Dipole Black Ring and Kaluza-Klein Bubbles Sequences

Petya Nedkova,

Stoytcho Yazadjiev

Department of Theoretical Physics, Faculty of Physics, Sofia University

5 James Bourchier Boulevard, Sofia 1164, Bulgaria

Black Hole and Singularity Workshop at TIFR, 3 – 10 March 2006

Outline

We will consider an exact static axisymmetric solution to the Einstein-Maxwell equations in 5D Kaluza-Klein spacetime (M4 × S1)

Related solutions:

R. Emparan, H. Reall (2002) H. Elvang, T.Harmark, N. A. Obers (2005); H. Iguchi, T. Mishima, S. Tomizawa (2008a);

S.Tomizawa, H. Iguchi, T. Mishima (2008b).

Spacetime Bubbles

Bubbles are minimal surfaces that represent the fixed point set of a spacelike Killing field;

They are localized solutions of the gravitational field equations → have finite energy; however no temperature or entropy;

Example: static Kaluza-Klein bubbles on a black hole

Elvang, Horowitz (2002)

Vacuum Kaluza-Klein bubble and black hole sequences Rod structure:

Solution:

Elvang, Harmark, Obers (2005)

Vacuum Kaluza-Klein bubble and black hole sequence Properties:

Conical singularities can be avoided;

Bubbles hold the black holes apart →

multi-black hole spacetimes without conical singularities;

Small pieces of bubbles can hold arbitrary large black holes in equilibrium;

Generalizations:

Rotating black holes on Kaluza-Klein Bubbles (Iguchi, Mishima, Tomizawa (2008));

Boosted black holes on Kaluza-Klein Bubbles (Tomizawa, Iguchi, Mishima (2008)).

Charged Kaluza-Klein bubble and black hole sequences Further generalization: charged Kaluza-Klein bubble and black hole

sequences

Field equations:

2 spacelike + 1 timelike commuting hypersurface orthogonal Killing fields

Static axisymmetric electromagnetic field

Gauge field 1-form ansatz

Charged Kaluza-Klein bubble and black hole sequence Reduce the field equations along the Killing fields Introduce a complex functions E - Ernst potential ;

(H. Iguchi, T. Mishima, 2006; Yazadjiev, 2008)

→→ Field equations :

Ernst equation

Charged Kaluza-Klein bubble and black hole sequences The difficulty is to solve the nonlinear Ernst equation → 2-soliton

Bäcklund transformation to a seed solution to the Ernst equation E0

Natural choice of seed solution → the vacuum Kaluza-Klein sequences metric function gφφ

Charged Kaluza-Klein bubble and black hole sequence Solution:

gE is the metric of the seed solution

Charged Kaluza-Klein bubble and black hole sequences Electromagnetic potential:

α, β, A0φ are constants

Charged Kaluza-Klein bubble and black hole sequences W and Y are regular functions of ρ, z, provided that:

the parameters of the 2-soliton transformation k1 and k2 lie on a bubble rod;

the parameters α, β satisfy

→ The rod structure of the seed solution is preserved

Charged Kaluza-Klein bubble and black hole sequences It is possible to avoid the conical singularities by applying the

balance conditions

on the semi-infinite rods

on the bubble rods

L is the length of the Kaluza-Klein circle at infinity, (ΔΦ)E is the period for the seed solution

Physical Characteristics: Mass The total mass of the configuration MADM is the gravitational energy

enclosed by a 2D sphere at spatial infinity of M4

ξ = ∂/∂t, η= ∂/∂φ

To each bubble and black hole we can attach a local mass, defined as the energy of the gravitational field enclosed by the bubble surface or the constant φ

slice of the black hole horizon;

→ The same relations hold for the seed solution

Physical Characteristics: Tension Spacetimes that have spacelike translational Killing field which is

hypersurface orthogonal possess additional conserved charge – tension.

Tension is associated to the spacelike translational Killing vector at infinity in the same way as Hamiltonian energy is associated to time translations.

Tension can be calculated from the Komar integral:

Explicit result:

Physical Characteristics: Charge The solution possesses local magnetic charge defined as

The 1-form A is not globally defined → Q is not a conserved charge;

The charge is called dipole by analogy, as the magnetic charges are opposite at diametrically opposite parts of the ring;

Dipole charge of the 2s-th black ring:

Physical Characteristics: Dipole potential There exists locally a 2-form B such that

We can define a dipole potential associated to the 2s-th black ring

Explicit result:

Conclusion

We have generated an exact solution to the Maxwell-Einstein equations in 5D Kaluza-Klein spacetime describing sequences of dipole black holes with ring topology and Kaluza-Klein bubbles.

The solution is obtained by applying 2-soliton transformation using the vacuum bubble and black hole sequence as a seed solution.

We have examined how the presence of dipole charge influences the physical parameters of the solution.

Work in progress: derivation of the Smarr-like relations and the first law of thermodynamics.