General relativity in a (2+1)-dimensional space-time: An electrically charged solution

Abstract

We have found a static electrically charged solution to the Einstein-Maxwell equations in a (2+1)-dimensional space-time. Studies of general relativity in lower dimensional space-times provide many new insights and a simplified arena for doing quantum mechanics. In (2+1)-dimensional space-time, solutions to the vacuum field equations are locally flat (point masses are conical sigularities), but when electromagnetic fields are presentT ab≠O and the solutions are curved. For a static chargeQ we find\(\mathop E\limits^ \to = Q\hat r/r\) andds 2= −(kQ 2/2π)In(r c /r)dt 2 + (2π/kQ 2[ln(r c /r)]−1 dr 2 +r 2 dφ 2 wherer c is a constant. There is a horizon atr =r c like the inner horizon of the Reisner-Nordström solution. We have produced a Kruskal extension of this metric which shows two static regions (I and III) withr <r c and two dynamical regions (II and IV) withr>r c . A spacelike slice across regions I and III shows a football-shaped universe with chargeQ at one end and −Q at the other. Slices in the dynamical regions (II and IV) show a cylindrical universe that is expanding in region II and contracting in region IV. Electromagnetic solutions to the Einstein-Maxwell field equations in lower dimensional space-times can be used to provide new insights into Kaluza-Klein theories. In terms of the Kaluza-Klein theory, for example, electromagnetic radiation in a (2+1)-dimensional space-time is really gravitational radiation in the associated (3+1)-dimensional Kaluza-Klein space-time. According to Kaluza Klein theory the absence of gravitational radiation in (2+1)-dimensional space-time implies (correctly) the absence of electromagnetic radiation in (1+1)-dimensional space-time.