Curvature using exterior differential forms

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SUMMARY

The discussion centers on the calculation of curvature using exterior differential forms as outlined in Charles Misner's work, specifically on pages 354-363. Participants explore the effectiveness of the "guess and check" method for finding connection forms (Eq. (14.31)) when dealing with metrics that contain only diagonal terms (g_{nn}). The conversation raises the question of whether this method can be applied to metrics with off-diagonal terms (g_{mn}, m ≠ n) or if one must resort to the systematic approach detailed in Equations (14.32) and (14.33). Additionally, a reference to Misner's 1962 article in the Journal of Math Physics is mentioned as a clearer alternative for calculating the Ricci tensor.

PREREQUISITES
  • Understanding of exterior differential forms
  • Familiarity with curvature calculations in differential geometry
  • Knowledge of the Einstein field equations
  • Experience with metrics in general relativity
NEXT STEPS
  • Study the systematic method for calculating connection forms as described in Equations (14.32) and (14.33)
  • Review Charles Misner's 1962 article in the Journal of Math Physics for clarity on exterior calculus
  • Explore the implications of off-diagonal terms in metrics on curvature calculations
  • Investigate the relationship between Ricci tensors and Einstein's equations in greater detail
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, and graduate students specializing in general relativity, particularly those interested in advanced curvature calculations and the application of exterior differential forms.

emma83
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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 [tex]g_{nn}[/tex], 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 ([tex]g_{mn}, m \neq n[/tex]), or is one then compelled to use the systematic way (Eq. (14.32) and (14.33)) ?

Thanks a lot for your help!
 
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emma83 said:
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 [tex]g_{nn}[/tex], 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 ([tex]g_{mn}, m \neq n[/tex]), 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.
 

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