转盘圆周率佯谬与非欧几何

Paradox About Circumference Ratio of Rotating Disk and Non-Euclidean Geometry

分析了非欧几何中的转盘圆周率佯谬,指出运动时间与运动长度的相对论胀缩根源于它们的参考系判据:本体参考系的同时间判据将导致反常运动长度膨胀,而实验室参考系的同位置判据将导致反常运动时间收缩.分别用狭义和广义相对论计算了转盘的非欧几何,并用本体局部瞬时惯性参考系对二者进行了衔接.得出的结果是,转盘圆周长与直径之比与参考系有关,这个比值在本体非惯性系大于π,正好是转盘的圆周率,对应于罗巴切夫斯基几何.进一步分析了弯曲时空的非欧几何特性,给出了在施瓦希外解与罗伯森-沃克(R-W)度规中的空间圆周率.最后通过普朗克黑体辐射谱的协变性,讨论了"长度变换"对"温度变换"的影响.

The paradox in non-Euclidean geometry about circumference ratio of rotating disk is analyzed. It is pointed out that relativistic constriction and expansion about length and time originate in criterion of their reference frame, simultaneity criterion of proper reference frame leads to expansion of abnormal motion length and comprovincial criterion of laboratory reference frame leads to constriction of abnormal motion time. The non-Euclidean geometry on rotating disk is respectively computed by special relativity and general relativity, and the connecting between both methods is made up. The obtained results are that the ratio of perimeter of rotating disk with respect to its diameter is associated with reference frame, this ratio is greater than π in proper non-inertial frame and is exactly circumference ratio of rotating disk which is corresponded with Lobachevsky＇s geometry. Furthermore, the characteristic of non-Euclidean geometry in curved space-time is analyzed and circumference ratio in the space of Schwarzschild outer metric and in the space of Robertson-Walker metric is obtained. Finally, the length transformation＇s influence on transformation about temperature is discussed by using covariability of Planck black body radiation spectrum.