World Library  
Flag as Inappropriate
Email this Article

Weyl scalar

Article Id: WHEBN0006381782
Reproduction Date:

Title: Weyl scalar  
Author: World Heritage Encyclopedia
Language: English
Subject: Vaidya metric, Construction of a complex null tetrad, Ricci scalars (Newman–Penrose formalism), Petrov classification, Newman–Penrose formalism
Collection: General Relativity
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Weyl scalar

In the Newman–Penrose (NP) formalism of general relativity, Weyl scalars refer to a set of five complex scalars \{\Psi_0, \Psi_1, \Psi_2,\Psi_3, \Psi_4\} which encode the ten independent components of the Weyl tensors of a four-dimensional spacetime.

Contents

  • Definitions 1
  • Alternative derivations 2
  • Physical interpretation 3
  • See also 4
  • References 5

Definitions

Given a complex null tetrad \{l^a, n^a, m^a, \bar{m}^a\} and with the convention \{(-,+,+,+); l^a n_a=-1\,,m^a \bar{m}_a=1\}, the Weyl-NP scalars are defined by[1][2][3]

\Psi_0 := C_{\alpha\beta\gamma\delta} l^\alpha m^\beta l^\gamma m^\delta\ ,
\Psi_1 := C_{\alpha\beta\gamma\delta} l^\alpha n^\beta l^\gamma m^\delta\ ,
\Psi_2 := C_{\alpha\beta\gamma\delta} l^\alpha m^\beta \bar{m}^\gamma n^\delta\ ,
\Psi_3 := C_{\alpha\beta\gamma\delta} l^\alpha n^\beta \bar{m}^\gamma n^\delta\ ,
\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 \bar{m}_a=-1\}, the definitions of \Psi_i should take the opposite values;[4][5][6][7] that is to say, \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 derive the five Weyl-NP scalars via the following NP field equations,

\Psi_0=D\sigma-\delta\kappa-(\rho+\bar{\rho})\sigma-(3\varepsilon-\bar{\varepsilon})\sigma+(\tau-\bar{\pi}+\bar{\alpha}+3\beta)\kappa\,,
\Psi_1=D\beta-\delta\varepsilon-(\alpha+\pi)\sigma-(\bar{\rho}-\bar{\varepsilon})\beta+(\mu+\gamma)\kappa+(\bar{\alpha}-\bar{\pi})\varepsilon\,,
\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\,,
\Psi_3=\bar{\delta}\gamma-\Delta\alpha+(\rho+\varepsilon)\nu-(\tau+\beta)\lambda+(\bar{\gamma}-\bar{\mu})\alpha+(\bar{\beta}-\bar{\tau})\gamma\,.
\Psi_4=\delta\nu-\Delta\lambda-(\mu+\bar{\mu})\lambda-(3\gamma-\bar{\gamma})\lambda+(3\alpha+\bar{\beta}+\pi-\bar{\tau})\nu\,.

where \Lambda (used for \Psi_2) refers to the NP curvature scalar \Lambda:=\frac{R}{24} which could be calculated directly from the spacetime metric g_{ab}.

Physical interpretation

Szekeres (1965)[8] gave an interpretation of the different Weyl scalars at large distances:

\Psi_2 is a "Coulomb" term, representing the gravitational monopole of the source;
\Psi_1 & \Psi_3 are ingoing and outgoing "longitudinal" radiation terms;
\Psi_0 & \Psi_4 are ingoing and outgoing "transverse" radiation terms.

For a general asymptotically flat spacetime containing radiation (Petrov Type I), \Psi_1 & \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 \Psi_4. It can be shown to describe outgoing gravitational radiation (in an asymptotically flat spacetime) as

\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\ .

Here, h_+ and h_\times are the "plus" and "cross" polarizations of gravitational radiation, and the double dots represent double time-differentiation.

There are, however, certain examples in which the interpretation listed above fails.[9] 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 \Psi_2, but also non-vanishing "transverse wave"-components \Psi_0 and \Psi_4. Furthermore, purely outgoing Einstein-Rosen waves have a non-zero "incoming transverse wave"-component \Psi_0.

See also

References

  1. ^ Jeremy Bransom Griffiths, Jiri Podolsky. Exact Space-Times in Einstein's General Relativity. Cambridge: Cambridge University Press, 2009. Chapter 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. ^ Peter O'Donnell. Introduction to 2-Spinors in General Relativity. Singapore: World Scientific, 2003.
  8. ^ .
  9. ^
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.