THE MYSTERY OF THE TWIN PARADOX

Time dilation is mutual in special relativity, which means that the travelling twin does see the sedentary twin's clock running slow:

http://www.einstein-online.info/dict.../time-dilation
"Time dilation can be mutual: When two inertial observers speed past each other, each will find that the other's clocks go slower."

However if one wants to both prove and calculate the slowness of the sedentary twin's clock, one would have to consider a scenario in which that clock commutes between two clocks belonging to the travelling twin's spaceship:

http://www.amazon.com/Relativity-Its.../dp/0486406768
Relativity and Its Roots, Banesh Hoffmann, p. 105: "In one case your clock is checked against two of mine, while in the other case my clock is checked against two of yours, and this permits us each to find without contradiction that the other's clocks go more slowly than his own."

Here lies the mystery of the twin paradox. Discussions in Divine Albert's world are restricted to a misleading scenario implicitly deprived of the equipment "two clocks on the travelling twin's spaceship against which the sedentary twin's clock can be checked" and so there is no way to demonstrate the slowness of the sedentary twin's clock. Unless they introduce and use that equipment, antirelativists would never be able to show that the paradox is in fact an absurdity.

The following scenario allows either twin's clock to be checked against two of the other twin's clocks. Two long inertial systems each carrying synchronous clocks pass one another:

..........Inertial system A moving to the right..........
..........Inertial system B moving to the left..........

The systems are so designed that, the moment they stop moving relative to one another, all clocks on both systems stop ticking. In this final static configuration clock A2 faces clock B1 and clock A1 faces clock B2:

..........A2..........A1..........
..........B1..........B2..........

Before reaching clock A2, clock B1 passed clock A1 and the difference in their readings, (A1then - B1then), was then registered. *Now*, in the final static configuration, clock B1 faces clock A2 and the difference in their readings is (A2now - B1now). Clearly clock B1 has been checked against two of Inertial system A's clocks so, according to special relativity, the following inequality holds:

(A2now - B1now) (A1then - B1then) /1/

Before reaching clock B2, clock A1 passed clock B1 and the difference in their readings, (B1then - A1then), was then registered. *Now*, in the final static configuration, clock A1 faces clock B2 and the difference in their readings is (B2now - A1now). Clearly clock A1 has been checked against two of Inertial system B's clocks so, according to special relativity, the following inequality holds:

(B2now - A1now) (B1then - A1then)

This inequality easily becomes:

(A1then - B1then) (A1now - B2now)

Since clocks on Inertial system A were synchronous and stopped ticking simultaneously, A1now = A2now. For the same reason B2now = B1now. So the last inequality becomes:

(A1then - B1then) (A2now - B1now) /2/

Inequalities /1/ and /2/ are contradictory and both are consequences of Einstein's 1905 light postulate. Reductio ad absurdum par excellence. The light postulate is false. Einstein should not have "introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether":

http://books.google.com/books?id=JokgnS1JtmMC
"Relativity and Its Roots" By Banesh Hoffmann, p.92: "There are various remarks to be made about this second principle. For instance, if it is so obvious, how could it turn out to be part of a revolution - especially when the first principle is also a natural one? Moreover, if light consists of particles, as Einstein had suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the speed of the particle is not independent of the motion of the object emitting it. And if we take light to consist of particles and assume that these particles obey Newton's laws, they will conform to Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null result in terms of particles of light and simple, familiar Newtonian ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether. If it was so obvious, though, why did he need to state it as a principle? Because, having taken from the idea of light waves in the ether the one aspect that he needed, he declared early in his paper, to quote his own words, that "the introduction of a 'luminiferous ether' will prove to be superfluous."

Pentcho Valev

The mystery of Einstein's rotating disc:

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.

http://gjl038.g.j.pic.centerblog.net/3fea2faf.jpg

Pentcho Valev