Solutions to Geodesic equations in AdS3 space

Question

I am struggling with an assignment from my lecturer, he has asked me to numerically (mathematica) and analytically compute solutions for the trajectory of a point particle in AdS3 space with global coordinates given by metric $$ds^2=R^2(-\cosh^2(\rho)dt^2+d\rho^2+\sinh^2(\rho)d\phi^2)$$ I calculated three geodesic equations of motion

$$\ddot{\rho}+\sinh(\rho)\cosh(\rho)(\dot{t}\dot{t}-\dot{\phi}\dot{\phi})=0$$

$$\ddot{\phi}+\coth(\rho)\dot{\rho}\dot{\phi}=0$$

$$\ddot{t}+\tanh(\rho)\dot{\rho}\dot{t}=0$$

I realise they're coupled but I honestly have no clue how I go finding $\rho(\tau)$, $t(\tau)$ and $\phi(\tau)$.

Answer 1

Assuming your equations are correct (I have not checked), but coupled ODEs such as this (and geodesic equations in general) must be solved numerically. The trick is to employ the standard Runge-Kutta solvers, it is most convenient to deal with first-order systems. In general, given the geodesic equations:

$\ddot{x}^{a} + \Gamma^{a}_{bc} \dot{x}^{b} \dot{x}^{c} = 0$,

we can write these as a first-order system:

$\dot{x}^{a} = v^{a}$, $\dot{v}^{a} = - \Gamma^{a}_{bc} v^{b} v^{c}$

So, for your system above, define $x^{a} = \dot{\rho}^{a}$, $y^{a} = \dot{\phi}^{a}$, and $z^{a} = \dot{t}^{a}$. Then, coupled with the remaining equations you have a 6-D system, first-order, and the solutions will depend on specifying 6 such initial conditions.

Hope this helps.