1. BLACK HOLE MATHEMATICAL THEORY – DUBNA 17 DECEMBER 2011 NEAR HORIZON PARTICLE DYNAMICS IN EXTREMAL KERR BLACK HOLE S. BELLUCCI INFN-LABORATORI NAZIONALI DI FRASCATI, ITALY bellucci@lnf.infn.it

2. Introduction and motivation The Kerr solution, describing rotating neutral black holes, plays a fundamental role in General Relativity, as well as in modern theoretical physics in general. Particularly special are its thermodynamic properties and connection to string theory, allowing one to expect that quantum gravity should be closely related to these objects. A very particular case of black hole solution, when the Cauchy and event horizons coincide is called extremal black hole solution. Having much larger symmetry, such solutions play a distinguished role in supergravity (for review, see Riccardo D'Auria, Pietro Fre', [arXiv:hep-th/9812160v2]). As a first step for the investigation of these objects one can consider a test particle moving in such a field. The investigation of a test particle system is important for many reasons. It may help to reveal some important symmetries or non-trivial constructions related to the field. For example the construction of Killing tensor for Kerr space-time is related to the discovery of a quadratic integral of motion of the massive particle moving in that field (B. Carter, Phys. Rev. 174 (1968) 1559; M. Walker, R. Penrose, Commun. Math. Phys.18 (1970) 265).

3. Introduction and motivation On the other hand, the direct interpretation of the purely mechanical problem is also motivated, since there are known objects with a set of parameters close to those in extremal Kerr's black hole (Jeffrey E. McClintock, Rebecca Shafee, Ramesh Narayan, Ronald A. Remillard, Shane W. Davis, Li-Xin Li, The Spin of the Near-Extreme Kerr Black Hole GRS 1915+105, Astrophys.J. 652, 518-539,2006,[arXiv:astro-ph/0606076]). In particular, a nearly extremal Kerr BH has been observed in our Galaxy (15/8/1992), with M BH =14M SUN. Its extremality parameter a*=J/GM BH 2 >0.98 (its spin reads J=10 78 hbar). Such a BH has an exact CFT dual M. Guica, T. Hartman, W. Song and A. Strominger, "The Kerr/CFT correspondence,“ Phys. Rev. D 80, 124008 (2009) [arXiv:0809.4266 [hep-th]], with a central charge connected to a*.

4. Introduction and motivation In Anton Galajinsky, Kirill Orekhov,[arXiv:1103.1047v2], conformal mechanics related to the near horizon extreme Kerr-Newman-AdS-dS black hole is studied. In this talk, we investigate the “spherical'' part of that conformal mechanics, constructing action-angle variables. Such an approach is motivated for several reasons. Except for a very simple form of the solution of motion equations, because of the uniqueness among all other canonic variables, action-angle variables allow us to establish a correspondence/discrepancy between different systems at least on the classical level. On the other hand the quantization in these variables Is very simple. In fact, it is very similar to the Bohr-Sommerfeld quantization.

5. KERR’S METRICS

6. EXTREMAL KERR’S BLACK HOLE

7. CONFORMAL MECHANICS

8. CONFORMAL ALGEBRA SO(1,2)

9. ACTION ANGLE VARIABLES

11. INTEGRATION RANGE

12. FINAL EXPRESSION FOR ACTION VARIABLES

13. FINAL EXPRESSION FOR ANGLE VARIABLES

14. CRITICAL POINT

15. GRAPHICS

16. QUANTIZATION

17. Discussion and Outlook We constructed the action-angle variable of the angular sector of the (near-horizon) dynamics of the particle moving near the horizon of the extreme black hole solution. These variables are expressed via initial ones in terms of elliptic functions, so they are not very convenient for analyzing the system. Nevertheless, they allowed us to indicate the existence of two regimes, with |p Ф | 2mM, separated by the critical point |p Ф | =2mM, where the particle motion becomes effectively 1d. Due to the dynamical conformal symmetry, the presented angular system accumulates the whole information on the initial dynamics of the system. It could be done in terms of the so-called “AdS basis", and in the “conformal" one, where the Hamiltonian takes a form of conventional “non-relativistic" quantum mechanics. Respectively, for negative values of the angular Hamiltonian the effective radial dynamics corresponds to the falling on the center, and for positive values it corresponds to the scattering problem. Hence, the proposed description provides us with the complete semiclassical description of the particle moving near the horizon of an extreme Kerr black hole.

18. Discussion and Outlook The given formulation allows us to immediately answer the question, whether it is possible to construct the N=4 superconformal extension of the near- horizon Kerr particle.Notice that with the N=4 supersymmetric extension of the angular Hamiltonian I at hand one can easily construct the D(1,2|α) superconformal extension of the whole conformal mechanics. However, one can check that the 2d spherical system does not belong to the family systems admitting N=4 superextensions in terms of existing linear and non-linear supermultiplets. Hence, the common opinion that the near-horizon Kerr particle does not admit a N= 4 superconformal extension is correct. However, we can construct a (formal) N=4 superextension of the system in the action-angle variables. Thus, one can obtain a physically relevant supersymmetric Hamiltonian. The proposed structure is just the analog of the well-known freedom in the the N=2$ supersymmetrization, which was used in past works.

19. Finally, the action-angle variables define the adiabatic invariants of the system, and yield a ground for the developing of classical perturbation theory. From this viewpoint our consideration is important for describing the dynamics of the particle near non-extreme Kerr black holes, which seemingly have been observed recently. Discussion and Outlook

20. Acknowledgements ERC Advanced Grant no. 226455, “Supersymmetry, Quantum Gravity and Gauge Fields'‘ (SUPERFIELDS), for partial financial support. Armen Nersessian and Vahagn Yeghikyan, for precious collaboration. Pietro Fre and the organizers of ROUND TABLE 4 ITALY-RUSSIA@DUBNA, Black Holes in Mathematics and Physics, for invitation. You all, for kind attention