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

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

Oct 1986

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1986gregr..18.1019g&link_type=abstract

General Relativity and Gravitation, Volume 18, Issue 10, pp.1019-1035

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31

General Relativity:Electromagnetic Fields

Scientific paper

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 findmathop Elimits^ to = Qhat 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 . 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.

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