Newman–Penrose formalism

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 ${\displaystyle {\mathrm{\nabla }}_a}$ is expressed using four separate symbols (${\displaystyle D,\mathrm{\Delta },\delta ,\overline{\delta }}$) which name a directional covariant derivative operator for each tetrad direction. Given a linear combination of tetrad vectors, ${\displaystyle X^a=\mathrm{a}{\ell }^a+\mathrm{b}{n}^a+\mathrm{c}{m}^a+\mathrm{d}{\overline{m}}^a}$, the covariant derivative operator in the ${\displaystyle X^a}$ direction is ${\displaystyle {\mathrm{\nabla }}_X=X^a{\mathrm{\nabla }}_a=(\mathrm{a}D+\mathrm{b}\mathrm{\Delta }+\mathrm{c}\delta +\mathrm{d}\overline{\delta })}$.

The operators are defined as

$${\displaystyle \begin{array}{cccc}D& :={\mathrm{\nabla }}_{\mathit{\ell }}={\ell }^a{\mathrm{\nabla }}_a,& \mathrm{\Delta }& :={\mathrm{\nabla }}_{\mathit{n}}=n^a{\mathrm{\nabla }}_a,\\ \delta & :={\mathrm{\nabla }}_{\mathit{m}}=m^a{\mathrm{\nabla }}_a,& \overline{\delta }& :={\mathrm{\nabla }}_{\overline{\mathit{m}}}={\overline{m}}^a{\mathrm{\nabla }}_a,\end{array}}$$

which reduce to ${\displaystyle D={\ell }^a{\mathrm{\partial }}_a,\mathrm{\Delta }=n^a{\mathrm{\partial }}_a,\delta =m^a{\mathrm{\partial }}_a,\overline{\delta }={\overline{m}}^a{\mathrm{\partial }}_a}$ when acting on scalar functions.

Twelve spin coefficients

In NP formalism, instead of using index notations as in orthogonal tetrads, each Ricci rotation coefficient ${\displaystyle {\gamma }_{ijk}}$ in the null tetrad is assigned a lower-case Greek letter, which constitute the 12 complex spin coefficients (in three groups),

$${\displaystyle \begin{array}{cccc}\kappa & :=-m^{a}D{\ell }_a=-m^a{\ell }^b{\mathrm{\nabla }}_b{\ell }_a,& \tau & :=-m^a\mathrm{\Delta }{\ell }_a=-m^a{n}^b{\mathrm{\nabla }}_b{\ell }_a,\\ \sigma & :=-m^a\delta {\ell }_a=-m^a{m}^b{\mathrm{\nabla }}_b{\ell }_a,& \rho & :=-m^a\overline{\delta }{\ell }_a=-m^a{\overline{m}}^b{\mathrm{\nabla }}_b{\ell }_a;\\ \pi & :={\overline{m}}^{a}D{n}_a={\overline{m}}^a{\ell }^b{\mathrm{\nabla }}_b{n}_a,& \nu & :={\overline{m}}^a\mathrm{\Delta }{n}_a={\overline{m}}^a{n}^b{\mathrm{\nabla }}_b{n}_a,\\ \mu & :={\overline{m}}^a\delta n_a={\overline{m}}^a{m}^b{\mathrm{\nabla }}_b{n}_a,& \lambda & :={\overline{m}}^a\overline{\delta }{n}_a={\overline{m}}^a{\overline{m}}^b{\mathrm{\nabla }}_b{n}_a;\end{array}}$$ $${\displaystyle \begin{array}{cc}\epsilon & :=-\frac{1}{2}\left(n^{a}D{\ell }_a-{\overline{m}}^{a}D{m}_a\right)=-\frac{1}{2}\left(n^a{\ell }^b{\mathrm{\nabla }}_b{\ell }_a-{\overline{m}}^a{\ell }^b{\mathrm{\nabla }}_b{m}_a\right),\\ \gamma & :=-\frac{1}{2}\left(n^a\mathrm{\Delta }{\ell }_a-{\overline{m}}^a\mathrm{\Delta }{m}_a\right)=-\frac{1}{2}\left(n^a{n}^b{\mathrm{\nabla }}_b{\ell }_a-{\overline{m}}^a{n}^b{\mathrm{\nabla }}_b{m}_a\right),\\ \beta & :=-\frac{1}{2}\left(n^a\delta {\ell }_a-{\overline{m}}^a\delta m_a\right)=-\frac{1}{2}\left(n^a{m}^b{\mathrm{\nabla }}_b{\ell }_a-{\overline{m}}^a{m}^b{\mathrm{\nabla }}_b{m}_a\right),\\ \alpha & :=-\frac{1}{2}\left(n^a\overline{\delta }{\ell }_a-{\overline{m}}^a\overline{\delta }{m}_a\right)=-\frac{1}{2}\left(n^a{\overline{m}}^b{\mathrm{\nabla }}_b{\ell }_a-{\overline{m}}^a{\overline{m}}^b{\mathrm{\nabla }}_b{m}_a\right).\end{array}}$$

