In the general case of a rotating black hole it is possible but very cumbersome to use the geodesic equations (13) directly to obtain the equations of motion of a test particle. Fortunately it is possible to use a Hamiltonian approach that also has the advantage that the constants of the motion are found directly. The procedure will not be described further here, but an excellent treatment can be found in Misner, Thorne & Wheeler (1973). The result, for the coordinate system chosen here, are the following
Here 
is the proper time of the star in units of 
, 
is
the energy in units of 
, 
is the z component
of the orbital angular momentum in units of 
and 
is
Carter's fourth constant (Carter 1968) in units of 
,
The equations of motion for the non-rotating case is found by setting
and since the motion takes place in one plane, the coordinate
system can be chosen such that the polar angle is 
. The result
is then
The existence of the constants of motion and follows
from the fact that the metric does not depend on t and 
respectively.
There are no more symmetries in the metric but still another constant of
motion exist. This is somewhat surprising and it is difficult to find a
physical interpretation of this constant. What can be said is, in the case of
parabolic motion with 
, that the relation between the total angular
momentum at infinity, 
, the projection of the angular
momentum on the z-axis, , and Carter's fourth constant,
, is
This WWW-exhibition contains an interactive geodesics page, where you can choose parameters for specific geodesics and view 3-D plots of the trajectory using any viewer capable of displaying VRML files or (if you are using Silicon Graphics) Inventor files (ivview).
Here is how to do it:
First you choose the rotation parameter of the black hole, 
.
Secondly you choose orbital parameters that you think might give an
interesting geodesic.
This can be done in two ways. You can either specify the initial
position and 4-velocity (form 1) or specify the initial position and the
constants of motion (form 2). In the second case you also have to specify
whether 
and 
are positive or negative. Thirdly
you specify how long (in proper time) a segment of the trajectory to
visualize. There is a build in maximum of the number of points used for
the trajectory. If more points are needed to evolve the equations of
motion the specified amount of proper time, the last part of the
trajectory is ignored. Finally you can press the `calculate
trajectory' button and the calculation will commence. The amount of time
used here is of course dependent on the load of the server but is
typically around 30 seconds. Now 1 of 3 things will happen
