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optical scalars : ウィキペディア英語版
optical scalars
In general relativity, optical scalars refer to a set of three scalar functions \ describing the propagation of a geodesic null congruence.〔Eric Poisson. ''A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics''. Cambridge: Cambridge University Press, 2004. Chapter 2.〕〔Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt. ''Exact Solutions of Einstein's Field Equations''. Cambridge: Cambridge University Press, 2003. Chapter 6.〕〔Subrahmanyan Chandrasekhar. ''The Mathematical Theory of Black Holes''. Oxford: Oxford University Press, 1998. Section 9.(a).〕〔Jeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Section 2.1.3.〕〔P Schneider, J Ehlers, E E Falco. ''Gravitational Lenses''. Berlin: Springer, 1999. Section 3.4.2.〕


In fact, these three scalars \ can be defined for both timelike and null geodesic congruences in an identical spirit, but they are called "optical scalars" only for the null case. Also, it is their tensorial predecessors \\,, \hat\omega_ \} that are adopted in tensorial equations, while the scalars \ mainly show up in equations written in the language of Newman–Penrose formalism.
==Definitions: expansion, shear and twist==


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