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Vector perturbations can be decomposed into harmonic modes
of the Laplacian in the same manner as the scalars,
| ![\begin{displaymath}
\nabla^2 Q^{(\pm 1)}_{i} = -k^2 Q^{(\pm 1)}_{i} ,\end{displaymath}](img134.gif) |
(20) |
which satisfy a divergenceless condition
| ![\begin{displaymath}
{\nabla}^i {Q}_i^{(\pm 1)}= 0 \, .\end{displaymath}](img135.gif) |
(21) |
A
velocity field based on vector perturbations thus represents vorticity,
whereas scalar objects such as density perturbations are entirely
absent.
The corresponding symmetric tensor is constructed out of derivatives
as
| ![\begin{displaymath}
Q_{ij}^{(\pm 1)}= - {1 \over 2k}
(\nabla_i Q_j^{(\pm 1)}+ \nabla_j Q_i^{(\pm 1)})\, .\end{displaymath}](img136.gif) |
(22) |
A convenient representation is
| ![\begin{displaymath}
{Q}_i^{(\pm 1)}= - {i \over \sqrt{2}}
(\hat{e}_1 \pm i \hat{e}_2)_i \exp(i\vec{k}
\cdot \vec{x}) \, .\end{displaymath}](img137.gif) |
(23) |
Notice that
and
.
Thus vector perturbations stimulate
the
modes in the radiation.
Wayne Hu
9/9/1997