04-06-2010, 11:09 PM
Hi,
Consider a simplified version of the paradox:
Two travelers, each in his own spaceship (i.e., 'reference frame') pass each other at a high relative velocity. By mutual arrangement, they synchronize their (perfect) clocks at the moment of closest approach. After some time, which has aged more? The question is indeterminate. Each 'sees' the other's clock running slow by the same amount. Since a universal 'now' does not exist, there is no way of comparing their relative ages that all observers would agree on. Only by bringing the two back together can we compare their ages in their local 'now'. Since they are moving apart, the only way to bring them back together is for one or both to accelerate. Unlike velocity, acceleration can be measured locally -- i.e., within the reference frame. If both accelerate by the same amount and in opposite directions, then when they pass each other, both will have experienced the same elapsed time. If one accelerates more than the other, then that one will experience a smaller elapsed time. It is the difference in the accelerations, not the velocity itself, that leads to the difference in elapsed time. Note that looking at it from this point of view, it doesn't matter if the accelerations are infinite or small. By assuming they are infinite, all the calculations can be done in special relativity.
--Pete
Quote:Why would "instantaneous" be necessary.Because the special theory of relativity cannot handle accelerations. That's why Einstein went on to develop the general theory, because the special theory was incomplete. Only by reducing the period of acceleration to zero can you stay completely within the special theory. So, either the twin paradox requires the general theory (contradicting the quoted statement) or it requires infinite accelerations (contradicting physical reality).
Quote:I thought the Hafele-Keating experiment verified the relativistic effects of velocity.There is no question about relativistic effects, either special or general. The details may turn out to need refinement, but within experimental error, both theories are well supported. The question is simply whether the special theory could explain the twin paradox. And the answer is 'no'.
Consider a simplified version of the paradox:
Two travelers, each in his own spaceship (i.e., 'reference frame') pass each other at a high relative velocity. By mutual arrangement, they synchronize their (perfect) clocks at the moment of closest approach. After some time, which has aged more? The question is indeterminate. Each 'sees' the other's clock running slow by the same amount. Since a universal 'now' does not exist, there is no way of comparing their relative ages that all observers would agree on. Only by bringing the two back together can we compare their ages in their local 'now'. Since they are moving apart, the only way to bring them back together is for one or both to accelerate. Unlike velocity, acceleration can be measured locally -- i.e., within the reference frame. If both accelerate by the same amount and in opposite directions, then when they pass each other, both will have experienced the same elapsed time. If one accelerates more than the other, then that one will experience a smaller elapsed time. It is the difference in the accelerations, not the velocity itself, that leads to the difference in elapsed time. Note that looking at it from this point of view, it doesn't matter if the accelerations are infinite or small. By assuming they are infinite, all the calculations can be done in special relativity.
--Pete
How big was the aquarium in Noah's ark?