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.

 

[Entropia]

Hello,

Thank you for your question. The "guess and check" method using exterior differential forms is a useful tool for calculating curvature in certain cases, as you have mentioned. However, when dealing with metrics that have off-diagonal terms, it may not be as effective and may require the use of the systematic method. This is because the orthonormal basis derived from the components of the metric may not accurately reflect the curvature of the space in the presence of off-diagonal terms. Therefore, it is recommended to use the systematic method (Eq. (14.32) and (14.33)) to ensure accurate calculations of curvature.

Additionally, the systematic method allows for a more general approach that can be applied to any metric, regardless of its form. So while the "guess and check" method may work well for certain cases, the systematic method is a more reliable and comprehensive approach. I hope this helps clarify your question. Thank you for your interest in exterior differential forms and their applications in calculating curvature.