A simple example

As a first simple example we consider light-like geodesics in the Schwarzschild space-time defined in isotropic coordinates (t,x,y,z). The line element is given by

where the Schwarzschild radius and .

Contents

  1. Start the GeodesicViewer...
  2. Choose the space-time metric...
  3. Set the initial position and the local reference frame...
  4. Play around with the initial direction...
  5. Place objects for a better orientation...
  6. Rotate the camera...
  7. Save parameters...

  1. Start the GeodesicViewer...

    When starting the GeodesicViewer, the initial OpenGL 3D view is empty. There is only a small axes of coordinates in the lower left corner. On the right hand side, there are the parameter windows for choosing the space-time metric, the integrator for the geodesics, the local tetrad, etc.

  2. Choose the space-time metric...

    The first step is to choose the space-time metric of interest. At the beginning, there is no space-time defined. This is indicated by the name 'unknown' in the Metric tab of the Metric/Integrator/Constants window. Here, we will study the Schwarzschild space-time in isotropic coordinates. For that, click on the combo box in the Metric tab and choose 'SchwarzschildIsotropic'. Below the combo box, there is a list of parameters of the current space-time. Here, we have only the mass of the black hole as free parameter which is set to one (in geometric units). Furthermore, there is a yellow line (the geodesic) in the OpenGL 3D view.

  3. Set the initial position and the local reference frame...

    When choosing a space-time metric, the GeodesicViewer shows a light-like geodesic with a predefined initial position and initial direction. The initial position is given in the Position tab of the Local tetrad window. In the line edit fields, set the initial position to

    t=0.0    x=10.0    y=6.0    and    z=0.0.

    The local reference frame is given by the default natural local tetrad of the space-time. In the case of the Schwarzschild metric in isotropic coordinates, the natural local tetrad reads
    Hence, the orientation of the local reference frame coincides with the coordinate axes of the OpenGL 3D view. The initial four-vector of a null geodesic is then defined by the two initial angles ξ and χ:

  4. Play around with the initial direction...

    One of the strengths of the GeodesicViewer is its interactivity. To obtain an impression of the space-time geometry, it is crucial to study the behaviour of null geodesics. So, play around with the initial direction ξ in the Direction Angles tab of the Geodesic window. You can change the angle ξ by either using the dial with dragging the mouse (left mouse button pressed) or by editing the angle directly in the line edit field. If your mouse cursor is above the line edit field, you can also change the angle by using the mouse wheel. The step size of the mouse wheel is given by the value in the 'step' line edit field.

  5. In the bottom right corner you can read that the maximum number of points has been exceeded. That is because the geodesic integrator is initially limited to 3000 points. You can change this value in the Integrator tab of the Metric/Integrator/Constants window. Please enter a value of

    max points = 6000

  6. Place objects for a better orientation...

    A useful feature of the GeodesicViewer is to place objects into the scenary for a better orientation. Here, we will indicate the horizon of the Schwarzschild black hole by a sphere. For that, click on the Objects menu and choose Show objects. The objects dialog appear.

    To add a sphere into the scenary, choose 'sphere3d' in the 'Add object' combo box and press the '+' button. The 'text field' now looks like this:

    The meaning of the different values is given in the 'Syntax' line. After the object name, here 'sphere3d', the Cartesian coordinates of the center of the sphere and the radius have to be defined. Because the sphere is represented by a mesh, the number of 'slices' and 'stacks' must be given. The last three values determine the colour of the sphere in rgb format, where 0 ≤ r,g,b ≤ 1. Because the event horizon has size rs=2 and we would like to have a red sphere, modify the line as follows:

    sphere3d    0.0000    0.0000    0.0000      2.0000    40.0    40.0      0.50    0.00    0.00

    and press 'Apply'. In the 'OpenGL 3D' view, a red sphere appears.

    Beside the sphere, let us also add a box into the scenary. For that, select 'box3d' in the 'Add object' combo box and press the '+' button. Then, change the parameters as follows:

    box3d      0.0000    0.0000    0.0000      20.0000    12.0000    10.0000      1.0    1.00   1.00   1.00

    and press 'Apply'. When you are finished adding objects, press the 'Close' button. Your OpenGL 3D view should now look like this:

  7. Rotate the camera...

    The projection of the three-dimensional scenary in the OpenGL 3D view can be set either to 'perspective' (default) or to 'orthographics'. The 'perspective' projection is realized by a pinhole camera whose parameters can be set in the DrawHandling window. 'eye x,y,z' give the position of the camera, 'poi x,y,z' the point of interest (focus point), and 'FoV' is the camera's field of view.

    For a simple mouse navigation, select 'rotate on sphere' in the 'mouse' combo box in the 3D tab. By dragging the mouse while pressing the left mouse button, the camera position moves on a sphere around the point of interest. The distance can be modified by dragging the mouse while pressing the middle/right mouse button.

  8. Save parameters...

    To save the current parameters and settings, click on the File menu and select 'Write protocol'.

    First, choose the directory where the files configuration files shall be stored. If you press on the '...' button, a directory dialog pops-up where you can choose the directory as usual. Next, enter a base-filename for the configuration files in the 'Name' line. (Details about the checkboxes can be found here). At last, press the 'Write' button. The resulting files can be seen here:


    If you like to reload all parameters and settings, click on the File menu and select 'Load all'. A file dialog opens where you can choose the desired configuration file.

    Good luck!