2
votes
1answer
89 views

Computing the Christoffel symbols with the geodesic equation

I would like to compute the Christoffel symbols of the second kind using the geodesic equation. To practice, I have tried the Schwarzschild Ansatz $$ g_{00} = \mathrm e^\nu,\quad g_{11} = - \mathrm ...
14
votes
4answers
371 views

The Lagrangian as a metric

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
3
votes
1answer
130 views

Schwarzschild geodesics

I've found on Wikipedia that energy $E$ and angular momentum $L$ of a particle are conserved quantities in Schwarzschild metric. It's written: $$L=mr^2 \frac {d\phi} {d\tau},$$ ...
10
votes
3answers
411 views

Equation of motion of a photon in a given metric

I have this metric: $$ds^2=-dt^2+e^tdx^2$$ and I want to find the equation of motion (of x). for that i thought I have two options: using E.L. with the Lagrangian: $L=-\dot t ^2+e^t\dot x ^2 $. ...
3
votes
2answers
881 views

Null geodesic given metric

I (desperately) need help with the following: What is the null geodesic for the space time $$ds^2=-x^2 dt^2 +dx^2?$$ I don't know how to transform a metric into a geodesic...! There is no need to ...