The attractor mechanism, C-functions and aspects of holography in Lovelock gravity Mohamed M. Anber...

The attractor mechanism,The attractor mechanism, C-functions and aspects of C-functions and aspects of

holography in Lovelock gravityholography in Lovelock gravity

Mohamed M. AnberMohamed M. AnberNovember 27 2007November 27 2007

HET bag-lunchHET bag-lunch

OutlineOutline

Introduction: the attractor mechanismIntroduction: the attractor mechanism Lovelock gravityLovelock gravity The attractor mechanism in Gauss-Bonnet The attractor mechanism in Gauss-Bonnet

gravity in 5-Dgravity in 5-D Entropy and C-function in Einstein gravityEntropy and C-function in Einstein gravity Entropy and C-functions in Lovelock gravityEntropy and C-functions in Lovelock gravity Covariant formulation and Raychadhuri’s Covariant formulation and Raychadhuri’s

equationequation

IntroductionIntroduction

Black holes radiate: Bekenstein-Hawking entropyBlack holes radiate: Bekenstein-Hawking entropy

Many approaches to count the number of states: still Many approaches to count the number of states: still open question open question

Quantum gravity may be the resolution to this problemQuantum gravity may be the resolution to this problem String theory may be the way toward quantum gravity: String theory may be the way toward quantum gravity:

black holes in string theoryblack holes in string theory


Black holes in string theory: string theory or M-theory Black holes in string theory: string theory or M-theory Compactified to lower dimensions, Torous, Calabi Yau Compactified to lower dimensions, Torous, Calabi Yau

The right entropy for SUSY black holes!!The right entropy for SUSY black holes!! Calabi Yau Moduli fields: massless and Calabi Yau Moduli fields: massless and

dangerous. dangerous. Far from the BH, moduli can take a range of Far from the BH, moduli can take a range of

continuous values . continuous values . CAN ENTROPY DEPEND ON CAN ENTROPY DEPEND ON THIS BIZARRE BEHAVIOUR?THIS BIZARRE BEHAVIOUR?

The answer is No!!, Resolution is the attractor The answer is No!!, Resolution is the attractor mechanismmechanism

Near horizon Near horizon


What is the attractor mechanism? What is the attractor mechanism? All moduli fields are attracted to the same value at the All moduli fields are attracted to the same value at the

horizon irrespective of their values at asymptotic infinity.horizon irrespective of their values at asymptotic infinity. Entropy depends only on few parameters : Mass , angular Entropy depends only on few parameters : Mass , angular

momentum and not on the value of these moduli at infinitymomentum and not on the value of these moduli at infinity

Attractor position

Damped pendulum


Is there a similar behavior for the non-supersymmetric Is there a similar behavior for the non-supersymmetric case? Yes!!case? Yes!!

Proven for classical Einstein gravity in 4-D and 5-D.Proven for classical Einstein gravity in 4-D and 5-D.

OutlineOutline

Introduction: the attractor mechanismIntroduction: the attractor mechanism Lovelock gravityLovelock gravity The attractor mechanism in Gauss-Bonnet The attractor mechanism in Gauss-Bonnet


equationequation

Lovelock gravityLovelock gravity

Possibility for higher dimensional space!!Possibility for higher dimensional space!! The most general second order gravity in higher The most general second order gravity in higher

dimensional space.dimensional space.

It contains Gauss-Bonnet term: the result of It contains Gauss-Bonnet term: the result of compactifying certain string theories.compactifying certain string theories.


Pure Lovelock of order mPure Lovelock of order m

Einstein GravityEinstein Gravity

Not all terms survive in a given dimension: D=5, only Not all terms survive in a given dimension: D=5, only m=2 (Gauss-Bonnet) survive, m=3 is a topological m=2 (Gauss-Bonnet) survive, m=3 is a topological termterm


Equation of motionEquation of motion

General Lovelock gravityGeneral Lovelock gravity: : sumsum over all mover all m

OutlineOutline

Introduction: the attractor mechanismIntroduction: the attractor mechanismLovelock gravityLovelock gravity The attractor mechanism in Gauss-Bonnet The attractor mechanism in Gauss-Bonnet


equationequation

The attractor mechanism in The attractor mechanism in Gauss-Bonnet gravityGauss-Bonnet gravity

