(This document is also available in PostScript format.)
In a recent issue of this journal, Singham
poses
a question that can be summarized as follows. Given that electromagnetic
information propagates at the speed of light, how does the information
about the charge of a black hole propagate across the event horizon
in order to produce an electric field on the outside?
Singham poses the question in quantum-mechanical terms, asking, ``Is the electric force not mediated by photons?'' However, the problem is just as puzzling when viewed from the perspective of classical electrodynamics. Let us therefore address the question classically and see if the answer sheds any light on the quantum version.
In order to make the statement of the puzzle precise, recall that the electromagnetic field at any point x in spacetime can be calculated from the retarded four-potential
where j is the four-current density.
The retarded Green's function G(x';x) has support only on the past
light cone of x. If x is a spacetime point outside the horizon
of a charged black hole, then the past light cone consists
only of points outside the event horizon. How, then, does
the charge inside the black hole produce an electromagnetic
field at x?
Consider first the case of a charged black hole that formed
by gravitational collapse (as opposed to one that has
existed for all time).
At early times (before the hole formed),
all of the charged
material that would eventually constitute the black hole was outside
the event horizon. The integral in (1) over the past light cone
contains contributions from this matter. In this case, then,
the mystery is solved. The electromagnetic field at x has as its
source the charged matter that would eventually form the black
hole.
For completeness, we should also consider the less physically realistic
case of a black hole that has existed for all time. Specifically,
let us assume a static,
isolated, charged black hole embedded in asymptotically flat
spacetime. Such a black hole is described by
the Reissner-Nordstrøm geometry.
In this
vacuum solution to the Einstein-Maxwell equations,
there is no infalling charged matter,
so the above resolution of the paradox does not work.
However,
the Reissner-Nordstrøm solution contains a white-hole singularity
in the past of every external observer.
The integral (1) always
hits this singularity, so the retarded potential is ill-defined.
In this situation, we should probably say that the electric field at
x has as its ``source'' the white-hole singularity on x's past
light cone.
Having dispensed with the classical problem, let us now regard the electromagnetic field as quantized. (From now on, we consider only black holes that formed via collapse, not eternally-existing Reissner-Nordstrøm holes.) Then, as Singham observes, the electromagnetic force on our test charge is mediated by virtual photons. We could in principle calculate the transition amplitude to any particular final state of the test charge by considering virtual photon exchange between the test charge and the charged particles that collapsed to form the black hole. Since the force is determined by virtual photon exchanges with the infalling charged matter, we would be justified in saying that the infalling matter is the ``source'' of the electric field, just as in the classical case.
There is, however, an important difference between the classical and
quantum treatments. Virtual photons, unlike real ones, need not
propagate along the light cone. When we compute transition amplitudes
for our test charge, we must include the contributions of all
virtual photons, including
off-shell
photons that propagate out from inside the event horizon. In
other words, the electromagnetic interaction between a test charge and the
black hole is mediated by both horizon-crossing and
non-horizon-crossing virtual photons.
It is tempting to suppose that, in some appropriate semiclassical
limit, ``most'' of the contribution to the external electric field
comes from nearly-on-shell, non-horizon-crossing virtual photons.
However, such a statement is difficult to make precise, since the
amount a particular virtual photon contributes to the final field is a
gauge-dependent quantity. In particular, in the Coulomb gauge, no
such statement is true. On the contrary, at times long after the
formation of the hole, the electric field is computed entirely from
the static Coulomb potential 
. Since
is determined by data on a slice at constant t,
one might somewhat fancifully say
that the field in Coulomb gauge is produced by ``virtual photons of
infinite speed.'' In a covariant gauge, one might find that the
dominant contribution to the field comes from photons near the light
cone, but as far as we know nobody has done such a calculation.
Emory F. Bunn
Department of Physics and Astronomy
44 Campus Ave.
Bates College
Lewiston, ME 04240
Electronic mail: ebunn@abacus.bates.edu
Matthew McIrvin
105 Trowbridge St. Apt. #6B
Cambridge, MA 02138
Electronic mail: mmcirvin@world.std.com