Distances on the tropical line determined by two points

Let $p',q'\in R^n$. Write $p'\sim q'$ if $p'-q'$ is a multiple of $(1,\ldots,1)$. Two different points $p$ and $q$ in $R^n/\sim$ uniquely determine a tropical line $L(p,q)$, passing through them, and stable under small perturbations. This line is a balanced unrooted semi--labeled tree on $n$ leaves. It is also a metric graph. If some representatives $p'$ and $q'$ of $p$ and $q$ are the first and second columns of some real normal idempotent order $n$ matrix $A$, we prove that the tree $L(p,q)$ is described by a matrix $F$, easily obtained from $A$. We also prove that $L(p,q)$ is caterpillar. We prove that every vertex in $L(p,q)$ belongs to the tropical linear segment joining $p$ and $q$. A vertex, denoted $pq$, closest (w.r.t tropical distance) to $p$ exists in $L(p,q)$. Same for $q$. The distances between pairs of adjacent vertices in $L(p,q)$ and the distances \dd(p,pq), \dd(qp,q) and \dd(p,q) are certain entries of the matrix $|F|$. In addition, if $p$ and $q$ are generic, then the tree $L(p,q)$ is trivalent. The entries of $F$ are differences (i.e., sum of principal diagonal minus sum of secondary diagonal) of order 2 minors of the first two columns of $A$.Comment: New corrected version. 31 pages and 9 figures. The main result is theorem 13. This is a generalization of theorem 7 to arbitrary n. Theorem 7 was obtained with A. Jim\'enez; see Arxiv 1205.416

Tropical conics for the layman

We present a simple and elementary procedure to sketch the tropical conic given by a degree--two homogeneous tropical polynomial. These conics are trees of a very particular kind. Given such a tree, we explain how to compute a defining polynomial. Finally, we characterize those degree--two tropical polynomials which are reducible and factorize them. We show that there exist irreducible degree--two tropical polynomials giving rise to pairs of tropical lines.Comment: 19 pages, 4 figures. Major rewriting of formerly entitled paper "Metric invariants of tropical conics and factorization of degree--two homogeneous tropical polynomials in three variables". To appear in Idempotent and tropical mathematics and problems of mathematical physics (vol. II), G. Litvinov, V. Maslov, S. Sergeev (eds.), Proceedings Workshop, Moscow, 200

Classical resolution of black hole singularities via wormholes

In certain extensions of General Relativity, wormholes generated by spherically symmetric electric fields can resolve black hole singularities without necessarily removing curvature divergences. This is shown by studying geodesic completeness, the behavior of time-like congruences going through the divergent region, and by means of scattering of waves off the wormhole. This provides an example of the logical independence between curvature divergences and space-time singularities, concepts very often identified with each other in the literature.ns of curvature divergences in the context of space-time singularities.Comment: 6 pages, 4 figures; several improvements in main body and abstract; final version to appear in Eur. Phys. J.