Construction of a complex null tetrad

Nonholonomic tetrad

The primary method to construct a complex null tetrad is via combinations of orthonormal bases. For a spacetime g a b with an orthonormal tetrad { ω 0 , ω 1 , ω 2 , ω 3 } ,

g a b = − ω 0 ω 0 + ω 1 ω 1 + ω 2 ω 2 + ω 3 ω 3 ,

the covectors { l a , n a , m a , m ¯ a } of the nonholonomic complex null tetrad can be constructed by

l a d x a = ω 0 + ω 1 2 , n a d x a = ω 0 − ω 1 2 ,
m a d x a = ω 2 + i ω 3 2 , m ¯ a d x a = ω 2 − i ω 3 2 ,

and the tetrad vectors { l a , n a , m a , m ¯ a } can be obtained by raising the indices of { l a , n a , m a , m ¯ a } via the inverse metric g a b .

Remark: The nonholonomic construction is actually in accordance with the local light cone structure.

la (na) aligned with null radial geodesics

In Minkowski spacetime, the nonholonomically constructed null vectors { l a , n a } respectively match the outgoing and ingoing null radial rays. As an extension of this idea in generic curved spacetimes, { l a , n a } can still be aligned with the tangent vector field of null radial congruence. However, this type of adaption only works for { t , r , θ , ϕ } , { u , r , θ , ϕ } or { v , r , θ , ϕ } coordinates where the radial behaviors can be well described, with u and v denote the outgoing (retarded) and ingoing (advanced) null coordinate, respectively.

Tetrads adapted to the spacetime structure

At some typical boundary regions such as null infinity, timelike infinity, spacelike infinity, black hole horizons and cosmological horizons, null tetrads adapted to spacetime structures are usually employed to achieve the most succinct Newman–Penrose descriptions.

Newman-Unti tetrad for null infinity

For null infinity, the classic Newman-Unti (NU) tetrad is employed to study asymptotic behaviors at null infinity,

l a ∂ a = ∂ r := D ,
n a ∂ a = ∂ u + U ∂ r + X ∂ ς + X ¯ ∂ ς ¯ := Δ ,
m a ∂ a = ω ∂ r + ξ 3 ∂ ς + ξ 4 ∂ ς ¯ := δ ,
m ¯ a ∂ a = ω ¯ ∂ r + ξ ¯ 3 ∂ ς ¯ + ξ ¯ 4 ∂ ς := δ ¯ ,

where { U , X , ω , ξ 3 , ξ 4 } are tetrad functions to be solved. For the NU tetrad, the foliation leaves are parameterized by the outgoing (advanced) null coordinate u with l a = d u , and r is the normalized affine coordinate along l a ( D r = l a ∂ a r = 1 ) ; the ingoing null vector n a acts as the null generator at null infinity with Δ u = n a ∂ a u = 1 . The coordinates { u , r , ς , ς ¯ } comprise two real affine coordinates { u , r } and two complex stereographic coordinates { ς := e i ϕ cot ⁡ θ 2 , ς ¯ = e − i ϕ cot ⁡ θ 2 } , where { θ , ϕ } are the usual spherical coordinates on the cross-section Δ ^ u = S u 2 (as shown in ref., complex stereographic rather than real isothermal coordinates are used just for the convenience of completely solving NP equations).

Also, for the NU tetrad, the basic gauge conditions are

κ = π = ε = 0 , ρ = ρ ¯ , τ = α ¯ + β .

Adapted tetrad for exteriors and near-horizon vicinity of isolated horizons

For a more comprehensive view of black holes in quasilocal definitions, adapted tetrads which can be smoothly transited from the exterior to the near-horizon vicinity and to the horizons are required. For example, for isolated horizons describing black holes in equilibrium with their exteriors, such a tetrad and the related coordinates can be constructed this way. Choose the first real null covector n a as the gradient of foliation leaves

n a = − d v ,
where v is the ingoing (retarded) Eddington–Finkelstein-type null coordinate, which labels the foliation cross-sections and acts as an affine parameter with regard to the outgoing null vector field l a ∂ a , i.e.

D v = 1 , Δ v = δ v = δ ¯ v = 0 .
Introduce the second coordinate r as an affine parameter along the ingoing null vector field n a , which obeys the normalization

n a ∂ a r = − 1 ⇔ n a ∂ a = − ∂ r .

Now, the first real null tetrad vector n a is fixed. To determine the remaining tetrad vectors { l a , m a , m ¯ a } and their covectors, besides the basic cross-normalization conditions, it is also required that: (i) the outgoing null normal field l a acts as the null generators; (ii) the null frame (covectors) { l a , n a , m a , m ¯ a } are parallelly propagated along n a ∂ a ; (iii) { m a , m ¯ a } spans the {t=constant, r=constant} cross-sections which are labeled by real isothermal coordinates { y , z } .

Tetrads satisfying the above restrictions can be expressed in the general form that

l a ∂ a = ∂ v + U ∂ r + X 3 ∂ y + X 4 ∂ z := D ,
n a ∂ a = − ∂ r := Δ ,
m a ∂ a = Ω ∂ r + ξ 3 ∂ y + ξ 4 ∂ z := δ ,
m ¯ a ∂ a = Ω ¯ ∂ r + ξ ¯ 3 ∂ y + ξ ¯ 4 ∂ z := δ ¯ .

The gauge conditions in this tetrad are

ν = τ = γ = 0 , μ = μ ¯ , π = α + β ¯ ,

Remark: Unlike Schwarzschild-type coordinates, here r=0 represents the horizon, while r>0 (r<0) corresponds to the exterior (interior) of an isolated horizon. People often Taylor expand a scalar Q function with respect to the horizon r=0,

Q = ∑ i = 0 Q ( i ) r i = Q ( 0 ) + Q ( 1 ) r + ⋯ + Q ( n ) r n + …

where Q ( 0 ) refers to its on-horizon value. The very coordinates used in the adapted tetrad above are actually the Gaussian null coordinates employed in studying near-horizon geometry and mechanics of black holes.