Daily Archives: December 30, 2007

The Painleve Equations of Motion

In the general theory of relativity, the orbits are given by geodesics. A geodesic is a path that extremizes the path length. The path length is defined as the integral of ds over the path, where ds^2 is the metric. For the case of Painleve coordinates on the Schwarzschild metric, ds^2 is given by:
Painleve coordinates ds2. Let’s let our path start at time t=0 and end at time t=1. For the path to be a geodesic, we must extremize the following integral (I’ll quickly sneak in a minus sign to make the path be timelike instead of spacelike):
ds integral for Painleve coordinates
To make life easier for us, we will make the assumption that the orbital motion is in the \theta = \pi/2 plane so there’s no \theta dependence. That turns the angular part of the square root into r^2\;(d\phi/dt)^2 . Furthermore, since the simulation is going to use Cartesian, (x,y) coordinates, we might as well replace r^2\;(d\phi/dt)^2 with (x\;dy/dt-y\;dx/dt)^2 , and dr/dt with x\;dx/dt + y\;dy/dt , their Cartesian equivalents. And put M=1, we can always fix it later by dimensional analysis.
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