Niels Obers- Charged Blackfolds

8/3/2019 Niels Obers- Charged Blackfolds

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Charged Blackfolds

Solvay Workshop, Brussels, May 20, 2010

Niels Obers, Niels Bohr Institute

0912.2352 (JHEP), 0910.1601 (JHEP) & 0902.0427 (PRL) + to appear

(with R. Emparan, T. Harmark, V. Niarchos)

0708.2181 (JHEP) (with R. Emparan, T. Harmark, V. Niarchos, M.J. Rodriguez)


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Basic idea and results

I basic idea:

take the fundamental black branes (or intersections/bound states)of ST/M-Theory and curve them into black holes with compacthorizon topologiesHere: blackfold limit (test-brane)

equilibrium: spinning, non-trivial backgrounds,

can include backreaction in perturbative expansion

For simplicity: focus on single-charge, asymptotically flatF1, NS5, Dp (type II) M2, M5 (M-theory)

but method is very general:

- generalize to multi-charge blackfolds- interesting extremal limits- other backgrounds

J more generally: can apply to any branes in effective SUGRA theories with

various p-form potentials


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Charged blackfolds in string theoryEmparan,Harmark,Niarchos,NO (in progress)


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Plan

Separation of scales for BHs in higher dimensional gravity

Effective worldvolume theory: Blackfold approach

Novel stationary charged blackfolds in ST/M-theory

Instabilities and correlated stability conjecture in blackfolds

Discussion and outlook


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Higher-dimensional gravity + separation of scales

J difficult to obtain exact black hole solutions in higher-dimensional gravity

theories (unless high degree of symmetry, SUSY, etc.)

! Dynamics of BHs in D 5 much richer than four dimensions D=4 : black hole uniqueness (in EM)

D=5: besides spherical black holes (S3) there are (dipole)black rings (S2 S1), black Saturn,

- 4D inspired techniques successful(assuming 2 axial Killing vector fields integrability )

D 6: MP black holes (SD-2 ) are only known exact solutions in Einstein grav.

- full dynamics too complex to be captured by conventional approaches

novel feature of higher D black holes:I in some regimes horizons are characterized by (at least) two separate scales

D=5R

ultraspinning black ringsR = radius of S1

r0 = radius of S2r0

radius of ring thickness of ring

Emparan,Reall


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Higher D black holes organized according to scales

I dynamics of higher-dimensional black holes naturally organized

in relative value of scales

single length scale: Kerr BH behavior

regime of mergers and connections between phaseswhen two horizon scales meet r0 R

- not accessible to effective methods;requires extrapolation or numerics

separation of scales allows effective descriptionof long-wavelength description physics

Based on idea that when black hole is locally a flat (possibly boosted)black brane (cf. known examples)

Effective theory describes how to bend black brane wv in background spacetime(similar to effective theories for other extended objects: cosmic strings, D-branes)

Blackfold = Black brane whose worldvolume extends alonga curved submanifold of background spacetime

- to leading order in : `test blackfold (neglect backreaction)


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Main ingredients + resultsEmparan,Harmark,Niarchos,NO

Ingredients:

- classical brane dynamics (Carter)- long wavelengths: dynamics of fluid that lives on dynamical worldvolume- (charged) black branes correspond to specific type of fluid

! to leading order: (charged) perfect fluid

blackfold equations

intrinsic (Euler equations of fluid

+ charge current conservation)extrinsic (generalized geodesic eqn. for

brane embedding)

gives novel stationary black holes + allows study of time evolution possible in principle to incorporate higher-derivative corrections

cf. closely related precedents of mappings black holes to fluid dynamics- membrane paradigm- fluid/AdS-gravity correspondence

notation: spacetime

worldvolume


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Neutral blackfoldsEmparan,Harmark,Niarchos,NO

Recently applied to neutral black branes of higher dim gravity:

Quick overview of results

-new helical black strings and rings

- odd-branes wrapped on odd-spheres

(generalizes 5D black ring)

- even-branes wrappend on even-ballscorrectly reproduce MP BHs inultraspinning (pancaked) limit

- non-uniform black cylinders

- static minimal blackfolds

(non-compact)

R


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Effective worldvolume theory collective coords

I similar to effective theories for other extended objects: cosmic strings, D-branes

difference: - short-distance d.o.f. = gravitational short-wavelength modes

- extended objects posses black hole horizon

J main clue: known black holes in limit ) flat black branes

- need collective field dynamics black boosted p-brane

- effective action from integrating out short-wavelength d.o.f.

- are `collective coordinates

- coordinates a

= (t,zi) span brane worldvolume

D collective coordinates (depending on wv coords )

positions in directionstransverse to worldvolume

horizon thickness velocity

charge


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Effective stress tensor

J validity of effective field approximation

length scale of theworldvolume Wp+1:

size-scale of the brane

I integrating out short-distance dynamics:= SUGRA eqs. solved at distances r R + effects at distances r r0encoded in a stress tensor (+charge current) that depends on collective coords.

introduce slow variation of collective coordinates:

perfect fluid form:

expected for anykind of brane


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Black p-branes in ST/M-theory

J action

black brane solutions:

characteristic length scale:

I (charged) perfect fluid with


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Blackfold dynamics

J general effective theory of classical brane dynamics

= theory of fluid on dynamical worldvolume

assume:- effective stress tensor derives from underlying conservative dynamics (GR)- spacetime diffeomorphism invariance

(consistent couping of wv. to long-wave length gravitational field

(stress tensor supported on worldvolume)

(using embedding tensors, extrinsiccurvature tensor etc. etc.)

