in equation (26) it is obvious
that the expression in the square root must be positive and
that the roots of this expression determines the turning points of the
radial motion. The expression can be perceived as a second degree polynomial
in 
with coefficients that are functions of 
, 
and 
, x. Setting equal to 0 and solving for gives the
effective potential
where
For a non-rotating black hole the result reduces to
At the horizon, where 
, the coefficients become
From this it can be seen that the determinant in equation (34)
vanishes at the horizon.
Thus, the formal solution consists of two branches that coincides at the
horizon and with one larger than the other for all values of 
.
This is shown in Figure 1 for the parameters 
,
and 
.
Figure: The effective potential for , and
. The dotted line marks the energy 
A particle will move with constant energy indicated by the horizontal dotted line in Figure 1. Where this line crosses the effective potential the radial motion will change direction. Particle motion is possible whenever its energy is either larger or smaller than both branches of the effective potential. No motion is possible if the energy is between the two branches of the effective potential. In figure 1 motion is possible for radial coordinates between x=2.52316 and x=7.03926 (bound motion) and for radial coordinates between the horizon and 0.59731. In the last case the particle is created (or placed by external forces) near the black hole and is captured by the black hole.
In the non-rotating case the two branches will be symmetric around 
but here the lower branch is unphysical since the energy of the particle has to
be positive, so this branch is ignored. In the rotating case it is possible to
have motion (in the ergosphere) with negative energy, but in order not to
complicate things I have chosen to plot the lower branch only when it is larger
than 0.
This WWW-exhibition contains an interactive effective potential page, where
you can investigate the effective potential for different values of the
parameters , and . This is faster than
using the interactive geodesics page and is meant as a tool for finding
interesting parameters before visualizing the geodesics themselves.
Here is how to do it:
First you choose the rotation parameter of the black hole, .
Then you choose values for and . This determines the
effective potential. You can also choose a specific value of to
see if motion is possible with this energy. If it is you will also get
information about the ranges of radial coordiates and polar angles that are
allowed. Typically after 10 seconds you will get a plot of the effective
potential, with the chosen energy plotted as a horizontal dotted line. Under
the plot some useful information will be printed. Firstly there will be
information about the position and value of local minima and maxima for the
effective potential or if none exist you will be notified that the effective
potential is monotonic. Note that the minima and maxima are found by numeric
methods, that may fail. If you find any inconsistencies please send me an
e-mail containing the parameters used and I will look into the problem.
Secondly information will be printed about the allowed ranges of radial
coordinates and polar angles where motion with the specified energy can occur.
When you find some interesting parameters go to the interactive geodesics page and try them out.
Have fun!