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In the [[Newman-Penrose formalism|Newman-Penrose (NP) formalism]] of [[general relativity]], independent components of the [[Ricci tensor]]s  of a four-dimensional [[spacetime]] are encoded into seven (or ten) '''Ricci scalars''' which consist of three real [[Scalar (physics)|scalars]]  <math>\{\Phi_{00}, \Phi_{11}, \Phi_{22}\}</math>, three (or six) complex scalars <math>\{\Phi_{01}=\overline{\Phi_{10}}\,,\Phi_{02}=\overline{\Phi_{20}}\,,\Phi_{12}=\overline{\Phi_{21}}\}</math> and the NP curvature scalar <math>\Lambda</math>. Physically, Ricci-NP scalars are related with the energy-momentum distribution of the spacetime due to [[Einstein's field equation]].
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==Definitions==
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Given a complex null tetrad <math>\{l^a, n^a, m^a, \bar{m}^a\}</math> and with the convention <math>\{(-,+,+,+); l^a n_a=-1\,,m^a \bar{m}_a=1\}</math>, the Ricci-NP scalars are defined by<ref name=refNP1>Jeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Chapter 2.</ref><ref name=refNP2>Valeri P Frolov, Igor D Novikov. ''Black Hole Physics: Basic Concepts and New Developments''. Berlin: Springer, 1998. Appendix E.</ref><ref name=refNP3>Abhay Ashtekar, Stephen Fairhurst, Badri Krishnan. ''Isolated horizons: Hamiltonian evolution and the first law''. Physical Review D, 2000, '''62'''(10): 104025. Appendix B. [http://arxiv.org/abs/gr-qc/0005083 gr-qc/0005083]</ref>  (where overline means [[complex conjugate]])
 
<math>\Phi_{00}:=\frac{1}{2}R_{ab}l^a l^b\,, \quad \Phi_{11}:=\frac{1}{4}R_{ab}(\,l^a n^b+m^a\bar{m}^b)\,, \quad\Phi_{22}:=\frac{1}{2}R_{ab}n^a n^b\,, \quad\Lambda:=\frac{R}{24}\,;</math>
 
<math>\Phi_{01}:=\frac{1}{2}R_{ab}l^a m^b\,, \quad\; \Phi_{10}:=\frac{1}{2}R_{ab}l^a \bar{m}^b=\overline{\Phi_{01}}\,,</math><br /> <math>\Phi_{02}:=\frac{1}{2}R_{ab}m^a m^b\,, \quad \Phi_{20}:=\frac{1}{2}R_{ab}\bar{m}^a \bar{m}^b=\overline{\Phi_{02}}\,,</math><br /> <math>\Phi_{12}:=\frac{1}{2}R_{ab}\bar{m}^a n^b\,, \quad\; \Phi_{21}:=\frac{1}{2}R_{ab}m^a n^b=\overline{\Phi_{12}}\,.</math>
 
Remark I: In these definitions, <math>R_{ab}</math> could be replaced by its [[Trace-free Ricci tensor|trace-free]] part <math> Q_{ab}=R_{ab}-\frac{1}{4}g_{ab}R</math><ref name="refNP2" /> or by the [[Einstein tensor]] <math> G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R</math> because of the normalization (i.e. inner product) relations that
 
:<math>l_a l^a = n_a n^a = m_a m^a = \bar{m}_a \bar{m}^a=0\,,</math>
:<math>l_a m^a = l_a \bar{m}^a = n_a m^a = n_a \bar{m}^a=0\,.</math>
 
Remark II: Specifically for [[Electrovacuum solution|electrovacuum]], we have <math>\Lambda=0</math>, thus
 
<math>24\Lambda\,=0=\,R_{ab}g^{ab}\,=\,R_{ab}\Big(-2l^a n^b+2m^a\bar{m}^b \Big)\; \Rightarrow \; R_{ab}l^a n^b\,=\,R_{ab}m^a\bar{m}^b\,,</math>
 
and therefore <math>\Phi_{11}</math> is reduced to
 
<math>\Phi_{11}:=\frac{1}{4}R_{ab}(\,l^a n^b+m^a\bar{m}^b)=\frac{1}{2}R_{ab}l^a n^b=\frac{1}{2}R_{ab}m^a\bar{m}^a\,.</math>
 
Remark III: If one adopts the convention <math>\{(+,-,-,-); l^a n_a=1\,,m^a \bar{m}_a=-1\}</math>, the definitions of <math>\Phi_{ij}</math> should take the opposite values;<ref>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.</ref><ref>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.</ref><ref name=NP3>Subrahmanyan Chandrasekhar. ''The Mathematical Theory of Black Holes''. Chicago: University of Chikago Press, 1983.</ref><ref name=ODonnell>Peter O'Donnell. ''Introduction to 2-Spinors in General Relativity''. Singapore: World Scientific, 2003.</ref> that is to say, <math>\Phi_{ij}\mapsto-\Phi_{ij}</math> after the signature transition.
 
