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
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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).

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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)

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Vacuum Kaluza-Klein bubble and black hole
sequences

• Rod structure
• Solution
• Elvang, Harmark, Obers (2005)

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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)).

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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

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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

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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 gff

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Charged Kaluza-Klein bubble and black hole
sequence

• Solution
• gE is the metric of the seed solution

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Charged Kaluza-Klein bubble and black hole
sequences

• Electromagnetic potential
• a, ß, A0f are constants

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Charged Kaluza-Klein bubble and black hole
sequences

• W and Y are regular functions of ?, z, provided
  that
• the parameters of the 2-soliton transformation
  k1 and k2 lie on a bubble rod
• the parameters a, ß satisfy
• ? The rod structure of the seed solution is
  preserved

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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, (?F)E is the period for the seed
  solution

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Physical Characteristics Mass

• The total mass of the configuration MADM is the
  gravitational energy
• enclosed by a 2D sphere at spatial infinity
  of M4
• 
  
  
• 
  
  ? ?/?t, ? ?/?f
  
  
  
  
• 
  
• 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 f slice of the black hole
  horizon
• ? The same relations hold for the seed solution

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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

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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

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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

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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.