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

Outline  Introduction: the attractor mechanism  Lovelock gravity  The attractor mechanism in Gauss-Bonnet gravity in 5-D  Entropy and C-function in Einstein gravity  Entropy and C-functions in Lovelock gravity  Covariant formulation and Raychadhuri’s equation

Introduction  Black holes radiate: Bekenstein-Hawking entropy  Many approaches to count the number of states: still open question  Quantum gravity may be the resolution to this problem  String theory may be the way toward quantum gravity: black holes in string theory

Introduction  Black holes in string theory: string theory or M-theory Compactified to lower dimensions, Torous, Calabi Yau  The right entropy for SUSY black holes!!  Calabi Yau Moduli fields: massless and dangerous.  Far from the BH, moduli can take a range of continuous values. CAN ENTROPY DEPEND ON THIS BIZARRE BEHAVIOUR?  The answer is No!!, Resolution is the attractor mechanism  Near horizon

Introduction  What is the attractor mechanism?  All moduli fields are attracted to the same value at the horizon irrespective of their values at asymptotic infinity.  Entropy depends only on few parameters : Mass, angular momentum and not on the value of these moduli at infinity Attractor position Damped pendulum

Introduction  Is there a similar behavior for the non-supersymmetric case? Yes!!  Proven for classical Einstein gravity in 4-D and 5-D.

Outline Introduction: the attractor mechanism Introduction: the attractor mechanism  Lovelock gravity  The attractor mechanism in Gauss-Bonnet gravity in 5-D  Entropy and C-function in Einstein gravity  Entropy and C-functions in Lovelock gravity  Covariant formulation and Raychadhuri’s equation

Lovelock gravity  Possibility for higher dimensional space!!  The most general second order gravity in higher dimensional space.  It contains Gauss-Bonnet term: the result of compactifying certain string theories.

Lovelock gravity  Pure Lovelock of order m  Einstein Gravity  Not all terms survive in a given dimension: D=5, only m=2 (Gauss-Bonnet) survive, m=3 is a topological term

Lovelock gravity  Equation of motion  General Lovelock gravity: sum over all m

Outline Introduction: the attractor mechanism Introduction: the attractor mechanism Lovelock gravity Lovelock gravity  The attractor mechanism in Gauss-Bonnet gravity in 5-D  Entropy and C-function in Einstein gravity  Entropy and C-functions in Lovelock gravity  Covariant formulation and Raychadhuri’s equation

The attractor mechanism in Gauss-Bonnet gravity M. Anber and D. Kastor JHEP 0710:084,2007  Phenomenological Lagrangian  Spherically symmetric solution

The attractor mechanism in Gauss-Bonnet gravity  Equations of motion  Point like electric charge

The attractor mechanism in Gauss-Bonnet gravity  Effective potential  Moduli field equation  A solution: constant V_eff \phi

The attractor mechanism in Gauss-Bonnet gravity  Attractor: positive  The procedure for testing the attractor 1-Start with 2-Find black hole solution using

The attractor mechanism in Gauss-Bonnet gravity 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 the Correction of a and b

The attractor mechanism in Gauss-Bonnet gravity 5-Use numerical technique to test if the solution is singularity free up to infinity

The attractor mechanism in Gauss-Bonnet gravity  Black hole solution at  Extremal : near horizon  Specific model of the potential

The attractor mechanism in Gauss-Bonnet gravity  Perturbation of  Same attractor behavior for a(r) and b(r)

The attractor mechanism in Gauss-Bonnet gravity  Numerical Results

The attractor mechanism in Gauss-Bonnet gravity

 Non-Extremal black hole: No Attractor !!

