Friday, 18 September 2015

How can I convert an action in terms of differential forms to tensors?


I have an action of a gravitational theory wich is in terms of differential forms. Now, I need to transform this action (including wedge product and exterior derivative of tetrad and metric ) to an action in terms of tensors.i mean converting it into component notation. How can I do this calculation? Or can anyone introduce me a book or paper in this way?


for example the action of a theory is


$S = \int g_{ij}(x) \partial_{\mu} X^i \, \partial^{\mu}X^j$


but it is possible to write this action as


$\bar{\delta}_XS[X,\gamma]=-\int_\Sigma \eta_{\mu \nu} \bar{\delta}X^{\mu}\wedge d \star_\gamma d X^\nu=0$


Now I want to know that how the first form can be achieved from the second one.




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...