Electrodynamics Around Schwarzschild and Reissner Nordstrom Black Holes

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

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

Key Points 1. 2. 3. 4. 5. 6. 7. 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 BH 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 + + - + -

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 (event horizon) • Obtaining

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 → ∞

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

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

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.

• 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

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

3. Linet’s Interpretation Linet adopts the convention that Copson)→ and are symmetric (from Claim: there are 2 charges! To fix this he adds a 2 nd charge inside the horizon And in standard coordinates (with )

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 → 2 m , V → 0 and we get back the Schwarzschild solution

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

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

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

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

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

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

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

Thank you!