Calculations in the Newman–Penrose (NP) formalism of general relativity normally begin with the construction of a complex null tetrad
These tetrad vectors respect the following normalization and metric conditions assuming the spacetime signature
gets constructed can one move forward to compute the directional derivatives, spin coefficients, commutators, Weyl-NP scalars
The primary method to construct a complex null tetrad is via combinations of orthonormal bases.
of the nonholonomic complex null tetrad can be constructed by
Remark: The nonholonomic construction is actually in accordance with the local light cone structure.
respectively match the outgoing and ingoing null radial rays.
As an extension of this idea in generic curved spacetimes,
can still be aligned with the tangent vector field of null radial congruence.
denote the outgoing (retarded) and ingoing (advanced) null coordinate, respectively.
Example: Null tetrad for Schwarzschild metric in Eddington-Finkelstein coordinates reads
so the Lagrangian for null radial geodesics of the Schwarzschild spacetime is
Also, the remaining two tetrad (co)vectors are constructed nonholonomically.
With the tetrad defined, one is now able to respectively find out the spin coefficients, Weyl-Np scalars and Ricci-NP scalars that
Example: Null tetrad for extremal Reissner–Nordström metric in Eddington-Finkelstein coordinates reads so the Lagrangian is For null radial geodesics with
, there are two solutions and therefore the tetrad for an ingoing observer can be set up as With the tetrad defined, we are now able to work out the spin coefficients, Weyl-NP scalars and Ricci-NP scalars that
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.
For null infinity, the classic Newman-Unti (NU) tetrad[3][4][5] is employed to study asymptotic behaviors at null infinity,
For the NU tetrad, the foliation leaves are parameterized by the outgoing (advanced) null coordinate
(as shown in ref.,[5] 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
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.
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
as an affine parameter along the ingoing null vector field
and their covectors, besides the basic cross-normalization conditions, it is also required that: (i) the outgoing null normal field
spans the {t=constant, r=constant} cross-sections which are labeled by real isothermal coordinates
Tetrads satisfying the above restrictions can be expressed in the general form that
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.
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.