==Alternative derivations==
{{see also|Newman–Penrose formalism#NP field equations|Newman-Penrose field equations}}
 
According to the definitions above, one should find out the [[Ricci tensor]]s 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 [[Newman–Penrose_formalism#Twelve_spin_coefficients|spin coefficients]] and then derive the Ricci-NP scalars <math>\Phi_{ij}</math> via relevant [[Newman–Penrose_formalism#NP_field_equations|NP field equations]] that<ref name="refNP2" /><ref name="ODonnell" />
 
:<math>\Phi_{00}=D\rho -\bar{\delta}\kappa-(\rho^2+\sigma\bar{\sigma})-(\varepsilon+\bar{\varepsilon})\rho+\bar{\kappa}\tau+\kappa(3\alpha+\bar{\beta}-\pi)\,,</math>
:<math>\Phi_{10}=D\alpha-\bar{\delta}\varepsilon-(\rho+\bar{\varepsilon}-2\varepsilon)\alpha-\beta\bar{\sigma}+\bar{\beta}\varepsilon+\kappa\lambda+\bar{\kappa}\gamma-(\varepsilon+\rho)\pi\,,</math>
:<math>\Phi_{02}=\delta\tau-\Delta\sigma-(\mu\sigma+\bar{\lambda}\rho)-(\tau+\beta-\bar{\alpha})\tau+(3\gamma-\bar{\gamma})\sigma+\kappa\bar{\nu}\,,</math>
:<math>\Phi_{20}=D\lambda-\bar{\delta}\pi-(\rho\lambda+\bar{\sigma}\mu)-\pi^2-(\alpha-\bar{\beta})\pi+\nu\bar{\kappa}+(3\varepsilon-\bar{\varepsilon})\lambda\,,</math>
:<math>\Phi_{12}=\delta\gamma-\Delta\beta-(\tau-\bar{\alpha}-\beta)\gamma-\mu\tau+\sigma\nu+\varepsilon\bar{\nu}+(\gamma-\bar{\gamma}-\mu)\beta-\alpha\bar{\lambda}\,,</math>
:<math>\Phi_{22}=\delta\nu-\Delta\mu-(\mu^2+\lambda\bar{\lambda})-(\gamma+\bar{\gamma})\mu+\bar{\nu}\pi-(\tau-3\beta-\bar{\alpha})\nu\,,</math>
:<math>2\Phi_{11}=D\gamma-\Delta\varepsilon+\delta\alpha-\bar{\delta}\beta-(\tau+\bar{\pi})\alpha-\alpha\bar{\alpha}-(\bar{\tau}+\pi)\beta-\beta\bar{\beta}+2\alpha\beta+(\varepsilon+\bar{\varepsilon})\gamma-(\rho-\bar{\rho})\gamma+(\gamma+\bar{\gamma})\varepsilon-(\mu-\bar{\mu})\varepsilon-\tau\pi+\nu\kappa-(\mu\rho-\lambda\sigma)\,,</math>
 
while the NP curvature scalar <math>\Lambda</math> could be directly and easily calculated via <math>\Lambda=\frac{R}{24}</math> with <math>R</math> being the ordinary [[scalar curvature]] of the spacetime metric <math>g_{ab}=-l_a  n_b - n_a  l_b +m_a  \bar{m}_b +\bar{m}_a  m_b</math>.
 
==Electromagnetic Ricci-NP scalars==
 
According to the definitions of Ricci-NP scalars <math>\Phi_{ij}</math> above and the fact that <math>R_{ab}</math> could be replaced by <math>G_{ab}</math> in the definitions, <math>\Phi_{ij}</math> are related with the energy-momentum distribution due to Einstein's field equations <math>G_{ab}=8\pi T_{ab}</math>. In the simplest situation, i.e. vacuum spacetime in the absence of matter fields with <math>T_{ab}=0</math>, we will have <math>\Phi_{ij}=0</math>. Moreover, for electromagnetic field, in addition to the aforementioned definitions, <math>\Phi_{ij}</math> could be determined more specifically by<ref name="refNP1" />
 
<br />
<math>\Phi_{ij}=\,2\,\phi_i\,\overline{\phi_j}\,,\quad (i,j\in\{0,1,2\})\,,</math>
 
where <math>\phi_i</math> denote the three complex Maxwell-NP scalars<ref name="refNP1" /> which encode the six independent components of the Faraday-Maxwell 2-form  <math>F_{ab}</math> (i.e. the [[Electromagnetic tensor|electromagnetic field strength tensor]])
 
<br />
<math>\phi_0:= -F_{ab}l^a m^b \,,\quad \phi_1:= -\frac{1}{2} F_{ab}\big(l^an^a-m^a\bar{m}^b \big)\,, \quad \phi_2 := F_{ab} n^a \bar{m}^b\,.</math>
 
Remark: The equation <math>\Phi_{ij}=2\,\phi_i\, \overline{\phi_j}</math> 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 <math>\Phi_{ij}=\,\text{Tr}\,(\digamma_i \,\bar{\digamma}_j)</math> where <math>\digamma_i (i\in\{0,1,2 \})</math> are Yang-Mills-NP scalars.<ref>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.</ref>
 
==See also==
*[[Newman-Penrose formalism]]
*[[Weyl scalar]]
 
==References==
<references/>
 
[[Category:General relativity]]

Latest revision as of 20:00, 11 September 2014

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