Weyl scalar

Given a complex null tetrad $\backslash \{l^a,\; n^a,\; m^a,\; \backslash bar\{m\}^a\backslash \}$ and with the convention $\backslash \{(-,+,+,+);\; l^a\; n\_a=-1\backslash ,,m^a\; \backslash bar\{m\}\_a=1\backslash \}$, the Weyl-NP scalars are defined byJeremy Bransom Griffiths, Jiri Podolsky. Exact Space-Times in Einstein's General Relativity. Cambridge: Cambridge University Press, 2009. Chapter 2.Valeri P Frolov, Igor D Novikov. Black Hole Physics: Basic Concepts and New Developments. Berlin: Springer, 1998. Appendix E.{{cite journal | arxiv=gr-qc/0005083 | doi=10.1103/PhysRevD.62.104025 | title=Isolated horizons: Hamiltonian evolution and the first law | date=2000 | last1=Ashtekar | first1=Abhay | last2=Fairhurst | first2=Stephen | last3=Krishnan | first3=Badri | journal=Physical Review D | volume=62 | issue=10 | page=104025 [Appendix B]}}

:$\backslash Psi\_0\; :=\; C\_\{\backslash alpha\backslash beta\backslash gamma\backslash delta\}\; l^\backslash alpha\; m^\backslash beta\; l^\backslash gamma\; m^\backslash delta\backslash \; ,$

:$\backslash Psi\_1\; :=\; C\_\{\backslash alpha\backslash beta\backslash gamma\backslash delta\}\; l^\backslash alpha\; n^\backslash beta\; l^\backslash gamma\; m^\backslash delta\backslash \; ,$

:$\backslash Psi\_2\; :=\; C\_\{\backslash alpha\backslash beta\backslash gamma\backslash delta\}\; l^\backslash alpha\; m^\backslash beta\; \backslash bar\{m\}^\backslash gamma\; n^\backslash delta\backslash \; ,$

:$\backslash Psi\_3\; :=\; C\_\{\backslash alpha\backslash beta\backslash gamma\backslash delta\}\; l^\backslash alpha\; n^\backslash beta\; \backslash bar\{m\}^\backslash gamma\; n^\backslash delta\backslash \; ,$

:$\backslash Psi\_4\; :=\; C\_\{\backslash alpha\backslash beta\backslash gamma\backslash delta\}\; n^\backslash alpha\; \backslash bar\{m\}^\backslash beta\; n^\backslash gamma\; \backslash bar\{m\}^\backslash delta\backslash \; .$

Note: If one adopts the convention $\backslash \{(+,-,-,-);\; l^a\; n\_a=1\backslash ,,m^a\; \backslash bar\{m\}\_a=-1\backslash \}$, the definitions of $\backslash Psi\_i$ should take the opposite values;{{cite journal |doi=10.1063/1.1724257 |title=An Approach to Gravitational Radiation by a Method of Spin Coefficients |date=1962 |last1=Newman |first1=Ezra |last2=Penrose |first2=Roger |journal=Journal of Mathematical Physics |volume=3 |issue=3 |pages=566–578 }}{{erratum |checked=yes |doi=10.1063/1.1704025}}Subrahmanyan Chandrasekhar. The Mathematical Theory of Black Holes. Chicago: University of Chicago Press, 1983.Peter O'Donnell. Introduction to 2-Spinors in General Relativity. Singapore: World Scientific, 2003. that is to say, $\backslash Psi\_i\backslash mapsto-\backslash Psi\_i$ after the signature transition.

{{see also|Newman–Penrose field equations}}

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{{citation needed|reason=I don't immediately find a reference where I can check these equations|date=May 2019}}

:$\backslash Psi\_0=D\backslash sigma-\backslash delta\backslash kappa-(\backslash rho+\backslash bar\{\backslash rho\})\backslash sigma-(3\backslash varepsilon-\backslash bar\{\backslash varepsilon\})\backslash sigma+(\backslash tau-\backslash bar\{\backslash pi\}+\backslash bar\{\backslash alpha\}+3\backslash beta)\backslash kappa\backslash ,,$

:$\backslash Psi\_1=D\backslash beta-\backslash delta\backslash varepsilon-(\backslash alpha+\backslash pi)\backslash sigma-(\backslash bar\{\backslash rho\}-\backslash bar\{\backslash varepsilon\})\backslash beta+(\backslash mu+\backslash gamma)\backslash kappa+(\backslash bar\{\backslash alpha\}-\backslash bar\{\backslash pi\})\backslash varepsilon\backslash ,,$

