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)

• solitons (3+1 dimensions; finite energy, localized, regular, particle-like configurations)

• 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

(approximate constructions, however: Carter’s formalism) L1>>L2

• No exact solutions (D=5 special: Weyl formalism, Laplace equation)• Approximate construction only:

thin black rings ~ circular boosted black strings• one expects similar properties to D=5

• Numerical solutions: (Kleihaus, Kunz, Radu)• nonperturbative approach: one can find solutions beyond the

blackfold approximation• The simplest solutions are found within the metric ansatz:

Equations:

• the result of numerical integration: new D>5 solutions with nonspherical horizon topology

• rod structure (useful in D=4,5):- generalization to D>5 => boundary conditions for PDEs

Schwarzschild black hole generalized black ring

• Results arXiv: 0904.2723, 1010.2898:• higher dimensional generalizations of static black

rings, black saturn and black diring• the properties of the D=5 configurations are

generic• all static solutions have conical singularities• generalizations with U(1)+scalars: the same

pathologies

However, rotation may lead to regular configurations (see D=5)

• arXiv: 1010.2898 : an asatz for rotating solutions+numerical results

• much more difficult numerical problem• dependence of at least three coordinates• no balance solutions yet..

D>5 static solutions

Conclusions & Open issues• black hole physics in D>4: new features

-- unexpected configurations (no black hole uniqueness ! )– the black ring

• Black ring construction in KG theory => vortons

the dictionary: possible new tool

• black ring—vorton in a different background- black rings with negative cosmological constant :

• rings: approximate solutions only • vortons in AdS space (to appear)

(some different properties )- static rings/vortons in dS space?- toroidal solutions in Melvin universe

• “blackfold approach” for solitons?

• Real physics: vortons in the standard model?Weinberg-Salam model:

with

there are:• vortex solutions• (bi)sphalerons:

two complex scalar fields

vortons: stable solutions?

Numerical approach:• input: position of the horizon(s)

+ boundary conditions• output: metric functions

=> relevant quantities

test the numerics: • reproduce known D=5 solutions• Smarr law

scalar functions for a typical vortons

Outline: GR in five dimensions:

The black ring

Toroidal solutions in D=4 flat space:

Vortons

Towards a dictionary

Higher dimensional generalizations

Conclusions and open problems