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

Introduction and motivationThe Kerr solution, describing rotating neutral black holes, plays a fundamentalrole in General Relativity, as well as in modern theoretical physics in general.Particularly special are its thermodynamic properties and connection to stringtheory, allowing one to expect that quantum gravity should be closely relatedto these objects.

A very particular case of black hole solution, when the Cauchy and event horizonscoincide is called extremal black hole solution. Having much larger symmetry, suchsolutions 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 particlemoving in such a field. The investigation of a test particle system is important formany reasons. It may help to reveal some important symmetries or non-trivialconstructions related to the field. For example the construction of Killing tensor forKerr space-time is related to the discovery of a quadratic integral of motion of themassive particle moving in that field (B. Carter, Phys. Rev. 174 (1968) 1559;M. Walker, R. Penrose, Commun. Math. Phys.18 (1970) 265).

Introduction and motivationOn the other hand, the direct interpretation of the purely mechanicalproblem is also motivated, since there are known objects with a set ofparameters 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 HoleGRS 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 MBH=14MSUN. Its extremality parameter a*=J/GMBH

2>0.98(its spin reads J=1078 hbar).

Such a BH has an exact CFT dual

M. Guica, T. Hartman, W. Song and A. Strominger, "The Kerr/CFTcorrespondence,“ Phys. Rev. D 80, 124008 (2009)[arXiv:0809.4266 [hep-th]],

with a central charge connected to a*.

Introduction and motivation

In Anton Galajinsky, Kirill Orekhov,[arXiv:1103.1047v2], conformalmechanics related to the near horizon extreme Kerr-Newman-AdS-dSblack hole is studied.

In this talk, we investigate the “spherical'' part of that conformalmechanics, constructing action-angle variables.

Such an approach is motivated for several reasons. Except for a verysimple form of the solution of motion equations, because of the uniquenessamong all other canonic variables, action-angle variables allow us toestablish a correspondence/discrepancy between different systems at leaston the classical level. On the other hand the quantization in these variablesIs very simple. In fact, it is very similar to the Bohr-Sommerfeld quantization.

KERR’S METRICS

EXTREMAL KERR’S BLACK HOLE

CONFORMAL MECHANICS

CONFORMAL ALGEBRA SO(1,2)

ACTION ANGLE VARIABLES

INTEGRATION RANGE

FINAL EXPRESSION FOR ACTION VARIABLES

FINAL EXPRESSION FOR ANGLE VARIABLES

CRITICAL POINT

GRAPHICS

QUANTIZATION

Discussion and OutlookWe 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 arenot very convenient for analyzing the system. Nevertheless, they allowed us toindicate the existence of two regimes, with |pФ| <2mM and |pФ| > 2mM, separatedby the critical point |pФ| =2mM, where the particle motion becomes effectively 1d.Due to the dynamical conformal symmetry, the presented angular systemaccumulates 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" quantummechanics.Respectively, for negative values of the angular Hamiltonian the effective radialdynamics corresponds to the falling on the center, and for positive values itcorresponds to the scattering problem.Hence, the proposed description provides us with the complete semiclassicaldescription of the particle moving near the horizon of an extreme Kerr black hole.

Discussion and OutlookThe 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, onecan check that the 2d spherical system does not belong to the family systemsadmitting N=4 superextensions in terms of existing linear and non-linearsupermultiplets.

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 systemin 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.

Finally, the action-angle variables define the adiabatic invariants of thesystem, and yield a ground for the developing of classical perturbationtheory.

From this viewpoint our consideration is important for describing thedynamics of the particle near non-extreme Kerr black holes, whichseemingly have been observed recently.

Discussion and Outlook

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