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| In the [[Newman-Penrose formalism|Newman-Penrose (NP) formalism]] of [[general relativity]], '''Weyl scalars''' refer to a set of five complex [[Scalar (physics)|scalars]] <math>\{\Psi_0, \Psi_1, \Psi_2,\Psi_3, \Psi_4\}</math> which encode the ten independent components of the [[Weyl tensor]]s of a four-dimensional [[spacetime]].
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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 Weyl-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>
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| :<math>\Psi_0 := C_{\alpha\beta\gamma\delta} l^\alpha m^\beta l^\gamma m^\delta\ , </math>
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| :<math>\Psi_1 := C_{\alpha\beta\gamma\delta} l^\alpha n^\beta l^\gamma m^\delta\ , </math>
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| :<math>\Psi_2 := C_{\alpha\beta\gamma\delta} l^\alpha m^\beta \bar{m}^\gamma n^\delta\ , </math>
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| :<math>\Psi_3 := C_{\alpha\beta\gamma\delta} l^\alpha n^\beta \bar{m}^\gamma n^\delta\ , </math>
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| :<math>\Psi_4 := C_{\alpha\beta\gamma\delta} n^\alpha \bar{m}^\beta n^\gamma \bar{m}^\delta\ . </math>
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| Note: If one adopts the convention <math>\{(+,-,-,-); l^a n_a=1\,,m^a \bar{m}_a=-1\}</math>, the definitions of <math>\Psi_i</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>Peter O'Donnell. ''Introduction to 2-Spinors in General Relativity''. Singapore: World Scientific, 2003.</ref> that is to say, <math>\Psi_i\mapsto-\Psi_i</math> after the signature transition.
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| ==Alternative derivations==
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| {{see also|[[Newman–Penrose_formalism#NP_field_equations|Newman-Penrose field equations]]}}
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| According to the definitions above, one should find out the [[Weyl tensor]]s 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 [[Newman–Penrose_formalism#Twelve_spin_coefficients|spin coefficients]] and then derive the five Weyl-NP scalars via the following [[Newman–Penrose_formalism#NP_field_equations|NP field equations]],
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| :<math>\Psi_0=D\sigma-\delta\kappa-(\rho+\bar{\rho})\sigma-(3\varepsilon-\bar{\varepsilon})\sigma+(\tau-\bar{\pi}+\bar{\alpha}+3\beta)\kappa\,,</math> | |
| :<math>\Psi_1=D\beta-\delta\varepsilon-(\alpha+\pi)\sigma-(\bar{\rho}-\bar{\varepsilon})\beta+(\mu+\gamma)\kappa+(\bar{\alpha}-\bar{\pi})\varepsilon\,,</math>
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| :<math>\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\,,</math>
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| :<math>\Psi_3=\bar{\delta}\gamma-\Delta\alpha+(\rho+\varepsilon)\nu-(\tau+\beta)\lambda+(\bar{\gamma}-\bar{\mu})\alpha+(\bar{\beta}-\bar{\tau})\gamma\,.</math>
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| :<math>\Psi_4=\delta\nu-\Delta\lambda-(\mu+\bar{\mu})\lambda-(3\gamma-\bar{\gamma})\lambda+(3\alpha+\bar{\beta}+\pi-\bar{\tau})\nu\,.</math>
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| where <math>\Lambda</math> (used for <math>\Psi_2</math>) refers to the NP curvature scalar <math>\Lambda:=\frac{R}{24}</math> which could be calculated directly from the spacetime metric <math>g_{ab}</math>.
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| ==Physical interpretation==
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| Szekeres (1965)<ref>{{cite journal | author= P. Szekeres | title=The Gravitational Compass | journal=Journal of Mathematical Physics | year=1965 | volume=6 | issue=9 | pages=1387–1391 | doi=10.1063/1.1704788 |bibcode = 1965JMP.....6.1387S }}.</ref> gave an interpretation of the different Weyl scalars at large distances:
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| :<math>\Psi_2</math> is a "Coulomb" term, representing the gravitational monopole of the source;
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| :<math>\Psi_1</math> & <math>\Psi_3</math> are ingoing and outgoing "longitudinal" radiation terms;
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| :<math>\Psi_0</math> & <math>\Psi_4</math> are ingoing and outgoing "transverse" radiation terms. | |
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| For a general asymptotically flat spacetime containing radiation ([[Petrov Type]] I), <math>\Psi_1</math> & <math>\Psi_3</math> can be transformed to zero by an appropriate choice of null tetrad. Thus these can be viewed as gauge quantities.
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| A particularly important case is the Weyl scalar <math>\Psi_4</math>.
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| It can be shown to describe outgoing [[gravitational radiation]] (in an asymptotically flat spacetime) as
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| :<math>\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\ .</math>
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| Here, <math>h_+</math> and <math>h_\times</math> are the "plus" and "cross" polarizations of gravitational radiation, and the double dots represent double time-differentiation.
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| == See also ==
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| *[[Newman–Penrose_formalism#Weyl-NP_and_Ricci-NP_scalars|Weyl-NP and Ricci-NP scalars]]
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| ==References==
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| <references/>
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| [[Category:General relativity]]
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