Analysis of half-spin particle motion in static Reissner-Nordström and Schwarzschild fields М.V.Gorbatenko, V.P.Neznamov, Е.Yu.Popov (DSPIN 2015), Dubna, 2015г. Russian Federal Nuclear Center Sarov

In the conventional quantum mechanics, the very form of potential, in which particle moves, makes it clear whether there are bound states and whether there is falling to the center. In Ya. B. Zel’dovich and V.S. Popov’s paper (English version: Sov. Phys. Usp. 14, 673 (1972) ), the analysis of the Dirac equation with the Coulomb potential was converted to that of the well studied Schroedinger equation problem. The Zel’dovich and Popov’s approach develops the effective potential techniques which could be usefully applied to the corresponding problems in curved space- time backgrounds, such as those with the Schwarzschild and Reissner- Nordstrom metrics. We would like to show that in general in the Schwarzschild and Reissner-Nordstrom metrics the ½-spin particle falls to horizon. The exception is an extremal Reissner-Nordstrom black hole satisfying some conditions which will be formulated below.

Spin 1/2 particle in the Schwarzschild space-time Consider spin ½ particle governed by Dirac equation in Schwarzschild space-time. Metric is where is gravitational radius or radius of event horizon.

M – Black hole mass, m – fermion mass, с – velocity of light, h – Planck’s constant, Radial part of the Dirac equation in Schwarzschild metric

Rewrite the system of the first order differential equations as of the second order differential equation

Using substitution we obtain a Schroedinger-type equation The effective potential U has a complicate analytical form, but our concern is not with the precise analytical expression for U. It is sufficient to know the behavior of the effective potential in the vicinity of event horizon It is well known that quantum-mechanical particle falls to the center if the singular effective potential behaves as with, otherwise it does not fall to the center. The above expression for effective potential shows that the conditions holds.

Behavior of the effective potential of the Dirac equation in the Schwarzschild background ( )

Dirac equation in a charged black hole background Reissner-Nordstrom metric is where

In contrast to Schwarzschild’s black holes a charged black holes have two event horizons If М=|Q|, then two horizons merged together. This is the so-called «extremal» black hole

Dirac’s particle in background of charged black hole The system of radial Dirac’s equations is

Just as the radial part of the Dirac equation can be rewritten as a Schroedinger-type equation in the Schwarzschild metric, so can this procedure be done in the charged black hole case Effective potentials in the vicinity of outer and inner horizons are:

The singular effective potentials behaves as in the vicinities of horizons, with the numerators of these expressions being negative, and their absolute values are greater than 1/8. It means that the energy spectrum of bound states is not bounded below. In other words, Dirac’s particles fall to the horizons (either inner or outer)

Dirac’s particle in the background of extremal black holes

Consider the effective potential in the case of an extremal black hole, M=|Q| or At first sight, the effective potential in the vicinity of horizon is more singular than but if we take into account the Dokuchaev and Eroshenko solution (JETP Vol.117, No 1, p.72-77, 2013 ) then the first and second terms are vanishing, and expression for the effective potential becomes

In the vicinity of horizon According to conventional quantum-mechanical analysis, see e.g. Landau and Lifschitz, fall to the center (which is the horizon in the present case) is prevented provided that However the eigenvalue with the convergent normalization integral for the wave function to exist, a more strong condition should be imposed (Dokuchaev and Eroshenko)

Contrary to the Dokuchaev and Eroshenko claim, we state that there is no stable bound states of Dirac particle in the exterior domain of extremal black holes. Indeed, the condition that a bound state to exist is E<m, which implies that. However, the found effective potential has no extremal point.

Behavior of the effective potential of the Schrödinger-type equation in the field of the RN extreme black hole at,,,.

If we assume that and take into account the DE conditions then the effective potential changes its form and acquires a local minimum Behavior of the effective potential of the Schrödinger-type equation in the field of the RN extreme black hole at

Can the stationary level form in the case ? Let us first assume that the black hole and particle have like signs of electric charges, i.e. such a stationary level can clearly not exist, because, in this case and energy level belongs to the continuum part of spectrum We then assume that the black hole and the particle have opposite signs of electric charges Then which, formally, does not prevent to the existence of a discrete energy level with, which would be interpreted as that corresponding an antiparticle state

Consider the interior of an extreme black hole, The effective potential is schematically displayed in this figure Behavior of the effective potential of the Dirac equation in the RN extreme black-hole field inside the event horizon

In the interior of an extremal black hole, the effective potential is positive. In the vicinity of origin and in the vicinity of the horizon Can a discrete state exist there? The answer is positive provided that the particle and the extremal black hole have the same sign of electric charge. Otherwise the answer is No.

EP method allows us to obtain qualitative results consisting in the following: first, we show that in all the explored cases but one the condition of a particle “fall” to appropriate event horizons is fulfilled. The exception is one of the solutions for the RN extreme field with the single event horizon. For this solution, inside the event horizon the possibility of existence of stationary bound states of spin ½ particles is shown only at like signs of the particle and the black hole. At that, the region outside the event horizon turns out to be away from the interior by an infinitely high potential barrier. Secondly, the results were obtained to be considered as a confirmation of the cosmic censorship conjecture. The confirmation consists in existence of an infinitely high potential barrier in origin not allowing quantum–mechanical particles approach to the singularity. This is shown both for the RN metric with event horizons (horizon) and for the RN naked singularity.

Thank you