# Circular Orbit in Schwarzschild: Orbital Period

• I
• epovo
In summary, Schutz finds that the orbital period for a circular orbit in Schwarzschild can be calculated using the expression P = 2π√(r^3/M). This is derived from the equations dt/dφ = (dt/dτ)/(dφ/dτ), with previous equations for dφ/dτ = L/r^2 and dt/dτ = E/(1-2M/r), where L^2 = Mr/(1-3M/r) and E = (1-2M/r)^2/(1-3M/r). However, there may be a typo in the expression for E and therefore the resulting expression for P may be incorrect. This can be found in chapter 11, section 1
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!

Where's this in Schutz?

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.

berkeman, epovo and vanhees71

Definitely a typo. Thank you!

Ibix and berkeman
epovo said:
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.

vanhees71 and PeterDonis

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