Outline Introduction: the attractor mechanism Introduction: the attractor mechanism Lovelock gravity Lovelock gravity The attractor mechanism in Gauss-Bonnet gravity in 5-D The attractor mechanism in Gauss-Bonnet gravity in 5-D  Entropy and C-function in Einstein gravity  Entropy and C-functions in Lovelock gravity  Covariant formulation and Raychadhuri’s equation

Entropy: Revisited  Entropy in Lovelock gravity (Myers and Jacobson 1993)  Any possible connection with quantum field theory?  ‘t Hooft and Susskind, Holographic principle in Einstein gravity ( Given a closed surface, we can represent all that happens inside it by degrees of freedom on this surface itself.)  Manifestation of the holographic principle AdS/CFT (Maldacena 1998)

Entropy: Revisited  Conformal description of horizon’s states (Solodukhin 1999) Use the near horizon coordinates (x-x_h) 5- The resulting near horizon theory is conformal

Entropy: Revisited 6-Use the light cone coordinates 7- Define Virasoro generators 8- Calculate Poisson’s bracket 9- quantize the calculations extension to Lovelock gravity (Cvitan, Pallua and Prester 2002)

C-functions in 2-D field theories  C-functions in the renormalization group flow in 2- D quantum field theories (Zamolodchikov 1986)  C-function is a function of the coupling of the theory that is monotonically increasing with energy.  For fixed points of the flow, corresponding to the extrema of this function, the C-function reduces to the central charge of Virasoro algebra E C

Holographic C-functions  AdS/CFT (Avarez, Gomez 1999, Susskind and Witten 1998) r AdS C(r )

C-functions in asymptotically flat Einstein gravity  C-functions in spherically symmetric and asymptotically flat spacetime (Goldstein et al 2006)  C-function (null energy condition is satisfied)

C-functions in asymptotically flat Einstein gravity  Conditions for the C-function 1-It can be evaluated on any spherical surface concentric with The horizon 2-When evaluated on the horizon of a black hole it equals its entropy 3-If certain physical conditions and certain boundary conditions are satisfied, then C is a non-decreasing function along the outward radial direction Can we find similar functions in Lovelock gravity?

Outline Introduction: the attractor mechanism Introduction: the attractor mechanism Lovelock gravity Lovelock gravity The attractor mechanism in Gauss-Bonnet gravity in 5-D The attractor mechanism in Gauss-Bonnet gravity in 5-D Entropy and C-function in Einstein gravity Entropy and C-function in Einstein gravity  Entropy and C-functions in Lovelock gravity  Covariant formulation and Raychadhuri’s equation

C-function in Lovelock gravity (pure) (M. Anber and D. Kastor, in progress)  Spherically symmetric metric in D=n+2 dimensions  Particular combination

C-function in Lovelock gravity (pure)  we obtain  Constraints : only local maxima, asymptotically flat.  Result: b(r) is monotonic

C-function in Lovelock gravity (pure)  But the C-function has to reduce to entropy when evaluated on horizon  C-function of the first kind

C-function in Lovelock gravity (pure)  C-function of the second kind!!  Proof outline: 1- take the derivative w.r.t r and use equations of motion to simplify the result 2- Existing of extrema require that one finds a solution for dC/dr 3-There is no solution (m=even!!)

C-function in Lovelock gravity (general)  General C-functions of the first kind  Proof of monotonicity: No solution for C’=0.

C-function in Lovelock gravity (general)  General C-functions of the second kind: Difficult to prove the monotonocity for general theory (general polynomial)  We can proove the monotonicity for Gauss-Bonnet gravity

C-function in Lovelock gravity (general)  Physical interpretation of two different C-functions!!  More C-Functions are possible??

Outline Introduction: the attractor mechanism Introduction: the attractor mechanism Lovelock gravity Lovelock gravity The attractor mechanism in Gauss-Bonnet gravity in 5-D The attractor mechanism in Gauss-Bonnet gravity in 5-D Entropy and C-function in Einstein gravity Entropy and C-function in Einstein gravity Entropy and C-functions in Lovelock gravity Entropy and C-functions in Lovelock gravity  Covariant formulation and Raychadhuri’s equation

Raychadhuri’s equation  Einstein gravity Raychadhuri’s equation Covariant holography Singularity theorems Covariant C-function Second law of thermo- dynamics n

Raychadhuri’s equation  Can we find appropriate to generalize Raychadhuri’s equation to the Lovelock gravity?

Summary and Conclusion  We have discussed the Attractor mechanism: Gauss-Bonnet gravity (Many other theories are investigated). What about brane-wrold scenarios?  C-functions in Lovelock gravity: two kinds!!. Physical interpretation (CFT??)  C-functions in Randall-Sundrum model ( with Gauss-Bonnet term)?  Covariant formulation of holographic principle in Lovelock gravity and generalized Raychadhuri’s equation.

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