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Carroll introduces in page 106 of his book "Spacetime and Geometry" the variational method to derive the geodesic equation.

I have a couple of questions regarding his derivation.

First, he writes:" it makes things easier to specify the parameter to be the proper time τ instead of the general parameter λ". Why does he do that? How does it make things easier? And why CAN he do that? is this just a variable substitution? I don't get it.

Second, he makes a taylor expansion of the metric in page 107. He writes down:

[tex] g_{\mu \nu} \rightarrow g_{\mu \nu} +(\partial_\sigma g_{\mu \nu}) \delta x^\sigma. [/tex]

Now, how can I make this taylor expansion? It seems to be a taylor expansion of a matrix but I never did that before. And what does the arrow mean? "Substitute by"?

Third and last, why is putting the term [tex] g_{\mu \nu} \frac{dx^\mu}{\tau} \frac{dx^\nu}{d \tau} [/tex] in the Euler lagrange equations equivalent to making the substitution mentioned earlier?

Thanks for reading, and any help regarding one of these questions would be really appreciated!

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# Derivation of the Geodesic equation using the variational approach in Carroll

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