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 O'Donnell: $${\displaystyle \begin{array}{cc}D{\ell }^a& =\left(\epsilon +\overline{\epsilon }\right){\ell }^a-\overline{\kappa }{m}^a-\kappa {\overline{m}}^a,\\ \mathrm{\Delta }{\ell }^a& =\left(\gamma +\overline{\gamma }\right){\ell }^a-\overline{\tau }{m}^a-\tau {\overline{m}}^a,\\ \delta {\ell }^a& =\left(\overline{\alpha }+\beta \right){\ell }^a-\overline{\rho }{m}^a-\sigma {\overline{m}}^a,\\ \overline{\delta }{\ell }^a& =\left(\alpha +\overline{\beta }\right){\ell }^a-\overline{\sigma }{m}^a-\rho {\overline{m}}^a;\end{array}}$$ $${\displaystyle \begin{array}{cc}D{n}^a& =\pi m^a+\overline{\pi }{\overline{m}}^a-\left(\epsilon +\overline{\epsilon }\right)n^a,\\ \mathrm{\Delta }{n}^a& =\nu m^a+\overline{\nu }{\overline{m}}^a-\left(\gamma +\overline{\gamma }\right)n^a,\\ \delta n^a& =\mu m^a+\overline{\lambda }{\overline{m}}^a-\left(\overline{\alpha }+\beta \right)n^a,\\ \overline{\delta }{n}^a& =\lambda m^a+\overline{\mu }{\overline{m}}^a-\left(\alpha +\overline{\beta }\right)n^a;\end{array}}$$ $${\displaystyle \begin{array}{cc}D{m}^a& =\left(\epsilon -\overline{\epsilon }\right)m^a+\overline{\pi }{\ell }^a-\kappa n^a,\\ \mathrm{\Delta }{m}^a& =\left(\gamma -\overline{\gamma }\right)m^a+\overline{\nu }{\ell }^a-\tau n^a,\\ \delta m^a& =\left(\beta -\overline{\alpha }\right)m^a+\overline{\lambda }{\ell }^a-\sigma n^a,\\ \overline{\delta }{m}^a& =\left(\alpha -\overline{\beta }\right)m^a+\overline{\mu }{\ell }^a-\rho n^a;\end{array}}$$ $${\displaystyle \begin{array}{cc}D{\overline{m}}^a& =\left(\overline{\epsilon }-\epsilon \right){\overline{m}}^a+\pi {\ell }^a-\overline{\kappa }{n}^a,\\ \mathrm{\Delta }{\overline{m}}^a& =\left(\overline{\gamma }-\gamma \right){\overline{m}}^a+\nu {\ell }^a-\overline{\tau }{n}^a,\\ \delta {\overline{m}}^a& =\left(\overline{\alpha }-\beta \right){\overline{m}}^a+\mu {\ell }^a-\overline{\rho }{n}^a,\\ \overline{\delta }{\overline{m}}^a& =\left(\overline{\beta }-\alpha \right){\overline{m}}^a+\lambda {\ell }^a-\overline{\sigma }{n}^a.\end{array}}$$

