Ricci scalars (Newman–Penrose formalism)

In the Newman–Penrose (NP) formalism of general relativity, independent components of the Ricci tensors of a four-dimensional spacetime are encoded into seven (or ten) Ricci scalars which consist of three real scalars {Φ00,Φ11,Φ22}, three (or six) complex scalars {Φ01=Φ‾10,Φ02=Φ‾20,Φ12=Φ‾21} and the NP curvature scalar Λ. Physically, Ricci-NP scalars are related with the energy–momentum distribution of the spacetime due to Einstein's field equation.

## Contents

* 1 Definitions
* 2 Alternative derivations
* 3 Electromagnetic Ricci-NP scalars
* 4 See also
* 5 References

## Definitions

Given a complex null tetrad {la,na,ma,m¯a} and with the convention {(−,+,+,+);lana=−1,mam¯a=1}, the Ricci-NP scalars are defined by[1][2][3] (where overline means complex conjugate)

Φ00:=12Rablalb,Φ11:=14Rab(lanb+mam¯b),Φ22:=12Rabnanb,Λ:=R24;

Φ01:=12Rablamb,Φ10:=12Rablam¯b=Φ‾01,
Φ02:=12Rabmamb,Φ20:=12Rabm¯am¯b=Φ‾02,
Φ12:=12Rabmanb,Φ21:=12Rabm¯anb=Φ‾12.

Remark I: In these definitions, Rab could be replaced by its trace-free part Qab=Rab−14gabR[2] or by the Einstein tensor Gab=Rab−12gabR because of the normalization (i.e. inner product) relations that

lala=nana=mama=m¯am¯a=0,
lama=lam¯a=nama=nam¯a=0.

Remark II: Specifically for electrovacuum, we have Λ=0, thus

24Λ=0=Rabgab=Rab(−2lanb+2mam¯b)⇒Rablanb=Rabmam¯b,

and therefore Φ11 is reduced to

Φ11:=14Rab(lanb+mam¯b)=12Rablanb=12Rabmam¯b.

Remark III: If one adopts the convention {(+,−,−,−);lana=1,mam¯a=−1}, the definitions of Φij should take the opposite values;[4][5][6][7] that is to say, Φij↦−Φij after the signature transition.

## Alternative derivations

According to the definitions above, one should find out the Ricci tensors before calculating the Ricci-NP scalars via contractions with the corresponding tetrad vectors. However, this method fails to fully reflect the spirit of Newman–Penrose formalism and alternatively, one could compute the spin coefficients and then derive the Ricci-NP scalars Φij via relevant NP field equations that[2][7]

Φ00=Dρ−δ¯κ−(ρ2+σσ¯)−(ε+ε¯)ρ+κ¯τ+κ(3α+β¯−π),
Φ10=Dα−δ¯ε−(ρ+ε¯−2ε)α−βσ¯+β¯ε+κλ+κ¯γ−(ε+ρ)π,
Φ02=δτ−Δσ−(μσ+λ¯ρ)−(τ+β−α¯)τ+(3γ−γ¯)σ+κν¯,
Φ20=Dλ−δ¯π−(ρλ+σ¯μ)−π2−(α−β¯)π+νκ¯+(3ε−ε¯)λ,
Φ12=δγ−Δβ−(τ−α¯−β)γ−μτ+σν+εν¯+(γ−γ¯−μ)β−αλ¯,
Φ22=δν−Δμ−(μ2+λλ¯)−(γ+γ¯)μ+ν¯π−(τ−3β−α¯)ν,
2Φ11=Dγ−Δε+δα−δ¯β−(τ+π¯)α−αα¯−(τ¯+π)β−ββ¯+2αβ+(ε+ε¯)γ−(ρ−ρ¯)γ+(γ+γ¯)ε−(μ−μ¯)ε−τπ+νκ−(μρ−λσ),

while the NP curvature scalar Λ could be directly and easily calculated via Λ=R24 with R being the ordinary scalar curvature of the spacetime metric gab=−lanb−nalb+mam¯b+m¯amb.

## Electromagnetic Ricci-NP scalars

According to the definitions of Ricci-NP scalars Φij above and the fact that Rab could be replaced by Gab in the definitions, Φij are related with the energy–momentum distribution due to Einstein's field equations Gab=8πTab. In the simplest situation, i.e. vacuum spacetime in the absence of matter fields with Tab=0, we will have Φij=0. Moreover, for electromagnetic field, in addition to the aforementioned definitions, Φij could be determined more specifically by[1]

Φij=2ϕiϕ‾j,(i,j∈{0,1,2}),

where ϕi denote the three complex Maxwell-NP scalars[1] which encode the six independent components of the Faraday-Maxwell 2-form Fab (i.e. the electromagnetic field strength tensor)

ϕ0:=−Fablamb,ϕ1:=−12Fab(lana−mam¯b),ϕ2:=Fabnam¯b.

Remark: The equation Φij=2ϕiϕ‾j for electromagnetic field is however not necessarily valid for other kinds of matter fields. For example, in the case of Yang–Mills fields there will be Φij=Tr(ϝiϝ¯j) where ϝi(i∈{0,1,2}) are Yang–Mills-NP scalars.[8]

## See also

* Newman–Penrose formalism
* Weyl scalar

## References

1. ↑ 1.0 1.1 1.2 Jeremy Bransom Griffiths, Jiri Podolsky. Exact Space-Times in Einstein's General Relativity. Cambridge: Cambridge University Press, 2009. Chapter 2.
2. ↑ 2.0 2.1 2.2 Valeri P Frolov, Igor D Novikov. Black Hole Physics: Basic Concepts and New Developments. Berlin: Springer, 1998. Appendix E.
3. ↑ Abhay Ashtekar, Stephen Fairhurst, Badri Krishnan. Isolated horizons: Hamiltonian evolution and the first law. Physical Review D, 2000, 62(10): 104025. Appendix B. gr-qc/0005083
4. ↑ Ezra T Newman, Roger Penrose. An Approach to Gravitational Radiation by a Method of Spin Coefficients. Journal of Mathematical Physics, 1962, 3(3): 566-768.
5. ↑ Ezra T Newman, Roger Penrose. Errata: An Approach to Gravitational Radiation by a Method of Spin Coefficients. Journal of Mathematical Physics, 1963, 4(7): 998.
6. ↑ Subrahmanyan Chandrasekhar. The Mathematical Theory of Black Holes. Chicago: University of Chikago Press, 1983.
7. ↑ 7.0 7.1 Peter O'Donnell. Introduction to 2-Spinors in General Relativity. Singapore: World Scientific, 2003.
8. ↑ E T Newman, K P Tod. Asymptotically Flat Spacetimes, Appendix A.2. In A Held (Editor): General Relativity and Gravitation: One Hundred Years After the Birth of Albert Einstein. Vol (2), page 27. New York and London: Plenum Press, 1980.

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