Weyl scalar

In the Newman–Penrose (NP) formalism of general relativity, Weyl scalars refer to a set of five complex scalars

{

Ψ

0

,

Ψ

1

,

Ψ

2

,

Ψ

3

,

Ψ

4

}

{\displaystyle \{\Psi _{0},\Psi _{1},\Psi _{2},\Psi _{3},\Psi _{4}\}}

which encode the ten independent components of the Weyl tensor of a four-dimensional spacetime.

Definitions

Given a complex null tetrad

{

l

a

,

n

a

,

m

a

,

m

¯

a

}

{\displaystyle \{l^{a},n^{a},m^{a},{\bar {m}}^{a}\}}

and with the convention

{

(

−

,

+

,

+

,

+

)

;

l

a

n

a

=

−

1

,

m

a

m

¯

a

=

1

}

{\displaystyle \{(-,+,+,+);l^{a}n_{a}=-1\,,m^{a}{\bar {m}}_{a}=1\}}

, the Weyl-NP scalars are defined by

Ψ

0

:=

C

α

β

γ

δ

l

α

m

β

l

γ

m

δ

,

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

Ψ

1

:=

C

α

β

γ

δ

l

α

n

β

l

γ

m

δ

,

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

Ψ

2

:=

C

α

β

γ

δ

l

α

m

β

m

¯

γ

n

δ

,

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

Ψ

3

:=

C

α

β

γ

δ

l

α

n

β

m

¯

γ

n

δ

,

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

Ψ

4

:=

C

α

β

γ

δ

n

α

m

¯

β

n

γ

m

¯

δ

.

{\displaystyle \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

m

¯

a

=

−

1

}

{\displaystyle \{(+,-,-,-);l^{a}n_{a}=1\,,m^{a}{\bar {m}}_{a}=-1\}}

, the definitions of

Ψ

i

{\displaystyle \Psi _{i}}

should take the opposite values; that is to say,

Ψ

i

↦

−

Ψ

i

{\displaystyle \Psi _{i}\mapsto -\Psi _{i}}

after the signature transition.

Alternative derivations

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 use the NP field equations to derive the five Weyl-NP scalars

Ψ

0

=

D

σ

−

δ

κ

−

(

ρ

+

ρ

¯

)

σ

−

(

3

ε

−

ε

¯

)

σ

+

(

τ

−

π

¯

+

α

¯

+

3

β

)

κ

,

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

Ψ

1

=

D

β

−

δ

ε

−

(

α

+

π

)

σ

−

(

ρ

¯

−

ε

¯

)

β

+

(

μ

+

γ

)

κ

+

(

α

¯

−

π

¯

)

ε

,

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

Ψ

2

=

δ

¯

τ

−

Δ

ρ

−

(

ρ

μ

¯

+

σ

λ

)

+

(

β

¯

−

α

−

τ

¯

)

τ

+

(

γ

+

γ

¯

)

ρ

+

ν

κ

−

2

Λ

,

{\displaystyle \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 \,,}

Ψ

3

=

δ

¯

γ

−

Δ

α

+

(

ρ

+

ε

)

ν

−

(

τ

+

β

)

λ

+

(

γ

¯

−

μ

¯

)

α

+

(

β

¯

−

τ

¯

)

γ

.

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

Ψ

4

=

δ

ν

−

Δ

λ

−

(

μ

+

μ

¯

)

λ

−

(

3

γ

−

γ

¯

)

λ

+

(

3

α

+

β

¯

+

π

−

τ

¯

)

ν

.

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

where

Λ

{\displaystyle \Lambda }

(used for

Ψ

2

{\displaystyle \Psi _{2}}

) refers to the NP curvature scalar

Λ

:=

R

24

{\displaystyle \Lambda :={\frac {R}{24}}}

which could be calculated directly from the spacetime metric

g

a

b

{\displaystyle g_{ab}}

.

Physical interpretation

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

Ψ

2

{\displaystyle \Psi _{2}}

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

Ψ

1

{\displaystyle \Psi _{1}}

&

Ψ

3

{\displaystyle \Psi _{3}}

are ingoing and outgoing "longitudinal" radiation terms;

Ψ

0

{\displaystyle \Psi _{0}}

&

Ψ

4

{\displaystyle \Psi _{4}}

are ingoing and outgoing "transverse" radiation terms.

For a general asymptotically flat spacetime containing radiation (Petrov Type I),

Ψ

1

{\displaystyle \Psi _{1}}

&

Ψ

3

{\displaystyle \Psi _{3}}

can be transformed to zero by an appropriate choice of null tetrad.