Curvature using exterior differential forms

emma83
Messages
31
Reaction score
0
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!
 
Physics news on Phys.org
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 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.
 

Thread 'I don't see how a black hole's event horizon can be crossed'

OK, so this has bugged me for a while about the equivalence principle and the black hole information paradox. If black holes "evaporate" via Hawking radiation, then they cannot exist forever. So, from my external perspective, watching the person fall in, they slow down, freeze, and redshift to "nothing," but never cross the event horizon. Does the equivalence principle say my perspective is valid? If it does, is it possible that that person really never crossed the event horizon? The...
View full post »

Thread 'Synchronizing clocks in an inertial frame if light is anisotropic'

ASSUMPTIONS 1. Two identical clocks A and B in the same inertial frame are stationary relative to each other a fixed distance L apart. Time passes at the same rate for both. 2. Both clocks are able to send/receive light signals and to write/read the send/receive times into signals. 3. The speed of light is anisotropic. METHOD 1. At time t[A1] and time t[B1], clock A sends a light signal to clock B. The clock B time is unknown to A. 2. Clock B receives the signal from A at time t[B2] and...
View full post »

Thread 'Is the variation of the metric ##\delta g_{\mu\nu}## a tensor?'

From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation ##\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}## should be a tensor...
View full post »

Similar threads

What is the Theory of Elasticity?
Replies
35
Views
28K

Hot Threads

Recent Insights

Back
Top