The Rotational Kinematic Formula Viewed from a Tetrad-based Anholonomic Frame

Abstract

We give a fresh look into the age-old rotational kinematic formula which was originally devised by L. Euler in the eighteenth century. Instead of some verbose explanations for its logical validity, an argument of a covariant differentiation with respect to coordinate time with a tetrad-based relativistic account in an anholonomic frame will be given as a viewpoint that is mathematically sound and self-explanatory. The familiar “ω(t) × q(t)” term is replaced by a linear combination of space-frame fields of a tetrad with ‘Ricci’s connection coefficients’ of infinitesimal generators, so as to be expressed by sumnolimits_{i,j}^3 { = 1} e(i)c{gamma ^i}j0mathop qlimits^{oj} (t) with one subscript index set to zero, ‘o’, for the time coordinate. This recognition gives a new interpretation of the “time derivative for the space set of rotated axes” as the formal covariant time derivative in this tetrad-based coordinate transformation of the four-dimensional space-time that is curved due to an implicit Galilean transformation.