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I am quite new to GR and I have a question regarding the construction of the action to find the geodesic equation.

In pretty much every book, you'll find:

##S=-m ∫ dS##

using: ## dS=dS\frac{d\tau}{d\tau}=\sqrt{g_{\mu \nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}d\tau## with ##dS=\sqrt{g_{\mu \nu}dx^{\mu}dx^{\nu}}##

one gets: ##dS= -m∫\sqrt{g_{\mu \nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}d\tau##

Minimisation of this quantity gives rise to the geodesic equations.

I understand that contruction: given a certain metric, the particule moves in a "straight" line, minimizing proper time.

Here's my problem, when dealing with the Schwarzchild metric, my prof. uses:

##S= -m∫g_{\mu \nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}d\tau##

Which makes the computation much easier, I've also been able to re-find the geodesic equation starting from this expression.

My questions would be:

What is the diffence between ##d\tau## and ##dS## and what is the difference between these to actions ? I feel like I'm confusing proper time, interval ##dS## and the action differential...

It would be great if someone could explain me the meaning of this!

Thank you very much