4
votes
2answers
114 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
353 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
378 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 ...