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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 $\{{\mathrm{\Phi }}_{00},{\mathrm{\Phi }}_{11},{\mathrm{\Phi }}_{22}\}$, three (or six) complex scalars $\{{\mathrm{\Phi }}_{01}=\overline{{\mathrm{\Phi }}_{10}},{\mathrm{\Phi }}_{02}=\overline{{\mathrm{\Phi }}_{20}},{\mathrm{\Phi }}_{12}=\overline{{\mathrm{\Phi }}_{21}}\}$ and the NP curvature scalar $\mathrm{\Lambda }$. 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 $\{l^a,n^a,m^a,{\overline{m}}^a\}$ and with the convention $\{(-,+,+,+);l^a{n}_a=-1,m^a{\overline{m}}_a=1\}$, the Ricci-NP scalars are defined by (where overline means complex conjugate)

${\mathrm{\Phi }}_{00}:=\frac{1}{2}{R}_{ab}{l}^a{l}^b,{\mathrm{\Phi }}_{11}:=\frac{1}{4}{R}_{ab}(l^a{n}^b+m^a{\overline{m}}^b),{\mathrm{\Phi }}_{22}:=\frac{1}{2}{R}_{ab}{n}^a{n}^b,\mathrm{\Lambda }:=\frac{R}{24};$

${\mathrm{\Phi }}_{01}:=\frac{1}{2}{R}_{ab}{l}^a{m}^b,{\mathrm{\Phi }}_{10}:=\frac{1}{2}{R}_{ab}{l}^a{\overline{m}}^b=\overline{{\mathrm{\Phi }}_{01}},$
${\mathrm{\Phi }}_{02}:=\frac{1}{2}{R}_{ab}{m}^a{m}^b,{\mathrm{\Phi }}_{20}:=\frac{1}{2}{R}_{ab}{\overline{m}}^a{\overline{m}}^b=\overline{{\mathrm{\Phi }}_{02}},$
${\mathrm{\Phi }}_{12}:=\frac{1}{2}{R}_{ab}{\overline{m}}^a{n}^b,{\mathrm{\Phi }}_{21}:=\frac{1}{2}{R}_{ab}{m}^a{n}^b=\overline{{\mathrm{\Phi }}_{12}}.$

Remark I: In these definitions, $R_{ab}$ could be replaced by its trace-free part $Q_{ab}=R_{ab}-\frac{1}{4}{g}_{ab}R$ or by the Einstein tensor $G_{ab}=R_{ab}-\frac{1}{2}{g}_{ab}R$ because of the normalization (i.e. inner product) relations that

$l_a{l}^a=n_a{n}^a=m_a{m}^a={\overline{m}}_a{\overline{m}}^a=0,$

$l_a{m}^a=l_a{\overline{m}}^a=n_a{m}^a=n_a{\overline{m}}^a=0.$

Remark II: Specifically for electrovacuum, we have $\mathrm{\Lambda }=0$, thus

$24\mathrm{\Lambda }=0=R_{ab}{g}^{ab}=R_{ab}(-2l^a{n}^b+2m^a{\overline{m}}^b)\Rightarrow R_{ab}{l}^a{n}^b=R_{ab}{m}^a{\overline{m}}^b,$

and therefore ${\mathrm{\Phi }}_{11}$ is reduced to

${\mathrm{\Phi }}_{11}:=\frac{1}{4}{R}_{ab}(l^a{n}^b+m^a{\overline{m}}^b)=\frac{1}{2}{R}_{ab}{l}^a{n}^b=\frac{1}{2}{R}_{ab}{m}^a{\overline{m}}^a.$

Remark III: If one adopts the convention $\{(+,-,-,-);l^a{n}_a=1,m^a{\overline{m}}_a=-1\}$, the definitions of ${\mathrm{\Phi }}_{ij}$ should take the opposite values; that is to say, ${\mathrm{\Phi }}_{ij}\mapsto -{\mathrm{\Phi }}_{ij}$ after the signature transition.

