A curious error in Einstein’s Relativity

Taking a break from thermodynamics, a subject which deeply fascinated Einstein and to which he made significant contributions.

Just to be clear, the headline does not refer to any of Einstein’s academic papers in which he presented the theory of relativity. The error occurs in the popular account Einstein later wrote under the title Über die spezielle und die allgemeine Relativitätstheorie; it then reappears in curiously modified form in the authorised English translation Relativity – the special and the general theory.

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The German original

Part 1 of Einstein’s book deals with Special Relativity. Having discussed coordinate systems, the Principle of Relativity (in the restricted sense), the Lorentz Transformation and the behavior of measuring rods and clocks in motion, Einstein presents in §15 the general results of the theory.

It is at this point that he gives the classical expression for kinetic energy, then the new expression for it according to relativity theory, and then employs a binomial expansion to extend this new expression into a series

From the 24th edition, 2009

And this is where the error occurs.

The binomial expansion of the first term in the parentheses is

Einstein seems to have neglected the “–1” term in the parentheses which cancels out the first term in the expansion. The actual result is

This is what you would expect the relativistic kinetic energy expression to look like, since when v<<c all terms other than the first can be neglected and it reduces to the classical expression ½mv².

Einstein’s result, on the other hand, reduces to mc² + ½mv² (*). Since this includes the non-zero rest-mass energy it cannot be a purely kinetic energy expression which he states it to be.

(*) the mass m in these expressions indicates the rest mass m₀

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The English translation

In 1920 an authorised English translation of Einstein’s popular book was published. The curious feature of this translation is that the parenthetic “–1” term in the relativistic kinetic energy expression has been removed, as can be seen below

From the 15th edition, reprinted 1979

This does not rectify the original error, but simply transfers it from one expression to the other. For while the series expression is a correct extension, the relativistic kinetic energy expression from which it is obtained is incorrect as it now contains the non-zero rest-mass energy.

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