The Weyl Components

The $\Psi$'s are defined as components of the Weyl tensor $C_{abcd}$ which in the vacuum case (assumed by PsiKadelia) is identical to the antisymmetrised Riemann tensor $R_{abcd}$.

$$\Psi_0 = C_{abcd}\,l^a\,m^b\,l^c\,m^d$$

$$\Psi_1 = C_{abcd}\,l^a\,n^b\,l^c\,m^d$$

$$\Psi_2 = C_{abcd}\,l^a\,n^b\,m^c\,\bar m^d$$

$$\Psi_3 = C_{abcd}\,n^a\,l^b\,n^c\,\bar m^d$$

$$\Psi_4 = C_{abcd}\,n^a\,\bar m^b\,n^c\,\bar m^d$$

If the tetrad has the right fall-off condition near infinity (the ``peeling property''), then the Weyl scalars $\Psi$ have the following meaning:

Scalar	Falloff	Physics
$\Psi_0$	$1/r$	outgoing gravitational (transverse) radiation
$\Psi_1$	$1/r^2$	outgoing gauge (longitudinal) radiation
$\Psi_2$	$1/r^3$	static gravitational (``Coulomb'') field
$\Psi_3$	$1/r^4$	ingoing gauge (longitudinal) radiation
$\Psi_4$	$1/r^5$	ingoing gravitational (transverse) radiation

Note: This is a different convention that usually chosen. Usually, the meanings of $\Psi_0$ and $\Psi_4$, and of $\Psi_1$ and $\Psi_3$ are exchanged. This difference comes essentially from the choice of the tetrad vectors $l^a$ and $n^a$; the choice above has $l^a$ pointing inwards and $n^a$ pointing outwards, which exchanges the notion of ``outgoing'' and ``ingoing''.

With a tetrad of the form (D18.1), these components can be expressed directly in terms of spatial quantities.

$$\Psi_0 = C_{ab}\,m^a\,m^b$$

$$\Psi_1 = \frac{1}{\sqrt2}C_{ab}\,m^a\,\hat{v}_{\scriptscriptstyle(1)}{}^b$$

$$\Psi_2 = \frac{1}{2}C_{ab}\,\hat{v}_{\scriptscriptstyle(1)}{}^a\,\hat{v}_{\scriptscriptstyle(1)}{}^b$$

$$\Psi_3 = \frac{-1}{\sqrt2}C_{ab}\,\bar m^a\,\hat{v}_{\scriptscriptstyle(1)}{}^b$$

$$\Psi_4 = C_{ab}\,\bar m^a\,\bar m^b$$

where $m^a$ is as defined above. $C_{ab}$ is the symmetric, trace-free, complex-valued tensor

$$C_{ab}=R_{ab}-K\,K_{ab}+K_a^{\,c}\,K_{cb}-i\epsilon_a^{\,cd}\,\nabla_d\,K_{bc} \nonumber$$

given in terms of the Ricci curvature $R_{ab}$ of the spatial metric and the extrinsic curvature $K_{ab}$.