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In the Newman-Penrose (NP) formalism of general relativity, Weyl scalars refer to a set of five complex scalars $\{\Psi _{0},\Psi _{1},\Psi _{2},\Psi _{3},\Psi _{4}\}$ which encode the ten independent components of the Weyl tensors of a four-dimensional spacetime.

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 Weyl-NP scalars are defined by

$\Psi _{0}:=C_{\alpha \beta \gamma \delta }l^{\alpha }m^{\beta }l^{\gamma }m^{\delta }\ ,$

$\Psi _{1}:=C_{\alpha \beta \gamma \delta }l^{\alpha }n^{\beta }l^{\gamma }m^{\delta }\ ,$

$\Psi _{2}:=C_{\alpha \beta \gamma \delta }l^{\alpha }m^{\beta }{\bar {m}}^{\gamma }n^{\delta }\ ,$

$\Psi _{3}:=C_{\alpha \beta \gamma \delta }l^{\alpha }n^{\beta }{\bar {m}}^{\gamma }n^{\delta }\ ,$

$\Psi _{4}:=C_{\alpha \beta \gamma \delta }n^{\alpha }{\bar {m}}^{\beta }n^{\gamma }{\bar {m}}^{\delta }\ .$

Note: If one adopts the convention $\{(+,-,-,-);l^{a}n_{a}=1\,,m^{a}{\bar {m}}_{a}=-1\}$, the definitions of $\Psi _{i}$ should take the opposite values; that is to say, $\Psi _{i}\mapsto -\Psi _{i}$ after the signature transition.

Alternative derivations

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According to the definitions above, one should find out the Weyl tensors before calculating the Weyl-NP scalars via contractions with relevant tetrad vectors. This method, however, does not fully reflect the spirit of Newman-Penrose formalism. As an alternative, one could firstly compute the spin coefficients and then derive the five Weyl-NP scalars via the following NP field equations,

$\Psi _{0}=D\sigma -\delta \kappa -(\rho +{\bar {\rho }})\sigma -(3\varepsilon -{\bar {\varepsilon }})\sigma +(\tau -{\bar {\pi }}+{\bar {\alpha }}+3\beta )\kappa \,,$

$\Psi _{1}=D\beta -\delta \varepsilon -(\alpha +\pi )\sigma -({\bar {\rho }}-{\bar {\varepsilon }})\beta +(\mu +\gamma )\kappa +({\bar {\alpha }}-{\bar {\pi }})\varepsilon \,,$

$\Psi _{2}={\bar {\delta }}\tau -\Delta \rho -(\rho {\bar {\mu }}+\sigma \lambda )+({\bar {\beta }}-\alpha -{\bar {\tau }})\tau +(\gamma +{\bar {\gamma }})\rho +\nu \kappa -2\Lambda \,,$

$\Psi _{3}={\bar {\delta }}\gamma -\Delta \alpha +(\rho +\varepsilon )\nu -(\tau +\beta )\lambda +({\bar {\gamma }}-{\bar {\mu }})\alpha +({\bar {\beta }}-{\bar {\tau }})\gamma \,.$

$\Psi _{4}=\delta \nu -\Delta \lambda -(\mu +{\bar {\mu }})\lambda -(3\gamma -{\bar {\gamma }})\lambda +(3\alpha +{\bar {\beta }}+\pi -{\bar {\tau }})\nu \,.$

where $\Lambda$ (used for $\Psi _{2}$) refers to the NP curvature scalar $\Lambda :={\frac {R}{24}}$ which could be calculated directly from the spacetime metric $g_{ab}$.

Physical interpretation

Szekeres (1965) gave an interpretation of the different Weyl scalars at large distances:

$\Psi _{2}$ is a "Coulomb" term, representing the gravitational monopole of the source;

$\Psi _{1}$ & $\Psi _{3}$ are ingoing and outgoing "longitudinal" radiation terms;

$\Psi _{0}$ & $\Psi _{4}$ are ingoing and outgoing "transverse" radiation terms.

For a general asymptotically flat spacetime containing radiation (Petrov Type I), $\Psi _{1}$ & $\Psi _{3}$ can be transformed to zero by an appropriate choice of null tetrad. Thus these can be viewed as gauge quantities.

A particularly important case is the Weyl scalar $\Psi _{4}$. It can be shown to describe outgoing gravitational radiation (in an asymptotically flat spacetime) as

$\Psi _{4}={\frac {1}{2}}\left({\ddot {h}}_{{\hat {\theta }}{\hat {\theta }}}-{\ddot {h}}_{{\hat {\phi }}{\hat {\phi }}}\right)+i{\ddot {h}}_{{\hat {\theta }}{\hat {\phi }}}=-{\ddot {h}}_{+}+i{\ddot {h}}_{\times }\ .$

Here, $h_{+}$ and $h_{\times }$ are the "plus" and "cross" polarizations of gravitational radiation, and the double dots represent double time-differentiation.

See also

• Weyl-NP and Ricci-NP scalars