You're using the wrong equations. I.e. you're using the time dilation formula for a clock accelerating relative to an inertial frame when in fact you want the expression for a non-accelerating clock in a non-inertial frame. In that case you have to use general relativity.Hi I'm working on how to resolve the twin paradox in special relativity for my 4th-year dissertation. I've found several different equations which supposedly allow for the calculation of the time experienced by each twin at set distance intervals from Earth, but they don't seem to work during the decelerations.
The travelling twin's flight consists of 4 stages; acceleration towards a distant planet for 3 lightyears, deceleration towards the planet for 3 lightyears, acceleration back to Earth for 3 lightyears, and finally, decelerating towards Earth for 3 lightyears. The absolute acceleration experienced by the travelling twin is 1 ly/yr^2 (approximately 1g).
The equations are as follows;
t = (c/a) sinh(aT/c) = sqrt[(d/c)2 + 2d/a] (time experienced by stationary twin),
d = (c2/a) [cosh(aT/c) − 1] = (c2/a) (sqrt[1 + (at/c)2] − 1) (distance travelled),
v = c tanh(aT/c) = at / sqrt[1 + (at/c)2] (velocity),
T = (c/a) asinh(at/c) = (c/a) acosh[ad/c2 + 1] (time experienced by travelling twin),
γ = cosh(aT/c) = sqrt[1 + (at/c)2] = ad/c2 + 1 (Lorentz factor).
Specifically, I'm interested in the 't' and 'T' equations. They seem to work fine for the 1st phase (acceleration towards the distant planet), but break during the 2nd phase (deceleration towards the distant planet).
I'm unsure as to whether I should plug in distance travelled or displacement from Earth for 'd', and when 'a' becomes negative (during stages 2 and 3), the acosh functions don't seem to work. Does anyone have any ideas? Any help would be appreciated!