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BERTOTTI{ROBINSON GEOMETRY and SUPERSYMMETRY S. Ferrara*) ABSTRACT

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BERTOTTI{ROBINSON GEOMETRY and SUPERSYMMETRY S. Ferrara*) ABSTRACT CERN-TH/97-10 hep-th/9701163 BERTOTTI{ROBINSON GEOMETRY AND SUPERSYMMETRY ) S. Ferrara Theoretical Physics Division, CERN CH - 1211 Geneva 23 ABSTRACT The role of Bertotti{Robinson geometry in the attractor mechanism of extremal black holes is describ ed for the case of N =2 sup ersymmetry. Its implication for a mo del- indep endent derivation of the Bekenstein{Hawking entropy formula is discussed. Talk given at the 12th Italian Conference on General Relativity and Gravitational Physics Rome, September 1996 CERN-TH/97-10 January 1997 BERTOTTI{ROBINSON GEOMETRY AND SUPERSYMMETRY Sergio FERRARA Theory Division, CERN 1211 Geneva 23, Switzerland E-mail: [email protected] ABSTRACT The role of Bertotti{Robinson geometry in the attractor mechanism of extremal black holes is describ ed for the case of N = 2 sup ersymmetry. Its implication for a mo del-indep endent derivation of the Bekenstein{Hawking entropy formula is discussed. 1. Extremal Black Holes and Attractors In this rep ort, I will discuss some recent work on the macroscopic determination of the Bekenstein{Hawking entropy-area formula for extremal black holes, using du- ality symmetries of e ective string theories enco ded in the sup ergravity low-energy actions. A new dynamical principle emphasizes the role played by \ xed scalars" and Bertotti{Robinson typ e geometries in this determination. Sup ersymmetry seems to be related to dynamical systems with xed p oints describing the equilibrium and a stability. The particular prop erty of the long-range b ehavior of dynamical ows in dissipative systems is the following: in approaching the attractors the orbits lose practically all memory of their initial conditions, even though the dynamics is strictly deterministic. The rst known example of such attractor b ehavior in the sup ersymmetric system 1;2 was discovered in the context of N = 2 extremal black holes . The corresp onding motion describ es the b ehavior of the mo duli elds as they approach the core of the 1 black hole. They evolve according to a damp ed geo desic equation (see eq. (20) in ) until they run into the xed p oint near the black hole horizon. The mo duli at xed 1 p oints were shown to b e given as ratios of charges in the pure magnetic case . It was further shown that this phenomenon extends to the generic case when b oth electric 2 and magnetic charges are present . The inverse distance to the horizon plays the role of the evolution parameter in the corresp onding attractor. By the time mo duli reach the horizon they lose completely the information ab out the initial conditions, i.e. ab out their values far away from the black hole, which corresp ond to the values 3 of various coupling constants, see Fig. 1. The recent result rep orted here is the derivation of the universal prop erty of the stable xed p oint of the sup ersymmetric a A p oint x where the phase velo city v (x )isvanishing is named a xedpoint and represents the x x system in equilibrium, v (x )=0. The xed p oint is said to b e an attractor of some motion x(t)if x lim x(t)=x (t): t!1 x 1 20 2 e 15 10 5 2 4 6 8 10 12 14 r Figure 1: Evolution of the dilaton from various initial conditions at in nity to a common xed p ointatr=0. b attractors: xed p oint is de ned by the new principle of a minimal central charge and the area of the horizon is prop ortional to the square of the central charge, computed at the p oint where it is extremized in the mo duli space. In N = 2, d = 4 theories, which is the main ob ject of our discussion here, the extremization has to b e p erformed in the mo duli space of the sp ecial geometry and is illustrated in Fig. 1. This results in the following formula for the Bekenstein{Hawking entropy S , which is prop ortional to the quarter of the area of the horizon: A 2 S = = jZ j ; d =4 : (1) x 4 This result allows generalization for higher dimensions, for example, in ve-dimensional space-time one has A 3=2 S = jZ j ; d =5 : (2) x 4 There exists a b eautiful phenomenon in the black hole physics: according to the c no-hair theorem, there is a limited number of parameters which describ e space and physical elds far