(for particle worldline: )

extrinsic equations (D-p-1 )

intrinsic equations (p+1)

generalizedgeodesic equation

I stress tensor conservation:

I

charged case: current conservation:


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Blackfold equations + stationary blackfolds

I blackfold equations (stress tensor + current conservation):

blackfolds represent objects with horizon- reflected in effective theory in entropy and local thermodynamics

assume regularity of event horizon under long-wave length perturbations whenblackfold equations are satisfied

no rigorous derivation but significant evidence:- thin black rings, black tori(first correction computed using MAE = matched asymptotic expansion)

- cf. black branes in AdS

J equilibrium configurations stationary in time = stationary black holes

can solve blackfold equations explicitly for thickness, velocity + charge! only need to solve extrinsic equations for the embedding

velocityfield

blackfolds with boundaries: fluid approaches speed of light at bdry. (horizon closes off !)

determine the D+1 collective brane coords.


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Action principle for stationary blackfolds and 1st law

J for stationary blackfolds: extrinsic blackfold eqns can be intergrated to action

I for any embedding (not nec. solution) this action is prop. to Gibbs free energy:

varying G ) 1st law of thermodynamics

1st law of thermo , blackfold equations for stationary configurations

can also use Smarr relation to show that:

total tension vanishes for (asymptotically flat) stationary blackfolds

compute mass and angular momentum by integrating appropriatestress tensor components over brane worldvolume + entropy from total area


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Novel solutions: products of odd-spheres

horizon topology:

Novel family of charged blackfolds with horizon topology:

J wrap brane wv. on products of (round) odd spheres and spin equally in all directions

can be non-trivially fibered if blackfold has boundaries

I example: wrap F1-string on a circle of radius R \

black dipole ring (carries no net charge, but dipole charge)

(compare: exact 5D dipole rings (Emparan))

R


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Summary: odd-sphere charged blackfolds

new stationary black holes in ST/M-theory, w. novel horizon topology

new type of charge (generalizing dipole charge of ring) entering1st law of thermo (cf. Copsey, Horowitz)

(presumably) stable for sufficiently high charge

Generalizes 5D neutral black ring of Emparan,ReallUpper bound on charge for solution to exist

Smarr

For sufficiently high : positive specific heat


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Gregory-Laflamme instability

J blackfold approach captures perturbative dynamics of BH when

- can be intrinsic variations (thickness, local velocity) or extrinsic (embedding)! generally coupled

simple case: ! worldvolume looks flat

I for a general perfect fluid:

- decoupling between intrinsic/extrinsic

transverse (elastic) perturbations

longitudinal (soundmode) perturbations

charged blackfolds have:

J sound mode instability is long-wavelength part of GL instability !good agreement withslope of GL curve

cf. correlated stability conjecture (Gubser, Mitra)

for sufficiently

high charge


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Comparison to earlier effective prescriptions

extrinsic part: generalization to p-branes of effective world-line formalism forsmall black holes Poisson/Gralla,Wald/Goldberger,Rothstein/Kol,Smolkin

intrinsic part: similar to fluid-dynamical formalisms for horizon fluctuations- membrane paradigm, fluid/AdS gravity correspondence

J can apply blackfold formalism to near-extremal D3-brane

small scale is charge radius rq of D3-brane- in overlap-zone: metric is flat up to corrections rq/R

can take Maldacena limit to decouple far-zone effects from near-horizon excitations

region is asymptotic to AdS5 S5 + far-zone effects absent

(would change boundary geometry)

integrating d.o.f. in asymptotic AdS region ! intrinsic (pure hydrodynamic)

collective modes only

I no extrinsic worldvolume dynamics, but:

- simpler to compute higher derivative corrections to perfect fluid dynamics- charge near extremality eliminates sound mode instability

- describes hydrodynamic regime of strongly-coupled YM


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Further Outlook

charged blackfolds with multiple charges/extremal limits- develop general theory of anistropic p-form charged fluids

- opens up new interesting dynamics (in progress)(cf. supertubes)

method can also be applied to blackfolds in other backgrounds (AdS, dS)+ turning on other fields- black rings in (A)dS

stability analysis

relation with DBI

relation to fluid/gravity correspondence

Caldarelli,Empran,Rodriguez

duality of higher D black holes to plasma balls + rings in SS AdS

(cf. Lahiri,Minwalla) many similar features

Emparan,Harmark,Niarchos,NO

higher-order analysis (via MAE/ClEFT)

SUSY blackfolds ?- extremal black holes and black rings

cf. 5D supersymmetric black ringElvang,Emparan,Mateos,Reall

Figueras,Kunduri,Lucetti,Rangamani

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Niels Obers- Charged Blackfolds