论文标题

空间各向同性所隐含的世界观变换群

Groups of Worldview Transformations Implied by Isotropy of Space

论文作者

Madarász, Judit X., Stannett, Mike, Székely, Gergely

论文摘要

考虑到任何欧几里得有序的字段,$ q $,以及(1+3)的任何“合理”组,$ g $,二维时空对称,我们展示了如何构建一个模型$ m_ {g} $ m_ {g} $ of Kinematics,set $ w $ w $ w $ w $ w $ w = g $ w = g $ w = g $ w = g $ w = g $。这尤其适用于所有相关的$ GAL $,$ CPOI $和$ CEUCL $(分别是Galilean,Poincaré和Euclidean Transformations的组,其中Q $中的$ C \是POINCARé变换的速度)。 在这样做的过程中,通过基本的几何证明,我们证明了我们的主要贡献:空间各向同性足以使Worldview Transformations的$ W $ w \ subseteq gal $,$ w \ subseteq cpoi $,或$ w \ subseteq cpoi $,或$ w \ subSeteq CEUCL $ for Some $ c> c> 0 $ 0 $ 0 $。因此,假设空间各向同性足以证明只有3种可能的情况:世界是经典的(惯性观察者之间的世界观变换是伽利略的转变);世界是相对论的(世界观的转变是庞加莱的转变);或世界是欧几里得人(对欧几里得几何形状提供了非标准的运动学解释)。该结果大大扩展了该领域的先前结果,该领域的相对性(严格强度更强)的特殊原则也将$ Q $的选择限制为Reals领域。 作为这项工作的一部分,我们还证明了一个令人惊讶的结果,对于任何包含翻译和轮换的$ g $修复了时间轴$ t $ $ g [t] = t $,然后$ g $是一种微不足道的转换(即$ g $是一种线性变换,可保留欧几里得长度并修复时间轴套装)。

Given any Euclidean ordered field, $Q$, and any 'reasonable' group, $G$, of (1+3)-dimensional spacetime symmetries, we show how to construct a model $M_{G}$ of kinematics for which the set $W$ of worldview transformations between inertial observers satisfies $W=G$. This holds in particular for all relevant subgroups of $Gal$, $cPoi$, and $cEucl$ (the groups of Galilean, Poincaré and Euclidean transformations, respectively, where $c\in Q$ is a model-specific parameter orresponding to the speed of light in the case of Poincaré transformations). In doing so, by an elementary geometrical proof, we demonstrate our main contribution: spatial isotropy is enough to entail that the set $W$ of worldview transformations satisfies either $W\subseteq Gal$, $W\subseteq cPoi$, or $W\subseteq cEucl$ for some $c>0$. So assuming spatial isotropy is enough to prove that there are only 3 possible cases: either the world is classical (the worldview transformations between inertial observers are Galilean transformations); the world is relativistic (the worldview transformations are Poincaré transformations); or the world is Euclidean (which gives a nonstandard kinematical interpretation to Euclidean geometry). This result considerably extends previous results in this field, which assume a priori the (strictly stronger) special principle of relativity, while also restricting the choice of $Q$ to the field of reals. As part of this work, we also prove the rather surprising result that, for any $G$ containing translations and rotations fixing the time-axis $t$, the requirement that $G$ be a subgroup of one of the groups $Gal$, $cPoi$ or $cEucl$ is logically equivalent to the somewhat simpler requirement that, for all $g\in G$: $g[t]$ is a line, and if $g[t]=t$ then $g$ is a trivial transformation (i.e. $g$ is a linear transformation that preserves Euclidean length and fixes the time-axis setwise).

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