ON (2+2)-DIMENSIONAL SPACE–TIMES, STRINGS AND BLACK HOLES

Abstract

We study black hole-like solutions (space–times with singularities) of Einstein field equations in 3+1 and 2+2 dimensions. We find three different cases associated with hyperbolic homogeneous spaces. In particular, the hyperbolic version of Schwarzschild's solution contains a conical singularity at r = 0 resulting from pinching to zero size r = 0 the throat of the hyperboloid [Formula: see text] and which is quite different from the static spherically symmetric (3+1)-dimensional solution. Static circular symmetric solutions for metrics in 2+2 are found that are singular at ρ = 0 and whose asymptotic ρ→∞ limit leads to a flat (1+2)-dimensional boundary of topology S1 × R2. Finally we discuss the (1+1)-dimensional Bars–Witten stringy black hole solution and show how it can be embedded into our (3+1)-dimensional solutions. Black holes in a (2+2)-dimensional "space–time" from the perspective of complex gravity in 1+1 complex dimensions and their quaternionic and octonionic gravity extensions deserve furher investigation. An appendix is included with the most general Schwarzschild-like solutions in D ≥ 4.