Eugen Radu Dublin Institute for Advanced Studies & National University of Ireland Maynooth, Ireland Based on: • Phys. Rep. 468, 101, 2008 (with M. Volkov) • JHEP 1102:058, 2011 (with B. Kleihaus and J. Kunz) • + work in progress... Supersymmetry in Integrable Systems - SIS'11
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Eugen RaduDublin Institute for Advanced Studies
& National University of Ireland Maynooth, Ireland
Based on: • Phys. Rep. 468, 101, 2008 (with M. Volkov)
• JHEP 1102:058, 2011 (with B. Kleihaus and J. Kunz)• + work in progress...
Supersymmetry in Integrable Systems - SIS'11
4D vs. higher D Black Holes(asymptotically Minkowski solutions only!)
GR in four dimensions• The topology of the horizon: a sphere S2
• “no hair” theorems• Kerr(-Newman) black hole (uniqueness)
why study GR in higher dimensions?• Dimension - a parameter of GR: interesting mathematical problem• String theory• Large extra dimensions
novel feature –non-uniqueness of black objects• Uniqueness is special to 4D
D=5 solutions in GR• Myers-Perry black holes (1986):
- the horizon is a sphere S3 (Sd-2 in the general case)- natural counterparts of the Kerr black hole
however...
• Emparan-Reall black ring (2001):- known only in D=5 (approximate construction for D>5)- the horizon is a ring: S2xS1
- however, S3 at infinity (analogy with caged black holes in Kaluza-Klein theory)
- the most important exact solution found after Kerr metric
The forging of the D=5 ring:
• there's an exact solution (Emparan-Reall hep-th/0110260)• explicit realisation of the heuristic construction! (large ring radius)• static black rings: unphysical (conical singularities => extra-source etc)
Schwarzschild black hole in 3+1 dimensions: black string in (3+1)+1 dimensions
Black rings:
The Emparan-Reall solution (ring coordinates):
(no conical singularities)
One-black hole phases in D=5:
•three different black holes with the same value of (M,J)•non uniqueness!•minimal J
(single J)
Black rings: generalizations
• Einstein-Maxwell black rings: dipoles as global charges • Einstein-Maxwell-Chern-Simons black rings:
- supersymmetric configurations- D=10 supertubes
• Black rings with J1,J2 (Pomeransky-Senkov)generic feature:
- no relevant static solutions! (tipical: conical singularities)
(active field of research – “the Fellowship of the Ring“)
Multi-black objects: • black hole+ black ring = black Saturn
•black ring+ black ring = black diring etc- Weyl formalism: exact solutions, balanced by rotation- still work to be done...
• also nonextremal solutions! (different from D=5)
Solitons and Vortices: D=4, no gravity
Field theory solutions in flat space (stationary; four dimensions, no quantization)
• vortex/string solutions ( (2+1)+1 dimensions;infinite extend; finite energy per unit length)
examples: • vortices in Abelian-Higgs theory • vortices and Q-balls in Klein-Gordon theory• sphalerons in standard model• monopoles in Georgi–Glashow model
Vortex+soliton = vorton
Vortons in field theory:heuristic construction
- applications: astrophysics, condensed matter, nuclear physics (Skyrme model)- internal structure of particles: - knots (old idea – lord Kelvin)
general formalism (large radius): before black rings – B. Carter
The simplest model (Witten 1985):
• Two interactingscalar fields:
•Potential:
numerical solutions only!(Radu and Volkov; Battye and Sutcliffe; Grandclement; Garaud)
- solve a set of coupled, nonlinear PDEs with suitable BCs- Witten’s model: three PDEs- test the numerics: virial relations
The energy density and a surface of constant energy densityfor a typical vorton
- experimental detection of vortons? Standard model?
Black rings-vortons: a comparison• The same heuristic construction• Both objects supported by rotation (minimal J)• Replace rotation by U(1) interaction?
- no balanced static black rings (conical sings.)
- no finite energy static vortons with Maxwell fields (to appear)
• Other matter fields/interactions? (e.g. black rings in EGB theory; nonabelian vortons)
black ring – vorton dictionary ?
qualitative picture…
• vorton equation of state = Smarr law for black rings• domain of existence• vortons: scalar potential => more parameters
Common features:•Nonuniqueness: two branches of solutions•Minimal value of J
horizon = maximal value of E
two length scales R1, R2
However..
• some black rings may be stable (no proof yet…)- stable vortons?- Gregory-Laflamme instability?
• new branches of non-axisymmetric solutions?• vortons?
new D=4 field theory composite solutions?
( vortices + solitons):
(analogy with gravity!)
+ =
,
• However, there are differences:- the vorton’s angular momentum is quantized- inner horizon structure of a black ring: no analogue
orthogonal rings:
• Blackfolds (Emparan 2009): D>5 BH solutions with more complicated horizon topology