Interpretation of κ, ε, ν, γ from Dℓa and Δna

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 ${\displaystyle T^a}$ is tangent to an affinely parameterized null geodesic if ${\displaystyle T^b{\mathrm{\nabla }}_b{T}^a=0}$, which is to say if the vector is unchanged by parallel propagation or transportation in its own direction.

${\displaystyle D{\ell }^a=(\epsilon +\overline{\epsilon }){\ell }^a-\overline{\kappa }{m}^a-\kappa {\overline{m}}^a}$ shows that ${\displaystyle {\ell }^a}$ is tangent to a geodesic if and only if ${\displaystyle \kappa =0}$, and is tangent to an affinely parameterized geodesic if in addition ${\displaystyle (\epsilon +\overline{\epsilon })=0}$. Similarly, ${\displaystyle \mathrm{\Delta }{n}^a=\nu m^a+\overline{\nu }{\overline{m}}^a-(\gamma +\overline{\gamma })n^a}$ shows that ${\displaystyle n^a}$ is geodesic if and only if ${\displaystyle \nu =0}$, and has affine parameterization when ${\displaystyle (\gamma +\overline{\gamma })=0}$.

(The complex null tetrad vectors ${\displaystyle m^a=x^a+i{y}^a}$ and ${\displaystyle {\overline{m}}^a=x^a-i{y}^a}$ would have to be separated into the spacelike basis vectors ${\displaystyle x^a}$ and ${\displaystyle y^a}$ before asking if either or both of those are tangent to spacelike geodesics.)

Commutators

The metric-compatibility or torsion-freeness of the covariant derivative is recast into the commutators of the directional derivatives,

$${\displaystyle \begin{array}{cc}\mathrm{\Delta }D-D\mathrm{\Delta }& =\left(\gamma +\overline{\gamma }\right)D+\left(\epsilon +\overline{\epsilon }\right)\mathrm{\Delta }-\left(\overline{\tau }+\pi \right)\delta -\left(\tau +\overline{\pi }\right)\overline{\delta },\\ \delta D-D\delta & =\left(\overline{\alpha }+\beta -\overline{\pi }\right)D+\kappa \mathrm{\Delta }-\left(\overline{\rho }+\epsilon -\overline{\epsilon }\right)\delta -\sigma \overline{\delta },\\ \delta \mathrm{\Delta }-\mathrm{\Delta }\delta & =-\overline{\nu }D+\left(\tau -\overline{\alpha }-\beta \right)\mathrm{\Delta }+\left(\mu -\gamma +\overline{\gamma }\right)\delta +\overline{\lambda }\overline{\delta },\\ \overline{\delta }\delta -\delta \overline{\delta }& =\left(\overline{\mu }-\mu \right)D+\left(\overline{\rho }-\rho \right)\mathrm{\Delta }+\left(\alpha -\overline{\beta }\right)\delta -\left(\overline{\alpha }-\beta \right)\overline{\delta },\end{array}}$$

