Mathematical logic, algebra and number theory

On random choice of elliptic and hyperbolic rotations of the Lorentz spaces

V. A. Churkin^ab, A. I. Ilin<sup>bc

a</sup> Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
^b Novosibirsk State University, st. Pirogova, 2, 630090, Novosibirsk, Russia
^c National Research University Higher School of Economics, Russian Federation

Abstract: Elliptic and hyperbolic rotations of the $(n+1)$-dimensional Lorentz space can be represented as exponential of rank $2$ matrices of the real Lie algebra $\mathfrak{so}(1, n)$. We shown that the ratio of the volumes of the corresponding sets of matrices Euclidean norm $\leqslant r$ is equal to $(\sqrt2)^{n-1}-1$ for all $r > 0$. Consequently the portion of hyperbolic rotations near identity decreases exponentially with increasing $n$. Another corollary is that in case of Minkovski space of special relativity choose of elliptic and hyperbolic rotations near identity is equiprobable.

Keywords: elliptic rotation, hyperbolic rotation, random matrix.

UDC: 512.865.3

MSC: 22E15, 22E43, 15B52

Received February 9, 2016, published November 8, 2016

DOI: 10.17377/semi.2016.13.077

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