Issues with equations derived in the PublicNPGHP.nb file?
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Sid
Aug 8, 2017, 5:02:59 PM8/8/17
to xAct Tensor Computer Algebra
Hi
I recently went through the file(PublicNPGHP.nb) which works out the NP and the GHP formalism via the spinor approach. To me it seems that some of the equations derived are incorrect. In particular, all the Ricci equations seems to have an incorrect sign associated with the curvature components when compared with the Newman and Penrose's original paper (or some other standard reference). For example, if one compares equation 4.2(o) in the NP paper to the corresponding equation in NPRicci, then one would find that curvature component, PhiCDe12Dyad[] has a sign opposite to the one produced in the paper. This seems to be the case for all the Ricci equations. Also some of the Bianchi Identities do not quite match those produced in standard references, for instance I used Chandrasekhar's "The Mathematical Theory of Black Holes" for comparison. To be precise equations 320 (c) , 322 (i) ,(j) and (k) seem to be missing from NPBianchi. Am I missing something? It'll be helpful if someone else could also check these equations.
Regards
Sid
I recently went through the file(PublicNPGHP.nb) which works out the NP and the GHP formalism via the spinor approach. To me it seems that some of the equations derived are incorrect. In particular, all the Ricci equations seems to have an incorrect sign associated with the curvature components when compared with the Newman and Penrose's original paper (or some other standard reference). For example, if one compares equation 4.2(o) in the NP paper to the corresponding equation in NPRicci, then one would find that curvature component, PhiCDe12Dyad[] has a sign opposite to the one produced in the paper. This seems to be the case for all the Ricci equations. Also some of the Bianchi Identities do not quite match those produced in standard references, for instance I used Chandrasekhar's "The Mathematical Theory of Black Holes" for comparison. To be precise equations 320 (c) , 322 (i) ,(j) and (k) seem to be missing from NPBianchi. Am I missing something? It'll be helpful if someone else could also check these equations.
Regards
Sid
Thomas Bäckdahl
Aug 9, 2017, 3:32:28 AM8/9/17
to xa...@googlegroups.com
Hi!
The sign for the curvature components comes from the fact that the NP paper, Penrose-Rindler and some other references uses a different sign convention for the Riemann tensor. If you want to use that sign convention please add
$RiemannSign = -1;
to the notebook.
The set "NPBianchi" is slightly different organized from Chandra's presentation. The contracted Bianchi identities (eq 322) can be found by solving the system "NPBianchi" for derivatives of Ricci scalar (called Lambda in NP formalism). Note that these only appear in the contracted Bianchi system. Equation (321c) (I guess you meant 321 instead of 320) which has D\Psi_3 in it also appears in "NPBianchi", e.g. on the RHS of the last equation.
If you want to double check the equivalence of the two sets, type in Chandra's form and insert "NPBianchi", that should give zero.
We would also like to point out that we developed a package SpinFrames for NP and GHP calculations.
http://xact.es/SpinFrames/index.html
All possible things you might want to do are not implemented, but it should make your NP and GHP calculation notebooks shorter. For the main features see the SpinFramesExamples notebook. All the commutators, Ricci relations are implemented so you don't have to re-derive them every time you want to use them.
Kindest regards
Thomas and Steffen
The sign for the curvature components comes from the fact that the NP paper, Penrose-Rindler and some other references uses a different sign convention for the Riemann tensor. If you want to use that sign convention please add
$RiemannSign = -1;
to the notebook.
The set "NPBianchi" is slightly different organized from Chandra's presentation. The contracted Bianchi identities (eq 322) can be found by solving the system "NPBianchi" for derivatives of Ricci scalar (called Lambda in NP formalism). Note that these only appear in the contracted Bianchi system. Equation (321c) (I guess you meant 321 instead of 320) which has D\Psi_3 in it also appears in "NPBianchi", e.g. on the RHS of the last equation.
If you want to double check the equivalence of the two sets, type in Chandra's form and insert "NPBianchi", that should give zero.
We would also like to point out that we developed a package SpinFrames for NP and GHP calculations.
http://xact.es/SpinFrames/index.html
All possible things you might want to do are not implemented, but it should make your NP and GHP calculation notebooks shorter. For the main features see the SpinFramesExamples notebook. All the commutators, Ricci relations are implemented so you don't have to re-derive them every time you want to use them.
Kindest regards
Thomas and Steffen
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Sid
Aug 9, 2017, 4:35:48 PM8/9/17
to xAct Tensor Computer Algebra, sabharwal...@gmail.com
Hi Alfonso
This is amazing. Thank you so much for this.
Regards
Sid
Sid
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