Electrodynamics Around
Schwarzschild and ReissnerNordstrom 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
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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
(event horizon)
(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 2nd 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 → 2m , 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!