The twins of Langevin
Special relativity
The basic idea of the Langevin twins is to have, within our coordinate system, one observer at rest, represented by a curve $\gamma_1$, and a second observer in motion, represented by a curve $\gamma_2$, such that $\gamma_1(\tau_1) = \gamma_2(\tau_1)$ and $\gamma_1(\tau_2) = \gamma_2(\tau_2)$, $\tau_1 < \tau_2$. In other words, the second observer starts out next to the first, and eventually joins back with him.
The first observer's path does not have to particularly change. With no loss of generality, we can assume it to remain at the origin of the coordinates,
\begin{equation} x_1^\mu(\tau) = (\tau, 0, 0, 0) \end{equation}The second observer's motion may vary a lot. To limit the possible types of motions, we'll require that :
- The motion happens entirely along a single axis of the coordinate system
- The trip be separated in two steps, the first one in which the motion will always have positive velocity, the second one in which it will always be negative.
In other words, we have $\gamma_2([\tau_1, \tau_e]) = \gamma_{i}$, and $\gamma_2([\tau_e, \tau_2]) = \gamma_{f}$. We can also ask that $\gamma_i$ and $\gamma_f$ be symmetric, but this is not strictly necessary.
Straight trip
The simplest case to consider is two trips of constant speed, with the ship turning around instantaneously for the second part.
\begin{eqnarray} x_i^\mu(\tau) &=& (\gamma , \beta, 0, 0)\\ x_f^\mu(\tau) &=& () \end{eqnarray}