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Thursday, 2 May 2013

Possibility of Charge Addition to the Reissner-Nordstrom Black Hole


     The Reissner-Nordstrom black hole has background field of metric
gμν=diag{(1rsr+rQ2r2),(1rsr+rQ2r2)1,r2,r2sin2θ}
where
rs=2GM;rQ2=GQ24π
and of electromagnetic field
At=Q4πr
Then, the geodesic is determined by the action
S=dτ[m(1rsr+rQ2r2)t˙211rsr+rQ2r2r˙2r2ϕ˙2+qQ4πrt˙]
assuming the motion is on the plane θ=π2,
where the normalisation of the proper time is determined by
(1rsr+rQ2r2)t˙211rsr+rQ2r2r˙2r2ϕ˙2=1

     We may, first, obtain the equation of motion in angle.
δSδϕ=ddτmr2ϕ˙=ddτmr2ϕ˙=0
So, we have found a constant of motion, angular momentum, which should be defined as
Lmr2ϕ˙
Now, we may take variation in direction of time.
δSδt=ddτ[m()t˙+qQ4πr]=ddτ[m()t˙+qQ4πr]=0
with the shorten expression
()(1rsr+rQ2r2)
This equation of motion is also cyclic so we may define the constant of motion
E=m()t˙+qQ4πr

     The only remaining equation of equation is related to r, but we may alternate the equation to the normalisation condition, conserving the degree of freedom.
1==()t˙2()1r˙2r2ϕ˙2=1m2()(qQ4πrE)2()1r˙2L2m2r2
We may manipulate the equation as similar to the classical mechanics.
12mr˙2+L22mr2(1rsr+rQ2r2)12mq2Q216π2r2+1mqQE4πr=E22mm2()
By special relativity, you may find 12mx˙2=E22mm2 where E is the total energy of a particle, so E is the asymtotic energy of the test particle, in this problem.
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 L=0.
Then, this seems to be a system with potential V(r)=q2Q232π2mr2+qQE4πmr, but this term itself contains the total energy E so we have to consider more carefully. First, from the shape of the potential, we may guess that the test particle may fall to the centre if the particle just overcome the threshold. i.e. V(r)=E22mm2 should have no solution. However, the solution is r=qQ4π(Em)which always exists, considering the absolute condition Em. Thus, the particle always bounce back to the infinity at the outer critical point r=qQ4π(Em)
Not to enter inside of the horizon, the criterion is thus
EmqM2+Q24πGMQ


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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