Using the Schwarzschild solution is there a simple differential equation describing the four position of a particle influenced by a Schwarzschild metric using the geodesic equation. How would the simplest form look like?

  • $\begingroup$ I would actually suggest solving for the motion of the particle with the line element instead of the geodesic equation in this case, it turns out to be much easier imo. You should use the geodesic equations to solve for constants of motion like angular momentum which are needed. $\endgroup$ – Thatpotatoisaspy Apr 7 at 0:28
  • $\begingroup$ en.wikipedia.org/wiki/Schwarzschild_geodesics $\endgroup$ – safesphere Apr 7 at 13:44
  • $\begingroup$ @Thatpotatoisaspy How would you solve it with a line element and would it give you all components of the four-vector? $\endgroup$ – Joshua Pasa Apr 7 at 22:26
  • $\begingroup$ @Joshua Pasa Generally you can solve the line element for one of the variables by remembering the fact that ds^2 = -c^2dτ^2 and dividing both sides by dτ^2, then subbing in the constants of motion. What you do after that depends on which variable you’re solving for. I warn you though, general analytic solutions are pretty hard, so a perturbative solution is preferred. I’ve only ever got it for the radius variable however, so i’m not sure about the whole four-vector. $\endgroup$ – Thatpotatoisaspy Apr 8 at 1:18
  • $\begingroup$ @Joshua Pasa i’m pretty sure i could find the four-velocity pretty easy though $\endgroup$ – Thatpotatoisaspy Apr 8 at 1:30

In the Scharzschild solution, we can write the geodesic equation in the form of the equations of motion $$ r^2 \dot\phi = h = \mathrm {const}$$ $$ {\dot r}^2 = {2\mu\over r} - \bigg(1-{2m\over r}\bigg){h^2\over r^2} $$ Einstein showed that solving these equations perturbs the Newtonian equations and results in orbital precession.

For comparison, the first equation is also true in Newtonian gravity (Kepler's second law). The equation in Newtonian gravity corresponding to the second is

$$ {\dot r}^2 = {2\mu\over r} - {h^2\over r^2} + k$$

where $k$ is a constant.

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  • $\begingroup$ Isn't the first equation equivalent to newtons law of gravity? Is there also a way of finding how time changes in a Scharzchild geodesic to give the form $x^\mu$ instead of just $x^i$ $\endgroup$ – Joshua Pasa Apr 7 at 22:24

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