away from the black hole. In application to the recently studied black holes in string theory, these parameters include the mass, the electric and magnetic charges, and the asymptotic values of the scalar elds. It app ears that for sup ersymmetric black holes one can prove a new, stronger version of the no-hair theorem: black holes lose all their scalar hair near the horizon. b We are assuming that the extremum is a minimum, as it can b e explicitly veri ed in some mo dels. However for the time b eing we cannot exclude situations with di erent extrema or even where the equation D Z = 0 has no solutions. i c This numb er can b e quite large, e.g. for N = 8 sup ersymmetry one can have56charges and 70 mo duli. 2 Black hole solutions near the horizon are characterized only by those discrete param- eters which corresp ond to conserved charges asso ciated with gauge symmetries, but not by the values of the scalar elds at in nity which maychange continuously. A simple example of this attractor mechanism is given by the dilatonic black holes 4;5 of the heterotic string theory . The mo dulus of the central charge in question which is equal to the ADM mass is given by the formula 1 0 0 M = jZ j = (e jpj +e jqj) : (3) AD M 2 In application to this case the general theory, develop ed in this pap er gives the fol- lowing recip e to get the area i) Find the extremum of the mo dulus of the central charge as a function of a 2 2 0 dilaton e = g at xed charges ! @ 1 1 1 @ jZ j(g ; p; q )= jpj + gjqj = jpj + jqj =0: (4) 2 @g 2@g g g ii) Get the xed value of the mo duli p 2 g = : (5) x q iii) Insert the xed value into your central charge formula (3), get the xed value of the central charge: the square of it is prop ortional to the area of the horizon and de nes the Bekenstein{Hawking entropy A 2 S = = jZ j = jpq j : (6) x 4 5;6 This indeed coincides with the result obtained b efore by completely di erent metho ds . In general sup ersymmetric N = 2 black holes have an ADM mass M dep ending on charges (p; q )aswell as on mo duli z through the holomorphic symplectic sections 710 X (z );F (z) . The mo duli present the values of the scalar elds of the theory far away from the black hole. The general formula for the mass of the state with one half of unbroken sup ersymmetry of N = 2 sup ergravity interacting with vector 7 10 multiplets as well as with hyp ermultiplets is 2 2 M = jZ j ; (7) 7 where the central charge is K(z;z) 2 Z (z; z; q; p)= e (X (z )q F (z ) p )= (L q M p ) ; (8) so that 2 2 2 M = jZ j = M (z; z ; p; q ) : (9) AD M AD M 3 The area, however, is only charge dep endent: A = A(p; q ) : (10) This happ ens since the values of the mo duli near the horizon are driven to the xed 1 p oint de ned by the ratios of the charges. This mechanism was explained b efore in 2 8 and on the basis of the conformal gauge formulation of N = 2 theory . This attractor mechanism is by no means an exclusive prop erty of only N = 2 theory in four dimensions. Our analysis suggests that it may be a quite universal phenomenon in any sup ersymmetric theory. It has in fact b een extended to all N > 2 11 theories in four dimensions and to all theories in ve dimensions . Further p ossible extensions to higher dimensions and to higher extended ob jects (p-branes) have also 12 b een discussed in recent literature . In this rep ort we will use the \co ordinate free" formulation of the sp ecial geo- 9;7;10 metry which will allow us to present a symplectic invariant description of the system. We will b e able to show that the unbroken sup ersymmetry requires the xed p oint of attraction to b e de ned by the solution of the duality symmetric equation 1 D Z =(@ + K )Z(z; z ; p; q )=0 ; (11) i i i 2 which implies, @ jZ j =0 (12) i @z at Z = Z = Z L (p; q );M (p; q );p;q : (13) x Equation @ jZ j = 0 exhibits the minimal area principle in the sense that the area i is de ned by the extremum of the central charge in the mo duli space of the sp ecial geometry, see Fig. 2 illustrating this p oint. Up on substitution of this extremal values of the mo duli into the square of the central charge we get the Bekenstein{Hawking entropy, A 2 S = = jZ j : (14) x 4 The area of the black hole horizon has also an interpretation as the mass of the 13 Bertotti{Robinson universe describing the near horizon geometry.
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