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Dipole Black Ring and KaluzaKlein 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.