Orbit velocity in Schwarzschild metric?

[MrTictac]

Hi, I'm trying to deduce orbit velocity of a particle with mass from Schwarzschild metric. I know for Newtonian gravity it is:
$$v^2=GM\left(\frac{2}{r}-\frac{1}{a}\right)$$
The so called vis-viva equation. Where $a$ is the length of the semi-major axis of the orbit. For Schwarzschild metric it should be similar, except I guess for a perihelion term. I understand that perihelion term complicates maths, however I'm not interested on that term, so it would be fine if it's ignored. Would you know about any bibliography/video where it is explained? Or would you like to help? Thanks!

 

[Ibix]

Sean Carroll's lecture notes cover geodesics in Schwarzschild spacetime in chapter 7. Equations 7.43, 7.44, 7.47 and 7.48.

https://www.preposterousuniverse.com/grnotes/

You'll have to decide what you mean by orbital velocity in this context.

 

[MrTictac]

Thanks for your answer! I couldn't find the solution in that bibliography. Eq. 7.43 and 7.44 are related to conserved quantities (total energy and angular momentum), while 7.47 and 7.48 are the differential equation to find the position of the particle. What I was looking for is the solution that can be derived from there. I'm not sure about the final comment, I was expecting something similar to the equation I posted, but for GR. Anyway, I've looked for it in many books, and nothing, so maybe my starting point is wrong.

 

[Ibix]

The fundamental problem you have is that velocity isn't generally well defined in curved spacetime. That will be why an expression for velocity is difficult to find. So first you need to decide what you want to call velocity. Is it the velocity as measured by a local hovering observer? Or the total distance around the orbit divided by the time taken? If the latter, time as measured by the orbiting observer, an observer hovering at the same altitude, one at infinity, or something else?

Once you've decided on that, you can proceed. Off the top of my head, I think you need to find the radius for circular orbits in terms of E and L (that's later on in chapter 7 - don't remember the equation number) and require $dr/d\lambda=0$ in 7.47. Together, that ought to allow you to calculate L in terms of the radius. Then 7.44 will give you $d\phi/d\lambda$ which, assuming you set $\epsilon=1$, is the angular velocity as measured by the orbiting observer. Note that Carroll uses units where c=1. If you use other units you will need to introduce factors of c where necessary so that the units work.

What you do next depends on how you chose to define velocity.

 

[MrTictac]

I think I understand now. What I need is the velocity related to proper time (I understand that's the point of view of the particle), and tangent to the orbit curve. The deduction, more than the velocity itself. I'll try to derive it from your clues, but I'm not good at that. Thanks again!

 

[Ibix]

I would say which thing you want to know depends on why you want to know it. Why did you want to know?

Happy to help with the maths if you get stuck. I note you marked this thread as "A" meaning you have postgraduate level of knowledge in this area. I'm guessing that's not the case. It's easier to pitch answers at a level you can follow if you mark threads you start appropriately.

 

[pervect]

I think I understand now. What I need is the velocity related to proper time (I understand that's the point of view of the particle), and tangent to the orbit curve. The deduction, more than the velocity itself. I'll try to derive it from your clues, but I'm not good at that. Thanks again!

Specifying proper time yields what's known as a "proper velocity". One can have proper velocities in special relativity as well as General relativity, proper velocities can increase without limit and are different from the usual notion of velocity which is always lower than c. Are you really sure you want to measure the proper velocity?

You also still need to specify a frame in which the distance is measured. The usual choice is a frame of reference of a static observer, one with constant Schwarzschild coordinates.

 

[MrTictac]

Thanks for your answers! I have postgraduate studies, but not in physics. I thought the A level was for the subject, not the person. My interest is just curiosity, I don't want to waste your time, I thought there was a class note somewhere.

I'm thinking in a static frame of reference, with an observer at distance R from the origin (curvilinear coordinates), I don't know if I'm missing something. Anyway, I believe I've figured out what to do, as eq. 7.47 and 7.48 from the bibliography provided are the same than Newtonian ones, except for the $\lambda$ instead of $t$, and the $GML^2/R^3$ term, so if I ignore that term, I can use the same deduction than classical mechanics' orbits, right?

 

[PeterDonis]

I thought the A level was for the subject, not the person.

The thread level is intended to tell potential responders at what level of knowledge their replies should be given. So if you mark a thread as "A" level, you should expect replies that assume that you have a postgraduate level of knowledge. Technically, you can still mark a thread "A" level because of the subject matter even if you don't have that level of knowledge; but if you do, you're likely to not be able to understand the replies.

 

[stevebd1]

While you may be looking at elliptical orbits, the following is the equation for stable orbit velocity in Schwarzschild metric-

$$v_s=\sqrt{\frac{M}{r\left(1-\frac{2M}{r}\right)}}$$

where $M=Gm/c^2$ (note the answer would be a fraction of the speed of light, multiply by c for S.I. units). The above reduces to $v_s=\sqrt(M/r)$, the Newtonian version, when at a great distance from the black hole. You can see that if you introduce $r=3M$, which is the photon sphere (the radius at which light is predicted to orbit the BH), then $v_s=1$ (for an object with mass, the innermost stable circular orbit is 6M for a Schwarzschild black hole).

The time dilation (and redshift) for an object in stable orbit would be the sum of the gravitational dilation and dilation due to velocity- $d\tau/dt=\sqrt{1-2M/r}\cdot\sqrt{1-v_s^2}$ or simply put $d\tau/dt=\sqrt{1-3M/r}$.

 

[Dale]

I'm thinking in a static frame of reference, with an observer at distance R from the origin (curvilinear coordinates), I don't know if I'm missing something.

At each point along the orbit you can hypothesize a static observer. In the Schwarzschild spacetime there is a timelike Killing vector which describes a family of such observers. You can measure your orbital velocity with respect to them. Their notion of time will generally be different from the orbiting objects notion of time, but they will agree on their relative velocity in their respective local inertial frames.

 

[MrTictac]

While you may be looking at elliptical orbits, the following is the equation for stable orbit velocity in Schwarzschild metric-

$$v_s=\sqrt{\frac{M}{r\left(1-\frac{2M}{r}\right)}}$$

That would be useful for me. May I ask how did you get it from Schwarzschild´'s? Thanks!

 

[stevebd1]

That would be useful for me. May I ask how did you get it from Schwarzschild´'s? Thanks!

I've not seen the equation for stable orbit explicitly expressed in Schwarzschild terms, only in Kerr terms (probably due to the fact that rotating black holes are more realistic than absolutely static ones). I simply took the Kerr version and removed the spin parameter $a$ which gives you the static (Schwarzshild) version. An example of the equation for stable orbit in Kerr metric in the equatorial plane is-

$$v_{s\pm}=\frac{r^2+a^2\mp 2a\sqrt{Mr}}{\sqrt{\Delta} \left[a\pm r\sqrt{r/M}\right]}$$

where $\Delta= r^{2}+a^{2}-2Mr$, which is referenced in this paper (equation 25) though there are a number of sources. The paper also provides an equation for gravity for an object with tangential velocity around/near a Kerr BH which reduces to the following for a static black hole-

$$a(U)=\gamma^2\left[\frac{M}{r^2\sqrt{1-2M/r}}-v^2\frac{\sqrt{1-2M/r}}{r}\right]$$

where $\gamma$ is the Lorentz transformation, which in turn reduces to the equation for stable tangential velocity for a static solution as in post #10

 

[MrTictac]

Thanks @stevebd1, I'll check that paper.