How to expand into tensor spherical harmonics?
Zi Su
I have some tensors (rank 2) which depend on a unit direction
(\hat{n}) and I want to expand it into tensor spherical harmonics. I
took a look of the Mathematica file "Harmonics" but have no idea how
to do it. Could anyone give me some illustrative examples to do so?
Thank you so much!
Best wishes
Simon
Leo Stein
I don't think that xAct is set up to do this right now, but I am not
an expert on the Harmonics package.
However, the principle is that you have to be able to integrate
tensors over the 2-sphere [see e.g. Thorne 1980, specifically for your
problem Eqs. (4.10a-b) and (4.11a-b)]. However, in xTensor,
calculations are always local. When you write something like T[-a,-b]
in xTensor, you are working at only a single manifold point.
Integrating of course requires being able to combine tensor quantities
from different points in the manifold.
Even though xTensor is not designed to do this, if you have coordinate
component expressions in some basis, and a way to compare tensors at
different points (a unique prescription for a bivector of transport;
e.g. in a flat spacetime, simply using rectangular coordinates), you
can use xCoba and then perform the integrations as regular
integrations in Mathematica.
Alternatively, if you are working on a flat spacetime, you can
implement Thorne's Eq. (2.3b) to do the integrations in terms of
tensors, rather than component values. I'm not sure that all of the
tensorial expressions you have can actually be written like that, so
it might not be useful. It's actually a bit of work to implement this
-- but if it would be helpful, I can provide some of the code I've
written that does just this (for post-Newtonian calculations).
Good luck,
Leo
Zi Su
Thank you so much for your quick response!
Actually, what I have to calculate is the polarization tensor of
photons (kind of generalized distribution function). So the tensor
itself depends on its position in the manifold and also the 4-vector
of momentum of the photons. What I am trying to get the expansion of
the polarization tensor in terms of the spacial unit direction vector
(\hat{n}) from the 4-vector of momentum. So it should be OK to be
local. I tried to use Thorne's paper for some of the calculations
(this is an excellent paper by the way). However, as you said, the
equations in Thorne's paper are not that enough. There are papers for
such an multipole expansion of the polarization tensor but I would
like to do it once by myself and extend it later.
Could you please send me some of the code you worked? It will be a
great help! Million thanks!
Best wishes,
Simon
Zi Su
Perfect! Thank you so much for your great help:)
Best wishes,
Simon
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