Overspinning an electrically charged sphere
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____ / \ Q _____ E electric field
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<--- rotation
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Let us assume that the electric field outside the sphere contains mass-energy and inertial mass such that the energy density D is
D = 1/2 ε₀ E²,
where E is the electric field strength and ε₀ is vacuum permittivity.
It is like a carousel where a substantial mass is outside the sphere. The far field of the sphere will move at the speed of light, or almost at the speed of light. There is a substantial "centrifugal" force which tries to pull the charge Q out from the sphere.
We can increase the centifugal force by adding more energy into the rotation of the field, since then the mass-energy of the field will be larger. Eventually, the centrifugal force will pull the charge Q out from the sphere.
If our sphere is analogous to a black hole in gravity, then over-spinning a black hole is a way to disassemble the black hole.
A spinning black hole
In an extremal black hole, orbits to the same direction as the black hole spins (prograde orbits) can touch the horizon. A "centrifugal" force seems to be at play.
The Kerr metric in the usual Boyer-Lindquist oblate spheroidal coordinates is:
We can move to a rotating coordinate system to diagonalize the metric:
φ' = φ - Ω t.
Let us assume that c = G = M = 1, and calculate the values for an extremal spinning black hole. Then r_s = 2, the horizon is at r = 1, and a = 1.
The angular speed in the equatorial plane (θ = 90 degrees), relative to the coordinate time t is then
Ω = 1 / 2.
We can map the oblate spheroidal coordinates to cartesian coordinates:
The cartesian (x, y, z) coordinate radius of r = 1 is
sqrt(r² + a²) = sqrt(2).
The cartesian coordinate speed of the rotation at r = 1 is then
sqrt(2) / 2.
It is 71% of the speed of light, as seen by a faraway observer.
Let us then slow down the black hole to a standstill, so that a = 0. The new mass is only sqrt(2) / 2, because 29% was "reducible" mass. The cartesian radius of the event horizon is then
2 * sqrt(2) / 2 = sqrt(2).
The radius stayed the same when we slowed down the rotation.
The circumference of an extremal black hole
Francesco Sorge (2021) calculated the rotating, diagonalized metric explicitly:
Let us determine the equatorial circumference of the event horizon of the extremal black hole in these rotating coordinates.
In the metric, the ratio
A / Σ = 4
holds the usual place of the square of the radius. The radius, if defined from the circumference, is 2.
The radius is 2 in the nonrotating Boyer-Lindquist coordinates, too.
But in the cartesian coordinates, the radius is only sqrt(2).
Which is the "true" radius? We can stop the rotation, and the black hole becomes a Schwarzschild black hole whose radius is sqrt(2) in the Schwarzschild coordinates.
The Ehrenfest paradox concerns the circumference of a fast rotating cylinder.
The speed of the horizon is c / sqrt(2) in the cartesian coordinates. The length contraction factor is
1 / γ = sqrt(1 - v² / c²)
= 1 / sqrt(2).
The proper length of the spinning horizon would correspond to a radius 2. This matches the value in the Boyer-Lindquist coordinates. Maybe we should call sqrt(2) the "true" radius?
This paper by Edwin F. Taylor gives the speed of prograde light in "bookkeeper" coordinates as 1.0, which presumably means the speed of light in those coordinates.
Accelerating a Schwarzschild black hole: a new Ehrenfest paradox?
The speed of light close to the horizon of a Schwarzschild black hole is extremely slow. If we start rotating the black hole, observers close to the horizon do not learn about the spinning until much later. How can the metric stay unchanged close to the horizon, while the metric is length-contracted a little farther away? Do we have here a new Ehrenfest paradox?
A similar process happens when we linearly accelerate a black hole. Points close to the event horizon do not know about the acceleration until much later – but a faraway observer sees the entire black hole to length-contract. How is this possible?
If we accelerate a ruler, observers sitting on the ruler feel an acceleration. The spatial metric measured by the observers does not contract significantly, while static observers see it to contract. There is no paradox there.
Let us use comoving coordinates inside the black hole, and static coordinates at some distance from it. There does not need to be any paradox. Observers inside the black hole are "frame-dragged", and do not feel any acceleration. Their comoving spatial metric does not change in any way.
Conclusions
We now have a grasp of what happens when we start to rotate a Schwarzschild black hole.
We still need to analyze if we really can over-spin the black hole using ropes. Then analyze if the black hole starts to disassemble through over-spinning.
Also: how fast does the new metric propagate toward the horizon? If the propagation never reaches the falling matter, then we cannot eject the matter out. However, this would be strange, since how could the black hole store excess angular momentum in that case?
If a neutron star contains too much angular momentum to form a black hole, a computer simulation showed that it will form a torus. The crucial question is if an Oppenheimer-Snyder collapse is irreversible. There is no proof that it would be. We conjecture that over-spinning will reverse an Oppenheimer-Snyder collapse.
Albert Einstein's 1939 paper on rapidly spinning Einstein clusters may offer a clue. Over-spinning a black hole may turn it into an Einstein cluster.
Ted Jacobson and Thomas P. Sotiriou (2010) write about destroying a black hole by over-spinning it. They correctly note that it is hard to deduce what is the dynamical process like if we start to over-spin an extremal Kerr black hole.
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