# Deaccelerating to the Speed of Light

The trajectory of an observer with a uniform proper acceleration $$a$$ (Rindler) in an inertial frame $$(t,z)$$ can be described by the hyperbola $$$$\left(z+\frac{\gamma_{0}}{a}\right)^{2} - \left(t+\frac{\gamma_{0}\beta_{0}}{a}\right)^{2} = \frac{1}{a^2}\,,$$$$ where $$\beta_{0}$$ is the initial velocity and $$\gamma_{0} = 1/\sqrt{1-\beta_{0}^{2}}$$ the initial Lorentz factor. Plotting this gives the following figure.

There is, of course, another hyperbola when you rotate this hyperbola by $$90$$ degrees and I'm interested in studying this situation. Purely mathematically, this hyperbola gets described by $$$$\left(z+\frac{\gamma_{0}}{a}\right)^{2} - \left(t+\frac{\gamma_{0}\beta_{0}}{a}\right)^{2} = -\frac{1}{a^2}\,,$$$$ where there is now a minus sign in the right hand side and is plotted below.

However, this makes not much sense since in this (superluminal) regime, the initial velocity $$\beta_{0}$$ is larger than one such that the initial Lorentz factor $$\gamma_{0}$$ becomes imaginary.

My question is if there is a way to work around this. How can we describe this regime such that it physically also makes sense?

This regime does not make physical sense as a worldline. This would be a worldline of a particle moving faster than $$c$$.