The Meissner effect for rotating black holes describes the expulsion of stationary axisymmetric magnetic fields from the outer horizon when the black hole reaches its maximum angular momentum.
June 23, 2017—Roberto Oliveri
History and References
The Meissner effect, originally discovered by Meissner and Ochsenfeld in 1933, describes the expulsion of magnetic fields from a superconductor when it is cooled below a critical temperature
T
c
, marking the transition to the superconducting state.
In[]:=
Pictorial representation of the Meissner effect for superconductors. [Wikipedia]
In this topic exploration, we would like to explain the Meissner-like effect for rotating black holes.
In the picture below, the circumference depicts the event horizon of the black hole, and the spacetime outside the horizon is permeated by the electromagnetic field, which does not deform the geometry of the spacetime (i.e. the electromagnetic field is a test field). The plot on the left shows that the field lines of the magnetic field penetrate the event horizon, while the plot on the right shows that the magnetic field lines are expelled out from the event horizon for a black hole rotating with its maximum angular momentum: this is the Meissner-like effect for rotating black holes.
In[]:=
Meissner-like effect for rotating black holes.
[Bičák, J., Karas, V., & Ledvinka, T. (2006). Black holes and magnetic fields. Proceedings of the International Astronomical Union, 2(S238), 139-144.]
The story of the Meissner-like effect starts with an elegant paper by R. Wald in 1974 [Phys. Rev. D, 10, 1680]. By using a theorem of Papapetrou’s [Ann. Inst. Henri Poincare, Sect. A 4, 83], i.e. that a Killing vector in a vacuum spacetime generates a vacuum (i.e. source-free) solution to Maxwell’s equations in that spacetime, Wald constructed a stationary, axisymmetric solution to vacuum Maxwell’s equations around Kerr spacetime.
One year later, in 1975, King, Lasota and Kundt [Phys. Rev. D, 12, 3037] computed the flux of the magnetic field across the upper hemisphere of the event horizon as a function of the angular momentum of the Kerr black hole. They concluded that the magnetic flux is vanishing for maximally rotating black holes, establishing the Meissner-like effect for rotating black holes. Ten years later, Bičàk and Janiš [Mon Not R Astron Soc (1985) 212 (4): 899-915] confirmed this effect for back-reacting Maxwell's fields, by explicitly solving the vacuum Maxwell’s equations around Kerr spacetime.
Geometric Background: The Kerr Spacetime
Papapetrou–Wald Construction of Test Electromagnetic Fields
Computation of the Magnetic Flux
Visualization of the Meissner-Like Effect
FURTHER EXPLORATIONS
Study the orbit of a test particle around Kerr spacetime
Study the gravitational radiation emitted by a particle spiraling around Kerr spacetime