Formula: velocity of circular orbit wrt Schwartzschild metric

[Buzz Bloom]

TL;DR

I need help to figure out how the formula for the velocity of a circular orbit can be derived from the Schwartzschild metric .

Below are equations/formulas/text from

https://en.wikipedia.org/wiki/Schwarzschild_geodesics​

https://hepweb.ucsd.edu/ph110b/110b_notes/node75.html​

​

I apologize for not remembering the source for the "v=" equation, or for my inability to find it again.

For a circular orbit, the distance r and proper distance s are both constant. Therefore

ds = dr = 0.​

Assume the coordinates are such that the orbit has:

sin ϑ = 1, and dϑ = 0.​

Therefore the metric an be written
${r^2 {dθ}^2} = {c^2} /sqrt {1-r_s/r}} {dt}^2$
$r = 1$

 

[Buzz Bloom]

Below are equations/formulas/text from

https://en.wikipedia.org/wiki/Schwarzschild_geodesics​

https://hepweb.ucsd.edu/ph110b/110b_notes/node75.html​

(I have no idea why the attachments do not get displayed.)

View attachment 264281
View attachment 264282

I apologize for not remembering the source for the "v=" equation, or for my inability to find it again.

For a circular orbit, the distance r and proper distance s are both constant. Therefore

ds = dr = 0.​

Assume the coordinates are such that the orbit has:

sin ϑ = 1, and dϑ = 0.​

The metric can now be written:

$r {\frac {dθ} {dt}} = c {\sqrt {{1-{r_s}/r}} }$ .​

 

[Ibix]

See equation 7.52 in Carroll's GR lecture notes, which relates the radius $r_c$ of a circular orbit to the specific angular momentum $L$, and $L=r_c^2(d\phi/d\tau)$.

Once you have $d\phi/d\tau$ you can use the fact that the four velocity is normalised to 1 to write $1=g_{00}\left(\frac{dt}{d\tau}\right)^2+g_{33}\left(\frac{d\phi}{d\tau}\right)^2$ with the metric components evaluated at $r=r_c, \theta=\pi/2$. Now you have the only two non-zero components of the four velocity of your orbiting object.

Now you write down the four velocity of a hovering observer. This is just $(1/\sqrt{g_{00}},0,0,0)$. The inner product of this with the four velocity of your orbiting object is the Lorentz gamma factor associated with that orbit. That is, $\gamma=\sqrt{g_{00}}\frac{dt}{d\tau}$. You can calculate the velocity from here, I think.

 

[Buzz Bloom]

See equation 7.52 in Carroll's GR lecture notes, which relates the radius of a circular orbit to the specific angular momentum ...
You can calculate the velocity from here, I think.

Hi Ibix:

Thank you very much for this post. Although I know what tensors are, and I have a very tiny amount of understanding about the mathematical manipulation of them, I think I might be able to follow your suggestions and make the calculation I want.

I am a bit confused by the equation for L. L is called "momentum", but the units appear to be length.

However, I am primarily interested in the result from the viewpoint of an observer at infinity.

Regards,
Buzz

 

[Ibix]

I am a bit confused by the equation for L. L is called "momentum", but the units appear to be length.

$L$ isn't momentum, it's specific angular momentum, the angular momentum per unit mass. The units are correct for that. It'll drop out of the maths anyway.

However, I am primarily interested in the result from the viewpoint of an observer at infinity.

The process I outlined will get you the velocity measured by a hovering observer at the same altitude. An observer at infinity would measure it to travel slower by a factor of $1/\sqrt{1-r_s/r_c}$ due to gravitational time dilation.

You commented in your other thread that you were having trouble with LaTeX, and the error above is an unbalanced bracket. My recommendation is to write the brackets for anything first, then come back and fill in the contents. So to write $\frac{1}{3+\sqrt{2}}$ I'd write \frac{}{}, and I know the brackets for the frac command are correct. Then I go back and fill in the numerator, \frac{1}{}. Then I start to fill in the denominator, \frac{1}{3+}, until I notice I need a command here too. So I do it and its brackets, \frac{1}{3+\sqrt{}}, and finally fill in its brackets, \frac{1}{3+\sqrt{2}}.

Another tip is not to try to do the whole thing at once. So if writing the Schwarzschild line element, write ds^2=\left(\right)dt^2 (note the balanced brackets!) and preview it. If it works, fill in the 1-\frac{r_S}{r} and preview. If it works, move on to the $dr^2$ term. If any step doesn't work you know that whatever you did wrong is in the little bit you just added, so you know where to look.

 

[Buzz Bloom]

You commented in your other thread that you were having trouble with LaTeX, and the error above is an unbalanced bracket.

Hi Ibix:

Thank you for your advice.

While I was working with Latex to express the simplification of the metric, some very strange things happened while I was looking at a Preview. I do not know exactly what I did on seveal occassions, but I think I must have accidently hit a key that invoked some mysterious function. The result was that the post I was working on vanished from my screen, and I was instead seeing some other aspect of PF, like for example (1) post#1 of the thread, or (2) a list of forums. From this mess, I later found out that I had created two different copies of the same thread. I am now more than ever fearful of messing something up when I use Preview.

I have decided to put a message at the top of any post with Latex saying the post is a work in progress and not ready for review. When I decide the post is ready for review, I will remove the message at the top. What do you think about this process?

BTW, one of the odd results is a new thread in the Astronomy and Astrophysics forum at the top of the list of threads with my icon identifying it as my thread, but having no Prefix, no Title, no Summary, and no Text.

Regards,
Buzz

 

[Ibix]

There do seem to have been some issues with LaTeX over the last few days. However, changing to different pages suggests to me that your browser had taken focus away from the edit box. For example, at least some browsers use the backspace key as a back button. Try deliberately clicking somewhere else (e.g. one click somewhere in the middle of this post) then hitting backspace and see if you go back to your previous page. I've had similar happen to me. The solution is to always click on the edit box before you start typing again.

BTW, one of the odd results is a new thread in the Astronomy and Astrophysics forum at the top of the list of threads with my icon identifying it as my thread, but having no Prefix, no Title, no Summary, and no Text.

You mean above the pinned links? That's been there for a while now (although I admit I looked straight past it for some time), and should be present in all forums. It's not a thread. It's another way of creating a new thread in case clicking the "post thread" button is too much effort for you. Start typing a title and it'll pop up an edit box where you can put in the body of the first post of a new thread.

 

[Buzz Bloom]

Try deliberately clicking somewhere else (e.g. one click somewhere in the middle of this post) then hitting backspace and see if you go back to your previous page.

It's another way of creating a new thread in case clicking the "post thread" button is too much effort for you. Start typing a title and it'll pop up an edit box where you can put in the body of the first post of a new thread.

Hi "Ibix:

I highlighted a character in your post and hit the space bar. Your post disappeared, and the following list that usually appears at the bottom of a thread appeared.

Related Threads on Formula: velocity of circular orbit wrt Schwartzschild metric​

Thanks for the explanation about starting a new thread. I have not noticed that before when I started a new thread. I guess that it is either a new PF feature, or I am just careless about seeing what is before my eyes.

Regards,
Buzz