@ecterrab SKMHH27_37.mw SKMHH27_37_Modified.mw signature.mw
Many thanks for your post responding to my query. As a result I have resolved most of my issues, though it has raised more questions for me. In your post you put in the metric (27,37) by hand and specified the order of the coordinates as in Stephani et al (hereafter Steph). Note that Setup(metric=[27,37,1]) does not give Steph's ordering of the coordinates, but rather specifies (u,r,z,zb) (as in SKMHH27_37.mw).This is why my definiiton of null tetrad (co)vectors looks different to what one might expect from Steph. I did have a couple of extraneous minus signs, and removing those has helped. Nevertheless, my .mw files (all three start with a call of [27,37,1]) and your metric are the same, only trhe coordiantes are ordered differently. Yet, when I use Setup(tetrad=null) in SKMHH27_37 the tetrad Maple returns is different to the one you get using the same command. Why would Maple give different results in these two cases, that seemingly only differ by the ordering of the coordinates?
Continuing in my SKMHH27_37 worksheet, I did get TransformTetrad(canonicalform) to output a result, though it's not the null tetrad of Steph. and still quite complicated.
I then defined the null tetrad of Steph directly, in covector form, and Maple now recognizes it as a null tetrad and returns the expressions for the Weyl scalars I expected. So, my remaining question on this worksheet is why Setup(tetrad=null) gave a different result to what you obtained? Despite the non-uniqueness of null tetrads, I would assume that this command applies some default procedure to the metric to construct a null tetrad and so, given the same metric I would have expected the same result from the same command, even though the coordinates are ordered differently.
In KSMHH27_37_Modified, I started by calling [27,37,1] and tried to follow your procedure for introducing a null tetrad, defining the contravariant vectors. It seemed to start okay, but when I wrote the equivalent of your line 'Define(16)' (my line 'Define(4)') the output I get differs from yours, and seems a bit bizzare, and then when I ask for the components of the covariant forms as you did I don't get an output. So something has gone wrong in my worksheet, even though I believe I have followed your syntax. Why is the line 'Define(16)' even ncecessary? Doesn't the previous 'Define' command define l, n, m, mb as tensor objects and then the following line specifies their components; what purpose does 'Define(16)' serve?
I then resorted to the same syntax I used in SKMHH27_37, introducing a matrix whose rows are the covector components and then specifying that as a tetrad, and checking it's null. So my question here is, why does my 'Define(4)' not do what your 'Define(16)' does? For some reason, my syntax has failed to define the four vectors l, n, m, mb I assume, which is why the comamnd l[ ]; n[ ];m[ ];mb[ ]; didn't return the covector components. Yet my equaitlon (4) looks okay.
My next question is about the commands l_[ ] etc in Tetrads. Am I right that calling those commands just returns the current null tetrad, which will be the same as Setup(tetrad=null) if no other null tetrad has already been specified? When I called l_[ ] etc at the end of SKMHH27_37_Modified, the null tetrad I had defined was returned, correclty labelled. I am surprised, however, in your worksheet that the first element of the null tetrad created by Setup(tetrad=null) is labelled n, the second m, the third mb, and the fourth l, with m clearly not the complex conjugate of mb. In fact, the n and l of your (8) and (13) would be the complex conjugate pair for Steph. I wonder if this seemingly odd labelling has something to do with the question I ask in the last paragrpah below?
In your worksheet you chose the default spacetime signature (---+), whereas the default signature of Steph is (+++-). One can set the signature with Setup, but when Itried to do so after calling [27,37,1] I got an error message that I could find no help on despite searching the Help and forrums. It's in the signature.mw worksheet. I get this error whether I place the Setup command to specify the signature before or after calling the metric. I couldn't find an actual example of the use of this comamnd in Help. So my last question is how does this command work? Can it be applied after the metric is called or only before? And why am I getting this error message.
Adding to my confusion is your choice of the default signature (---+) together with Steph's form of the metric. In Steph, the coordinates z and zb are complex conjugates and their coordinate plane is positive definite while the u coordinate is negative definite, in accord with their signature of (+++-). In your worksheet Maple states that all the coordinates are real (equation (2)). If z and zb are interperted as real, their coordinate plane would be of signature (+-), given the metric. The coordinate plane of (u,r) is orthogonal to that of (z,zb) according to the metric, and contains a null vector (the coordinate vector of r) while, generically, the coordinate vector of u is non-null and not orthogonal to that of r, which would make the signature of this coordinate plane also (+-), whence the metric would be of neutral signatuer (+-+-). I know from experience with the Physics package that one can set up a metric of, say, neutral signature, and Maple will compute all the ususal differential geometry quantities correctly without complaint (which is a good thing since neutral signature is not an option in Setup). So I'm assuming these symbolic computations are performed independently of the specified signature, i.e., Maple doesn't assume a signature for the purposes of calculating any metric-dependent quanity, it just uses the metric. Though the specified signature would affect Maple's interpretation of the coordinates?