Electrodynamics Around Schwarzschild and Reissner-Nordstrom Black Holes

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Electrodynamics Around Schwarzschild and
Reissner-Nordstrom Black Holes

• Maya Watanabe and Anthony Lun
• Centre for Stellar and Planetary Astrophysics
• School of Mathematical Sciences
• Monash University

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Introduction

• This is a work in progress on the electromagnetic
  phenomena around black holes

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Key Points

1. Copsons electric point solution ? point charge
   outside the black hole
2. Linet wrote down Copsons solution in
   Schwarzschild coordinates in an attempt to
   clarify the physics
3. Linet interpreted Copsons single charge solution
   ? two charges of same sign, one in the black
   hole, the other outside the black hole, refer to
   point 7
4. Extend on Copsons solution for 2 charges of
   opposite sign
5. Solution for 2 point charges of opposite signs at
   antipodal points
6. The isotropic form of the Reissner-Nordstrom
   metric
7. Use isotropic form to derive solution for a point
   charge outside Reissner-Nordstrom black hole c.f.
   point 3

BH
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1. Isotropic Coordinates

• Schwarzschild metric in isotropic coordinates
  (c.f. for example Adler et al, 1965)

• Normalizing r in natural unit size of the black
  hole
• Obtaining

(event horizon)
(event horizon)
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Important Characteristics of Isotropic Coordinate

• For every value of R there are two values for R
• When R ? 8, ? 8 and R ? 0
• Thus as R ? 8, R ? 8 and R ? 0, R ? 8

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2. Copsons Solution

• Copson places a single charge, q, at a point
  in the isotropic
  coordinate which, by virtue of the coordinate
  system, creates an image at
• The Laplace Equation for this configuration is

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To solve this, Copson uses the following method
(the Copson-Hadamard method) Where And
His solution is a fundamental solution of the
Laplace equation
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• As we are considering the situation with only a
  single point charge we let
• And the Copson solution can be written as
• And the boundary condition is satisfied.

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• Linet finds a discrepancy, saying that there are
  clearly 2 charges in the Copson solution when
  there should be only one
• He was both correct and incorrect

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A

• isotropic coordinates standard
  Schwarzschild coordinates,
• two charges single charge
• This IS the case, following Linet we transform
  Copsons potential into Schwarzschild coordinates
  
• And when r ? 8

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3. Linets Interpretation

• Linet adopts the convention that and are
  symmetric (from Copson)?
• Claim there are 2 charges!
• To fix this he adds a 2nd charge inside the
  horizon
• And in standard coordinates (with
  )

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4. Extending Linets Solution

• Extend Linets solution for charges of
  opposite sign residing outside the horizon at
  and inside the horizon at the physical
  singularity
• And we have chosen

• In standard coordinates this is
• When a ? 2m , V ? 0 and we get back the
  Schwarzschild solution

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5. Dipole Solution
BH

• The Einstein-Maxwell equation
• Using Copson solution, by virtue of
  superposition
• Where
• This coincides with Israels 1968 expansion

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6. Reissner-Nordstrom Metric in
Isotropic Coordinates

• The Reissner-Nordstrom metric in isotropic
  coordinates is

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• Normalizing this by the natural unit size of the
  black hole
• where
• and
• Thus the metric can be written as

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7. Single charge outside Reissner-Nordstrom Black
Hole

• If we define the electric field as
• Then the Einstein-Maxwell equation is

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• Solving this for a charge, q, at
  
• We use the Copson-Hadamard method
• Where
• The Laplace equation becomes
• Same as Schwarzschild case!

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• Solving this for F gives
• Finally, choosing
  the potential of a single charge, q, situated
  outside a Reissner-Nordstrom black hole of
  charge, e, is

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8. Conclusion

• Copsons 1928 solution was for single point
  charge
• Linets 1976 solution was for two charges of same
  sign
• We found solution for two charges of opposite
  sign, inside and outside horizon
• We found solution for two charges of opposite
  sign at antipodal points to create a dipole
• We found analytical solution for single charge
  outside Reissner-Nordstrom black hole in
  isotropic coordinates which corresponds with
  Copsons 1928 solution

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Thank you!