Linked Questions

2
votes
1answer
814 views

Lagrangian, metric and geodesics: obtention of equation of motion in GR [duplicate]

I would like to check my global understanding on G.R: how we determine the trajectories of particles. Everything starts from the equations of Einstein. We have to find a metric that satisfies ...
2
votes
0answers
440 views

General relativistic Euler Lagrange equation [duplicate]

I am learning general relativity and at some point in the lecture notes it introduces the variational form of the equations of motion, i.e. minimise the action: $$S = \int_{A}^{B} d\tau = \frac{1}{c}\...
19
votes
4answers
17k views

What is the physical meaning of the affine parameter for null geodesic?

For time-like geodesic, the affine parameter is the proper time $\tau$ or its linear transform, and the geodesic equation is $$\frac{\mathrm d^{2}x^{\mu}}{\mathrm d\tau^{2}}+\Gamma_{\rho\sigma}^{\mu}...
6
votes
2answers
9k views

Geodesic equation from Euler - Lagrange

There are several ways to derive the geodesic equation. One of which is the variational method which I seemed to understand it because it was written in great details. Then it was mentioned that the ...
10
votes
2answers
4k views

Geodesics equations via variational principle

I would like to recover the (timelike) geodesics equations via the variational principle of the following action: $$ \mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}} $$ ...
5
votes
2answers
1k views

Finding geodesics: Lagrangian vs Hamiltonian

I have a question referring to how to compute geodesics of a given spacetime (say, Kerr). I know that the direct way is via the geodesic equation $$\frac{d^{2}x^{\mu}}{d\lambda^{2}}+\Gamma^{\mu}_{\...
2
votes
1answer
1k views

Why a timelike geodesic maximizes path length?

I'm studying some GR and my book says that in Pseudo-Riemannian manifolds geodesics may even maximize the path locally. That's what happen to the timelike geodesics, for example. My first question: Is ...
7
votes
1answer
1k views

How is the Lagrangian defined in GR?

Reading about the Schwarzschild metric in general relativity I see that sometimes $$L=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}$$ and sometimes $$L=\sqrt{g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}.$$ Which is ...
8
votes
1answer
756 views

Action of a massive free point-particle in relativistic mechanics

I was reading about the formulation of mechanics in special relativity and found that the action for a massive free point-particle as $$ S = -mc\int_a^b ds $$ So, I did a few observations, ie. $$ S =...
5
votes
1answer
828 views

Why two different Lagrangians to derive geodesic equations?

I'm trying (very early stages) to understand the derivation of the geodesic equation $$\frac{d^{2}x^{\alpha}}{d\lambda^{2}}+\Gamma_{\gamma\beta}^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{...
2
votes
2answers
640 views

From Euler-Lagrange equation to non-affine geodesic equation

I have some problems to demonstrate the non-affine geodesic equation from Euler-Lagrange's equations. I start defining the square root Lagrangian $$L=\sqrt{ g_{ij}(x) \dot{x}^i \dot{x}^j},$$ but then ...
2
votes
1answer
747 views

Geodesic equation from the proper time integral

This is something that has been bothering me for a little while. The usual procedure that I've seen is to write the proper time as the line integral $$\tau=\int_\gamma d\tau$$ along some curve $\gamma$...
1
vote
2answers
492 views

Do the same equations of motion imply the same Lagrangians? [duplicate]

If two Lagrangian (densities) $\mathcal{L}$ give the same equations of motion, are they equivalent?
1
vote
2answers
318 views

Why can we choose affine parameterization?

In general relativity when deriving the geodesic equation $$\ddot{x}^\mu + \Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta = 0\tag{1}$$ from the action $$S = \int d\tau \sqrt{|g_{\mu\nu} \dot{x}...
2
votes
1answer
249 views

Euler-Lagrange equation in General Relativity

In Relativity the Lagrangian of a free particle is \begin{align} \mathcal L=\sqrt{g_{ab}\frac{dx^a}{d\tau}\frac{dx^b}{d\tau}}\end{align} Since $\mathcal L$ is parameterization invariant we can always ...

15 30 50 per page