At the beginning of his career Einstein enjoys, thanks to Lorentz, the time dilation factor 1/γ=(1-v²/c²)^1/2. Years later he decides to show that the frequency shift factor, 1+Φ/c², can validly be deduced within his theory. So he replaces 1/γ with the Taylor approximation 1-v²/2c², then substitutes Φ for -v²/2 and the dream comes true (see Appendix 3 in his "Relativity"). The feat involves the intuition of a brilliant juggler but also some luck. The form of the factor 1/γ may have proved entirely different from that of 1+Φ/c². Also, it may have turned out that no justification existed, not even a spurious one, for some of the steps in the "deduction". If that had been the case, "the theory of Einstein" would have remained "the theory of Lorentz-Poincare", or, even more realistically, "the strange theoretical deviation caused by the strange assumption that, although light is proved to move as a particle in the blackbody cavity, it still moves as a wave elsewhere".

Among the tricks performed in accordance with Rule 3 the conversion of 1/γ into 1+Φ/c² is perhaps the most difficult and risky. On one hand, 1/γ belongs to Lorentz transforms which are consistent with the constancy of the speed of light (the speed of light is relativistic in the c sense). On the other, the frequency shift factor 1+Φ/c² is a corollary of the assumption that the speed of light is variable (the speed of light is relativistic in the c+v sense – see http://www.wbabin.net/valev/valev3.htm ). Under these circumstances, to infer the frequency shift factor from Lorentz transforms is a task that only people like Einstein can set themselves. The success would be guaranteed if the inference were logical – in such cases the absence of rigorous logical rules allows jugglers to infer anything from anything. However in this particular case the inference is mathematical. Einstein will have to build a sequence of formulas connecting Lorentz transforms with the frequency shift factor, and this sequence will have to look like a valid inference. No juggler, not even the most brilliant one, can do this in the absence of luck. This luck requires both that the factor 1/γ and the factor 1+Φ/c² be similar in form, and that the sequence of formulas leading from the former to the latter look reasonable.

Einstein certainly knows that the speed of light in a gravitational field is variable (see Section 22 in his "Relativity"). Therefore, by using this knowledge and by applying the equivalence principle, he must have obtained the frequency shift factor 1+Φ/c² quite easily. On the other hand, the fact that the speed of light is variable implies that Lorentz transforms are totally irrelevant. Einstein is not very sad – he has already extracted all possible profits from those transforms. However at the moment he badly needs their component 1/γ. So in Sections 22 and 23 in his “Relativity” Einstein wages a fierce campaign designed to convince the reader that Lorentz transforms are irrelevant only for the non-inertial system (the one experiencing the gravitational field) but remain perfectly relevant for the inertial one. In protest, clocks in the non-inertial system start demonstrating time contraction with respect to inertial clocks (a property incompatible with Lorentz transforms) but Einstein does not care. He is just happy that the factor 1/γ is saved and its conversion into 1+Φ/c² can begin. (Needless to say, the grotesque "semiirrelevance" of Lorentz transforms can only be acceptable if it is announced by a god-like creature.)

The salvation of 1/γ, the factor so closely related to inertial conditions, is the major problem and Einstein has successfully resolved it. The rest is relatively easy (see Appendix 3 in his “Relativity”). First, 1/γ is replaced by the Taylor approximation, 1-v²/2c². This implies that the inference is not valid for high relative speeds but who cares. Then, since -v²/2 is equal to the work that must be done on a unit of mass against the centrifugal force in order to transport it from the periphery to the centre of a rotating disc, -v²/2 is replaced by the potential Φ corresponding to this type of work. So the Taylor approximation is converted into 1+Φ/c². Then, through the fallacy of affirming the consequent (typical of Einstein - see an explanation of this logical trick in http://www.wbabin.net/valev/valev3.htm ), the result is generalised (“the result we have obtained will hold quite generally for gravitational fields”). Finally, since 1/γ is the factor that determines the magnitude of the time dilation according to the “semiirrelevant” Lorentz transforms, nothing can discourage Einstein from conferring the same “time-dilational” function on 1+Φ/c² as well. And since things occur in a gravitational field, the old "inertial" time dilation can now be called "gravitational". Atoms immediately obey and become clocks whose rate of running (identified with the frequency of the emitted light) conforms to the new world order. (Note that, in accordance with the rotating disk thought experiment, the magnitude of the gravitational time dilation is independent of the gravitational field but depends on the gravitational potential – a wisdom comparable to that attributing a "semiirrelevance" to Lorentz transforms.)