We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in Maudlin ( 2012 ) and Malament ( unpublished ). It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of Tarski ( 1959 ): a predicate of betwenness and a four place predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional ‘vector space’ that can be decomposed everywhere into a spacelike hyperplane—which obeys the Euclidean axioms in Tarski and Givant ( The Bulletin of Symbolic Logic , 5 (2), 175–214 1999 )—and an orthogonal timelike line. The length of other ‘vectors’ are calculated according to Pythagoras’ theorem. We conclude with a Representation Theorem relating models M $\mathfrak {M}$ of our system M 1 ${\mathscr{M}}^{1}$ that satisfy second order continuity to the mathematical structure 〈 ℝ 4 , η a b 〉 $\langle \mathbb {R}^{4}, \eta _{ab}\rangle $ , called ‘Minkowski spacetime’ in physics textbooks.

中文翻译：

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闵可夫斯基时空公理系统

我们提出了闵可夫斯基时空几何学的基本公理系统。它在一组简单和流线型的公理与试图在 Maudlin (2012) 和 Malament (未发表) 中对 Minkowski 时空的标准解释的一阶逻辑中直接形式化之间取得平衡。它旨在将来用于 Minkowski 时空物理理论的形式化。原语的选择是本着 Tarski (1959) 的精神：一个介于两者之间的谓词和一个四位谓词来比较相对论区间的平方。闵可夫斯基时空被描述为一个四维“向量空间”，它可以在任何地方分解成一个类空间的超平面——它遵守 Tarski 和 Givant 中的欧几里得公理（符号逻辑公报，5 (2), 175–214 1999）——和一条正交的类时线。其他“向量”的长度根据毕达哥拉斯定理计算。我们以一个表示定理来结束我们的系统 M $\mathfrak {M}$ 的模型 M $\mathfrak {M}$ 的系统 M 1 ${\mathscr{M}}^{1}$ 满足数学结构 〈 ℝ 4 , η ab 〉的二阶连续性$\langle \mathbb {R}^{4}, \eta _{ab}\rangle $ ，在物理教科书中称为“闵可夫斯基时空”。