Talk:Schwarzschild geodesics

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Proposal: more concrete examples

This is a humble proposal from a non-academic to facilitate numerical simulations. I know this is almost the opposite of Anon who proposed more high-level concepts. Here are specific proposals.

Explicitly write out the differential equations for at least one system of coordinates, such as (tau * c, radius, angle) or (t * c, radius, angle), or even just (radius, angle) if that is illustrative enough. Explicitly write out a few sets of system parameters (length scales a, b, and r_s) which demonstrate interesting classes of geodesics. For example, one orbit is spiraling inwards, one orbit is flower-shaped with precession on the order of 30 degrees, as shown in one existing animation, one simple scattering, one scattering with one orbit, and one scattering with two orbits (https://i.stack.imgur.com/Zfksm.jpg currently). These should be essentialy ready to plug into a numerical DiffEq solver such as SciPy's solve_ivp.

Note that the existing equation in the article (dr/dphi)^2 = F(r, a, b, r_s) (where a, b, r_s are constants) is not suitable for a standard solver because the solver has to take the correct sign of the square root, and behavior at the states where (dr/dphi) = 0 is not well behaved. My understanding is that in order to implement this, one needs to take another derivative with respect to phi:

2 (dr/dphi) (ddr/ddphi) = [d/dr F(r, a, b, r_s)] (dr/dphi) = (dr/dphi) (4r^3(1/b^2-1/a^2) + 3r^2(r_s)/a^2 - 2r + r_s)

dr/dphi should be canceled out, (even though dr/dphi = 0 occurs in the desired solution, I don't have the exact theoretical reason why). This yields a second order ODE which can be implemented as a system of first order ODEs in the two variables (r, dr/dphi) with indpt variable phi [note: not 100% sure of terminology]. If this is correct, this algebraic manipulation should be emphasized.

Here is an example of an explicit set of parameters and initial conditions: (r_s = 1, a = 2.5, b = 2.51684542, r = 10.75693909, dr_dphi = sqrt(F(r, a, b, r_s)) (calculate to floating point precision, 8.10043608...). I am trying to come up with more interesting parameters, but this at least seems to give a highly elliptical orbit with precession angle around 75 degrees.

38.73.247.37 (talk) 16:29, 19 May 2022 (UTC)

What you are asking for is the "geodesic equation"; it appears half-way down this article (and also in other articles). You can plug this into any differential equation solver that you wish, with no modifications whatsoever, and let them crank away solutions. To be clear, the Christoffel symbols $\Gamma _{\rho }^{\mu \nu }$ are unique functions of $r,\theta ,\phi ,t$. That is, $\Gamma _{\rho }^{\mu \nu }=\Gamma _{\rho }^{\mu \nu }(r,\theta ,\phi ,t)$ and are explicitly given in this article. The position of the particle is $x^{\mu }=x^{\mu }(q)$ and you can pick any position that you want. The parameter $q$ is a single real number, it is what your diffeq solver will increment for each "time step", and so you will get a new $x^{\mu }=x^{\mu }(q+dq)$ for "time step" $dq$. The only confusion might be due to the index $\mu$. It runs from 0 to 3, so that $x^{\mu }$ is an ordinary vector (python, scipy vector). The confusion arises because the four vector components 0,1,2,3 are given "special names". By abuse of notation, 0 is called t, 1 is called r, 2 is called theta and 3 is called phi. But these are not the same t,r,theta,phi as the coordinate. Instead, these are "infinitessimal vectors" pointing in the direction t,r,theta,phi. Thus, you will sometimes see the notation $\partial _{t},\partial _{r},\partial _{\theta },\partial _{\phi }$ to denote these infinitessimal (tangent) vectors. So that's it. Nothing more needs to be done. Just cut-n-paste, plug it in, without any modifications at all, no translations, no reinterpretation, no conversions. Just code and go. Does this answer your question?

If it does, then perhaps the above paragraph could be the starting point for a new article, called "Numerical methods for geodesics in general relativity", which could maybe spell things out in greater detail. But it could be argued that such articles are not suitable for wikipedia, and should instead be published in some scipy handbook or module. Its really not at all hard, if you have a good theoretical understanding of what's going on. That's why building a good theoretical understanding is important; all the plug-it-in-and-go details become easy to do, and don't need explanation or demonstration. 67.198.37.16 (talk) 17:37, 1 June 2023 (UTC)

No circular orbits without radiation input

At the end of the history section, it says "there is a minimum radius for the circular orbit to be stable in Schwarzschild metric". This is false. If we assume that there is no incoming radiation (gravitational or electromagnetic), the existence of a circular orbit would cause outgoing gravitational radiation. That would result in a loss of energy and angular momentum and thus a shrinkage of the orbit. Eventually the orbiting object would fall into the central mass.

Probably, the author intended to say that there is a value for the radius such that an object at or beyond that radius but with no radial motion would not fall inward immediately if it had sufficiently large tangential velocity. OK? JRSpriggs (talk) 23:19, 3 August 2023 (UTC)

The stability of orbits does not take into account gravitational radiation. Otherwise there would be no stable orbits at all, as you always emit gravitational radiation and eventually collide. Tercer (talk) 05:51, 4 August 2023 (UTC)

Sounds to me, like you are basically agreeing with me. What criteria would you use to define a "stable" orbit in your sense? JRSpriggs (talk) 16:47, 5 August 2023 (UTC)

I'm agreeing with you about the facts, not about how to describe them. As for the definition of stable, I'd start with stable under Newtonian gravity. For this article specifically, I'd look in a GR textbook where they discuss the minimum radius of a stable orbit in a Schwarzschild metric. Tercer (talk) 18:59, 5 August 2023 (UTC)