Weyl scalar

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In General Relativity, the Weyl scalars are a set of five complex scalar quantities,

$\Psi_0, \ldots, \Psi_4$ ,

describing the curvature of a four-dimensional spacetime. They are the expression of the ten independent degrees of freedom of the Weyl tensor $C a b c d$ in the Newman-Penrose Formalism for general relativity. Given a null tetrad ( $l^a, n^a, m^a, \bar{m}^a$ ), the scalars are given (up to an overall conventional sign) by

$\Psi_0 = - C_{\alpha\beta\gamma\delta} l^\alpha m^\beta l^\gamma m^\delta\ ,$

$\Psi_1 = -C_{\alpha\beta\gamma\delta} l^\alpha n^\beta l^\gamma m^\delta\ ,$

$\Psi_2 = -C_{\alpha\beta\gamma\delta} l^\alpha m^\beta \bar{m}^\gamma n^\delta\ ,$

$\Psi_3 = -C_{\alpha\beta\gamma\delta} l^\alpha n^\beta \bar{m}^\gamma n^\delta\ ,$

$\Psi_4 = -C_{\alpha\beta\gamma\delta} n^\alpha \bar{m}^\beta n^\gamma \bar{m}^\delta\ .$

[edit] Physical Interpretation

Szekeres (1965)^[1] gave an interpretation of the different Weyl scalars at large distances:

Ψ 2

is a "Coulomb" term, representing the gravitational monopole of the source;

Ψ 1

Ψ 3

are ingoing and ougoing "longitudinal" radiation terms;

Ψ 0

Ψ 4

are ingoing and ougoing "transverse" radiation terms.

For a general asymptotically flat spacetime containing radiation (Petrov Type I), $Ψ 1$ & $Ψ 3$ 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 $Ψ 4$ . 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.