twinship
Space under construction: The twin ship problem
The basic problem is that one of a set of twins leaves Earth to travel at near-light speed, encounters time dilation due to relativity, and returns to find the other twin has experienced more time and is now "older." I wish to discover how to calculate this problem under variable conditions of acceleration. I have never worked through this problem before and am not sure of what mathematics is required, but hope to gather the information to solve the problem and present it here. Any helpful comments are solicited.
I will specify flat space and two engines, one, the thruster, providing forward acceleration, and the other, the "kicker," providing lateral thrust so that the ship moves in a circle.
The ship leaves Earth, travels in a circle, then returns to pass by near-Earth space at a high velocity.
For simplicity, I will assume that space is flat and that the entire trip can be accomplished without correction for extraneous mass or charge.
I will further assume that the thrusters are not limited by fuel considerations, but can continue to provide thrust within their frame of reference at a constant rate for any necessary length of time.
The problem now is to find the correct formulas, and then to determine how to calculate them.
I have the following formulae from DW, who tried to help me with this problem in another thread at
https://www.physicsforums.com/showthread.php?t=15826
v = ctanh(\frac{\alpha \tau}{c})in the ship
v = \frac{\alpha t}{\sqrt{1 + \frac{\alpha ^{2}t^{2}}{c^2}}} from the Earth
where:
v = velocity
c = speed of light
\alpha = proper acceleration (as experienced by accelerated frame)
\tau = time elapsed in accelerated frame
t = time elapsed in rest frame
and for convenience, the term
(\frac{\alpha t}{c}) = \theta
DW says \theta is known as rapidity.
I notice that \theta is a dimensionless quantity, with terms for velocity above and below the fraction, so they cancel.
The two formulae can then be written as follows:
v = ctanh(\theta)in the ship
v = \frac{\alpha t}{\sqrt{1 + \theta^2}} from the Earth
.