:$\backslash Psi\_2=\backslash bar\{\backslash delta\}\backslash tau-\backslash Delta\backslash rho-(\backslash rho\backslash bar\{\backslash mu\}+\backslash sigma\backslash lambda)+(\backslash bar\{\backslash beta\}-\backslash alpha-\backslash bar\{\backslash tau\})\backslash tau+(\backslash gamma+\backslash bar\{\backslash gamma\})\backslash rho+\backslash nu\backslash kappa-2\backslash Lambda\backslash ,,$

:$\backslash Psi\_3=\backslash bar\{\backslash delta\}\backslash gamma-\backslash Delta\backslash alpha+(\backslash rho+\backslash varepsilon)\backslash nu-(\backslash tau+\backslash beta)\backslash lambda+(\backslash bar\{\backslash gamma\}-\backslash bar\{\backslash mu\})\backslash alpha+(\backslash bar\{\backslash beta\}-\backslash bar\{\backslash tau\})\backslash gamma\backslash ,.$

:$\backslash Psi\_4=\backslash delta\backslash nu-\backslash Delta\backslash lambda-(\backslash mu+\backslash bar\{\backslash mu\})\backslash lambda-(3\backslash gamma-\backslash bar\{\backslash gamma\})\backslash lambda+(3\backslash alpha+\backslash bar\{\backslash beta\}+\backslash pi-\backslash bar\{\backslash tau\})\backslash nu\backslash ,.$

where $\backslash Lambda$ (used for $\backslash Psi\_2$) refers to the NP curvature scalar $\backslash Lambda:=\backslash frac\{R\}\{24\}$ which could be calculated directly from the spacetime metric $g\_\{ab\}$.

Szekeres (1965){{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 | doi-access=free }}. gave an interpretation of the different Weyl scalars at large distances:

:$\backslash Psi\_2$ is a "Coulomb" term, representing the gravitational monopole of the source;

:$\backslash Psi\_1$ & $\backslash Psi\_3$ are ingoing and outgoing "longitudinal" radiation terms;

:$\backslash Psi\_0$ & $\backslash Psi\_4$ are ingoing and outgoing "transverse" radiation terms.

For a general asymptotically flat spacetime containing radiation (Petrov Type I), $\backslash Psi\_1$ & $\backslash Psi\_3$ 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 $\backslash Psi\_4$.

It can be shown to describe outgoing gravitational radiation (in an asymptotically flat spacetime) as

:$\backslash Psi\_4\; =\; \backslash frac\{1\}\{2\}\backslash left(\; \backslash ddot\{h\}\_\{\backslash hat\{\backslash theta\}\; \backslash hat\{\backslash theta\}\}\; -\; \backslash ddot\{h\}\_\{\backslash hat\{\backslash phi\}\; \backslash hat\{\backslash phi\}\}\; \backslash right)\; +\; i\; \backslash ddot\{h\}\_\{\backslash hat\{\backslash theta\}\backslash hat\{\backslash phi\}\}\; =\; -\backslash ddot\{h\}\_+\; +\; i\; \backslash ddot\{h\}\_\backslash times\backslash \; .$

Here, $h\_+$ and $h\_\backslash times$ are the "plus" and "cross" polarizations of gravitational radiation, and the double dots represent double time-differentiation.{{clarify|date=August 2014|reason=Does this hold at linear order in h where h is the deviation of the metric from Minkowski spacetime?}}

There are, however, certain examples in which the interpretation listed above fails.{{cite journal|last1=Hofmann|first1=Stefan|last2=Niedermann|first2=Florian|last3=Schneider|first3=Robert|title=Interpretation of the Weyl tensor|journal= Physical Review D|date=2013|volume= 88|issue=6|page=064047|doi=10.1103/PhysRevD.88.064047|arxiv = 1308.0010 |bibcode = 2013PhRvD..88f4047H |s2cid=118647223}} These are exact vacuum solutions of the Einstein field equations with cylindrical symmetry. For instance, a static (infinitely long) cylinder can produce a gravitational field which has not only the expected "Coulomb"-like Weyl component $\backslash Psi\_2$, but also non-vanishing "transverse wave"-components $\backslash Psi\_0$ and $\backslash Psi\_4$. Furthermore, purely outgoing Einstein-Rosen waves have a non-zero "incoming transverse wave"-component $\backslash Psi\_0$.

• Weyl-NP and Ricci-NP scalars

Category:General relativity