The Reissner-Nordstrom black hole has background field of metric
where
and of electromagnetic field
Then, the geodesic is determined by the action
assuming the motion is on the plane
where the normalisation of the proper time is determined by
We may, first, obtain the equation of motion in angle.
So, we have found a constant of motion, angular momentum, which should be defined as
Now, we may take variation in direction of time.
with the shorten expression
This equation of motion is also cyclic so we may define the constant of motion
The only remaining equation of equation is related to
We may manipulate the equation as similar to the classical mechanics.
By special relativity, you may find
Thus, this equation is in analogy to the energy conservation law in classical mechanics.
To simplify the situation and ease to enter the black hole, we may consider the situation
Then, this seems to be a system with potential
Not to enter inside of the horizon, the criterion is thus
For further study: I'll note here http://arxiv.org/abs/1304.6474 which is a part of my project here. I may list tasks to determine charged black hole stability and radiation
*particle generation in strong electric field
*decay rate of particle-anti-particle pair in terms of gas density
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