On (2+2)-dimensional space-times, strings and black holes

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 H-2 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 p = 0 and whose asymptotic p -> infinity limit leads to a flat (1 + 2)-dimensional boundary of topology S-1 x R-2. 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.