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. Outline.

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Dipole Black Ring and Kaluza-Klein Bubbles Sequences

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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

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

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

Vacuum Kaluza-Klein bubble and black hole sequences

  • Rod structure:

  • Solution:

    Elvang, Harmark, Obers (2005)


Vacuum kaluza klein bubble and black hole sequence

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

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

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 sequences1

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 sequence1

Charged Kaluza-Klein bubble and black hole sequence

  • Solution:

  • gE is the metric of the seed solution


Charged kaluza klein bubble and black hole sequences2

Charged Kaluza-Klein bubble and black hole sequences

  • Electromagnetic potential:

  • α, β, A0φare constants


Charged kaluza klein bubble and black hole sequences3

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 sequences4

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

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

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

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

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

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.


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