http://www.bartleby.com/173/23.html Albert Einstein: "An observer who is sitting eccentrically on the disc K' is sensible of a force which acts outwards in a radial direction... (...) The observer performs experiments on his circular disc with clocks and measuring-rods. In doing so, it is his intention to arrive at exact definitions for the signification of time- and space-data with reference to the circular disc K', these definitions being based on his observations. What will be his experience in this enterprise? To start with, he places one of two identically constructed clocks at the centre of the circular disc, and the other on the edge of the disc, so that they are at rest relative to it. We now ask ourselves whether both clocks go at the same rate from the standpoint of the non-rotating Galileian reference-body K. As judged from this body, the clock at the centre of the disc has no velocity, whereas the clock at the edge of the disc is in motion relative to K in consequence of the rotation. According to a result obtained in Section XII, it follows that the latter clock goes at a rate permanently slower than that of the clock at the centre of the circular disc, i.e. as observed from K."
Einstein refers to Section XII but this Section does not contain any results explaining why the (inertial) clock at the centre of the rotating disc should run FASTER than the (non-inertial) clock placed on the edge of the disc. Rather, the results in Section XII are all based on the Lorentz transformation which predicts MUTUAL time dilation for two INTERTIAL clocks: either INERTIAL clock (rather, the observer in this clock's system) sees the other INERTIAL clock running SLOW by a factor of 1/gamma = sqrt(1-(v/c)^2). The Lorentz transformation does not predict anything about a system of two clocks one of which (in this case the one on the edge of the disc) is not inertial. Yet in the above text Einstein suggests that, IN ACCORDANCE WITH THE LORENTZ TRANSFORMATION, the inertial K-clock (at the center of the disc) is running FASTER than the non-inertial K'-clock (on the edge of the disc) by a factor of gamma = 1/sqrt(1-(v/c)^2). What makes him lie so blatantly? What does he fear?
By increasing the perimeter of the disc while keeping the linear speed of the periphery constant, one can convert clocks fixed on the periphery into VIRTUALLY INERTIAL clocks (the "gravitational field" they experience is reduced to zero). Now, IN ACCORDANCE WITH THE LORENTZ TRANSFORMATION, the (VIRTUALLY INERTIAL) observer "sitting eccentrically" on the edge of the disc (K'-observer) sees the clock at the center of the disc (K-clock) run SLOWER than clocks fixed on the periphery (K'-clocks).
We have reductio ad absurdum par excellence - the clock at the center runs both FASTER than clocks on the periphery (as observed from K) and SLOWER than clocks on the periphery (as observed from K'). The consequent (mutual time dilation) is absurd, therefore the antecedent (Einstein's 1905 constant-speed-of-light postulate) is false.