which imply that

$${\displaystyle \begin{array}{cc}\mathrm{\Delta }{\ell }^a-D{n}^a& =\left(\gamma +\overline{\gamma }\right){\ell }^a+\left(\epsilon +\overline{\epsilon }\right)n^a-\left(\overline{\tau }+\pi \right)m^a-\left(\tau +\overline{\pi }\right){\overline{m}}^a,\\ \delta {\ell }^a-D{m}^a& =\left(\overline{\alpha }+\beta -\overline{\pi }\right){\ell }^a+\kappa n^a-\left(\overline{\rho }+\epsilon -\overline{\epsilon }\right)m^a-\sigma {\overline{m}}^a,\\ \delta n^a-\mathrm{\Delta }{m}^a& =-\overline{\nu }{\ell }^a+\left(\tau -\overline{\alpha }-\beta \right)n^a+\left(\mu -\gamma +\overline{\gamma }\right)m^a+\overline{\lambda }{\overline{m}}^a,\\ \overline{\delta }{m}^a-\delta {\overline{m}}^a& =\left(\overline{\mu }-\mu \right){\ell }^a+\left(\overline{\rho }-\rho \right)n^a+\left(\alpha -\overline{\beta }\right)m^a-\left(\overline{\alpha }-\beta \right){\overline{m}}^a.\end{array}}$$

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 ${\displaystyle \{{\ell }^a,n^a,m^a,{\overline{m}}^a\}}$ 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,

$${\displaystyle \begin{array}{cccc}{\mathrm{\Psi }}_0& :=C_{abcd}{\ell }^a{m}^b{\ell }^c{m}^d,& {\mathrm{\Psi }}_1& :=C_{abcd}{\ell }^a{n}^b{\ell }^c{m}^d,\\ {\mathrm{\Psi }}_2& :=C_{abcd}{\ell }^a{m}^b{\overline{m}}^c{n}^d,& {\mathrm{\Psi }}_3& :=C_{abcd}{\ell }^a{n}^b{\overline{m}}^c{n}^d,\\ {\mathrm{\Psi }}_4& :=C_{abcd}{n}^a{\overline{m}}^b{n}^c{\overline{m}}^d.\end{array}}$$

The 10 independent components of the Ricci tensor are encoded into 4 real scalars ${\displaystyle \{{\mathrm{\Phi }}_{00}}$, ${\displaystyle {\mathrm{\Phi }}_{11}}$, ${\displaystyle {\mathrm{\Phi }}_{22}}$, ${\displaystyle \mathrm{\Lambda }\}}$ and 3 complex scalars ${\displaystyle \{{\mathrm{\Phi }}_{10},{\mathrm{\Phi }}_{20},{\mathrm{\Phi }}_{21}\}}$ (with their complex conjugates),

$${\displaystyle \begin{array}{cccc}{\mathrm{\Phi }}_{00}& :=\frac{1}{2}{R}_{ab}{\ell }^a{\ell }^b,& {\mathrm{\Phi }}_{11}& :=\frac{1}{4}{R}_{ab}\left({\ell }^a{n}^b+m^a{\overline{m}}^b\right),\\ {\mathrm{\Phi }}_{22}& :=\frac{1}{2}{R}_{ab}{n}^a{n}^b,& \mathrm{\Lambda }& :=\frac{1}{24}R;\end{array}}$$$${\displaystyle \begin{array}{cccc}{\mathrm{\Phi }}_{01}& :=\frac{1}{2}{R}_{ab}{\ell }^a{m}^b,& {\mathrm{\Phi }}_{10}& :=\frac{1}{2}{R}_{ab}{\ell }^a{\overline{m}}^b=\overline{{\mathrm{\Phi }}_{01}},\\ {\mathrm{\Phi }}_{02}& :=\frac{1}{2}{R}_{ab}{m}^a{m}^b,& {\mathrm{\Phi }}_{20}& :=\frac{1}{2}{R}_{ab}{\overline{m}}^a{\overline{m}}^b=\overline{{\mathrm{\Phi }}_{02}},\\ {\mathrm{\Phi }}_{12}& :=\frac{1}{2}{R}_{ab}{m}^a{n}^b,& {\mathrm{\Phi }}_{21}& :=\frac{1}{2}{R}_{ab}{\overline{m}}^a{n}^b=\overline{{\mathrm{\Phi }}_{12}}.\end{array}}$$

In these definitions, ${\displaystyle R_{ab}}$ 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 relations. Also, ${\displaystyle {\mathrm{\Phi }}_{11}}$ is reduced to ${\mathrm{\Phi }}_{11}=\frac{1}{2}{R}_{ab}{\ell }^a{n}^b=\frac{1}{2}{R}_{ab}{m}^a{\overline{m}}^b$ for electrovacuum (${\displaystyle \mathrm{\Lambda }=0}$).

