Download Stefano Bellucci (INFN-Laboratori Nazionali di Frascati)

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
[email protected]
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).
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 MBH=14MSUN. Its extremality parameter a*=J/GMBH2>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/CFT
correspondence,“ 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], 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.
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 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 |pФ| <2mM and |pФ| > 2mM, separated
by the critical point |pФ| =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.
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 nearhorizon 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.
Discussion and Outlook
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.
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