Dipole Black Ring and Kaluza-Klein Bubbles Sequences

1. 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

2. 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).

3. 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)

4. Vacuum Kaluza-Klein bubble and black hole sequences • Rod structure: • Solution: Elvang, Harmark, Obers (2005)

5. 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)).

6. 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

7. 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

8. 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

9. Charged Kaluza-Klein bubble and black hole sequence • Solution: • gE is the metric of the seed solution

10. Charged Kaluza-Klein bubble and black hole sequences • Electromagnetic potential: • α, β, A0φare constants

11. 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

12. 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

13. 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

14. 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:

15. 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:

16. 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:

17. 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.