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 is identical.

$${\displaystyle \begin{array}{cc}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 }}_{00},\\ D\sigma -\delta \kappa & =(\rho +\overline{\rho })\sigma +(3\epsilon -\overline{\epsilon })\sigma -(\tau -\overline{\pi }+\overline{\alpha }+3\beta )\kappa +{\mathrm{\Psi }}_0,\\ D\tau -\mathrm{\Delta }\kappa & =(\tau +\overline{\pi })\rho +(\overline{\tau }+\pi )\sigma +(\epsilon -\overline{\epsilon })\tau -(3\gamma +\overline{\gamma })\kappa +{\mathrm{\Psi }}_1+{\mathrm{\Phi }}_{01},\\ 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 }}_{10},\\ D\beta -\delta \epsilon & =(\alpha +\pi )\sigma +(\overline{\rho }-\overline{\epsilon })\beta -(\mu +\gamma )\kappa -(\overline{\alpha }-\overline{\pi })\epsilon +{\mathrm{\Psi }}_1,\\ D\gamma -\mathrm{\Delta }\epsilon & =(\tau +\overline{\pi })\alpha +(\overline{\tau }+\pi )\beta -(\epsilon +\overline{\epsilon })\gamma -(\gamma +\overline{\gamma })\epsilon +\tau \pi -\nu \kappa +{\mathrm{\Psi }}_2+{\mathrm{\Phi }}_{11}-\mathrm{\Lambda },\\ 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 }}_{20},\\ D\mu -\delta \pi & =(\overline{\rho }\mu +\sigma \lambda )+\pi \overline{\pi }-(\epsilon +\overline{\epsilon })\mu -(\overline{\alpha }-\beta )\pi -\nu \kappa +{\mathrm{\Psi }}_2+2\mathrm{\Lambda },\\ D\nu -\mathrm{\Delta }\pi & =(\pi +\overline{\tau })\mu +(\overline{\pi }+\tau )\lambda +(\gamma -\overline{\gamma })\pi -(3\epsilon +\overline{\epsilon })\nu +{\mathrm{\Psi }}_3+{\mathrm{\Phi }}_{21},\\ \mathrm{\Delta }\lambda -\overline{\delta }\nu & =-(\mu +\overline{\mu })\lambda -(3\gamma -\overline{\gamma })\lambda +(3\alpha +\overline{\beta }+\pi -\overline{\tau })\nu -{\mathrm{\Psi }}_4,\\ \delta \rho -\overline{\delta }\sigma & =\rho (\overline{\alpha }+\beta )-\sigma (3\alpha -\overline{\beta })+(\rho -\overline{\rho })\tau +(\mu -\overline{\mu })\kappa -{\mathrm{\Psi }}_1+{\mathrm{\Phi }}_{01},\\ \delta \alpha -\overline{\delta }\beta & =(\mu \rho -\lambda \sigma )+\alpha \overline{\alpha }+\beta \overline{\beta }-2\alpha \beta +\gamma (\rho -\overline{\rho })+\epsilon (\mu -\overline{\mu })-{\mathrm{\Psi }}_2+{\mathrm{\Phi }}_{11}+\mathrm{\Lambda },\\ \delta \lambda -\overline{\delta }\mu & =(\rho -\overline{\rho })\nu +(\mu -\overline{\mu })\pi +(\alpha +\overline{\beta })\mu +(\overline{\alpha }-3\beta )\lambda -{\mathrm{\Psi }}_3+{\mathrm{\Phi }}_{21},\\ \delta \nu -\mathrm{\Delta }\mu & =({\mu }^2+\lambda \overline{\lambda })+(\gamma +\overline{\gamma })\mu -\overline{\nu }\pi +(\tau -3\beta -\overline{\alpha })\nu +{\mathrm{\Phi }}_{22},\\ \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 }}_{12},\\ \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 }}_{02},\\ \mathrm{\Delta }\rho -\overline{\delta }\tau & =-(\rho \overline{\mu }+\sigma \lambda )+(\overline{\beta }-\alpha -\overline{\tau })\tau +(\gamma +\overline{\gamma })\rho +\nu \kappa -{\mathrm{\Psi }}_2-2\mathrm{\Lambda },\\ \mathrm{\Delta }\alpha -\overline{\delta }\gamma & =(\rho +\epsilon )\nu -(\tau +\beta )\lambda +(\overline{\gamma }-\overline{\mu })\alpha +(\overline{\beta }-\overline{\tau })\gamma -{\mathrm{\Psi }}_3.\end{array}}$$

