Orbit description in Schwarzschild metric

Question

Suppose to have a restricted 2-body system (BH + star with $M_{BH}\gg M_{\mathrm{star}}$) and you want to describe the orbit of the star relative to the BH, i.e. in the Schwarzschild metric.

Usually, the orbit, which is an ellipse, is parametrically described by:

\begin{equation}\tag{1} r_{\mathrm{star}}= \dfrac{a(1-e^2)}{1+e\cos{\phi}} \end{equation}

where $a$ is the semi-major axis, $e$ the eccentricity and $\phi$ the true anomaly. Once chosen $a$ and $e$, I obtain a certain orbit. My question is how this orbit changes in different coordinate systems.

A) If I identify $r_{\mathrm{star}}$ above with the radial coordinate $r$ of the Schwarzschild metric, $r=r_{\mathrm{star}}$, then I am implicitly implying that the actual (physical) distance between the star and the BH is not given by $r_{\mathrm{star}}$, since we know that radial coordinate in Schwarzschild metric does not correspond to a radial distance. For example, If I have chosen $a=5$ and $e=0$ in $r_{\mathrm{star}}$, this means that the real orbit of the star has a radius slightly different than 5. Right?

B) Suppose now the contrary: I know that a star moves on a circular orbit around a BH at a distance $R$. How do I translate such an orbit in the Schwarzschild spacetime?

C) The distance between star and BH should be the same in every coordinate system (e.g. Schwarz. metric with B.L. coordinates vs. Post-Newtonian metric with harmonic coordinates)?

Answer 1

1. Orbits in Schwarzschild spacetime are not closed ellipses in general. Therefore they cannot be described by ellipses. Since you are treating the star as a massive test particle, you have to solve the geodesic equation in order to obtain the orbit. You will find that orbits of specific angular momentum $L$ obey an effective radial potential $$-\frac{GM}{r} + \frac{L}{2r^2} - \frac{GML^2}{r^3}$$ following which circular orbits satisfy $$GMr^2 - L^2 r + 3GML^2 = 0.$$

2. The Newtonian equation for circular orbits $$v=\sqrt{\frac{GM}{r}}$$ continues to hold in Schwarzschild spacetime provided that $v$ is the spatial velocity as observed at infinity, cf this post. Note that the innermost stable circular orbit for massive particles is at three times the Schwarzschild radius.

3. The whole idea of relativity is that spatial distance depends on the observer, i.e. on the choice of coordinates. The invariant distance between two events is the spacetime distance rather than the spatial distance.

Answer 2

The real radius is the Schwarzschild radius, because that is what is measured in reality. The other radius is the Newtonian/Euclidean expectation which is imaginary. The Newtonian radius can be defined in a number of different ways:

1. The measured circumference of a circle around the gravitational body, divided by $2 \pi$

2. By $R = \sqrt{\frac{GMm}{F}}$ using the Newtonian equation for gravitational force.

3. By $R = \frac{GM}{v^2}$ where v is the velocity of a body in circular orbit at the same height of the observer.

4. By $a = \sqrt{\frac{GMT^2}{4\pi^2}}$ where a is the semimajor axis of an elliptical orbit, using the Keplerian relationship between the orbital period (T) and the radius.

5. By literally measuring the ruler distance from the observer's location to the centre of the gravitational body.

and there are other ways too. The Newtonian radius is not explicitly defined by any of the above methods because they are all equivalent. In reality, all but the first method would disagree with what would be measured in real experiments.

Below is what we see or measure locally in a Schwarzschild gravitational field compared to the Newtonian expectation, for the respective methods above:

1b) The Schwarschild and Newtonian expectations agree when this methd is used.

2b) The proper force measured by a stationary observer at any given radius in the Schwarzschild metric is given by to $F = \frac{-GMm}{R^2\sqrt{1-2M/R}}$ and so the radius calculated by this method would be different from the Newtonian expectation.

3b) The Schwarzschild radius obtained from measuring the local velocity of a satellite in a circular orbit at the height of the observer is $R = GM(2+1/v^2)$. Compare to eq (3). Note that when v=1, the radius is equal to to 3GM, the "photon orbit".

4b) For the Schwarzschild radius calculated from the orbital period of an elliptical orbit from perihelion to successive perihelion, things get more complicated because the orbit is precessing and satellite passes through regions where the gravitational gamma factor is constantly changing, so some calculus is probably involved. I refer you to the references attached below for further research.

5b) have a look at this geometrical shape called Flamm's paraboloid.

The Newtonian/Euclidean radius is represent by A, B, C etc and occur at regular interval's while the ruler distance measured in a Schwarzschild gravitational field, varies between successive radius measurements and is proportional to the distance along the red curve between intervals. It can be seen that the distance along the curve between A and H is significantly greater than between A and B, but as we go further out the corresponding distance intervals start to agree more. This is a consequence of geometry of space around a black hole being non-Euclidean.

For further research on precessing elliptical orbits see this excellent page by Fourmilab:

"Orbits in Strongly Curved Spacetime" which gives a very good description of the maths and physics involved and an app that plots the orbits and effective gravitational potential in real time.

A more in depth analysis of the maths and history of calculating the advance the perihelion of Mercury is given by Mathpages here.

A bit more about orbits in the Schwarzschild metric also from Mathpages here.

P.S. I just realised you can also measure the Newtonian radial distance using
$R = \frac{T}{2\pi}\sqrt{\frac{GM}{L}}$
where T is the period of a simple pendulum with arm length L.

Answer 3

It's quite easy to get the necessary Schwarzschild orbit for Mercury if you take gravitational and kinematic time dilation/length contraction into account. I don't do it parametrically. In fact, I use Euclidean 3-D space with Cartesian coordinates.

The Newtonian acceleration vector is: \begin{equation} \vec{g}_n = \frac{\hat{d} G M}{{\lvert\lvert \vec{d} \rvert\rvert}^2}. \end{equation}

One important value is closely related to the kinematic time dilation: \begin{equation} \label{eq_kinematic} \alpha = 2 - \sqrt{1 - \frac{\lvert\lvert \vec{v}_{o}\rvert\rvert^2}{c^2}}. \end{equation} Another important value is the gravitational time dilation: \begin{equation} \beta = \sqrt{1 - \frac{R_{s}}{\lvert \lvert \vec{d} \rvert \rvert}}. \end{equation}

Finally, the semi-implicit Euler integration is: \begin{align} \vec{v}_{o}(t + \delta_t) &= \vec{v}_{o}(t) + \delta_{t} \alpha \vec{g}_n, \\ \ell_{o}(t + \delta_t) &= \ell_{o}(t) + \delta_{t} \beta \vec{v}_{o}(t + \delta_t). \end{align}

I won't pass on my code quite yet, but here's a beginner's C++ source code that calculates strictly Newtonian orbit. Adding in the required code isn't all that tough, but I just don't know if my code has bugs, so I need independent validation.

Please let me know if there are still questions.