I recently decided for some reason to re-read some of the first part, the kinematical part, of Einstein’s initial paper on relativity, “The Electrodynamics of Moving Bodies”, in which he develops the conceptual framework of special relativity and derives the Lorentz transformations. I must have done it more carefully than the first time I looked at it many years ago, because I noticed what seems to be a rather glaring contradiction, not in the derivation of the Lorentz transformations themselves, but rather in Einstein’s interpretation of them, which he gives in section 4, “The Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Clocks.”(adsbygoogle = window.adsbygoogle || []).push({});

In this section he expresses the traditional and well verified interpretations now known as space contraction and time dilation. He first describes the modified length values applicable to the moving frame, where the values of a length unit moving with velocity v along the axis of the length unit, measured in the stationary reference frame, give a contraction according to the standard Lorentz formula.

He then gives the Lorentz transformation for time, and eliminates the position dependence of the expression by substituting x=vt, where v is the relative velocity, and x,t are the values observed in the laboratory reference frame, at rest. He then gives the well-known retardation of √(1-v^2/c^2), which he reduces to the first approximation of ½ (v^2/c^2), a value which as far as I know has been repeatedly verified, and accounts for the change in observed decay rates in unstable nuclei and particles at high velocities. So far, no surprises.

But then he makes an astonishing (to me) remark about “peculiar consequences” of this derivation. He gives an example, which seems to me to differ in no fundamental way from any previous one used in the arguments by which he derived the Lorentz transformations, where two synchronized clocks are at rest some distance from each other. One is then moved at velocity v towards the other until it reaches it. At this point, Einstein gives us a new formula for the time retardation, one different from the Lorentz transformation, one for which he gives no mathematical or logical justification, which is the value of the Lorentz retardation at velocity v, multiplied by the time of transit of the moving clock to the position of measurement at the laboratory clock (!!!). He seems, for some reason I don’t understand, to have integrated the Lorentz value over the time path, to take from t1 to t2, ∫(v^2/c^2)dt, giving a retardation of ½ t(v^2/c^2), one which I don’t believe is ever observed.

Einstein then extends the example to that of two clocks starting at rest at the same point, and the moving clock traveling by any closed path back to the starting point, then showing a cumulative retardation given by this new formula. He proceeds to give a physical case of a clock stationed at the equator giving such a retardation when compared to a clock at the pole.

But this would mean that a satellite clock in stationary orbit at the equator, if compared to a resting clock at the pole, would show at the launch a value more or less that given by the Lorenz transformation, something on the order of 10^-10, but after a year that retardation would have increased by a factor of 3.15x10^7, the number of seconds in a year, and would increase by a similar amount each year thereafter!!! If that were the case, satellite clocks would be essentially unusable over much of the earth’s surface, and as far as I know is completely inaccurate.

I am at loss to understand what was Einstein’s thinking in this remark. It makes no sense to me whatsoever. Obviously I must be missing something.

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# Inconsistency in relativity paper

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