Hyperbolic motion

Example

Fix the speed of light $c > 0$ and a proper acceleration $\alpha > 0$. Hyperbolic motion is the world line in Minkowski space given, as a function of proper time $\tau \in \R$, by $$ct(\tau) = \frac{c^2}{\alpha} \sinh\frac{\alpha \tau}{c}, \qquad x(\tau) = \frac{c^2}{\alpha} \cosh\frac{\alpha \tau}{c}, \qquad y = z = 0.$$ Its points satisfy $x^2 - c^2 t^2 = c^4 / \alpha^2$, so the world line is one branch of a hyperbola.

Check that $\tau$ is proper time: the four-velocity is $$U = \frac{d(ct, x)}{d\tau} = \left( c \cosh\frac{\alpha\tau}{c}, \ c \sinh\frac{\alpha\tau}{c} \right),$$ and $\eta(U, U) = -c^2 \cosh^2 + c^2 \sinh^2 = -c^2$. The four-acceleration is $$A = \frac{dU}{d\tau} = \left( \alpha \sinh\frac{\alpha\tau}{c}, \ \alpha \cosh\frac{\alpha\tau}{c} \right),$$ with $\eta(A, A) = \alpha^2 (\cosh^2 - \sinh^2) = \alpha^2$. So the proper acceleration is the constant $\alpha$: this is the motion of a rocket that feels a constant acceleration $\alpha$.

The ordinary velocity is $dx/dt = c \tanh(\alpha \tau / c)$, so the rapidity is $\alpha \tau / c$, growing linearly in proper time, while the speed stays below $c$. In Rindler coordinates this world line is the curve $\rho = c^2 / \alpha$, $\eta = \alpha \tau / c$.