Alternative derivations

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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 ${\mathrm{\Phi }}_{ij}$ via relevant NP field equations that

${\mathrm{\Phi }}_{00}=D\rho -\overline{\delta }\kappa -({\rho }^2+\sigma \overline{\sigma })-(\epsilon +\overline{\epsilon })\rho +\overline{\kappa }\tau +\kappa (3\alpha +\overline{\beta }-\pi ),$

${\mathrm{\Phi }}_{10}=D\alpha -\overline{\delta }\epsilon -(\rho +\overline{\epsilon }-2\epsilon )\alpha -\beta \overline{\sigma }+\overline{\beta }\epsilon +\kappa \lambda +\overline{\kappa }\gamma -(\epsilon +\rho )\pi ,$

${\mathrm{\Phi }}_{02}=\delta \tau -\mathrm{\Delta }\sigma -(\mu \sigma +\overline{\lambda }\rho )-(\tau +\beta -\overline{\alpha })\tau +(3\gamma -\overline{\gamma })\sigma +\kappa \overline{\nu },$

${\mathrm{\Phi }}_{20}=D\lambda -\overline{\delta }\pi -(\rho \lambda +\overline{\sigma }\mu )-{\pi }^2-(\alpha -\overline{\beta })\pi +\nu \overline{\kappa }+(3\epsilon -\overline{\epsilon })\lambda ,$

${\mathrm{\Phi }}_{12}=\delta \gamma -\mathrm{\Delta }\beta -(\tau -\overline{\alpha }-\beta )\gamma -\mu \tau +\sigma \nu +\epsilon \overline{\nu }+(\gamma -\overline{\gamma }-\mu )\beta -\alpha \overline{\lambda },$

${\mathrm{\Phi }}_{22}=\delta \nu -\mathrm{\Delta }\mu -({\mu }^2+\lambda \overline{\lambda })-(\gamma +\overline{\gamma })\mu +\overline{\nu }\pi -(\tau -3\beta -\overline{\alpha })\nu ,$

$2{\mathrm{\Phi }}_{11}=D\gamma -\mathrm{\Delta }\epsilon +\delta \alpha -\overline{\delta }\beta -(\tau +\overline{\pi })\alpha -\alpha \overline{\alpha }-(\overline{\tau }+\pi )\beta -\beta \overline{\beta }+2\alpha \beta +(\epsilon +\overline{\epsilon })\gamma -(\rho -\overline{\rho })\gamma +(\gamma +\overline{\gamma })\epsilon -(\mu -\overline{\mu })\epsilon -\tau \pi +\nu \kappa -(\mu \rho -\lambda \sigma ),$

while the NP curvature scalar $\mathrm{\Lambda }$ could be directly and easily calculated via $\mathrm{\Lambda }=\frac{R}{24}$ with $R$ being the ordinary scalar curvature of the spacetime metric $g_{ab}=-l_a{n}_b-n_a{l}_b+m_a{\overline{m}}_b+{\overline{m}}_a{m}_b$.

Electromagnetic Ricci-NP scalars

According to the definitions of Ricci-NP scalars ${\mathrm{\Phi }}_{ij}$ above and the fact that $R_{ab}$ could be replaced by $G_{ab}$ in the definitions, ${\mathrm{\Phi }}_{ij}$ are related with the energy-momentum distribution due to Einstein's field equations $G_{ab}=8\pi T_{ab}$. In the simplest situation, i.e. vacuum spacetime in the absence of matter fields with $T_{ab}=0$, we will have ${\mathrm{\Phi }}_{ij}=0$. Moreover, for electromagnetic field, in addition to the aforementioned definitions, ${\mathrm{\Phi }}_{ij}$ could be determined more specifically by

${\mathrm{\Phi }}_{ij}=2{\varphi }_i\overline{{\varphi }_j},(i,j\in \{0,1,2\}),$

where ${\varphi }_i$ denote the three complex Maxwell-NP scalars which encode the six independent components of the Faraday-Maxwell 2-form $F_{ab}$ (i.e. the electromagnetic field strength tensor)

${\varphi }_0:=-F_{ab}{l}^a{m}^b,{\varphi }_1:=-\frac{1}{2}{F}_{ab}(l^a{n}^a-m^a{\overline{m}}^b),{\varphi }_2:=F_{ab}{n}^a{\overline{m}}^b.$

Remark: The equation ${\mathrm{\Phi }}_{ij}=2{\varphi }_i\overline{{\varphi }_j}$ 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 ${\mathrm{\Phi }}_{ij}=\text{Tr}({ϝ}_i{\overline{ϝ}}_j)$ where ${ϝ}_i(i\in \{0,1,2\})$ are Yang-Mills-NP scalars.

See also

• Newman-Penrose formalism
• Weyl scalar