We study black-hole-like solutions ( spacetimes with singularities ) of Einstein field equations in 3+1 and 2+2-dimensions. In the 3+1-dim case, it is shown how the horizon of the standard black hole solution at r = 2GNM can be displaced to the location r = 0 of the point mass M source, when the radial gauge function is chosen to have an ultra-violet cutoff R(r = 0) = 2GNM if, and only if, one embeds the problem in the Finsler geometry of the spacetime tangent bundle (or in phase space) that is the proper arena where to incorporate the role of the physical point mass M source at r = 0. 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 H2 and which is quite different from the static spherically symmetric 3+1-dim 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-dim boundary of topology S1 x R2. Finally we discuss the 1+1-dim Bars-Witten stringy black-hole solution and show how it can be embedded into our 3 + 1-dimensional solutions with a displaced horizon at r = 0 and discuss the plausible stringy nature of a point-mass, along with the maximal acceleration principle in the spacetime tangent bundle (maximal force in phase spaces). Black holes in a 2 + 2-dimensional "spacetime" 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.
Comments: 41 Pages. This article appeared in the Int. J. Mod. Phys. A vol 22, no. 11 (2007) 2021.
[v1] 25 Sep 2009
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