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The Schwarzschild orbit is $U'' + U = 1 + \varepsilon \cdot U^2$. I've only seen turning-point analysis that demonstrates shift of perihelion. Is there an exact solution?

Mauricio
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LiveProton
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  • Would [math.se] be a better home for this question? While you've made the physics context clear, the question is clearly about analytic solutions to a particular differential equation. – rob Nov 01 '23 at 23:17
  • If there is a closed, exact solution, that is a complex number and no real solution is possible, this indicates GR is NOT!! The correct representation of gravitational interactions.. OK,, I'm fishing bc there is such a solution... perhaps a better way to ask, but it is a critical issue for GR. – LiveProton Nov 01 '23 at 23:27
  • I'm about 70% sure that I found an exact solution on this site recently, albeit one involving some obscure special function (so it wasn't terribly illuminating). I'll see if I can dig it up but no promises. – Michael Seifert Nov 01 '23 at 23:42
  • Michael: thanks... it's an old and important issue for GR... in which I do not believe. My solution is in complex exponentials or sines and cosines. Pretty straightforward... – LiveProton Nov 01 '23 at 23:48
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    You need elliptic integrals / functions. I realise that you're asking about "closed" orbits, but for "hyperbolic" trajectories, rob has a great analysis for a massive particle here: https://physics.stackexchange.com/a/774043/123208 I have some info for photon trajectories here: https://physics.stackexchange.com/a/680961/123208 & https://physics.stackexchange.com/a/766180/123208 – PM 2Ring Nov 01 '23 at 23:55
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    Since you apparently have (or think you know) an answer to this question, you should post an answer. – ProfRob Nov 02 '23 at 08:01
  • What kind of exakt solution are you looking for? You want to find the acceleration as a function of position and velocity? If so this basically the same question as this: https://physics.stackexchange.com/questions/784415/ – Agerhell Nov 02 '23 at 12:37
  • Did you even check Wikipedia ? Before saying you are sure you solved the equation, test it against the others' solution. – Jeanbaptiste Roux Nov 03 '23 at 12:22
  • I solved the DE... what does Wikipedia have to do with it But.. – LiveProton Nov 04 '23 at 16:59
  • Yes my advisors and I have peer reviewed this solution... they do not know of an exact solution.. – LiveProton Nov 04 '23 at 17:00
  • Except the Phi(r) integral, which one comment addressed the use of elliptical integrals. Trust me, there is an exact solution – LiveProton Nov 04 '23 at 17:02

2 Answers2

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Closed-form analytic solutions for bound timelike geodesics around a Schwarzschild black hole were first (to the best of my knowledge) given by Charles G. Darwin (grandson of) in 1959 in this paper (unfortunately paywalled).

In terms of $u =1/r$, and making use of the fact that energy $E$ and angular momentum $L$ are constants of motion, the Schwarzchild geodesic equation can be written in first order form:

\begin{align} \left( \frac{du}{d\phi}\right)^2 &= 2M u^3 -u^2 + \frac{2M}{L}u + \frac{E^2-1}{L^2}\\ &=2M (u-u_1)(u-u_2)(u-u_3), \end{align}

where $M$ is the mass of the Schwarzschild solution (also units such that $G=c=1$).

For bound orbits, $0<u_1\leq u_2\leq u_3$, and the solution is given by

$$ u= u_1 + (u_2-u_3) sn^2(\xi(\phi),k),$$

where $sn$ is the Jacobi elliptic sine function, elliptic parameter $k= (u_2-u_1)/(u_3-u_1)$, and

$$\xi(\phi) = \sqrt{\frac{M(u_3-u_1)}{2}}\phi + \xi_0.$$

TimRias
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  • You can also examine the Schwarzschild solution in Mino time as presented here in equations (32)-(53), if you set spin to zero: "Analytic solutions for the motion of spinning particles near spherically symmetric black holes and exotic compact objects", https://arxiv.org/abs/2308.00021 – Void Nov 03 '23 at 13:37
  • @Void For Schwarzschild geodesics $\phi = L \lambda+ \lambda_0$. So Mino time and $\phi$ are essentially the same thing. – TimRias Nov 03 '23 at 18:08
  • Sure, just an alternative unpaywalled source. – Void Nov 04 '23 at 12:47
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Exact solutions for this equation are generally not available, and the orbits near a Schwarzschild black hole usually rely on numerical methods and perturbation theory.

beens_answers
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    It is a fact that I solved the orbit, though it turns out to be a complex number and by the same method, show no real solution is possible. Paper available by email . – LiveProton Nov 02 '23 at 01:34