Newman–Penrose formalism
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 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—formula_1 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: 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, formula_6 and formula_7 are just a pair of standard (real) null vectors such that formula_8. For example, we can think in terms of spherical coordinates, and take formula_9 to be the outgoing null vector, and formula_10 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: formula_12 and formula_13. 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 formula_13 for a systematic review of the NP formalism (see also refs .). It's important to note that, when switching from formula_15 to formula_16, definitions of the spin coefficients, Weyl-NP scalars formula_17 and Ricci-NP scalars formula_18 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 formula_19 and two complex null (co)vectors formula_20. Being "null" (co)vectors, "self"-normalization of formula_19 naturally vanishes, formula_22, so the following two pairs of "cross"-normalization are adopted formula_23 while contractions between the two pairs are also vanishing, formula_24. Here the indices can be raised and lowered by the global  metric formula_25 which in turn can be obtained via formula_26 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 formula_27 is expressed using four separate symbols (formula_28) which name a directional covariant derivative operator for each tetrad direction. Given a linear combination of tetrad vectors, formula_29, the covariant derivative operator in the formula_30 direction is formula_31. The operators are defined as formula_32 formula_33 which reduce to formula_34 when acting on "scalar" functions. Twelve spin coefficients. In NP formalism , instead of using index notations as in orthogonal tetrads, each Ricci rotation coefficient formula_35 in the null tetrad is assigned a lower-case Greek letter, which constitute the 12 complex "spin coefficients" (in three groups), formula_36 formula_37 formula_38 formula_39 formula_40 formula_41 formula_42 formula_43 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: covariant derivatives of tetrad vectors. The sixteen directional covariant derivatives of tetrad vectors are sometimes  called the "transportation/propagation equations," perhaps because the derivatives are zero when the tetrad vector is parallel propagated or transported in the direction of the derivative operator. These results in this exact notation are given by ODonnell: formula_44 formula_45 formula_46 formula_47 formula_48 formula_49 formula_50 formula_51 formula_52 formula_53 formula_54 formula_55 formula_56 formula_57 formula_58 formula_59 Interpretation of formula_60 from formula_61 and formula_62. The two equations for the covariant derivative of a real null tetrad vector in its own direction indicate whether or not the vector is tangent to a geodesic and if so, whether the geodesic has an affine parameter. A null  tangent vector formula_63 is tangent to an affinely parameterized null geodesic if formula_64, which is to say if the vector is unchanged by parallel propagation or transportation in its own direction. formula_65 shows that formula_9 is tangent to a geodesic if and only if formula_67, and is tangent to an affinely parameterized geodesic if in addition formula_68. Similarly, formula_69 shows that formula_10 is geodesic if and only if formula_71, and has affine parameterization when formula_72. Commutators. The metric-compatibility or torsion-freeness of the covariant derivative is recast into the "commutators of the directional derivatives", formula_77 formula_78 formula_79 formula_80 which  imply that formula_81 formula_82 formula_83 formula_84 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 formula_85 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, formula_86 formula_87 formula_88 formula_89 formula_90 The 10 independent components of the Ricci tensor are encoded into 4 "real" scalars formula_91, formula_92, formula_93, formula_94 and 3 "complex" scalars formula_95 (with their complex conjugates ), formula_96 formula_97 formula_98 formula_99 In these definitions, formula_100 could be replaced by its trace-free part formula_101 or by the Einstein tensor formula_102 because of the normalization relations. Also, formula_92 is reduced to formula_104 for electrovacuum (formula_105). Einstein–Maxwell–NP equations. 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), These equations in various notations can be found in several texts. The notation in  Frolov and Novikov formula_106 formula_107 formula_108 formula_109 formula_110 formula_111 formula_112 formula_113 formula_114 formula_115 formula_116 formula_117 formula_118 formula_119 formula_120 formula_121 formula_122 formula_123 Also, the Weyl-NP scalars formula_17 and the Ricci-NP scalars formula_18 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) formula_126 can be encoded into three complex Maxwell-NP scalars formula_127 and therefore the eight real Maxwell equations formula_128 and formula_129 (as  formula_130) can be transformed into four complex equations, formula_131 formula_132 formula_133 formula_134 with the Ricci-NP scalars formula_18 related to Maxwell scalars by formula_136 It is worthwhile to point out that, the supplementary equation formula_137 is only valid for electromagnetic fields; for example, in the case of Yang-Mills fields there will be formula_138 where formula_139 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 formula_1 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 formula_142). In empty space, the Einstein Field Equations reduce to formula_143. From the definition of the Weyl tensor, we see that this means that it equals the Riemann tensor, formula_144. 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 formula_150 direction. Combining these, and using the definition of formula_1 above, we  can write Far from a source, in nearly flat space, the fields formula_153 and formula_154 encode everything about gravitational radiation propagating in a given direction. Thus, we see that formula_1 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 formula_158 and formula_159 are the mass and current multipoles, respectively. formula_160 is the  spin-weight -2 spherical harmonic.