スライド 1 - I C R A
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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.