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

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

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

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

$l_{a}l^{a}=n_{a}n^{a}=m_{a}m^{a}={\bar {m}}_{a}{\bar {m}}^{a}=0\,,$

$l_{a}m^{a}=l_{a}{\bar {m}}^{a}=n_{a}m^{a}=n_{a}{\bar {m}}^{a}=0\,.$

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

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

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

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

Remark III: If one adopts the convention $\{(+,-,-,-);l^{a}n_{a}=1\,,m^{a}{\bar {m}}_{a}=-1\}$, the definitions of $\Phi _{ij}$ should take the opposite values; that is to say, $\Phi _{ij}\mapsto -\Phi _{ij}$ 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 $\Phi _{ij}$ via relevant NP field equations that

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

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

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

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

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

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

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

while the NP curvature scalar $\Lambda$ could be directly and easily calculated via $\Lambda ={\frac {R}{24}}$ with $R$ being the ordinary scalar curvature of the spacetime metric $g_{ab}=-l_{a}n_{b}-n_{a}l_{b}+m_{a}{\bar {m}}_{b}+{\bar {m}}_{a}m_{b}$.

Electromagnetic Ricci-NP scalars

According to the definitions of Ricci-NP scalars $\Phi _{ij}$ above and the fact that $R_{ab}$ could be replaced by $G_{ab}$ in the definitions, $\Phi _{ij}$ 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 $\Phi _{ij}=0$. Moreover, for electromagnetic field, in addition to the aforementioned definitions, $\Phi _{ij}$ could be determined more specifically by

$\Phi _{ij}=\,2\,\phi _{i}\,{\overline {\phi _{j}}}\,,\quad (i,j\in \{0,1,2\})\,,$

where $\phi _{i}$ 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)

$\phi _{0}:=-F_{ab}l^{a}m^{b}\,,\quad \phi _{1}:=-{\frac {1}{2}}F_{ab}{\big (}l^{a}n^{a}-m^{a}{\bar {m}}^{b}{\big )}\,,\quad \phi _{2}:=F_{ab}n^{a}{\bar {m}}^{b}\,.$

Remark: The equation $\Phi _{ij}=2\,\phi _{i}\,{\overline {\phi _{j}}}$ 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 $\Phi _{ij}=\,{\text{Tr}}\,(\digamma _{i}\,{\bar {\digamma }}_{j})$ where $\digamma _{i}(i\in \{0,1,2\})$ are Yang-Mills-NP scalars.

See also

• Newman-Penrose formalism
• Weyl scalar