Ricci scalars (Newman–Penrose formalism)
In the Newman–Penrose (NP) formalism of general relativity, independent components of the Ricci tensors of a four-dimensional spacetime are encoded into seven (or ten) Ricci scalars which consist of three real scalars formula_1, three (or six) complex scalars formula_2 and the NP curvature scalar formula_3. Physically, Ricci-NP scalars are related with the energy–momentum distribution of the spacetime due to Einstein's field equation. Definitions. Given a complex null tetrad formula_4 and with the convention formula_5, the Ricci-NP scalars are defined by (where overline means complex conjugate) formula_6 formula_7 formula_8 formula_9  Remark I: In these definitions, formula_10 could be replaced by its trace-free part formula_11 or by the Einstein tensor formula_12 because of the normalization (i.e. inner product) relations that Remark II: Specifically for electrovacuum, we have formula_15, thus formula_16 and therefore formula_17 is reduced to formula_18 Remark III: If one adopts the convention formula_19, the definitions of formula_20 should take the opposite values; that is to say, formula_21 after the signature transition. Alternative derivations. According to the definitions above, one should find out the Ricci tensors before calculating the Ricci-NP scalars via contractions with the corresponding tetrad vectors . However, this method fails to fully reflect the spirit of Newman–Penrose formalism and alternatively, one could compute the spin coefficients and then derive the Ricci-NP scalars formula_20 via relevant NP field equations that while the NP curvature scalar formula_3 could be directly and easily calculated via formula_31 with formula_32 being the ordinary scalar curvature of the spacetime metric formula_33. Electromagnetic Ricci-NP scalars. According to the definitions of Ricci-NP scalars formula_20 above and the fact that formula_10 could be replaced by formula_36 in the definitions, formula_20 are related with the energy–momentum distribution due to Einstein's  field equations formula_38. In the simplest situation, i.e. vacuum spacetime in the absence of matter fields with formula_39, we will have formula_40. Moreover, for electromagnetic field, in addition to the aforementioned definitions, formula_20 could be determined more specifically by formula_42 where formula_43 denote the three complex Maxwell-NP scalars which encode the six independent components of the Faraday-Maxwell 2-form formula_44 (i.e. the electromagnetic field strength tensor) formula_45 Remark: The equation formula_46 for electromagnetic field is however not necessarily valid for other kinds of matter fields. For example, in the case of Yang–Mills fields there will be formula_47  where formula_48 are Yang–Mills-NP scalars.