ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 14 Aug 2019 09:51:40 -0500Source for Principal Null Directions Kerr, Example Worksheethttp://ask.sagemath.org/question/47468/source-for-principal-null-directions-kerr-example-worksheet/ In the Worksheet "Walker-Penrose Killing tensor in Kerr spacetime" of the SageManifolds project, the sheet gives the principal null directions for Kerr spacetime in Boyer-Lindquist coordinates. I have difficulties verifying (with Mathematica) that this is indeed the case and would appreciate a source. Tue, 13 Aug 2019 07:43:40 -0500http://ask.sagemath.org/question/47468/source-for-principal-null-directions-kerr-example-worksheet/Answer by eric_g for <p>In the Worksheet "Walker-Penrose Killing tensor in Kerr spacetime" of the SageManifolds project, the sheet gives the principal null directions for Kerr spacetime in Boyer-Lindquist coordinates. I have difficulties verifying (with Mathematica) that this is indeed the case and would appreciate a source. </p>
http://ask.sagemath.org/question/47468/source-for-principal-null-directions-kerr-example-worksheet/?answer=47483#post-id-47483The vectors $\ell$ and $k$ introduced in this notebook correspond to Eqs. (12.3.5) and (12.3.6) of Wald's textbook *General Relativity* (1984).
You can check that they do define repeated principal null directions, i.e. that they obey the following identity (where $C^a_{\ \ bcd}$ stands for the Weyl tensor) $$ C^a_{\ \ mn[b} k_{c]} k^m k^n = 0,\qquad\qquad\mbox{(1)}$$ with SageMath itself (no need of Mathematica ;-)). It suffices to run (same notations as in the [original notebook](https://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Worksheets/v1.3/SM_Kerr_Killing_tensor.ipynb))
C = g.weyl()
(C.contract(1, 2, k*k, 0, 1) * k.down(g)).antisymmetrize(1, 2).display()
which yields `0`.
If you prefer to use index notations (passed as strings in LaTeX format) instead of `contract()` and `antisymmetrize()`, SageMath allows for it as well. The check of (1) is then equivalent to
A = C['^a_{mnb}'] * (k*k)['^{mn}']
kf = g['_{am}'] * k['^m']
(A*kf)['^a_{[bc]}'].display()
the outcome of which is `0`.
A Jupyter notebook implementing the check of (1) by both methods is posted [here](https://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Notebooks/SM_Kerr_principal_null.ipynb).Wed, 14 Aug 2019 09:51:40 -0500http://ask.sagemath.org/question/47468/source-for-principal-null-directions-kerr-example-worksheet/?answer=47483#post-id-47483