Dipole Black Ring and Kaluza Klein Bubbles Sequences

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 n We will consider an exact static axisymmetric solution to the Einstein-Maxwell equations in 5 D Kaluza-Klein spacetime (M 4 × S 1) n Related solutions: Ø Ø Ø R. Emparan, H. Reall (2002) H. Elvang, T. Harmark, N. A. Obers (2005); H. Iguchi, T. Mishima, S. Tomizawa (2008 a); S. Tomizawa, H. Iguchi, T. Mishima (2008 b).

Spacetime Bubbles n Bubbles are minimal surfaces that represent the fixed point set of a spacelike Killing field; n 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 n Elvang, Horowitz (2002)

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

Vacuum Kaluza-Klein bubble and black hole sequence n Properties: Ø Conical singularities can be avoided; Ø Bubbles hold the black holes apart → multi-black hole spacetimes without conical singularities; Ø Small pieces of bubbles can hold arbitrary large black holes in equilibrium; n Ø Ø 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 n Further generalization: charged Kaluza-Klein bubble and black hole sequences n Field equations: 2 spacelike + 1 timelike commuting hypersurface orthogonal Killing fields Static axisymmetric electromagnetic field n Gauge field 1 -form ansatz

Charged Kaluza-Klein bubble and black hole sequence n n 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 n The difficulty is to solve the nonlinear Ernst equation → 2 -soliton Bäcklund transformation to a seed solution to the Ernst equation E 0 n Natural choice of seed solution → the vacuum Kaluza-Klein sequences metric function gφφ

Charged Kaluza-Klein bubble and black hole sequence n Ø Solution: g. E is the metric of the seed solution

Charged Kaluza-Klein bubble and black hole sequences n Electromagnetic potential: n α, β, A 0φ are constants

Charged Kaluza-Klein bubble and black hole sequences n W and Y are regular functions of ρ, z, provided that: Ø the parameters of the 2 -soliton transformation k 1 and k 2 lie on a bubble rod; Ø the parameters α, β satisfy → The rod structure of the seed solution is preserved

Charged Kaluza-Klein bubble and black hole sequences Ø It is possible to avoid the conical singularities by applying the balance conditions on the semi-infinite rods on the bubble rods Ø L is the length of the Kaluza-Klein circle at infinity, (ΔΦ)E is the period for the seed solution

Physical Characteristics: Mass n The total mass of the configuration MADM is the gravitational energy enclosed by a 2 D sphere at spatial infinity of M 4 ξ = ∂/∂t, η= ∂/∂φ n 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 n Spacetimes that have spacelike translational Killing field which is hypersurface orthogonal possess additional conserved charge – tension. n Tension is associated to the spacelike translational Killing vector at infinity in the same way as Hamiltonian energy is associated to time translations. n Tension can be calculated from the Komar integral: n Explicit result:

Physical Characteristics: Charge n The solution possesses local magnetic charge defined as n The 1 -form A is not globally defined → Q is not a conserved charge; n The charge is called dipole by analogy, as the magnetic charges are opposite at diametrically opposite parts of the ring; n Dipole charge of the 2 s-th black ring:

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

Conclusion n We have generated an exact solution to the Maxwell-Einstein equations in 5 D Kaluza-Klein spacetime describing sequences of dipole black holes with ring topology and Kaluza-Klein bubbles. n The solution is obtained by applying 2 -soliton transformation using the vacuum bubble and black hole sequence as a seed solution. n We have examined how the presence of dipole charge influences the physical parameters of the solution. n Work in progress: derivation of the Smarr-like relations and the first law of thermodynamics.