3+1 decomposition of Christoffel symbols

[Alex Vañó Viñuales]

Dear all,

I am trying to perform a 3+1 decomposition of gauge equations with aims to a numerical implementation. The expression I'm working with is first defined on a 4D manifold and includes a contracted Christoffel symbol of the form m4tric[a,b]Christoffelcd4[c,-a,-b] (where m4tric is the 4D metric, cd4 its associated covariant derivative and Christoffelcd4 the connection related to the latter).

Introducing a 3+1 decomposition in the usual way (with lapse and shift) allows to express the 4D contracted Christoffel in terms of the 3D contracted one, lapse, shift and the trace of the extrinsic curvature (see for instance eqs (B.13-14) on page 409 in Alcubierre's Introduction to 3+1 Numerical Relativity). 

I'm trying to reproduce this same decomposition in xTensor, but cannot seem to find the appropriate functions to apply --- I have already looked into the "ADM-type calculations" slide in the "Advanced concepts" Mathematica slide show in the documentation. I am also wondering if it's possible to do it purely in xTensor, or if one needs to rely on xCoba to do it, partially substituting some of the components. 

Any ideas or suggestions will be much appreciated. Thanks very much! 

Best wishes,

Alex

[Leo Stein]

Hi Alex,

Hope everything is going well! I don't remember if this has been automated or not... but using commands that I remember, here's one way to do it. This ought to be packed into a rule and automated. Also worth checking by hand to make sure I haven't flubbed anything. Here CD is the ambient covariant derivative, and cd is the one compatible with the induced metric. I thought the easiest approach would be to break the induced Christoffel into PD's, break all the induced metrics into g's and n's, and then go back to the Christoffel of the ambient space plus derivatives of n. Please see the attached notebook, and the result is summarized in this screenshot:

Best

Leo

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[Alex Vañó Viñuales]

Dear Leo,

Thanks, hope you are well too! Thank you very much for the example notebook, it is indeed very useful. I must have been applying the functions in the wrong order, because I could not find the result you arrive at. I think that just tweaking your recommendation slightly should allow me to get exactly what I want. Thank you! 

Best,

Alex