- #1
Demon117
- 165
- 1
For pure interest I have been trying to solve for the geodesics of the Schwarzschild metric. To do so I know I need to find the explicit Lagrangian for the variational principle for geodesics in this spacetime in Schwarzschild coordinates. How do I derive this lagrangian?
I know that the proper time along a timelike world line between two points in spacetime is
[tex]\sqrt {[-g_{{\alpha \beta }} \left( x \right) {{\it dx}}^{\alpha}{{
\it dx}}^{\beta}]} \left( B-A \right)[/tex]
But how do I use this and what does it end up telling me?
I know that the proper time along a timelike world line between two points in spacetime is
[tex]\sqrt {[-g_{{\alpha \beta }} \left( x \right) {{\it dx}}^{\alpha}{{
\it dx}}^{\beta}]} \left( B-A \right)[/tex]
But how do I use this and what does it end up telling me?