The acceleration misconception has been notorious in the world of relativistic physics education. Gruber and Price make the logical conclusion that the best way to deal with the dilemma is to design a situation where there is either acceleration or differential aging but not both. They begin by citing a colleague's previous attempt at this crux. His scenario entails both twins having identical histories of acceleration yet differential aging still occurs. The authors thought this to be a noble effort but they still saw the allure for students to place the necessity of acceleration on the differential aging. To dispel the myth they designed a scenario where the traveler is constantly accelerating relative to the earthbound observer yet no differential aging occurs.
Let's consider a rocket that is demonstrating pseudo-periodic motion (pseudo because of relativistic effects):
where x is the position and t is the time both measured by the earthbound observer. Now if we differentiate eq. 6 to arrive at velocity vs. time, substitute the results into eq. 5 for
u and finally separate the variables we arrive at this:
Now we can define the duration of the observation to begin at t = 0 and lasting until
Dt
= n
p/
w. If we use this for our limits of integration and attempt to solve the ODE we find ourselves here:
If we substitute
q for
wt and
Dt/n
p for 1/
w then divide both sides by
Dt and finally integrate over half of the original limits and multiply by two (you will need some paper and a pencil for this one) we can arrive at the following manipulation:
Because this integral is a wee bit on the difficult side, we can settle for a numerical approximation.
The important thing to glean from the plot is that differential aging can be shown as a function of Vmax only. Any reference to acceleration would have to include some configuration of Vmax
w. Indeed Gruber and Price provide a numerical example in their text that demonstrates this point even further. They provide an example where the time dilation actually contracts when the acceleration increases. The implication here is that one could arrange a situation with a given time dilation and a given acceleration without considering their correlation.