XVI European Workshop on Strings Theory

Madrid– 14 June 2010

arXiv:0909.0008 [hep-th]

arXiv:0909.3852 [hep-th]

Dumitru Astefanesei, MJR, Stefan Theisen

Dumitru Astefanesei, Robert B. Mann, MJR, Cristian Stelea

Maria J. Rodriguez

Thermodynamic instabilityThermodynamic instability

ofof

doubly spinning black objectsdoubly spinning black objects

&1003.2421[hep-th]

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Motivation

Black holes are the most elementary and fascinating

objects in General Relativity

In their presence the effects of the space-time curvature are dramatic

In string theory, mathematics and recent cutting edge experiments black objects are also relevant.

The study of the properties of higher dimensional black

holes is essential to understand the

dynamics of space-time

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Asymptotically flat

On the BH species (by means of natural selection)

Vacuum Einstein´s equation

Boundary conditions

EquilibriumStationary – no time dependence

Regular solutions on and outside the event horizon

Rµν=0

[*]

[*] We start from a five dimensional continuum which is x1,x2,x3,x4,x0. In it there exists a Riemannian metric with a line element ds2= gμν dxμdxν

μ, ν = 1,2,…,D

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On the BH species (by means of natural selection)

S5

SD-2

S4

S3

S2

The boundary stress tensor satisfies a local conservation law

Summary of neutral D-dim BHs classified by its horizon topologies

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In D>4 many black holes have been found:

Motivation

Galloway+SchoenTopology: Rigidity : Hollands et al

In D=4 stationary black holes are spherical and unique

The main feature of high-D is the richer rotation dynamics

Study thermodynamical properties of spinning BH in D-dim to learn how these solutions connect.

Our goal:

NEW FEATURE

NEW FEATURE

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One&Two angular momenta + Vacuum + Asymtotically flat

Phase diagram of black objects

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What do we know about black

objects?

In D=4 dimensions

In D=5 dimensions

-Kerr black hole

In D>5 dimensions

BH w/ one J in D-dim

-the Myers-Perry black hole

- the Myers-Perry black hole

aH

j

It seems that there is an infinite number of BHs.

aH

j2

1

aH

j1

-black ring

- thin black ring and black saturn

j12

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The generalization of the black hole solution with ANY # of angular momenta is the Myers-Perry (MP) solution.

singular

The gray curve is the phase of zero temperature BH’s

Representative phase of MP-BHs with one of the two angular momenta fixed

BH w/ two J in D-dim

D=5

The dashed lines show MP for fixed values of =0.1,0.3,0.5 right to leftj2

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The generalization of the black ring with TWO angular momenta is known.

BR w/ two J in D-dim D=5

The dark gray curve is the phase ofzero temperature BR’s

Representative phase of the doubly spinning BR with the S2 angular momenta fixed

The dashed lines show BR for fixed values of (right towards left)

The black dashed curve is the phase ofzero temperature MP BH’s

The angular momenta are bounded The fat ring branch disappears for

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What do we know about

these black objects?

In D=5 dimensions

In D>5 dimensions

-Myers-Perry Black Hole (BH)

-Black Ring (BR)

-Myers-Perry Black Hole -Black Ring (BR)

BH w/ two J in D-dim

-Helical BH-Black Saturn-Bicycling BR

j2 is fixed

-Helical BH-Black Saturn-Bicycling BR-Blackfolds

aH

j1

j2 is fixed

aH

j11

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Why are we interested doubly spinning solutions?

Black Holes with T=0 are interesting because they can teach us

about the microscopic origin of their physical properties

Black Holes with T=0 are interesting because they can teach us

about the microscopic origin of their physical properties

SUSY

Asymptotically flat

Non SUSY

T=0

BH w/ two J in D-dim

Doubly spinning black rings, in contrast to the singly spinning black rings, can be extremal.

Doubly spinning black rings, in contrast to the singly spinning black rings, can be extremal.

