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I'm reading Zee's new GR book right now and ran across an action I am having trouble "varying". It's the first term in Eq (9), page 244. Looks like this:

[itex]S=-m\int d\tau \sqrt{-\eta _{\mu \nu }\frac{dx^{\mu }}{d\tau }\frac{dx^{\nu }}{d\tau }}[/itex]

I'm familiar with the trick that gets rid of the square root as outlined by guys like Carrol, as well as Zee. But, I want to tackle this thing head on the hard way w/o the trick ;-)

But, I'm getting stuck reproducing what Zee has in Eq (10)

[itex]\delta \left ( -m\int d\tau \sqrt{-\eta _{\mu \nu }\frac{dx^{\mu }}{d\tau }\frac{dx^{\nu }}{d\tau }} \: \right )=m\int d\tau \; \eta _{\mu \nu }\frac{dx^{\mu }}{d\tau }\frac{d\delta x^{\nu }}{d\tau }[/itex]

I get the same thing, but with a 1/L included, because the root won't go away.

Anybody know the trick? Or does Zee have a typo? Can't be me, right? ;-)

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# Varying an Action

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