"Show that if a space time metric admits three linearly independent 4 vector fields with vanishing covariant derivatives then Rabcd = 0"(adsbygoogle = window.adsbygoogle || []).push({});

We can set the three vectors as (1,0,0,0), (0,1,0,0) and (0,0,1,0). Use covariant derivative of vector field X^b is:

d(X^b)/d(x^a) + (Christoffel symbol with superscript b and subscripts a, c)* (X^c)

where the derivative above is partial.

Therefore the following Christoffel symbols are zero:

(superscript b, subscripts a,0)

(superscript b, subscripts a,1)

(superscript b, subscripts a,2)

Assume that the Christoffel symbols are symmetric (for a symmetric gab), therefore we know that only the Christoffel symbol with both subscripts equal to 3 can be non zero, i.e

(superscipt b, subscripts 3,3)

At this point I get stuck.

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# Homework Help: Rabcd = 0 proof

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