I Circular Orbit in Schwarzschild: Orbital Period

[epovo]

TL;DR Summary

I followed Schutz derivation and I don't get his result

Schutz finds that the orbital period for a circular orbit in Schwarzschild is

$$ P = 2 \pi \sqrt {\frac { r^3} {M} }$$

He gets this from
$$ \frac {dt} {d\phi} = \frac {dt / d\tau} {d\phi/d\tau} $$
Where previously he had $\frac {d\phi}{d\tau} = \tilde L / r^2$ and $\frac {dt}{d\tau} = \frac {\tilde E} { 1 - 2M/r}$ and where

$\tilde L^2= \frac {Mr } { 1-3M/r}$ and $\tilde E = \frac {(1- 2M/r)^2} {1-3M/r}$

After doing the algebra I don't get that expression for the period (I get a much more complicated expression).
I punched in some numbers for M and r in a spreadsheet and the period given by the expression above does not match the calculations I have done. It does not even seem to be a very good approximation. Help, please!

 

[Ibix]

Where's this in Schutz?

 

[epovo]

It's in ch 11 section 1 (page 280 in my edition) under Perihelion Shift

 

[Ibix]

Your expression for $\tilde{E}$ is wrong - it's the correct expression for $\tilde{E}^2$. Schutz has it correct in equation 11.21 on p287 in my edition, and I think his result for $P$ follows.

You may have made a transcription error, or there may be a typo in your edition. Either is possible - I've commented before that I think Schutz needed a better editor.

 

[epovo]

Definitely a typo. Thank you!

 

[Ibix]

Definitely a typo.

I have the second edition, so I hope you have the first edition... This particular text does seem to have more than usual stuff like this, so I would say that when you can't make sense of Schutz, "my textbook is wrong" (or at least confusingly written) should be a bit higher up your probability list than normal.