Solving Geodesic Equations with Godel Metric

[space-time]

I have been working with the Godel metric (- + + + signature). I wanted to derive the geodesics for the metric, so I took to the geodesic equation:

(d²x^m/ds²) + Γ^m_ab(dx^a/ds)(dx^b/ds) = 0

In the case of the Godel metric, the geodesic equations that I was able to derive after deriving the Christoffel symbols are as follows:

(d²x⁰/ds²) + 2(dx⁰/ds)(dx¹/ds) + e^x(dx¹/ds)(dx³/ds) = 0

(d²x¹/ds²) + e^x(dx⁰/ds)(dx³/ds) + ( e^2x / 2 )(dx³/ds)(dx³/ds) = 0

(d²x²/ds²) = 0 (This one is easy to solve. It is just a straight line x²(s) = As + B where A and B are constants).

(d²x³/ds²) - (2 / e^x)(dx⁰/ds)(dx¹/ds) = 0Now can anyone either direct me to some free or cheap software that I could use to solve these equations, or give me a method that would commonly be used to solve these?

Thank you.

 

[pervect]

The best way to solve the geodesic equations is when you can find symmetries, called Killing vectors, that lead to conserved quantities.

The Wiki article on Killing vector field, <<link>>, might help.

If you write out the line element for the godel metric that you're using, I might be able to do a bit better writeup.

You might also look at the wiki (or a textbook) on the Schwarzschild geodesics, <<link>>. The point is that $\xi^a = \delta^a_0$ is a Killing vector, because the metric is independent of $x^0$, where $x^0$ represent time. We can lower the index by writing $\xi_b = g_{ab} \xi^a = g_{ab} \delta^a_0$.

This yields the result that $g_{00} \frac{dt}{d\tau}$ is constant along the geodesic, here t is a synonym for $x^0$. The constant E is basically interpretable as a constant energy along the geodesic , it's constant because the metric doesn't depend on time. Wiki writes

$$(1 - \frac{r_s}{r}) \frac{dt}{d\tau} = E$$

where E is some constant. Wiki writes this constant that I call E as E/mc^2 . But c^2 is just another constant for unit conversions, and m is another constant, the mass of the test particle following the geodesic. If we're just interested in the geodesic curve, we don't really need to sepcify m an can set it to unity.

 

[space-time]

Wow. Thank you for your reply. You asked me to write out the line element, so here you go:

ds² = (1/2ω²)[-(cdt + e^xdz)² + dx² + dy² + (1/2)e^2xdz²]

 

[pervect]

Wow. Thank you for your reply. You asked me to write out the line element, so here you go:

ds² = (1/2ω²)[-(cdt + e^xdz)² + dx² + dy² + (1/2)e^2xdz²]

This has at least 3 Killing vectors, due to the independence of the metric from t,y,z. Physically, this should correspond to a conserved energy and two conserved momenta, P_y and P_z.

I get, for the line element you gave

$$E(\tau) = -\frac{1}{2} \omega^2 *(c^2 \frac{dt}{d\tau} + c \,e^x\,\frac{dz}{d\tau} )$$
$$P_y(\tau) = \frac{1}{2} \omega^2 \frac{dy}{d\tau}$$
$$P_z(\tau)= -\frac{1}{2} \omega^2 (c \, e^x \frac{dt}{d\tau} + \frac{1}{2} e^{2x} \frac{dz}{d\tau} ) $$

So three of the geodesic equations should reduce to $dE/d\tau = 0$ , $dP_y / d\tau = 0$, and $dP_z / d\tau = 0$.

Wiki gives two more Killing vectors for this metric in <<link>>, the simpliest which is $\xi^a = \frac{\partial}{\partial x} - z \frac{\partial}{\partial z}$. This should give you the fourth independent equation you need, namely

$$P_k(\tau) = \frac{1}{2} \omega^2 (c\, z \, e^x \frac{\partial t}{\partial \tau} + \frac{\partial x}{\partial \tau} + \frac{1}{2} \,z\,e^{2x} \frac{\partial z}{\partial \tau} )$$

This should give you enough to solve the geodesic equations. I'd suggest a symbolic algebra package, the only free one I know of is maxima <<link>>.

I tried not to make typos, but - no guarantees.

There are some some much better symbolic algebra packages out there than Maxima with support for differential geometry and general relativity, but I don't know of any free ones other than Maxima. Maple (esp. with GRTensor) and Mathematica are two of the popular but expensive ones. GRTensor is rather nice, but it seems difficult to get it to run under modern versions of Maple. GRTensor is free, but it requires Maple, which is not free, to work. Maple also has native differential geometry support in it's latest version.

 

[Vick]

Sagemath (SageManifold) is free and powerful enough for said calculations. You could look at their example worksheets: Sagemath