 2 := F a b m ¯ a n b ,

and therefore the eight real Maxwell equations d F = 0 and d ⋆ F = 0 (as F = d A ) can be transformed into four complex equations,

D ϕ 1 − δ ¯ ϕ 0 = ( π − 2 α ) ϕ 0 + 2 ρ ϕ 1 − κ ϕ 2 ,
D ϕ 2 − δ ¯ ϕ 1 = − λ ϕ 0 + 2 π ϕ 1 + ( ρ − 2 ε ) ϕ 2 ,
Δ ϕ 0 − δ ϕ 1 = ( 2 γ − μ ) ϕ 0 − 2 τ ϕ 1 + σ ϕ 2 ,
Δ ϕ 1 − δ ϕ 2 = ν ϕ 0 − 2 μ ϕ 1 + ( 2 β − τ ) ϕ 2 ,

with the Ricci-NP scalars Φ i j related to Maxwell scalars by

Φ i j = 2 ϕ i ϕ j ¯ , ( i , j ∈ { 0 , 1 , 2 } ) .

It is worthwhile to point out that, the supplementary equation Φ i j = 2 ϕ i ϕ j ¯ is only valid for electromagnetic fields; for example, in the case of Yang-Mills fields there will be Φ i j = Tr ( ϝ i ϝ ¯ j ) where ϝ i ( i ∈ { 0 , 1 , 2 } ) are Yang-Mills-NP scalars.

To sum up, the aforementioned transportation equations, NP field equations and Maxwell-NP equations together constitute the Einstein-Maxwell equations in Newman–Penrose formalism.

Applications of NP formalism to gravitational radiation field

The Weyl scalar Ψ 4 was defined by Newman & Penrose as

(note, however, that the overall sign is arbitrary, and that Newman & Penrose worked with a "timelike" metric signature of ( + , − , − , − ) ). In empty space, the Einstein Field Equations reduce to R α β = 0 . From the definition of the Weyl tensor, we see that this means that it equals the Riemann tensor, C α β γ δ = R α β γ δ . We can make the standard choice for the tetrad at infinity:

In transverse-traceless gauge, a simple calculation shows that linearized gravitational waves are related to components of the Riemann tensor as

assuming propagation in the r ^ direction. Combining these, and using the definition of Ψ 4 above, we can write

Far from a source, in nearly flat space, the fields h + and h × encode everything about gravitational radiation propagating in a given direction. Thus, we see that Ψ 4 encodes in a single complex field everything about (outgoing) gravitational waves.

Radiation from a finite source

Using the wave-generation formalism summarised by Thorne, we can write the radiation field quite compactly in terms of the mass multipole, current multipole, and spin-weighted spherical harmonics:

Here, prefixed superscripts indicate time derivatives. That is, we define

The components I l m and S l m are the mass and current multipoles, respectively. − 2 Y l m is the spin-weight -2 spherical harmonic.