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Calculation of "Ricci rotation coefficients" OR "Spin connections"

The expression of Ricci rotation coefficient is mentioned below. I want to write a Mathematica program to calculate the Ricci rotation coefficients (or, so-called the Spin connections) in the Kerr or Schwarzschild spacetime. Can anyone help me to write the program? enter image description here

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Thanks for your reply. Please, find the attached notebook. I have written down the desired output at the end of the notebook. However, I'm getting a different result for omega.

Please provide an explicit input (in copy-pastable format) and desired output.

POSTED BY: Daniel Lichtblau
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