General relativity in a \((2+1)\)-dimensional space-time: An electrically charged solution
The authors solve the Einstein-Maxwell equation in \(2+1\) dimensions, in the case that the only nonvanishing components of the electromagnetic field are \(F^{01}=-F^{10}=E(r)\) (stationary and radially-symmetric case). They obtain \(E(r)=Q/r\) and the metric \[ ds^ 2=-A(r)dt^ 2+[A(r)]^{-1}dr^ 2+r^ 2d\Phi^ 2, \] where \(A(r)=(kQ^ 2/2\pi)\ln (r_ c/r)\), Q is the electrical charge and \(r_ c\) an integration constant, independent of Q. The singularities will be at \(r=0\), \(r=r_ c\) and \(r\to \infty\). The case \(r=r_ m\), in which \(A(r_ m)=1\), is of a particular interest. The authors also study the Kruskal extension of this solution and arrive at some conclusions comparing the above solution with the Kaluza-Klein formalism.
scientific article; zbMATH DE number 3975814