There are several packages for tensor and spinor calculus, e.g. very powerful package xAct available at

http://www.xact.es/

However, I usually don't use these packages because I want to have everything under my control :) If you are interested, I can send you some my notebooks.

But I don't understand what you mean by calculating the tetrad from metric. The tetrad is not related to the coordinates: you can have null tetrad, orthonormal tetrad, etc. Do you have metric tensor in particular coordinate system and want to construct, say, orthonormal tetrad? Or do you have already metric components with respect to some chosen tetrad?

If you can be more specific about the problem, I can send you the solution in Mathematica...

Hello Martin Scoltz. Thank you for the information. I also dont like using predefined packages. I downloaded the "xAct" packages and didnt enjoy using it.

My problem isnt complex, i think. may be its because am a biginner in Mathematica. I have a lagrangian that contains Spin-connection', i.e Omega_i=gamma_a gamma_b e_j^a(del_ik e_l^b - Gamma_ik^l e_l^b)g^kl.

The metric from which I wish to compute this thing is ds^2=A(t, r, theta)dt^2-B(t, r, theta)dr^2 - C(t, r, theta)dtheta^2.

In a geometry like this, not only the vielbeins(also called tetrad fields) but also the gamma matrices depends on coordinates.

I do hope this piece will help clarify things. I shall be very grateful to have your notebook on this stuff. Thank you for your kindness.

Dear Musah,

thank you for your reply, now I understand the problem. I will send you the answer (if I will be able to solve it, but I think so) during next few days, OK? Best regards

Ok! Thanks

Dear Musah, sorry, I was busy during the weekend, but finally I attach the calculation of the spin connection in Mathematica. I have used slightly different definition of the spin connection, but you can adapt it to your needs. Moreover, I don't know your representation of gamma-matrices, but if you tell me what representation you use, I can extend the notebook. I hope that this is what you need, please, le me know.

Dear Dr. Scholtz, thank you very much for your great and wonderful assistance. The notebook perfectly answered my problem. I really, appreciate that.

The spin-connection is perfect, I can easily adapt it to my problem. I need some little clarifications, please. Can the Viebein be computed directly from the Meric? This is particularly a problem when the metric is not just diagonal.

Regarding the curved Gamma Matrices, I am using the Planer Representation where the 0, 1, 2 matrices are diagonal in the pauli sigma matrices; i.e sigma3, i sigma1 and i sigma2 respectively.

Dear Musah, I am glad that I could help. Please, call me Martin rather than dr. Scholtz :) I also hope that Musah is your first name :)

Yes, you can calculate vielbeins using Mathematica for non-diagonal metrics. The problem is that the vielbein is not unique. Independently of coordinates, you can construct infinitely many tetrads which are all related by the Lorentz transformation. So, the usual procedure is to adapt the tetrad to your coordinates in some way.

For example, if you have a spacetime with radiation and you want to construct a null tetrad (not the orthonormal one), you choose one of your vectors to be tangent to the null geodesics. I just want to say that given the metric, you cannot construct the vielbein uniquely because you always have the freedom - Lorentz rotation.

However, I have created an example how to construct orthonormal tetrad for simple non-diagonal metric tensor. This procedure applies also to more complicated things but as you will see in the notebook, you must fix the Lorentz freedom at least partially.

With the best regards,

Martin

Dear Martin, thank you for the notebook. Again it wad very helpful. Frakly, I was stucked at these stuff in my work.

Yes! My first name is Rabiu.

Please, I have used the code "LinearSolve[{{1,0,0},{0,1,0},{0,0,1}}, MatrixPower[gu,1/2]]" but obtain diffetent results. I believe this is still due to the gauge freedom of the viebeins?

Is it also possible for the metric "gu" to be visualise graphically? We can take examples of Cosmic String Space metric or Robertson-Walker geometry. I am sorry for asking too much questions!

Best wishes

Dear Martin. I just realized I did not include the representation of my Gamma matrices. It is in a 2D fundamental Representation; Gamma3 = Sigma3, gamma_i = Sigma_i.

Thank you

Thank you both for your discussion. I also downloaded your notebook Martin and plan to adapt it to my work as well. Truly appreciate it.

Dear All,

Nice discussion.

These days i am trying to solve Dirac equation in the Re-No background.

I need to spin connection so i downloaded the worksheet by Martin.

Thanks.

Mohsen

Dear Martin Scholtz , then how can I calculate the tetrad fields (Eq. 4 of Article Schwarzschild horizon dynamics and SU(2) Chern-Simons theory