s-wave scattering of charged fermions by a magnetic black hole.

We argue that, classically, s-wave electrons incident on a magnetically charged black hole are swallowed with probability one: the reflection coefficient vanishes. However, quantum effects can lead to both electromagnetic and gravitational backscattering. We show that, for the case of extremal, magnetically charged, dilatonic black holes and a single flavor of low-energy charged particles, this backscattering is described by a perturbatively computable and unitary S-matrix, and that the Hawking radiation in these modes is suppressed near extremality. The interesting and much more difficult case of several flavors is also discussed. To many physicists it has seemed natural that extremal black holes, stabilized by their charge, should interact with their surroundings very much as elementary particles. In particular, they should scatter incident particles in a manner described by a unitary, quantum-mechanical S-matrix. This contrasts with another view, according to which the black hole absorbs a particle and then Hawking radiates[1] back to its extremal state. The Hawking radiation is in a mixed state, so there is no S-matrix. In support of the “elementary particle” picture, it has recently been shown[2,3,4] that in many cases potential barriers arise around near-extremal dilatonic black holes[5,2], which reflect incident particles before they reach the horizon. In this paper we examine the threat posed to this picture by an exception to that result: the s-wave charged fermions in the presence of a magnetic black hole, which classically sees no barrier. We show that these modes are reflected from extremal (a = 1) dilatonic black holes by an infinite quantum-mechanical barrier, and hence s-wave fermion-hole scattering is indeed described by a unitary S-matrix. Let us first consider the classical theory consisting of gravity, electromagnetism and a massless left-handed “electron”. Of course this theory is anomalous at the quantum level, but this does not prevent us from studying classical scattering solutions of the Dirac equation. In particular we are interested in the scattering of charge e electrons from a black hole of magnetic charge g such that eg = 2. In the study of wave equations in a fixed black hole background, one expands the field Φ into modes Φ(`, ω) of energy ω and angular momentum `. The wave equation for a bosonic mode Φ(`, ω) may then be written in the general form ( −ω2 + ( ∂ ∂r∗ )2 + V (`) ) Φ(`, ω) = 0 (1) where r∗ is a convenient radial coordinate. For fermions there is a similar first 2 order differential equation. The presence of a non trivial potential V – in general non-vanishing even for ` = 0 – implies the possibility of backscattering. Some component of a matter s-wave impinging on a black hole will in general backscatter off the gravitational field, while the rest will continue on across the event horizon. However something rather special can occur for an electrically charged fermion in the presence of a magnetically charged black hole. The total angular momentum ~ J = ~s+ ~̀+ r̂eg (2) contains spin (~s), orbital (~̀) and field (egr̂) contributions. If eg = 2 (or more generally n + 2), then there is a ~ J = 0 s-wave mode for which the spin and field contributions cancel. (These are of course the Callan-Rubakov modes which lead to monopole catalysis of proton decay [6,7].) An incoming, ~̀= 0, left handed electron has its spin aligned with its (radial) momentum, so that ~ J = 0; but in a theory with only left-handed electrons, there are no outgoing modes with ~ J = 0. Therefore angular momentum conservation forbids an incoming electron from backscattering off of a black hole. This will continue to be true classically, even if we include the other chirality and allow Dirac mass terms. We conclude that an incoming fermion will continue across the horizon and into the singularity with unit amplitude, and the analog of the potential V must therefore vanish. In reference [3,8] this was confirmed by explicit calculation, and in [3] it was pointed out that the absence of a potential barrier would seem to contradict the “elementary particle” picture of extremal black holes, and allow catastrophic Hawking radiation in these modes (at least in the regime a little above extremality, where back-reaction can be neglected). Quantum mechanically the situation differs. We are forced to include both chiralities in order to avoid anomalies. There are then outgoing ~ J = 0 fermion 3 modes. If backreaction is included, as the s-wave fermion impinges on the black hole large electric fields are produced in its wake. These large electric fields are unstable due to Schwinger pair production. This is a quantum mechanical form of backscattering. It is essentially an electromagnetic effect, and is distinct from Hawking emission. This problem is difficult to analyze in general. We shall consider only a special case which turns out to be particularly simple: the backscattering of an s-wave electron incident on an extremal, magnetic charge Q dilatonic black hole[5,2]. These black holes are extrema of the action: S = ∫ d4x √−g ( e−2φ ( R + 4(∇φ)2 − 1 2 F 2 ) + iψ̄ / Dψ ) (3) where φ is the scalar dilaton field, ψ is a charged fermion and D the covariant derivative*. A key feature of this type of black hole is the following. As pointed out in [2] and described in detail in [4], the geometry consists of three regions. The first is the asymptotically flat (AF) region far from the black hole. Nearer the black hole the curvatures begin to rise and one enters the mouth region. The mouth leads into an infinitely long throat region. Well into the throat region, the metric is approximated by the flat metric on two-dimensional Minkowski space times the round metric on the two-sphere with radius Q (in Planck units). The spacetime has no horizons or singularities. The dilaton field φ increases linearly with the proper distance into the throat. The electromagnetic field strength is tangent to and integrates to 4πQ over the two-sphere. At scales large relative to the radius Q of the two-sphere, dynamics within the throat region are described by a two-dimensional effective action [9,4]. Regarding * The power of eφ appearing in front of the fermionic part of the action is irrelevant since it may be eliminated (modulo quantum anomalies which could be important for gravitational backreaction) by a rescaling of ψ. 4 the two-sphere as a “compactification manifold” this effective action can be derived with standard Kaluza-Klein technology. The result is [9,4]: S = ∫ d2σ √−g ( e−2φ ( R+ 4(∇φ)2 + 1 Q2 − 1 2 F 2 ) + iψ̄ / Dψ ) (4) where g, φ and F = dA are relics of four dimensional fields taking constant values on the two-sphere and the charged Dirac fermion ψ results from a zero mode of the charged Dirac equation on a two-sphere threaded with magnetic flux. The fourdimensional extremal black hole corresponds to the two-dimensional linear dilaton vacuum: ds2 = −dτ2 + dσ2 φ(σ) = φ0 + σ/2Q