Physics:Peeling theorem
In general relativity, the peeling theorem describes the asymptotic behavior of the Weyl tensor as one goes to null infinity. Let [math]\displaystyle{ \gamma }[/math] be a null geodesic in a spacetime [math]\displaystyle{ (M, g_{ab}) }[/math] from a point p to null infinity, with affine parameter [math]\displaystyle{ \lambda }[/math]. Then the theorem states that, as [math]\displaystyle{ \lambda }[/math] tends to infinity:
- [math]\displaystyle{ C_{abcd} = \frac{C^{(1)}_{abcd}}{\lambda}+\frac{C^{(2)}_{abcd}}{\lambda^2}+\frac{C^{(3)}_{abcd}}{\lambda^3}+\frac{C^{(4)}_{abcd}}{\lambda^4}+O\left(\frac{1}{\lambda^5}\right) }[/math]
where [math]\displaystyle{ C_{abcd} }[/math] is the Weyl tensor, and we used the abstract index notation. Moreover, in the Petrov classification, [math]\displaystyle{ C^{(1)}_{abcd} }[/math] is type N, [math]\displaystyle{ C^{(2)}_{abcd} }[/math] is type III, [math]\displaystyle{ C^{(3)}_{abcd} }[/math] is type II (or II-II) and [math]\displaystyle{ C^{(4)}_{abcd} }[/math] is type I.
References
- Wald, Robert M. (1984), General Relativity, University of Chicago Press, ISBN 0-226-87033-2
External links
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