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c (6a) and (6b), let us consider the coordinate transformation [22] to the uniformly rotating disk, in terms of Newton’s notion of “absolute time” as follows: x = x’ cos Wt – y’ sin Wt, (7a) y = x’ sin Wt + y’ cos Wt, z = z’ where W is the angular velocity; or in cylindrical coordinates, r = r’ , z = z’. f = f’ + Wt. (7b) Then, the resulting metric has the following form, ds2 = (c2 - W2r’2) dt2 - 2Wr’2 df’dt – dr’2 - r’2 df’2 – dz’2 (6b’) However, the mathematical coordinate system K*(x’, y’, z’, t) is not a physical space-time coordinate system for the uniformly rotating disk K’ because the time coordinate t remains associating with the inertial frame of reference K. In other words, metric (6b’) together with its coordinates is not a space-time c oordinate system that can be used for physical interpretation. For instance, it follows from ds2 = 0 that the coordinate light speed produced by (6b’) could be larger than c (a problem of Newton’s notion of absolute time). Since a physical principle is violated, the equivalence principle would not be applicable in the coordinate system K*(x’, y’, z’, t). This example of Einstein’s demonstrates the necessity that, for physical interpretation, you have, not only just a mathematical coordinate system, but also a physical coordinate system. Nevertheless, as shown by Zel’dovich & Novikov [22], metric (6b’) alone can be used to recover metric (6b). This is expected since the metric is transformed from a physical metric. (For an arbitrary Lorentz manifold, however, it has been shown that the hope of finding a valid space-time coordinate system cannot be guaranteed [23].) To obtain a physical coordinate system including the time t’ of the rotating disk, a comparison of (6b) and (6b’) leads to, df’ = df - Wdt ; (8a) and cdt’ = [cdt - (rW/c)rdf][1 – (rW/c)2]-1 . (8b) Thus, it is necessary to modify the time coordinate t’. An interesting fact of this local coordinate transformation is that (8a) can be obtained directly from (7b) and looks like a Galilean transformation. The inverse transformation is as follows: df = df’[1 – (rW/c)2]-1 + Wdt’ ; and cdt = cdt’ + (rW/c)rdf’[1 – (rW/c)2]-1 . (8c) It would be difficult to guess the factor [1 – (rW/c)2]-1, which seems to be incompatible with time dilation and spatial contraction manifested in metric (6b). But, the time dilation and the spatial contraction are results due to comparisons with a clock and a measuring rod in relatively rest at the beginning of a free fall. According to Einstein’s equivalence principle (see sections 5 and 6), such a coordinate system is locally Minkowski. To verify this, consider the Lorentz coordinate transformation, rdf = [1 – (rW/c)2]-1/2 [dX+ rWdT] ; (9a) and cdt = [1 – (rW/c)2]-1/2 [cdT + (rW/c)dX] . (9b) Then, rdf’ = [1 – (rW/c)2]1/2 dX ; and cdt’ = [1 – (rW/c)2]-1/2 cdT (9c) These are exactly the time dilation and s