WOLFRAM NOTEBOOK

Insert

Sample

Meissner effect for rotating black holes

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.

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

, marking the transition to the superconducting state.

◼

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.

◼

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 [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

Convention:1. Newton constant G = 1, speed of light c = 1 (geometrized units); 2. Indices of covariant tensors are denoted by lower-case letters, e.g., Tdddd stands for

αβγδ

, while indices of contravariant tensors are denoted by upper-case letters, e.g. TUUUU stands for

αβγδ

;3. Tensor contraction is written as TUUU . TddU and it means

αβγ

γδ

A rotating black hole is described by the Kerr spacetime.

In order to have a physical intuition of it, let us consider a sphere rotating around an axis:

In[1]:=

Graphics3D[{{Green,Opacity[.35],Sphere[{0,0,0},1]},Text["rotation axis",{3,1.3,0}],{Blue,Arrow[{{0,0,-1.5},{0,0,2}}],Line[Append[0]/@Append[#,First@#]&[CirclePoints[1,100]]]}},Lighting"Neutral",BoxedFalse]

Out[1]=

It exhibits an axial symmetry, because it looks the same if it is rotated by any angle around the rotation axis and it is constant in time.

The Kerr black hole is a stationary (constant in time) and axial-symmetric (it exhibits rotational symmetry around its axis of rotation) solution to the vacuum Einstein’s field equations.

It is parametrized by its mass m and its angular momentum J or, equivalently, by its angular momentum per unit mass a = J/m.

We define the coordinates to describe our spacetime in four dimensions:

In[2]:=

coordinates={t,r,θ,ϕ}

Out[2]=

{t,r,θ,ϕ}

In[3]:=

dimension=Length[coordinates]

Out[3]=

The spacetime is described by a symmetric metric tensor as follows:

In[4]:=

gdd={{gtt[r,θ],0,0,gtϕ[r,θ]},{0,grr[r,θ],0,0},{0,0,gθθ[r,θ],0},{gtϕ[r,θ],0,0,gϕϕ[r,θ]}};gdd//MatrixForm

Out[5]//MatrixForm=

gtt[r,θ]	0	0	gtϕ[r,θ]
0	grr[r,θ]	0	0
0	0	gθθ[r,θ]	0
gtϕ[r,θ]	0	0	gϕϕ[r,θ]

where

In[6]:=

gtt[r_,θ_]:=-

Σ[r,θ]Δ[r,θ]

A[r,θ]

Σ[r,θ]

ΩZ^2Sin[θ]^2

In[7]:=

gtϕ[r_,θ_]:=-

A[r,θ]

Σ[r,θ]

ΩZSin[θ]^2

In[8]:=

gϕϕ[r_,θ_]:=

A[r,θ]

Σ[r,θ]

Sin[θ]^2

In[9]:=

grr[r_,θ_]:=

Σ[r,θ]

Δ[r,θ]

In[10]:=

gθθ[r_,θ_]:=Σ[r,θ]

and

In[11]:=

Σ[r_,θ_]:=r^2+a^2Cos[θ]^2

In[12]:=

Δ[r_,θ_]:=r^2-2mr+a^2

In[13]:=

A[r_,θ_]:=(r^2+a^2)^2-a^2Δ[r,θ]Sin[θ]^2

In[14]:=

ΩZ=

2mar

A[r,θ]

;

We notice that the function grr(r, θ) blows up at a specific radial location. Let us set m = 1, a = 0.3 m, and we look at the profile of

(r,θ=π/2)

In[15]:=

Plot[grr[r,π/2]/.m1/.a0.3,{r,1,3},FrameTrue,FrameLabel{"r","

(r, θ)"}]

Out[15]=

Obviously, the location where the function

blowsup

depends on the mass parameter m and tha angular momentum per unit mass a.
Such a radius, where our coordinates description breaks down, is the radial location of the event horizon. Therefore, we solve Δ(r, θ) = 0 and we get:

In[16]:=

Solve[Δ[r,θ]0,r][[2]]

Out[16]=

rm+



Hence, let

the event horizon location:

In[17]:=

=m+

;

It is a two dimensional sphere with radius

In[18]:=

Graphics3D[{{Green,Opacity[.35],Sphere[{0,0,0},1]},{Blue,Arrow[{{0,0,0},{1,0,0}}],Text[Subscript["r","h"],{.65,0,0}],Line[Append[0]/@Append[#,First@#]&[CirclePoints[1,100]]]}},Lighting"Neutral",BoxedFalse]

Out[18]=

Papapetrou-Wald construction of test electromagnetic fields

Killing vectors: definition

As claimed before, the Kerr black hole is a stationary and axial-symmetric spacetime. To express the concept of stationarity and axial-symmetric in a precise mathematical way, we need to introduce the Killing vectors.

Let us define the Lie derivative operation from the second equality in the expression above:

Killing vectors of Kerr Spacetime

Kerr spacetime is a stationary and axisymmetric spacetime. This implies that it has two commuting Killing vectors:

It is easy to check that both η and ψ are two Killing vectors:

Construction of electromagnetic fields

Here, we define the Christoffel symbols:

and we construct the following linear combination

Here, B0 is the asymptotic value (at r -> ∞) of the magnetic field strength and Q is the charge of the black hole. In the following, we simply set Q = 0.

We might check that this electromagnetic field tensor comes from the vector potential

In the next section, we aim at computing the magnetic flux though the event horizon.

Computation of the magnetic flux

which can be rewritten in a more readable way as

and we plot the near-extremal range [0.9, 1]:

As it is clear from the last two plots, when the black hole rotates with its maximal angular momentum, a = m = 1, the magnetic flux is zero:

This is the Meissner-like effect for rotating black holes! Notice that this effect is not present if the black hole is charged Q ≠ 0.

Visualization of the Meissner-like effect

We set

For a < 1, the magnetic flux penetrates the event horizon:

For a = 1, the magnetic flux is expelled out, according to our analysis in the previous section:

We explore the whole range of the angular momentum and do an animation to highlight the Meissner-like effect:

Further Explorations

Study the orbit a test particle around Kerr spacetime

Study of the gravitational radiation emitted by a particle spiralling around Kerr spacetime

Authorship information

Roberto Oliveri

Friday, June 23rd 2017

roliveri@ulb.ac.be

You are using a browser not supported by the Wolfram Cloud

Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.

I understand and wish to continue anyway »