Strings and Quantum Gravity








RESEARCH > Scattering by black holes > Multiple black holes

The study of moduli space of a system of maximally charged black holes has recently attracted a lot of attention [5]. It was first discussed by Ferrell and Eardley [6] in four spacetime dimensions. It was subsequently extended to five dimensions [5]. In the near-horizon limit, an $SL(2, \mathbb{R})$ conformal symmetry was discovered that generalized the case of two black holes. Consequently, the general system inherited the pathologies of the system of two black holes: the Hamiltonian possessed no well-defined vacuum state.

This problem was studied a long time ago by de Alfaro, Fubini and Furlan (DFF) [7]. The simplest quantum mechanical system with conformal symmetry is described by the Hamiltonian

\begin{displaymath}H = {p^2\over 2} + {g\over 2x^2} \end{displaymath} (1)

This Hamiltonian possesses a continuous spectrum down to zero energy and there is no well-defined ground state. DFF suggested a solution to this problem. They proposed the redefinition of the Hamiltonian by the addition of a harmonic oscillator potential which is also the generator of special conformal transformations, K = x2/2. The modified Hamiltonian has a well-defined ground state.

In the case of two slowly-moving maximally-charged black holes, the near-horizon geometry is AdS2 x Sn. The isometries of the AdS2 space are conformal symmetries. As a result, the moduli (spatial distance between the two black holes) has dynamics governed by (super)conformal non-relativistic quantum mechanics of the form discussed above. The DFF redefinition of the Hamiltonian has a nice interpretation in this case as a redefinition of the time coordinate. The DFF Hamiltonian corresponds to a globally defined time coordinate whereas the conformally invariant definition does not. Thus, the DFF trick appears plausible on physical grounds.

Dr. Siopsis approached the problem through the path integral and the Faddeev-Popov gauge-fixing procedure. He showed that the DFF trick can be understood in terms of the standard Faddeev-Popov procedure [8]. He started with a discussion of the quantization of a particle in the presence of a background metric field as well as an external vector potential. He performed the standard Faddeev-Popov procedure in the canonical formalism and showed its connection to commutation relations through Dirac brackets. He then applied the procedure to the case of an extreme Reissner-Nordström black hole [5]. He showed that the naïve identification of time coordinate (which leads to a Hamiltonian system with no well-defined ground state) corresponds to a gauge which is not ``good." He found that in this gauge the Faddeev-Popov procedure encounters an obstruction at the boundary of spacetime introducing an additional constraint there. This alters the standard commutation relations and the eigenvalue problem for the attendant Hamiltonian system. He did not calculate the effects of this obstruction explicitly. This would require the introduction of a regulator which would break gauge invariance explicitly and therefore alter the commutation rules. Instead, he exhibited another set of gauges where no obstruction existed on the boundary. He showed that this set of gauges led to a Hamiltonian system with a well-defined vacuum, equivalent to the one obtained through the DFF trick [7].

Dr. Siopsis then applied his procedure to multiple black hole scattering [9]. He noted that the underlying theory is a gauge theory, so the Faddeev-Popov procedure should be applicable. He discussed a systematic implementation of the quantization procedure which correctly accounted for the singularities of moduli space. Each black hole is described by moduli (position vector) $X^\mu (\tau)$ and the action is reparametrization invariant. This gauge invariance necessitated the introduction of gauge-fixing conditions equal in number to the number of black holes. By identifying $X^0$ with time for all black holes, one arrives at the standard Hamiltonian that possesses no well-defined ground state. This pathology comes from a subtlety in the Faddeev-Popov quantization procedure that does not take into account the singularities of moduli space. To properly account for these singularities would be tedious (entailing the introduction of a regulator) and would lead to a modification of the quantization rules. Instead, Dr. Siopsis introduced gauge-fixing conditions that did not suffer from this pathology. The resultant Hamiltonian differed from the pathological one by the addition of the potential K (generator of special conformal transformations), in accordance with the DFF prescription.

The method is generalizable to any system of black holes and more general solutions of the Einstein-Maxwell equations. It would be interesting to apply the Faddeev-Popov procedure to these systems, such as near-extreme black holes. This should enable one to move away from the AdS limit.

Dr. Siopsis and his student S. Musiri also extended these results to continuous distributions of maximally-charged matter [10]. They found that the DFF trick is needed to ensure the existence of a ground state only in matter distributions of dimensionality less than the dimensionality of space. It is interesting to note that it is in these cases that the action for the matter possesses gauge (reparametrization) invariance. Therefore, the DFF trick can be understood as a consequence of a gauge-fixing procedure in this more general setting. It would be desirable to understand the implications of these results to strings and other extended objects.

next up previous


^Department of Physics and Astronomy  --- e-mail: siopsis AT tennessee DOT edu

Voice: (865) 974-7859  -- Fax: (865) 974-7843

401 Nielsen Physics Bldg., The University of Tennessee, Knoxville, TN 37996-1200, U.S.A.