某些线性Finsler空间中的距离模和速度模概念
The Notions of Distance and Velocity Modulus in the Some Linear Finsler Spaces
摘要：
在Berwald-Moor度量下，得到了四维线性空间中三维距离和速度模的计算公式。所使用的算法既适用于Minkowski空间，也适用于任意多线性Finsler空间，其中类时分量可以选择。这里构造的具有Berwald-Moor度规的空间中的三维速度的模在小（非相对论性）速度下与伽利略空间中的相应表达式一致，而在最大速度下，即对于位于未来圆锥表面上的世界线，该模等于1。为了构造三维距离，使用了相对对称性表面的概念，这类似于狭义相对论中的相应推测。得到了从一个惯性系到另一个惯性系的速度变换公式。当两个速度都沿所选三条直线中的一条直线运动时，所得到的关系式与狭义相对论的类似关系式是一致的，但在其它情况下则不同。此外，还得到了与闵可夫斯基空间中的洛伦兹变换具有相同作用的变换的表达式。我们发现，如果三个空间坐标轴是直线，速度沿着直线相加，就像狭义相对论中的那样，那么取新惯性系的速度与其中一个坐标轴共线，可以看到这个坐标和时间坐标的变换与洛伦兹变换一致，而两个横坐标的变换不同于相应的洛伦兹变换。
英文摘要：
The formulas for the 3-dimensional distance and the velocity modulus in the 4-dimensional linear space with the Berwald-Moor metrics are obtained. The used algorithm is applicable both for the Minkowski space and for the arbitrary poly-linear Finsler space in which the time-like component could be chosen. The constructed here modulus of the 3-dimensional velocity in the space with the Berwald-Moor metrics coincides with the corresponding expression in the Galilean space at small (non-relativistic) velocities, while at maximal velocities, i.e. for the world lines lying on the surface of the cone of future, this modulus is equal to unity. To construct the 3-dimensional distance, the notion of the surface of the relative simultaneity is used which is analogous to the corresponding speculations in special relativity. The formulas for the velocity transformation when one pass from one inertial frame to another are obtained. In case when both velocities are directed along one of the three selected straight lines, the obtained relations coincide with the analogous relations of special relativity, but they differ in other cases. Besides, the expressions for the transformations that play the same role as Lorentz transformations in the Minkowski space are obtained. It was found that if the three space coordinate axis are straight lines along which the velocities are added as in special relativity, then taking the velocity of the new inertial frame collinear to the one of these coordinate axis, one can see that the transformation of this coordinate and time coordinate coincide with Lorentz transformations, while the transformations of the two transversal coordinates differ from the corresponding Lorentz transformations.
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