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SpECTRE
v2025.08.19
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The complex m-mode field \((\Psi_m)_{ab}\). More...
#include <Tags.hpp>
Public Types | |
| using | type = tnsr::aa< ComplexDataVector, 3 > |
The complex m-mode field \((\Psi_m)_{ab}\).
Defined by the m-mode decomposition of the metric perturbation \(h_{ab}\):
\begin{equation} \bar{h}_{ab}(t,r,\theta,\phi) = \sum_{m=-\infty}^{\infty} \bar{h}^m_{ab}(r,\theta) e^{im(\phi - \Omega t)} \end{equation}
and further decomposition with convenient prefactors:
\begin{align*} &(\Psi_m)_{tt} = r \bar{h}^m_{tt} \\ &(\Psi_m)_{tr} = \frac{\Delta}{r} \bar{h}^m_{tr} \\ &(\Psi_m)_{t\theta} = \bar{h}^m_{t\theta} \\ &(\Psi_m)_{t\phi} = \frac{1}{\sin\theta} \bar{h}^m_{t\phi} \\ &(\Psi_m)_{rr} = \frac{\Delta^2}{r^3} \bar{h}^m_{rr} \\ &(\Psi_m)_{r\theta} = \frac{\Delta}{r^2} \bar{h}^m_{r\theta} \\ &(\Psi_m)_{r\phi} = \frac{\Delta}{r^2\sin\theta} \bar{h}^m_{r\phi} \\ &(\Psi_m)_{\theta\theta} = \frac{1}{r} \bar{h}^m_{\theta\theta} \\ &(\Psi_m)_{\theta\phi} = \frac{1}{r\sin\theta} \bar{h}^m_{\theta\phi} \\ &(\Psi_m)_{\phi\phi} = \frac{1}{r\sin^2\theta} \bar{h}^m_{\phi\phi} \\ \end{align*}