## On 2 + 2-Dimensional Spacetimes, Strings, Black-Holes and Maximal Acceleration in Phase Spaces

**Authors:** Carlos Castro, J. A. Nieto

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 = 2G_{N}M 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) = 2G_{N}M 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
H^{2} 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 S^{1} x R^{2}. 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.

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### Submission history

[v1] 25 Sep 2009

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