Scholarlink

In the same way as spherical geometry arises from Euclidean geometry on the unit sphere, hyperbolic geometry ensues from Minkowskian geometry on the unit mass shell, the set of 4-velocitiesVelocity. The relations of hyperbolic geometry correspond to the ones of spherical geometry by the substitution of trigonometric functions \(\cos \) and \(\sin \) by their hyperbolic counterparts ch and sh, and by some signs. Lorentz boosts accelerate 4-velocities and define by their successive application a kind of addition \(\hat {+}\) which, however, is neither commutative nor associative, because the deficit angle of nondegenerate triangles does not vanish. We determine this angle by comparing products of elementary complex 2 × 2-matrices.

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Hyperbolic and Spherical Geometry

  • 章节

  • 发表日期：

    2025年

查看全文
  • Norbert Dragon

摘要

In the same way as spherical geometry arises from Euclidean geometry on the unit sphere, hyperbolic geometry ensues from Minkowskian geometry on the unit mass shell, the set of 4-velocitiesVelocity. The relations of hyperbolic geometry correspond to the ones of spherical geometry by the substitution of trigonometric functions \(\cos \) and \(\sin \) by their hyperbolic counterparts ch and sh, and by some signs. Lorentz boosts accelerate 4-velocities and define by their successive application a kind of addition \(\hat {+}\) which, however, is neither commutative nor associative, because the deficit angle of nondegenerate triangles does not vanish. We determine this angle by comparing products of elementary complex 2 × 2-matrices.