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One&Two angular momenta + Vacuum + Asymtotically flat

Ultra-spinning black objects

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Ultra-spinning black objects

R

Balance conditionParameters in the solution

where

S1 x SD-3

R

Thin Black ring

R ∞

SD-2

+

-

+

-

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Thin Blackring/fold

Recently the matched asymptotic expansion has been applied to solve Einstein´s equations to find thin Black ring/folds in D>4 dimensions

The basic idea

Black ring Black string

≡R ∞

R

roro

ro

Emparan + Harmark + Niarchos + Obers + MJR 0708.2181 [hep-th]

Having a better understanding of the properties of BO may be useful to construct

new solutions

Thin

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Black Holes and black rings in ultra-spinning regime will inherit the instabilities.

In certain regimes black holes and black rings behave like black strings and black p-branes.

Ultra-spinning black objects

Black strings and branes exhibit Gregory-Laflamme instability

Gubser + WisemanBranch of static lumpy black strings

A black hole solution which is thermally unstable in the grand-canonical ensemble will develop a classical instability.Gubser + Mitra

Emparan + Myers

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Q1: If black objects are thermally unstable in the grand-canonical ensemble for jth does this imply that there they are classically unstable?

Instabilities from thermodynamics

But to investigate this and where the threshold of the classical instability is one has to perform a linearized analysis of the perturbations.

Q2c: Is there any relation between zeros of eigenvalues of Hess(G) and jm?

Q2: What information can we get from the study of the thermodynamical instabilities?

We can establish a membrane phase signaled by the change in its thermodynamical behavior which could imply the classical instability.

Q2b: Which is the threshold of the membrane phase, jm?

We can study the zeros of the Hess(G) which seem to be linked to the classical instabilities

Q2a: How to establish the membrane phase?

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Thermodynamics of black objects

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Which ensemble is the most suitable for this analysis?

Entropy – microcanonical ensemble

Thermal ensembles

Gibbs potential – grand canonical ensemble

Enthalpy Helmholtz free energy – canonical ensemble

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Due to the equivalence principle, there is no local definition of the energy in gravitational theories

Basic idea of the quasilocal energy: enclose a region of space-time with some surface and compute the energy with respect to that surface – in fact all thermodynamical quantities can be computed in this way

For asymptotical flat space-time, it is possible to extend the quasilocal surface to spatial infinity provided one incorporates appropriate boundary (counterterms) in the action to remove

divergences from the integration over the infinite volume of space-time.

r =co

nst.

Brown + York gr-qc/9209012

Mann + Marolf

Quasilocal thermodynamics

Compute directly the Gibbs-Duhem relation

by integrating the action supported with counterterms.

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Instabilities from Thermodynamics

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Thermal stability

In analogy with the definitions for thermal expansion in the liquid-gas system, the specific heat at a constant angular velocity, the isothermal compressibility, and the coefficient of thermal expansion can be defined

The conditions for thermal stability in the grand-canonical ensemble

or

What do we know about the thermal stability of black objects?

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Black hole thermal stability

Monterio + Perry + Santos 0903.3256[gr-qc]

The response functions are positive for different values of the parameters implying there is no region in parameters space where both are simultaneously positive.

The black holes is thermally unstable, both in the canonical and grand-canonical ensembles.

CompressibilityHeat capacity

Singly rotating Myers-Perry black hole

For doubly spinning MP-BH the response functions are positive for different (complementary) regions of the parameter space implying its

instability.

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Black ring thermal stability

The black ring is thermally unstable, both in the canonical and grand-canonical ensembles.

The CΩ→0 as T→0 which is expected and can be drawn from Nernst

theorem.

Heat capacity Compressibility

Singly spinning black ring

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We investigated the stability of the doubly spinning black ring

The doubly spinning black hole

and the singly spinning black ring

are thermally unstable in the grand-canonical

ensembles.

A second rotation could help to stabilize the solution

Doubly spinning black ring

What about the thermal stability of the doubly spinning black ring?

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Doubly spinning black ring

The grand canonical potential for doubly spinning black ring (using the quasi local formalism)

The Hessian should be negatively defined

The doubly spinning black ring is local thermally unstable.

where

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Critical points & turning points

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These points should not be considered as a sign for an instability or a new branch but a transition to an infinitesimally nearby solution along the same family of solutions.

