Can't remember where I read this (when using the proper-time parametrization)

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ShayanJ
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A while ago, I read a proof in a book on GR that when using the proper-time parametrization, the two conditions ## \delta \int_{\lambda_1}^{\lambda_2} \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu} d\lambda=0## and ## \delta \int_{\lambda_1}^{\lambda_2} (-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu) d\lambda=0## are equilvalent. But then I forgot which book was it. I was able to reconstruct the proof but now I'm really curious what book was that! Does anyone know? I've checked several books so I know that not many books on GR contain such a proof.
Thanks
 
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Page 45 https://www.maths.tcd.ie/~fionnf/dg/dg.pdf mentions this, and you can prove it yourself thinking about it, but it feels weird and dirty...
 
bolbteppa said:
Page 45 https://www.maths.tcd.ie/~fionnf/dg/dg.pdf mentions this, and you can prove it yourself thinking about it, but it feels weird and dirty...
As I said in the OP, I have no problem with the proof. I just don't remember which GR book I read the proof in! So if anyone knows a GR book that contains the proof, I'll appreciate it if they let me know.
 
Shyan said:
A while ago, I read a proof in a book on GR that when using the proper-time parametrization, the two conditions ## \delta \int_{\lambda_1}^{\lambda_2} \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu} d\lambda=0## and ## \delta \int_{\lambda_1}^{\lambda_2} (-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu) d\lambda=0## are equilvalent. But then I forgot which book was it. I was able to reconstruct the proof but now I'm really curious what book was that! Does anyone know? I've checked several books so I know that not many books on GR contain such a proof.
Thanks
I found it! :woot:

R. Adler, M. Bazin, M. Schiffer, Introduction to General Relativity (1975)
Sec. 4.2, Eqs. (4.101)-(4.102)
 
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Demystifier said:
I found it! :woot:

R. Adler, M. Bazin, M. Schiffer, Introduction to General Relativity (1975)
Sec. 4.2, Eqs. (4.101)-(4.102)
Thanks, seems a nice book!
But I didn't know it at the time I read that proof. Also the book I read, precisely proves the statement by introducing ## F=\sqrt{-g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu} ## and then showing that if ## \int_{\tau_1}^{\tau_2} F d\tau ## is extermized, ## \int_{\tau_1}^{\tau_2} \frac{F^2}{2} d\tau ## is extermized too.(About the notation, I'm only sure about the F!)

Demystifier said:
Misner Thorne Wheeler - Gravitation also has it:
Page 322-323, Box 13.3, Eqs. (1)-(2).
Well, that's a pretty big book and hard to search. But I remembered it wasn't the book I read anyway.

I have two reasons for posting this thread, first is that I like books that do such calculations and I want to read it more carefully and second is that I hate it when from time to time it comes to my mind "God, what was that book?"!
 
Shyan said:
Thanks, seems a nice book!
But I didn't know it at the time I read that proof. Also the book I read, precisely proves the statement by introducing ## F=\sqrt{-g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu} ## and then showing that if ## \int_{\tau_1}^{\tau_2} F d\tau ## is extermized, ## \int_{\tau_1}^{\tau_2} \frac{F^2}{2} d\tau ## is extermized too.(About the notation, I'm only sure about the F!)Well, that's a pretty big book and hard to search. But I remembered it wasn't the book I read anyway.

I have two reasons for posting this thread, first is that I like books that do such calculations and I want to read it more carefully and second is that I hate it when from time to time it comes to my mind "God, what was that book?"!

It was such an emphatic "No!" that I decided not to post further. My third choice was Adler Bazin and Schiffer as these are the three books in my library but then figured the Adler book was too old for Shyan to consider. Also, I think Zee may have it as well but since its too big to search...

In any event, I'm glad the mystery is solved. Thanks Demystifier!
 
jedishrfu said:
It was such an emphatic "No!" that I decided not to post further. My third choice was Adler Bazin and Schiffer as these are the three books in my library but then figured the Adler book was too old for Shyan to consider. Also, I think Zee may have it as well but since its too big to search...

In any event, I'm glad the mystery is solved. Thanks Demystifier!
Looks like I should have listened to my instinct that was telling me to accompany that "no" with some excess stuff! Sorry if it didn't seem friendly.
Thanks for the help to all.