M. Anber and D. Kastor M. Anber and D. Kastor JHEP 0710:084,2007JHEP 0710:084,2007

Phenomenological Lagrangian Phenomenological Lagrangian

Spherically symmetric solutionSpherically symmetric solution


Equations of motionEquations of motion

Point like electric chargePoint like electric charge


Effective potentialEffective potential

Moduli field equationModuli field equation

A solution: constant A solution: constant

V_eff

\phi


Attractor: positive Attractor: positive

The procedure for testing the attractorThe procedure for testing the attractor

1-Start with

2-Find black hole solution using


3-Use perturbation theory to find the perturbed solution for the

moduli fields about

near the horizon

4-Use the perturbed Value of the moduli as

a source to theCorrection of a and b


5-Use numerical technique to test if the solution is

singularity free up to infinity


Black hole solution atBlack hole solution at

ExtremalExtremal : near horizon : near horizon

Specific model of the potentialSpecific model of the potential


Perturbation of Perturbation of

Same attractor behavior for a(r) and b(r)Same attractor behavior for a(r) and b(r)


Numerical ResultsNumerical Results



Non-ExtremalNon-Extremal black hole: black hole: No Attractor !!No Attractor !!

OutlineOutline

Introduction: the attractor mechanismIntroduction: the attractor mechanismLovelock gravityLovelock gravityThe attractor mechanism in Gauss-Bonnet The attractor mechanism in Gauss-Bonnet


equationequation

Entropy: Revisited Entropy: Revisited

Entropy in Lovelock gravity (Myers and Jacobson 1993)Entropy in Lovelock gravity (Myers and Jacobson 1993)

Any possible connection with quantum field theory?Any possible connection with quantum field theory? ‘‘t Hooft and Susskind , Holographic principle in t Hooft and Susskind , Holographic principle in Einstein Einstein

gravitygravity ( ( Given a closed surface, we can represent all that Given a closed surface, we can represent all that happens inside it by degrees of freedom on this surface happens inside it by degrees of freedom on this surface itself.)itself.)

Manifestation of the holographic principle AdS/CFT Manifestation of the holographic principle AdS/CFT (Maldacena 1998)(Maldacena 1998)

Entropy: RevisitedEntropy: Revisited

Conformal description of horizon’s statesConformal description of horizon’s states (Solodukhin 1999)(Solodukhin 1999)

1-1-

2-2-

3-3-

4- Use the near horizon coordinates (x-x_h)4- Use the near horizon coordinates (x-x_h)

5- The resulting near horizon theory is conformal5- The resulting near horizon theory is conformal

Entropy: RevisitedEntropy: Revisited

6-Use the light cone coordinates6-Use the light cone coordinates

7- Define Virasoro generators7- Define Virasoro generators

8- Calculate Poisson’s bracket8- Calculate Poisson’s bracket

9- quantize the calculations 9- quantize the calculations

10-10-

11-extension to Lovelock gravity (Cvitan, Pallua and Prester 2002)11-extension to Lovelock gravity (Cvitan, Pallua and Prester 2002)

C-functions in 2-D field theoriesC-functions in 2-D field theories

C-functions in the renormalization group flow in 2-C-functions in the renormalization group flow in 2-D quantum field theoriesD quantum field theories (Zamolodchikov 1986)(Zamolodchikov 1986)

C-function is a function of the coupling of the theory that is C-function is a function of the coupling of the theory that is monotonically increasing with energy.monotonically increasing with energy.

For fixed points of the flow, corresponding to the extrema For fixed points of the flow, corresponding to the extrema of this function, the C-function reduces to the central of this function, the C-function reduces to the central charge of Virasoro algebracharge of Virasoro algebra

E

C

Holographic C-functionsHolographic C-functions

AdS/CFT (Avarez, Gomez 1999, Susskind and Witten AdS/CFT (Avarez, Gomez 1999, Susskind and Witten 1998)1998)

r

AdS

C(r )

C-functions in asymptotically flat C-functions in asymptotically flat Einstein gravityEinstein gravity

C-functions in spherically symmetric and C-functions in spherically symmetric and asymptotically flat spacetimeasymptotically flat spacetime (Goldstein et al 2006) (Goldstein et al 2006)

C-function (null energy condition is satisfied)C-function (null energy condition is satisfied)

C-functions in asymptotically flat C-functions in asymptotically flat Einstein gravityEinstein gravity

Conditions for the C-functionConditions for the C-function

11-It can be evaluated on any spherical surface concentric with -It can be evaluated on any spherical surface concentric with

The horizonThe horizon

22-When evaluated on the horizon of a black hole it equals its -When evaluated on the horizon of a black hole it equals its entropyentropy

33-If certain physical conditions and certain boundary -If certain physical conditions and certain boundary conditions are satisfied, then C is a non-decreasing conditions are satisfied, then C is a non-decreasing function along the outward radial directionfunction along the outward radial direction

Can we find similar functions in Lovelock gravity?Can we find similar functions in Lovelock gravity?

