1. 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

2. Introduction
This is a work in progress on the electromagnetic phenomena around black holes

3. Key Points
Copson’s electric point solution → point charge outside the black hole
Linet wrote down Copson’s solution in Schwarzschild coordinates in an attempt to clarify the physics
Linet interpreted Copson’s single charge solution → two charges of same sign, one in the black hole, the other outside the black hole, refer to point 7
Extend on Copson’s solution for 2 charges of opposite sign
Solution for 2 point charges of opposite signs at antipodal points
The isotropic form of the Reissner-Nordstrom metric
Use isotropic form to derive solution for a point charge outside Reissner-Nordstrom black hole c.f. point 3 + + BH + - + -

4. 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)

5. Important Characteristics of Isotropic Coordinate
For every value of R there are two values for R
When R → ∞, → ∞ and R → 0
Thus as R → ∞, R → ∞ and R → 0, R → ∞

6. 2. Copson’s 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

7. To solve this, Copson uses the following method (the Copson-Hadamard method)
Where
And
His solution is a fundamental solution of the “Laplace” equation

8. And the boundary condition is satisfied.
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.

9. 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

10. isotropic coordinates standard Schwarzschild coordinates,
two “charges” single charge
This IS the case, following Linet we transform Copson’s potential into Schwarzschild coordinates
And when r → ∞

11. 3. Linet’s 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 )

12. 4. Extending Linet’s Solution
Extend Linet’s 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

13. 5. Dipole Solution The Einstein-Maxwell equation+ -
5. Dipole Solution BH
The Einstein-Maxwell equation
Using Copson’ solution, by virtue of superposition
Where
This coincides with Israel’s 1968 expansion

14. 6. Reissner-Nordstrom Metric in Isotropic Coordinates
The Reissner-Nordstrom metric in isotropic coordinates is

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

16. 7. Single charge outside Reissner-Nordstrom Black Hole
If we define the electric field as
Then the Einstein-Maxwell equation is

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

18. 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

19. 8. Conclusion Copson’s 1928 solution was for single point charge
Linet’s 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 Copson’s 1928 solution

20. Thank you!