Logarithmic parameters have been been found useful for describing the translational and gravitational states of some systems. These parameters are additive, and may serve to push unwanted singularities off to infinity. The best known of these is rapidity, a speedlike parameter used in special relativity. Another is the so-called “linear time” of Misner and Lévy-Leblond, which applies to general relativistic cosmology. We point out that these two parameters are both logarithmic measures of frequency shift and represent a quantity we call “vibracy.” Majernik’s trigonometric variant of rapidity is found to be an aberration angle.

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The unattainability of infinity is not always obvious and may need to be investigated carefully. There are notorious examples in the history of science, where a certain limit was falsely assumed unreachable. Perhaps the earliest recorded are Zeno’s paradoxes of motion. As emphasized by Misner, an old argument points out that in the race between the tortoise and the hare, the paradox results from assuming the total elapsed time is infinite because the geometric series of distance-halving times has infinitely many terms to sum. Another more modern blunder, in the early years of general relativity, was the assumption that the Schwarzschild surface could never be reached by a test particle because the coordinate time diverged to infinity. Relativists eventually realized, however, that the proper time of the particle is more relevant, and the latter remains finite. Hence the particle will reach and even cross the event horizon, though only “privately” since distant observers can never witness it in finite time. In these examples, we encounter certain variables—number of terms, coordinate time—whose divergence to infinity does not preclude the attaining of certain limits.

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*Basic Relativity*(Springer-Verlag, New York, 1994)]. Teachers of general relativity, however, are probably apt to regard the use of*ict*as a quaint old trick that merely postpones the inevitable, i.e., the necessary slogging to acquire elements of tensor analysis and metric geometry3.

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We use “Doppler shift” loosely here to denote either the familiar velocity effect or the cosmological redshift.

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See Ref. 9.

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*x*axis to the photon velocity, which is the complement of Majernik’s angle, $\alpha =\pi /2\u2212\phi .$
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