@prldb

You asked a very similar (basically the same) question in 2020. The answer today would be the same, only **emphasizing** that the ordering of the NP null vectors that conform a tetrad depends on the signature, basically on where you put the timelike component in the metric. For the signature used in Maple, *which is also the one used in Stephanie's book*, the ordering is n, m, mb and l also in the book. And yes the convention used by Maple for the NP null vectors is the existing one (standard) - also used in Stephanie's book. All this is shown in that answer of 2020, also explained in the link I mentioned in the previous reply "What to take care of when entering a tetrad".

Now, the tetrad is defined up to an arbitrary rotation in 4 dimensions, from which you can mix everything and write a new tetrad with the imaginary unit anywhere you want. That doesn't change the fact that n, m, mb, l form a tetrad when you put them in the correct order according to the signature, and doesn't change the definitions of these NP vectors that you can see, for example with n_, via

Try **TensorArray([%])** with some simplfier and you see all their defining equations are satisfied, regardless of where you put the imaginary unit.

Regarding the optimized tetrads shown in Stephanie's book: Maple is not using those tetrads - as said in a previous reply above, those book's tetrads are coded, but won't be used until they are reviewed. Instead, Maple computes a tetrad from scratch as explained in *?Tetrads:-IsTetrad*, and as said there you could also enter any other different tetrad that you may prefer and the system will work with it the same way. Or if you prefer to perform transformations to optimize your tetrad in any particular way, see *?Tetrads:-TransformTetrad.*

Regarding Maple's tetrad, which is constructed from the NP vectors, the standard ones, in the right ordering according to the signature used in Stephanie's book, or regarding any other tetrad, you can always test whether it is or not a correct tetrad using Tetrad:-IsTetrad, or checking whether it satisfies the tetrad equations, basically input **TensorArray(e_[definition])** and you see, by eye, if it is or not a correct tetrad, regardless of where you put complex components. Here is for the metric you are mentioning:

In summary: the tetrad computed by Maple is correct, the NP null vectors are correct too, they are the standard ones, and the conventions for how to order the NP vectors to form a tetrad are those shown in the book too, as shown step-by-step in the answer I gave to you during 2020. The only thing that is not a match to Stephanie's book is that Maple computes a tetrad that is not as the one shown there, where the case is that tetrads are defined up to an arbitrary 4D rotation.

I'm sorry but also need to move to other topics.

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft