Hello Everybody,(adsbygoogle = window.adsbygoogle || []).push({});

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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**