How to obtain the geodesic equation from the action?

Question

Hi! I would be grateful if someone tells me how to obtain the geodesic equation from, for example, the cadabra notebook https://cadabra.science/notebooks/einstein_equations.html

I mean: is there a command such as "geodesic(_);" to readily obtain it? Thanks a lot! Mario

Comments

I don't understand your question. Do you want to obtain the equation from a variational principle or obtaining the differential equations from a set of connection components?

---

Hi doxdrum, Thank you for paying attention to this possibly dumb question. I am new to cadabra, slowly figuring out how to use it. My question is if there is a simple instruction of the kind "geodesic(action,...)" to immediately give the well-known geodesic equation, or the equation of motion usually presented in the form

x"^a+\Gamma^a_{bc} x'^b x'^c=0

where \Gamma is the metric connection of g_{ab} could be directly obtained from the action. Thank you very much!

Answer 1

So far there is no build-in command that spits the geodesic equations.

However, I'd like to clarify: the geodesic equation can be obtain by extremising (one half of) the Riemannian line element for a given metric, and it is given by the expression stated by @skyfold in the comments of the question.

It is, nonetheless, straightforward to obtain the differential equations for the geodesic---once one knows the metric---. The only nontrivial ingredient is to define the (Levi-Civita) connection in terms of the metric, and calculate the coefficients. This has been shown in the notebook about Schwarzschild spacetime.

Comments

Once again, thanks a lot!

---

No problem... We are a community!