0
votes
0answers
53 views

Variation of a lagrangian, least action, metric, covariant derivate [closed]

I am doing the variation of a Lagrangian, but I am having problem with a particle terminus. My action is: $$S=\int d^4x \sqrt{-g}[ (\nabla_\mu A^\mu)^2] $$ I want to get the $T_{\mu\nu} $ ...
2
votes
1answer
65 views

Getting the Lagrangian from the action in curved spacetime

Suppose I have this action: $$ S = \int \mathrm d^4 x\sqrt{-g}\times \text{something}$$ where $g$ is the determinant of the metric. Should I take the Lagrangian to be: $$ \mathcal L = \sqrt{-g} ...
2
votes
1answer
328 views

Why vary the action with respect to the inverse metric?

Whenever I have read texts which employ actions that contain metric tensors, such as the Nambu-Goto, Polyakov or Einstein-Hilbert action, the equations of motion are derived by varying with respect to ...
3
votes
2answers
371 views

Polyakov action: difference induced metric and dynamical metric

The Polyakov action is given by: $$ S_p ~=~ -\frac{T}{2}\int d^2\sigma \sqrt{-g}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu} ~=~ -\frac{T}{2}\int d^2\sigma ...