Taken from my lecturer's notes on GR:(adsbygoogle = window.adsbygoogle || []).push({});

I'm trying to understand what goes on from 2nd to 3rd line:

[tex]N^\beta \nabla_\beta (T^\mu \nabla_\mu T^\alpha) - N^\beta \nabla_\beta T^\mu \nabla_\mu T^\alpha = -T^\beta \nabla_\beta N^\mu \nabla_\mu T^\alpha [/tex]

Using commutator relation ## T^v \nabla_v N^\alpha = N^v\nabla_v T^\alpha ## we swap the 'N' for the 'T' in the last term of the second line:

[tex] N^\beta \nabla_\beta (T^\mu \nabla_\mu T^\alpha) - T^\beta \nabla_\beta N^\mu \nabla_\mu T^\alpha = -T^\beta \nabla_\beta N^\mu \nabla_\mu T^\alpha [/tex]

[tex] N^\beta \nabla_\beta \left(\frac{D T^\alpha}{D\lambda}\right) - T^\beta \nabla_\beta N^\mu \nabla_\mu T^\alpha = -T^\beta \nabla_\beta N^\mu \nabla_\mu T^\alpha [/tex]

This means that ## \nabla_\beta \left(\frac{D T^\alpha}{D\lambda}\right) = 0 ##. Why is this so?

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# Quick one-line working on Geodesic Deviation Equation

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