Instabilities from thermodynamics

The instabilities and the threshold of the membrane phase of the singly spinning MP BH are 0

D=5

D=10

D=6

D=5D=10

D=6

Numerical evidence supports this connection with the zero-mode perturbation of the solution.

Note that the relation between ensembles is not in general valid.

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Indicates where the transitionto the black membrane phase.

More general black holes with N spins ultra-spin iff

Critical points: MP BH

jm

Black holes with one spin

0 where

Are there other ultra spinning MP black holes?

And for

and

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The existence and location of the threshold of this regime is signaled by the minimum of the temperature and the maximum

angular velocity as functions of the angular momentum.

The transition to a membrane-like phase of the rapidly spinning black holesis established from the study of the thermodynamics of the system.

where for the ultra spinning MP BH

Critical points: MP BH

while the angular velocity reaches its maximum value.

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But let´s take a closer look to the Hessian, which has to be negatively defined,

Do the zeros of the eigenvalues of this Hessian have any physical interpretation?

We´ve checked that at least one of the eigenvalues of the Hess[G] is zero.

Critical points: MP BH

And also checked that the Ruppeiner curvature pinpoints thezero of the determinant of the Gibbs potential’s hessian

These points seem to be related to the classical instabilities.

is the so called Ruppeiner metric

The Ruppeiner metric measures the complexity of the underlying statistical mechanical model

A curvature singularity is a signal of critical behavior.

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Turning points : BR

We´ve checked that at least one of the eigenvalues of the Hess[G] is zero there.

λ=0.5 At the cusp in s vs j

In this case the temperature does not have a minimum, but there exists a turning point and

plays a similar roleas the minimum of the temperature for the BH

The Ruppeiner curvature diverges.

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Indicate where the transition to the thermodynamical black membrane phase.

λ [ν] At the cusp in s vs j

No eigenvalue of the Hess[G] is zero there.

Turning points : BR

Particular BR solutions with jψ > 1/5 fall into the same category as other black holes with no membrane phase as the four dimensional Kerr black

hole and the five dimensional Myers-Perry black hole.

I

III

I

III

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Summary and outlook

It would be interesting to investigate numerically whether these correspond to the zero-mode perturbations.

We showed ,in parameter space, that doubly spinning black rings are thermally unstableFound the thresholds of the transition to the black membrane phase of black holes and black rings with at least two spins.Identified particular cases of doubly spinning BR with no membrane phase

Study the ultraspinning behavior of multi black holes, such as the bicycling black ring and saturn, which can be relevant in finding new higher dimensional multi

black hole solutions.

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In five dimensions stationary implies axisymmetric

To calculate the physical quantities we employ the complex instanton method

The ADM decomposition of the full spacetime

We can write (B) in the (A) form

The Wick transformation changes the intensive variables

but not the extensive ones

(B)

(A)

Lapse function Shift function

Angular velocity Temperature

Quasilocal thermodynamics

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Black String

=0The stress energy tensor is

conserved for any value of the parameters

Observe that

=0Corresponds to the thin black ring limit

Boundary stress energy tensor for black strings

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To compare solutions we need to fix a common scale Classical GRTo compare solutions we need to fix a common scale Classical GR

We'll fix the mass M equivalently and factor it out to get dimensionless quantities

We'll fix the mass M equivalently and factor it out to get dimensionless quantities

aHaH j2j2 ωω ……

Disconnected compact horizons: multi horizon black hole solutions

One compact horizons: uni horizon black hole solutions

On the number of angular momenta

On the number of horizons

Maximum # angular momenta:

On how we compare solutions

Compare by drawing diagrams i.e. a phase diagram

aH

j2

j1 j2

Jargon and reminder

j[(D-1)/2]. . .

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Outline

Introduction

Thermodynamics Instabilities from Thermodyn.

Summary and outlook

Motivation

- Ultra-spinning BH

- BH solutions in D-dim

- Membrane phase

- Critical points & turning points

- Thermal ensembles

- Thermodynamic stability

- Grand-canonical ensemble

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Ultra spinning multi BH