Also, the Weyl-NP scalars ${\displaystyle {\mathrm{\Psi }}_i}$ and the Ricci-NP scalars ${\displaystyle {\mathrm{\Phi }}_{ij}}$ 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) ${\displaystyle F_{ab}}$ can be encoded into three complex Maxwell-NP scalars

$${\displaystyle {\varphi }_0:=F_{ab}{\ell }^a{m}^b,{\varphi }_1:=\frac{1}{2}{F}_{ab}\left({\ell }^a{n}^b+{\overline{m}}^a{m}^b\right),{\varphi }_2:=F_{ab}{\overline{m}}^a{n}^b,}$$

and therefore the eight real Maxwell equations ${\displaystyle d\mathbf{F}=0}$ and ${\displaystyle d^{\star }\mathbf{F}=0}$ (as ${\displaystyle \mathbf{F}=dA}$) can be transformed into four complex equations,

$${\displaystyle \begin{array}{cc}D{\varphi }_1-\overline{\delta }{\varphi }_0& =(\pi -2\alpha ){\varphi }_0+2\rho {\varphi }_1-\kappa {\varphi }_2,\\ D{\varphi }_2-\overline{\delta }{\varphi }_1& =-\lambda {\varphi }_0+2\pi {\varphi }_1+(\rho -2\epsilon ){\varphi }_2,\\ \mathrm{\Delta }{\varphi }_0-\delta {\varphi }_1& =(2\gamma -\mu ){\varphi }_0-2\tau {\varphi }_1+\sigma {\varphi }_2,\\ \mathrm{\Delta }{\varphi }_1-\delta {\varphi }_2& =\nu {\varphi }_0-2\mu {\varphi }_1+(2\beta -\tau ){\varphi }_2,\end{array}}$$

with the Ricci-NP scalars ${\displaystyle {\mathrm{\Phi }}_{ij}}$ related to Maxwell scalars by

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

It is worthwhile to point out that, the supplementary equation ${\displaystyle {\mathrm{\Phi }}_{ij}=2{\varphi }_i\overline{{\varphi }_j}}$ is only valid for electromagnetic fields; for example, in the case of Yang-Mills fields there will be ${\displaystyle {\mathrm{\Phi }}_{ij}=\text{Tr}({ϝ}_i{\overline{ϝ}}_j)}$ where ${\displaystyle {ϝ}_i(i\in \{0,1,2\})}$ 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 the NP formalism to gravitational radiation field

