Problems with taking covariant derivatives of scalar functions

[Vincent Lee]

Hi!

I have been trying to explicitly evaluate covariant derivatives on some general scalar function \Phi. I used DefScalarFunction[\Phi] to define a scalar function and tried evaluating the derivatives. 

I found that when the result is a scalar, then xact gives me the correct answer. For example cd[-a][cd[a][\[CapitalPhi][t[], z[]]]] works perfectly fine (here I am using AdS2 metric with t[] and z[] being the coordinates). However, if the result is some tensor, then xact gives me answer that I do not understand. For example, cd[-a][cd[-b][\[CapitalPhi][t[], z[]]]] does not work for me.

I have attached a minimal notebook that demonstrates the problem. Ideally the final answer should be some rank-2 tensor (since I am taking two covariant derivatives on a scalar) with 4 components (tt, tz, zt and zz). So the final answer should just be a 2x2 matrix.

Please let me know if you have any thoughts on this. Many thanks in advance!

Best regards,

Vincent

[Alex Vañó Viñuales]

Hi Vincent,

Try adding 

% // ToBasis[ads]
ComponentArray[%]

to your last cell. ToBasis[basis] translates abstract tensor indices into component indices in the chosen basis and ComponentArray writes the result as a matrix. 

Hope this helps! 

Best wishes,

Alex

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