@ecterrab 

The method works great. Thanks.

Michael

@ecterrab 

You are correct. The adjustments work. I typed in the wrong values.  We can close this post. Thanks.

Michael

@ecterrab 

I have uploaded the new version of Physics. It seems to help, however.....

Now eqn1 and eqn2 are simple, but eqn 3 has difficulty. I also rearranged the variables because the zeta differential was not showing up. Thanks for the help I will keep working with it.

Michael

@Torre 

thanks. It is working.

@ecterrab 

By your example, the code is working corrctly and you are correct about curvature in the metric.  However, my metric (g_mn=η_mn + l_ml_n) allows l^μ = g^μνl_ν = η^μνl_ν. Where η is the Cartesian Minkowski metric and l^μ is null vector.

Therefore, l^a_,bl_al^b = (η^acl_c),_bl_al^b = l_c,bη^acl_al^b = l_c,bl^cl^b . 

If the code is using g^mnl_n=(η^mn + l^mlⁿ) internally for the Simplify command, then that may lead to my confusion. Because I need g^mn=η^mn - l^mlⁿ. Although, I compared the Christoffels to my notes and got the same results. Anyway, I am glad the problem is resolved. I will be more careful with the substitution.

Thank you.

Michael

@ecterrab 

Download Maple_Question_7.8.14.mw

@ecterrab 

Got it.  Thanks.

@ecterrab 

Thanks.  I looked it over and there was just confusion on how the indices are applied. The internal treatment by the software replaces the dummy indices in order to do the summation.  I am okay to move on.  I appreciate your help.

Michael

@ecterrab 

I updated the program, so I don't get the wierd alpha1's anymore. I did the test and it worked.  I found the error that I was making by hand. (Which is why I am learning to use Maple). I am think the Simplify is working corrctly, but I still have the issue with the indices.

SubstituteTensorIndices changes all of the indices. For (18), I need kappa in the 1st term to be mu and kappa in the 2nd term to be nu.

Taking the result of (17) and relabelling the indices to correspond to the indices in (15) is diffficult. Terms 1 & 2 indicate that beta -> rho and kappa -> mu.  However, this substitution affects the kappa in the 2nd term. So I get a l[~mu]^2 which is clearly wrong.  The third term is also incorrect after substitution, but the fourth term is correct.

Download Vacuum_Solutions_(Kerr-Schild)_3.mwVacuum_Solutions_(Kerr-Schild)_3.mw

@ecterrab 

Thanks.  I appreciate your help.  I think that I have a much better idea on how to proceed.  We can close this Question section.

I just realized that my last comment was in eror.  I did not set me signature to `+`.  Also I want the  g_[mu,nu] to have a determinant where -g = 1.  These changes are not going to fundamentally change the problem. I am just making sure that I incorporate the details of the paper.

Thank you, again.

Michael

P.S. - SubstituteTensor does not show in the Maple Help search. The help document is only accessible from the Physics[Library] page, but not as a Physics package command.

@ecterrab 

I sent two fiiles. TestfileA uses the Christoffel of the First Kind and TestfileB uses the Second Kind.  The inverse of the metric is not being created properly.

Thank you.

Michael

@ecterrab 

Okay I think this should work.  I provided link to MW and PDF.

Michael

testfile.mw

testfile.pdf

@ecterrab 

Thank you.  I incorporated your sugestions.  I almost have the result by using algsub to enforce the l[~nu]*l[nu] = 0 condition.  I have attached the file. The file is Maple 18 compatible.

Vacuum_Solutions_(Kerr-Schild).mw

@ecterrab 

Thank you. This helped a lot!

I forgot to add the coordinates, which I am making arbitrary. The hard part is translating things I do by hand into Maple. The convert command can be used to display the Christoffel and Riemann in terms of $g_{\alpha,\beta}$. However, is there a way to write them in terms of my $l_\mu$? For example, By hand I use the Christoffel:
\[
\Gamma^\alpha_{\beta\gamma} = (1/2) g^{\alpha \mu} (g_{\beta \mu,\gamma} + g_{\gamma \mu,\beta} - g_{\beta \alpha,\mu})
\]
Let
\[
g^{\alpha \mu} = \eta^{\alpha \mu} - l^\alpha l^\mu
\]
Plug and chug:
\[
\Gamma^\alpha_{\beta \gamma} = (1/2) (\eta^{\alpha \mu} - l^\alpha l^\mu)((l_\beta l_\mu)_{,\gamma}) + (l_\gamma l_\mu)_{,\beta}) -(l_\beta l_\gamma)_{,\mu})
\]
And so on......
Plugging into
\[
R_{\mu \nu} l^\mu l^\nu =0
\]
After ridiculously tedious algebra, I get the relationship
\[
l_{\mu,\nu}l^\nu l^\mu_{,\rho} l^\rho = 0
\]

Now if I type: Ricci$[1, 2]*l[\sim1]*l[\sim 2]$ in Maple, I get an answer which is not as elegant as the result above. :-)

Yes, I am working the steps in the Kerr and Schild conference paper. The plan is to modify the steps and the metric, but I need to make sure that Maple can reproduce the same steps I use with pencil and paper. Thank you again for your help.

Thank you.  That is what I get for coding at 2 am.

Michael