Curvature using exterior differential forms

[emma83]

Hello,

I have a question related to the calculation of curvature using exterior differential forms (Misner, pp. 354-363). In all the examples given in the book (i.e. Friedmann, Schwarzschild, pulsating star metrics), the "guess and check" method used to find the connection forms (Eq. (14.31)) work well because the metric has only diagonal terms $$g_{nn}$$, so an orthonormal basis such as the one given in Eq. (1) in Box 14.5 (p. 355) can be derived in a straightforward manner from the components of the metric.

Now is there any chance to apply this method when there are also off-diagonal terms ($$g_{mn}, m \neq n$$), or is one then compelled to use the systematic way (Eq. (14.32) and (14.33)) ?

Thanks a lot for your help!

 

[AEM]

Hello,

I have a question related to the calculation of curvature using exterior differential forms (Misner, pp. 354-363). In all the examples given in the book (i.e. Friedmann, Schwarzschild, pulsating star metrics), the "guess and check" method used to find the connection forms (Eq. (14.31)) work well because the metric has only diagonal terms $$g_{nn}$$, so an orthonormal basis such as the one given in Eq. (1) in Box 14.5 (p. 355) can be derived in a straightforward manner from the components of the metric.

Now is there any chance to apply this method when there are also off-diagonal terms ($$g_{mn}, m \neq n$$), or is one then compelled to use the systematic way (Eq. (14.32) and (14.33)) ?

Thanks a lot for your help!

I don't know if this will help you, or not, but I used the exterior calculus to calculate the Ricci tensor, and thus the source free Einstein's equations in my Ph.D. thesis a long time ago. Charles Misner published an article in 1962 in the Journal of Math Physics in the appendix of which he gave a very nice description of the procedure. I always thought his version there was more clear than the procedure given in MTW. I will look in my attic and see if I can find my copy of the article and post the exact reference.