Solving Geodesic Equations with Godel Metric

  • I
  • Thread starter space-time
  • Start date
  • Tags
    Geodesic
In summary: SageManifold) is free and powerful enough for said calculations. You could look at their example worksheets: <<link>>In summary, the conversation discusses the process of deriving geodesic equations for the Godel metric and the use of Killing vectors to find conserved quantities. The conversation also mentions different software options, such as Maxima and Sagemath, for solving the equations.
  • #1
space-time
218
4
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:

(d2xm/ds2) + Γmab(dxa/ds)(dxb/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) + ex(dx1/ds)(dx3/ds) = 0

(d2x1/ds2) + ex(dx0/ds)(dx3/ds) + ( e2x / 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 / ex)(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.
 
Physics news on Phys.org
  • #2
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.
 
  • Like
Likes space-time
  • #3
Wow. Thank you for your reply. You asked me to write out the line element, so here you go:

ds2 = (1/2ω2)[-(cdt + exdz)2 + dx2 + dy2 + (1/2)e2xdz2]
 
  • #4
space-time said:
Wow. Thank you for your reply. You asked me to write out the line element, so here you go:

ds2 = (1/2ω2)[-(cdt + exdz)2 + dx2 + dy2 + (1/2)e2xdz2]

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.



 
  • #5
Sagemath (SageManifold) is free and powerful enough for said calculations. You could look at their example worksheets: Sagemath
 
  • Like
Likes vanhees71

FAQ: Solving Geodesic Equations with Godel Metric

1. What are Geodesic Equations?

Geodesic Equations are a set of mathematical equations that describe the shortest path between two points on a curved surface, such as a sphere or a curved space-time. They are used in the field of physics to calculate trajectories of objects in a curved space.

2. What is the Godel Metric?

The Godel Metric is a solution to Einstein's field equations in general relativity that describes a rotating universe. It is named after the mathematician Kurt Godel who first proposed this solution in 1949.

3. How are Geodesic Equations solved with the Godel Metric?

To solve Geodesic Equations with the Godel Metric, one must first set up the equations of motion for the object in the given space-time. Then, using the Godel Metric, the equations can be solved to determine the path of the object.

4. What are the applications of solving Geodesic Equations with the Godel Metric?

The solutions obtained from solving Geodesic Equations with the Godel Metric can be used to study the behavior of objects in a rotating universe, such as the motion of planets and stars. It also has applications in understanding the structure of space-time and the effects of gravity on objects in a rotating universe.

5. Are there any limitations to using the Godel Metric to solve Geodesic Equations?

Yes, there are limitations to using the Godel Metric to solve Geodesic Equations. It is a highly simplified model of a rotating universe and does not take into account other factors such as the presence of matter and energy. Additionally, the Godel Metric is only applicable to certain types of space-time, so it cannot be used to solve Geodesic Equations in all scenarios.

Similar threads

Replies
10
Views
2K
Replies
15
Views
2K
Replies
17
Views
2K
Replies
18
Views
2K
Replies
38
Views
4K
Replies
2
Views
2K
Replies
11
Views
907
Replies
1
Views
1K
Back
Top