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 thoose equation: this is the first step (which is generally really hard). Once it is done we have two different ways to proceed.

First method

We want to find the trajectories of particles. We know that thoose trajectories will be geodesics. It mean that they will minimise the length:

$$ L=\int ds = \int \sqrt{g_{\mu, \nu} \frac{dx^{\mu}}{dt} \frac{dx^{\nu}}{dt}} dt.$$

Little question: how do we know that the particles will follow geodesics? Is it a postulate of GR?

To find the minimum of this quantity is equivalent as finding the minimum of:

$$ \int g_{\mu \nu} \frac{dx^{\mu}}{dt} \frac{dx^{\nu}}{dt} dt. $$

Thus, we can say that the Lagrangian describing our moving particle is in fact:

$$ \mathcal{L}=g_{\mu \nu} \frac{dx^{\mu}}{dt} \frac{dx^{\nu}}{dt}.$$

Of course this Lagrangian has "no link" with the lagrangian of GR. The last one describes the equation of the fields and lead to the einstein equation, whereas this one is the lagrangian of a moving particle in the theory of general relativity.

Second method :

We also could use the Levi-Civita connexion. This connexion ensure the conservation of scalar product for 2 parallel vectorial field along a given curve.

Using this connexion, a geodesic is a curve which tangent vector is parallel along the curve.

Then we have a differential equation for the geodesics (which will be the same as the one derived from the lagrangian written above).

Am I right with everything I wrote?

In fact I have a general relativity class this year which is supposed to be a part II and I haven't followed the part I so I need to catch up fast. This is why my questions are probably basics ones.


I didn't see anything obviously wrong with your description, so just some further remarks:

In a way, the first method is a generalization of Fermat's principle (the particle follows the path that makes its eigentime stationary under variation), whereas the second method is a generalization of Newton's first law (the particle moves with constant velocity along a straight line, but through a curved spacetime).

Because the Levi-Civita connection is torsion-free, the methods agree, and you can make either one a postulate.

Also note that under certain assumptions, the equations of motion for test particles actually follow from the field equations as an approximation (cf Einstein, Albert; Grommer, J., Allgemeine Relativitätstheorie und Bewegungsgesetz).


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