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References
C. Møller:The Theory of Relativity (Oxford, 1952), p. 247.
We takex 4 to be the «time» co-ordinate and choose the metric to beg ij =δ ij ,g 44=−1 in inertial co-ordinates, to agree with the notation of ref. (1)C. Møller:The Theory of Relativity (Oxford, 1952), p. 247. Greek indices run from 1 to 4, Latin ones from 1 to 3; repeated indices are to be summed over.
See ref. (1)C. Møller:The Theory of Relativity (Oxford, 1952), p. 247., Sect.89, p. 237, for a discussion of this normalization in terms of measuring rods.
Here, of course, is where the connection betweenc and the speed of light is made.
The sign ambiguity in the solution of the quadratic is resolved by requiringw to be positive. With the other sign choice,w would be negative andn i would point opposite to the direction of propagation. Our eq. (7) differs from eq. (70) in ref. (1)C. Møller:The Theory of Relativity (Oxford, 1952), p. 247., p. 240, because the latter uses co-ordinate time rather than proper time in the definition of velocity.
Note that eq. (7) implies thatc is also the average speed of light back and forth over an infinitesimal path. An alternative derivation of eq. (5) follows from a definition of distance based on this result. SeeL. Landau andE. Lifshitz:The Classical Theory of Fields (Cambridge, Mass., 1951), p. 257.
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Markley, F.L. Rate of a moving clock in general relativity. Lett. Nuovo Cimento 7, 133–134 (1973). https://doi.org/10.1007/BF02727487
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DOI: https://doi.org/10.1007/BF02727487