There is an object in free fall above the surface of some planet. This planet has no significant atmosphere so there is no significant air resistance on the object. Also there is no significant electric force acting on this object, and any electric charge of the planet is negligible. Also the angular velocity of the planet is negligible. Also the reference frame that we are using to define the objects four velocity and spacetime coordinates is one that is on the planets surface close to the object in free fall.

Given the rest mass of the planet $M_0$, and the proper radius of the planet $r_0$, the initial four velocity of the object, and the initial spacetime coordinates of the object, and the proper time $\tau$ of the object, how would I calculate the final four velocity and final spacetime coordinates of the object using General Relativity.


1 Answer 1


If an object is in free fall it will simply follow a geodesic. This means that you simply have to work out the geodesic equations $$ {d^2 x^\mu \over ds^2} =- \Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}\, $$ for the coordinates on a Schwarzschild background $$ c^2 \,{d \tau}^{2} = \left(1 - \frac{r_\mathrm{s}}{r} \right) c^2 \,dt^2 - \left(1-\frac{r_\mathrm{s}}{r}\right)^{-1} \,dr^2 - r^2 \left(d\theta^2 + \sin^2\theta \, d\varphi^2\right). $$ where $r_s = 2GM_0 / c^2$. So look up the Christoffel symbols $\Gamma^\mu {}_{\alpha \beta}$ on this geometry and work out the four differential equatiosn this gives you for $x^\mu$. Then insert the initial position $(0, r_0, 0, 0)$ and initial 4-velocity $(v_t, v_r v_\theta, v_\phi)$ and you have the differential equation that describes the motion of your particle.

  • $\begingroup$ Is $\varphi$ an angle and if so what is the difference between $\varphi$ and $\theta$? $\endgroup$ Commented Jan 26, 2018 at 0:43
  • $\begingroup$ Yes, $\varphi$ is an angle as the Schwarzschild metric here is in angular coordinates. There are two angular directions on a sphere, latitude, and longitude. $\endgroup$
    – Jeroen
    Commented Jan 29, 2018 at 19:36

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