OutlineOutline


gravity in 5-Dgravity in 5-DEntropy and C-function in Einstein gravityEntropy and C-function in Einstein gravity Entropy and C-functions in Lovelock gravityEntropy and C-functions in Lovelock gravity Covariant formulation and Raychadhuri’s Covariant formulation and Raychadhuri’s

equationequation

C-function in Lovelock gravity C-function in Lovelock gravity (pure)(pure)

(M. Anber and D. Kastor, in progress)(M. Anber and D. Kastor, in progress)

Spherically symmetric metric in D=n+2 dimensionsSpherically symmetric metric in D=n+2 dimensions

Particular combinationParticular combination


we obtainwe obtain

Constraints : only local maxima, asymptotically flat.Constraints : only local maxima, asymptotically flat. Result: b(r) is monotonicResult: b(r) is monotonic


But the C-function has to reduce to entropy when But the C-function has to reduce to entropy when evaluated on horizonevaluated on horizon

C-function of the first kindC-function of the first kind


C-function of the second kind!!C-function of the second kind!!

Proof outline:Proof outline:

1- take the derivative w.r.t r and use equations of motion 1- take the derivative w.r.t r and use equations of motion to simplify the resultto simplify the result

2- Existing of extrema require that one finds a solution for 2- Existing of extrema require that one finds a solution for dC/drdC/dr

3-There is no solution (m=even!!)3-There is no solution (m=even!!)

C-function in Lovelock gravity C-function in Lovelock gravity (general)(general)

General C-functions of the first kindGeneral C-functions of the first kind

Proof of monotonicity:Proof of monotonicity:

No solution for C’=0.No solution for C’=0.


General C-functions of the second kind: Difficult to prove General C-functions of the second kind: Difficult to prove the monotonocity for general theory (general polynomial)the monotonocity for general theory (general polynomial)

We can proove the monotonicity for Gauss-Bonnet gravityWe can proove the monotonicity for Gauss-Bonnet gravity


Physical interpretation of two different C-functions!!Physical interpretation of two different C-functions!! More C-Functions are possible??More C-Functions are possible??

OutlineOutline


gravity in 5-Dgravity in 5-DEntropy and C-function in Einstein gravityEntropy and C-function in Einstein gravityEntropy and C-functions in Lovelock gravityEntropy and C-functions in Lovelock gravity Covariant formulation and Raychadhuri’s Covariant formulation and Raychadhuri’s

equationequation

Raychadhuri’s equationRaychadhuri’s equation

Einstein gravityEinstein gravity

Raychadhuri’s equation

Covariant holography Singularity

theorems

CovariantC-functionSecond law

of thermo-dynamics

n

Raychadhuri’s equationRaychadhuri’s equation Can we find appropriate to generalize Raychadhuri’s Can we find appropriate to generalize Raychadhuri’s

equation to the Lovelock gravity?equation to the Lovelock gravity?

Summary and ConclusionSummary and Conclusion

We have discussed the Attractor mechanism: Gauss-Bonnet gravity We have discussed the Attractor mechanism: Gauss-Bonnet gravity (Many other theories are investigated). What about brane-wrold (Many other theories are investigated). What about brane-wrold scenarios?scenarios?

C-functions in Lovelock gravity: two kinds!!. Physical interpretation C-functions in Lovelock gravity: two kinds!!. Physical interpretation (CFT??)(CFT??)

C-functions in Randall-Sundrum model ( with Gauss-Bonnet term)?C-functions in Randall-Sundrum model ( with Gauss-Bonnet term)?

Covariant formulation of holographic principle in Lovelock gravity and Covariant formulation of holographic principle in Lovelock gravity and generalized Raychadhuri’s equation.generalized Raychadhuri’s equation.

Thank YouThank You

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The attractor mechanism, C-functions and aspects of holography in Lovelock gravity Mohamed M. Anber...

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