The Weyl scalar ${\displaystyle {\mathrm{\Psi }}_4}$ was defined by Newman & Penrose as $${\displaystyle {\mathrm{\Psi }}_4=-C_{\alpha \beta \gamma \delta }{n}^{\alpha }{\overline{m}}^{\beta }{n}^{\gamma }{\overline{m}}^{\delta }}$$ (note, however, that the overall sign is arbitrary, and that Newman & Penrose worked with a "timelike" metric signature of ${\displaystyle (+,-,-,-)}$). In empty space, the Einstein field equations reduce to ${\displaystyle R_{\alpha \beta }=0}$. From the definition of the Weyl tensor, we see that this means that it equals the Riemann tensor, ${\displaystyle C_{\alpha \beta \gamma \delta }=R_{\alpha \beta \gamma \delta }}$. We can make the standard choice for the tetrad at infinity: $${\displaystyle {\ell }^{\mu }=\frac{1}{\sqrt{2}}\left(\hat{t}+\hat{r}\right),}$$ $${\displaystyle n^{\mu }=\frac{1}{\sqrt{2}}\left(\hat{t}-\hat{r}\right),}$$ $${\displaystyle m^{\mu }=\frac{1}{\sqrt{2}}\left(\hat{\theta }+i\hat{\varphi }\right).}$$

In transverse-traceless gauge, a simple calculation shows that linearized gravitational waves are related to components of the Riemann tensor as $${\displaystyle \frac{1}{4}\left({\ddot{h}}_{\hat{\theta }\hat{\theta }}-{\ddot{h}}_{\hat{\varphi }\hat{\varphi }}\right)=-R_{\hat{t}\hat{\theta }\hat{t}\hat{\theta }}=-R_{\hat{t}\hat{\varphi }\hat{r}\hat{\varphi }}=-R_{\hat{r}\hat{\theta }\hat{r}\hat{\theta }}=R_{\hat{t}\hat{\varphi }\hat{t}\hat{\varphi }}=R_{\hat{t}\hat{\theta }\hat{r}\hat{\theta }}=R_{\hat{r}\hat{\varphi }\hat{r}\hat{\varphi }},}$$ $${\displaystyle \frac{1}{2}{\ddot{h}}_{\hat{\theta }\hat{\varphi }}=-R_{\hat{t}\hat{\theta }\hat{t}\hat{\varphi }}=-R_{\hat{r}\hat{\theta }\hat{r}\hat{\varphi }}=R_{\hat{t}\hat{\theta }\hat{r}\hat{\varphi }}=R_{\hat{r}\hat{\theta }\hat{t}\hat{\varphi }},}$$ assuming propagation in the ${\displaystyle \hat{r}}$ direction. Combining these, and using the definition of ${\displaystyle {\mathrm{\Psi }}_4}$ above, we can write $${\displaystyle {\mathrm{\Psi }}_4=\frac{1}{2}\left({\ddot{h}}_{\hat{\theta }\hat{\theta }}-{\ddot{h}}_{\hat{\varphi }\hat{\varphi }}\right)+i{\ddot{h}}_{\hat{\theta }\hat{\varphi }}=-{\ddot{h}}_{+}+i{\ddot{h}}_{\times }.}$$ Far from a source, in nearly flat space, the fields ${\displaystyle h_{+}}$ and ${\displaystyle h_{\times }}$ encode everything about gravitational radiation propagating in a given direction. Thus, we see that ${\displaystyle {\mathrm{\Psi }}_4}$ 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: $${\displaystyle {\mathrm{\Psi }}_4(t,r,\theta ,\varphi )=-\frac{1}{r\sqrt{2}}\sum _{\ell =2}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }\left[{}^{(\ell +2)}{I}^{\ell m}(t-r)-i{}^{(\ell +2)}{S}^{\ell m}(t-r)\right]{}_{-2}{Y}_{\ell m}(\theta ,\varphi ).}$$ Here, prefixed superscripts indicate time derivatives. That is, we define $${\displaystyle {}^{(\ell )}G(t)={\left(\frac{d}{dt}\right)}^{\ell }G(t).}$$ The components ${\displaystyle I^{\ell m}}$ and ${\displaystyle S^{\ell m}}$ are the mass and current multipoles, respectively. ${\displaystyle {}_{-2}{Y}_{\ell m}}$ is the spin-weight −2 spherical harmonic.