In Carroll's derivation of the geodesic equations (page 69, http://preposterousuniverse.com/grnotes/grnotes-three.pdf), he starts with τ=∫(−gμνdxμdλdxνdλ)1/2dλ
δτ=∫(−12∂σgμνdxμdτdxνdτδxσ−gμνdxμdτd(δxν)dτ)dτ.
I cannot see how that substitution works. I've been told it uses the chain rule, but I just can't see it. Can anyone help? Thanks.
Answer
Basically think of it this way. Take the original equation τ=∫f(x)dλ
dτ=f(x)dλ
after a little rearranging gives
dλdτ = (f(x))−1--------(3)
with the function f(x) in this case being equal to
f(x) = (−gμνdxμdλdxνdλ)1/2--------(4)
as was demonstrated
EDIT:
Using eq (3)
dλdτ = (f(x))−1 = (−gμνdxμdλdxνdλ)−1/2
Substitute into
δτ=∫ (−gμνdxμdλdxνdλ)−1/2 (−12gμν,σdxμdλdxνdλδxσ−gμνdxμdλd(δxν)dλ) dλ
gives
δτ=∫ dλdτ (−12gμν,σdxμdλdxνdλδxσ−gμνdxμdλd(δxν)dλ) dλ
δτ=∫ dλdτ (−12gμν,σdxμdτdxνdτδxσ−gμνdxμdτd(δxν)dτ) dτdλdτdλ dλ
Use chain rule to get
δτ=∫ (−12gμν,σdxμdτdxνdτδxσ−gμνdxμdτd(δxν)dτ) dτ
Here I use , to represent the partial derivative with respect to xσ.
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