Peeling theorem

In general relativity, the peeling theorem describes the asymptotic behavior of the Weyl tensor as one goes to null infinity. Let ${\displaystyle \gamma }$ be a null geodesic in a spacetime ${\displaystyle (M,g_{ab})}$ from a point p to null infinity, with affine parameter ${\displaystyle \lambda }$. Then the theorem states that, as ${\displaystyle \lambda }$ tends to infinity:

${\displaystyle C_{abcd}={\frac {C_{abcd}^{(1)}}{\lambda }}+{\frac {C_{abcd}^{(2)}}{\lambda ^{2}}}+{\frac {C_{abcd}^{(3)}}{\lambda ^{3}}}+{\frac {C_{abcd}^{(4)}}{\lambda ^{4}}}+O\left({\frac {1}{\lambda ^{5}}}\right)}$

where ${\displaystyle C_{abcd}}$ is the Weyl tensor, and abstract index notation is used. Moreover, in the Petrov classification, ${\displaystyle C_{abcd}^{(1)}}$ is type N, ${\displaystyle C_{abcd}^{(2)}}$ is type III, ${\displaystyle C_{abcd}^{(3)}}$ is type II (or II-II) and ${\displaystyle C_{abcd}^{(4)}}$ is type I.

References

• Wald, Robert M. (1984), General Relativity, University of Chicago Press, ISBN 0-226-87033-2

External links