Magnetic solutions to 211 gravity

Studying physics in spacetimes with dimensions less than four has often proved useful. While the study of such systems is of intrinsic interest, one usually has the hope that the properties of these lower-dimensional systems will mimic the properties of some corresponding four-dimensional system. Witten’s discovery @1# of a black hole solution in twodimensional string theory has sparked renewed interest in lower-dimensional gravity. This solution has been used to study problems which have been intractable in four dimensions such as black hole information loss. Another lower-dimensional black hole solution that has generated a great deal of interest is the three-dimensional black hole discovered by Bañados, Teitelboim, and Zanelli ~BTZ! @2,3#. This spacetime is a solution to Einstein gravity with a negative cosmological constant; it is also known that this solution can be formulated as a string theory solution @4#. Like the two-dimensional black hole, this solution has been studied with the hope of shedding light on problems in four-dimensional gravity. This hope is supported by the fact that there are striking similarities between some of these recently discovered three-dimensional solutions and their fourdimensional counterparts. However, despite these similarities, one should bear in mind that there are some important differences, not the least of which is that the universe we live in is four dimensional. Nevertheless, in the more simplified realm of three dimensions, we might reasonably hope to obtain some insight into the nature of gravity and quantum gravity in particular @5#. The discovery of the BTZ black hole has spawned efforts to find other solutions to the three-dimensional Einstein equations as well as solutions to various generalizations of them coupled to a variety of matter fields. One such solution is the static electrically charged black hole originally discussed by BTZ @2#. This solution is specified by three parameters, a mass parameter M , a charge Q , and a ‘‘radial parameter’’ r0 . To see that one needs this radial parameter, it is sufficient to observe that while the energy density in the electromagnetic field approaches zero asymptotically the rate at which it does is sufficiently slow so that the total energy in