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 ${\displaystyle \{\Phi _{00},\Phi _{11},\Phi _{22}\}}$, three (or six) complex scalars ${\displaystyle \{\Phi _{01}={\overline {\Phi }}_{10}\,,\Phi _{02}={\overline {\Phi }}_{20}\,,\Phi _{12}={\overline {\Phi }}_{21}\}}$ and the NP curvature scalar ${\displaystyle \Lambda }$. 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 ${\displaystyle \{l^{a},n^{a},m^{a},{\bar {m}}^{a}\}}$ and with the convention ${\displaystyle \{(-,+,+,+);l^{a}n_{a}=-1\,,m^{a}{\bar {m}}_{a}=1\}}$, the Ricci-NP scalars are defined by^[1][2][3] (where overline means complex conjugate)

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

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

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

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

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

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

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

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

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

Remark III: If one adopts the convention ${\displaystyle \{(+,-,-,-);l^{a}n_{a}=1\,,m^{a}{\bar {m}}_{a}=-1\}}$, the definitions of ${\displaystyle \Phi _{ij}}$ should take the opposite values;^[4][5][6][7] that is to say, ${\displaystyle \Phi _{ij}\mapsto -\Phi _{ij}}$ 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 ${\displaystyle \Phi _{ij}}$ via relevant NP field equations that^[2][7]

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

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

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

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

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

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

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

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

Electromagnetic Ricci-NP scalars

According to the definitions of Ricci-NP scalars ${\displaystyle \Phi _{ij}}$ above and the fact that ${\displaystyle R_{ab}}$ could be replaced by ${\displaystyle G_{ab}}$ in the definitions, ${\displaystyle \Phi _{ij}}$ are related with the energy–momentum distribution due to Einstein's field equations ${\displaystyle G_{ab}=8\pi T_{ab}}$. In the simplest situation, i.e. vacuum spacetime in the absence of matter fields with ${\displaystyle T_{ab}=0}$, we will have ${\displaystyle \Phi _{ij}=0}$. Moreover, for electromagnetic field, in addition to the aforementioned definitions, ${\displaystyle \Phi _{ij}}$ could be determined more specifically by^[1]

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

where ${\displaystyle \phi _{i}}$ denote the three complex Maxwell-NP scalars^[1] which encode the six independent components of the Faraday-Maxwell 2-form ${\displaystyle F_{ab}}$ (i.e. the electromagnetic field strength tensor)

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

Remark: The equation ${\displaystyle \Phi _{ij}=2\,\phi _{i}\,{\overline {\phi }}_{j}}$ 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 ${\displaystyle \Phi _{ij}=\,{\text{Tr}}\,(\digamma _{i}\,{\bar {\digamma }}_{j})}$ where ${\displaystyle \digamma _{i}(i\in \{0,1,2\})}$ are Yang–Mills-NP scalars.^[8]

See also

• Newman–Penrose formalism
• Weyl scalar

References