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Dipole Black Ring and Kaluza-Klein Bubbles Sequences

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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

- 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).

- 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)

- Rod structure:
- Solution:
Elvang, Harmark, Obers (2005)

- 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)).

- 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

- 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

- 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

- Solution:
- gE is the metric of the seed solution

- Electromagnetic potential:
- α, β, A0φare constants

- 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

- 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

- 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

- 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:

- 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:

- There exists locally a 2-form B such that
- We can define a dipole potential associated to the 2s-th black ring
- Explicit result:

- 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.